EQUILIBRIUM STRATEGIES IN AN M/M/1 QUEUE WITH SETUP TIMES AND A SINGLE VACATION POLICY

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1 EQUILIBRIUM STRATEGIES IN AN M/M/1 QUEUE WITH SETUP TIMES AND A SINGLE VACATION POLICY Dequan Yue 1, Ruiling Tian 1, Wuyi Yue 2, Yaling Qin 3 1 College of Sciences, Yanshan University, Qinhuangdao , China 2 Department of Intelligence and Informatics, Konan University, Kobe , Japan 3 College of Liren, Yanshan University, Qinhuangdao , China ydq@ysu.edu.cn, tianrl@ysu.edu.cn, yue@konan-u.ac.jp, @qq.com Keywords: Queueing systems, single vacation, setup times, balking, equilibrium strategies. Abstract We consider equilibrium strategies in an Markovian queue with setup times and a single vacation policy. Arriving customers decide either to enter the system or to balk based on their desire for service and their unwillingness for waiting. We derive equilibrium strategies for both cases of fully observable queue and fully unobservable queue. Then, for fully unobservable queue, we consider socially optimal strategy and illustrate the effects of several parameters on equilibrium strategy and socially optimal balking strategy using numerical examples. 1 Introduction In recent years, there is an emerging tendency in the literature to study queueing systems from an economic viewpoint. In such studies, a certain reward-cost structure is imposed on the system that reflected the customers desire for service and their unwillingness for waiting. Customers is allowed to make decisions about their actions in the system. For example, they need to decide whether to join or to balk, to wait or to abandon, to retry or not, etc. These decisions result a symmetric game among the customers, where the basic problem is to find equilibrium strategies and socially optimal strategies. The study of queueing systems under a gametheoretic perspective was initiated by Naor [1] who studied M/M/1 queueing system with a simple linear reward-cost structure. Naor [1] assumed that an arriving customer observed the number of customers and then made his decision whether to join or balk. His study was complemented by Edelson and Hildebrand [2], who considered the same M/M/1 queueing system, but assumed that the customers made their decisions without being informed about the state of the system. Their results were further refined and extended by several authors. Many results on this topic can be found in a comprehensive monograph written by Hassin and Haviv [3]. Queueing systems with vacations have been extensively studied. Detailed surveys are referred to the two monographs written by Takagi [4] and Tian and Zhang [5]. Equilibrium customer strategies for a single-server Markovian queue with setup times were first analyzed by Burnetas and Economou [6]. They derived equilibrium strategies for the customers under the various levels of information and analyzed the stationary behavior of the system under these strategies. Recently, Sun and Tian [7] considered the equilibrium and socially optimal strategies for a partially observable queue with vacations. Guo and Hassin [8] studied a Markovian vacation queue with N-policy and exhaustive service. They presented equilibrium strategies and socially optimal strategies for both cases of unobservable queue and observable queue. Similar to [6], Tian and Yue [9] considered an M/M/1 queue with a single vacation policy. They studied equilibrium strategies and socially optimal strategies for case of unobservable queue. In this paper, we extend the queuing models in [6] and [9] to a queuing model with a single vacation and setup times. We classify the queues into two categories: the fully observable queue and the fully unobservable queue, according to the information levels regarding system states. The customers dilemma is whether to join the system or to balk, based on a linear rewardcost structure that incorporates their desire for service, as well as their unwillingness for waiting. We consider equilibrium customer strategies for both cases of observable queue and unobservable queue. We also study socially optimal strategies for unobservable queue. This paper is organized as follows: In Section 2, we describe the system model. In Section 3, we determine equilibrium threshold strategy for fully observable queue. In Section 4, we determine the equilibrium balking strategy and the socially optimal strategy for fully unobservable queue. Some numerical results are also provided. Conclusions are given in Section 5. 2 Model Descriptions Consider an M/M/1 queueing system with an infinite waiting room, where customers arrive according to a Poisson process with rate λ and service times are as ISORA IET 159 Huangshan, China, August 23 25, 2013

