Remaining loads in a PH/M/c queue with impatient customers

TL;DR

This paper extends existing results on remaining loads in queues with impatient customers from Coxian to phase type distributions, providing new insights into waiting time distributions and their relations.

Contribution

It generalizes the result for GI/M/c queues with impatient customers from Coxian to phase type distributions, enabling broader applicability.

Findings

Distribution of remaining loads is valid for phase type distributions.

Derived expressions for virtual and actual waiting times.

Discussed relations to alternative waiting time distributions.

Abstract

There are two contributions to the literature in this note. Firstly, we point out that under some weak conditions the result in Swensen (1986) on the remaining loads in a GI/M/$c$ queue with impatient customers, derived under the assumption that the distribution of inter-arrival times is Coxian, is also valid for the more general phase type distribution. From this result the distributions of the virtual and the actual waiting time can easily be obtained. Secondly, the relation to an alternative expression for the distribution of the virtual waiting time derived by Kawanishi and Takine (2015) is also discussed.