Robert B. Cooper

Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations

This paper considers a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. As has been noted by other researchers, for several specific models of this type, the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables, one of which is the stationary number of customers present in the standard M/G/1 queue i.e., the server is always available at a random point in time. In this paper we demonstrate that this type of decomposition holds, in fact, for a very general class of M/G/1 queueing models. The arguments employed are both direct and intuitive. In the course of this work, moreover, we obtain two new results that can lead to remarkable simplifications when solving complex M/G/1 queueing models.

Relating polling models with zero and nonzero switchover times

We consider a system ofN queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues), and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zero-switchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzero-switchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [Queueing Systems 11 (1992) 109—120] and Cooper, Niu, and Srinivasan [to appear in Oper. Res.].

The average time until bucket overflow

It is common for file structures to be divided into equal-length partitions, called buckets, into which records arrive for insertion and from which records are physically deleted. We give a simple algorithm which permits calculation of the average time until overflow for a bucket of capacity n records, assuming that record insertions and deletions can be modeled as a stochastic process in the usual manner of queueing theory. We present some numerical examples, from which we make some general observations about the relationships among insertion and deletion rates, bucket capacity, initial fill, and average time until overflow. In particular, we observe that it makes sense to define the stable point as the product of the arrival rate and the average residence time of the records; then a bucket tends to fill up to its stable point quickly, in an amount of time almost independent of the stable point, but the average time until overflow increases rapidly with the difference between the bucket capacity and the stable point.

BENES'S FORMULA FOR M/G/1-FIFO 'EXPLAINED' BY PREEMPTIVE-RESUME LIFO

We provide a term-by-term interpretation of Beness well-known but mysterious inversion of the Pollaczek-Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/I-FIFO with remaining work in M/G/1-LIFo-preemptive resume. In the process, we give a new and simple

Duality and other results for M/G/1 and Gl/M/1 queues, via a new ballot theorem

We generalize the classical ballot theorem and use it to obtain direct probabilistic derivations of some well-known and some new results relating to busy and idle periods and waiting times in M / G /1 and GI / M /1 queues. In particular, we uncover a duality relation between the joint distribution of several variables associated with the busy cycle in M / G /1 and the corresponding joint distribution in GI / M /1. In contrast with the classical derivations of queueing theory, our arguments avoid the use of transforms, and thereby provide insight and term-by-term “explanations” for the remarkable forms of some of these results.

Queues with Service in Random Order

We consider two models, the GI/M/s queue and the M/G/1 queue, in which waiting customers are served in random order. For each model we derive expressions for the calculation of the stationary waiting-time distribution function. Our methods differ from those of previous authors in that we do not use transforms, and consequently our results may be better suited for calculation. We illustrate our methods by deriving previously known results for the M/M/s and M/D/1 random-service queues, and by making sample calculations for the M/Ek/1 random-service queue for various values of the utilization factor and the index k.

ON THE RELATIONSHIP BETWEEN THE DISTRIBUTION OF MAXIMAL QUEUE LENGTH IN THE M/G/1 QUEUE AND THE MEAN BUSY PERIOD IN THE M/G/1/n QUEUE

Takacs has shown that, in the M/G/1 queue, the probability P(k j i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k i) = Qk ,/Qk, where the Qs are easily calculated by recurrence in terms of an arbitrary QO,, 0. We augment Tak~icss theorem by showing that P(k I i) = bk_,/bk, where b,, is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take ,, equal to the mean service time, then O,, = b,,. MAXIMAL OQUEUE LENGTH DISTRIBUTION QOUEUES WITH FINITE WAITING ROOM

A duality relation for busy cycles in GI/G/1 queues

Using a generalization of the classical ballot theorem, Niu and Cooper [7] established a duality relation between the joint distribution of several variables associated with the busy cycle inM/G/1 (with a modified first service) and the corresponding joint distribution of several related variables in its dualGI/M/1. In this note, we generalize this duality relation toGI/G/1 queues with modified first services; this clarifies the original result, and shows that the generalized ballot theorem is superfluous for the duality relation.

Performance and control of network systems

This Special Issue contains a selection of contributions based on the papers presented at the conference, Performance and Control of Network Systems II, held in Boston, Massachusetts, USA, on 2–4 November 1998. The conference was part of the International Symposium on Voice, Video, and Data Communications, sponsored by SPIE — The International Society for Optical Engineering. Its purpose was to promote discussion on the development of performance evaluation techniques, traffic control principles, and traffic engineering methods and practices. The eight papers in this Special Issue cover a spectrum of topics, including both theory and practice. All are extended versions of papers originally presented at the conference and included in Proceedings of SPIE, vol. 3530. The papers appearing here were nominated after the conference by the Session Chairs and the conference participants. Then they underwent a rigorous review process by anonymous referees, including both participants and non-participants of the conference. We would like to express our appreciation for the support and encouragement of Werner Bux, Editor-inChief, in publishing these papers in Performance Evaluation. We also want to thank the referees, whose efforts are reflected in the high quality of the papers presented here.

On the convergence of jacobi and gauss-seidel iteration for steady-state probabilities of finite-state continuous-time markov chains

We consider two nonsingular versions of the problem described in the title. For one of these versions, we show by example that neither Jacobi nor Gauss-Seidel iteration is guaranteed to converge; for the other version, we outline a proof that both methods are guaranteed to converge.