Queue Decomposition and its Applications in State-Dependent Queueing Systems

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1 Submitted to Manufacturing & Service Operations Management manuscript Please, provide the mansucript number! Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Queue Decomposition and its Applications in State-Dependent Queueing Systems Hossein Abouee-Mehrizi Department of Management Sciences, University of Waterloo, Waterloo, ON, N2L 3G, CANADA, HAboueeMehrizi@UWaterloo.ca Opher Baron Joseph L. Rotman School of Management, University of Toronto, Toronto, M5S 3E6, CANADA, Opher.Baron@Rotman.Utoronto.Ca A main idea in the analysis of Markovian queueing systems is that at each point in time the system can be decomposed into two systems: in one system there are n or less people and in the other one there are n + or more people. In this paper we introduce the idea of Queue Decomposition QD into non-markovian systems. Specifically, we consider an M n/g n/ state-dependent single server queue. We allow the arrival rate of customers to depend on the number of people in the system. Service times are also state-dependent and service rates can be modified at both arrivals and departures of customers as is the case in many applications. We show that the steady-state solution of this system at arbitrary times can be derived by QD using the supplementary variable method, and that the system s state at arrival epochs can be decomposed using an embedded Markov chain. For the queueing system with infinite buffer size, we first obtain an expression for the steady-state distribution of the number of customers in the system at both arbitrary and arrival times. Then, we derive the rate at which the number of customers in the system decreases by one i.e., the system moves from state n to n at both arbitrary times and arrival epochs. We show that using these transition rates, our state-dependent queueing system is equivalent to a Markovian birth-and-death process. This equivalency demonstrates our main insight that the M n/g n/ system can be decomposed at any given state as a Markovian queue. Thus, many of the existing results for systems modeled as M/M/ queue can be carried through to the much more practical M/G/ model with state dependence. We discuss several service and inventory problems that can be modeled as a state-dependent queueing system with either finite or infinite buffer size and demonstrate our results for these applications. Key words : M n/g n/ queue, birth-and-death process, state-dependent service times, state-dependent arrivals

2 2 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number!. Introduction Markovian queues have been used to model and analyze the congested queuing systems in a vast body of literature. The beauty of the single stage Markovian queues is that the system can be modeled as a Birth-and-Death B&D process and decomposed to new Markovian queues at any given state. These properties make the problem tractable even when the arrivals and service rates are state-dependent or when the control of the queue dynamically changes. This fact made the M/M/ queue a preferable model for many theoretical studies in management science. However, in many applications assuming Markovian service times is not realistic. Thus, much attention has been given to analysis of queues with generally distributed service times. An important limitation of such queueing systems however is that their analysis is not straightforward. This limitation is especially true when the arrival rates and the service times depend on the state of the system. Moreover, since such systems are not regenerative at arrival epochs, the solution of the system is not simple when the control of the queue is adjusted at such epochs. This apparent difficulty in the analysis of such queueing systems limited their study, even when they are a more appropriate model for the application considered. We believe that this difficulty therefore limited the insights generated by many papers. In this paper we consider a queueing system with state-dependent arrival and service rates, denoted as M n /G n /. We show that such state-dependent queueing systems can be decomposed at any given state of the system and modeled and solved as a B&D process. We next explain the statedependent M n /G n / queueing model we consider and discuss several interesting applications of this model in the control of service systems, security screening, healthcare, and inventory management. These applications can be analyzed using our results. Therefore our results shed light on these applications and can be further used to study these and other systems... Model and Several Applications We study a single server queue with state-dependent arrival rates and service times. Customers arrive to the system according to a Poisson process with a rate of λ n when there are n customers in the system. The service times are also state-dependent. Specifically, when there are n customers in the system and a new service time starts, it is generally distributed with a mean of /µ n, a density function of b n, and Laplace Transform LT of b n. We assume that b n is absolutely continuous with finite moments. The density function b n may be completely different than b n+, e.g., b n+ can be uniformly distributed while b n is exponentially distributed. We also allow the service rate to change when a new customer arrives to the system as follows: when there are n customers in the system and a new arrival occurs, the rate of the service increases by a factor of α n+ > 0, so that the residual service time of the customer in service