2 sumed to be exponentially distributed with rate µ. The service discipline is first-come first-served (FCFS). Whenever the system becomes empty, the server takes a vacation at the completion instant of a service. When the vacation ends, if there are customers waiting in the queue, the server begins to serve the customers. Otherwise, the server shuts down for a period of time called close-down time. If a customer arrives during a close-down time, then the close-down time ends, and a setup time is needed for the server before being able to provide a service. The setup times and the vacation times are assumed to be exponentially distributed with rate ξ and θ, respectively. It is assumed that inter-arrival times, service times, vacation times and setup times are mutually independent. We suppose that a customer s utility consists of a reward for receiving service minus a waiting cost. It is assumed that every customer receives a reward of R units for completing a service. This may reflect his (or her) satisfaction and the added value of being served. On the other hand, there exists a waiting cost of C units per time unit when the customer remains in the system. We are interested in the customers strategic response as they can decide whether to join or to balk upon arrival. Customers are risk neutral and maximize their expected net benefit. From now on we assume the condition ( 1 R > C µ + min{1 θ, 1 ) ξ }. (1) This condition ensures that the reward for a service exceeds the expected cost for an arriving customer who finds the server is in a close-down time or a vacation time and decides to enter the system. Otherwise, no customer will enter the system during a close-down time or a vacation time. Under the above framework, we develop a symmetric game among the customers who are all indistinguishable. Let S be the common set of strategies, and let F (a, b) be the payoff of a customer who selects strategy a when everyone else chooses strategy b. A strategy s e is an (symmetric Nash) equilibrium strategy if F (s e, s e ) F (s, s e ) for every s S. Intuitively, an equilibrium strategy is a best response against itself, so that if all customers agree to follow it, no one can benefit by deviating from it. A strategy s 1 is said to dominate strategy s 2 if F (s 1, s) F (s 2, s) for every s S and at least one s the inequality is strict. A strategy s is said to be weakly dominant if it dominates all other strategies in S. It is assumed that the customers decisions are irrevocable: retrials of balking customers and reneging of entering customers are not allowed. 3 Fully Observable Queue In this section, we derive threshold equilibrium strategies for the fully observable queue where arriving customers observe both the number of customers in the system and the state of the server. We represent the state at time t by the pair {(N(t), I(t)), t 0}, where N(t) denotes the number of customers in the system and I(t) denotes the state of the server, which is defined as follows: I(t) = 0 denotes that the server is taking on vacation, I(t) = 1 denotes that the server is in close-down time or setup time, and I(t) = 2 denotes that the server is busy. A threshold strategy is specified by a vector n e = (n e (0), n e (1), n e (2)), which means that a customer who arrives at time t and observes (N(t), I(t)), enter the system if N(t) n e (I(t)) and balk otherwise. Let s e (i)(i = 0, 1, 2) be the mean sojourn time of an tagged arriving customer who joins the system and observes state (n, i), n 0, i = 0, 1, 2. It is easy to see that ( n + 1 (s e (0), s e (1), s e (2)) = µ + 1 θ, n + 1 