We consider the waiting time (delay) W in a FCFS c-server queue with arrivals which are either renewal or governed by Neuts' Markovian arrival process, and (possibly heterogeneous) service time distributions of general phase-type Fi, with mi phases for the ith server. The distribution of W is then again phase-type, with m1⋅⋅⋅mc phases for the general heterogeneous renewal case and ${\pmatrix{{m+c-1}\\{c}}}$ phases for the homogeneous case FieF, miem. We derive the phase-type representation in a form which is explicit up to the solution of a matrix fixed point problem; the key new ingredient is a careful study of the not-all-busy period where some or all servers are idle. Numerical examples are presented as well.
['Søren Asmussen', 'Jakob R. Møller']
Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues