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      • Department of Electrical and Electronics Engineering
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      Solving the single server semi-Markov queue with matrix exponential kernel matrices for interarrivals and services

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      Author
      Akar, Nail
      Sohraby, K.
      Date
      2006
      Source Title
      Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools, VALUETOOLS 2006
      Publisher
      ACM
      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.
      Keywords
      Lindley equation
      Markov renewal processes
      Matrix exponential distribution
      Ordered schur decomposition
      Markov processes
      Matrix algebra
      Quality of service
      Queueing networks
      Permalink
      http://hdl.handle.net/11693/27165
      Published Version (Please cite this version)
      http://dx.doi.org/10.1145/1190095.1190109
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      • Department of Electrical and Electronics Engineering 3619
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