µ + 1 ξ, n + 1 ) µ (2) where n+1 µ is the mean service times of the tagged customer and the other n customers preceding the tagged arriving customer, and 1 θ is the mean residual vacation time, and 1 ξ is the mean residual setup time. Obviously, the expected net benefit of the tagged arriving customer is r e (i) = R Cs e (i), i = 0, 1, 2. (3) Theorem 1. For the fully observable queue, there exists an unique equilibrium strategy in the class of threshold strategies n e = ( Rµ C µ θ 1, Rµ C µ 1, ξ ) Rµ 1 C where x denotes the maximum integer not exceeding x. Moreover, it is a weakly dominant strategy. Proof. Let S be the set of threshold strategies, which is given by (4) S = {n = (n(0), n(1), n(2)) n(i) = 0, 1, 2,, i = 0, 1, 2}. Let F (s, m) = (F 0 (s, m), F 1 (s, m), F 2 (s, m)) be the payoff of a customer who selects strategy s when everyone else chooses strategy m, where s, m S. Then, we have { re (i), n s(i), F i (s, m) = 0, n > s(i). We note that r e (i) is decreasing in n. It is easy to observe that r e (i) > 0 for all n < n e (i) and r e (i) 0 for all n n e (i) ISORA IET 160 Huangshan, China, August 23 25, 2013

3 If s(i) < n e (i), then we have that: F i (s, m) = F i (n e, m) = r e (i) > 0, n < s(i), 0 = F i (s, m) < F i (n e, m) = r e (i), m(i) n < n e (i), F i (s, m) = F i (n e, m) = 0, n n e (i). If s(i) > n e (i), then we have that: F i (s, m) = F i (n e, m) = r e (i) > 0, n < n e (i), r e (i) = F i (s, m) < F i (n e, m) = 0, n e (i) n < m(i), F i (s, m) = F i (n e, m) = 0, n n e (i). If s(i) = n e (i), then we have that F i (s, m) = F i (n e, m) for all s, m S. Thus, we prove that F i (s, m) F i (n e, m) for all m S, i = 0, 1, 2. Therefore, n e = (n e (0), n e (1), n e (2)) is an unique equilibrium strategy in the class of threshold strategies.a weakly dominant strategy. Moreover, it is a weakly dominant strategy. 4 Fully Unobservable Queue In this section, we derive the equilibrium balking strategy and the socially optimal balking strategy for fully unobservable queue where we assume that arriving customers can not observe both the number of the customers in the system and the states of the server. So, arriving customers may join the queue with the probability q, or may balk with probability 1 q, 0 q 1. A balking strategy is characterized by the joining probability q. q = 0 and q = 1 are called pure strategies, and q(0 < q < 1) is called a mixed balking strategy. 4.1 Equilibrium Balking Strategies In this subsection, we consider the customers equilibrium balking strategies. If all arriving customers follow a strategy q, then the arrival stream of customers who join the system follows a Poisson process with an effective arrival rate λq. This results in an M/M/1 queue with a single vacation and setup times, where the arrival rate is given by λq, and the other assumptions on the distribution of the service times, the vacation times and the setup times are same as that described in Section 2. Xu et al. [10] considered M/M/1 queue with working vacation and setup time. If we choose the server s service rate during the working vacation to be zero, then the model in [10] becomes our model considered in this paper. Under steady-state condition λ/µ < 1, Xu et al. [10] obtained the expected sojourn time given as follows: E(W ) = 1 µ λ + K 1 r (5) where r is the root of the equation and K = µ 0 r 2 (λ + θ + µ 0 )r + λ = 0 (6) µ µ 0 (1 µ 0r θ λ θ(µ λ) λ ) + θµ(1 r)(λ + ξ) λξ 2 + θµ ξ + θ + µ µ 0r 1 r. (7) Let µ 0 = 0 in Eq. (6), then we get r = λ/(λ + θ). Let µ 0 = 0, r = λ/(λ + θ), and replace λ by λq in Eq. (7), we get the mean sojourn time denoted by W (q) for our model, given by the following lemma. Lemma 1. Assume that qλ/µ < 1. If all arriving customers follow a common strategy q. Then, the mean sojourn time of a customer who joins the system is given by W (q) = 1 µ λq + 1 θ λξ2 q 2 + λθ(ξ 2 + θ 2 )q + ξθ 3 λξ 2 q 2 + λθ(ξ 2 + ξθ)q + ξ 2 θ 2. (8) If all arriving customers follow a strategy q, then a tagged customer who decides to join the system will get the expected net benefit U(q) = R CW (q), where W (q) is given by Eq. (8). Lemma 2. If θ < ξ, then U(q) is a decreasing function of q for 0 q 1. Proof. The detail of the proof is omitted. In order to derive the equilibrium balking strategies, we consider the following three cases: (1) θ = ξ, (2) θ < ξ and (3) θ > ξ. Theorem 2. (1) For the case of θ = ξ, there exists an unique equilibrium balking strategy q e = min{q0, 1}, where q0 = µ λ 1 λ ( R C ). 