ANALYTICAL AND EMPIRICAL INVESTIGATIONS OF TICKET QUEUES: IMPLICATIONS FOR SYSTEM PERFORMANCE, CUSTOMER PERCEPTIONS AND BEHAVIOR

Transcription

1 The Pennsylvania State University The Graduate School The Smeal College of Business ANALYTICAL AND EMPIRICAL INVESTIGATIONS OF TICKET QUEUES: IMPLICATIONS FOR SYSTEM PERFORMANCE, CUSTOMER PERCEPTIONS AND BEHAVIOR A Dissertation in Business Administration by Kaan Kuzu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2010

2 The thesis of Kaan Kuzu was reviewed and approved* by the following: Susan H. Xu Professor of Management Science and Supply Chain Management, Director of PhD Program Dissertation Advisor, Chair of Committee Christopher W. Craighead Assistant Professor of Supply Chain Management Terry P. Harrison Professor of Supply Chain and Information Systems Zan Huang Assistant Professor of Supply Chain and Information Systems Tao Yao Assistant Professor of Industrial & Manufacturing Engineering *Signatures are on file in the Graduate School.

3 Abstract This dissertation investigates the system performance, customer perceptions of and behavior in ticket queues, the new generation of queuing systems that issue tickets to customers upon their arrival, using analytical and empirical approaches. Ticket queues are widely used by service providers to manage customer flows. In this dissertation, we present three studies with the goals of generalizing system representations of ticket queues, investigating the validity of the assumptions used in the analytical models, and providing empirical support for the analytical model results. The first essay proposes a Markov chain (MC) model for multi-server ticket queues with state-dependent balking and state-independent reneging customers. This study aims at analyzing the system performance of a multi-serve ticket queue and benchmarking ticket and physical (stand-in-line) queue performances. We develop exact analytical results for key system performances with customer abandonment (including both customer balking and reneging behavior), and present efficient and accurate heuristics to analyze large systems. The heuristics prove to be robust and accurate approximations. A ticket queue differs from a corresponding physical queue in terms of the amount of information available to customers. The lack of information for the number of existing customers in the ticket queue system results in naive customer abandonment behavior and worsens system performances compared to those in the corresponding physical queue. In order to close the performance gap, we propose to provide different types of information to ticket queue customers. We find that for each type of information provided, the customer balking probability decreases while the customer reneging probability increases, and the overall service completion rate improves. Therefore, providing information does not necessarily guarantee an improvement in each system iii

4 performance measures. The queuing system managers should consider this trade-off when they are providing information to customers. The second essay generalizes the model developed in the first essay by considering state-dependent reneging behavior of ticket queue customers. The reneging behavior of a customer depends on his/her ticket position in the system. The literature on progress towards the goal and on the escalation of commitment suggests that the reneging rate will be an increasing function of one s ticket position. On the other hand, the view that the time spent in line makes customers more impatient will suggest the reneging rate should be a decreasing function of the ticket position. Our analysis can incorporate either an increasing or a decreasing function of the reneging rate in the model. In conclusion, the second essay provides the framework to analyze multi-server ticket queues with state-dependent abandonment and provides insight into developing more realistic ticket queue models. Although certain system performances deteriorate in ticket queues compared to those in physical queues, we observe that ticket queues are popular systems in both private and public sectors. This dilemma motivates us to obtain insights into the customer perceptions of and behavior in different queue arrangements, and to investigate whether the assumptions and results of our analytical models hold. The third essay investigates the perceptions of and behavior in ticket queues empirically using a series of surveys. Our results indicate that subjects generally prefer a ticket queue arrangement over a physical queue arrangement, and the level of preference is amplified in high traffic intensity or high service value settings. In addition, we find that the subjects are willing to wait longer in the ticket queue setting than they are in the physical queue setting, which contradicted one of the assumptions we used in the first two essays. On the other hand, we validated the existence of the sunk cost effect in waiting scenarios, which provides support for our assumption in the analytical pieces that customers become more patient after they join and spend some time in line. Finally, we identify the factors effective in the subjects decisions to prefer iv

5 the ticket queue setting over the physical queue setting and show the impact of the context of the wait on the preference of and behavior in different waiting lines. Our results suggest that the analytical models used in the literature should incorporate the abandonment behavior of customers to reflect reality. We also identify and recommend the type of queues to be implemented under different service settings as a decision support to system managers. In summary, we analyze, obtain insights into and compare the performances of ticket and the physical queues from both the analytical and empirical points of view in this dissertation. Our findings could assist system managers in designing better queuing arrangements, and matching an appropriate type of queue with the services provided. v

6 Contents List of Figures List of Tables Acknowledgments ix xi xiii 1 Introduction 1 2 A Multi-server Ticket Queue with Customer Abandonment Introduction Literature Review A Markov Chain Model for the Ticket Queue with Abandonment The Stationary Distribution of the Ticket Queue The Aggregated Markov Chain and its Stationary Distribution Steady-State Probabilities of States in Super-States Key Performance Measures of the Ticket Queue Heuristics The 2-Digit Heuristic (2DH) The 3-Digit Heuristic (3DH) Performance of Heuristics Service Improvement of the Ticket Queue Conclusions and Future Studies A Markov Chain Model for the Ticket Queue with State-Dependent Abandonment Methodology vi

7 3.2 The Stationary Distribution of the Ticket Queue The Aggregated Markov Chain and its Stationary Distribution Steady-State Probabilities of States in Super-States Key Performance Measures of the Ticket Queue Heuristic Approximation The 2-Digit Heuristic (2DH) Conclusion Comparisons of Perceptions and Behavior in Ticket and Physical Queues Introduction Waiting Experience in Queuing Systems Literature Review Hypotheses Methodology Results Hypotheses on Queue Preference: Hypotheses on Willingness to wait (Patience) in Queues: Hypothesis for Sunk Cost (Escalation of Commitment) Effect in Queues: Factors for Queue Preference, Patience and Time Pressure Scales Discussion and Conclusion Bibliography 119 Appendix A. Proof of Theorem Appendix B. An Example of Generator Q with s = 2 and K = Appendix C. Proof of Theorem vii

8 Appendix D. Proof of Theorem Appendix E. Aggregated MC - Infinitesimal Generator Matrices 144 Appendix F. Proof of Theorem Appendix G. An Example of Generator Q G with s = 2 and K = Appendix H. Proof of Theorem Appendix I. General Aggregated MC Infinitesimal Generator Matrices161 Appendix J. Physical Queue and Ticket Queue images in the survey 166 Appendix K. Factorial Design for Treatments 168 Appendix L. Treatments used in the survey 169 Appendix M. Escalation of Commitment Scenario 172 viii

9 List of Figures 2.1 Transition Rate Diagram of a Ticket Queue with s = 2 and K = Transition Diagram of the Aggregated Ticket Queue MC with s = 2 and K = The 2DH Representation of the Markov Chain of the Ticket Queue with s = 2 and K = Markov chain for a 3-Digit Heuristic Ticket Queue with s = Distributions of N, N 2D, and N 3D for K = 10, θ = 2.5, δ = Distributions of N, N 2D, and N 3D ; K = 25, θ = 0.75, δ = δ vs. MAD; K = 10, θ = δ vs TAD; K = 10, θ = Expected number of E-customers in TQ, 2DH and 3DH; K = 10, 20, 30, δ = 0.8θ Balking Probabilities in TQ, 2DH and 3DH: ρ = 0.75, 1.00, 1.25, δ = 0.8θ Reneging Probabilities in TQ, 2DH and 3DH; ρ = 0.75, 1.25, δ = 0.8θ Effects of δ and ρ on Service Completion Probability in TQ; K = 10, θ = Comparison of E[N] in Ticket and Physical Queues; K = 10, δ = 0.8θ Comparison of Service Completion Probabilities in Ticket and Physical Queues K = 10, δ = 0.8θ Effects of Information on Service Completion Probability (K = 50, δ = 0.8θ) Effects of Information on Balking Probability; (K = 50, δ = 0.8θ) Effects of Information on Reneging Probability; (K = 50, δ = 0.8θ).. 57 ix

10 3.1 Transition Diagram of a Ticket Queue with state-dependent reneging for s = 2 and K = Aggregated Ticket Queue Markov chain with state-dependent reneging for s = 2 and K = The 2DH Markov chain of the state-dependent Ticket Queue with s = 2 and K = Balking Patience and Wait Context Sunk Cost in Waits J.1 Physical Queue Arrangement for a bank with a single representative. 166 J.2 Ticket Queue Arrangement for a bank with a single representative x

11 List of Tables 2.1 State space reduction through heuristics Comparison of Deviations for P(N) and P(D), ρ = 0.75, δ = 0.8 θ Comparison of Deviations for P(N) and P(D), ρ = 1, δ = 0.8 θ The types of waiting time information provided Subjects Queue Preference Traffic Intensity and Subjects Queue Preference Value of Service and Subjects Queue Preference Availability of Alternatives and Subjects Queue Preference Ticket Queue Preference vs. Traffic Intensity and Value of Service Descriptive Statistics for Balking Patience in Physical and Ticket Queues Paired Samples t-test for Hypothesis Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Traffic Intensity Independent Samples t-test for Hypothesis Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Value of Service Independent Samples t-test for Hypothesis Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Existence of Alternatives Independent Samples t-test for Hypothesis Willingness to Wait vs. Time Already Spent Waiting in PQ & TQ Nonparametric Friedman s Test for Hypothesis Results of the Binary Logistic Regression Summary for Hypotheses Test Results xi

12 4.18 Recommended Queue System K.1 The set of survey treatments M.1 Escalation of Commitment Table xii

13 Acknowledgments I am heartily thankful and deeply indebted to my dissertation advisor, Susan Xu, for her guidance, support and encouragement throughout the course of my Ph.D. studies. She has continuously inspired me to appreciate and conduct better research. Her support and patience has given me the strength to push my dissertation forward, realize my own potential and improve myself. The guidance I received from her made my Ph.D. purposeful and enlightening. I am also very grateful to Christopher Craighead for supporting me to explore and develop new concepts and models, for encouraging me to improve my research further, and for guiding me through my professional development. I would also like to thank my other committee members, Terry Harrison, Zan Huang and Tao Yao, for their support during my studies, for taking the time to read and evaluate my work, and for their insightful comments. I would like to show my gratitude to Edward Glantz, who encouraged me to pursue my Ph.D. degree, and demonstrated me how to become an effective teacher. I would also like to thank, from the bottom of my heart, Alice Young, Beth Bower, Teresa Lehman, Kitty Riley, Marilyn Blanco, Norman Aggon and all my friends for their continuing encouragement, support and advice. Most importantly, none of this would have been possible without the unconditional love and support of my family in all aspects of my life. I would like to thank my wife, my parents, my brother and my parents in-law for being the source of encouragement, patience and strength throughout my research studies. The education I received from my family helped me to become the person I am today. Finally, I am also proud that Eren has become a part of this journey as well. xiii

14 Chapter 1 Introduction Managing customer flows in service systems is becoming more and more critical in today s competitive environment, which values time as one of the most important resources. One of newer queueing technologies to manage customer flows is ticket queues (Xu et al. 2007). In a ticket queue system, each arriving customer is issued a numbered ticket from a ticket kiosk. The ticket numbers of the customers currently being served are continuously updated through display panels, and waiting customers are called to service counters in the order of the numbered tickets. The ticket queue arrangement has several advantages over the physical queue arrangement, in which customers form physical waiting lines to wait behind each other. From a customer point of view, a ticket queue setting enhances customer experience by eliminating waiting lines and liberating customers from physical discomfort and mental boredom of standing in queues (Xu et al. 2007). They also provide a tidy and relaxing environment for customers, which improves their satisfaction and increases sales potential. From a system manager point of view, a ticket queue arrangement adds values to businesses by collecting vital information on customer flow patterns, abandonment behavior, waiting times, transaction times, and agents service performance for different service categories. This information assists system managers in making real-time decisions on customer routing and staff allocation. These advantages give rise to the wide use of ticket queue arrangements in the public sector, such as in government offices, post offices, motor vehicle offices, embassies and admission offices, as well as in private sectors, such as in banks, retail stores, pharmacies, and travel agencies. In the customer flow management industry, 1

15 2 companies like Q-MATIC Corporation, TASKE Technology, Genesys Telecommunications, and Aspect Software are involved in the implementation and monitoring of ticket queues. For example, The Q-Matic Corporation, the biggest ticket queue management solution provider, manages ticket queues for banks (in Europe and Hong Kong) and health service providers. In any queueing system, an arriving customer s decision to abandon the system is a critical aspect of customer behavior to those managing queueing systems, and can significantly affect the performance of the system (Warren 1981). An arriving customer will immediately start receiving service if any of the servers is idle upon her arrival. If all of the servers are busy, the customer might either join the line to start waiting for service or abandon the system immediately (balk). A customer joining the line might still abandon the line (renege) after spending some time waiting. In a physical queue, a customer can make the abandonment decision based on her queueing position (i.e., the number of customers ahead of her), which is directly observable by the customer. In contrast, the customers in a ticket queue do not form a physical line. In particular, when a customer abandons her ticket queue position either by balking or reneging, this action will not be known to other customers until the abandoned ticket is called for service and immediately discarded. Therefore, a customer in a ticket queue only has full information of how many uncalled tickets exist ahead of her, but not of how many existing customers are ahead of her. This information loss can severely impede a customer s ability to make sound join/abandon decision. Principally, a customer who bases her balking decision on how many tickets exist in front of her will overestimate the actual number of existing customers in front of her. From the management perspective, such naive abandonment behavior will lead to low service rates, poor customer satisfaction, and lost revenues. In general, we assume that an arriving customer carries a balking patience (the time the customer is initially willing to wait before starting service) with her. If the arriving customer realizes all of the servers are busy, she makes the decision to join

16 3 the system or balk by comparing her balking patience with the expected total service time of the tickets ahead of her ticket in the line. If the customer joins the line to wait for service, she forms her reneging patience (the time she is willing to wait after joining the line). We assume a joining customer becomes more patient in a stochastic order sense (i.e. her reneging patience time is stochastically longer than her balking patience time) and continuously compares her reneging patience time with the time spent in line to determine whether she will continue to wait in line or renege the system. As mentioned above, a customer arriving to a ticket queue system does not consider the abandoning behavior of customers in front of her and assumes that her ticket position represents the actual number of existing customers in front of her. Therefore, we can infer that the different information available to customers in the ticket and the physical queues provide different service completion and abandonment probabilities, even when the customer s balking as well as reneging patience times follow the same distribution in both queuing arrangements. To compare the performances of the ticket and the physical queues, Xu et al. (2007) proposed a Markov chain (MC) model of a single server ticket queue with balking as the only form of abandonment, and showed that the ticket queues have significantly lower service rates due to a higher percentage of customers balking the system, especially when customer patience is low and the traffic intensity in the system is high. In this dissertation, we present the following three studies with the goals of generalizing system representations of the ticket queues, investigating the validity of the assumptions used in analytical models, and providing empirical support for analytical model results. A Multi-server Ticket Queue with Customer Abandonment: In this study, we generalize the single-server model developed by Xu et al. (2007) by proposing a Markov chain (MC) model of a multi-server ticket queue system with both forms of customer abandonment. In the study, we use state-dependent

17 4 balking and state-independent reneging rates to develop the analysis framework. A Markov Chain Model for the Ticket Queue with State-Dependent Abandonment: This study provides the methodology for analyzing a multiserver ticket queue with state-dependent abandonment using exact and heuristic analysis. We generalize the model in Section 2 by considering state-dependent balking and reneging rates. Comparisons of Perceptions and Behavior in Ticket and Physical Queues: In this study, we present the findings of a survey conducted to obtain insights into customer perceptions of and behavior in different queue arrangements, and to investigate whether the assumptions used in Sections 2 and 3 hold. The analytical models developed in Sections 2 and 3 provide steady-state distributions for the ticket queue system when customers have low levels of balking patience, which results in a fewer number of customers joining the system. When customers have higher levels of balking patience, which results in a large number of customers joining the system, the state space of our Markov chain model grows exponentially and it becomes impractical to derive the steady-state system performances due to the curse of dimensionality. Therefore, we develop heuristics to evaluate system performances. We also validate our analytical models using simulation. The heuristics prove to be accurate and robust approximations to the original ticket queue models developed in the first two essays. In the first essay of this dissertation, we show that system performance measures (e.g., the service completion probability or the expected number of existing customers in the system) in the ticket queue are smaller than their counterparts in the physical queue. To close the gap between the ticket and physical queue performances, we provide four types of information and find the existence of a trade-off between the balking and reneging probabilities. Although information improves the service

18 5 completion rate, a decrease in the customer balking probability results in an increase in the customer reneging probability. Managers should consider this trade-off in their information-providing decisions. In the second essay, we use the same technique in our first essay to model and provide the stationary distribution of multi-server ticket queues with state-dependent balking and reneging. We also develop a heuristic approximation for this generalized ticket queue model. The generalized model allows us to model the high-dimensional Markov chain of the ticket queue more realistically. Finally, in our survey, we find out that the ticket queue arrangement is the preferred system setting compared to the physical queue arrangement under different service environments. Moreover, we find support for the existence of escalation of commitment effect in both queueing settings. The probability that the subjects are willing to wait more than their initial balking patience time after they joined the system increases with the amount of time they have already spent in the system. On the other hand, we also find out that customers are more willing to wait in ticket queues than they are in physical queues, which contradicts to one of our assumptions in the analytical models. These results should be incorporated into future models to develop more realistic system representations. In summary, we analyze, obtain insights into and compare the performances of the ticket and the physical queues from an analytical and an empirical point of view in this dissertation. Our findings could assist system managers in designing better queuing arrangements, and in matching the appropriate type of queues with the service provided. The remainder of this dissertation is organized as follows. Section 2 provides the analytical model for a ticket queue and compares the system performances of the ticket and physical queues. Section 3 extends the model developed in Section 2 by incorporating state-dependent reneging. Finally, Section 4 presents a survey study designed to obtain insight into customer perceptions of and behavior in different queuing arrangements.