3 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! 3 decreases by this factor. Note that if α n+ < the service rate decreases and the residual service time of the customer in service actually increases. Specifically, an arrival that sees n customers in the system and a remaining service time of η for the customer in service, causes the service rate to immediately change such that the remaining service time becomes η α n+ assuming no future arrivals before the end of the current service. That is, if the residual service time when a new customer arrives, η, has a density fη, the remaining service time after this arrival will change to α n+ η and therefore it will have a density of f η. α n+ α n+ We assume that both µ n and α n are finite and greater than zero, which implies that the mean of all service times are finite. We further assume a non-idling policy, i.e., the server starts working as soon as there is a customer in the system and is only idle when the system is empty. The M n /G n / system that we consider is very flexible and therefore can be used to model many applications. We next discuss several such applications and their respective M n /G n / model. We will derive solutions to several of these applications in Section Service Systems Parkinsons Law: Work expands to fill the time available for its completion Parkinson 955 Since this paper there have been several empirical support for this law, three examples are given next. Marin et al empirically analyze the behavior of security screening at an airport and show that service times depend on the queue length. They recommend to include such service time dependency in queueing models. The stylized M/G/ has been established as a good approximation for the real world security-check queues in Zhang et al. 20. Thus, the M n /G n / we consider allows the inclusion of such service time dependency in security screening systems. Hasija et al. 200 observe a similar behavior in a contact center. In another study, Zhao et al. 202 conduct an experimental study in a supermarket modeled as a single server queue and observe that the service rate depends on the number of customers in the queue. They notice that the service rate is non-monotone and it converges. In these cases it appears that arrival rate should also converge. Indeed, considering the changes of arrival and service rates λ n and α n in the M n /G n / allows to simultaneously examine customers and the servers behaviors as a function of the queue length. This leads to our first M n /G n / model: Model : the M n /G n / system when state-dependence is for a finite number of states. In this model we assume that there exists a k < such that for any i k, arrival rates and service times are independent of the number of customers in the system, λ i = λ k, b i = b k, α i =, and λ i µ i+ < for every i k. This model is appropriate for analyzing cases where server s rate depends on the number of people in the system as in the security system and supermarket cases discussed above. We analyze this model in Section 4..

4 4 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! Arrival Rates and Service Times Control in Service Systems In recent years dynamic control mechanisms have been studied in the literature as an important tool for achieving competitive advantage, cost reduction, and added value for service systems. For example, George and Harrison 200 consider an M/M/ queueing system with adjustable service rate with the objective of minimizing the long-run average holding congestion and service level costs. Our results allow the arrival and service rates in an M/G/ queueing system to be dynamically controlled. Therefore, we can analyze the M/G/ queueing system with adjustable arrival and service rates under different control policies. This leads to our second and third models as follow: Model 2: M n /G n //K queues. In this model we assume that there exists a K < such that for any i K the arrival rate λ i to the system is zero. Alternatively, we can allow λ K > 0, but assume that the manager rejects all customers as long as there are K people in the system. We further let α K = and 0 < α j < for j = 0,..., K. We analyze this model in Section 4.2 and obtain the closed form expression for the steady-state probability of having n customers in the system, P F n, and the steady-state service rate ˆµ n when there are n people in the system. These closed form results enable the effective performance evaluation of any service and arrival rate control policy in the presence of general service times...2. Performance Evaluation Model 3: Control of arrival and service rates in M/G/ queues Ata and Shneorson 2006 consider an M/M/ service facility where the system manager makes service and arrival rates decisions at both arrivals and service terminations epochs. They model the system as an M/M/ queueing system. They show that it is optimal to operate the system as one with a finite buffer. They mention that Arguably, the M/G/ queue would be a better model, but it is not amenable to dynamic analysis. As the third application, we apply the results obtained for the M/ n /G n / queueing systems to control the arrival and service rates in an M/G/ queueing system. This model combines optimization with performance evaluation based on Model...3. Inventory Systems: Multi-Class of Customers with Priority and Brute-Force Optimization Multi-Echelon Inventory Management Multi-echelon inventory management with ample supply has been considered since Clark and Scarf 960 under different settings; see Zipkin 2000, Axsater 2006, and references therein. But, few papers in the literature consider multi-echelon inventory systems where the supplier faces congestion; see Abouee-Mehrizi et al. 203 and references therein. The main reason is that to obtain the distribution of the number of orders and inventory level at each echelon, the distribution of the number of orders and inventory level at the supplier is required, see e.g., Svoronos and Zipkin 99. Using the Queue Decomposition

5 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! 5 QD approach given in this paper, we can extend the results of Svoronos and Zipkin 99 to systems with general production and transportation times under different allocation policies as in Abouee-Mehrizi et al Moreover, we can examine multi-echelon systems where some arriving customers may be rejected based on the congestion in the system see e.g., Ceryan et al. 202 for systems with the customer rejection. In addition, the QD approach can be used to analyze multi-echelon inventory systems with several priorities. Multi-Class Make-to-Stock Systems with Priority and Backlogs Multilevel Rationing MR inventory policy has been studied in multi-class make-to-stock queueing systems with priority since Ha 997a. Based on this policy, the inventory control changes dynamically over time. More specifically, under the MR policy in a system with n classes of customers, there are n + rationing levels, R,..., R n+ where customers of classes,..., r are served as long as the inventory level is between R r and R r+. During this period when inventory is between R r and R r+ the customers of other classes are backlogged. To obtain the total cost of a M/M/ make-to-stock system under the MR policy, Dynamic Programming DP has been used see e.g., de Vericourt et al., But, when production times are not exponentially distributed, it is not practical to use DP. Fortunately, the QD approach given in this paper can be used to capture the dynamic changes of the control policy at any given inventory level see Abouee-Mehrizi et al., 202. Moreover, the QD approach can be used to extend these results to a system where the arrival rate decreases or the production rate increases as the congestion in the system increases. Multi-Class Make-to-Stock Systems with Lost Sales Multilevel Rationing MR inventory policy has been also considered in multi-class make-to-stock queueing systems with lost sales. Ha 997 shows that for an M/M/ make-to-stock system with multi-class of customers and lost sales, the MR policy is optimal in which it minimizes the total cost. Then, he provides a static method to obtain the total cost of the system with two classes of customers. Ha 2000 consider an M/E/ make-to-stock system with lost sales and show that the MR policy is optimal in this setting. However, finding the optimal rationing levels when the number of customers type is large i.e., there are more than three using DP is very hard due to curse of dimensionality. To the best of our knowledge, the analysis of this problem for M/G/ make-to-stock queues with lost sales has not been addressed in the literature before. This is the third model we consider: Model 4: MR policy in multi-class make-to-stock systems with lost sales. In Section 4.4 we model the make-to-stock inventory system with several classes of customers where the customers have different priority levels as a state-dependent M n /G n / queueing system. Here, λ n is the sum of demands of all customers types that are served when the shortfall is of n customers there are n orders in the system, and α j = for j =,..., K. This application demonstrates that the QD results provided in this paper can be used to analyze systems with several classes of customers

