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Lattice path counting andM/M/c queueing systems

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Abstract

We apply the lattice path counting method to the analysis of the transientM/M/c queueing system. A closed-form solution is obtained for the probability of exactlyi arrivals andj departures within a time interval of lengtht in anM/M/c queueing system that is empty at the initial time. The derivation of the probability is based on the counting of paths from the origin to(i,j) on thexy-plane, that have exactly rd x-steps whose depth from the liney=x isd (d=0,1,...,c−1). The closed-form solution has an expression useful for numerical calculation.

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Muto, K., Miyazaki, H., Seki, Y. et al. Lattice path counting andM/M/c queueing systems. Queueing Syst 19, 193–214 (1995). https://doi.org/10.1007/BF01148946

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Keywords

  • M/M/c queue
  • transient solution
  • lattice path counting
  • depth of path