19 Chapter 2 A Multi-server Ticket Queue with Customer Abandonment 2.1. Introduction This chapter develops a framework for the analysis of a multi-server ticket queue system with two forms of customer abandonment-customer balking and reneging. We first present an analytical model to evaluate key system performance measures of the multi-server ticket queue. Using the model developed, we benchmark the system performance of the ticket queue with that of the corresponding physical queue, and show that the ticket queue tends to result in worse system performances than the physical queue does. Finally, we propose providing different types of information to the ticket queue customers to improve service performance. The customers arrive to the multi-server ticket queue system in a Poisson process (with rate λ) and the service times are independently and identically distributed (i.i.d.) exponential random variables (with mean 1/µ). We assume that a customer has a balking patience time before arrival, and she will estimate her waiting time in the queue as the total service time of the tickets ahead of her ticket. Therefore, she will join the queue if her estimate of waiting time in queue is less than her balking patience time. We also assume that if all the servers are busy and a new arrival observes that the number of the tickets ahead of her ticket is greater than or equal to a balking limit K, she will balk with probability 1. If the new arrivals join the queue, we let their reneging patience time be i.i.d. exponential random variables (with mean 1/δ). A joining customer becomes more patient in a stochastic order sense (i.e. her 6

20 7 reneging patience time is stochastically longer than her balking patience time) after she starts waiting in line. Moreover, the waiting customers continuously compare their reneging patience time with the time spent in line to determine whether they will continue to wait or renege the system. Using these assumptions, we first develop a Markov chain model for the multiserver ticket queue with both forms of customer abandonment. This Markov chain is formed with the assumption that an arriving customer will balk if the number of tickets ahead of her ticket in the system is greater than or equal to a balking threshold, K. The resulting MC is high-dimensional and is too complex to yield a closed-form solution of the steady-state distribution even for small values of balking patience, K. As the MC model is neither of a GI/M/1 type (whose infinitesimal generator is a lower Hessenberg matrix), an M/G/1 type (whose infinitesimal generator is an upper Hessenberg matrix), nor their intersection, the quasi-birth-death (QBD), we can not use efficient matrix-analytic techniques developed for the exact analysis of those types of systems. Therefore, we develop a two-step solution procedure to obtain the steady-state distribution of the Markov chain model of the ticket queue. In the first step, we repartition the state space so that we can use the quasi-birth-death process framework. We aggregate the states with the same number of existing customers and with ticket positions greater than equal to K into super-states and reduce the infinite state space of the Markov chain into a finite one through this aggregation. We solve the resulting quasi-birth-death process that admits matrix product form solution for the steady-state distribution. In the second step, we disaggregate the aggregated states into individual states to compute their steady-state probabilities. This solution procedure for the exact analysis of our multi-server ticket queue, unfortunately, can only be used for small K values due to the complexity of the MC model. The cardinality of the state space is O(K s)2 K s after state aggregation, and the solution procedure requires exponentially increasing computing time and memory space in K. Although we validate our solution procedure through simulation analysis,

21 8 we also develop two heuristic solutions to analyze the systems with larger K values. The heuristics are based on the idea that the abandoning customers tend to appear in consecutive sequences in the system, and can be discarded from the system as a group when their ticket numbers are called in the system. Our 2-Digit Heuristic (2DH) assumes all the existing customers form a group, and all the abandoned customers form another group, and gives the existing customer group the first service priority (i.e. all the abandoned tickets are called and discarded simultaneously after all the existing customers are served). On the other hand, our 3-Digit Heuristic (3DH) assumes the existing customers and the abandoned customers can be clustered in three groups and assigned service priority based on the group index: the first group is the first set of consecutive existing customers (with the first service priority); the second group is the set of all the abandoned customers (with the second service priority); and the third group is the rest of the existing customers (with the last service priority). These two heuristics not only drastically reduce the state space of the original ticket queue system, but also yield the Markov chains of a GI/M/1 type, enabling the application of matrix-geometric solution procedures. The computation time and the memory space required are decreased significantly using the heuristics. We numerically display that the heuristics prove to be accurate and robust approximations to the original ticket queue system, especially when the balking limit K takes low values. For large values of K, the performance measures slightly deteriorate, but remain to be close approximations. The 3DH provides more accurate approximations to the system, whereas the 2DH provides more efficient solutions computationally. The exact (or simulation) analysis of the original ticket queue as well as the heuristic solutions provide us the framework to benchmark the system performance of the ticket and physical queues. As mentioned above, one drawback for the ticket queue is that the number of existing customers in the system is not known either by management or customers when the number of tickets in the system is larger than the number of servers. This lack of information results in customers naive

22 9 abandonment behavior, and can adversely affect the system performance. In general, using the same set of system parameters, a physical queue provides a higher service completion probability and a higher expected number of existing customers in the system than its counterpart ticket queue does. Therefore, we investigate whether it is possible to match, or even surpass, the performance of the physical queue by providing extra information to ticket queue customers about their waiting times in the system. We also examine the impact of providing additional information on the balking and reneging probabilities. To achieve our goals, we propose providing four types of waiting time information in the ticket queue setting. As a first level, we classify the waiting time information into the conservative and aggressive types. In the conservative case, we provide an estimate of the waiting time of a new arrival, conditioning on her ticket position and ignoring the reneging behavior of the existing customers. In the aggressive case, we provide an estimate of the waiting time of a new arrival, conditioning on her ticket position and considering the reneging behavior of the existing customers. As a second level, we categorize both of the information types above into the expected value case and the random value case. In the expected value case, we provide the expected waiting time for the customers who completed service and had the same ticket position with the new arrival. In the random value case, we provide the waiting time information for the last customer who completed service and had the same ticket position with the new arrival. We developed simulation models to investigate the effects of waiting time information on system performance measures in ticket queue arrangements. Our numerical analyses reveal that the service completion probability in the ticket queue improves by providing information. Although the overall abandonment probability decreases with additional information, a decrease in the balking probability is accompanied by an increase in the reneging probability. The ticket queue system managers should consider the above trade-off between the two types of abandonment probabilities and

23 10 determine the long run costs of losing customers through balking opposed to reneging when they decide which type of information to provide to the customers. The remainder of this chapter is organized as follows: Section 2.2 will present a literature review for the analysis of multi-server queuing systems with abandonment behavior. Section will explain the quasi-birth-death model used in the study along with the transition matrices. Section will present the methodology to develop the stationary distributions in the QBD model, and will introduce the system performance measures. We develop the approximation heuristics for analyzing large systems in Section 2.3 and analyze the performance of the heuristics in Section 2.4. Section 2.5 will focus on how to improve the service performance measures by providing different types of information. Finally, we will provide conclusions and managerial insights in Section Literature Review Queuing theory is one of the most popular research areas in operations management, marketing and psychology disciplines. In this section, we will review the existing research in queuing theory, focusing on waiting duration and customers responses to waiting. Traditionally, researchers have preferred to combine balking and reneging probabilities under the single term - abandonment - to ease the difficulties of modeling these two different customer behaviors. Although modeling of dynamic balking and reneging is difficult, doing so will provide qualitatively different managerial insights, and structurally different control policies (Armony et al. 2005). In the queuing literature, the balking probability is modeled either as a constant, independent of the state of the system, or as a state-dependent probability that increases in the number of customers ahead of a newly arrived customer. The reneging phenomenon appears for the first time in Palm (1937), in the context of impatient telephone switchboard customers. In an early study of a multi-server queue with impatience, Barrer (1957)

24 11 determined the loss probability (due to impatience) in multi-server queues with general abandonment-time distribution (M/M/m + G queues). As outlined in Brown et al. (2005) the time that a customer (with infinite patience for waits) needs to wait before starting service can be referred to as virtual waiting time, and the time that a customer is willing to wait before abandoning the system can be referred to as patience. Baccelli and Hebuterne (1981) derive equations for the virtual waiting time distribution and the probability of having different number of customers for the M/M/m + G queue. Bhattacharya and Ephremides (1991) show that the number of successful departures and the number of customers lost over a time interval are (stochastically) monotone functions of the arrival, service and deadline processes. Most of the early research assumes the simplest structure for reneging (customers renege after an exponentially distributed amount of time) or balking (balk with probability p if there is any wait, and with probability (1 p) wait until the service is provided). Whitt (1999a, 1999b) provides exceptions to these assumptions by analyzing the situations where the service provider may or may not provide information to customers about the system state, and compares how much time the customers spend in the system under provided information and no information cases. In Whitt (1999a), he assumes that when system state (the number of customers in front) information is not provided, the arriving customers would balk with a constant probability, and renege after an exponential time. When the system state and the remaining service times of all the customers is provided as information, Whitt assumes that all the reneging behavior is replaced by the balking behavior, and shows that the number of customers in the system is smaller when state information is provided than that when state information is not provided, in the likelihood-ratio stochastic ordering. Considering a limited waiting space area, Whitt also shows that customers are more likely to be blocked when no state information is provided and are more likely to be served without waiting when state information is provided. In Whitt (1999b), the author estimates the cumulative distribution function of customers remaining service time,

25 12 and uses it to estimate the remaining waiting times for customers. Nakibly (2002) also discusses estimation methods for M/M/s, G/M/s, G/G/s and G/M/s queues with abandonment. Two other notable exceptions are Ward and Glynn (2003) and Zeltyn and Mandelbaum (2006). Both papers allow general distributions for patience time, and the authors perform asymptotic analysis of these systems under conventional heavy traffic and the many-servers heavy traffic regimes, respectively. Mandelbaum and Shimkin (2000) derive complex dynamic customer behavior from primitives on valuation and waiting costs, for an M/M/s queue with congestion/failure shocks, assuming customers cannot observe the queue length. Some of the literature on queueing models with reneging focus especially on performance evaluation. We refer the reader to Ancker and Gafarian (1962a, 1962b and 1962c), and Garnett et al. (2002), and references therein for simple models assuming exponential reneging times. Zohar et al. (2002) investigates the relationship between customer reneging and the experience of waiting in queue. Other papers allow reneging to follow a general distribution. Related studies include Baccelli and Hebuterne (1981), Brandt and Brandt (2002), Ward and Glynn (2003), and references therein. The continued growth of both importance and complexity of modern call centers has also led to an extensive and growing literature in queueing models. Due to the uncertainty governing the call center environment (customer and agent behaviors), there are studies similar to this paper in the call center management literature. Queueing models incorporating impatient customers have received a lot of attention in the call center literature. Gans et al. (2003) and Mandelbaum and Zeltyn (2006) underlined the importance of abandonment modeling in the call centers, and gave some numerical examples that point out the effects of abandonment on system performance. Armony et al. (2005) show that failure to account for duplicate ordering and reneging can cause either over or under investment in capacity. In their study, they used sample path arguments to derive convexity properties of an M/M/s queue with impatient customers that balk and renege. They assumed that the balking prob-

26 13 ability and reneging rate are increasing and concave in the total number of customers in the system (head-count), and proved that the expected head-count is convex decreasing in the service rate. And finally, the only analytical study about the ticket queues is provided by Xu et al. (2007), where the authors analyze a single-server ticket queue with balking behavior of customers. In Xu et al. (2007), the authors assume a balking threshold, K, which forces a customer to balk (with probability 1) if the difference between her ticket number and the displayed ticket number of the customer in service exceeds the threshold. Our paper will be an extension of this paper by focusing on the multi-server ticket queue systems, considering state-dependent balking probability and including the reneging behavior of customers as a secondary form of abandonment. On the other hand, the literature on the effects of providing delay information to customers begins with Naor (1969), who studies a system with identical customers and linear waiting cost. He states that an arriving customer who stays imposes delays on later customers, but ignores them in making his decision, and, too many customers stay. The abandonment behavior of customers will be determined by the customers expectations for their waiting time, and is affected by their psychology. Maister (1984) developed a theory of queue psychology that focused on a combination of perceptions and expectations management. According to Maister, If you expect a certain level of service, and perceive the service received to be higher, you will be a satisfied customer. There are two main directions in which customer satisfaction can be influenced: by working on what the customer expects and what the customer perceives. Katz et al. (1991) show (in a field study) that people tend to overestimate the amount of time they spend waiting in line. Katz et al. (1991) found that providing customers with information about how long they must wait reduces customers perceived wait estimates. Providing information about the expected waiting time with continuous updates has been used widely to reduce customer dissatisfaction with waiting (Larson 1987, Maister 1984, Warren, 1981). Taylor (1994) shows that

27 14 delays affect customers overall service evaluations. Hui and Tse (1996) study the relationship between information and customer satisfaction. Gavish and Schweitzer (1973) study a full-information system under assumptions similar to Naor s. Hassin and Haviv (2003) summarize additional research along these lines. Whitt (1999a) studies two systems corresponding to providing no information and partial information. The author studies the effect of announcing delays on the performance of a single class call center, and shows that information always reduces both waiting and throughput. Guo and Zipkin (2007) contradict Whitt s findings such that, although the authors have found sufficient conditions to ensure that more information helps the provider or the customers, they show that, in certain cases, more information can actually hurt one or the other. Armony and Maglaras (2004a, 2004b) analyze systems where an arriving customer learns some delay information and then can choose to balk, wait, or leave a message, in which case the provider calls back within a guaranteed time. In this study, I provide four different types of information to the ticket queue customers and analyze the effects of information on service performance measures A Markov Chain Model for the Ticket Queue with Abandonment We make the following assumptions to build the Markov chain model of the ticket queue with customer abandonment: Customers arrive in a Poisson process with rate λ. There are s identical servers and the service times are independent and identically distributed (i.i.d.) exponential random variables with mean 1/µ. When a customer arrives to the ticket queue, she receives a ticket from a kiosk. If any of the s servers is idle, the customer will always join and start service immediately; otherwise, the customer will balk with a certain state-dependent probability (this probability will be specified later). The ticket queue display monitors spreaded in the facility announce

28 15 the ticket numbers of the customers currently in service and the system will call waiting customers for service following the First-Come, First-Served (FCFS)) rule. If an arriving customer joins the queue, she will renege the system if her waiting time in queue exceeds her reneging patience time, an exponentially distributed time with mean 1/δ, independent of the number of tickets ahead of her ticket. A customer will not renege after her service begins. We call the customers currently in the system, who are either receiving service or waiting in line, as the existing customers (called E-customers hereafter), and the customers who have abandoned (either balked or reneged) the system but whose numbers have not been called as the abandoned customers (called A-customers hereafter). We denote N as the stationary number of E-customers in the system. From PASTA (Poisson-arrivals-see-time-average), N is also the number of E-customers in front of a newly arriving customer, i.e., the queueing position of the new customer. We denote D as the stationary number of tickets in the system, which is also the number of tickets (held by both E- and A-customers) ahead of a newly arriving customer, i.e., the ticket position of the new customer. Note that N cannot be observable by a new arrival, since she has no knowledge of other customers abandonment actions. Nevertheless, she can compute D by D = her ticket number the largest ticket number in service + (s 1). (2.1) For example, in a three server (s = 3) ticket queue, if a customer s ticket number is 25 and she observes that the ticket numbers shown on the display panels are 15, 16, and 19, respectively, then she can infer from available information that the number of tickets ahead of her ticket in the system is D = 25 max{15, 16, 19} + (3 1) = 8. Since D contains the tickets of both E- and A-customers, D N almost surely. We now specify customers balking and reneging behavior, which is similar to the assumptions of the M/M/s + M queue with abandonment studied by Whitt (1999a).

29 The M/M/s + M queue can be considered as the physical queue counterpart of our ticket queue, in which a customer has full information of her queuing position N. Suppose that a customer has a balking patience time, T b 16 (an exponential random variable with mean 1/θ), and she will join the queue if and only if her estimated waiting time in queue is less than her balking patience time T b. Since an arriving customer can only observe her ticket position D but not her actual queueing position N, we assume that the customer will use D as a proxy of N. In other words, the customer will treat each ticket ahead of her own ticket as an E-customer, ignoring the possibility that some customers may have already abandoned their positions. This assumption is supported by some empirical evidence in Section 3: a survey of customer behavior in ticket queues shows that only 19.4% of 588 survey participants take other customers abandonment behavior into consideration when determining their queueing positions. Based on this assumption, a new customer with ticket position D = d estimates her time in queue as the total service time of d s + 1 tickets ahead of her ticket, i.e., S d s+1 = d s+1 i=1 Y i, D s, where Y i are i.i.d. exponential random variables with rate sµ. Therefore, a customer with ticket position D = d will balk with probability P (T b < S d s+1 ). Note that S d s+1 has an Erlang density function, denoted by f Sd s+1 with parameters d s + 1 and sµ. Denote the balking probability of a customer who observes her ticket position d by P b (d). Then, we can work out P b (d) = P (T b < S d s+1 ) = = 0 0 P (T b < S d s+1 S d s+1 = y)f Sd s+1 (y)dy (1 e θy )f Sd s+1 (y)dy = 1 ( ) sµ d s+1, d s. sµ + θ (2.2) Let P b (d) = 1 P b (d) be the probability that a customer observing d tickets joins the system. Now, for a prespecified infinitesimal value ϵ > 0, let K, s K <, be

30 17 the smallest integer satisfying ( ) sµ K s+1 P (T b < S K s+1 ) = 1 1 ϵ. (2.3) sµ + θ We assume that a customer will balk with probability 1 as long as her ticket position d K, and call K the balking limit of customers. Clearly, the number of E-customers in the system, N, cannot be greater than the balking limit K. Next, let us consider the reneging behavior of customers who joined the system initially. Similar to Whitt (1999a), we let a joining customer s reneging patience time be an exponential random variable T r with mean 1/δ, independent of her balking patience time T b. If the actual waiting time of a customer in queue exceeds T r, the customer, who initially joined the system, abandons her ticket without receiving service. Let δ < θ, meaning that a joining customer becomes more patient in a stoachastic order sense. Our model includes two types of abandonment as special cases: if θ = 0 and δ > 0, then a customer always joins the system but may renege in the future; if θ > 0 and δ = 0, then a customer may balk but never renege once she joins the line. Our objective is to construct a Markov chain model that enables us to extract information of the number of existing customers N and the number of existing tickets D. Because neither N nor D individually nor N and D jointly forms a Markov chain, we must depict the dynamics of the ticket queue with more detailed information, that is, define the system state that records, for each ticket in the system, whether or not its holder has abandoned her position in queue. For example, the current system state could be (E, E, E, A, A, E). Such a state can be tedious to represent when the number of tickets in the system is large. To develop a more compact notation for system states, we note, a system state can be viewed as a sequence of consecutive E-customers followed by a sequence of consecutive A-customers, then followed by the second group of consecutive E-customers and then the second group of consecutive

31 18 A-customers, and so on. Therefore, for any system state, if E-customers are separated by A-customers into l groups, then we can represent the system state, denoted by x, as a 2l-dimensional vector (or l pairs of numbers) as follows: x = ( n E 1,n A 1 ; n E 2,n A 2 ;... ; n E l,na l ) = ( n E i,n A i : 1 i l ), 1 l K s 2 + 1, where n E i 1 is the number of E-customers in the ith E-group, i = 1,..., l, and n A i 1 is the number of A-customers in the ith A-group, i = 1,..., l 1, and n A l number of A-customers, if any, following the last group of E-customers (n A l 0 is the = 0 if no A-customers follow the last E-customer). We require 1 l K s since the maximum number of pairs in each state cannot exceed K s 2 + 1, which occurs when the first 2l 1 groups of E- and A-customers take their respective minimum numbers, n E 1 = s, and na 1 = ne 2 = na 2 = = na l 1 = ne l = 1. In our aforementioned example, the representation of state (E, E, E, A, A, E) is the 2l = 2 2 dimensional vector (3, 2; 1, 0). Let x = l n E i and x = i=1 l (n E i + n A i ), respectively, be the total number of E-customers and the total number of E- and A-customers (i.e. i=1 the total number of tickets) in state x, that is, x and x are realizations of N and D. As an example, if x = (3, 2; 1, 0), then x = = 4 and x = = 6. If the number of E-customers is less than the number of servers, i.e., 0 x < s, then we must have x = x, hence all the E-customers are in a single group, and the state of the system becomes x = (n E 1, na 1 ) = ( x, 0), 0 x < s, where state (0,0) represents an empty system. Clearly, our state representation gives complete information of the numbers of E- and A-customers in the system. It can be shown that the Markov chain that can accommodate at most K of the existing customers is ergodic, regardless of the traffic intensity ρ = λ sµ. The state space S of

32 19 the ticket queue can be represented as follows: S = x = (0, 0) and x = ( n E i,na i : 1 i l ) : n E i 1, 1 i l and n A i 1 for 1 i l 1, n A l 0, x n A l K, 1 l K s (2.4) The constraints on n E i and n A i ensure that every group of E- or A- customers is at least of size 1, except the last group of A-customers, which can be of size 0; the constraint x n A l K means that the total ticket count of any state x, excluding the last group of A-customers, is no more than K, since a customer with ticket position greater than or equal to K would not join the system. Figure 2.1 displays the transition rate diagram of a ticket queue with s = 2 and K = 4, where states (0,0) and (1,0) correspond to the system with no customers and 1 E-customer, respectively; column 1 entries correspond to the system with two E- customers; column 2 and 3 entries correspond to the system with three E-customers; and, column 4 entries correspond to the system with four E-customers. In Figure 2.1, an arc labeled λ or λ P b (d) (λp b (d)) corresponds to a new arrival who observes her ticket position d and joins (balks) the system, an arc labeled kµ, k = 1, 2, corresponds to a service completion, and an arc labeled δ corresponds to a customer reneging. For example, suppose the current state is x = (3, 0), meaning that the first two E- customers are currently being served, the third E-customer is waiting in queue, and there are no abandoned tickets in the system. A new customer arriving in state x = (3, 0) will infer her ticket position as x = = 3 (i.e., 3 tickets ahead of her ticket). With rate λ P b ( x ) = λ P b (3), this customer will join the queue and move the system to state x = (4, 0); with rate λp b (3), this customer will balk and move the system to state x = (3, 1). On the other hand, a service completion in state x = (3, 0) occurs with rate 2µ, which moves the system to state x = (2, 0). Finally, if the waiting E-customer in state (3,0) reneges, the system will move to state x = (2, 1) with rate δ.

