Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are non-empty, a customer of queue 1 is served with probability β and a customer of queue 2 is served with probability 1 − β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating functionU(z1, z2) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z1, z2) in β. The first coefficient of this power series corresponds to the priority case β = 0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.