Monotonicity properties in various retrial queues and their applications

We consider several multi-server retrial queueing models with exponential retrial times that arise in the literature of retrial queues. The effect of retrial rates on the behavior of the queue length process is investigated via sample path approach. We show that the number of customers in orbit and in the system as a whole are monotonically changed if the retrial rates in one system are bounded by the rates in second one. The monotonicity results are applied to show the convergence of generalized truncated systems that have been widely used for approximating the stationary queue length distribution in retrial queues. AMS subject classifications: Primary 60K25     

Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs

Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. It is of basic importance to study reliability of retrial queues with server breakdowns and repairs because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system. However, so far the repairable retrial queues are analyzed only by queueing theory. In this paper we give a detailed analysis for reliability of retrial queues. By using the supplementary variables method, we obtain the explicit expressions of some main reliability indexes such as the availability, failure frequency and reliability function of the server. In addition, some special queues, for instance, the repairable M/G/1 queue and repairable retrial queue can be derived from our results. These results may be generalized to the repairable multi-server retrial models.     

A matrix continued fraction approach to multiserver retrial queues

We consider basic M/M/c/c (c≥1) retrial queues where the number of busy servers and that of customers in the orbit form a level-dependent quasi-birth-and-death (QBD) process with a special structure. Based on this structure and a matrix continued fraction approach, we develop an efficient algorithm to compute the joint stationary distribution of the numbers of busy servers and retrial customers. Through numerical experiments, we demonstrate that our algorithm works well even for M/M/c/c retrial queues with large value of c.     

A survey on retrial queues

Queueing systems in which arriving customers who find all servers and waiting positions (if any) occupied may retry for service after a period of time are called retrial queues or queues with repeated orders. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication networks, computer networks and computer systems. In this paper, we discuss some important retrial queueing models and present their major analytic results and the techniques used. Our concentration is mainly on single-server queueing models. Multi-server queueing models are briefly discussed, and interested readers are referred to the original papers for details. We also discuss the stochastic decomposition property which commonly holds in retrial queues and the relationship between the retrial queue and the queue with server vacations.     

Dynamic scheduling of a single-server two-class queue with constant retrial policy

We analyze the non-preemptive assignment of a single server to two infinite-capacity retrial queues. Customers arrive at both queues according to Poisson processes. They are served on first-come-first-served basis following a cost-optimal routing policy. The customer at the head of a queue generates a Poisson stream of repeated requests for service, that is, we have a constant retrial policy. All service times are exponential, with rates depending on the queues. The costs to be minimized consist of costs per unit time that a customer spends in the system. In case of a scheduling problem that arise when no new customers arrive an explicit condition for server allocation to the first or the second queue is given. The condition presented covers all possible parameter choices. For scheduling that also considers new arrivals, we present the conditions under which server assignment to either queue 1 or queue 2 is cost-optimal.     

Matrix analytic methods for a multi-server retrial queue with buffer

We present numerical methods for obtaining the stationary distribution of states for multi-server retrial queues with Markovian arrival process, phase type service time distribution with two states and finite buffer; and moments of the waiting time. The methods are direct extensions of the ones for the single server retrial queues earlier developed by the authors. The queue is modelled as a level dependent Markov process and the generator for the process is approximated with one which is spacially homogeneous above some levelN. The levelN is chosen such that the probability associated with the homogeneous part of the approximated system is bounded by a small tolerance and the generator is eventually truncated above that level. Solutions are obtained by efficient application of block Gaussian elimination.     

Analysis of Markov Multiserver Retrial Queues with Negative Arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes.     

A retrial queue with redundancy and unreliable server

Retrial queues are useful in the stochastic modelling of computer and telecommunication systems amongst others. In this paper we study a version of the retrial queue with variable service. Such a point of view gives another look at the unreliable retrial queueing problem which includes the redundancy model.By using the theory of piecewise Markovian processes, we obtain the analogue of the Pollaczek-Khintchine formula for such retrial queues, which is useful for operations researchers to obtain performance measures of interest.     

Stochastic decomposition for retrial queues

Summary This paper deals with the stochastic decomposition property for retrial queues. This property is connected with similar results for vacation models. As applications, the moments of the number of customers in orbit and the rate of convergence under high retrial intensity can be obtained. This work was supported under Grant PR161/93-4777     

Algorithmic approximations for the busy period distribution of the M/M/c retrial queue

In this paper we deal with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/c retrial queue.     

Queues with Service Rate Controlled by a Delayed Feedback

We consider a single queue with a Markov modulated Poisson arrival process. Its service rate is controlled by a scheduler. The scheduler receives the workload information from the queue after a delay. This queue models the buffer in an earth station in a satellite network where the scheduler resides in the satellite. We obtain the conditions for stability, rates of convergence to the stationary distribution and the finiteness of the stationary moments. Next we extend these results to the system where the scheduler schedules the service rate among several competing queues based on delayed information about the workloads in the different queues.     

M/M/c Retrial Queue with Multiclass of Customers

We consider the M/M/c retrial queues with multiclass of customers. We show that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/M/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity. Approximation formulae for the distributions of the number of customers in service facility, the mean number of customers in orbit and the sojourn time distribution of a customer are presented. The approximations are compared with exact and simulation results.     

Approximation of multiserver retrial queues by means of generalized truncated models

It is well-known that an analytical solution of multiserver retrial queues is difficult and does not lead to numerical implementation. Thus, many papers approximate the original intractable system by the so-called generalized truncated systems which are simpler and converge to the original model. Most papers assume heuristically the convergence but do not provide a rigorous mathematical proof. In this paper, we present a proof based on a synchronization procedure. To this end, we concentrate on theM/M/c retrial queue and the approximation developed by Neuts and Rao (1990). However, the methodology can be employed to establish the convergence of several generalized truncated systems and a variety of Markovian multiserver retrial queues. J.R. Artalejo thanks the support received from DGES 98-0837.     

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

A bibliographical guide to the analysis of retrial queues through matrix analytic techniques

This paper provides a bibliographical guide to researchers who are interested in the analysis of retrial queues through matrix analytic methods. It includes an author index and a subject index of research papers written in English and published in journals or collective publications, as well as some papers accepted for a forthcoming publication.     

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

The paper deals with a research of bivariate Markov process \(\{X(t), t\ge 0\}\) whose state space is a lattice semistrip \(S(X)=\{0,1,{\ldots },c\} \times Z_{+}\). The process \(\{X(t), t\ge 0\}\) describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.     

A discrete-time retrial queue with multiplicative repeated attempts

The repeated attempts have been always made individually by each unsatisfied customer in discrete-time retrial queues. However, the time between two consecutive repeated requests can be independent of the number of customers applying for service. This paper considers a new retrial discipline, that we call multiplicative, which extends both types of repeated attempts.     

Strategic behavior and optimization in a hybrid M/M/1 queue with retrials

Queueing Systems - In standard queues, when there are waiting customers, service completions are followed by service commencements. In retrial queues, this is not the case. In such systems,...