33 20 Note that if a state has more than s + 1 E-customers, the reneging of the customers with different ticket positions will move the system to different states. For example, if x = (4, 0), then the abandonments of the third and fourth E-customers will move the system to states (2,1;1,0) and (3, 1), respectively. Figure 2.1: Transition Rate Diagram of a Ticket Queue with s = 2 and K = The Stationary Distribution of the Ticket Queue Our multi-server ticket queue with abandonment is a high-dimensional Markov chain. In this section, we propose matrix-analytical techniques (Neuts 1981) for the station-

34 21 ary distribution of the Markov chain. In the matrix analytical solution procedure, the state space of a two-dimensional Markov chain is partitioned into different levels, where each level has a finite number of states (called phases). Efficient matrix-analytic techniques have been developed for the exact analysis of the steady state distribution of a Markov chain if it is either of a GI/M/1 type (whose infinitesimal generator is a lower Hessenberg matrix), an M/G/1 type (whose infinitesimal generator is an upper Hessenberg matrix), or their intersection, the Quasi-Birth-Death process (whose infinitesimal generator has zero matrices except for the matrices on the tridiagonal of the generator). For the M/G/1 type of Markov chains, matrix analytic methods have been proposed to solve a set of equations π Q M/G/1 = 0, where π is the stationary probability vector of the states in the Markov chain (Riska and Smirni 2002). The GI/M/1 type of Markov chains, on the other hand, have significantly simpler algorithms to solve using the matrix geometric relation that exists among the stationary probabilities of the set of states that correspond to the repetitive portion of the chain. Quasi-Birth-Death processes can be solved with any of the two techniques, but the matrix geometric solution is preferred because of its simplicity (Riska and Smirni 2002). In our Markov chain model, we define level n as the number of the last group of A-customers in the system, and the phases in level n as the collection of states T n = { x = (n E i, n A i ; 1 i l) S : n A l = n }, n = 0, 1,.... (2.5) Then S = {T n, n 0}. Within T n, we order any two states x and x in a lexicographical order, i.e., x = ( n E i,na i : 1 i l ) ( is listed ahead of x = ñ E i,ña i : 1 i l ) if (1) x < x, (2) x = x and l < l, or (3) x = x, l = l, and x is majorized by x in the sense that the sum of the first k digits in x is no more than its counterpart in x for k = 1,..., 2l. In Figure 2.1, T 0 = {00, 10, 20, 30, 2110, 40}, in our stated lexicographical order corresponds to the states in the first row, and T n = {2n, 3n, 211n, 4n} corresponds

35 22 to the states in the row with the last digit being n, n 1. It can be verified that T n, the cardinality of T n, is (2 K s + s) for n = 0 and 2 K s for n 1. Our objective is to derive the steady-state probability distribution p = {p n, n 0}, where p n is the probability vector associated with block T n. For this purpose, let us consider the structure of the Markov chain. For n = 0, the Markov chain can remain within T 0 due to either a service completion or a reneging by a waiting customer other than the last waiting customer (e.g., in Figure 2.1, the Markov chain can visit from (2, 0) T 0 to (1, 0) T 0 if one of the two servers completes service, and from (4, 0) T 0 to (2, 1; 1, 0) T 0 if the third E-customer in (4,0), i.e., the first waiting customer, reneges). Also, the Markov chain can visit T i from T 0, 1 i K s, due to a new abandonment (e.g., in Figure 2.1, the Markov chain can visit from (2, 0) T 0 to (2, 1) T 1 if a new arrival, who observes her ticket position d = 2, immediately abandons her ticket, and from (2, 1; 1, 0) T 0 to (2, 2) T 2 if the only customer in the second E-group reneges from her position). In general, for any n 1, the Markov chain can visit from (s, n) T n to (s 1, 0) T 0 due to a service completion, or to blocks T n+i, 0 i K s, due to a new balking or reneging customer. Referring to Figure 2.1, from block T 2, the Markov chain can visit blocks T 0, T 2, T 3 and T 4 in the next transition. The following theorem specifies the infinitesimal generator of the Markov chain under partition {T n, n 0}. The proof is given in Appendix A. Theorem The block-partitioned infinitesimal generator of the Markov chain for the ticket queue under partition {T n, n 0}, defined in (2.5), is given by

36 23 Q = T 0 T 1 T 2 T K s 1 T K s T K s+1 T K s+2 T 0 L 0 F 0,1 F 0,2... F 0,K s 1 F 0,K s T 1 B 1 L F 1,1 F K s 2 F K s 1 F K s T 2 B 2 L F K s 3 F K s 2 F K s 1 F K s T K s 1 B K s 1 L F K s 1,1 F 2 F 3 T K s B L F 1 F 2 T K s+1 B L F 1... T K s+2 B L..... (2.6) where L 0 is a (2 K s + s) (2 K s + s) dimensional matrix, B i, 1 i K s 1, and B are 2 K s (2 K s + s) dimensional matrices, F 0,j, 1 j K s, are (2 K s + s) 2 K s dimensional matrices, and all others are 2 K s 2 K s dimensional matrices. These matrices are defined in expressions (A.1)-(A.9). In Appendix B, we illustrate the construction of the matrices in Q by a system with K = 4 and s = 2. Since the Markov chain with the partition {T n, n 0} can skip to non-neighbouring blocks on both left and right, the Markov chain is neither of a skip-free-to-the-right type (a transition from level n to level n is impossible if n > n + 1) nor a skip-freeto-the-left type (a transition from level n to level n is impossible if n < n 1). As a result, the block-partitioned infinitesimal generator Q does not have a lower or an upper Hessenberg form, such a form is generally considered essential to enable the well-known matrix-analytical techniques popularized by (Neuts 1981). To proceed, we observe that, for any x T n, n K s, a new arrival in state x always abandons the system immediately, and the matrices start to exhibit a repetitive pattern, independent of n. Recall that p = {p n, n 0} is the equilibrium distribution

37 24 of the Markov chain under partition T n, n 0. It can be seen from the structure of generator Q that the probability vector p n associated with the block T n, n K s, depends only on p n (K s), p n (K s 1),..., and p n 1, the probability vectors associated with the preceding K s blocks of T n. This motivates us to carry out the solution procedure of the Markov chain in two steps. In the first step, our objective is to compute p(x), for all x T n, 0 n K s 1. To achieve this, we repartition the state space so that we can use the quasi birth-death (QBD) process framework. In a QBD process, the state space of a two-dimensional Markov chain is partitioned into different levels, where each level has a finite number of states, and transitions are allowed only to the neighboring levels or in the same level (Neuts 1981). In our Markov chain model, we can define level j as the total number of E-customers in the system, and the phases in level j, S j, as the collection of states, S j = {x S : x = j}, 0 j K. Since the number of E-customers can only increase or decrease by 1 or remain the same in each transition, the Markov chain can only visit one of its neighboring levels or remain in the same level. However, because the number of states in level j, s j K, is infinite (e.g., each column in Figure 2.1 has an infinite number of states), our Markov chain is not a QBD process, which requires the number of states in each level to be finite (Neuts 1981). Therefore, we apply a state aggregation-disaggregation procedure, as follows. Step 1 Aggregate all the states x with x = j and at least (K s) A-customers in the last group into a single super-state S j : S j = {x S : x = j, n A l K s}. The Markov chain with the super-states (called the aggregated Markov chain) has a finite number of states in each level and thus fits into the framework of a QBD process. The matrix analytical solution for the aggregated Markov chain is derived in Section

38 25 Step 2 Disaggregate super-states S j, j s, and partition the original state space S according to {T n, n 0}, given in (2.5). Since we have already obtained p n, 0 n < K s, and p(s j ), j = s,..., K, in Step 1, we can compute the steadystate probabilities of the individual states in each super-state recursively. The solution procedure is derived in Section The Aggregated Markov Chain and its Stationary Distribution The aggregation of the original Markov chain is based on the idea that if the number of tickets in the system is greater than or equal to the balking limit K, a new customer will automatically balk, so the transition rates of the states in super states show a repetitive pattern, as seen in Figure 2.1. For example, in the Markov chain with s = 2 and K = 4, all the shaded states in Figure 2.1 have the number of tickets greater than or equal to K. Through state aggregation via super states S j, we reduce the infinite state space S into a finite state space all the states x with j = x : S j = S = K j=0 S j, where the finite set S j contains {(j, 0)}, 0 j < s, { x S : x = j, n A l K s 1 } {S j }, s j K. For example, for a 2 server system with K = 4, we have: {(1, 0)}, S 2 = {(2, 0), (2, 1)} {S 2 }, S 4 = {(4, 0), (4, 1)} {S 4 }. S 0 = {(0, 0)}, S 1 = S 3 = {(3, 0), (3, 1), (2, 1; 1, 0), (2, 1; 1, 1)} {S 3 }, and Let us specify state transitions in the aggregated Markov chain. We let a state transition between two non-super states in the aggregated Markov chain be the same as its counterpart in the original Markov chain, with the same rate. We also let a state transition from a non-super state to a super state in the aggregated Markov chain (such a transition corresponds to either a new balking customer with rate λ

39 26 or a new reneging customer with rate δ) be the same as the state transition from the non-super state to a state within the super state, again with the same rate. In terms of state transitions between super states S j, we note, first, since a new arrival in S j will result in an automatic balking, the number of E-customers will remain the same, and the Markov chain will remain in super-state S j ; second, for super-state S s, a service completion in S s will move the Markov chain to state (s 1, 0), since all A-customers in queue will leave the system simultaneously along with the customer who just got served. For s < j K, a service completion in S j or reneging will reduce the number of E-customers to j 1 and the Markov chain will move to super state S j 1. An aggregated Markov chain with s = 2 and K = 4 is displayed in Figure 2.2. Figure 2.2: Transition Diagram of the Aggregated Ticket Queue MC with s = 2 and K = 4 S turns out to be a QBD pro- The aggregated Markov chain with state space cess. Let S j denote the cardinality of x and x in a lexicographical order, i.e., x = ( n E i,na i S j. Within S j, we order any two states : 1 i l ) is listed ahead of

40 x = ( ñ E i,ña i : 1 i l ) if (1) x < x, (2) x = x and l < l, or (3) x = x, l = l, and x is majorized by x in the sense that the sum of the first k digits in x is no more than its counterpart in x for k = 1,..., 2l. The infinitesimal generator of the QBD process has the tridiagonal form: 27 Q = A 0,0 A 0,1 A 1,0 A 1,1 A 1, A s 1,s 2 A s 1,s 1 A s 1,s A s,s 1 A s,s A s,s A K 1,K 2 A K 1,K 1 A K 1,K, A K,K 1 A K,K where the matrices in the first s rows are scalars with A j,j+1 = λ, A j,j 1 = jµ and A j,j = (λ+jµ), for 0 j s 1 (with the exception for A s 1,s, a 1 S s matrix where A s 1,s (1, 1) = λ, A s 1,s (1, k) = 0, 2 k S s ), corresponding to state transitions when the system has idle servers; and the matrices in the last K s + 1 rows, A j,j 1, A j,j, and A j,j+1, are of S j S j 1, S j S j, S j S j+1 dimensions, respectively, corresponding to state transitions when the system has j existing customers in the system, for s j K. A j,j 1, A j,j, and A j,j+1 matrices are defined in expressions E.1-E.4 in Appendix E. For example, for s = 2 and K = 4, we have A 0,0 = λ,

41 A 0,1 = λ, A 1,0 = µ, A 1,1 = (λ + µ), A 1,2 = ( ) λ 0 0 and, 28 2µ (λ + 2µ) λp b (2) 0 A 2,1 = 2µ, A 2,2 = 0 (λ + 2µ) λp b (3), 2µ 0 0 2µ 2µ δ 0 λ P b (2) µ δ A 2,3 = 0 0 λ P b (3) 0 0, A 3,2 = 2µ 0 δ, µ δ 0 0 2µ + δ (λ + 2µ + δ) λp b (3) (λ + 2µ + δ) 0 0 λ A 3,3 = 0 0 (λ + 2µ + δ) λ 0, (λ + 2µ + δ) λ (2µ + δ) λ P b (3) µ δ δ 0 0 A 3,4 = 0 0 0, A 4,3 = 0 2µ 0 δ δ, (µ + δ) (λ+2µ+2δ) λ 0 A 4,4 = 0 (λ+2µ+2δ) λ (µ+δ) Let p j be the vector of the steady-state probabilities of the states in S j, 0 j K. For 0 j s 1, S j = {(j, 0)} and p j = p(j, 0). For s j K, and in general, p j is a S -dimensional j row vector. The steady state probabilities can be computed by

42 29 solving the balance equations ( p 0, p 1,..., p K ) Q = 0, which translates to: λ p 0 + µ p 1 = 0, p j 1 A j 1,j + p j A j,j + p j+1 A j+1,j = 0, j = 1,..., K 1, p K 1 A K 1,K + p K A K,K = 0. Recursively, we can write the solution to the above equations as follows: p j = λ j j!µ j p 0, 1 j s 1, p j 1 R j 1,j, s j K, (2.7) by where R j 1,j, a S j 1 S j matrix, can be found recursively, starting from j = K, R K 1,K = A K 1,K (A K,K ) 1, R j 1,j = A j 1,j (A j,j + R j,j+1 A j+1,j ) 1, s j K 1. Define R j = j R i 1,i. Then, p j = p s 1 R j, s j K, where p s 1 = i=s (2.7). Now, using the normalization equation p 1 T = p 0 ( s 1 i=0 λ i i!µ i + λs 1 (s 1)!µ s 1 p 0 by ) λs 1 K (s 1)!µ s 1 j=s R j 1 T = 1, where 1 T is the transpose of a unit vector 1 of appropriate dimension, we finally obtain p j = ( 1 s 1 λ i i=0 i!µ i + λs 1 K (s 1)!µ s 1 j=s R ( j ) s 1 λ i i=0 i!µ i + λs 1 K (s 1)!µ s 1 R j 1 T j=s R j 1 T ) λj j!µ j, 0 j < s, λs 1 (s 1)!µ s 1, s j K. (2.8) The next theorem relates the steady-state distribution of the aggregated Markov

43 chain { p j, 0 j K} to the steady-state distribution of the original Markov chain. Theorem Let { p(x), x 30 S } be the steady-state distribution of the aggregated Markov chain, and {p(x), x S } be the steady-state distribution of the original Markov chain. Then p(x) = p(x) for x = x and x S j, s j K, and p(s j ) = x S : x =j,n A p(x), for any s j K. K s l The above theorem implies that the aggregated Markov chain provides complete information for E-customers in the system, N, through the expression P (N = j) = x S p(x), j = 0, 1,..., K. However, {p(x), x S } provides neither the marginal j distribution of D nor the joint distribution of (N, D), as both depend on the individual probabilities of the states aggregated into super-states. In the next section, we provide the solution procedure for the individual probabilities of the states in super-states, using the partition of the state space S = {T n, n 0} defined in (2.5) Steady-State Probabilities of States in Super-States This subsection outlines the second step of the solution procedure, which derives the steady-state probabilities of the individual states in super-states. After disaggregation, we recover our original Markov chain with state space S. We go back to our original partition of the state space S = {T n, n 0}, defined in (2.5), where T n is the collection of the states with the number of A-customers in the last group being n. In Section 2.2.3, we have already derived the steady-state probabilities of the individual states (j, 0), 0 j s 1, and T n, n < K s. Our objective is to derive the steady state probabilities for blocks T n, n K s. Using the structure of generator Q given in Theorem 2.2.1, for n K s, we can write the balance equations as:

44 31 K s 2 p 0 F 0,K s + p j F K s j + p K s 1 F K s 1,1 + p K s L = 0, n = K s, or, equivalently, j=1 K s j=1 p n j F j + p n L = 0, n K s + 1, K s 2 p K s = p 0 ( F 0,K s L 1 ) + p j ( F K s j L 1 ) + p K s 1 ( F K s 1,1 L 1 )(2.9) p n = K s j=1 j=1 p n j ( F j L 1 ), n K s + 1. (2.10) Using (2.9), we first obtain p K s, whose solution can be expressed by the known probability vectors p n, 0 i < K s. Recursively, we can compute p n for n > K s, using (2.10) Key Performance Measures of the Ticket Queue Given the stationary distribution p(x), x S, we can compute the key performance measures. The marginal distributions of N and D are, respectively, p(n, 0), n = 0,..., s 1, P (N = n) = p(x), n = s,..., K, x Sn and p(d, 0), d = 0,..., s 1, P (D = d) = p(x), d = s, s + 1,.... x: x =d From a customer s perspective, a key interest is her estimated queuing position given she has the ticket position d, i.e., N D = d. The conditional distribution of

45 32 N D = d is given by: P (N = n D = d) = P (N = n, D = d) P (D = d) = 1, 0 n = d s x: x Sn, x =d x: x =d p(x), s n d, 0, otherwise. (2.11) Therefore, a customer s ticket position provides complete information of her queuing position only when she observes no more than s tickets upon arrival. We can also compute the conditional mean, d, 0 d s, E[N D = d] = (2.12) d n=s np (N = n D = d), d > s. Let γ n be the probability that a new customer with the queueing position n s reneges as the next departure event. This event occurs if the customer s reneging time is smaller than the next service completion time and the minimum reneging time of other (n s) waiting E-customers. Therefore, γ n = δ, s n. (2.13) sµ + (n + 1 s)δ This implies that n (1 γ i ) is the probability that the new customer with queuing i=s position n eventually receives service. The probability that a new customer with ticket position d eventually receives service, denoted by Γ d, can be obtained by conditioning on N = n D = d : Γ d = 1, d < s, d P (N = n D = d) n (1 γ i ), s d K 1, n=s i=s 0 d K. (2.14)

46 33 Using Γ d, we can compute P (B), P (R), and P (S), the probabilities that a new arrival will balk, renege and complete service, respectively. Conditioning on a new arrival observes d tickets, we obtain P (B) = d s P (D = d)p b (d), (2.15) P (R) = d s P (D = d) P b (d)(1 Γ d ), (2.16) P (S) = s 1 d=0 P (D = d) + d s P (D = d) P b (d)γ d. (2.17) It can be verified that P (B) + P (R) + P (S) = 1. Next, we compute E[W S D = d] and E[W R D = d], the expected waiting time of a customer in queue who eventually completes service and reneges, respectively, conditioned on that she observes s d K 1 tickets upon arrival and joins the system. Consider E[W S D = d] first. Given N = n D = d, s n d K 1, the expected time until this customer receives service is n j=s m j, where m j = 1 sµ+(j s)δ is the expected time until either a customer is served or one of the j s waiting E-customers, excluding herself, reneges. Therefore, E[W S D = d] = d n=s n P (N = n D = d) m j, s d K 1. (2.18) j=s Unconditioning on D, we obtain the expected waiting time of a joining customer who eventually receives service as E[W S ] = K 1 d=s P (D = d)e[w S D = d] = K 1 d=s P (D = d) d n=s P (N = n D = d) n m j. j=s (2.19) Next consider E[W R D = d]. Recall that given N = n s, γ n is the probability

47 that a new customer reneges at the next departure event. If this customer reneges at the k th departure event among the n+1 customers, including herself, 1 k n+1 s, which occurs with probability k 2 j=0 (1 γ n j)γ n (k 1) (where k 2 j=0 (1 γ n j) = 1 if k = 1 by convention), her expected waiting time will be k 1 j= sµ+(n+1 s j)δ. Therefore, the expected waiting time of the joining customer, who has ticket position d, s d K 1, and eventually reneges, is E[W R D =d]= d n+1 s k 1 P (N =n D =d) 1 k 2 (1 γ n j )γ sµ+(n+1 s j)δ n (k 1). n=s k=1 j=0 j=0 (2.20) Unconditioning on D, the expected waiting time of a joining customer who eventually reneges is E[W R ] = K 1 d=s P (D = d)e[w R D = d]. (2.21) Finally, we obtain the expected waiting time of a customer in queue, regardless whether she balks immediately, reneges after waiting awhile in queue, or receives service: E[W ] = E[W S ]P (S) + E[W R ]P (R). (2.22) 2.3. Heuristics The aggregation-disaggregation approach discussed in Sections and is an effective solution procedure for small values of K s. Unfortunately, the cardinalities of S j and T n grow exponentially in K s. The cardinality of T n is 2 K s, n 1, and the cardinality of S j is S j = (K s) ( K s of states in the aggregated Markov chain is S = K ) j s +1, s j K. Therefore, the total number S j = (K s) 2 K s + K + 1. As such, the dimension of the state space S is mainly determined by O(K s)2 K s, which grows at an exponential rate as K s increases. For example, ignoring the linear term K + 1, for K s = 10, S 10, 240 and T n = 1, 024. As K s increases to 15, j=0

48 S 491, 520 and T n = 32, 768. As a result, we can only manage to numerically carry out the two-step procedure for K s = 8. In this section, to overcome this difficulty we develop two heuristics, the 2-Digit Heuristic (2DH) and the 3-Digit Heuristic (3DH). Each heuristic not only drastically reduces the cardinality of the state space, but also yields a Markov chain that is a GI/M/1 type process with a skip-free-to-the-right infinitesimal generator, leading to a polynomial-time algorithm. In the 2-Digit Heuristic (2DH), we approximate the system by combining all E- customers into one group (called the E-group) and all A-customers into another group (called the A-group), and giving the E-group the first service priority (i.e., after all E- customers are called and an idle server becomes available, all A-customers are called and depart simultaneously). We term the heuristic 2DH because the state of the system can be represented by two scalars, representing the cardinalities of the E- and A-groups. For example, state (3,2;1,4) in the original system is represented in 2DH as (4, 6). Because A-customers may delay their departure times under 2DH, this approximation tends to inflate the number of A-customers (and hence the total number of tickets) in the system as compared with that in the original system. As a result, it may cause a new arrival to observe a larger ticket position and hence a higher likelihood of immediate abandonment. Consequently, it is plausible that the joining probability as well as the expected number of E-customers in 2DH are, respectively, the lower bounds of their counterparts in the original model. On the other hand, in the 3-Digit Heuristic (3DH), we approximate the system by combining all E- and A-customers into three groups: the first group consists of the first set of consecutive E-customers and they will be called first; the second group consists of all A-tickets and they will be called afterward; and the last group consists of the rest of the E- customers and they will be called after the A-customers are evacuated. This heuristic is termed 3DH because the system can be represented by the cardinalities of the three groups. For example, state (3,2;1,4) in the original system would be represented in 3DH as (3, 6, 1). Since the A-customers in 3DH tend to be called earlier than the 35

49 36 A-customers in the original system, a new arrival is more likely to experience a smaller ticket position, and hence a higher likelihood of joining the system. Therefore, we expect that the joining probability as well as the expected number of E-customers in 3DH constitute the upper bounds of their counterparts in the original system. These bounds are indeed confirmed in our numerical results. Our results also demonstrate that while 2DH is computationally more efficient, 3DH turns out to be more accurate, resulting in a trade-off between computation time and solution accuracy. In the next two subsections, we discuss the solution procedures for 2DH and 3DH The 2-Digit Heuristic (2DH) For a given state ( n E 1, na 1 ;... ; ne l, ) na l in the original system, the 2DH representation of the state is x = ( n E,n A) ( l = i=1 ne i, ) l i=1 na i. The state space S 2D of 2DH can be represented by S 2D = { x = ( n E, n A), 0 n E K, n A 0 }. Under the 2-digit heuristic, a customer s abandonment behavior mimics that under the original system: upon arrival, she can only observe n E +n A, but not their individual values, and makes the balking-joining decision accordingly. Once joined (balked), however, she will be placed at the end of the E-group (A-group). We again assume the reneging time of a joining customer is exponentially distributed with rate δ. Upon reneging, she will be removed from the E-group and added to the A-group (i.e., n E will be reduced by 1 and n A increased by 1). Figure 2.3 displays the Markov chain with s = 2 and K = 4, using 2DH. Hereafter, the Markov chain under 2DH will be referred to as the 2D-Markov chain. We partition the 2D-Markov chain according to the number of A-customers in the system: T 2D n = {x = (n E, n A ) S 2D, n A = n}, n = 0, 1,....

