Abstract
During the past few years, batch service systems have attracted considerable attention due to their wide area of applications. In this present paper, we study a special batch service polling system (the socalled Israeli queue) with priorities. Different from the previous papers which focus on the performance analysis, we aim to investigate the strategic behavior of customers and optimal design for the underlying queueing model. By considering two levels of information (observable and unobservable) provided upon customers’ arrival, we, respectively, derive the equilibrium strategies of highpriority and lowpriority customers, regarding the joining or balking dilemma. We also present some numerical examples to reveal the impacts of several parameters on the equilibrium strategies, together with some intuitive explanations. Finally, we formulate the revenue function of the service provider and present the Particle Swarm Optimization algorithm to seek the optimal service prices for the highpriority and lowpriority customers to maximize the service provider’s revenue under the two levels of information.
1. Introduction
Nowadays, queueing models with batch services frequently encountered in practice and studies addressing the batch service queueing systems have been investigated extensively owing to their wide area of applications, such as the transportation systems, tourist services, and telecommunication systems. So far, the literature on this topic can be roughly divided into two categories. The first category is to analyse the batch service models with fixed batch size. For example, at the airports, the airplane departs once all seats are occupied. In tourism services, the tour guide starts his/her work as soon as a fixed number of travelers have been accumulated. Another example is that, in tourist areas, sightseeing bus will start off while the bus is full of travelers. The second category is to deal with the batch service models with infinite batch size. For example, at the bus stations and metro stations, the bus and the train can remove all present passengers. Such batch service models can be also referred to as stochastic clearing models.
Due to the wide range of applications on batch service models, in this paper, we will consider a special polling system with batch service of an unlimited size, i.e., the socalled “Israeli queue,” which has been widely used in practice and extensively studied in the literature. As pointed out by Perel and Yechiali [1], Israeli queue stands for a real situation of a physical waiting queue for buying tickets at stations and it is a multiqueue, singleserver polling system with unlimitedsize batch service; after completion of a visit at a queue, the next queue to be served is the one whose first customer has been waiting for the longest time. For example, when buying ticket at a railway station, a new arriving customer first searches for the acquaintance already standing in the waiting line; if he/she finds one, he/she will ask for the acquaintance to buy the tickets along with him/her. Since the service time of a batch does not depend on its size, the waiting time of customers is not affected by the customers that join the queue in front of them. Actually, this setting can be also seen in real life especially in transportation systems. At a bus station or a metro station, there may be multiple routes through the station. While a bus or a train visits the station, it can pick up all passengers going to the same destination. Therefore, the passengers may be served prior to the passengers who have already arrived at the station before them, since each arriving bus goes to a specific destination, and passengers waiting to go to different destinations have to wait for their dedicated bus. Therefore, for the Israeli queue, this may not be a real interjection, since the waiting time of customers is not affected by the customers who join the queue in front of them; nonetheless, we could come to an agreement that the queueing phenomenon no longer follows the firstcome, firstserved (FCFS) discipline and the customers may be served prior to the customers who have already arrived at the system before them.
However, the majority of the papers on batch service systems are devoted to the stationary distribution and the performance evaluation. It seems that only limited attention has been paid to the issue of the batch service systems arising from the customers perspective; i.e., there are few works that consider such batch service systems from the viewpoint of customer behavior (see, e.g., Economou and Manou [2], Manou et al. [3], and Bountali and Economou [4, 5]). Actually, the optimization problems related to the strategic customer behavior in batch service polling systems could not only provide important managerial insights, but also have a wide range of applications in real life. Therefore, it seems important to give a gametheoretic research to the special batch service polling system.
Apart from the gametheoretic research to batch service queueing systems, providing priority access is a widespread method for improving the work efficiency; meanwhile, designing an appropriate priority system is also an efficient scheme for reducing congestion. Therefore, the problems of priority queueing models are frequently encountered in practice and they have been extensively studied due to their wide applications in some healthcare systems, production systems, communication systems, etc. For more studies on priority queues, we may refer the interested readers to Drekic and Grassmann [6], Jouini and Roubos [7], Takagi [8], Atencia [9], and the reference therein.
To this end, we incorporate customers’ joining/balking decisions into a special batch service polling system with priorities. Different from the classical batch service queueing systems, the queueing phenomenon in consideration no longer follows FCFS discipline. Therefore, we are interested in the following research questions: Q1: How do the strategic customers (highpriority and lowpriority customers) make their joining/balking decisions under different levels of information (observable and unobservable cases)? Q2: How do the system parameters affect the customers’ joining/balking decisions? Q3: How does the revenuemaximizing decision maker choose the optimal pricing decision to maximize his/her revenue?