6 6 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! where customers have different priorities. Moreover, the optimal controls of such systems can be obtained using the QD results. Although this optimization is done using a brute force approach, it is still effective since the solution of the M n /G n //K using QD is computationally efficient. Moreover, the QD approach can be used to model systems where customers may decide not to join the system when the number of people in the system is large, or the supplier may outsource some customers to manage the congestion in the system. The former can be modeled by having λ n nonincreasing in n with a possible of λ n = 0 for large n...4. Summary of Models The models discussed above show that the QD approach can be used to model and analyze different queueing systems with finite and infinite buffer sizes as well as single and multi-class customers under the FCFS and priority policies. Moreover, they demonstrate that the results can be used to find the optimal controls of a system such as rationing levels in the multi-class inventory system for a given policy..2. Contributions and Results In this paper, we introduce the Queueing Decomposition QD approach for the analysis of the M n /G n / system with infinite and finite buffer sizes. Since the arrival process depends on the number of customers in the system, the Poisson Arrival Sees Time Average PASTA property does not hold. Therefore, we consider the system in both continuous time and at arrival epochs. To analyze the system in continuous time, we use the supplementary variable method introduced by Cox 955 to model the system as a continuous time Markov Chain MC. Using this method, we derive a closed form expression for the steady-state distribution of the number of customers in the system assuming, of course, that the system is stable. Then, we obtain the steady-state rate at which the M n /G n / system moves from state n to state n where n denotes the number of people in the system. We show that these transition rates can be used to decompose the statedependent queueing systems as a B&D process. That is, we show that the M n /G n / system can be decomposed to several new M n /G n / queues at every given state, and we characterize these queues and their solutions. We also analyze the system at arrival epochs. For this analysis, we define an embedded MC and obtain the transition probabilities in these MCs. Then, we derive the steady-state distribution of the number of customers in the systems observed by an arrival. We show that the probability of having n customers in the system at an arbitrary time and the probability of observing n people in the system by an arrival are closely related. Specifically, the ratio between these two distributions is identical to the ratio of the arrival rate when there are no customers in the system to the arrival rate when there are n customers in the system.

7 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! 7 Similar to the continuous time analysis, we obtain the steady-state rate at which the M n /G n / system moves from state n to state n and show that these rates can be used to decompose the state-dependent queueing systems at arrival epochs as a B&D process. The results obtained using our QD approach allow to study very general single server queues with a similar fashion to the study of the tractable M/M/. Thus, our results are helpful to generalize the insights from earlier studies to more practical cases. To demonstrate the application of QD and the steady-state transition rates in the M n /G n / system, we consider in detail four models: The M n /G n / system where the arrival and service processes depend on the number of people in the system for a finite number of states. Using QD we provide exact closed form expression for the distribution of the number of people in this system. 2 The state-dependent queueing system with a finite buffer, M n /G n //K. We show that the distribution of the number of people in the system is similar to the one in the M n /G n / system and derive the exact closed form expression for number of people in this queue. 3 control of the arrival and service rates in the M/G/ system to maximize the total welfare of the system. And 4 a multi-class M/G/ make-to-stock system with lost sales where the inventory is managed under the MR policy. We model this problem as a state-dependent queueing problem and present some numerical examples. To summarize the main contribution of this paper is the introduction of the QD approach to systematically study systems that can be adequately modeled as a single server queue. For this we: i provide the exact analysis of the M n /G n / state-dependent queueing systems with general service times distributions. We analyze this system at both arbitrary times and arrival epochs and obtain closed form expressions for the distribution of the number of customers in the system. ii derive the steady-state transition rates at which the state-dependent queueing systems move from state n to state n. We show that the M n /G n / system can be decomposed at any given state similar to the Markovian systems. And iii demonstrate the use of the QD results to analyze several single server models. We show that the results provided in this paper can be used to model and analyze queueing systems with single and multiple classes of customers. The analyses can be used to derive the objective functions and obtain the optimal controls of these systems..3. Literature Review Queueing systems with state-dependent arrival and service rates have been studied in the literature since Harris 967. He provides the probability distribution of the number of people in the system for M/M n / queueing systems where the rate of the service is µ n = nµ. He also derives the probability distribution of the number of people in two-state M/M/ systems where the service time of a customer depends on whether there are any other customers in the system or not at the onset of their service. Shanthikumar 979 considers a two-state state-dependent M/G/ queueing system

