A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distributionwas also shown to be phase-type of orderm, but no derivation for the queue length was provided in the general case. This paper introduces an order m2 phase-type representation (κ, K) for the queue length distribution in the general case and proves that the order m2 of the distribution cannot be further reduced in general. A matrix geometric representation (κ, K , ν) is also established for the number of type τ ⊆ {1, . . . ,m} customers in the system, where a customer is of type τ if the phase inwhich it completes service belongs to τ .Wederive these results in both discrete and continuous time and also discuss the numerical procedure to compute (κ, K , ν). When the arrivals have aMarkovian structure, the numerical procedure is reduced to solving a Quasi–Birth–Death (for the discrete time case) or fluid queue (for the continuous time case). Finally, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi–Birth–Death Markov chain. © 2012 Elsevier B.V. All rights reserved.