Steady State Analysis of an M~x/(G_1,G_2)/1 Queue with Restricted Admissibility and Random Setup Time

摘要

We consider a two stage heterogeneous service batch arrival queue with a Bernoulli schedule server vacation, where after completion of two stages of heterogeneous service in succession the server either goes for a vacation of random length with probability r (0 ≤ r ≤ 1) or may continue to serve the next unit, if any, with probability (1 - r). Further, we assume restricted admissibility of arriving batches as introduced by Madan et al. [30], [32]. According to this policy, not all batches are allowed to join the system at all times. We also assume that after termination of a busy period, as soon a customer or a batch of customers arrives, the server needs a random setup time (or the warming up time) before actually starting service of the first customer. We derive the steady state queue size distribution at a random epoch as well as at a departure point of time and we also obtain some important performance measures of this model. In this paper we not only generalize the results of recent papers by Madan et al. [30] [32], but also we make an attempt to unify results of several classes of related batch arrival queueing systems.