Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1119–1149 | Cite as

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

  • Xiaoyuan Liu
  • Brian FralixEmail author


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.


Er/M/1 queue Generalized binomial series Lattice path counting M/Er/1 queue Markovian queues 

Mathematics Subject Classification (2010)

60J27 60J28 60K25 90B22 


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BF gratefully acknowledges the support of the National Science Foundation, via grant NSF-CMMI-1435261.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA

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