Mohan L. Chaudhry

Modelling and analysis of M/G^{a,b}/1/N queue – A simple alternative approach

In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed. The queue is analyzed using both the supplementary variable and imbedded Markov chain techniques. The relations between state probabilities at departure and arbitrary epochs have been presented in explicit forms.

Computational analysis of steady-state probabilities of M/G a,b /1 and related nonbulk queues

In their book, Chaudhry and Templeton [6] present a unified approach to many problems on bulk queues. Using their analytical approach, we show how to numerically evaluate steady-state probabilities and moments for number in system (or queue) at each of three time epochs — postdeparture, prearrival and random — for several bulk and nonbulk queues. The approach can be used for other problems in queueing theory, and for similar problems in the theories of dams, inventories, etc. The present study extends the computational results available in tables, such as those produced by Hillier and Yu [12], and has several potential applications. The method proposed is computationally efficient, accurate, and stable. It accommodates high values of the queueing parameters. Sample numerical results and graphs are also presented.

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.

Analysis of the discrete-time bulk-service queue Geo/GY/1/N+B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G^Y/1/N+B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.

On Numerical Computations of Some Discrete-Time Queues

Since Erlang did pioneering work in the application of queues to telephony, a lot has been written about queues in continuous time (see, for example [As-mussel, 1987, Bacelli and Bremaud, 1994, Bhat and Basawa, 1992, Boxma and Syski, 1988, Bunday, 1986, Bunday, 1996, Chaudhry and Templeton, 1983, Cohen, 1982, Cooper, 1981, Daigle, 1992, Gnedenko and Kovalenko, 1989, Gross and Harris, 1985, Kalashnikov, 1994, Kashyap and Chaudhry, 1988, Kleinrock, 1975, Lipsky, 1992, Medhi, 1991, Prabhu, 1965, Robertazzi, 1990, Srivastava and Kashyap, 1982, Tijms, 1986, White et al., 1975]). In comparison to that large body of literature, not much has been written about queues in discrete time.

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.

Modeling and Analysis of Discrete-Time Multiserver Queues with Batch Arrivals: GIX/Geom/m

Multiserver queues are often encountered in telecommunication systems and have special importance in the design of ATM networks. This paper analyzes a discrete-time multiserver queueing system with batch arrivals in which the interbatch and service times are, respectively, arbitrarily and geometrically distributed. Using supplementary-variable and embedded-Markov-chain techniques, the queue is analyzed only for the early arrival system. Since the late arrival system can be discussed similarly, it is not considered here. In addition to developing relations among state probabilities at prearrival, arbitrary, and outside observers observation epochs, the numerical evaluation of state probabilities is also discussed. It is also shown that, in the limiting case, the relations developed here tend to continuous-time counterparts. Further, the waiting-time distribution of a random customer of a batch is obtained. Finally, in some cases simulation experiments have been performed to validate our results.

Computational analysis of single-server bulk-arrival queues M x / G /1

This paper deals with numerical computations for the bulk-arrival queueing modelGIX/M/1. First an algorithm is developed to find the roots inside the unit circle of the characteristic equation for this model. These roots are then used to calculate both the moments and the steady-state distribution of the number of customers in the system at a pre-arrival epoch. These results are used to compute the distribution of the same random variable at post-departure and random epochs. Unifying the method used by Easton [7], we have extended its application to the special cases where the interarrival time distribution is deterministic or uniform, and to cases whereX has a given arbitrary distribution. We also improved on the various root-finding methods used by several previous authors so that high values of the parameters, in particular large batch sizes, can be investigated as well.

A complete and simple solution for a discrete-time multi-server queue with bulk arrivals and deterministic service times

A complete distribution for the system content of a discrete-time multi-server queue with an infinite buffer is presented, where each customer arriving in a group requires a deterministic service time that could be greater than one slot. In addition, when the service time equals one slot, a complete distribution for the delay is also presented.

Queue-length and waiting-time distributions of discrete-time GI^{X}/Geom/1 queueing systems with early and late arrivals

In this paper, we give a unified approach to solving discrete-time GIX/Geom/ 1 queues with batch arrivals. The analysis has been carried out for early- and late-arrival systems using the supplementary variable technique. The distributions of numbers in systems at prearrival epochs have been expressed in terms of roots of associated characteristic equations. Furthermore, distributions at arbitrary as well as outside observers observation epochs have been obtained using the relation derived in this paper. We also present delay analyses for both the systems. Numerical results are presented for various interarrival-time and batch-size distributions.