To address the research questions, we consider a special batch service polling system with priorities, which can be modelled as a twoclass, singleserver, preemptive priority queueing model, in which the highpriority customers form an M/M/1 queue, while the lowpriority customers form the socalled Israeli queue with at most N different groups. The main contribution of our study can be summarized as follows:(1)In the existing literature, steadystate analysis of the Israel queue has been well studied; however, only limited attention has been paid to the issue of the queueing service from the viewpoint of customer behavior. Such a treatment (which ignores customer behavior) could be sometimes unrealistic especially when customers’ strategic behaviors affect service system performance and system revenue. Based on this situation, we aim to investigate the strategic customers’ joining/balking decisions of the underlying Israel queue, and this has not been studied in the open literature.(2)Different from the classical batch service queueing systems with homogeneous strategic customers, the underlying special batch service polling system adopts classification of service through setting priority due to the heterogeneity of sensitive customers. Therefore, under the two levels of information, we distinguish the highpriority and lowpriority customers, respectively, derive their joining/balking strategies, and present the equilibrium rates with respect to various system parameters by considering different scenarios.(3)In order to achieve optimal pricing strategies and maximize the service provider’s revenue by distinguishing the observable and unobservable cases, we establish a new revenue function and use the Particle Swarm Optimization (PSO) algorithm to search for the global numerical solution (feasible combination of the service prices for the highpriority and lowpriority customers) and show the difference between the two informational cases.
Some interesting insights can be found from the derived results. First, the joining rates of highpriority and lowpriority customers are monotonically decreasing in the arriving rate of highpriority customers. The numerical result shows that if the arriving rate of highpriority customers is high, the lowpriority customers are inclined to balk. Conversely, if the arriving rate of highpriority customers is low, both of the highpriority and lowpriority customers decide to join the system. Therefore, if the service provider wants to ensure his/her revenue, he/she should effectively control the number of highpriority customers. Second, the lowrate service of highpriority customers could cause the highpriority customers to balk, which could also reduce the waiting time of lowpriority customers and lead them to join the Israeli queue with a higher probability. And then, as the service rate of highpriority customers gradually increases, the highpriority customers are more willing to join the system due to the reduction of their waiting time, which indirectly affects the joining strategies of lowpriority customers. When the service rate of highpriority customers is large enough, the service time of highpriority customers is quite short, and their expected waiting time tends to be zero; that is, the waiting time of highpriority customers has a smaller effect on the lowpriority customers, which exactly promotes the lowpriority customers to join the Israeli queue. In short, the joining probability of lowpriority customers is sensitive to the service rate of highpriority customers. Therefore, under a fixed cost, determining a suitable service rate of highpriority customers is vitally important for the service provider.
The rest of the paper is organized as follows. A detailed literature review on related works is given in Section 2. In Section 3, we describe the batch service polling system with priorities and construct the rewardcost structure. Section 4 and Section 5 are devoted to deriving the individual optimal threshold strategies and the mixed equilibrium joining strategies under the two informational cases (observable and unobservable). In Section 6, we construct a revenue function of the service provider and employ the PSO algorithm to solve the optimization problem of service provider’s revenue under the observable and unobservable settings. We conclude the paper and summarize the main findings of our work in Section 7.
2. Literature Review
This work is closely related to three streams of literature: the Israeli queues, batch service queueing systems with strategic customers, and queueing models with customer service priorities.
Different from some batch service systems that the server serves customers in batches of fixed size, Israeli queue is a specific polling system with batch service of an unlimited size. Customers arrive at the system and form groups; each group is unrestricted in its size and served in one batch; after one group is served, the next group to be visited is the one whose group leader has been waiting for the longest time. This model was first introduced by Van der Wal and Yechiali [10], who considered a multiqueue, singleserver polling system with unlimitedsize batch, and customers in the same queue can be served together. Lately, Boxma et al. [11] studied the batch service polling system with unrestricted batch size and finite number of queues. Perel and Yechiali [12] considered a batch service queueing model with infinite number of groups and studied the M/M/c, M/M/1/Ntype queues, and a priority queueing model. Perel and Yechiali [1, 13, 14], respectively, investigated the Israeli queue with preemptive priorities, the Israeli queue with retrial, and the Israeli queue with a general groupjoining policy. By using the probability generating function and matrix analytic method, the authors provided an extensive probabilistic analysis and derived some key performance measures. Recently, Jiang et al. [15, 16], respectively, investigated the multiserver Israeli queue with dynamic service control and the Israeli queue operating in a multiphase random environment on the basis of Perel and Yechiali’s studies. Moreover, in [13], if each retrial customer in the orbit independently repeatedly tries for receiving service rather than only the first customer in the orbit tries for receiving service, an explicit closedform solution for the steadystate probability distribution may be difficult to derived. Therefore, Jiang [17] again analysed the tail asymptotics property of the Israeli queue with retrials and nonpersistent customers on the basis of Perel and Yechiali’s studies [13].
