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A stochastic directional convexity result and its application in comparison of queues. (English) Zbl 1125.60312
Summary: 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 $$(\lambda,-\mu)$$, where $$\lambda$$ is arrival rate and $$\mu$$ 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 $$(\lambda_1, \mu_1)$$ and $$(\lambda_2, \mu_2)$$, spending an exponentially distributed amount of time with rate $$c\alpha_i$$ in level $$i$$, $$i=1,2$$. When the system is in state $$i$$, the arrival rate is $$\lambda_i$$ and the service rate is $$\mu_i$$ . Applying the increasing directional convexity result we show that the blocking probability is decreasing in $$c$$, extending a result of S. Fond and S. M. Ross [Nav. Res. Logist. Q. 25, 483–488 (1978; Zbl 0393.60091)] for the case $$C=N=1$$.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60E15 Inequalities; stochastic orderings
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