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On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue

Author

Listed:
  • Xiaoyuan Liu

    (Clemson University)

  • Brian Fralix

    (Clemson University)

Abstract

We explain how lattice-path counting techniques can be used in conjunction with the random-product representations from Buckingham and Fralix (Markov Process Related Field 21:339–368 2015) to study both the stationary and time-dependent behavior of Markovian queueing systems, and continuous-time Markov chains in general. We illustrate how the approach works by applying it to both the Er/M/1 queue, and the M/Er/1 queue. Interestingly, through this approach we show that the stationary distributions, as well as the Laplace transforms of the transition functions associated with both the Er/M/1 queue and the M/Er/1 queue, can be expressed explicitly in terms of generalized binomial series from Chapter 5 of the text Concrete Mathematics of Graham, Knuth, and Patashnik.

Suggested Citation

  • Xiaoyuan Liu & Brian Fralix, 2019. "On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1119-1149, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9658-8
    DOI: 10.1007/s11009-018-9658-8
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    References listed on IDEAS

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    1. Benny Van Houdt & Johan S. H. van Leeuwaarden, 2011. "Triangular M/G/1-Type and Tree-Like Quasi-Birth-Death Markov Chains," INFORMS Journal on Computing, INFORMS, vol. 23(1), pages 165-171, February.
    2. Fralix, Brian, 2015. "When are two Markov chains similar?," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 199-203.
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