Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues
Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues
Faculty of Sciences. Mathematics and Computer Science

article

2014
New York, N.Y.
, 2014

Mathematics

Stochastic models. - New York, N.Y.

30(2014)
:4
, p. 554-575

1532-6349

000344371600007

E

English (eng)

University of Antwerp

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N-2, respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N-2 distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an order N distribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue.

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