A Stochastic Directional Convexity Result and Its Application in Comparison of Queues

Abstract

Second order properties of queues are important in design and analysis of service systems. In this paper we show that the blocking probability of M/M/C/N queue is increasing directionally convex in (λ,−μ), where λ is arrival rate and μ is service rate. To illustrate the usefulness of this result we consider a heterogeneous queueing system with non-stationary arrival and service processes. The arrival and service rates alternate between two levels (λ1,μ1) and (λ2,μ2), spending an exponentially distributed amount of time with rate cα i in level i, i=1,2. When the system is in state i, the arrival rate is λ i and the service rate is μ i . Applying the increasing directional convexity result we show that the blocking probability is decreasing in c, extending a result of Fond and Ross [7] for the case C=N=1.