8 8 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! and obtains the Laplace Transform LT of the steady-state waiting time distribution in such a system. Regterschot and Smit 986 analyze an M/G/ queueing system with Markov modulated arrivals and service times. Gupta and Rao 998 consider a queueing system with finite buffer where the arrival rates and service times depend on the number of people in the system. They assume that the service times can be adjusted only at the beginning of the service and obtain the distribution of the number of people in the system in continuous time and at arrival epochs. Kerner 2008 considers a state-dependent M n /G/ queueing system and provides a closed form expression for the probability distribution of the of the number of people in the system as a function of the probability that the server is idle. But in contrast to our results, he could not derive this probability and doesn t allow state-dependent service times. The derivations in both Gupta and Rao 998 and Kerner 2008 are special restrictive cases of our results. Abouee-Mehrizi et al. 202 consider a multi-class M/G/ make-to-stock system with backlogs where customers are prioritized according to their backlog costs. They use the QD for the special case of the M/G/ queue to obtain a closed form expression for the total cost of the system under the MR inventory policy. Abouee-Mehrizi et al. 203 follow the same approach to analyze a two-echelon make-to-stock inventory system with general production and transportation times. Workload-dependent queueing systems in which the arrivals and service times depend on the workload of the system rather than the number of people in the system have been also studied in the literature see Bekker and Boxma, 2007, and references therein. The rest of the paper is organized as follows. In Section 2 we explain the problem. In Section 3 we model and analyze the M n /G n / system at both arbitrary times and arrival epochs. Then, we demonstrate the results obtained for the three applications discussed above in Section 4. We summarize the paper in Section 5. All proofs not in the body of the paper appear in the Appendix. 2. State-Dependent Queueing System In this section we first explain some preliminaries of the state-dependent queueing system that we consider in this paper. Then, we present the known results of a Markovian system to highlight the parallelism between the results we obtain for the M n /G n / and those for the M n /M n /. 2.. Preliminaries Assuming the system detailed in Section. is stable, let P i denote the steady-state probability of having i customers in the M n /G n / system, and ˆµ i denote the expected service time given there are i customers in the system. Note that expressing ˆµ i in our setting is not trivial because of the service rate changes allowed. We will characterize it in Section 3. In principle ˆµ i is not equal to µ i. Note that assuming the system is stable is equivalent to assuming that the utilization of the

9 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! 9 system is less than or that the probability that the system is idle, P 0, is greater than zero e.g., Asmussen, 99 P 0 = i=0 P i λ i ˆµ i+ > 0. 2 Condition 2 is based on the steady-state probabilities P i that requires the stability condition to be obtained. For the M n /G n / system, we present the necessary and sufficient condition for the stability of this system. A sufficient condition for the stability of this system is to have every i C where C is a finite positive number see e.g., Wang, 994. The reason is that λ i ˆµ i+ < for λ i ˆµ i+ < ensures that the transient probability of being in state C of the system is positive since ˆµ i < and the arrival process is Poisson the transit probability of being in any state i < C is positive as well. This sufficient condition is relevant for systems where the arrival and service rates are state-dependent only for a finite number of states as in Model given in Section 4.. Note that the PASTA property does not hold in the M n /G n / system that we consider. The reason is that the rate of the arrival process depends on the number of people in the system, so that future arrivals are not independent of the past and present states of the system. Therefore, the Lack of Anticipation Assumption LAA required for PASTA does not hold in such systems For more detail of LAA assumption and its essentiality to PASTA, see Bertsimas, However, conditioning on the number of customers in the system, PASTA does hold. This property that is called conditional PASTA Doorn and Regterschot, 988 will help us to analyze the systems at arrival epochs Markovian Systems To highlight the parallelism between the M n /G n / and M n /M n / queues, we present the well known analysis of the standard M n /M n / queueing system in this section. Suppose that customers arrive to the system according to a Poisson process with rate λ j when there are j customers. Service times are exponentially distributed with a rate µ j whenever there are j customers and let α j = analyzed using the basic relation µ j µ j. This queue can be modeled as a standard B&D process and λ i P i = µ i+ P i + i 0, 3 see e.g., Gross and Harris 20. Leading to the solution: Observation. Consider the M n /M n / system with arrival rate λ j and service rate µ j when there are j people in the system. Suppose that i= λ 0 i λ j+ <. 4 λ i µ j+

