Calculation of the steady state waiting time distribution in GI/PH/c and MAP/PH/c queues

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 F-i, with m(i) phases for the i th server. The distribution of W is then again phase-type, with m(1). . .m(c) phases for the general heterogeneous renewal case and (c(m+c-1)) phases for the homogeneous case F-i = F, m(i) = m 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.