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1st International ICST Conference on Performance Evaluation Methodologies and Tools

Research Article

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

Cite
BibTeX Plain Text
  • @INPROCEEDINGS{10.1145/1190095.1190109,
        author={Nail  Akar and Khosrow  Sohraby},
        title={Solving the single server semi-Markov queue with matrix exponential kernel matrices for interarrivals and services},
        proceedings={1st International ICST Conference on Performance Evaluation Methodologies and Tools},
        publisher={ACM},
        proceedings_a={VALUETOOLS},
        year={2012},
        month={4},
        keywords={Lindley equation Markov renewal processes matrix exponential distribution ordered Schur decomposition},
        doi={10.1145/1190095.1190109}
    }
    
  • Nail Akar
    Khosrow Sohraby
    Year: 2012
    Solving the single server semi-Markov queue with matrix exponential kernel matrices for interarrivals and services
    VALUETOOLS
    ACM
    DOI: 10.1145/1190095.1190109
Nail Akar1,*, Khosrow Sohraby1,*
  • 1: Electrical and Electronics Eng., Bilkent University, Bilkent 06800, Ankara, Turkey.
*Contact email: akar@ee.bilkent.edu.tr, sohrabyk@umkc.edu

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.

Keywords
Lindley equation Markov renewal processes matrix exponential distribution ordered Schur decomposition
Published
2012-04-06
Publisher
ACM
http://dx.doi.org/10.1145/1190095.1190109
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