LATTICE PATH COUNTING AND M/M/C QUEUING-SYSTEMS

We apply the lattice path counting method to the analysis of the transient M/M/c queueing system. A closed-form solution is obtained for the probability of exactly i arrivals and j departures within a time interval of length t in an M/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 the xy-plane, that have exactly r(d) x-steps whose depth from the line y = x is d (d = 0, 1,..., c - 1). The closed-form solution has an expression useful for numerical calculation.