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

AbstractWe 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 $$\left( {\begin{array}{*{20}c} {m + c - 1} \\ c \\ \end{array} } \right)$$ 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.

[1]  Jacob Cohen.  On the M/G/2 queeing model , 1982 .

[2]  Ward Whitt,et al.  Exponential Approximations for Tail Probabilities in Queues II: Sojourn Time and Workload , 1996, Oper. Res..

[3]  Dimitris Bertsimas.  An exact FCFS waiting time analysis for a general class of G/G/s queueing systems , 1988, Queueing Syst. Theory Appl..

[4]  B. Sengupta.  Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue , 1989, Advances in Applied Probability.

[5]  Ward Whitt,et al.  The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution , 2000, Queueing Syst. Theory Appl..

[6]  David Perry,et al.  On cycle maxima, first passage problems and extreme value theory for queues , 1992 .

[7]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[8]  Bhaskar Sengupta.  Phase-type representations for matrix-geometric solutions , 1990 .

[9]  J.H.A. de Smit.  Explicit Wiener-Hopf Factorizations for the Analysis of Multi-Dimensional Queues , 1995 .

[10]  Walter L. Smith.  On the distribution of queueing times , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Søren Asmussen,et al.  Computational methods in risk theory: A matrix-algorithmic approach , 1992 .

[12]  D. Daley,et al.  Idle and busy periods in stable M/M/k queues , 1998, Journal of Applied Probability.

[13]  Colm Art O'Cinneide,et al.  Representations for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues , 1998 .

[14]  Søren Asmussen,et al.  Phase-Type Representations in Random Walk and Queueing Problems , 1992 .

[15]  J. D. Smit.  The queue GI/M/s with customers of different types or the queue GI/Hm/s , 1983 .

[16]  M. Neuts.  A Versatile Markovian Point Process , 1979 .

[17]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[18]  L. P. Seelen.  An algorithm for Ph/Ph/c queues , 1986 .