Abstract
A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in theMx/GI/1/kqueue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensityρ< 1 and for the virtual waiting time process inM/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality ofGI/MY/1/ktoMx/GI/1/k +1, we get a similar result for the system queue length distribution just before the arrival of a customer inGI/MY/1/k.We apply those results to prove that the loss probabilities ofMx/GI/1/kandGI/MY/1/kare increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.
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