Our model also belongs to the class of queues with strategic customers. The study of queueing systems under a gametheoretic perspective was first introduced by Naor [18] who studied the strategic behavior of customers in M/M/1 queue, where arriving customers know the number of customers in the system and decide whether to join or balk based on their surplus utilities. Edelson and Hildebrand [19] considered the strategic behavior of customers in M/M/1 queue by studying the corresponding unobservable case. Since then, there has been considerable attention paid to the strategic behavior of customers in queueing systems; interested readers may also refer to the books of Hassin and Haviv [20, 21], Hassin [22], recent papers Hassin and RoetGreen [23], Ibrahim [24], and the reference therein. The study of strategic customers in batch service queueing systems is a recent endeavor. For example, Economou and Manou [2] investigated an M/M/1 clearing system in an alternating environment, where all customers are served together at the completion epoch of a service cycle. Manou et al. [3] considered the strategic behavior of customers in a transportation station, where all present passengers can be removed together, once a vehicle reaches the station. Bountali and Economou [4, 5] investigated the strategic behavior of customers in queueing systems with fixed size batch services under the unobservable, observable, and partially observable cases. In [4, 5], the authors discussed the impact of different levels of information and the batch size on the joining strategies of customers and compared the joining strategies of customers under the incomplete information level with the corresponding strategies in the complete information level. Recently, Bountali and Economou [25] studied the strategic customer behavior in a twostage service system with batch processing. Wang et al. [26] studied passengers’ joining strategies in a batch transfer queueing system with gated policy under the full observable case. Adan et al. [27] considered a polling system with two queues and studied the optimal routing problem of customers under different levels of information. Chai et al. [28] studied a batch doublesided service system with impatient servers and boundedly rational customers, where each server could serve the customers with a fixed service capacity N.
The queueing model in consideration is also closely related to queueing models with customer service priorities. In queueing systems with customer service priorities, highpriority customers could skip over lowpriority customers; i.e., the highpriority customers are served prior to the lowpriority customers, which leads to a longer waiting time for the lowpriority customers who may be already in the queue. Therefore, queueing models with customer service priorities can be also regarded as a special queueing system with customer interjection. However, different from the other queueing systems with customer interjection, providing priority access is a widespread method for improving the work efficiency; meanwhile, designing an appropriate priority queueing system is also an efficient method for managing congestion. For more detailed studies on priority queues, readers are referred to Jouini and Roubos [7], Takagi [8], Drekic and Woolford [29], Afèche and Mendelson [30], Gavirneni and Kulkarni [31], Wang et al. [32], and references therein. Moreover, the study of priority queues from an economic viewpoint is also a recent endeavor. For example, Drekic and Woolford [29] analysed a twoclass, singleserver, preemptive priority queueing model, where the decision to balk or not of the two types of customers depends on the queue length. Brouns and van der Wal [33] studied the optimal threshold strategies in a preemptive priority queueing system with admission and termination control. Liu and Berry [34] considered a preemptive priority queueing model, where the authors focus their analysis on the price competition and social welfare in an unobservable M/G/1 queue with two types of customers. Later, Xu et al. [35, 36], respectively, obtained the optimal balking strategies of highpriority and lowpriority customers in an M/G/1 queue with preemptive priorities. In addition, in the literature of operations management, some authors considered the queueing systems with a payforpriority option. For example, Hassin and Haviv [20] considered the equilibrium threshold policies with preemptive priority customers. Afèche and Mendelson [30] investigated the pricing and priority auctions in queueing systems with a generalized delay cost structure. Gavirneni and Kulkarni [31] studied a selfselecting priority queue, where customers have heterogeneous waiting costs , and they decide whether to become priority customers or regular customers or balking customers on the basis of their different waiting costs levels. Wang et al. [32] studied the equilibrium strategies in an M/M/1 priority queue, in which the authors consider the system with a payforpriority option, and derive the customers’ joint decisions between joining/balking and payforpriority. For more details on this topic, the interested readers are referred to Adiri and Yechiali [37], Hassin and Haviv [21], and Hassin [22], in which the authors give comprehensive reviews on the topic of payforpriority option.
