Calculation of the steady state waiting time distribution in GI/PH/c and
MAP/PH/c queues
Søren Asmussen and Jakob R. Møller,
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
1999
ISSN 14039338

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
phasetype $F_i$, with $m_i$ phases for the $i$th server. The distribution
of $W$ is then again phasetype, with $m_1\cdots m_c$ phases for the general
heterogeneous renewal case and $(m+c1)!/(c!(m1)!)$ phases for the homogeneous
case $F_i=F$, $m_i=m$. We derive the phasetype representation in a form
which is explicit up to the solution of a matrix fixpoint problem; the key
new ingredient is a careful study of the notallbusy period where some
or all servers are idle. Numerical examples are presented as well.

Key words:

busy period, heterogenous servers, iteration, Kronecker product, Kronecker
sum, manyserver queue, Markovian arrival process, matrixanalytic methods,
nonlinear matrix equation, phasetype distribution, waiting time