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      System-theoretical algorithmic solution to waiting times in semi-Markov queues

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      Author
      Akar, N.
      Sohraby, K.
      Date
      2009-05-07
      Source Title
      Performance Evaluation
      Print ISSN
      0166-5316
      Publisher
      ELSEVIER
      Volume
      66
      Issue
      11
      Pages
      587 - 606
      Language
      English
      Type
      Article
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      Abstract
      Markov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.
      Keywords
      Correlated arrivals and services
      Lindley equation
      Matrix-analytical approach
      Schur decomposition
      Semi-Markov queues
      Permalink
      http://hdl.handle.net/11693/22577
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.peva.2009.05.001
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      • Department of Electrical and Electronics Engineering 3524
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