On the Characteristic Equation of General Queueing Systems

There are numerous papers in the literature, which analyze the behavior of specific queueing systems in terms of certain roots of a nonlinear equation. In this paper we show that a very general class of queueing systems, including queues with heterogeneous servers, multiple arrival streams, bulk queues and feedback, is described by a characteristic equation. We prove that the number in queue and the waiting time distribution is a sum of geometric and exponential terms respectively involving the roots of the characteristic equation. The sum is finite if the service time distributions belong to the class R of distributions with rational Laplace transform. We find explicitly the characteristic equation for a wide variety of queueing systems and furthermore, we provide an easy method to generate the characteristic equation for even more general systems. Our results are structural for the case of arbitrary distributions and aim to unify the field, but they can potentially lead to an algorithmic solution for queues with distributions in the class R. *Dimitris Bertaimas, Sloan School of Management and Operations Research Center, MIT, Cambridge, Ma 02139. The research of the author was partially supported by grants from the Leaders for Manufacturing program at MIT and from Draper Laboratory. tDaisuke Nakazato, Operations Research Center, MIT, Cambridge, MA 02139.