A stochastic directional convexity result and its application in comparison of queues.

*(English)*Zbl 1125.60312Summary: 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 \(C=N=1\).