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.
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Chao, X., Luh, H.P. A Stochastic Directional Convexity Result and Its Application in Comparison of Queues. Queueing Systems 48, 399–419 (2004). https://doi.org/10.1023/B:QUES.0000046583.57857.f1
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DOI: https://doi.org/10.1023/B:QUES.0000046583.57857.f1