Lattice path counting andM/M/c queueing systems

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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