Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues

Authors:
Søren Asmussen;Jakob R. Møller
Affiliations:
Mathematical Statistics, Centre for Mathematical Sciences, Lund University, Box 118, S-221 00 Lund, Sweden asmus@maths.lth.se;Mathematical Statistics, Centre for Mathematical Sciences, Lund University, Box 118, S-221 00 Lund, Sweden jrm@maths.lth.se
Venue:
Queueing Systems: Theory and Applications
Year:
2001

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Abstract

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 Fi=F, mi=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.