3. Model Description
We consider a twoclass, singleserver, preemptive priority queueing model in which the highpriority customers form a classical M/M/1 priority queue, while the lowpriority customers form the socalled Israeli queue with at most N different groups. We use “he” to represent a highpriority customer and “she” to represent a lowpriority customer for ease of exposition. We also assume that high (low)priority customers arrive at the system according to a Poisson process with rate . Service time of highpriority customers is exponentially distributed with rate . For the lowpriority customers, customers are served in an unlimitedsize batch mode; i.e., the lowpriority customers in a group are served together at a service period, and the service time of a batch independent of its size is assumed to be exponentially distributed with parameter . A new arriving lowpriority customer who determines to join the system will first search for the leader of each group (the searching sequence follows the agebased discipline; i.e., an arriving lowpriority customer first searches for the leader of each group who has waited for the longest time), and if she knows a group leader, she immediately joins the group and receives service in a batch mode along with the group leader. We further assume that a new arriving lowpriority customer knows a group leader with probability . That is, if , groups exist in the Israeli queue, then an arriving lowpriority customer who decides to enter the Israeli queue will join th group with probability for or will create a new group and become a group leader with probability . If N groups are present in the Israeli queue, an arriving lowpriority customer who decides to enter the Israeli queue will join Nth group even if she does not know any of the existing group leaders. We further assume that the highpriority customers have preemptive priority over the lowpriority customers, which indicates that the service for lowpriority customers may be interrupted immediately by highpriority customers.
For the recent study of Israeli queue with priorities in [1], the authors have obtained the stationary distribution and performance measures; however, in the present paper, we are interested in investigating the strategic behavior of customers in the batch service special polling system with priorities, where arriving customers have the option to determine whether or not to join the system according to their expected utilities. We further assume that they make their decisions based on the information of the system including the arrival rate, the service rate, the probability that a new arriving lowpriority customer knows a group leader, and the unit waiting cost. Then, they determine whether or not to join the system according to the natural linear rewardcost structure (service reward service valuewaiting cost), where is the expected utility of a customer, is the reward on completion of a service, is the average delay cost per unit of time for each customer, and is the expected sojourn time of a customer who decides to join the system. Then, an arriving customer decides whether or not to join the system based on the value of the expected utility . If , he/she will join the system; otherwise, he/she will balk.
In the following parts, we will investigate the strategic behavior of highpriority and lowpriority customers regarding their joining or balking dilemma by distinguishing the observable and unobservable cases, i.e., the arriving customers decide whether or not to join the system according to the information available at their arrival instants.
4. Analysis of the Observable Case
We first start our analysis by investigating the Israeli queue with priorities under the observable case. In the observable case, the customers could receive exact information about the Israeli queue upon their arrival; i.e., they could get informed about the statistical and economic parameters of the system, the number of different groups, and the number of highpriority customers in the system. In order to obtain the individual optimal pure strategy, we follow the example of Perel and Yechiali [1] to calculate the expected sojourn time of customers. According to [1], we have the following conclusions:(1)Since the highpriority customers form a classical M/M/1 queue, the expected sojourn time of an arbitrary highpriority customer is .(2)Define as the time of duration from the first moment when there are highpriority customers in the system until the first moment that no highpriority customers are present. We have the expected value .(3)Suppose that there are no highpriority customers in the system and there are lowpriority customers groups. Let be the time of duration that these groups are served and leave the system; then the expected value .(4)Assume that there are highpriority customers and lowpriority customers groups in the system and let denote the time of duration that the service of all those lowpriority customers groups is completed. We have the expected value .
4.1. The Individual Optimal Pure Strategy of HighPriority Customers
Next, based on the above results, we first determine the unique individual optimal pure strategy of highpriority customers. Under the observable case, since the highpriority customers form a classical queue ( is the threshold), it is easy to obtain the unique individual optimal pure strategy. For an arriving highpriority customer, when he finds highpriority customers present in the system, he joins if ; otherwise he balks, where represents the expected sojourn time of an arriving highpriority customer who finds highpriority customers and decides to enter the system.
