Analysis of a Single Server Queue with Interdependence of Arrival and Service Processes – A Semi-Markov Approach

In many queuing systems, the inter-arrival time and service distributions are dependent. In this paper we analyze such a system where the dependence is through a semi-Markov process. For this we assume that the arrival and service processes evolve in a finite number of phases/stages according to a Markov chain. So, the product space of the two finite sets of states (phases) is considered. The nature of transitions in the states of the combined process are such that transition rates at which the states of the combined process changes depend on the phase in which each ‘marginal’ process is currently in and (the phases of) the state to be visited next. We derive the stability condition and the effect of the interdependence on the stability of the system is brought out. A numerical investigation of the steady state characteristics of the system is also carried out.