10 0Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! Then, the steady-state distribution of the number of people in the system is P i = λ 0P 0 λ i i where from 5 and the fact that i=0 P i =, we obtain P 0 = + i λ 0 i= λ i λ j+ µ j+, 5. 6 λ j+ µ j+ Note that, due to memoryless, relation 3 can be interpreted as both i the rate of moving from state i to state i + at any time t while the B&D process is in state i, and ii the average rate of moving from state i to state i + while the B&D process is in state i. Note however that while the first interpretation is motivating the solution of B&D process, the second interpretation is sufficient to express the steady-state probabilities P i. Indeed from Level Crossing Theory LCT Perry and Posner, 990 for any stable system that its states change by jumps of, 0, and, we have average arrival rate while in state i P i = average service rate while in state i+ P i +. Equation 7 implies that the solution for any queueing systems where service and arrivals are done a single job at a time can be decomposed as in any B&D processes. Specifically, defining λ i as the average arrival rate while in state i and ˆµ i as the average service rate while in state i +, for the M n /G n / system 7 transforms into 3. In Section 3..2, we derive the steady-sate rate at which the M n /G n / system moves from state n to state n and demonstrate that the M n /G n / system can also be analyzed as a B&D process. Then, we show that the stability condition and the distribution of the number of customers in the M n /G n / system is similar to the one in the M n /M n / system given in Observation. An important implication of the B&D representation for Markovian queueing systems is that the system can be easily decomposed to new Markovian queues at any given state. The B&D structure implies that we can decompose the queue such that during time intervals with κ or more people in the system the queue is also an M n /M n /. Specifically, conditioning on that there are κ or more people in the original system, the decomposed queue has arrival rate λ κ+i and service rate µ κ+i whenever there are i people in this decomposed queue. We call this M n /M n / queue with arrival rate λ κ+i and service rate µ κ+i the Auxiliary queue. Let P A i denote the steady-state probability of having i people in the auxiliary queue. Let F i := i P j, then, Observation 2. QD in Markovian systems: The steady-state probability of having κ + i i 0 customers in this system and the steady-state probability of having i 0 customers in the auxiliary queue are related as: P κ + i = F κ P A i, i = 0,,

11 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! Observation 2 can be used to analyze queueing systems where the control of the queue dynamically changes. In the M/G/ setting this has been demonstrated in Abouee-Mehrizi et al We obtain a similar result for the M n /G n / system in Section Analysis of M n /G n / Queues In this section, we first analyze the state-dependent queueing systems with general service time distribution at both arbitrary times and arrival epochs. Then, we demonstrate that the number of customers in the M n /G n / system is identical in distribution to this number at a specific B&D process and obtain the transition rates of this process. We also show that the system can be decomposed to new queues at any given state. 3.. Time Average Analysis In this section, we analyze the M n /G n / system at an arbitrary time. We use the method of supplementary variable introduced by Cox 955, see also Chapter II.6 in Cohen 982 and Hokstad 975. To model the system using this method, we consider a pair of variables n t and η t where n t and η t denote the number of customers in the system and the remaining service time of the customer in service, respectively. Note that η t is called the supplementary variable; but as you will see next, it has an important role in characterizing the distribution of the number of customers in the system. Let p n η, t denote the probability-density of having n customers in the system when the residual service time of the customer in service is η at time t so that: p 0 t = P n t = 0, 9 p n η, tdη = P n t = n η < η t η + dη n =, 2, 3,... 0 We have: Theorem. The Chapman-Kolmogorov equations that describe the dynamic of the pair {n t, η t, t [0, } in our M n /G n / system are given by: p 0 t + dt = p 0 t λ 0 dt + p 0, tdt + odt. p η dt, t + dt = p η, t λ dt + p 2 0, tb ηdt + p 0 tλ 0 b ηdt + odt. 2 p j η dt, t + dt = p j η, t λ j dt + p j+ 0, tb j ηdt + α j p j ηα j, t λ j dt + odt. 3 To get the intuition behind the Chapman-Kolmogorov equations given in Theorem, consider 3. Suppose that there are j > customers in the system at time t + dt and the remaining service time is η. Then, at time t one of the following has happened: there were j customers in the system

12 2Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! and no new customer arrived during the next dt units of time with probability of λ j dt first term on the right side of 3; 2 there were j + customers in the system, service was completed, and the service time of the next customer was between η and η + dt with probability of b j ηdt second term on the right side of 3; 3 there were j customers in the system when the remaining service time was ηα j and a new customer arrived during the next dt units of time with probability of λ j dt third term on the right side of 3. Similar logic leads to and 2 and the rigorous derivation is provided in the proof. Theorem generalized the continuous time MC for the M n /G/ from Kerner 2008 to the M n /G n / system that we consider Distribution of Number of People in the System In this section, we investigate the MC presented in Theorem and express the steady-state distribution of the number of people in the corresponding M n /G n / system. Let P i denote the steady-state probability of having i customers in the M n /G n / system. In Theorem 2 we present a closed form expression for P i i 0 for each i 0 using the continuous MC obtained in Theorem. Assuming that a steady-state for the system exists, we let h j denote the LT of the steady-state residual service time given that there are j 0 customers in the system. Thus, for example h 0 = b. The solution below assumes that h j are given and for convenience sets µ 0 = µ and α =. Then, Theorem 2. Suppose that i= λ 0 i λ i h λj+ j α j+ bj+ λ j+ <. 4 Then, the steady-state distribution of the number of people in an M n /G n / queue is P i = λ 0P 0 λ i where from 5 and i=0 P i =, we have i h λj+ j α j+ bj+ λ j+. 5 P 0 = + λj+. 6 i λ 0 h j α j+ i= λ i bj+ λ j+ Note that condition 4 guarantees that the steady-state probability of having no customers in the system, P 0, is larger than zero. Therefore, it is the necessary and sufficient condition for the stability of our M n /G n / system. We next assume that the system is stable, and obtain h i recursively.