Lemma 1. (see [18]). In the observable Israeli queue with priorities, there exists a unique individual optimal pure strategy of highpriority customers which has the following forms:
Case 1. If , then an arriving highpriority customer always balks.
Case 2. If , the unique individual optimal strategy is the threshold strategy having the following form: when an arriving highpriority customer observes that there are highpriority customers present in the system, he joins if ; otherwise, he balks, where with .
4.2. The Individual Optimal Pure Strategy of LowPriority Customers
Let and , respectively, denote the number of highpriority and lowpriority customers in the system at time . Then, is a continuoustime Markov chain with state space . For the lowpriority customers, when an arriving lowpriority customer finds the system being in state , she joins the system if ; otherwise she balks, where represents the expected sojourn time of an arriving lowpriority customer who finds the system being in state . Next, we determine the unique individual optimal pure strategy of lowpriority customers. We first give the following theorem to show the individual optimal pure strategy of lowpriority customers under the observable case.
Theorem 1. In the observable Israeli queue with priorities, there exists a unique individual optimal pure strategy of lowpriority customers which has the following forms:wherewith .
Proof. We first derive the expected sojourn time of an arriving lowpriority customer who finds the system being in state . Based on the detailed list of results presented in Section 4 (obtained by Perel and Yechiali in [1]), the expected sojourn time of an arriving lowpriority customer has the following forms:After some calculations, we haveDifferentiating with respect to and , respectively, we then have and ; that is, is a decreasing function of and . Hence, an arriving lowpriority customer balks if ; otherwise, she prefers to join the Israeli queue. In order to obtain the unique individual optimal pure strategy of lowpriority customers, we consider the following cases.
Case 1. The arriving lowpriority customer finds no groups in the Israeli queue.(1)If , i.e., , then, an arriving lowpriority customer always balks.(2)If , i.e., , then, there exists a unique root , such that . Solving this equation, we have , since is a decreasing function of ; an arriving lowpriority customer determines to join the Israeli queue if and only if .
Case 2. The arriving lowpriority customer finds , groups in the Israeli queue.(1)If , i.e., , then, an arriving lowpriority customer always balks.(2)If , i.e., , then, there exists a unique root , such that . Solving this equation, we have , since is a decreasing function about ; an arriving lowpriority customer determines to join the Israeli queue if and only if .
Case 3. The arriving lowpriority customer finds N groups in the Israeli queue.(1)If , i.e., , then, an arriving lowpriority customer always balks.(2)If , i.e., , then, there exists a unique root , such that . Solving this equation, we have , since is a decreasing function of ; an arriving lowpriority customer determines to join the Israeli queue if and only if .According to the above analysis, we find that .
Remark 1. (1)If the individual optimal threshold strategy of highpriority customers is greater than the individual optimal threshold strategy of lowpriority customers , i.e., , then, the individual optimal threshold strategy of lowpriority customers is the result obtained in Theorem 1.(2)If there exists k, such that , then the result in Theorem 1 should be(3)If , then the result in Theorem 1 should beSince is the unique individual optimal pure strategy of highpriority customers, i.e., the number of highpriority customers is no more than , we could obtain the results of (6) and (7).
4.3. The Stationary Distribution under the Observable Case
In order to solve the optimization problem of subsequent equilibrium rewards, deriving the steadystate probabilities on the basis of the threshold strategies is quite crucial. According to the results in Lemma 1 and Theorem 1, the equilibrium joining strategies of both types of customers become threshold policies in the observable case. Each customer (either highpriority or lowpriority customer) could determine the individual optimal strategy once the state of the system is observable. Meanwhile, it is not difficult to find that when the highpriority customers always balk (), the model degenerates into a onedimensional finite Markov chain, and then the underlying queueing model reduces to an queue with only lowpriority customers under a threshold policy; otherwise forms a nonhomogeneous finite QuasiBirthandDeath (QBD) process. Considering the variety of situations caused by the threshold policies of highpriority and lowpriority customers, taking the situations and , for example, then the transition rate diagram can be shown in Figure 1.
Next, we mainly focus on the general case with the assumption that the highpriority customers do not balk. According to the transition rate diagram given in Figure 1, we refer to Latouche and Ramaswami in [38] to solve the nonhomogeneous finite QBD with level process and phase process . The generator of the underlying queueing system has the following forms:where the order of each submatrix is by and all of the block matrices can be easily obtained while the necessary parameters are given. Next, the following theorem indicates how to determine the steadystate probability vector.