50 In words, T0 2D contains all the states that have no abandoned tickets, and Tn 2D contains all the states that has n abandoned tickets, n 1. Given balking limit K, we have T 2D 0 = K+1 and T 2D n = K s+1 for any n, which increases linearly with K s. Recall that the cardinality of T n in the original Markov chain is T n = 2 K s, an exponential function of K s. For example, for K s = 30, T n in the original Markov chain has more than 1 billion states and T 2D n 37 in the 2D-Markov chain contains merely 31 states, which represents a % reduction of T n. Therefore, 2DH drastically reduces the number of states in each level when K s is large. Observe that the 2D-Markov chain can move from T 2D n completion (when n E = s), remains within T 2D n to T 2D 0 due to a service due to either a service completion or a new joining customer (when n < K s), or move to T 2D n+1 due to an abandonment, including both balking and reneging. In other words, unlike the original system, the 2D-Markov chain has skip-free-to-the-right transitions, i.e., it is a GI/M/1 type process. For this type of Markov chains, it is easy to calculate its steady-state distribution by forward recursion. As a result, we can avoid the aggregation step discussed in Section 2.2.3, and directly use forward recursion, analogous to Step 2 discussed in Section 2.2.4, to determine the steady-state distribution of the 2D-Markov chain. The following theorem, specifies the infinitesimal generator of the 2DH Markov chain under partition {Tn 2D, n 0}. The proof of Theorem can be found in Appendix C. Theorem The 2D-Markov chain has the lower Hessenberg block-partitioned

51 38 Figure 2.3: The 2DH Representation of the Markov Chain of the Ticket Queue with s = 2 and K = 4 generator given by: T 0 T 1 T 2 T 3 T K s 1 T K s T K s+1 T K s+2 T 0 L 2D 0 F0 2D T 1 B 2D L 2D 1 F1 2D T 2 B 2D L 2D 2 F2 2D Q 2D = T K s 1 B 2D L 2D K s 1 FK s 1 2D T K s B 2D L 2D F 2D T K s+1 B 2D L 2D F 2D T K s+2 B 2D L 2D , where the component matrices in Q 2D are provided in (C.1) (C.7) in Appendix C.

52 Let p 2D = {p 2D n, n 0} be the probability vector associated with the partition {Tn 2D, n 0}. Then, since the probability vectors must satisfy the balance equations p 2D Q 2D = 0, we have p 2D 0 L 2D 0 + ( n=1 p 2D n ) 39 B 2D = 0, (2.23) p 2D n 1F 2D n 1 + p 2D n L 2D n = 0, 1 n < K s, (2.24) p 2D K s 1F 2D K s 1 + p 2D K sl 2D = 0, (2.25) p 2D n 1F 2D + p 2D n L 2D = 0, n > K s. (2.26) To construct the solution of the above system of linear equations, we define matrices Rn 1,n 2D = F 2D n 1 (L2D n ) 1, 1 n < K s, F 2D K s 1 (L2D ) 1, n = K s, and R 2D = F 2D (L 2D ) 1. Note that R 2D 0,1 is a (K + 1) (K s + 1) dimensional matrix and R 2D n 1,n, n > 1, and R 2D are (K s + 1) (K s + 1) matrices. We also let R 2D n = n i=1 R2D i 1,i, for 1 n K s, which is a (K + 1) (K s + 1) dimensional matrix. Then, (2.24)-(2.26) can be written as p 2D n = = p 2D n 1 R2D n 1,n, 1 n K s, p 2D n 1 R2D, n > K s, p 2D 0 R2D (2.27) n, 1 n K s, p 2D ( K s R 2D ) n K+s = p 2D 0 RK s 2D ( R 2D ) (2.28) n K+s, n > K s. To determine p 2D 0, we first substitute the above equation into the normalization equation. Because the 2D-Markov chain is positive recurrent, n=k s+1 (R2D ) n K+s =

53 40 R 2D (I R 2D ) 1. Thus, p 2D 0 ( 1 T + K s n=1 R 2D n 1 T + R 2D K sr 2D (I R 2D ) 1 1 T ) = 1. (2.29) Furthermore, from (2.23) and (2.28), we also obtain p 2D 0 L 2D 0 + n=1 p 2D n B 2D = p 2D 0 ( L 2D 0 + ( K s n=1 R 2D n + R 2D K sr 2D (I R 2D ) 1 ) B 2D ) = 0. (2.30) Equations (2.29) and (2.30) form a system of linear equations that p 2D 0 must satisfy. After discarding any one of the equations in (2.30) (since we use the normalization condition), we obtain the unique solution of p 2D 0. We can calculate p 2D n, n 1 by using (2.28) The 3-Digit Heuristic (3DH) In the 3DH, we use x = ( n E 1,nA,n E ) 2 to represent the system state, with its digits representing the cardinalities of the first set of consecutive E-customers, all A-customers, and the rest of E-customers. The state space S 3D of the ticket queue under 3DH can be represented by S 3D = { x = (0, 0, 0) and x = ( n E 1,n A,n E ) 2 : n E 1 1, n A 0, n E 2 0 }. Under the 3-digit heuristic, a customer s abandonment decision is the same as that under the original system: upon arrival, she can only observe the total number of tickets in the system x = n E 1 +na +n E 2, but not their individual values, and makes the balking decision with probability P b ( x ). If she joins, she will be placed at the end of the first E-group if n A = 0 and at the end of the second E-group if n A > 0. If she balks, she will join the A-group. The reneging time of a joining customer again is exponentially distributed with rate δ. If she reneges from the first E-group with n E 1

54 41 existing customers, her action will trigger the following state transition: first, she will be added to the A-group; in addition, the original first E-group will be split into two groups, the E-customers before the the reneging customer form the new first E-group with ñ E 1 (< ne 1 ) customers, and the remaining (ne 1 ñe 1 1) E-customers are combined with the E-customers in the second E-group to form the new second E-group with (n E 1 ñe ne 2 ) E-customers. If she reneges while waiting in the second E-group, she will be removed from the second E-group and added to the A-group (i.e., n E 2 will be reduced by 1 and n A increased by 1). Figure 2.4 displays the Markov chain with s = 2 and K = 4, using 3DH. Figure 2.4: Markov chain for a 3-Digit Heuristic Ticket Queue with s = 2 We partition S 3D into disjoint subsets T n, n 0, where T0 3D = {x = (n E 1, 0, 0) S 3D : 0 n E 1 K}, and T 3D n = {x = (n E 1, na, n E 2 ) S 3D : s n E 1 K, n A = n, n E 1 + ne 2 K}, n 1. In words, T 3D 0 contains all the states without abandoned tickets and one group of E-customers, and T 3D n contains all the states that have n 1

55 abandoned tickets. The cardinality of T 0 is K +1, and that of T 3D n is K 1 42 (i s+1)+1 = (K s)(k s+1) 2 + 1, a quadratic function of K s. For example, when K s = 30, the number of states in T 3D n is 466 and the number of states in T n in the original system is more than 1 billion, and Tn 3D represents a % reduction of T n. For the rest of the paper, we denote the Markov chain under the 3DH as the 3D-Markov chain. The following theorem, specifies the infinitesimal generator of the 3DH Markov chain under partition {Tn 3D, n 0}. The proof of Theorem can be found in Appendix D. Theorem The 3D-Markov chain has the block partitioned generator: i=s Q 3D = T 0 T 1 T 2 T 3 T 4 T K s 1 T K s T K s+1 T K s+2 T 0 L 3D 0 F0 3D T 1 B 3D L 3D 1 F1 3D T 2 B 3D L 3D 2 F2 3D T K s 1 B 3D L 3D K s 1 FK s 1 3D T K s B 3D L 3D F 3D T K s+1 B 3D L 3D F 3D T K s+2 B 3D L 3D where the element matrices in Q 3D are specified in (D.1) (D.7) in Appendix D. Note that Q 3D has the exact structure as Q 2D, the generator of the 2D-Markov chain, although individual block matrices have different dimensions and specifications. Consequently, we can use the same solution procedure, as described in (2.23) (2.30) to solve the balance equations. Let p 3D = {p 3D n, n 0} be the probability vector associated with the partition {Tn 3D, n 0}. Then, since the probability vectors must satisfy the balance equations

56 43 p 3D Q 3D = 0, we have p 3D 0 L 3D 0 + ( n=1 p 3D n ) B 3D = 0, (2.31) p 3D n 1F 3D n 1 + p 3D n L 3D n = 0, 1 n < K s, (2.32) p 3D K s 1F 3D K s 1 + p 3D K sl 3D = 0, (2.33) p 3D n 1F 3D + p 3D n L 3D = 0, n > K s. (2.34) To construct the solution of the above system of linear equations, we define matrices Rn 1,n 3D = F 3D n 1 (L3D n ) 1, 1 n < K s, F 3D K s 1 (L3D ) 1, n = K s, and R 3D = F 3D (L 3D ) 1. Note that R0,1 3D is a (K + 1) ) dimen- ) ( ) (K s)(k s+1) sional matrix and R 3D n 1,n, n > 1, and R 3D are ( (K s)(k s+1) ( (K s)(k s+1) matrices. We also let Rn 3D = n i=1 R3D i 1,i, for 1 n K s, which is a (K + 1) ( ) (K s)(k s+1) dimensional matrix. Then, (2.32)-(2.34) can be written as p 3D n = = p 3D n 1 R3D n 1,n, 1 n K s, p 3D n 1 R3D, n > K s, p 3D 0 R3D (2.35) n, 1 n K s, p 3D ( K s R 3D ) n K+s = p 3D 0 RK s 3D ( R 3D ) (2.36) n K+s, n > K s. To determine p 3D 0, we first substitute the above equation into the normalization equation. Because the 3D-Markov chain is positive recurrent, n=k s+1 (R3D ) n K+s = R 3D (I R 3D ) 1. Thus, p 3D 0 ( 1 T + K s n=1 R 3D n 1 T + R 3D K sr 3D (I R 3D ) 1 1 T ) = 1. (2.37)

57 44 Furthermore, from (2.31) and (2.36), we also obtain p 3D 0 L 3D 0 + n=1 p 3D n B 3D = p 3D 0 ( L 3D 0 + ( K s n=1 R 3D n + R 3D K sr 3D (I R 3D ) 1 ) B 3D ) = 0. (2.38) Equations (2.37) and (2.38) form a system of linear equations that p 3D 0 must satisfy. After discarding any one of the equations in (2.38) (since we use the normalization condition), we obtain the unique solution of p 3D 0. We can calculate p 3D n, n 1 by using (2.36). In summary, we have introduced two heuristics that not only drastically reduce the state space of the original system, but also yield the Markov chains of a GI/M/1 type, enabling the application of matrix-analytical solution procedures without invoking the aggregation procedure discussed in Section Since Tn 2D = K s + 1 increases linearly, it can handle ticket queues with any practical size of K. On the other hand, Tn 3D = (K s)(k s+1) increases in K s in a quadratical order. When K s = 40, T 3D n contains 821 states, which can be readily managed in our numerical procedure Performance of Heuristics In this section, we compute the performance measures defined in Section by using the distributions of {p(x)}, {p 2D (x)} and {p 3D (x)}. We evaluate the accuracy of both heuristics by considering a ticket queues with s = 2 servers and investigate the impact of the queue parameters on system performance measures. To compare the performance of the original and the heuristic ticket queues, we use a total of 180 system scenarios as follows: 1) 12 levels of balking limit K = 5, 6, 7, 8, 9, 10, 15, 20, 25, 30, 35, 40, where for each given K, we use (2.3) to deduce the corresponding balking patience rate θ so that P b (K) ; 2) 3 levels of traffic intensity ρ = 0.75, 1.00, 1.25; we select the parameter value ρ 1 to investigate how the ticket queue performs in heavy traffic regimes; and 3) 5 levels of reneging rate δ = 0, 0.2 θ, 0.4 θ, 0.6 θ, 0.8 θ, where

58 45 δ = 0 corresponds to the scenario that customers do not renege once they join the system. Table 2.1 presents the state space reduction achieved through the use of heuristics. When the balking limit K increases beyond 10 (i.e. the customers become more patient), we reduce the state space by more than 99% using either heuristic. Table 2.1: State space reduction through heuristics # states # states %Reduced # states %Reduced K-s Original TQ 2DH by 2DH 3DH by 3DH % 13 0% % % % % % % % % % % 8 2, % % 9 4, % % 10 10, % % , % 1, % 20 20,971, % 4, % ,860, % 8, % For K s 8, we compute the exact steady-state distribution {p(x)} of the original ticket queue using the aggregation-disaggregation procedure discussed in Sections and For K s > 8, we rely on simulations of the original ticket queue since the exact solution of {p(x)} cannot be obtained due to the exponential growth of the state space. We compare the exact and the simulation steady-state distributions for the original ticket queue with the computed results of {p 2D (x)} and {p 3D (x)}, the approximate distributions under 2DH and 3DH, respectively. Figure 2.5 displays the snapshots for the distributions of N, N 2D, and N 3D for a system with K = 10, θ = 2.5, δ = 0.6 θ = 1.5, whereas Figure 2.6 displays that for a system with K = 25, θ = 0.75, δ = 0.8 θ = 0.6, under the traffic intensities ρ = 0.75 and ρ = 1. Those plots show that for the systems with less patient customers (K = 10), the heuristics prove to be accurate approximations. However, when the customers patience level increases (K = 25), the heuristic results start to slightly deviate from the exact (simulation) results, but still stay to be robust approximations. On the

59 46 other hand, Figures 2.5 and 2.6 also reveal that the heuristic approximations start deteriorating when the traffic intensity becomes extremely heavy (ρ = 1). Although the plots for the comparison of the distribution of D under original and heuristic ticket queues are not displayed in this paper, overall, the heuristic ticket queues behave extremely close to the original ticket queue measured by the distributions of N and D. Figure 2.5: Distributions of N, N 2D, and N 3D for K = 10, θ = 2.5, δ = 1.5 Figure 2.6: Distributions of N, N 2D, and N 3D ; K = 25, θ = 0.75, δ = 0.6 To compare the closeness of the exact and the heuristic solutions statistically,

60 47 we use the maximum absolute difference (MAD), which is the maximum of all the absolute differences between two compared sequences of values, and the total absolute difference (TAD), which is the sum of the absolute differences between two compared sequences of values. Tables 2.2 and 2.3 compare the TAD and MAD values for the distributions of N and N 2D, N and N 3D, D and D 2D, and, D and D 3D, for δ = 0.8 θ, ρ = 0.75 and ρ = 1.0. The largest TAD values for P (N) and P (D) ( and , respectively) are obtained for the difference between the 2DH and the original system steady-state distributions when K = 25, ρ = 1. The largest MAD values for P (N) and P (D) ( and , respectively) are also obtained for the difference between the 2DH and the original system steady-state distributions when K = 25, ρ = 1. These results confirm that both heuristics provide accurate and robust approximations, but show slight deteriorations when the customers have higher levels of patience and the traffic intensity becomes heavy (ρ 1). Table 2.2: Comparison of Deviations for P(N) and P(D), ρ = 0.75, δ = 0.8 θ Exact (Simulation) vs. 2DH Exact (Simulation) vs. 3DH P(N) P(D) P(N) P(D) K TAD MAD TAD MAD TAD MAD TAD MAD On the other hand, TAD and MAD values for the difference between the 3DH and the original ticket queue steady-state distributions are lower than those for the difference between the 2DH and the original ticket queue steady-state distributions. Therefore, the 3DH approximation is more accurate than the 2DH approximation. This result should follow as the 2DH eliminates a larger number of system states than the 3DH does, and one would expect that the 2DH will be less accurate than the 3DH. In conclusion, both heuristics provide high quality approximations in terms

61 48 Table 2.3: Comparison of Deviations for P(N) and P(D), ρ = 1, δ = 0.8 θ Exact (Simulation) vs. 2DH Exact (Simulation) vs. 3DH P(N) P(D) P(N) P(D) K TAD MAD TAD MAD TAD MAD TAD MAD of solution accuracy and robustness. Although the 2DH is computationally more efficient than the 3DH, the latter provides more accurate and robust approximations. We also investigate the effect of a change in the reneging rate δ on the heuristics performance. For a given θ, we vary delta from 0 to 0.8 θ and observe the MAD and the TAD values in Figures 2.7 and 2.8, respectively. As seen in these plots, the highest deviations occur for δ = 0 case where a new arrival has infinite reneging patience time and either joins the system to receive service without abandoning the system, or balks immediately. When the customers become less patient (reflected by increasing values of δ), the MAD and the TAD values tend to decrease. Figure 2.7: δ vs. MAD; K = 10, θ = 2.5 In our discussion of the two heuristics, we conjectured that: i) the expected number of existing customers in the original ticket queue will be bounded by its coun-

62 49 Figure 2.8: δ vs TAD; K = 10, θ = 2.5 terpart in the 2-Digit heuristic (a lower bound) and its counterpart in the 3-Digit heuristic (an upper bound); and, ii) the joining probability in the original ticket queue will be bounded by that in the 2-Digit heuristic (a lower bound), and by that in the 3-Digit heuristic (an upper bound). To validate these conjectures, we compared the E[N] and P (B) values of the original, 2DH and 3DH ticket queues. Figure 2.9 plots E[N] using steady-state distributions of the original, 2DH and 3DH ticket queues. For K > 10, simulation results are displayed rather than the exact analysis results. In general, the expected number of existing customers in the system increases in customers balking patience limit. Figure 2.9 also indicates that, the expected number of existing customers is the highest in the 3DH ticket queue, followed in order by that in the original and the 2DH ticket queue. These results confirm our first conjecture. In addition, Figure 2.10 provides a snapshot for the comparison of balking probabilities P (B), P (B 2D ), and P (B 3D ). First of all, systems with higher traffic intensity levels exhibit higher balking probabilities. Secondly, the balking probability decreases in the customers balking patience limit. Finally, the plots indicate that the balking probability is the highest in the 2DH ticket queues, followed in order by the original and the 3DH ticket queues. Therefore, we also provide support for our second

63 50 Figure 2.9: Expected number of E-customers in TQ, 2DH and 3DH; K = 10, 20, 30, δ = 0.8θ conjecture. On the other hand, Figure 2.11 provides a snapshot for the comparison of the system reneging probabilities obtained through exact or simulation analysis. In general, systems with higher traffic intensity levels exhibit higher reneging probabilities. The reneging probabilities exhibit a concave pattern such that the probability values increase up to a certain point (specific to a system parameter set), and then level off. This can be interpreted as follows: As the balking limit increases, the customers become more patient. Therefore, for the ticket queues, there might exist the threshold for reneging after which the service rate of the system will be larger than the combined reneging rate of the customers, and the reneging probabilities will start converging to a unique value. In general, the 3DH reneging probabilities are higher than their counterparts in the original TQ and 2DH systems. Finally, the overall abandonment probabilities for the original ticket queue system is bounded with the abandonment probabilities of the 2DH (upper bound) and the 3DH (lower bound) systems. We also investigated the effects of the changes in ρ and δ on the service completion

64 51 Figure 2.10: Balking Probabilities in TQ, 2DH and 3DH: ρ = 0.75, 1.00, 1.25, δ = 0.8θ Figure 2.11: Reneging Probabilities in TQ, 2DH and 3DH; ρ = 0.75, 1.25, δ = 0.8θ and abandonment probabilities. As an example, for an original ticket queue with balking limit of K = 10, Figure 2.12 indicates that the service completion probability decreases both in δ and ρ. As a summary, the heuristics provide accurate and robust approximations to the original ticket queues. In general, for a given level of balking patience (K), the service completion probability decreases as the traffic intensity (ρ) and the reneging rate (δ)

