Two Queues with Alternating Service

We consider a system consisting of two queues and a single server. The server always takes successive customers from different queues, if possible. The queues have Poisson arrivals and general service-time distributions. The mathematical description of this system leads to a single functional equation in two unknowns. By taking advantage of the analyticity of the unknown, functions, the functional equation is transformed into an integral equation on a complex domain, whose solution can be obtained. The method of solution is novel and quite interesting. Due to the difficult nature of the mathematics, however, some steps in the solution remain to be proven. The calculations have been carried through numerically by means of a computer program and some results are displayed in this paper.