Theorem 2. The stationary probability vector can be given bywhere satisfies
Proof. This theorem can be proved by expanding . We may first obtain in terms of by several iterations. Then, by normalizing the vector , we could determine . The detailed process is not shown here.
The detailed recursive algorithm for calculating the steadystate probability vectors can be summarized in Algorithm 1.

5. Analysis of the Unobservable Case
In this section, we will turn to the analysis of the unobservable case. In the unobservable case, the customers could not receive exact information about the Israeli queue except the statistical and economic parameters of the system upon their arrival; i.e., they do not receive any information about the number of different groups and the number of highpriority customers in the system. Before analysing the strategic behavior of customers under this case, we first derive the steadystate probabilities of the system. We assume that an arriving highpriority customer joins the system with probability and an arriving lowpriority customer joins the system with probability . Since is a continuoustime Markov chain with state space , then, by referring to the continuoustime Markov process, we can obtain the statetransitionrate matrix as follows:where
with , , , and
with , and is the identity matrix of order .
In order to analyse the system effectively by matrix analytic method, an important matrix in analysing the process is the rate matrix , which is the minimal nonnegative solution of
According to the special structure of matrices , we know that is an upper triangular matrix and has a closedform expression. The explicit calculation of can be seen in [1]. We denote the steadystate probability vector of the queueing model , where , , then, the steadystate probability vector can be obtained by solving the following linear equations:
and the steadystate probability vector , can be obtained by .
5.1. Equilibrium Strategy of HighPriority Customers
Next, according to the obtained steadystate probability vectors, we investigate the strategic behavior of customers under the unobservable case. It is readily seen that the system is stable if and only if . However, even if , the system may be stable. So, we will analyse this problem in two cases: and . We first derive the joining strategies of highpriority customers by the following lemma.
Lemma 2. In the unobservable Israeli queue with priorities under the condition , there exists a unique equilibrium mixed strategy of highpriority customers, which has the following forms:
Remark 2. For the case , the analog of Lemma 2 is that a unique equilibrium mixed strategy of highpriority customers exists, which has the following form:
5.2. Equilibrium Strategy of LowPriority Customers
Next, we aim to obtain the joining strategies of lowpriority customers. For the lowpriority customers, the expected utility of a lowpriority customer while all the other lowpriority customers follow the strategy iswhere represents the expected sojourn time of a lowpriority customer who decides to join the system, which is a function with regard to and . According to the equilibrium strategies of highpriority customers, we then derive the equilibrium strategies of lowpriority customers. Next, we first assume and consider the following three scenarios to investigate the equilibrium strategies of lowpriority customers.(1)Scenario 1: all of the highpriority customers balk, i.e., .(2)Scenario 2: a fraction of the highpriority customers joins the priority queue, i.e., .(3)Scenario 3: all of the highpriority customers join, i.e., .
Scenario 1. All of the highpriority customers balk, i.e., . Under this scenario, the system reduces to an M/M/1/N type Israeli queue without priorities; we then have the following theorem.
Theorem 3. When all of the highpriority customers balk , , then, there exists a unique equilibrium mixed strategy of lowpriority customers, which has the following forms:whereand is the unique solution of the following equation:
Proof. When all of the highpriority customers balk, i.e., , the system reduces to an M/M/1/N type Israeli queue without priorities.
Under this scenario, the stationary distribution is easily given by (we can also refer to Perel and Yechiali in [12])Then, we have the following expression of :whereNext, we will derive the variation trend of the function on . Define ; then the expression of can be rewritten asDifferentiating with respect to , we haveWe next focus on the analysis ofwhose sign is exactly contrary to (26) (see Wang and Sun in [39] and Wang et al. in [40]). Expanding the termsinto polynomial functions of respectively, we havewhere and .
Fixing the value of , for any feasible , it is not difficult to find thatwhich leads to ; thus we have . Then, the result shows that is a decreasing function of . Next, we consider the following three cases to obtain the equilibrium strategies of lowpriority customers under this scenario.
Case 1. If , i.e., , we have ; then is nonpositive for every ; the best response of an arriving lowpriority customer is balking and the unique equilibrium point is .
Case 2. If and , i.e.,then, there exists a unique solution that lies in , such that . By solving the following equation:we could obtain the unique solution .
