Analysis of the asymmetrical shortest two-server queueing model

This study presents the analytic solution for an asymmetrical two-server queueing model for arriving customers joining the shorter queue for the case of Poisson arrivals and negative exponentially distributed service times. The bivariate generating function of the stationary joint distribution of the queue lengths is explicitly determined. The determination of this bivariate generating function requires a construction of four generating functions. It is shown that each of these functions is the sum of a polynomial and a meromorphic function. The poles and residues at the poles of the meromorphic functions can be simply calculated recursively; the coefficients of the polynomials are easily found, in particular, if the asymmetry in the model parameters is not excessively large. The starting point for the asymptotic analysis for the queue lengths is obtained. The approach developed in the present study is applicable to a larger class of random walks modeling asymmetrical two-dimensional queueing processes.