Title
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
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
New York, N.Y. ,
Subject
Mathematics
Source (journal)
Stochastic models. - New York, N.Y.
Volume/pages
30(2014) :4 , p. 554-575
ISSN
1532-6349
ISI
000344371600007
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
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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