On equilibrium threshold strategies when choosing between observable and unobservable queues

We consider threshold equilibrium strategies in a two-server memoryless queueing system where customers inspect one queue before deciding which queue to join. We show that such an equilibrium may not exist. A numerical study indicates that if threshold equilibria exist, the effective arrival rate to the unobserved queue is higher when the regime there is last-come first-served rather than first-come first-served.     

Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs

We consider the Markovian single-server queue that alternates between on and off periods. Upon arriving, the customers observe the queue length and decide whether to join or balk. We derive equilibrium threshold balking strategies in two cases, according to the information for the server’s state.     

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 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.     

Optimal incentive-compatible pricing for M/G/1 queues

This paper extends the applicability of the pricing results of Mendelson and Whang (Oper. Res. 38 (5) (1990) 870) and Balachandran and Radhakrishnan (Management Sci. 40 (10) (1994) 353) for congested service facilities by considering general, class-dependent, service time distributions. Two theorems for nonpreemptive M/G/1 queues and preemptive-resume M/G/1 queues are presented.     

Batch arrival queues with vacations and server setup

We study a single server queueing system whose arrival stream is compound Poisson and service times are generally distributed. Three types of idle period are considered: threshold, multiple vacations, and single vacation. For each model, we assume after the idle period, the server needs a random amount of setup time before serving. We obtain the steady-state distributions of system size and waiting time and expected values of the cycle for each model. We also show that the distributions of system size and waiting time of each model are decomposed into two parts, whose interpretations are provided. As for the threshold model, we propose a method to find the optimal value of threshold to minimize the total expected operating cost.     

Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers

We consider a single server queueing system in which service shuts down when no customers are present, and is resumed when the queue length reaches a given critical length. We assume customers are heterogeneous on delay sensitivity and analyze customers’ strategic response to this mechanism and compare it to the overall optimal behavior. We provide algorithms to compute the equilibrium arrival rates and also derive the monotonicity of equilibrium and optimal arrival rates. We show that there may exist multiple equilibria in such a system and the optimal arrival rate may be larger or smaller than the decentralized equilibrium one.     

Equilibrium and optimal strategies to join a queue with partial information on service times

In this paper, we study customer equilibrium as well as socially optimal strategies to join a queue with only partial information on the service time distribution such as moments and the range. Based on such partial information, customers adopt the entropy-maximization principle to obtain the expectation of their waiting cost and decide to join or balk. We find that more information encourages customers to join the queue. And it is beneficial for decision makers to convey partial information to customers in welfare maximization but reveal full information in profit maximization.     

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.     

Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)),s time-dependent exponential unreliable/repairable servers and finite capacityc is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state-space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only (s + 1)(s + 6) differential equations for the queue length moments rather than the 2(c + 1)(s +1) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.This research was partially funded by National Science Foundation grant ECS-8404409.     

Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

带有N策略,启动时间和服务台故障的M/M/1排队的均衡策略

本文考虑带有N策略,启动时间和服务台故障的M/M/1排队的顾客的策略行为.当系统为空时服务台关闭,并且只有当系统中的顾客数达到一个给定的阈值时才会被激活,启动时间服从指数分布.服务台在工作时可能会故障,一旦发生故障,它立即被维修,维修的时间服从指数分布.我们得到了不同状态的均衡到达率并且给出了均衡社会收益函数.最后对均衡到达率和均衡社会收益进行了数值研究.     

Analyzing discrete-time D-BMAP/G/1/N queue with single and multiple vacations

This paper analyzes a single-server finite-buffer vacation (single and multiple) queue wherein the input process follows a discrete-time batch Markovian arrival process (D-BMAP). The service and vacation times are generally distributed and their durations are integral multiples of a slot duration. We obtain the state probabilities at service completion, vacation termination, arbitrary, and prearrival epochs. The loss probabilities of the first-, an arbitrary- and the last-customer in a batch, and other performance measures along with numerical aspects have been discussed. The analysis of actual waiting time of these customers in an accepted batch is also carried out.     

Equilibrium customer strategies in a single server Markovian queue with setup times

We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.      

Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.     

Transient analysis of an M/G/1 retrial queue subject to disasters and server failures

An M/G/1 retrial queueing system with disasters and unreliable server is investigated in this paper. Primary customers arrive in the system according to a Poisson process, and they receive service immediately if the server is available upon their arrivals. Otherwise, they will enter a retrial orbit and try their luck after a random time interval. We assume the catastrophes occur following a Poisson stream, and if a catastrophe occurs, all customers in the system are deleted immediately and it also causes the server’s breakdown. Besides, the server has an exponential lifetime in addition to the catastrophe process. Whenever the server breaks down, it is sent for repair immediately. It is assumed that the service time and two kinds of repair time of the server are all arbitrarily distributed. By applying the supplementary variables method, we obtain the Laplace transforms of the transient solutions and also the steady-state solutions for both queueing measures and reliability quantities of interest. Finally, numerical inversion of Laplace transforms is carried out for the blocking probability of the system, and the effects of several system parameters on the blocking probability are illustrated by numerical inversion results.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.