Case 3. If , i.e., , then is nonnegative for every and the best response is 1. So, joining the Israeli queue is the unique equilibrium strategy.
Based on the above analysis, we can derive the result of this theorem.
Next, we consider the other two scenarios that highpriority customers join the priority queue; i.e., and . Under the two scenarios, we first give the expression of by referring to the results in Perel and Yechiali in [1]. From [1], we haveAccording to the two results, the expression can be written aswhere is the function with regard to and , which can be rewritten as follows:, , and and are the functions with regard to . Due to the complexity of the expression , it is difficult to derive the variation trend of the function on as and . Due to the lack of analytic results on the monotonicity, we may use the numerical results to show the monotonicity of on . To this end, we use a numerical example to show the monotonicity of the function on . We assume that , , , , , , , and .
According to Figure 2, we find that while and , is an increasing function of . Intuitively, higher arrival rate leads to more lowpriority customers staying in the system, which will necessarily increase the sojourn time of lowpriority customers. Since , we obtain that is a decreasing function of . In order to derive the equilibrium mixed strategy of lowpriority customers, we next consider the other two scenarios.
(a)
(b)
Scenario 2. A fraction of the highpriority customers joins the priority queue; i.e., . Under this scenario, we have the following theorem.
Theorem 4. A fraction of the highpriority customers joins the priority queue ; if , then there exists a unique equilibrium mixed strategy of lowpriority customers, which has the following forms:where and is the unique solution of the following equation:
Proof. In this scenario, we consider the following three cases.
Case 1. ; i.e., ; then is nonpositive for every ; the best response of an arriving lowpriority customer is balking and the unique equilibrium point is .
Case 2. and ; i.e.,
; then, there exists a unique solution that lies in , such that . By solving the following equation:we could obtain the unique solution .
Case 3. ; i.e., ; then is nonnegative for every ; i.e., the best response is 1. So, joining the Israeli queue is the unique equilibrium strategy.
Scenario 3. All of the highpriority customers join the priority queue, i.e., .
The analysis of this scenario is similar to that of Scenario 2, and we will show the results directly in the following theorem.
Theorem 5. All of the highpriority customers join the priority queue ; if , then there exists a unique equilibrium mixed strategy of lowpriority customers, which has the following forms:where and is the unique solution of the following equation:
Remark 3. For the case , we can obtain the same results as in Theorem 2 and similar results as in Theorem 3; the difference of the results between Theorem 3 and the underlying case is that, under the case , the condition of Theorem 3 “a fraction of highpriority customers joins the priority queue if ” should be replaced by “a fraction of highpriority customers joins the priority queue if ” without changing the conclusion.
5.3. Sensitivity Analysis
In this subsection, we illustrate some numerical results to study the impact of different parameters on the equilibrium joining probabilities of both types of customers. Without loss of generality, we assume that and conduct the sensitivity analysis by setting a series of values of the target parameters. The exact numerical results are presented in Figure 3.
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(e)
(f)
According to the model assumption, highpriority customers have a preemptive priority than lowpriority customers; then, with the change of the parameters related to lowpriority customers, there are no impacts on the strategies of highpriority customers, and the exact results are confirmed in Figures 3(b), 3(d)–3(f). In Figure 3(a), we could easily find that both of and are monotonically decreasing in the parameter ; the difference is that decreases faster than , which can be attributed to the preemptive priority of highpriority customers. The observations also reveal that, for low values of , both of the highpriority and lowpriority customers have an incentive to join the system because of a shorter waiting time. For high values of , the system becomes congested; since the highpriority customers have preemptive priority than the lowpriority customers, it is plausible that the waiting time of lowpriority customers is significantly longer than the highpriority customers, and then the lowpriority customers are more willing to balk than the highpriority customers. In addition, Figures 3(b) and 3(d) vividly demonstrate the relationship between the equilibrium joining probability of lowpriority customers and their potential arrival rate as well as the corresponding service rate. Figure 3(c) illustrates the following facts: (1) for very low values of , is equal to one, and has a low value. (2) For intermediate values of , is equal to zero, and gradually increases with the increase of . (3) For very high values of , both of and are equal to one. In short, is not a monotonous function of , and is an increasing function of . The interesting results are clearly related to the waiting time of joining customers. The lowrate service of highpriority customers causes the highpriority customers to balk, which could reduce the waiting time of lowpriority customers and lead the lowpriority customers to join the system with a higher probability. And then, as the service rate of highpriority customers gradually increases, the highpriority customers are more willing to join the system due to the reduction of the waiting time, which indirectly affects the joining strategies of lowpriority customers. When the service rate of highpriority customers is large enough, the service time of highpriority customers is quite short, and their expected waiting time tends to be zero; that is, the waiting time of highpriority customers has a smaller effect on the lowpriority customers, which exactly promotes the lowpriority customers to join the system. From Figure 3(e), it is shown that as increases, an arriving lowpriority customer knows a group leader with a higher probability, and it is more likely to join the group which locates in front of the Israeli queue. So, based on this, there are fewer groups in the system, which also leads to the reduction of the lowpriority customers’ waiting time and the increase of . Figure 3(f) actually indicates that, with the increase of , the lowpriority customers may queue at the back of the Israeli queue, which may increase the expected waiting time of lowpriority customers (since more groups exist in the system); at the same time, the lowpriority customers are more reluctant to join the system.