65 52 Figure 2.12: Effects of δ and ρ on Service Completion Probability in TQ; K = 10, θ = 2.5 increases. The expected number of E-customers in the original ticket queue is bounded by that in the 2-Digit heuristic (a lower bound) and that in the 3-Digit heuristic (an upper bound); and, the joining probability for the original model is bounded by that in the 2-Digit heuristic (a lower bound) and that in the 3-Digit heuristic (an upper bound). The conjectures we proposed for the heuristics bounding behavior were validated with our results Service Improvement of the Ticket Queue Although the ticket queues provide efficient mechanisms to control customer flows in service systems, their drawback is the fact that the actual number of existing customers in the system is not known either by the system managers nor the customers. This lack of information results in customers naive abandonment behavior, where they base their abandonment decisions on the number of tickets ahead of their ticket. Predicting waiting times is difficult for customers when they do not have full information about the system state, and their predictions might result in overestimation

66 53 of their waiting time if they assume all the tickets in front of them represent existing customers. This behavior can adversely affect the system performance. On the other hand, in the physical queues, the customers observe the number of existing customers in front of them, and base their abandonment decisions on that number. Therefore, when we compare the performance measures of the ticket queue with those of the physical queue, we identify performance gaps for the ticket queues. For example, in a 2-server system, Figure 2.13 displays that the existing number of customers in the physical queue is bounded below by the existing number of customers in the ticket queue (assuming both systems have the same balking and reneging behavior of customers). Figure 2.13: Comparison of E[N] in Ticket and Physical Queues; K = 10, δ = 0.8θ Moreover, Figure 2.14 illustrates that the service completion probabilities are higher for the physical queue than those of the ticket queue under different traffic intensity regimes. Hence, the following questions remain to be answered for keeping the ticket queues as preferred systems: i) Is it possible to match (or even surpass) the performance of the physical queues by providing extra information to the ticket queue customers? ii) What types of information improve the system performance measures

67 in the ticket queues? iii) What are the effects of different types of information provided on the abandonment probabilities? Figure 2.14: Comparison of Service Completion Probabilities in Ticket and Physical Queues K = 10, δ = 0.8θ 54 To improve the system performance measures, such as improving the service completion probability, of the ticket queues, we provide conservative and aggressive types of waiting time information. In the conservative case, we provide an estimate of the waiting time of a new arrival, conditioning on her ticket position and ignoring the reneging behavior of the existing customers. In the aggressive case, we provide an estimate of the waiting time of a new arrival, conditioning on her ticket position and considering the reneging behavior of the existing customers. On the other hand, in each case, we further categorize the information into the expected value case and the random value case. In the expected value case, we provide the expected waiting time for the customers who completed service and had the same ticket position with the new arrival. In the random value case, we provide the waiting time information for the last customer who completed service and had the same ticket position with the new arrival. If we assume that a new arrival has ticket position d, Table 2.4 displays a summary for the types of information provided. In Table 2.4, the conservative random value case information is interpreted as follows: For a new arrival with ticket position d, we find the last customer who com-

68 55 Table 2.4: The types of waiting time information provided Expected Value Random Value E[N D=d] Conservative sµ S N D=d Aggressive E[W S D = d] W S D = d pleted service when she had the same ticket position (d) upon arrival, and determine the number of existing customers, N, (given the ticket position d) upon that last customer s arrival to the system. We then assume the current arrival has the same number of existing customers ahead of her and all those existing customers will complete service without reneging. Therefore, we provide the service completion time (Erlang density function with parameters N s + 1 and sµ) for those N existing customers given the ticket position d for the new arrival. Aggressive random value information, W S D = d, refers to the actual waiting time for the last customer who had the ticket position d upon arrival and completed service. As we ignore the reneging behavior of customers in the conservative type of information, the customers are provided with an overestimate of their expected waiting time. Conversely, by considering the reneging behavior in the aggressive information case, we provide the exact estimate of their expected value or a random observation of the waiting time. Our objective is to understand how the service completion and the abandonment probabilities are affected by each type of information availability. In order to point out the effects of different types of information, we compare the values of performance measures under different information regimes relative to the current ticket queue performance measures. In general, assuming a base value of 0 for the ticket queue performance measures, we determine how much of an increase (a decrease) we obtain by providing different information to the customers. Figures display the effects of different information on the service completion, balking and reneging probabilities for a system with s = 2, K = 50 and δ = 0.8θ. Figure 2.15 indicates that in systems with more patient customers, by providing all

69 56 Figure 2.15: Effects of Information on Service Completion Probability (K = 50, δ = 0.8θ) Figure 2.16: Effects of Information on Balking Probability; (K = 50, δ = 0.8θ) types of information, we can improve the service completion probabilities from the ticket queue level. By providing aggressive information, we obtain higher service completion probabilities than the physical queues. The expected value information also outperforms the random value information in the aggressive information case. On the other hand, providing conservative information improves the service comple-

70 57 Figure 2.17: Effects of Information on Reneging Probability; (K = 50, δ = 0.8θ) tion probabilities from the base ticket queue values, but can not capture the gap in between the ticket and physical queues. Contrary to the aggressive information case, the random value information outperforms the expected value information in the conservative case. On the other hand, Figure 2.16 reveals that the balking probabilities are reduced by all types of information. The highest reduction is obtained through the aggressive information expected value case, followed by the aggressive information random value case. The conservative random value case reduces the balking probability more than the conservative information expected value case does. Finally, Figure 2.17 displays that providing information increases the customer reneging probabilities for all the information cases. The aggressive information expected value case provides the highest increase followed by the aggressive information random value case. In conclusion, these numerical results indicate that providing information will in general improve the service completion probabilities (by reducing the overall abandonment probability), but a reduction in the balking probability is followed by an increase in the reneging probability. Therefore, when providing information, system managers have to con-

71 58 sider the trade-off between the change in the balking and the reneging probabilities, and determine the long run costs of losing the customers through balking opposed to reneging. In this analysis, I also show that providing information does not guarantee improvement in service completion probabilities (or even joining probabilities). We can improve the customer joining probabilities with higher percentages in the systems with less patient customers (lower K values) than in the systems with more patient customers (higher K values). In addition, for higher K values, the improvement for the service completion probabilities is less than that for lower K values. When the traffic intensity level (ρ) increases, the benefits we get by providing information also increase Conclusions and Future Studies Intelligent ticket queue systems are becoming more and more popular in the services systems. In this paper, we present an analytical model to quantify service performance in the multi-server ticket queues with balking and reneging customers. First, we develop the Markov chain model for the multi-server ticket queues and identify the key performance measures. As the model provides the stationary distribution only low levels of balking patience for customers, and we present efficient and accurate heuristics to analyze systems with higher levels of balking patience for customers. Second, we benchmark the performance measures of the ticket queues with those of the physical queues, and gain some insights on the impact of information loss in the ticket queues on key performance measures. And finally, we investigate how to improve the key performance measures by providing different types of information. The heuristics prove to be accurate and robust approximations for the original ticket queue model. The comparison of the system performance of the ticket and the physical queues indicates that the physical queues provide better performance

72 59 measures than the ticket queues do. To close the gap in between the system performances, we provide four different types of information to the ticket queue customers. Although we improve the overall service completion rates, an decrease in the system balking probability results in an increase in the system reneging probability. Therefore, we point out the trade-off between the improvement in balking probability and the degradation in reneging probability, and recommend the system managers to consider the effects of losing customers through balking or reneging. A future study will be to quantify the total costs of abandonment for the customers, and compare the overall system costs to determine which type of information improves the system performance the most. On the other hand, there is a section of literature that focuses on customer regret. The customer regret can simply be classified as: i) the regret for a customer who balks and would have received service before his balking patience time in case she had chosen to stay; ii) the regret for a customer who joins the queue, but her actual waiting time turns out to be larger than her balking patience time. As an additional study, we plan on analyzing customer regret in ticket queue systems. Certain ticket queue systems (such as Q-Matic ticket queues) also enable the system managers to collect invaluable information about customer arrival, service completion and abandonment behaviors. Using these information, this study can assist ticket queue system managers to quantify their system performance, and simulate the impact of their decisions on workforce allocations, workforce hirings and layouts. In this paper, we assume that all the customers demand the same type of service from identical servers. Another future project is to analyze the ticket queues with non-identical parallel servers and with service differentiation. We also would like to analyze the workforce allocation strategies for ticket queue systems.

73 Chapter 3 A Markov Chain Model for the Ticket Queue with State-Dependent Abandonment 3.1. Methodology In this chapter of the dissertation, we extend the ticket queue model developed in Section 2 by considering state-dependent reneging rates and provide the framework for the generalized ticket queue model. We make the following assumptions for the generalized ticket queue model: Customers arrive to the system in a Poisson process with rate λ. There are s identical servers, with service times being independent and identically distributed (i.i.d.) exponential random variables with mean 1/µ. The ticket queue system displays the ticket numbers of the customers currently in service, which assists the waiting customers in their decision to continue to wait or renege and will call waiting customers for service following the FCFS (First-Come, First-Served) rule. If one of the s servers is idle when a new customer arrives, the customer will always join and start service immediately; otherwise, the customer will balk with a certain state-dependent probability. We assume that a customer will base her balking decision on her ticket position, and the balking probability will be the same as outlined in 2.2. If the customer joins the queue, then her reneging time depends on her current ticket position: if her current ticket position is j, then she will renege from the system if her waiting time in this position exceeds an exponentially distributed time T r,j with mean 1/δ j. We assume T b and T r,j, j 1, are independent random variables. A 60

74 61 customer will not abandon the system after she starts receiving service. We assume δ j < θ, j 1, i.e. customers become more patient, in a stochastic order sense, once they join the system such that, for every ticket position, T r,j st T b. When a customer joins the system her reneging rate will be updated every time she advances in her ticket position. To understand how the reneging rate δ j depends on the current ticket position j of a customer, we offer two alternative views. First, people tend to focus more on progressing toward the goal (Meyer 1994) and will be willing to wait longer when they advance in their positions in the system. In addition, customers often exhibit the behavior known as the escalation of commitment, which states that the probability of a customer s willingness to wait is more than her initial reneging time increases with the amount of time she has already spent in the system. Bitran et al. (2008) state that there is no theory of sunk costs in the domain of time. Zhou and Soman (2003) found that the sunk cost effect is not observed for past investments of time, but reappears when the investments are expressed as monetary quantities. Despite these literature, in a survey study presented in Section 4 of this dissertation, we show that 87.2% of the survey participants would take the time they already spent waiting into account when they decided to continue to wait or abandon the system, and the escalation of commitment effect is observed in both the ticket and physical queue scenarios. From this perspective, we may assume that δ j is a increasing function of j. This implies that the system reneging rate, i.e., the rate that one of the waiting customers abandon her ticket, will be increasing and convex in the number of the tickets in the system. On the other hand, one could also argue that a waiting customer would take the time she already spent in line into account, and the more she waits, the more impatient she becomes. Since the customer with a smaller ticket position tends to have already waited longer than the customer with a larger ticket position, it implies that δ j should be a decreasing function of j. Therefore, from this perspective, the system s reneging rate is an increasing concave function of the number of tickets in the system (Plambeck 2009).

75 62 Our objective, again, is to construct a Markov chain model that enables us to extract information of the number of existing customers N and the number of existing tickets D. We use the same system state representation as in Chapter 2, denoted by x, as a 2l-dimensional vector (or l pairs of numbers) as follows: x = ( n E 1,n A 1 ; n E 2,n A 2 ;... ; n E l,na l ) = ( n E i,n A i : 1 i l ), 1 l K s 2 + 1, where n E i 1 is the number of E-customers in the ith E-group, i = 1,..., l, and n A i 1 is the number of A-customers in the ith A-group, i = 1,..., l 1, and n A l 0 is the number of A-customers, if any, following the last group of E-customers (n A l = 0 if no A-customers follow the last E-customer). Let x = l n E i and x = i=1 l (n E i + n A i ), i=1 respectively, be the total number of E-customers and the total number of E- and A-customers (i.e. the total number of tickets) in state x, that is, x and x are realizations of N and D. Also let x j = j n E i and x j = i=1 j (n E i + n A i ), i=1 respectively, be the total number of E-customers and the total number of E- and A-customers (i.e. the total number of tickets) in the first j pairs of state x. As an example, if x = (3, 2; 1, 0), then x = = 4, x = = 6, x 1 = 3, x 1 = = 5. If the number of E-customers is less than the number of servers, i.e., 0 x < s, then we must have x = x, hence all the E-customers are in a single group, and the state of the system becomes x = (n E 1, na 1 ) = ( x, 0), 0 x < s, where state (0,0) represents an empty system. Clearly, our state representation gives complete information of the numbers of E- and A-customers in the system. It can

76 be shown that the Markov chain that can accommodate at most K E-customers is ergodic, regardless of the traffic intensity ρ = λ sµ. The state space S of the ticket queue can be represented as follows: 63 S = x = (0, 0) and x = ( n E i,na i : 1 i l ) : n E i 1, 1 i l and n A i 1 for 1 i l 1, n A l 0, x n A l K, 1 l K s (3.1) The constraints on n E i and n A i ensure that every group of E- or A- customers is at least of size 1, except the last group of A-customers, which can be of size 0; the constraint x n A l K means that the total ticket count of any state x, excluding the last group of A-customers, is no more than K, since a customer with her ticket position greater than or equal to K would not join the system. Figure 3.1 displays the transition rate diagram of a ticket queue with s = 2 and K = 4, where states (0,0) and (1,0) correspond to the system with no customers and 1 E-customer, respectively; column 1 entries correspond to the system with two E- customers; column 2 and 3 entries correspond to the system with three E-customers; and, column 4 entries correspond to the system with four E-customers. In Figure 3.1, an arc labeled λ or λ P b (d) (λp b (d)) corresponds to a new arrival who observes her ticket position d and joins (balks) the system, an arc labeled kµ, k = 1, 2, corresponds to service completion, and an arc labeled δ j corresponds to the customer with the ticket position j reneging from her position. For example, suppose the current state is x = (3, 0), meaning that the first two E-customers are currently being served, the third E-customer is waiting in queue, and there are no abandoned tickets in the system. A new customer arriving in state x = (3, 0) will infer her ticket position as x = 3+0 = 3 (i.e., 3 tickets ahead of her ticket). With probability λ P ( x ) = λ P b (3), this customer will join the queue and move the system to state x = (4, 0); with probability λp b (3), this customer will balk and move the system to state x = (3, 1). On the other hand,

77 64 Figure 3.1: Transition Diagram of a Ticket Queue with state-dependent reneging for s = 2 and K = 4 a service completion in state x = (3, 0) occurs with rate 2µ, which moves the system to state x = (2, 0). Finally, if the waiting E-customer in (3,0) reneges, the system will move to state x = (2, 1) with rate δ 2. Note that if a state has more than s + 1 E- customers, the reneging of the customers with different ticket positions will move the system to different states with different rates. For example, in Figure 3.1, if x = (4, 0), then the abandonments of the third and fourth E-customers will move the system to states (2,1;1,0) and (3, 1) with rates δ 2 and δ 3, respectively.

78 3.2. The Stationary Distribution of the Ticket Queue Our multi-server ticket queue with state-dependent abandonment is a high-dimensional Markov chain, and as in Chapter 2, we propose matrix-analytical techniques (Neuts 1981) for the stationary distribution of this Markov chain. In our Markov chain model, we define level n as the number of the last group of A-customers in the system, and the phases in level n as the collection of states 65 T n = { x = (n E i, n A i ; 1 i l) S : n A l = n }, n = 0, 1,.... (3.2) Then S = {T n, n 0}. Within T n, we order any two states x and x in a lexicographical order, i.e., x = ( n E i,na i : 1 i l ) ( is listed ahead of x = ñ E i,ña i : 1 i l ) if (1) l < l, or (2) l = l, and x is majorized by x in the sense that the sum of the first k digits in x is no more than its counterpart in x for k = 1,..., 2l. In Figure 3.1, T n = {2n, 3n, 4n, 211n} corresponds to the states in the row with the last digit being n, n 0. It can be verified that T n, the cardinality of T n, is (2 K s + s) for n = 0 and 2 K s for n 1. Our objective is to derive the steady-state probability distribution p = {p n, n 0}, where p n is the probability vector associated with block T n. The following theorem specifies the infinitesimal generator of the Markov chain under partition {T n, n 0}. The proof is given in Appendix F. Also, in Appendix G, we illustrate the construction of the matrices by the example of K = 4 and s = 2. Theorem The block-partitioned infinitesimal generator of the Markov chain for the ticket queue under partition {T n, n 0}, defined in (3.2), is given by

79 66 Q G = T 0 T 1 T 2 T K s 1 T K s T K s+1 T K s+2 T 0 L 0 F 0,1 F 0,2... F 0,K s 1 F 0,K s T 1 B 1 L F 1,1 F K s 2 F K s 1 F K s T 2 B 2 L F K s 3 F K s 2 F K s 1 F K s T K s 1 B K s 1 L F K s 1,1 F 2 F 3 T K s B L F 1 F 2 T K s+1 B L F 1... T K s+2 B L..... (3.3) where L 0 is a (2 K s + s) (2 K s + s) dimensional matrix, B i, 1 i K s 1, and B are 2 K s (2 K s + s) dimensional matrices, F 0,j, 1 j K s, are (2 K s + s) 2 K s dimensional matrices, and all others are 2 K s 2 K s dimensional matrices. These matrices are defined in expressions (F.1)-(F.9). Since the Markov chain with the partition {T n, n 0} can skip to non-neighbouring blocks on both left and right, the Markov chain is neither of a skip-free-to-the-right type (a transition from level n to level n is impossible if n > n + 1) nor a skip-freeto-the-left type (a transition from level n to level n is impossible if n < n 1). As a result, the block-partitioned infinitesimal generator Q G does not have a lower or an upper Hessenberg form, such a form is generally considered essential to enable the well-known matrix-analytical techniques popularized by Neuts (1981). To proceed, we observe that, for any x T n, n K s, a new arrival in state x always abandons the system immediately, and the matrices start to exhibit a repetitive pattern, independent of n. Recall that p = {p n, n 0} is the equilibrium distribution of the Markov chain under partition T n, n 0. It can be seen from the structure of generator Q G that the probability vector p n associated with the block T n, n K s,

80 67 depends only on p n (K s), p n (K s 1),..., and p n 1, the probability vectors associated with the preceding K s blocks of T n. This motivates us to carry out the solution procedure of the Markov chain in two steps similar to Chapter 2. In the first step, our objective is to compute p(x), for all x T n, 0 n K s 1. To achieve this, we repartition the state space so that we can use the quasi birth-death (QBD) process framework. In a QBD process, the state space of a two-dimensional Markov chain is partitioned into different levels, where each level has a finite number of states, and transitions are allowed only to the neighboring levels or in the same level (Neuts 1981). In our Markov chain model, we can define level j as the total number of E-customers in the system, and the phases in level j, S j, as the collection of states, S j = {x S : x = j}, 0 j K. Since the number of E-customers can only increase or decrease by 1 or remain the same in each transition, the Markov chain can only visit one of its neighboring levels or remain in the same level. However, because the number of states in level j, s j K, is infinite (e.g., each column in Figure 3.1 has an infinite number of states), our Markov chain is not a QBD process, which requires the number of states in each level to be finite (Neuts 1981). Denote x = (n E 1, na 1 ;... ; ne l, na l ) by x = (x l, n A l ), where x l = (n E 1, na 1 ;... ; ne l 1, na l 1 ; ne l ). We apply a similar state aggregation-disaggregation procedure as in Chapter 2, as follows. Step 1 Aggregate all the states x with x = j, x = (x l, n A l ), x l = (n E 1, na 1 ;... ; ne l 1, na l 1 ; ne l ) and at least (K s) A-customers in the last group into a single super-state S j,xl : S j,xl = {x S : x = j, x = (x l, n A l ), na l K s}. The Markov chain with the super-states (called the aggregated Markov chain) has a finite number of states in each level and thus fits into the framework of a QBD process.

