# On Lattice Path Counting and the Random Product Representation, with Applications to the E_{r}/M/1 Queue and the M/E_{r}/1 Queue

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## 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 E_{r}/M/1 queue, and the M/E_{r}/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 E_{r}/M/1 queue and the M/E_{r}/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.

## Keywords

E_{r}/M/1 queue Generalized binomial series Lattice path counting M/E

_{r}/1 queue Markovian queues

## Mathematics Subject Classification (2010)

60J27 60J28 60K25 90B22## Preview

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## Notes

### Acknowledgments

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

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