A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

Abstract

We consider a batch arrival infinite-buffer single-server queue with generally distributed inter-batch arrival times with arrivals occurring in batches of random sizes. The service process is correlated and its structure is governed by a Markovian service process in continuous time. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at a pre-arrival epoch. We also obtain the steady-state probability distribution at an arbitrary epoch using the classical argument based on Markov renewal theory. Some important performance measures such as the average number of customers in the system and the mean sojourn time have also been obtained. Later, we have established heavy-and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System

“…A single server is serving the customers according to a continuous-time Markovian service process (C-) with matrix representation (L 0 , L 1 ). For a detailed discussion on -, the readers are referred to Chaudhry et al [5]. Let us denote L(…”

“…The two processes -and -are convenient representations of a versatile Markovian point process or N-process; see Neuts [3] and Ramaswami [4]. Chaudhry et al [5] obtain stationary system-length distributions for the infinite-buffer batch arrival / -/1 queues at various epochs using roots. Further, Singh et al [6] have shown that it is beneficial to use roots rather than the matrixgeometric/matrix-analytic methods in terms of computation time.…”

“…In this paper we obtain the explicit closed-form expressions for the Laplace-Stieltjes transform (LST) of waitingtime distribution. For analytic purpose, we start our work with associated prearrival epoch probability distribution; see Chaudhry et al [5] for details. After that we consider characteristic equations involved in the Laplace-Stieltjes transform's (LST's) waiting-time distribution for the first and an arbitrary customer of an arrival batch.…”

“…It is well-known that the determination of the "R" matrix can be done through the minimal nonnegative solution of a nonlinear matrix equation; see Neuts [10] for details. Finally, it may be remarked here that the same queueing model has been dealt with in [5]. But in both the queueing models, corresponding probability density functions of actual waiting time for the first/an arbitrary customer have not been derived.…”

“…It is needless to mention here that probability density functions of the waiting times are important for obtaining statistical measures of Quality of Service (QoS) requirement of a queueing system. Thus, the present work is a nontrivial extension of [5].…”