James G. C. Templeton

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.

A Poisson input queue under N -policy and with a general start up time

Abstract The paper examines the steady state behaviour of an M/G/1 queue under control operating policy and with a general start up time. Stochastic decomposition results of Fuhrmann and Cooper (Ops Res. 33, 1127–1129, 1985) for an M/G/1 queue with generalized vacation have been utilized in this study. The paper generalizes the results obtained by Borthakur et al. (Computers Ops Res. 14, 33–40, 1987)

An approximation method for the M/G/1 retrial queue with general retrial times

Abstract In this paper, we consider a single-server queueing system with no waiting space. Customers arrive according to a Poisson process with rate λ. An arriving customer receives immediate service if he finds the server idle; otherwise he will retry for service after a certain amount of time. Blocked customers will repeatedly retry for service until they get served. The times between consecutive attempts are assumed to be independently distributed with common distribution function T (·). The service times of customers are drawn from a common distribution function B (·). We show that the number of customers in the system in steady-state can be decomposed into two independent random variables: the number of customers in the corresponding ordinary M/G/1 queue (with unlimited waiting space) and the number of customers in the retrial queue given that the server is idle. Applying this decomposition property, an approximation method for the calculation of the steady-state queue size distribution is proposed and some properties of the approximation are discussed. It is demonstrated through numerical results that the approximation works very well for models of practical interest.

Exact and approximate numerical solutions of steady-state distributions arising in the queue GI/G/ 1

In this paper we first obtain, in a unified way, closed-form analytic expressions in terms of roots of the so-called characteristic equation (c.e.), and then discuss the exact numerical solutions of steady-state distributions of (i) actual queueing time, (ii) virtual queueing time, (iii) actual idle time, and (iv) interdeparture time for the queueGI/R/1, whereR denotes the class of distributions whose Laplace-Stieltjes transforms (LSTs) are rational functions (ratios of a polynomial of degree at mostn to a polynomial of degreen). For the purpose of numerical discussions of idle- and interdeparture-time distributions, the interarrival-time distribution is also taken to belong to the classR. It is also shown that numerical computations of the idle-time distribution ofR/G/1 queues can be done even ifG is not taken asR. Throughout the discussions it is assumed that the queue discipline is first-come-first-served (FCFS). For the tail of the actual queueing-time distribution ofGI/R/1, approximations in terms of one or more roots of the c.e. are also discussed. If more than one root is used, they are taken in ascending order of magnitude. Numerical aspects have been tested for a variety of complex interarrival- and service-time distributions. The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn), but also covers nonphase type distributions such as uniform (U) and deterministic (D). Some numerical results are also presented in the form of tables and figures. It is expected that the results obtained from the present study should prove to be useful not only to practitioners, but also to queueing theorists who would like to test the accuracies of inequalities, bounds or approximations.

On the relations among the distributions at different epochs for discrete-time GI/Geom/1 queues

In this paper we discuss the discrete-time GI/Geom/1 queue and derive relations among prearrival and random epochs as well as the outside observers distributions. Two variations of the model, namely late arrival system with delayed access and early arrival system, have been discussed along with the relations between prearrival-epoch probabilities for the two models.

Numerical methods in Markov chains and Bulk queues

I: Introduction.- 1.1 A Perspective.- 1.2 Earlier Work.- II: Mathematical Formulation of the Bulk Queuing Problem.- 2.1 A Class of Markov Recurrence Relations.- 2.2 The Abstract Formalism.- III: A Numerical Approach to Waiting Line Problems.- 3.1 Numerical Methods in Queuing Theory.- 3.2 The Basic Theory.- 3.3 The Simple Queue with Limited Waiting Room.- 3.4 A Bulk Queue with Limited Waiting Room.- 3.5 Queues with Variable Arrival Rate.- 3.6 A Heuristic Experiment.- 3.7 Operational Solutions vs. Exact Results.- IV: Conclusions and Directions for Further Research.- References.- Appendix A: Program Listings.- Appendix B: On the Analysis of Computational Errors.- Appendix C: On Analytic Approximations.

A NOTE ON THE Mx/GI/1, K BULK QUEUEING SYSTEM

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohens method to generalize his results to the Mx/GY/1, K queue.

The GR X n /G n /∞ system: system size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.

The M/G/1 retrial queue with nonpersistent customers

We consider anM/G/1 retrial queue in which blocked customers may leave the system forever without service. Basic equations concerning the system in steady state are established in terms of generating functions. An indirect method (the method of moments) is applied to solve the basic equations and expressions for related factorial moments, steady-state probabilities and other system performance measures are derived in terms of server utilization. A numerical algorithm is then developed for the calculation of the server utilization and some numerical results are presented.

Analysis of the discrete-time GI/Geom(n)/1/N queue

Abstract In this paper, we consider the discrete-time single-server finite-buffer late arrival with delayed access queue GI /Geom( n )/1/ N . Whereas the interarrival times are independently identically distributed random variables with arbitrary probability mass function, the service times are geometrically distributed random variables with probability of a service completion during a slot dependent on the number of customers present in the system. Using the supplementary variable technique, we obtain probability distributions of numbers of customers in the system at arbitrary and prearrival epochs as well as an outside observers distribution. In addition, we derive some important results which are used to develop relations between probabilities at prearrival and arbitrary epochs. Results obtained in this paper can be used in several areas such as performance evaluation of computer-communication systems.