13 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number!3 Theorem 3. Suppose that the M n /G n / system is stable. Then, the LT of the steady-state distribution of the residual service time given that there are i customers in the system can be calculated recursively from: where h 0 = b. h i s = λ i [ b i λ i h i s α i s λ i h i λ i α i b i s] i, 7 Note that to characterize h i we assume that the system is stable. But, the stability condition in 4 is a function of h i. To verify the stability condition, one should assume that h i obtained in Theorem 3 are well defined, calculate them recursively, and then check if 4 holds. If 4 holds, then the system is stable and therefore h i are well defined. Of course, 4 requires the evaluation of an infinite series of LTs, which can only be recursively calculated using 7; thus, in application, the value of 4 may be limited. Still, we show below that Theorems 2 and 3 can be used in several relevant cases Modeling the M n /G n / Queues as a Birth-and-Death Process In this section, we demonstrate that the state-dependent queueing systems with general service time distribution can be analyzed as a B&D process. The rate at which the number of customers in the system decreases in the M n /M n / system is equal to the service rate, µ i, because of the memoryless property. In the following observation, we obtain the steady-state rate at which the number of customers in the system decreases by one in the M n /G n / system. Let ˆµ i denote the steady-state rate at which the system moves from state i to i when i is the number of customers in the system. Then, Observation 3. The steady-state number of customers in the M n /G n / system has the same distribution as the steady-state number of customers in a B&D M n /M n / process with arrival rate λ i and service rate when there are i people in the system. ˆµ i = λ i b i λ i h i λ i α i 8 Using Observation 3 and 8 we can rewrite the stability condition 4 for the M n /G n / system as the probability that there are i customers in the system as i= λ 0 i λ j+ <, 9 λ i ˆµ j+ P i = λ 0P 0 λ i i λ j+ ˆµ j+, 20

14 4Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! and the probability that the M n /G n / system is idle as P 0 = +. 2 i λ 0 λ j+ i= λ i ˆµ j+ Comparing 9-2 with 4-6 we observe that the stability condition and the distribution of the number of people in the M n /G n / has the same structure as the one in the M n /M n /. This similarity indicates that the solution of the M n /G n / can be decomposed as in any B&D processes. Although expressing ˆµ i is not trivial whenever service times are not Markovian, fortunately, we could obtain it for the state-dependent M n /G n / system in Observation 3. Having ˆµ i and relation 3, we can decompose the M n /G n / system as in the B&D process and analyze the system. We will use these results to analyze several problems as a B&D process in Section Queue Decomposition at a Given State In Observation 2, we showed that the M n /M n / queueing system can be decomposed at any given state. In this section we extend this result to the M n /G n / system. More specifically, we show that the time intervals when there are κ or more people in the M n /G n / can be decomposed as in the M n /M n /. Specifically, as in Abouee-Mehrizi et al. 202a we define an auxiliary queue such that the steady-state probability of having κ + i customers in the original system given that there are κ or more people in this system is identical to the steady-sate distribution of having i jobs in this auxiliary queue. To distinguish between the original queue and the auxiliary queue, we use the term job in the auxiliary queue. We define the auxiliary queue as an M n /G n / queue with the following a arrival and b service processes: Step a: jobs arrive to the auxiliary queue according to a Poisson process with rate λ κ+i for i 0 when there are i people in the auxiliary queue. Step b: the distribution of the first service time in each busy period of the auxiliary queue is the conditional residual service time in the original queue given that there are κ customers in the system, i.e., the equilibrium remaining service times given that there are κ customers in the system. The distribution of the rest of the service times in each busy period of the auxiliary queue is identical to the original queue, i.e., b κ+i for i 0 when there are i people in the auxiliary queue. Moreover, when there are i customers in the auxiliary queue and a new arrival occurs, the rate of the service increases by a factor of α κ+i+ > 0, so that the residual service time of the customer in service decreases by this factor. To summarize, the auxiliary queue is a state-dependent queue with general service time distribution, M n /G n /. We next prove that the auxiliary queue is equivalent the original M n /G n / queue during the time intervals when there are κ or more people in the queue.