81 68 Step 2 Disaggregate super-states S j,xl, j s, and partition the original state space S according to {T n, n 0}, given in (3.2). Since we have already obtained p n, 0 n < K s, and p(s j,xl ), j = s,..., K, in Step 1, we can compute the steady-state probabilities of the individual states in each super-state recursively. The solution procedure is derived in Section The Aggregated Markov Chain and its Stationary Distribution The aggregation of the original Markov chain is based on the idea that if the number of tickets in the system is greater than or equal to the balking limit K, a new customer will automatically balk, so the transition rates of the states in super states show a repetitive pattern, as seen in Figure 3.1. For example, in the Markov chain with s = 2 and K = 4, all the shaded states in Figure 3.1 have the number of tickets greater than or equal to K. Through state aggregation via super states S j,xl, we reduce the infinite state space S into a finite state space x with j = x : S j = S = K j=0 S j, where the finite set S j contains all the states {(j, 0) : l = 1, x l = n E 1 = j}, 0 j < s, { x S : x = j, x = (xl, n A l ), na l K s 1 } {S j,xl ( x l : l i=1 ne i = j)}, s j K. In other words, for each column of the transition matrix in Figure 3.1, we will have a corresponding super state S j,xl to aggregate all the states with x = j, x l = (n E 1, na 1 ;... ; ne l 1, na l 1 ; ne l ), na l K s. For example, for a 2 server system with K = 4, we have: S 0 = {(0, 0)}, S 1 = {(1, 0)}, S 2 = {(2, 0), (2, 1)} {S 2,2 }, {(3, 0), (3, 1), (2, 1; 1, 0), (2, 1; 1, 1)} {S 3,3, S 3,211 } and S 4 = {(4, 0), (4, 1)} {S 4,4 }. S 3 = Let us specify the state transitions in the aggregated Markov chain. We let a state transition between two non-super states in the aggregated Markov chain be the

82 same as its counterpart in the original Markov chain, with the same rate. We also let a state transition from a non-super state to a super state in the aggregated Markov chain (such a transition corresponds to either a new balking customer with rate λ or a new reneging customer with rate δ j ) be the same as the state transition from the non-super state to a state within the super state, again with the same rate. In terms of state transitions between super states S j,xl, we note, first, since a new arrival in S j,xl will result in an automatic balking, the number of E-customers will remain the same, and the Markov chain will remain in super-state S j,xl ; second, for super-state S s,s, a service completion will move the Markov chain to state (s 1, 0), since all A-customers in queue will leave the system simultaneously along with the customer who just got served. For s < j K, a service completion or reneging will reduce the number of E-customers to j 1 and if the Markov chain is at a super state in system will move to another super state in s = 2 and K = 4 is displayed in Figure 3.2. S turns out to be a QBD pro- The aggregated Markov chain with state space cess. Let S j denote the cardinality of 69 S j, the S j 1. An aggregated Markov chain with S j. Within S j, we order any two states x and x in a lexicographical order, i.e., x = ( n E i,na i : 1 i l ) is listed ahead of ( x = ñ E i,ña i : 1 i l ) if (1) l < l, or (2) l = l, and x is majorized by x in the sense that the sum of the first k digits in x is no more than its counterpart in x for k = 1,..., 2l.

83 70 Figure 3.2: Aggregated Ticket Queue Markov chain with state-dependent reneging for s = 2 and K = 4 The infinitesimal generator of the QBD process has the tridiagonal form: Q G = A 0,0 A 0,1 A 1,0 A 1,1 A 1, A s 1,s 2 A s 1,s 1 A s 1,s A s,s 1 A s,s A s,s A K 1,K 2 A K 1,K 1 A K 1,K, A K,K 1 A K,K where the matrices in the first s rows are scalars with A j,j+1 = λ, A j,j 1 = jµ and A j,j = (λ+jµ), for 0 j s 1 (with the exception for A s 1,s, a 1 S s matrix where A s 1,s (1, 1) = λ, A s 1,s (1, k) = 0, 2 k S s ), corresponding to state transitions when the system has idle servers; and the matrices in the last K s + 1 rows, A j,j 1, A j,j, and A j,j+1, are of S j S j 1, S j S j, S j S j+1 dimensions, respectively,

84 corresponding to state transitions when the system has j existing customers in the system, for s j K. A j,j 1, A j,j, and A j,j+1 matrices are defined in expressions I.1-I.3. For example, for s = 2 and K = 4, we have A 0,0 = λ, A 0,1 = λ, A 1,0 = µ, ( ) A 1,1 = (λ + µ), A 1,2 = λ 0 0 and, 2µ (λ + 2µ) λp b (2) 0 A 2,1 = 2µ, A 2,2 = 0 (λ + 2µ) λp b (3), 2µ 0 0 2µ 2µ δ 2 0 λ P 0 2µ δ b (2) A 2,3 = 0 0 λ P b (3) 0 0 0, A 2µ 0 δ 3,2 = 3, µ δ µ + δ µ + δ 3 71 (λ+2µ+δ 2 ) λp b (3) (λ+2µ+δ 2 ) 0 0 λ (λ+2µ+δ A 3,3 = 3 ) λ 0 0, (λ+2µ+δ 3 ) 0 λ (2µ+δ 2 ) (2µ+δ 3 ) λ P b (3) µ δ δ A 3,4 =, A 4,3 = 0 2µ 0 δ δ 3 0, µ + δ 3 δ

85 (λ+2µ+δ 2 +δ 3 ) λ 0 A 4,4 = 0 (λ+2µ+δ 2 +δ 3 ) λ. 0 0 (2µ+δ 2 +δ 3 ) 72 Let p j be the vector of the steady-state probabilities of the states in S j, 0 j K. For 0 j s 1, S j = {(j, 0)}, and p j = p(j, 0). For s j K, and in general, p j is a S -dimensional j row vector. The steady state probabilities can be computed by solving the balance equations ( p 0, p 1,..., p K ) Q G = 0, which translates to: λ p 0 + µ p 1 = 0, p j 1 A j 1,j + p j A j,j + p j+1 A j+1,j = 0, j = 1,..., K 1, p K 1 A K 1,K + p K A K,K = 0. Recursively, we can write the solution to the above equations as follows: p j = λ j j!µ j p 0, 1 j s 1, p j 1 R j 1,j, s j K, (3.4) by where R j 1,j, a S j 1 S j matrix, can be found recursively, starting from j = K, R K 1,K = A K 1,K (A K,K ) 1, R j 1,j = A j 1,j (A j,j + R j,j+1 A j+1,j ) 1, s j K 1. Define R j = j R i 1,i. Then, p j = p s 1 R j, s j K, where p s 1 = i=s (2.7). Now, using the normalization equation p 1 T = p 0 ( s 1 i=0 λ i i!µ i + λs 1 (s 1)!µ s 1 p 0 by ) λs 1 K (s 1)!µ s 1 j=s R j 1 T = 1, where 1 T is the transpose of a unit vector 1 of appropriate dimension, we finally obtain

86 73 p j = ( 1 s 1 λ i i=0 i!µ i + λs 1 K (s 1)!µ s 1 j=s R ( j ) s 1 λ i i=0 i!µ i + λs 1 K (s 1)!µ s 1 R j 1 T j=s R j 1 T ) λj j!µ j, 0 j < s, λs 1 (s 1)!µ s 1, s j K. (3.5) The next theorem relates the steady-state distribution of the aggregated Markov chain { p j, 0 j K} to the steady-state distribution of the original Markov chain. Theorem Let { p(x), x S } be the steady-state distribution of the aggregated Markov chain, and {p(x), x S } be the steady-state distribution of the original Markov chain. Then p(x) = p(x) for x = x and x S j,xl, s j K, and p(s j,xl ) = x S : x=(x l,n A p(x), for any l ), x =j,na s j K. K s l The above theorem implies that the aggregated Markov chain provides complete information for E-customers in the system, N, through the expression P (N = j) = x S p(x), j = 0, 1,..., K. However, {p(x), x S } provides neither the marginal j distribution of D nor the joint distribution of (N, D), as both depend on the individual probabilities of the states aggregated into super-states. In the next section, we provide the solution procedure for the individual probabilities of the states in super-states, using the partition of the state space S = {T n, n 0} defined in (3.2) Steady-State Probabilities of States in Super-States This subsection outlines the second step of the solution procedure, which derives the steady-state probabilities of the individual states in super-states. After disaggregation, we recover our original Markov chain with state space S. We go back to our original partition of the state space S = {T n, n 0}, defined in (3.2), where T n is the collection of the states with the number of A-customers in the last group being n. In Section 3.2.1, we have already derived the steady-state probabilities of the

87 74 individual states (j, 0), 0 j s 1, and T n, n < K s. Our objective is to derive the steady state probabilities for blocks T n, n K s. Using the structure of generator Q G given in Theorem 3.2.1, for n K s, we can write the balance equations as: K s 2 p 0 F 0,K s + p j F K s j + p K s 1 F K s 1,1 + p K s L = 0, n = K s or, equivalently, j=1 K s j=1 p n j F j + p n L = 0, n K s + 1, K s 2 p K s = p 0 ( F 0,K s L 1 ) + p j ( F K s j L 1 ) + p K s 1 ( F K s 1,1 L 1 )(3.6) p n = K s j=1 j=1 p n j ( F j L 1 ), n K s + 1. (3.7) Using (3.6), we first obtain p K s, whose solution can be expressed by the known probability vectors p n, 0 i < K s. Recursively, we can compute p n for n > K s, using (3.7) Key Performance Measures of the Ticket Queue Given the stationary distribution p(x), x S, we can compute the key performance measures. The marginal distributions of N and D are, respectively, p(n, 0), n = 0,..., s 1, P (N = n) = p(x), n = s,..., K, x Sn

88 75 and p(d, 0), d = 0,..., s 1, P (D = d) = p(x), d = s, s + 1,.... x: x =d From a customer s perspective, a key interest is her estimated queuing position given she has the ticket position d, i.e., N D = d. The conditional distribution of N D = d is given by: P (N = n D = d) = P (N = n, D = d) P (D = d) = 1, 0 n = d s x: x Sn, x =d x: x =d p(x), s n d, 0, otherwise. (3.8) Therefore, a customer s ticket position provides complete information of her queuing position only when she observes no more than s tickets upon arrival. We can also compute the conditional mean, d, 0 d s, E[N D = d] = (3.9) d n=s np (N = n D = d), d > s. Let γ d be the probability that a new customer with ticket position d s reneges at the next departure event. This event occurs if the customer s reneging time T r,d is smaller than the next service completion time and the minimum reneging time of other waiting E-customers. Given state x with x = d, the total reneging rate of the E-customers ahead of the new customer satisfy n E 1 1 i=s δ j + l i=2 x i 1 +n E i 1 j= x i 1 δ j, where the first term is the total reneging rate of the waiting E-customers in the first group and the second term is the total reneging rate of the waiting customers in other E-groups.

89 Heuristic Approximation The aggregation-disaggregation approach discussed in Sections and is an effective solution procedure for small values of K s. Unfortunately, the cardinalities of S j and T n grow exponentially in K s. The cardinality of T n is 2 K s, n 1, and the cardinality of S is S = (K s + 1) ( K s states in the aggregated Markov chain is S = K the dimension of the state space ) j s, s j K. Therefore, the total number of S j = (K s+1) 2 K s +s. As such, j=0 S is mainly determined by O ( (K s + 1)2 K s), which grows at an exponential rate as K s increases. For example, ignoring the linear term s, for K s = 10, S 11, 264 and T n = 1, 024. As K s increases to 15, S 524, 288 and T n = 32, 768. As a result, we can only manage to numerically carry out the two-step procedure for K s = 8. In this section, to overcome this difficulty we develop the 2-Digit Heuristic (2DH), which not only drastically reduces the cardinality of the state space, but also yields a Markov chain that is a GI/M/1 type process with a skip-free-to-the-right infinitesimal generator, leading to a polynomialtime algorithm The 2-Digit Heuristic (2DH) For a given state ( n E 1, na 1 ;... ; ne l, ) na l in the original system, the 2DH representation of the state is x = ( n E,n A) ( l = i=1 ne i, ) l i=1 na i. The state space S 2D of 2DH can be represented by S 2D = { x = ( n E, n A), 0 n E K, n A 0 }. Under the 2-digit heuristic, a customer s abandonment behavior mimics that under the original system: upon arrival, she can only observe n E +n A, but not their individual values, and makes the balking-joining decision accordingly. Once joined (balked), however, she will be placed at the end of the E-group (A-group). We again assume

90 the reneging time of a joining customer (conditional on the position she reneges from) is exponentially distributed with rate δ j. Upon reneging, she will be removed from the E-group and added to the A-group (i.e., n E will be reduced by 1 and n A increased by 1). Figure 3.3 displays the Markov chain with s = 2 and K = 4, using 2DH. Hereafter, the Markov chain under 2DH will be referred to as the 2D-Markov chain. We partition the 2D-Markov chain according to the number of A-customers in the system: T 2D n = {x = (n E, n A ) S 2D, n A = n}, n = 0, 1,.... In words, T0 2D contains all the states that have no abandoned tickets, and Tn 2D contains all the states that has n abandoned tickets, n 1. Given balking limit K, we have T 2D 0 = K+1 and T 2D n = K s+1 for any n, which increases linearly with K s. Recall that the cardinality of T n in the original Markov chain is T n = 2 K s, an exponential function of K s. For example, for K s = 30, T n in the original Markov chain has more than 1 billion states and T 2D n in the 2D-Markov chain contains merely 31 states, which represents a % reduction of T n. Therefore, 2DH drastically reduces the number of states in each level when K s is large. Observe that the 2D-Markov chain can move from T 2D n completion, remains within T 2D n 77 to T 2D 0 due to a service due to either a service completion or a new joining customer (when n < K s), or move to T 2D n+1 due to an abandonment, including both balking and reneging. In other words, unlike the original system, the 2D-Markov chain has skip-free-to-the-right transitions, i.e., it is a GI/M/1 type process. For this type of Markov chains, it is easy to calculate its steady-state distribution by forward recursion. As a result, we can avoid the aggregation step discussed in Section 3.2.1, and directly use forward recursion, analogous to Step 2 discussed in Section 3.2.2, to determine the steady-state distribution of the 2D-Markov chain. Theorem The 2D-Markov chain has the lower Hessenberg block-partitioned

91 78 Figure 3.3: The 2DH Markov chain of the state-dependent Ticket Queue with s = 2 and K = 4 generator given by: T 0 T 1 T 2 T 3 T K s 1 T K s T K s+1 T K s+2 T 0 L 2D 0 F0 2D T 1 B 2D L 2D 1 F1 2D T 2 B 2D L 2D 2 F2 2D Q 2D G = T K s 1 B 2D L 2D K s 1 FK s 1 2D T K s B 2D L 2D F 2D T K s+1 B 2D L 2D F 2D T K s+2 B 2D L 2D , where the component matrices in Q 2D G are provided in Appendices (H.1) (H.7).

92 Let p 2D = {p 2D n, n 0} be the probability vector associated with the partition {Tn 2D, n 0}. Then, since the probability vectors must satisfy the balance equations p 2D Q 2D G = 0, we have p 2D 0 L 2D 0 + ( n=1 p 2D n ) 79 B 2D = 0, (3.10) p 2D n 1F 2D n 1 + p 2D n L 2D n = 0, 1 n < K s, (3.11) p 2D K s 1F 2D K s 1 + p 2D K sl 2D = 0, (3.12) p 2D n 1F 2D + p 2D n L 2D = 0, n > K s. (3.13) To construct the solution of the above system of linear equations, we define matrices Rn 1,n 2D = F 2D n 1 (L2D n ) 1, 1 n < K s, F 2D K s 1 (L2D ) 1, n = K s, and R 2D = F 2D (L 2D ) 1. Note that R 2D 0,1 is a (K + 1) (K s + 1) dimensional matrix and R 2D n 1,n,n > 1, and R 2D are (K s + 1) (K s + 1) matrices. We also let R 2D n = n i=1 R2D i 1,i, for 1 n K s, which is a (K + 1) (K s + 1) dimensional matrix. Then, (3.11)-(3.13) can be written as p 2D n = = p 2D n 1 R2D n 1,n, 1 n K s, p 2D n 1 R2D, n > K s, p 2D 0 R2D (3.14) n, 1 n K s, p 2D ( K s R 2D ) n K+s = p 2D 0 RK s 2D ( R 2D ) (3.15) n K+s, n > K s. To determine p 2D 0, we first substitute the above equation into the normalization equation. Because the 2D-Markov chain is positive recurrent, n=k s+1 (R2D ) n K+s =

93 80 R 2D (I R 2D ) 1. Thus, p 2D 0 ( 1 T + K s n=1 R 2D n 1 T + R 2D K sr 2D (I R 2D ) 1 1 T ) = 1. (3.16) Furthermore, from (3.10) and (3.15), we also obtain p 2D 0 L 2D 0 + n=1 p 2D n B 2D = p 2D 0 ( L 2D 0 + ( K s n=1 R 2D n + R 2D K sr 2D (I R 2D ) 1 ) B 2D ) = 0. (3.17) Equations (3.16) and (3.17) form a system of linear equations that p 2D 0 must satisfy. After discarding any one of the equations in (3.17) (since we use the normalization condition), we obtain the unique solution of p 2D 0. We can calculate p 2D n, n 1 by using (3.15) Conclusion In this study, we extended the ticket queue model developed in Section 2 by considering state-dependent reneging rates, and provided the framework for the analysis of generalized ticket queue models. The stationary distribution of the generalized ticket queue MC model can be obtained using the two-step solution procedure for the multi-server ticket queues outlined in Section 2. Because the solution procedure requires exponentially increasing computing time and memory space in K, we also developed the 2-Digit Heuristic for the generalized ticket queues to approximate the steady-state distributions of the generalized ticket queues. The reneging rate for an individual in the ticket queue system can be considered either as an increasing or a decreasing function of the customer s ticket position in the queue. In reality, the customers might base decisions by considering multiple factors, such as the progress toward the goal, the escalation of commitment effect or the time already spent waiting in line. Models that incorporate all these factors are

94 too complex to solve analytically, and could be studied through simulation models. 81

95 Chapter 4 Comparisons of Perceptions and Behavior in Ticket and Physical Queues 4.1. Introduction Operations Management has dominated the research and applications of theories related to queues. Most of the studies in Operations Research have focused on the representations of queueing systems and the methods for improving queueing systems efficiency. In general, the articles in mathematical queueing theory are normative in character and can be found frequently in journals in the area of management science (Pruyn and Smidts 1993). However, most of this research has focused on the physical aspects of the queueing systems, and has not considered the consumer behavior (e.g. perceptual factors that may affect consumers attitudes and preferences regarding queues) (Carmon 1991). Although the analytical models provide useful objective improvements in the queuing systems (e.g. reducing the actual waiting times) they fail to capture the psychological costs (Seawright and Sampson 2007), which affect customer satisfaction and perceptions of systems (e.g. perceived wait times can be as important as actual wait times). Marketing and Psychology researchers have provided consumer behavior studies in efforts to fill this gap. Earlier behavioral research in queueing theory has focused on using surveys, scenario reading sessions, and field experiments. Advances in technology and computers resulted in a shift for studies that also use computer-based experiments. Most of the existing research focused on Physical queues (PQs), arrangements where customers physically stand in line behind each other for their turn to receive 82

96 83 service. Recent technological advances also popularized the Ticket Queue systems, arrangements where arriving customers are issued numbered tickets from a ticket kiosk, and service is provided according to the order of the tickets. The ticket queues are widely used in the public sector (e.g. The Department of Motor Vehicles, embassies and admissions offices) as well as in the private sector (e.g. banks, retail stores and travel agencies). This common use of ticket queues motivated us to investigate the system performance the ticket queues in Section 2. In Section 2, we provided an analytical model to compare the performance of the ticket and the physical queues, and showed that the ticket queues provide lower performance measures (e.g. lower service completion rates) than the physical queues do. Although the ticket queues provide lower performance measures, surprisingly, the ticket queues are preferred systems by many service providers. Understanding the impact of the structure of a waiting line on customer preferences, patience and abandonment behavior could provide insight into the reasons behind the ticket queues becoming more popular. Although the ticket queue systems are becoming popular in different waiting environments, there is very limited analytical or empirical research on the comparison of the ticket queue and the physical queue performances, and on how the customers perceive and behave in these queuing arrangements. The queuing literature identifies the two forms of customer abandonment as balking (refusing to join a queue upon arrival) and reneging (joining a queue initially, but leaving the queue before receiving service). The different configurations of the waiting environment in the ticket and physical queue arrangements provide different levels of information to the waiting customers about other customers abandonment behavior. In a TQ, the waiting customers have only partial information about their positions in the queue because they cannot immediately observe abandoning customers behavior. Therefore, as outlined in Section 2 the TQ customers display naive abandonment behavior in the ticket queues. Our goals in this paper are to obtain insights into the customer perceptions of

97 84 the ticket queue and physical queue arrangements, to understand the effects of the queue arrangement on the willingness to wait in lines, investigate the effects of the wait context on queue arrangement preference, and investigate the validity of the theoretical assumptions used in the literature. Testing the results of Section 2 and obtaining insight into the perceptions of and behavior in a ticket and a physical queue of a real service provider is difficult as it requires finding two queueing environments in which the only difference is the structure of the queue. Therefore, we designed a series of surveys to achieve our goals. We conducted the surveys at a northeast U.S.A. university to illustrate waiting scenarios in the ticket and the physical queues by alternating three factors: Value (or type) of service, traffic intensity, and availability of alternative ways to complete service. Using these factors, we created eight different treatments for participants who volunteered to the study in exchange of extra credit. The value of service and the traffic intensity had high and low levels, whereas the alternative ways to complete service either existed or not. In our first analysis step, we examined whether the queue preference is context dependent or not, and identified the settings in which the subjects prefer the ticket queues over the physical queues. The second step in our analysis was to understand how subjects willingness to wait is adapted and adjusted before and during waits. In addition, the queuing literature suggests that people who judge themselves to be impatient are most negatively impacted by the queue-type delay in their satisfaction with service (or evaluation of queuing experience). We generated two personality scales, time pressure and patience, in the survey to investigate the effects of these scales on the subjects queue preference. As a third analysis step, we used a binary logistic regression model to identify the factors effective in preferring one type of queue over the other including the personality scales. The survey results indicate that the ticket queues are preferred arrangements across all the treatments. Although the preference towards ticket queues is overall

