Combinatorial approach to M/M/1 queues using hypergeometric functions

In this paper we use a well known reflection principle for lattice path countingto apply for the analysis of the M/M/1 queuing system. The joint distribution of iarrivals and j departures over a time interval of length t is obtained, when there are kcustomers in the beginning of the system. The derivation uses the lattice pathapproach between two points, in two dimensional x-y plane with certain restrictions.Finally a known result of queue length is verified from Satty (1961).