Retrial queueing systems have been extensively studied because of their applications in telephone systems, call centers, telecommunication networks, computer systems, and in daily life. This survey deals with various retrial queueing models. The main focus of this survey is to show analytic results for queue length distributions, waiting time distributions, and tail asymptotics for the queue length and waiting time distributions. This survey also considers the stability analysis of retrial queueing models.

A survey of retrial queueing systems

We describe main models and results of a new branch of the queueing theory, theory of retrial queues, which is characterized by the following basic assumption: a customer who cannot get service (due to finite capacity of the system, balking, impatience, etc.) leaves the service area, but after some random delay returns to the system again. Emphasis is done on comparison with standard queues with waiting line and queues with losses. We give a survey of main results for both single server M/G/1 type and multiserver M/M/c type retrial queues and discuss similarities and differences between the retrial queues and their standard counterparts. We demonstrate that although retrial queues are closely connected with these standard queueing models they, however, possess unique distinguished features. We also mention some open problems.

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Standard and retrial queueing systems: a comparative analysis

Single server retrial queues with priority calls

This paper deals with a single server retrial queueing system subject to active and independent breakdowns. The objective is to extend the results given independently by Aissani l1r and Kulkarni and Choi l15r. To this end, we introduce the concept of fundamental server period and an auxiliary queueing system with breakdowns and option for leaving the system. Then, we concentrate our attention on the limiting distribution of the system state. We obtain simplified expressions for the partial generating functions of the server state and the number of customers in the retrial group, a recursive scheme for computing the limiting probabilities and closed-form formulae for the second order partial moments. Some stochastic decomposition results are also investigated.

On the single server retrial queue subject to breakdowns

We consider the single-server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Equilibrium customer strategies and social–profit maximization in the single‐server constant retrial queue

In this paper we present information theoretic approximations for theM/G/1 queue with retrials. Various approximations for this model are obtained according to the available information about the service time probability density and the steady-state distribution of the system state. The results are well-suited for numerical computation.

Information theoretic approximations for the M/G /1 retrial queue

In teletraffic applications of retrial queues only the service zone is observable. Another part of a retrial queue, the orbit, which represents the delay before repeated attempts to get service, cannot be observed. Thus, it is very important to get general results about behavior of the orbit. We investigate two characteristics of the orbit, namely, the orbit busy period and the orbit idle period, which seem to be very useful from this point of view.

On the orbit characteristics of the M/G/1 retrial queue

We are concerned with the M/G/1 queue with repeated attempts where a customer who finds the server busy leaves the service area and repeats his request after a random amount of time. We concentrate on the study of the waiting time process. Its analysis in terms of Laplace transforms has been discussed in the literature. However, this solution has important limitations in practice. For instance, we cannot calculate the first moments of the waiting time, W, by direct differentiation. This paper supplements the existing work and provides a direct method of computation for the second moment of W. Then the maximum entropy approach is used to estimate the true waiting time distribution.

A second order analysis of the waiting time in the M/G/1 retrial queue

The paper deals with a special bivariate Markov process which can be considered as the joint process of the channel (server) state and the number of customers in the orbit of a Markovian single server retrial system with state dependent intensities Several retrial queues, analysed in the literature, are special cases in view of the versatility
of the process Necessary and sufficient conditions for the regularity, ergodicity and recurrence of the process are given For the stationary distribution a product formula
is derived Further the busy period, the number of served customers and other related quantities are studied

On a bivariate Markov process arising in the theory of single‐server retrial queues

Stochastic inequalities for M/G/1 retrial queues

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