6. Optimal Pricing Strategy
In service systems, it is evident that customers are usually heterogeneous in many aspects, such as the evaluation of delay and the waiting cost. Due to the heterogeneity of delaysensitive customers, the service provider often adopts classification of service through setting priority. However, the classification of service with uniform price will produce the difference of waiting time and service value, which may cause the unfairness psychology for the lowpriority customers with longer waiting time. Hence, the service provider needs to adopt the differential pricing so as to eliminate lowpriority customers unfairness psychology, where highpriority customers expect to be served as soon as possible and need to pay a high price to save their time in queueing and waiting. Under the specific application background, we will analyse the optimal pricing strategy for both of the highpriority and lowpriority customers, so that we could give some management suggestions for the special batch service polling system with priorities. Next, in this section, on the basis of a new assumption that the service provider can set the prices for the two kinds of service, we will deal with the service provider’s revenue maximization problem. Hence, the linear utility function of highpriority customers (lowpriority customers) () should be converted to (), where and are the setting prices for the highpriority and lowpriority service, respectively. That is to say, the symbols and in the previous sections should be replaced by and . In order to maximize the service provider’s revenue (denoted by ) in our model, the most common way for the revenuemaximizing service provider is to adjust the prices for both types of services. Owing to the preemptive priority of highpriority customers, setting will affect not only the arrival rate of highpriority customers, but also the arrival rate of lowpriority customers, and setting only affects the lowpriority customers’ joining decisions. So, the decision problem can be formulated aswhere and are the actual arrival rates. It should be noted that either or has different representations in the observable and unobservable cases. In the observable case, the two types of customers adopt each equilibrium threshold strategy; i.e., and are the effective arrival rates. However, in the unobservable case, both of the highpriority and lowpriority customers access the system with each equilibrium joining probability, i.e., and . Due to the monotonicity of lowpriority customers’ utility function, we apply dichotomy method to obtain when there exists an intersection between the utility function and axis. Meanwhile, could be obtained in accordance with Lemma 2 and Remark 2.
However, in most cases, due to the computational complexity, it is quite difficult to obtain the explicit expressions of the effective arrival rates with respect to the corresponding prices. Hence, we aim to search for the numerical optimal solution of through PSO algorithm which will be introduced briefly.
6.1. Particle Swarm Optimization Algorithm
PSO algorithm was firstly presented by Kennedy and Eberhart in [41], which is devoted to solving some continuous nonlinear optimization problems. The algorithm is exempt from the analyticity of the objective function; it does not use the gradient of the optimization problem; that is to say, unlike the classical gradient descent method and quasiNewton method, PSO algorithm does not require that the optimization problem is differentiable. Due to its simplicity and convenience, this algorithm is frequently applied to search for the global optimal solution. Certainly, PSO is undoubtedly applicable to this twodimensional optimization problem . To seek the optimal price combination , the procedure of applying PSO algorithm to this twodimensional optimization problem is presented as follows: (Algorithm 2)

6.2. Optimal Pricing in the Observable Case
Through Algorithm 1, we can obtain the steadystate probability vectors, and the optimization problem can be formulated aswhere and is an effective arrival rate with a complex expression while highpriority customers do not take the balk strategy.
Similarly, PSO algorithm can be also applied to solve the optimization problem (see, e.g., Wang et al. [42], Zhang et al. [43], and Yang et al. [44]). To keep with the above analysis, we consider a numerical example under the observable case with the assumption that