1 (9) θ (2) For the case of θ < ξ, we have: (a) If q 0 1, or q 0 < 1 and U(1) 0, there exists an unique equilibrium balking strategy q e = 1; (b) If q 0 < 1 and U(1) < 0, there exists an unique equilibrium balking strategy q e satisfy U(q e ) = 0 and q 0 < q e < 1. (3) For the case of θ > ξ, we have: (a) If R < C(1/µ + 1/ξ) and U(1) > 0, there exists at least one equilibrium balking strategy q e satisfy U(q e ) = 0 and 0 < q e < 1; (b) If R > C(1/µ + 1/ξ) and U(1) < 0, there exists at least one equilibrium balking strategy q e satisfy U(q e ) = 0 and 0 < q e < min{q 0, 1}. Proof. If a tagged customer who decides to enter the system, then the customer s expected net benefit is U(q) = R CW (q), where W (q) is given by Eq. (8). Then, the tagged customer prefers to enter if U(q) > 0, he is indifferent between entering and balking if U(q) = 2013 ISORA IET 161 Huangshan, China, August 23 25, 2013

4 0 and he prefers to balk if U(q) < 0. It is easy to see that ( 1 U(0) = R CW (0) = R C µ + 1 ) > 0. (10) ξ Let ( 1 φ(q) = R C µ λq + 1 ). (11) θ It is easy to see that φ(q) is a decreasing function of q for 0 q < µ/λ. From Eq. (1), we have φ(0) > 0. Thus, φ(q) = 0 has an unique root q0 in interval for (0, µ/λ), and q0 is expressed in Eq. (9). Consider the following three cases: (1) θ = ξ, (2) θ < ξ and (3) θ > ξ, we can obtain the equilibrium balking strategy given by Theorem 2. The details of the proof are omitted here. Remark 1. For the case of θ > ξ, we do not know if there exists any equilibrium balking strategy q (0, 1) for the following two cases: (1)R < C(1/µ + 1/ξ) and U(1) 0, (2)R > C(1/µ + 1/ξ) and U(1) 0. Remark 2. Theorem 2 shows that if θ ξ there exists an unique equilibrium balking strategy. However, if θ > ξ we are not able to show if there exists an unique equilibrium balking strategy for the two cases expressed in Theorem Socially Optimal Balking Strategies In this subsection, we derive socially optimal baking strategies. Clearly, the mean number of arriving customers that join the system per time unit is λq, and individual customer s expected net benefit is R CW (q). Thus, the total expected social benefit per time unit is given by S(q) = λq[r CW (q)], where W (q) is given by Eq. (8). The first two derivatives of S(q) are denoted by S (q) and S (q). They can be computed from the expression of S(q). We omitted their cumbersome expressions. Let q1 be the root of the equation S (q) = 0, and let q be the optimal joining probability which is called socially optimal balking strategy, i.e., S(q ) = max{s(q), q [0, 1]}. In general, it is difficult to get an analytical expression of q. However, if θ = ξ, we have [ S (q) = λ R C ] θ Cµ (µ λq) 2. Clearly, S (q) = 0 has unique root ( ) q1 = 1 Cµξ µ. λ Rξ C It is easy to see that S (q) < 0 for 0 < q < µ/λ, i.e., S(q) is a concave function of q for 0 < q < µ/λ. If 0 < q 1 < 1, then q = q 1. If q 1 1, then q = 1. For cases of θ < ξ and θ > ξ, we can obtain the socially optimal balking strategy numerically as shown in Fig. 1 and Fig. 2. In Fig. 1, we choose the parameters are as follows: R = 25, λ = 0.7, µ = 1, ξ = 0.1, θ = 0.05, C = 1. From Fig. 1, we find that the socially optimal balking strategy q = q 1 = 0.198, and S(q ) = In Fig. 2, we choose the parameters are as follows: R = 20, λ = 0.7, µ = 1, ξ = 0.4, θ = 0.8, C = 1. From Fig. 2, we find that q 1 = 1.093, and the socially optimal balking strategy q = 1, and S(q ) = In the following, we investigate the effect of the vacation time parameter and the setup time parameter on the joining probability and compare the equilibrium joining probability and social optimal joining probability. In Fig. 3, the system parameters are given as follows: λ = 0.7, µ = 1, R = 50, ξ = 0.8 and C = 1. The joining probability is plotted against the vacation rate θ. It is observed from Fig. 3 that q e increases quickly and then equals 1 with