The analysis of queues by state-dependent parameters by Markov renewal processes

Abstract

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinary M/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinary M/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of the M/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.