Solving the single server semi-Markov queue with matrix exponential kernel matrices for interarrivals and services

Date
2006
Advisor
Instructor
Source Title
Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools, VALUETOOLS 2006
Print ISSN
Electronic ISSN
Publisher
ACM
Volume
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Pages
Language
English
Type
Conference Paper
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Abstract

Markov renewal processes with semi-Markov kernel matrices that have matrix-exponential representations form a superset of the well-known phase-type renewal process, Markovian arrival process, and the recently introduced rational arrival process. In this paper, we study the steady-state waiting time distribution in an infinite capacity single server queue with the auto-correlation in interarrival and service times modeled with this general Markov renewal process. Our method relies on the algebraic equivalence between this waiting time distribution and the output of a feedback control system certain parameters of which are to be determined through the solution of a well known numerical linear algebra problem, namely the SDC (Spectral-Divide-and- Conquer) problem. We provide an algorithmic solution to the SDC problem and in turn obtain a simple matrix exponential representation for the waiting time distribution using the ordered Schur decomposition that is known to have numerically stable and efficient implementations in various computing platforms.

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Keywords
Lindley equation, Markov renewal processes, Matrix exponential distribution, Ordered schur decomposition, Markov processes, Matrix algebra, Quality of service, Queueing networks
Citation
Published Version (Please cite this version)