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

Abstract

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 distribution was also shown to be phase-type of order m, 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 in which it completes service belongs to τ. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (κ,K,ν). When the arrivals have a Markovian 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.