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

Date
2006
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Source Title
Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools, VALUETOOLS 2006
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ACM
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Language
English
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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
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Published Version (Please cite this version)