Queueing Systems 0 ( 2000 ) ? { ? 1 Calculation of the steady state waiting

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 m + c ? 1 c phases for the homogeneous case Fi = F, mi = m. We derive the phase{type representation in a form which is explicit up to the solution of a matrix xed 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.