Strategic joining in M/M/1 retrial queues

The equilibrium and socially optimal balking strategies are investigated for unobservable and observable single-server classical retrial queues. There is no waiting space in front of the server. If an arriving customer finds the server idle, he occupies the server immediately and leaves the system after service. Otherwise, if the server is found busy, the customer decides whether or not to enter a retrial pool with infinite capacity and becomes a repeated customer, based on observation of the system and the reward–cost structure imposed on the system. Accordingly, two cases with respect to different levels of information are studied and the corresponding Nash equilibrium and social optimization balking strategies for all customers are derived. Finally, we compare the equilibrium and optimal behavior regarding these two information levels through numerical examples.     

Equilibrium analysis of the observable queues with balking and delayed repairs

The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs.     

具有两类故障特性的M/M/1排队系统均衡分析

考虑顾客在具有两种故障特性的马尔科夫排队系统中的均衡策略.在该系统中,正常工作的服务台随时都可能发生故障.假设服务台只要发生故障就不再接收新顾客,并且可能出现的故障类型有两种：（1）不完全故障：此类故障发生时,服务台仍有部分服务能力,以较低服务率服务完在场顾客后进行维修;（2）完全故障：此类故障发生时,服务台停滞服务并且立即进行维修,维修结束后重新接收新顾客.顾客到达时为了实现自身利益最大化都有选择是否进队的决策,基于线性“收益-损失”结构函数,分析了顾客在系统信息完全可见和几乎不可见情形下的均衡进队策略,及系统的平均社会收益,并在此基础上,通过一些数值例子展示系统参数对顾客策略行为的影响.     

Equilibrium balking strategies in Markovian queues with working vacations

The equilibrium balking strategies are investigated in the paper for observable and unobservable single-server queues with working vacations. In such an M/M/1 queue with working vacations, the server undertakes the workload with a lower service rate rather than completely stops to work during the vacation period. Upon arrival, the customers decide whether to join or balk the queue based on observation of the queue length and the status of the server, along with the reward-cost structure of the system. Accordingly, four cases with respect to different levels of information are studied and the corresponding Nash equilibria are derived. Finally, the effect of the information levels as well as several parameters on the equilibrium threshold and equilibrium entrance probabilities is illustrated by numerical examples.     

Equilibrium threshold strategies in observable queueing systems under single vacation policy

This paper studies the equilibrium behavior of customers in continuous/discrete time queueing systems under single vacation policy. In the single vacation queueing system, the server can only take exactly one vacation when no customers exist in the system. This scheme is more practical under many specific circumstances. Based on the reward–cost structure, equilibrium behavior is considered in the fully observable and almost observable cases. The threshold strategies in equilibrium are obtained and the stationary system behavior is analyzed under the corresponding strategies. Finally, we illustrate the effect of the information level as well as several parameters on the equilibrium thresholds and social benefits via numerical experiments. The research results could instruct the customers to take optimal strategies and provide the managers with reference information on the pricing problem in the queueing system.     

不同价格水平下排队策略研究

在M/M/1排队中引入了不同的服务价格,基于"收益-成本"结构,以顾客和企业均追求利益最大化为出发点,在两种不可见情形下,研究了顾客均衡策略行为和企业最优服务定价决策,通过数值模拟,描述了休假期服务价格对顾客均衡策略的影响,以及几乎不可见情况下休假期服务价格对企业收益的作用和完全不可见情况下休假期服务价格随潜在到达率的变化情况,以及当企业获得最大收益时,正常工作期和休假期服务价格的关系.     

基于Geo/Geo/1(E,SV)排队系统的均衡止步策略

基于单重休假Geo/Geo/1排队系统,研究顾客的均衡止步策略,首次将休假服务机制引入到离散时间排队经济学模型中. 顾客基于“收入--支出”结构,自主决定去留. 利用拟生灭过程理论,运用差分方程求解技巧,对系统进行了稳态分析,得到了顾客的平均逗留时间;进而构造适当的函数,给出了寻找均衡止步策略的具体方法并证明之;而后分析了在均衡策略下, 系统的稳态行为和社会收益;最后通过数值实验讨论了系统参数对均衡行为的影响.     

Pricing and setup/closedown policies in unobservable queues with strategic customers

In this paper we study unobservable Markovian queueing systems with three types of setup/closedown policies: interruptible, skippable and insusceptible setup/closedown policies, respectively. For a system with the interruptible setup/closedown policy, service starts as soon as a customer arrives during a closedown time; However, for a system with the skippable setup/closedown policy, customers arriving in a closedown time (if any) can be served only after the closedown time finishes and the following setup time can be skipped; Then for a system with the insusceptible setup/closedown policy, customers arriving in a closedown time can??t be served until the following setup time finishes. We assume that customers need a price for service, and derive the equilibrium and socially optimal balking strategies for customers as well as the maximal social welfare. Then we make pricing control to motivate customers to adopt the optimal strategies and obtain an appropriate price that also maximizes server??s profit. Moreover, we numerically make some comparisons between the various performance measures.     

Strategic behavior in queues with the effect of the number of customers behind

In this paper, we investigate the strategic behavior in queues by considering the effect of the number of customers behind. The equilibrium joining strategy of customers is obtained and its implications for the service system are examined. We find that the complete queue transparency (i.e., disclosing the real-time system information) can have positive effect on customers, which might encourage more customers to join. Further, the follow-the-crowd (FTC) behavior can be observed, which results in multiple equilibria. By comparing the customer welfare under two different information levels, we demonstrate that, somewhat surprisingly, the queue transparency does not necessarily hurt the customer welfare, and a higher customer welfare can be obtained in the transparent case than that in opaque case when the demand volume is large.     

Naor’s model with heterogeneous customers and arrival rate uncertainty

This paper examines an extension of Naor’s observable queueing model in which the arrival rate is not known with certainty by either customers or the system manager. Further, customers are heterogeneous in terms of their service valuation and waiting time cost. We analyze the stability of the system and properties of the revenue maximizing fee and socially optimal fee.     

Equilibrium balking strategies for a clearing queueing system in alternating environment

We consider a Markovian clearing queueing system, where the customers are accumulated according to a Poisson arrival process and the server removes all present customers at the completion epochs of exponential service cycles. This system may represent the visits of a transportation facility with unlimited capacity at a certain station. The system evolves in an alternating environment that influences the arrival and the service rates. We assume that the arriving customers decide whether to join the system or balk, based on a natural linear reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium strategies under various levels of information.     

Customer equilibrium and optimal pricing in an M/G/1 queue with heterogeneous rewards and waiting cost rates

We consider an unobservable M/G/1 queue in which customers are allowed to join or balk upon arrival. The service provider charges the same admission fee to all joining customers. All joining customers receive a reward from completion of service and incur a waiting cost. The reward and waiting cost rate are random, however the customers know their own values upon arrival. We characterize the customer’s equilibrium strategy and the optimal prices associated with profit and social welfare maximization.     

Comparison of socially optimal and profit maximizing prices in an unobservable queue with heterogeneous waiting costs

We consider a general unobservable queueing model in which customers are allowed to join or balk upon arrival. The service provider charges the same admission fee to all joining customers. All joining customers receive the same reward and incur heterogeneous waiting cost rates. We show that the socially optimal arrival rate is greater than or equal to the profit maximizing arrival rate. Equivalently, the socially optimal admission fee is smaller than or equal to the profit maximizing admission fee.     

Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes

We consider a Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. We assume that the arriving customers decide whether to join the system or balk, based on a natural reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium and social optimal strategies.     

Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior

We consider an M/M/1 queueing system in which the queue length may or may not be observable by a customer upon entering the system. The “observable” and “unobservable” models are compared with respect to system properties and performance measures under two different types of optimal customer behavior, which we refer to as “selfishly optimal” and “socially optimal”. We consider average customer throughput rates and show that, under both types of optimal customer behavior, the equality of effective queue-joining rates between the observable and unobservable systems results in differences with respect to other performance measures such as mean busy periods and waiting times. We also show that the equality of selfishly optimal queue-joining rates between the two types of system precludes the equality of socially optimal joining rates, and vice versa.     

Analysis of multi-server queues with station and server vacations

In this paper, we consider GI/M/c queues with two classes of vacation mechanisms: Station vacation and server vacation. In the first one, all the servers take vacation simultaneously whenever the system becomes empty, and they also return to the system at the same time, i.e., station vacation is a group vacation for all servers. This phenomenon occurs in practice, for example, when the system consists of a set of machines monitored by a single operator, or the system consists of inseparable interconnected parallel machines. In such situations the whole station has to be treated as a single entity for vacation when the system is utilized for a secondary task. For the second class of vacation mechanisms, each server takes its own vacation whenever it complexes a service and finds no customers waiting in the queue, which occurs, for instance in the post office, when each server is a relatively independent working unit, and can itself be used for other purposes. For both models, we derive steady state probabilities that have matrix geometric form, and develop computational algorithms to obtain numerical solutions. We also analyze and make comparisons of these models based on numerical observations.     

Equilibrium threshold strategies in observable queueing systems with setup/closedown times

This paper considers two types of setup/closedown policies: interruptible and insusceptible setup/closedown policies. When all customers are served exhaustively in a system under the interruptible setup/closedown policy, the server shuts down (deactivates) by a closedown time. When the server reactivates since shutdown, he needs a setup time before providing service again. If a customer arrives during a closedown time, the service is immediately started without a setup time. However, in a system under the insusceptible setup/closedown policy, customers arriving in a closedown time can not be served until the following setup time finishes. For the systems with interruptible setup/closedown times, we assume both the fully and almost observable cases, then derive equilibrium threshold strategies for the customers and analyze the stationary behavior of the systems. On the other hand, for the systems with insusceptible setup/closedown times, we only consider the fully observable case. We also illustrate the equilibrium thresholds and the social benefits for systems via numerical experiments. As far as we know, there is no work concerning equilibrium behavior of customers in queueing systems with setup/closedown times.     

Non zero-sum stochastic games in admission,service and routing control in queueing systems

The purpose of this paper is to investigate situations of non-cooperative dynamic control of queueing systems by two agents, having different objectives. The main part of the paper is devoted to analyzing a problem of an admission and a service (vacation) control. The admission controller has to decide whether to allow arrivals to occur. Once the queue empties, the server goes on vacation, and controls the vacations duration (according to the state and past history of the queue). The immediate costs for each controller are increasing in the number of customers, but no convexity assumptions are made. The controllers are shown to have a stationary equilibrium policy pair, for which each controller uses a stationary threshold type policy with randomization in at most one state. We then investigate a problem of a non-zero sum stochastic game between a router into several queues, and a second controller that allocates some extra service capacity to one of the queues. We establish the equilibrium of a policy pair for which the router uses the intuitive Join the shortest queue policy.     

带有止步和中途退出的成批到达的M~x/M/R/N同步休假排队系统的性能分析

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,R个服务员立即进行同步多重休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.     

Equilibrium analysis in batch-arrival queues with complementary services

This paper studies a batch-arrival queue with two complementary services. The two services are complementary and any customer has no benefit from obtaining just one of them. To the best of the authors’ knowledge, there are no works contributed to the batch-arrival queues on analysis of the equilibrium behaviors in queueing systems by now. The properties of batch-arrival queues, which is more practical and universal in reality, induce different Nash equilibria under competition or monopoly compared with the single-arrival queues. We observe the joint effect of batch joining rate and cost structure on the behavior of customers and graphically interpret the equilibrium solutions under competition. Moreover, we discuss the model under three types of price structures and give comparisons from customer and server points.