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A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue 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 intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

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Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

A single-server queue with limited virtual waiting time

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2 results

Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

A single-server queue with limited virtual waiting time