Stochastic comparisons for bulk queues

Abstract

We consider two important classes of single-server bulk queueing models: M^(X)/G^(Y)/1 with Poisson arrivals of customer groups, and G^(X)/m^(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G^(s)/1,s ⩾ 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G^(s)/1,s ⩾ 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s⩾ 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.