the increasing of the vacation rate θ, while q increases quickly and then increases slowly with the increasing of the vacation rate θ and is less than q e. This can be explained by noting the fact that the increasing of the vacation rate θ may lead to a decreasing of the expected waiting time of the customers. In Fig. 4, the system parameters are given as follows: λ = 0.7, µ = 1, θ = 0.3, R = 35 and C = 1. The joining probability is plotted against the setup rate ξ. We observed from Fig. 4 that both q e and q increase with the increasing of the setup rate ξ. Also, we observe that the socially optimal strategy q is less than or equal to the equilibrium strategy q e. From Fig. 3 and Fig. 4, it can be shown that q q e regardless of the system parameters. This ordering is typical when customers make individual decisions maximizing their own profit. Then, they ignore those negative externalities that they impose on later arrivals, and they tend to overuse the system. It is clear that these externalities should be taken into account when we aim to maximize the total revenue. As a result, in the latter case, less customers should join the system, which in turn leads to less congested systems. 5 Conclusions In this paper, we studied the equilibrium customer behavior in an M/M/1 queue with a single vacation and setup times. We classified the queue into fully observable queue and fully unobservable queue with respect to the information provided to customers. For fully observable queue, we founded that there exists an unique equilibrium threshold balking strategy. For fully unobservable queue, we considered equilibrium balking strategy by considering three cases. We founded that if the vacation parameter θ is less than or equal to the setup time parameter ξ there exists an unique equilibrium balking strategy. In addition, we discussed the so ISORA IET 162 Huangshan, China, August 23 25, 2013

5 Social benefit S(q) Social benefit S(q) Probability q Probability q Figure 1: Social benefit S(q) vs. probability q. Figure 2: Social benefit S(q) vs. probability q. Joining probability q q e q * Vacation rate θ Joining probability q q e q * Setup rate ξ Figure 3: Joining probability q vs. vacation rate θ. Figure 4: Joining probability q vs. setup rate ξ. cially optimal strategies numerically for unobservable queue. There is a direction for future work considering the equilibrium customer behavior for almost observable queue and almost unobservable queue. For the first case, customers observe only the number of the customers in the system, while for the second case, customers observe only the state of the server. Acknowledgments This research was supported in part by the National Natural Science Foundation of China (No ) and the Natural Foundation of Hebei Province (No. G ) and is supported in part by MEXT, Japan. References [1] Naor, P., The regulation of queue size by levying tolls, Econometrica, 1969, 37, pp [2] Edelson, N.M., and Hildebrand, K., Congestion tolls for Poisson queueing processes, Econometrica, 43, pp , [3] Hassin, R., and Haviv, M., Equilibrium behavior in queueing systems: to queue or not to queue (Kluwer, 2003). [4] Takagi, H., Queuing analysis, a foundation of performance evaluation. vol. 1: vacation and priority (Elsevier, 1991). [5] Tian, N., and Zhang, Z.G., Vacation queueing models: theory and applications(springer, 2006). [6] Burnetas, A., and Economou, A., Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 2007, 56, pp [7] Sun, W., and Tian, N., Contrast of the equilibrium and socially optimal strategies in a queue with vacations, Journal of Computational Information Systems, 2008, 4, pp [8] Guo, P., and Hassin, R., Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 2011, 59, pp [9] Tian, R., and Yue, D., Optimal balking strategies in an Markvian queue with a single vacation, Journal of Information and Computational 2013 ISORA IET 163 Huangshan, China, August 23 25, 2013

6 Science, 2012, pp [10] Xu, X., Zhang, Z., and Tian, N., The M/M/1 queue with single working vacation and set-up times, International Journal of Operational Research, 2009, 6, pp ISORA IET 164 Huangshan, China, August 23 25, 2013