Stability Analysis of a Multi-server Model with Simultaneous Service and a Regenerative Input Flow

Abstract

We study the stability conditions of a multi-server queueing system in which each customer requires a random number of servers simultaneously. The input flow is assumed to be a regenerative one and random service times are identical for all occupied servers. The service time has a hypoexponential distribution which belongs to the class of phase-type distributions. We introduce an auxiliary queueing system in which there are always customers in the queue and define an auxiliary service process as the number of served customers in this system. Then we construct the sequence of common regeneration points for the regenerative input flow and the auxiliary service process. Based on the relationship between the real and the auxiliary service processes we obtain upper and lower estimates for the mean of the number of actually served customers during the common regeneration period. It allows us to deduce the stability criterion of the model under consideration. It turns out that the stability condition does not depend on the structure of the input flow. It only depends on the rate of this process.