98 85 dominant, the high traffic intensity and the high value of service scenarios provided higher percentage differences between the ticket and the physical queue preference ratios, whereas the availability of alternatives did not provide any statistically significant difference between the corresponding ratios. The binary logistic regression model identified the value of service, level of traffic intensity, salience of the queue movement and the comfort of the waiting environment as the effective factors in queue preference. The patience and time pressure scales appeared to have no effect on the queue preference. Additionally, we found that the subjects have higher initial willingness to wait (balking patience) in the ticket queues than they do in the physical queues. and they adjust their balking patience to higher levels in high traffic and high value of service scenarios. These results indicate that customer patience is context dependent and the customers adapt their patience according to the context of the wait. We also show that once the subjects join a line and spend some time waiting, they are willing to wait longer than they did initially. In other words, the subjects reneging patience time longer than their balking patience time, which is described as the sunk cost (or escalation of commitment) effect in queuing literature. This study also provides insight into the customers willingness to wait in the ticket and the physical queues. Contradicting the theoretical assumptions of Section 2, we demonstrate that the customers initial willingness to wait in the ticket queues is larger than its counterpart in the physical queues. Therefore, the structural features of a queue affects the level of patience the customers would have in that queue. In light of these findings, theoretical assumptions on the abandonment behavior of customers should be reconsidered to build more realistic models. Our results can also assist service providers in matching the correct type of queue arrangement with the service provided in their facilities. We recommend the use of ticket queues in all but low value of service low traffic intensity settings. The remainder of this paper is organized as follows: Section 4.2 will discuss and

99 86 compare the waiting experience in the ticket and the physical queue arrangements under different scenarios. In this section, we first present a literature review for the empirical studies in the queuing theory field and then develop the hypotheses tested in this study. Section 4.3 will present the methodology we used to test our hypotheses. In Section 4.4, we present our findings from the study. Finally, Section 4.5 will provide a discussion of the study focusing on managerial and research insight on designing queuing systems and will provide the limitations of the study Waiting Experience in Queuing Systems In their model of the Waiting-Profit Chain Bitran et al. (2008, p.64) consider servicescape (with ambient conditions, space/function and signs, symbols, and artifacts as the three dimensions) as an environmental moderator. In their model, the space/function dimension considers the physical arrangement of the queues in the service facility. There are various studies that present how controlling the actual and perceptual waiting duration modifies the overall evaluations for service, behavioral intentions and satisfaction of customers. However, controlling the environmental moderator servicescape can also be effective in achieving the above goals, and there is a limited number of studies focusing on this issue (Bitran et al. 2008). For example, Rafaeli et al. (2002) showed that when you manipulate the queue structure (the physical layout), customer perceptions of fairness of a queue will be different. In this section, we first will summarize the existing literature on waiting experience in queues by focusing on three areas. The first area is how the researchers have investigated the effects of different types of queues on the customer perceptions and behavior in service facilities. We also try to identify the studies about the impact of servicescape/environment on the queue preference. Secondly, we explore how willingness to wait in queues are defined and studied by the researchers and whether the service environment has an effect on the willingness to wait in queues. Finally,

100 87 we also would like to see how the individuals patience and time pressure have been measured in the literature using different scales. Following this literature review, we will present the hypotheses tested in the paper Literature Review Servicescape/Environment and Queue Preference Larson (1987) states that Queuing environment can influence customer attitudes and ultimately, a firm s market share (Larson 1987, p. 895) and demonstrates that the subjects disutility of waiting depends on the queuing environment. The effect of different types of queue on customer perceptions have been studied by several researchers. In one of the first studies, Hornik (1984) collected data through a survey on actual and perceived waiting times in multi-server multi queue, single queue and express queue arrangements of a chain supermarket, a department store and a bank. Although the results indicate that individuals, in general, exhibit a tendency to overestimate waiting time, the differences in perceived waiting times across the different queuing arrangements are not statistically significant. Rothkopf and Rech (1987) claim that although multi-server single queues analytically provide lower waiting times compared to the multi queues, using single queue structures might not be advantageous in service systems as it eliminates the jockeying (switching between lines) behavior and might increase the service times. Baker and Cameron (1996) suggest that customers could prefer queuing arrangements that provide higher levels of salience in their progress. Zhou and Soman (2003) demonstrate the impact of the number of people behind on abandonment decisions, and claim the queue factors that influence the ease with which social comparisons can be made moderate the effect of the number behind. They state that while linear queue systems offer unambiguous information about the relative queue positions of a consumer, queuing arrangements like ticket queues do not provide salient relative

101 88 queue position, hence social comparisons are less likely to occur in ticket queues. Rafaeli et al. (2002) designed a computer laboratory study in which participants experienced actual waits in simulated queues. Examining the relationship between the queue structure and the attitudes of people waiting, they show that people waiting in single queues feel more predictability and arousal than those waiting in multi queues. On the other hand, waits in multiple queues produce a sense of lack of justice, even when First in First Out (or FCFS - First Come First Serve) rule is not violated. In another study, Rafaeli et al. (2008) compare the participant perceptions in multi-server single queue, multi queue and ticket queue arrangements. Using a survey, they show that even though the expected waiting time was the shortest in the multiple queue and the longest in the ticket queue arrangement, the participants preferred the ticket queue the most and the multiple queue the least. Their explanation for this paradox is that the perceived fairness is significantly higher in the single and the ticket queue arrangements than that in the multiple queue arrangement (with no significant difference between the single and the ticket queue arrangements) even when the FIFO rule is not violated. As Bitran et al. (2008) state, there are no studies that investigated and crossculturally compared customer behavior in different queueing contexts. In addition, there are no generalizable claims about the impact of customer experience on the perception and evaluation of waits for services. To our knowledge, there are no studies that investigate the effects of the type of service provided, the traffic intensity or the availability of alternatives on the preference of queue system, either. The managers should consider a variety of factors before investing in the queue arrangements, and in this paper, we aim at providing more insight on how to choose the appropriate queue arrangement in the service facilities under different settings. We investigate the impact of the level of traffic intensity, the value of service and the availability of alternative ways to complete service on queue preferences, and determine the factors that affect the preferences.

102 89 Willingness to wait (patience) in Queues: How to model willingness to wait (patience) to better understand the customer behavior in queues is also a popular point of interest in queueing theory. If a customer visiting a service provider finds out that all of the servers are busy assisting other customers, she will make a decision to join the system or abandon. The literature considers balking and reneging as the two forms of abandonment, and all of us frequently make abandonment decisions in our lifetime. However, there is a lack of empirical and experimental studies that would assist the researchers in establishing analytical models that incorporate the psychological aspects of the abandonment decisions. In one of the earlier studies, Palm (1953) assumed the abandonment rate is proportional to the annoyance level of the customers. Alternatively, other researchers have also assumed that customers will abandon if their annoyance levels exceed certain thresholds, (or they do not have any more patience to wait). Most of these studies model abandonment behavior by considering the disutility of waiting, and aim at separating the economic cost and the psychological cost of waiting. We also find the focus of existing literature on managing the perceptions of waiting time, which is shown to affect the customer satisfaction levels. Osuna (1985) was the first to develop a mathematical model for stress, and Suck and Holling (1997) showed the dependence of stress on the remaining waiting time distribution. Patience is obviously influenced by numerous factors related to customer profiles and environment characteristics (see, for example, Maister 1985, Zakay and Hornik 1996, Levine 1997). Katz et al. (1991) showed that although customer satisfaction decreases as the perceptions of waiting time increases, the increased distractions during waits make the waiting experience more interesting and increases customer satisfaction. Taylor (1994) separates the reactions to waiting into two general types: uncertainty reactions and associated feelings of uneasiness, unsettledness, and anxiety; and anger reactions and associated feelings of annoyance, irritation, and frustra-

103 90 tion. Leclerc et al. (1995) show that the value of time is highly context dependent. Their study indicates that the marginal value of time is higher in short waits than in long ones, and the value of time is also higher for services with higher monetary value. Pruyn and Smidts (1993), Houston et al. (1998) and Cameron et al. (2003) also show that as postulated by Maister (1984), the more valuable the service is the longer people will wait. The traditional queuing theory assigns willingness to wait (patience) to individual customers independent of the system performance characteristics, and the patience is not updated by possible changes in traffic intensity. In general, patience is assumed to follow exponential distribution, and each customer is randomly assigned a value from this distribution. In addition, the normative models assume the customers abandonment decision is based on the comparison of the utility of receiving the service and the cost of the expected wait, specifically on the entire waitingtime distribution (Zohar et al. 2002, p.568). Carmon et al. (1995) incorporated the psychological costs of waiting into analytical queuing models and demonstrated that such an action could result in prescriptions that are inconsistent with those dictated by conventional queuing models. A normative, utility-maximizing model for abandonments has been considered in several recent papers (Hassin and Haviv 1995, Mandelbaum and Shimkin 2000, Haviv and Ritov 2001). The abandonment time of each customer is chosen to maximize a personal utility function, which balances the service utility and the expected cost of waiting. They note that in the basic form of these models, the customer choice relies on the entire distribution of the offered waiting time, rather than just on its average. On the contrary, Zohar et al. (2002) hypothesize that customers patience depends on their waiting time expectations, which are formed through factors like past experience, time perception and value of service. They focus on patience in tele-queues, which they state as invisible queues, and state that patience is influenced by factors related to customer profiles and environment characteristics, but most of these factors

104 91 could be taken as a priori given and fixed. Although there exists empirical support for willingness to wait in the tele-queues, the exponential patience distribution hypothesis is not sufficiently investigated in other queue arrangements and the field lacks empirical support. Brown et al. (2005) showed in an Israeli call center data that the willingness to wait does not follow exponential distribution, and provide support for the heterogeneity of patience among customers, which was shown before in Thierry (1994), Friedman and Friedman (1997), Diekmann et al. (1996). Alternatively, Zohar et al. (2000) claim that customers adapt their patience (modeled by an abandonment-time distribution) to their service expectations, in particular to their anticipated waiting time. They present empirical support for that hypothesis, and propose an M/M/m-based model that incorporates adaptive customer behavior. In this paper, we look into the impact of the service environment on the patience levels in the ticket and the physical queue arrangements. Again, to our knowledge there are no studies focusing on the levels of patience in ticket queue and physical queue settings, and how the patience levels are adapted in these systems. To simplify the analysis, we focused on single server systems and investigated the impact of the value of service, the level of traffic intensity and the availability of alternatives on the subjects willingness to wait. We also provide support for the existence of sunk cost effect in waits, which states that the customers are willing to wait longer than their initial willingness to wait after they join and spend some time in line. Patience and Time Pressure Scales and their impact on queue preference: Finally, we would like to outline the research on customer patience and time pressure scales in the psychology literature. Patience is defined as the capacity or habit of enduring evil,... courageous endurance,... calm self-possession in confronting obstacles or delays: steadfastness by the Webster s Third New International Dictionary (1981,

105 92 p 1655). There are a limited number of patience scales developed in the literature. Blount and Janicik (1999,2000) were the first to construct patience scales. In their model, patience is considered as a cognitive-emotional based process. For Blount and Janicik, patience initiates when an individual perceives a delay in situation, then attributes responsibility for the delay followed by determination of one s level of responsibility for reacting to the delay. In their explanation, patience occurs when a goal is delayed either from the postponing of a goal (i.e., anticipated achievement of the goal moves back as in delay of receiving service from a bank teller) or from the blocking of a goal (i.e., individual must wait for the conclusion of something as in a car alarm getting triggered and not disengaged). Using Blount and Janicik as a foundation, Dudley (2003) developed a scale of patience. He proposed an objective means of measuring patience through individuals responses to items relating to delay and the response to delay. Dudley s patience scale (2003) produced a means of counting and tallying a total patience score as well as six factors of patience (postponement, even-tempered, composure, time abundance, tolerance, and limits of patience). Schnitker and Emmons (2007) also developed a patience scale. According to Hornik and Zakay (1996), for people under time pressure, the perceived wait time is prospective and the waiting customers focus their attention mostly on time. Hornik claims that the pressure can affect consumer decision-making by forcing consumers to base their decisions on existing knowledge and experience rather than additional information, or by reducing the number of products/services a consumer initially planned to buy and the frequency of unplanned purchases. Hornik also claims Time scarcity may be considered as a form of continuous and long-term time pressure. In their model, Hornik and Zakay (1996) claim that while waiting in line, one is automatically occupied by prospective time estimates, which are based on an attentional mechanism. They also state that waiting times are typically overestimated because the customers focus their attention to the passing of time during

106 93 waits. Francis-Smythe and Robertson (1999) used a seven item impatience scale with questions like At work, I frequently feel like hurrying other people up. and I am quite impatient.. Patience related questions also exist in studies such as Jenkins Activity Survey and Framingham Type A scale, however, these studies also include questions about other personal characteristics in their methodology and are not only focused on patience scales. Katz et al. (1991) observed queues in a bank and concluded that patient customers tended to be entertained by events that swirled around them while impatient customers focused instead on the misery of the wait. In the paper, patience was also defined in terms of an individual s willingness to accept delays. Our main goal in this study is to analyze the perceptions, preferences and behavior of customers in different queue arrangements. Therefore, rather than using a long list of questions on patience and time pressure scales, we adapted a subset of the scales for the purposes of this study and investigated the impact of these personality traits on individual queue preference. Following this literature review, we also would like to focus on the validity of the three of the assumptions used in our analytical model in Section 2. In Section 2 of this dissertation, I provided the framework for the analysis of the multi-server ticket queues with both forms of customer abandonment. The first assumption we used was that every customer carries an initial willingness to wait for joining a line (balking patience) before arriving to a system facility and each customer s initial willingness to wait in the TQ is the same as that in the PQ. The second assumption was that customers in the TQ assume all the tickets ahead of their ticket represent existing customers in the system, such that they ignore the possibility that some customers with tickets ahead of their ticket in the system might have already abandoned their positions. Finally, we also assumed that if the customers join a line, they become more patient, i.e. their willingness to wait after joining (reneging patience) is higher

107 94 than their willingness to wait for joining (balking patience). On the other hand, we assumed that a customer s willingness to wait after joining the TQ is the same as its counterpart in the PQ. These assumptions will also be investigated in this study Hypotheses Queue Preference: Our first goal in this study is to obtain more insight into the customers preferences and perceptions of the ticket and the physical queue systems. In our first set of hypotheses, we investigate whether the subjects prefer one queuing arrangement over the other under various waiting scenarios. The ticket queues are known to enhance customer experience by eliminating the waiting lines and they provide a tidy and relaxing environment for the customers, which improves customer satisfaction and increases sales potential (Xu et al. 07). Supporting this claim, our survey indicates that 69.7% of the participants believe the ticket queue arrangement provides a more comfortable waiting environment than the physical queue arrangement does. In addition, as Rafaeli et al. (2008) display, perceived fairness in ticket queue arrangements is higher than that in physical queues. Considering the advantages presented for the ticket queues, we predict that, in general, the subjects will prefer the ticket queue arrangement over the physical queue arrangement. Hypothesis 1. Subjects prefer the ticket queue arrangement over the physical queue arrangement. Although we predict that the participants will, in general, prefer the ticket queues over the physical queues, we also want to investigate how the waiting context affects the preference of queues. First of all, an analysis of the service providers that implemented the ticket queue systems would reveal that the ticket queues are commonly used in high traffic regime (i.e. high customer volume) environments. In a low traffic service facility, a customer visiting the facility could find it less time consuming and

108 95 simpler to physically wait behind a small number of people rather than waiting in a ticket queue. A customer could view the time to receive a ticket, find a waiting area and wait for her ticket number to be called as activities to waste her time, if the traffic intensity is low for the service provider. Moreover, the customers might be more satisfied by recognizing her progress towards the goal while waiting in a physical queue. As a result, we predict that the difference between the percentage of participants who would prefer to use a ticket queue rather than a physical one would be higher in high traffic regime systems than that in low traffic regime systems. Hypothesis 2. Subjects prefer the ticket queues with a higher percentage margin over the physical queues in high traffic regimes than they do in the low traffic regimes. Another point of interest in waits is whether the subjects would prefer a certain queue arrangement over the other when the type of service provided is different (or the value of service provided is high vs. low). As Maister (1984)postulated, the more valuable the service is the longer people will wait. This hypothesis was confirmed with other studies as we summarized in Section As people are willing to wait longer when the value of service increases, they would like their wait to be less stressful and more comfortable as well. According to our survey, the subjects perceive the ticket queues as more comfortable waiting environments than the physical queues. Therefore, we predict the following hypothesis to hold. Hypothesis 3. Subjects prefer the ticket queues with a higher percentage margin over the physical queues in high value services than they do in the low value services. Finally, the availability of the availability of alternative ways to receive service could also affect the type of queue the customers would prefer. To our knowledge, other studies did not investigate the effect of alternatives on queue preference before. When a customer arrives to a facility and observes the system state, depending on

109 96 the existence of alternative methods to complete service, she might prefer to wait for service or abandon the system. If there are no alternative ways to complete the service, we assume that the subjects would prefer waits in ticket queues more as the ticket queues are perceived as more comfortable and tidy waiting environments. We test the effect of alternatives on queue preference using the following hypothesis. Hypothesis 4. Subjects prefer the ticket queues with a higher percentage margin over the physical queues in transactions where subjects have alternative ways to receive service than they do in the transactions where subjects do not have alternative ways to receive service. Willingness to wait (Patience) in Queues: In any waiting environment, a newly arrived customer will first observe the system state and if any of the servers are idle, will start receiving service upon arrival. However, if all the servers are busy, then the customer will have to decide whether to join the waiting line or balk (leave the system immediately). If the customer joins the line to wait, she could either wait until receiving service or abandon the system by reneging (leave the system after spending some time in line). We could denote the time the customers are initially willing to wait (balking patience) as T B, and the time the customers are willing to wait after joining the queue (reneging patience) as T R. Researchers commonly express patience as willingness to wait or reservation waiting time in the literature. Brandt et al. (1999) and Weinberg et al. (2003) analyzed models where they assumed customers would abandon if the time elapsed since their arrival to the service facility exceed a random maximal patience time and the reservation waiting time, respectively. Zohar et al. (2002, p.567) denoted patience as the time to abandon or its probability distribution, and showed that patience significantly depends on customers expectations regarding the waiting time in the system. These expectations, in turn, are formed through accumulated experience and are affected by subjective factors - time perception, the importance of the service

110 97 being sought, and so on. Bitran et al. (2008) state that although the psychological and behavioral decision making literature that model reneging based on the economic cost of time can be effective in the description and prediction of customer behavior, most of these studies kept the environmental moderators (e.g. space/function) or personal moderators (e.g. personality, culture, experience) constant, which results in subjects exhibiting similar balking rates or reneging patience. Therefore, the reliability of these empirical estimates depends on the modeler s ability to segment customers into groups with similar behavior. Leclerc et al. (1995) and Ryan and Valverde (2003) showed the value of consumer s time depends on contextual characteristics of the wait situation, rather than being a constant. The environment/servicescape has received significant attention in the literature focusing on reneging behavior in physical lines, call centers and online services. Zhou and Soman (2003) showed that reneging behavior for a person not only depends on the number of people in front, but also on the number of people behind that person. They explained that reneging is influenced by the ease of people engaging in social comparisons under queuing environments, and identify queue factors, individual factors, and situational factors as important categories of moderators. In telephones (call centers), there is very little research about customer behavior during telewaits (Gans et al. 2003). Munichor and Rafaeli (2007) investigated the impact of filling the customers waiting time with music, apologies, activities or queuing information and found that the queue position information reduced the reneging rate the most compared to the other time fillers. In the online systems, Ryan and Valverde (2003) identified nine factors that affect a customer s willingness to wait for Web pages to download. They showed that more experienced users have less tolerance and are more sensitive to delays. In Section 2 of this dissertation, we assume that a customer will base her decision to balk or join the line by comparing her balking patience (initial willingness to wait)

111 98 with the estimated time for all the customers ahead of her to complete service. Our surveys demonstrate that 74.1% of the respondents admitted they would use a balking patience time to make the join/balk decision. For the ticket queues, Section 2 also assumes the customers ignore the possibility that some customers with tickets ahead of their ticket in the system might have already abandoned their positions. Therefore, a new arrival bases her decision to join or abandon a system on the number of tickets ahead of them. We tested the above assumption in the survey and found that 80.6% of the participants would behave in this manner, which could lead to naive abandonment behavior. We also assume in Section 2 that once a customer decides to join a line, the time they are willing to wait in the line (reneging patience), T R, will be higher than the initial time they were willing to wait (balking patience), T B. We also assume that although the reneging patience is higher than the balking patience on an individual basis, the type of queue the customer is in does not have an effect on the level patience for that type of queue. Within the survey, we asked the subjects questions about their willingness to wait in different treatment scenarios. Our fifth hypothesis investigates whether the balking patience of participants is the same under the ticket queue and the physical queue arrangement. Hypothesis 5. The subjects initial willingness to wait in the ticket queues is larger than that in the physical queues. As mentioned above, several queuing studies reveal how customers (or subjects) adjust their patience levels in different waiting environments. We also would like to test how the context of the environment would affect the customer patience levels in the ticket and the physical queue systems. First of all, in heavy traffic systems, the customers would see the system to be busy and would adjust their patience levels accordingly considering all the other customers waiting in the system. In addition, in a ticket queue system, the waiting environment is more comfortable and tidier than that in the physical queue system, which could result in higher levels of patience in

112 99 ticket queues. On the other hand, in low traffic systems, the customers could be indifferent in between the waits in a physical and a ticket queue system, which could result in similar levels of patience in both the systems. Therefore, we predict that the difference between the balking patience levels between the ticket queue customers and the physical queue customers should be higher in heavy traffic systems, than it is in the low traffic systems. Hypothesis 6. The difference between the subjects initial willingness to wait in the ticket queue arrangement and that in the physical queue arrangement is greater in heavy traffic regimes than its counterpart in the low traffic regimes. The type (or value) of service provided could also cause the subjects to adjust their patience levels. The two scenarios we provided for different valued services were renewing a driver license in a DMV center and exchanging currency in a bank. The users who experienced waits in these two systems might have biases towards waits in DMV centers, which are often described as long, meaningless and painful waits. A ticket queue would provide more comfort in a DMV than a physical queue would, and this could lead to higher patience levels in the ticket queues. Bank lines, especially in the U.S.A., are not described as painful and long compared to DMV lines, and the customers could be indifferent in terms of the arrangement of the waiting line in bank systems. As we presented the DMV scenario as a high value and the Bank scenario as a low value service, we predict that the difference between patience levels in a ticket queue and a physical queue at a high value service would be higher than that in a low value service. Hypothesis 7. The difference between the subjects initial willingness to wait in the ticket queue arrangement and that in the physical queue arrangement is greater in high value transactions than its counterpart in low value transactions.

