Lattice path counting and M/M/c queueing systems

The authors 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 rd 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.