An analysis of the El/Ek/1 queueing system by restricted minimal lattice paths

The usual procedure for obtaining the equilibrium probability distribution of the queue length in a queueing system is by constructing and solving the difference-differential equations. In this paper, a new approach for deriving the equilibrium probability distributions of the queue length in the M/M/1, M/Ek/1 and El/Ek/1 queueing systems is presented, based on the generating function of the number of the minimal lattice paths. The proposed procedure obtains the equilibrium probability distribution more easily than the usual procedure, which solves difference-differential equations.