The MAP, M/G1,G2/1 queue with preemptive priority

Bong Dae Choi,Gang Uk Hwang

doi:10.1155/s1048953397000440

Abstract

We consider the MAP, M/G1,G2/1 queue with preemptive resume priority, where low priority customers arrive to the system according to a Markovian arrival process (MAP) and high priority customers according to a Poisson process. The service time density function of low (respectively: high) priority customers is g1(x) (respectively: g2(x)). We use the supplementary variable method with Extended Laplace Transforms to obtain the joint transform of the number of customers in each priority queue, as well as the remaining service time for the customer in service in the steady state. We also derive the probability generating function for the number of customers of low (respectively, high) priority in the system just after the service completion epochs for customers of low (respectively, high) priority.

Highlights

The Markovian arrival process (MAP) is a good mathematical model for input traffics which have strong autocorrelations between cell arrivals and high burstiness in broadband-integrated services digital networks (B-ISDNs)
To apply the supplementary variable method to the MAP/G/1 type queues, Choi et al [1] extended the notion of the Laplace Transform, which is suitable for dealing with matrix equations
We investigate the MAP, M/G1,G2/1 queue with preemptive resume priority by the supplementary variable method developed by Choi et al [1]

Summary

Introduction

The Markovian arrival process (MAP) is a good mathematical model for input traffics which have strong autocorrelations between cell arrivals and high burstiness in broadband-integrated services digital networks (B-ISDNs). With the help of the fundamental period of the PH-MRP/G/1 queue, he derived the distribution of the number of customers in the system at the service completion epochs for non-priority customers by the embedded Markov chain method. From our supplementary variable analysis, we derive the joint transform of the number of customers in each priority queue, as well as the remaining service time for the customer in service in the steady state. The overall organization of this paper is as follows: Section 2 reviews MAPs and the ELT; Section 3 derives the joint transform for the number of customers of each priority and the remaining service time in the steady state for our model; Section 4 derives the PGF (Probability Generating Function) for the number of customers of low (respectively, high) priority at the service completion epochs

Preliminaries

Analysis of Our Model

Marginal Queue Length Distributions