Remaining Loads in a PH/M/c Queue with Impatient Customers

There are three 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 clarified. Finally, we explain how the results on the remaining loads can be used to find distributions describing particular fixed servers and provide a couple of numerical examples of how this can be done.