Stochastic comparisons for bulk queues (Q1116559)

This paper deals with two single server bulk queueing models: \(M^{(x)}/G^{(y)}/1\) and \(G^{(x)}/M^{(y)}/1\). It is proved that a queueing system is more congested than the other if the basic random variables are stochastically ordered in a suitable way. The following models have been taken for illustrative purpose: (i) \(\{M/G^{(s)}/1\), \(s\geq 1\}\) with varying service capacity; (ii) \(\{G^{(s)}/M/1\), \(s\geq 1\}\) with varying bulk arrival sizes; (iii) \(\{\) M/D/s, \(s\geq 1\}\) with constant service and varying number of servers. In each family it is found that the system which is best for customers, turns out to be worst for managers (or vice versa). Furthermore upper bounds for the expected queue length in steady state and the expected number of batches served during a busy period are calculated. The idea of these results is based on the stochastic comparison of random walks which are inherent in these queueing systems.