For a class of discrete-time queueing systems, we present a new exact method of computing both the expectation and the distribution of the queue length. This class of systems includes the bulk-service queue and the fixed-cycle traffic-light (FCTL) queue, which is a basic model in traffic-control research and can be seen as a non-exhaustive time-limited polling system. Our method avoids finding the roots of the characteristic equation, which enhances both the reliability and the speed of the computations compared to the classical root-finding approach. We represent the queue-length expectation in an exact closed-form expression using a contour integral. We also introduce several realistic modifications of the FCTL model. For the FCTL model for a turning flow, we prove a decomposition result. This allows us to derive a bound on the difference between the bulk-service and FCTL expected queue lengths, which turns out to be small in most of the realistic cases.

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Many (discrete) stochastic systems are analyzed using the probability generating function (pgf) technique, which often leads to expressions in terms of the (complex) roots of a certain equation. In this paper, for a class of pgfs with a rational form, we show that it is not necessary to compute the roots in order to evaluate these expressions. Instead, one can use contour integrals, which is computationally a more reliable method than the classical root-finding approach. We also give the necessary and sufficient condition for the mean of the corresponding random variable, e.g., queue length, to be an additive function of the roots. In this case, the mean is found using one contour integral. Finally, we give the necessary and sufficient condition for the mean to be independent of the roots.

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