15 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number!5 Let P A i denote the steady-state probability of having i jobs in the Auxiliary queue. The steady-state distribution of the number of jobs in the auxiliary queue is similar to the one in an M n /G n / queue given in Theorem 2: i P A i = P A 0 h κ+j λ κ. 22 bκ λ κ Recall the in the original system we let F i := i P j for the original M n/g n / system. In the following observation, we show that the probability of having i jobs in the auxiliary queue, P A i, is identical to the probability of having κ+i customers in the original queue, P κ+i, given that there are more than κ customers in the M n /G n / system. Observation 4. QD in M n /G n / systems: The steady-state probability of having κ + i i 0 customers in the original M n /G n / system and the steady-state probability of having i 0 customers in the auxiliary queue are related as: P κ + i = F κ P A i, i = 0,, The intuition behind Observation 4 is that because the M n /G n / queue is similar to a B&D process, this queue can be decomposed at any state just as any Markovian queue. Indeed because in the M n /G n / settings the memoryless property does not hold, this decomposition requires a careful definition of the first service time in the auxiliary queue; however, then all other service and arrival times are defined as in the original queue. Observation 4 states that the steady-state number of jobs in the auxiliary queue is identical to the steady-state number of customers in the original state-dependent queue during the time intervals when there are more than κ customers in the original M n /G n / system. This observation can be used to decompose the M n /G n / queueing system at any given state. We use this observation and appropriate definitions of auxiliary queues in Section 4 to analyze several applications Analysis at Arrival Epochs In this section, we analyze the M n /G n / at arrival epochs. We remind that since in this system the arrival process are state dependent, the PASTA property does not hold. Therefore, the steadystate distribution of the number of customers seen by an arrival are typically different than the steady-state distribution of the number of customers in the system at an arbitrary time. Let P a n denote the steady-state probability that an Arrival observes n customers in the system. To obtain P a n, we consider that in the M n /G n / the distribution of the number of customers seen by an arrival is identical to the steady-state distribution of the number of customers seen by a departure this easily follows by a level crossing argument as in Buzacott and Shanthikumar, 993. Therefore, we analyze the system at departure epochs by defining an embedded MC.

16 6Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! M n Let M n denote the number of customers left behind by the n th departing customer in the system. can be found by the MC embedded at the Departures. Let P d n denote the steady-state distribution of being in state n of this MC. Then, P a n = P d n. 24 To derive P d n, we need to determine the one-step transition probabilities of M n, For state j, let v j k p jk = P M n+ = k M n = j. 25 denote the probability of having k 0 arrivals during the service time that starts when there are j customers in the system. For j = 0, the relevant service time starts when there is one customer in the system. So we let v 0 k = v k. Then, the probability that the next departure leaves k customers behind given that there are no customers in the system, p 0k, is v 0 k. Similarly, the probability that the next departure leaves k customers behind given that there are j customers in the system, p jk, is v j k j+. To summarize, we denote the one-step transition probabilities of M n, the embedded MC, as p jk = { v 0 k, j = 0; v j k j+, j. 26 Note that in the standard M/G/, these probabilities can be easily obtained since the distribution of the service times are identical for all customers. Let λ and b denote the arrival rate and the LT of the service time distribution in the standard M/G/, respectively. Then, considering that v j k is independent of j, the probability generating function of v k := v j k j = 0,,... is see e.g., Takagi 99 V z = v k z k = bλ z. 27 k=0 We next extend this results to the M n /G n / and derive the one-step transition probabilities of the embedded MC. Note that the probability of the number of arrivals between two departures cannot be obtained similar to the standard M/G/ as in 27 since the distribution of the service times are not identical, and any arrival may change the arrival and service rates. Therefore, instead of considering the distribution of the number of arrivals during a service time, we consider the probability of a new arrival during the residual service time observed by any customer upon arrival. Fortunately, this distribution possesses the conditional PASTA property: Corollary. The conditional steady-state distribution of the residual service time observed by arrivals that find j customers in the system is identical to the conditional steady-state distribution of the residual service time at an arbitrary time in the M n /G n / system that also observes j customers in the system.

17 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number!7 We next determine the transition probabilities, p jk, for j > 0. The derivation for p 0k is similar but more detailed and it is provided in the proof of Theorem 4. Consider p j,j. The probability that the next departing customer leaves one less customer behind given that the last departing customer left, P M n+ = j M n = j. This is equal to the probability of no arrival during the next service time, b j λ j e.g., Conway 967, page 7. Therefore, p j,j = b j λ j, j. 28 With similar logic, the probability that the next departing customer leaves k customers behind given that the last departing customer left j customers behind, P M n+ = k M n = j, is equal to the probability of k j + arrivals during the next service time. This probability is equal to the probability that one customer arrives after the next service time starts, b j λ j, and a customer arrives during the remaining service time of all arrivals that see i < k customers in the system, h i λ i+ α i+, and no arrival during the remaining service time once there are k customers in the system, h k λ k+. Therefore, α k+ p jk = h k λk+ α k+ b k j λ j Note that for the M/M/ queue we have h j λ j = b i λ i = bλ = i=j h λi+ i, j, k j. 29 α i+ λ µ+λ so that p jk = µ k j+ λ, j, k j. 30 µ + λ µ + λ As expected due to memoryless, PASTA, and that the minimum of two independent exponential variables is an exponential variable, in the M/M/ setting p jk has a binomial distribution with µ parameter. µ+λ Using the transition probabilities in 29 and these for p 0j, the steady-state distribution of the number of customers in the system observed by an arrival is as follows. Theorem 4. Suppose that i i= h λj+ j α j+ bj+ λ j+ <. 3 Then, the steady-state distribution of the number of people in an M n /G n / queue observed by an arrival is P a i = P i a 0 where from 32 and i=0 P a i =, we have P a 0 = + h λj+ j α j+ bj+ λ j+ i i= λj+ h j α j+ bj+ λ j+,

