Computation of the moments of queue length in the $BMAP∕SM∕1$ queue

Abstract

The $BMAP∕SM∕1$ queue is the most general single-server queueing model which can be analysed analytically. Problem of computation of stationary distributions of queue length is solved in the literature. However, the problem of computation of the moments of these distributions is not enough addressed. This problem is more complicated than its particular case when the service times are independent identically distributed random variables due to reducibility of some involved matrices. In this communication, we solve this problem.

Introduction

The $BMAP∕G∕1$ queue as a single server system with infinite buffer, Batch Markovian Arrival Process ($BMAP$) and independent identically arbitrarily distributed service time is very important for applications queueing model because the $BMAP$ (see [2], [11]) is an adequate mathematical model of bursty, correlated flows of information in modern telecommunication networks. Therefore its analysis was very important theoretical task. Initially, this task was solved long time ago by V. Ramaswami in [14] where essentially the same arrival process as the $BMAP$ was called as $N$ process. The $BMAP∕G∕1$ queue was then analysed in [11]. However, until now analysis of such a system attracts attention of researchers, see, e.g., very recent paper [17].

As it was mentioned above, the $BMAP∕G∕1$ type model assumes independence and identical distribution of service times of successive customers. In many real-world systems, service times of successive customers may be dependent and have different distributions. To take into account such a dependence, so called Semi-Markovian ($SM$) service process was considered, see, e.g., [3], [18]. Importance of consideration of queues with $SM$ service process for practical needs is stressed, e.g., in recent work [1] and references therein. In the paper [16], a very general model of the $SM∕SM∕1$ type with possible dependence of inter-arrival and service times was considered under assumption that the marginal distribution of service times is of phase type. The $BMAP∕SM∕1$ type queue without such an assumption and with batch arrivals was analysed in [7], [12]. The problem of computation of steady state distribution of queue length and waiting time distribution in the system was successfully solved. In this communication, we supplement results of [12] by the effective recursive procedures for computation of the moments of the queue length distributions. Moments have an important role for performance evaluation of various queueing systems. Sometimes, information about the mean value and variance of queue length is enough for managerial decisions. If this information is insufficient and the shape of queue length distribution is of a primary interest (e.g., to evaluate the probability that the queue length will exceed a certain important level) while this shape hardly can be found exactly, the shape can be estimated numerically based on the knowledge of the value of several moments of the distribution, see, e.g. [19]. Effective recursive procedures for computing the moments of the queue length at service completion epochs and arbitrary time for $BMAP∕G∕1$ queue were given in [6]. Direct extension of results from [6] appears not possible for the $BMAP∕SM∕1$ type queue because that results essentially exploit irreducibility of some matrix generating functions at the point $z=1$ while they are reducible when the service is of $SM$ type. In this communication we elaborate the recursive procedures for computation of the moments of the queue length in this $BMAP∕SM∕1$ type system.