113 100 Finally, we analyze the effects of the existence of alternative methods to complete service on the subjects initial willingness to wait. In our opinion, having alternative methods to complete a service would make a customer more impatient and increase the tendency to abandon the system. Considering the ticket queue customers would also be more patient than the physical queue customers, we test the impact of the existence of alternatives on the patience levels in different queue arrangements with the following hypothesis. Hypothesis 8. The difference between the subjects initial willingness to wait in the ticket queue arrangement and that in the physical queue arrangement is greater in transactions with alternatives than its counterpart in transactions without alternatives. Sunk Cost (Escalation of Commitment) Effect in Queues: On the other hand, we also would like to test whether the sunk cost effect is existing during waits. Sunk cost effect refers to the tendency to continue with an endeavor after an unrecoverable investment or commitment has been made. For example, after purchasing a ticket, a person may choose to see a movie even if she learns she will not like it. Utility-maximizing rational customers should ignore such investments or commitments when making decisions, but psychologists have established the sunk-cost effect by showing that this is not the case (Thaler 1980, Arkes and Blumer 1985). On the other hand, the sunk cost effect, also known as the escalation of commitment effect, in waiting lines states that after joining a waiting line, the likelihood of exceeding the initial willingness to wait increases with time spent in the line. If the sunk cost effect is present in queuing context, the amount of time customers have already waited should influence the amount they are still willing to wait. Bitran et al. (2008) state that time is not like money in terms of mental accounting and there is no theory of sunk costs in the domain of time. Leclerc et al. (1995, p.10) found that in decision making under conditions of risk, people seemed

114 101 to make risk-averse choices with respect to decisions in the domain of time in contrast to risk-seeking behavior often found with respect to decisions involving loss of money.. Soman (2001, p. 169) found that the sunk cost effect is not observed for past investments of time, but reappears when the investments are expressed as monetary quantities. and explains that the pseudo-rationality is due to the fact that individuals lack the ability to account for time in the same way they account for money. Brown et al. (2005) found that the time to abandonment hazard rate in an Israeli call center did not exhibit the memoryless property, and could not be following exponential distribution. In Section 2 of this dissertation, we assume the customers will become more patient after they join the line, which is a representation of the sunk cost effect. In this survey, we also investigate this assumption by providing the subjects with a waiting scenario and Table M to fill, which both can be found in Appendix 4.2. A particular question in the survey revealed that 62.6% of the participants would be willing to wait more than their initial balking patience time after they joined the system and spent some time waiting in line. On the other hand, if a customer decides to join the waiting line, then she could either wait until she receives service or renege the system by without receiving service. How the customers make this abandonment decision is also a popular point of interest in the literature. Our survey indicated that 87.2% of the respondents would take the time they already spent waiting into account when they decide to continue to wait for service or abandon without receiving service. With Table M, our goal is to identify whether the participants actually adjusted their willingness to wait after they spent some time waiting in the system. Our prediction is the more time spent in the system, the higher the likelihood to exceed the initial willingness to wait (balking patience) would be, which is tested using the following hypothesis.

115 102 Hypothesis 9. The escalation of commitment effect is observed in queuing systems. The probability of subjects willing to wait more than their initial patience increases with the amount of time the subjects have already spent in line Methodology To achieve our goals, we conducted a series of surveys at a northeast U.S.A. university. Participants of the survey were undergraduate students who volunteered in exchange for extra credit in their classes. The initial number of responses was 862. From this initial set, all the responses that included at least one missing field were omitted for the future analysis. This resulted in a set of 767 responses. Following this step, we eliminated the respondents who declared that they did not have any preference towards the ticket or the physical queues (which we identify as indifferent respondents), and ended up with a total of 588 respondents (242 female and 346 male) for the analysis data set. In this empirical study, subjects were asked to imagine themselves in waiting situations under different queuing settings. We used a factorial design to differentiate our treatments. The first factor is the value (type) of service provided. Considering the fact that the ticket queues are used commonly in the banks and government agencies, we decided to represent the high and the low value services from a DMV center and a Bank, respectively. The high value service at the DMV asked for renewing a driver s license at the local DMV center, whereas the low value service at the Bank required exchanging currency (US dollars with euros) at the local bank. The second factor is the level of traffic intensity in the service facility. We differentiated the high traffic intensity and low traffic intensity cases by associating an average of 30 and 5 customers waiting for service in the system, respectively. When

116 103 we look at the real-world implementations of ticket queues, we can identify the pattern that the ticket queues are preferred arrangements in heavy traffic systems, where the average number of customers waiting for service is high. In the survey, we also wanted to see whether the customers would prefer the ticket queues in heavy traffic regimes as well, considering the ticket queues might provide more relaxed waiting environments and alleviate the physical discomfort and mental boredom of standing behind other people (Xu et al. 2007). The third factor in our design is the availability of alternative ways for receiving service. We presented the subjects either with statements indicating whether alternative methods to complete service exist or not. Finally, per our design, we provided each participant both with the physical queue and the ticket queue version of the same service scenario. This allowed us to execute within subject analysis. The complete treatment set can be seen in Table K. Using this information, we presented the participants with scenarios as seen in Appendix L. As mentioned above, we presented each subject both with the ticket queue and the physical queue scenarios for the same type of service. In all of the scenarios, we also presented the subjects with the graphical representations of the physical and the ticket queue arrangements (see Appendix J). To obtain the responses, we either used a 5-point Likert scale to state their preferences, or asked the participants to fill tables to provide numerical values.

117 Results Hypotheses on Queue Preference: In order to test our first hypothesis, we first assume p i subjects that prefer type i queue arrangement. denote the proportion of H 0 : (p T Q p P Q ) = 0; H a : (p T Q p P Q ) > 0 We used the χ 2 test and compared the proportion of subjects who prefer the ticket queues with the proportion of subjects who prefer the physical queues. As seen in Table 4.1, 70.7% of the subjects prefer the ticket queue arrangement over the physical queue arrangement. The Chi-Square statistic is for the hypothesis, which indicates the test is significant. Therefore, we can reject the null hypothesis and conclude that the ticket queues indeed are preferred systems over the physical queues. Table 4.1: Subjects Queue Preference Observed responses Expected responses Residual Physical Queue Ticket Queue Total 588 On the other hand, as we see in Table 4.2, 73.9% of the participants preferred the ticket queues over the physical ones in the high traffic intensity scenarios, compared to 67.7% in the low traffic intensity scenarios. Table 4.2: Traffic Intensity and Subjects Queue Preference Traffic Intensity Physical Queue Ticket Queue High Traffic 26.1% 73.9% Low Traffic 32.3% 67.7%

118 105 For testing the second hypothesis, as the sample size is large enough for the survey, and we used Bernoulli test statistic with parameters p 1 and p 2 (with independent random samples of sizes n 1 and n 2 ). The null hypothesis H 0 : p 1 = p 2 can be tested using the test statistic Z = ˆp 1 ˆp 2 (p 1 p 2 ) ˆp1 ˆq 1 n 1 + ˆp 2 ˆq 2 n 2 N(0, 1) The test statistic, z has a value of 1.659, which is larger than the 95% α value of Therefore, we can reject the null hypothesis (the p-value is ) and conclude that the ticket queues are preferred with higher margins in the high traffic regimes than they are in the low traffic regimes. On the other hand, as we see in Table 4.3, 78.2% of the participants preferred the ticket queues over the physical ones in the high value of service scenarios, compared to 62.1% in the low value of service scenarios. Table 4.3: Value of Service and Subjects Queue Preference Value of Service Physical Queue Ticket Queue High Value (DMV) 21.8% 78.2% Low Value (Bank) 37.9% 62.1% For testing our third hypothesis, we again used the Bernoulli test statistic. The test statistic, z, has a value of 4.277, which is larger than the 95% α value of 1.645, and we can reject the null hypothesis (the p-value is ) and provide support for our third hypothesis. Finally, as we see in Table 4.4, 70.1% of the participants preferred the ticket queues over the physical ones in the scenarios where alternative ways to complete service existed, compared to 71.2% in the scenarios where no alternatives existed. Using the Bernoulli test statistic, we obtained a z value of 0.292, which is smaller

119 106 Table 4.4: Availability of Alternatives and Subjects Queue Preference Availability of Alternatives Physical Queue Ticket Queue Alternatives Exist 29.9% 70.1% No Alternatives Exist 28.8% 71.2% than the 95% α value of 1.645, and we cannot reject the null hypothesis (the p-value is ). Therefore, we conclude that the availability of alternatives does not affect the participants preferences of queuing arrangements. In our first four hypotheses, we test whether the queue preference is context dependent or not using the above mentioned three factors to differentiate the service environment. As outlined above, the level of traffic intensity and the value of service are factors affecting the type of queue preferred, whereas the availability of alternative methods to complete service is not a significant factor. Overall, the ticket queues are preferred systems in all the scenarios, but the level of ticket queue preference diminishes as we move to low traffic low value service scenarios (see Table 4.5). Therefore, we can conclude that the system managers should consider the nature of their service in designing their queue arrangement. Table 4.5: Ticket Queue Preference vs. Traffic Intensity and Value of Service High Value of Service Low Value of Service High Traffic Intensity 81.5% 64.9% Low Traffic Intensity 74.8% 59.4% Hypotheses on Willingness to wait (Patience) in Queues: Let T bi denote a subject s initial willingness to wait (balking patience) in type i queue arrangement (such that the subject will make the decision to join a type i queue or abandon immediately (balk) by comparing T bi with her estimated waiting time in

120 the queue), and T ri denote a customer s willingness to wait after she joins the type i queue. Using this notation, we can formulate our fifth hypothesis as follows: 107 H 0 : E[T bt Q ] E[T bp Q ] = 0; H a : E[T bt Q ] > E[T bp Q ] To test this fifth hypothesis, we used the paired t-test. Table 4.6 presents the descriptive statistics for the balking patience of subjects in physical and ticket queue arrangements. As seen in Table 4.7, the difference between the balking patience in PQ and that in TQ is , and the difference is statistically significant. Therefore, we conclude that the subjects have greater initial willingness to wait in the ticket queue arrangement than they do in the physical queue arrangement. This result contradicts the assumption we have in our MC model for the multi-server ticket queues outlined in Section 2. Table 4.6: Descriptive Statistics for Balking Patience in Physical and Ticket Queues Mean Std.Dev. Std. Error Mean Balking Patience in Physical Queue Balking Patience in Ticket Queue PQ and TQ Balking Patience Correlation.801 Sig..000 Table 4.7: Paired Samples t-test for Hypothesis 5 Pair Balking Patience in PQ - Balking Patience in TQ Paired Differences Mean Std.Deviation Std. Error Mean % Conf. Int. (-9.331, ) t-statistic Sig. (2-tailed).000 df 587 For testing our sixth hypothesis we let E[T bi,k ] denote the expected value of subjects initial willingness to wait in type i queue arrangement for level k traffic intensity.

121 108 We can formulate the sixth hypothesis as: H 0 : H a : ( ) ( ) E[T bt QHT ] E[T bp QHT ] E[T bt QLT ] E[T bp QLT ] = 0; ( ) ( ) E[T bt QHT ] E[T bp QHT ] E[T bt QLT ] E[T bp QLT ] > 0; We used the Independent Samples T-test for this hypothesis. The Independent Samples T-Test compares means for two groups of cases. In order to compare means, we first calculated the difference in between each subject s initial willingness to wait in the ticket queue and physical queue regimes. Afterwards, we used the Independent Samples T-test to compare the means of the differences. Table 4.8 provides the summary for the difference between the subjects balking patience in the ticket queue and the physical queue arrangements. Table 4.8: Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Traffic Intensity Traffic Intensity Mean Std.Dev. Std. Error Mean Low Traffic High Traffic Table 4.9: Independent Samples t-test for Hypothesis 6 Equal Variances Assumed Equal Variances not assumed Mean Difference Std. Error Difference % Conf. Int. (-4.524,.763) (-4.530,.768) t-statistic Sig. (2-tailed) Levene s Test for Equality of Variances F Sig..301 On the other hand, as seen in Table 4.9, the mean of differences of initial willing-

122 ness to wait in heavy traffic regimes is not statistically different than its counterpart in low traffic regimes. The t-test for equality of means does not provide statistically significant results. Therefore, we cannot reject the null hypothesis. Contrary to our predictions, the difference between the balking patience levels in the ticket queues and the physical queues under heavy and low traffic regimes are not statistically different. Let E[T bi,j ] denote the expected value of subjects initial willingness to wait in type i queue arrangement for type j service. We can use the following to test our seventh hypothesis: H 0 : H a : ( ) ( ) E[T bt QDMV ] E[T bp QDMV ] E[T bt QBANK ] E[T bp QBANK ] = 0; ( ) ( ) E[T bt QDMV ] E[T bp QDMV ] E[T bt QBANK ] E[T bp QBANK ] > 0; Again, we used the Independent Samples T-test for this hypothesis. 109 As seen in Table 4.11, the mean of differences of initial willingness to wait in high value transactions is not statistically different than its counterpart in low value transactions. The t-test for equality of means does not provide statistically significant results. Therefore, we cannot reject the null hypothesis. Contrary to our predictions, we did not find any statistically significant difference between the difference of balking patience levels in the ticket and the physical queues when the type of service is altered. Table 4.10: Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Value of Service Value of Service Mean Std.Dev. Std. Error Mean Low (Bank) High (DMV) Finally, let E[T bi,k ] denote the expected value of subjects initial willingness to wait in type i queue arrangement for level k availability of alternatives. We can formulate our eighth hypothesis as follows:

123 110 Table 4.11: Independent Samples t-test for Hypothesis 7 Equal Variances Assumed Equal Variances not assumed Mean Difference Std. Error Difference % Conf. Int. (-4.040, 1.266) (-3.959, 1.185) t-statistic Sig. (2-tailed) Levene s Test for Equality of Variances F Sig..000 H 0 : H a : ( ) ( ) E[T bt QALT ] E[T bp QALT ] E[T bt QNOALT ] E[T bp QNOALT ] = 0; ( ) ( ) E[T bt QALT ] E[T bp QALT ] E[T bt QNOALT ] E[T bp QNOALT ] > 0; Again, we used the Independent Samples T-test for the eighth hypothesis. As seen in Table 4.13, the mean of differences of initial willingness to wait in ticket and physical queues considering high value transactions is not statistically different than its counterpart considering low value transactions. The t-test for equality of means does not provide statistically significant results. Therefore, we cannot reject the null hypothesis. Contrary to our predictions, we did not find any statistically significant difference between the difference of balking patience levels in the ticket and the physical queues when the availability of alternatives is a factor in the service setting. In the above four hypothesis, we tested the subjects initial willingness to wait and how it changed under different service settings. A summary of the initial willingness to wait in different queue arrangements can be found in Figure 4.1. First of all, we can see that the ticket queue balking patience is larger than the physical queue balking

124 111 Table 4.12: Difference between Subjects Balking Patience in Ticket and Physical Queues - Effect of Existence of Alternatives Alternative Mean Std.Dev. Std. Error Mean Does not Exist Exists Table 4.13: Independent Samples t-test for Hypothesis 8 Equal Variances Assumed Equal Variances not assumed Mean Difference Std. Error Difference % Conf. Int. (-2.234, 3.082) (-2.153, 3.001) t-statistic Sig. (2-tailed) Levene s Test for Equality of Variances F Sig..010 patience in all the treatments. We also can infer from Figure 4.1 that subjects are willing to wait more for services that are of higher value to them. Besides, the higher the traffic intensity is, the longer the subjects are willing to wait. On the other hand, having alternative methods to complete the service results in lower patience levels for subjects both in the ticket and the physical queue cases. The subjects are most patient when the value of service and the traffic intensity is high and there are no alternative methods to receive the service. They are also the least patient when the value of service and the traffic intensity is low and they have alternative methods to complete the service. In summary, we can see that the customers adjust their balking patience to higher levels in high traffic and high value services, whereas they lower their balking patience levels when they have alternative methods to complete service. These results indicate that customer patience is context dependent and the customers adjust their patience according to the context of the wait. However, the difference between the mean

125 112 Figure 4.1: Balking Patience and Wait Context balking patience in ticket queues and physical queues when we isolate each one of the factors effect is not statistically significant as we displayed in Hypotheses 6 to Hypothesis for Sunk Cost (Escalation of Commitment) Effect in Queues: For our ninth hypothesis, we used the Nonparametric Friedman s test. The descriptive statistics are displayed in Table 4.14 and the test results are displayed in Table In Table 4.14, the total time the subjects are willing to wait in line is displayed under the mean column, and each row represents the time they already spent waiting in line. As seen in Table 4.15, the subjects willingness to wait increases with the time they spent in the system, and we observe the escalation of commitment effects. The subjects indicate that they would be willing to wait longer after they spent time

126 waiting in line, and would adjust their initial willingness to wait during their waits. 113 Table 4.14: Willingness to Wait vs. Time Already Spent Waiting in PQ & TQ Mean Std.Dev. Minimum Maximum PQ 15 Min Elapsed PQ 30 Min Elapsed PQ 45 Min Elapsed PQ 60 Min Elapsed TQ 15 Min Elapsed TQ 30 Min Elapsed TQ 45 Min Elapsed TQ 60 Min Elapsed Table 4.15: Nonparametric Friedman s Test for Hypothesis 9 PQ Case Mean Rank N Chi-Square Asymp. Sig. PQ 15 Min Elapsed PQ 30 Min Elapsed 1.57 PQ 45 Min Elapsed 1.57 PQ 60 Min Elapsed 1.57 TQ Case Mean Rank N Chi-Square Asymp. Sig. TQ 15 Min Elapsed TQ 30 Min Elapsed 2.10 TQ 45 Min Elapsed 2.61 TQ 60 Min Elapsed 3.69 To provide more support for our hypothesis, in Figure 4.2 we display the percentage of subjects willing to wait longer than their initial balking patience time considering the they spent in line. As seen in this figure, the more time the subjects spend in line, the more likely they are willing to wait more than their initial patience time. In conclusion, we observe the sunk cost effect in queues and provide support for the theoretical assumptions of Section 2.

127 114 Figure 4.2: Sunk Cost in Waits Factors for Queue Preference, Patience and Time Pressure Scales In this section, we present the results of the binary logistic regression model developed to identify the factors that cause the subjects to prefer one type of queue arrangement over the other. Before explaining the details of the model, we would like to discuss the scales used in the study. We used a set of five questions that represent the chronical time pressure scale to measure the level of chronical time pressure the subjects feel. Along with the set of questions in the time pressure scale, we also used a set of seven questions that represent the patience scale, and used the patience scale to identify the level of patience the participants have in general. The time pressure and the patience scales were slightly modified from previous study scales used in literature, and are customized for waiting scenarios. Our goals by using these scales is to understand the reasons behind queue preference as well as providing means to cluster the participants according to their relative positions in these two scales. Using the subject responses, we ran a series of factor analysis to determine which variables to keep and which variables to omit for the binary logistic regression model.