18 8Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number! Comparing 5 and 32, we can relate the steady-state probability of having i customers in the system at an arrival epoch to the steady-state probability of having i customers in the system at an arbitrary time. Observation 5. The steady-state probability that an arrival observes i customers in the system is equal to the fraction of intervals when there are i customers in the system given an arrival occurs. More specifically, in an M n /G n / system, the relation between the steady-state probability of having i customers in the system at arrival or departure epochs and at arbitrary times is given by: P a i = λ i P i k=0 λ kp k. 34 Note that when arrival rates are identical, λ i = λ for i = 0,,..., the PASTA property holds and therefore P a i = P i. Observation 5 indicates that P a i can be obtained using P i given in Theorem 2. More interestingly, Observation 5 together with Theorem 4 provide an alternative proof for Theorem 2 using that: Lemma. In an M n /G n / system, the relation between the steady-state probability of having i customers in the system at arbitrary times and at arrival or departure times is given by: P i = P 0 λ 0 P a i λ i P a 0 i Similar to the analysis in Section 3..2, we next obtain the rate at which the number of customers in the M n /G n / system decreases by one at arrival epochs. Let ˆµ a i denote the rate that the number of people in the system decreases from i to i. Then, Observation 6. The rate at which the M n /G n / system moves in steady-state from state i to state i considering the system at arrival epochs is ˆµ a i = λ i λ i ˆµ i = λ i b i λ i h i λ i α i. 36 Observation 6 relates the utilization of the system in steady-state with the utilization of the system as it would be measured by arrivals. Specifically, Observation 6 indicates that the steadystate utilization of the system observed by arrivals who find j customers in the system, λ j, is ˆµ a j identical to the steady-state utilization of the system when there are j customers in the system, λ j ˆµ j. 4. Applications of QD In this Section, we analyze the four models discussed in Section. using the results from Section 3: the M n /G n / system when state-dependence is for a finite number of states, 2 M n /G n //K queues, 3 control of arrival and service rates in the M/G/ queues, and 4 MR policy in multiclass make-to-stock systems with lost sales.

19 Article submitted to Manufacturing & Service Operations Management; manuscript no. Please, provide the mansucript number!9 4.. Application : Analysis of the M n /G n / System when State-Dependence is for a Finite Number of States In Section 3., we obtained the steady-state probability of having i > 0 customers in the system as a function of P 0, and provided an expression for P 0 that includes an infinite sum. In this section we obtain a closed form expression for the steady-state probability that there are no customers in the system, P 0, for the special case of Application. We assume that there exists a k < such that for any i k, λ i and b i are independent of the number of people in the system, i.e., k = min i {λ i, b i, α i : λ i = λ i+ =..., b i = b i+ =..., α i = α i+ =...}. This queue is state-dependent for i < k and has the same arrival rate and service time distribution for i k. We assume that λ k /µ k < to ensure that the system is stable. As we will soon show, in this case the infinite summation in 6 becomes finite; thus the probability that the system is idle can also be simplified see 39 below. To obtain P 0, we use QD. Consider the auxiliary queue defined in Section 3..3 for κ = k. Let ρ b and µ b denote the server utilization and the rate of the first exceptional service times in this auxiliary queue, respectively. Note that the service time distributions, b i, in this system are identical for i > k. Therefore, the rate of the service time in the auxiliary queue is µ k probability ρ b and µ b with probability ρ b, so that with leading to ρ b = ρ b λ k µ k + ρ b λ k µ b ρ b = λ k µ k µ b µ k + λ k µ k µ b. 37 To obtain the utilization of the auxiliary queue, ρ b, we derive the average rate of the exceptional first service times, µ b. is, Lemma 2. The average exceptional first service times in the busy period of the auxiliary queue = k + µ b µ k λ k k α i+ j= i=j µ j λ j bi+ λ i+ h i λ i+ α i+ µ b λ h 0 λ α k i= bi+ λ i+ α i h i λ i+ α i+. 38 Note that Sigman and Yechiali 2007 provide the expected conditional stationary remaining service time in an M/G/ queue. Lemma 2 extends their result to the M n /G n / queue. Using 5, 22, and 23 the probability of having no customers in the M n /G n / system of Application is: