On the application of Rouché's theorem in queueing theory

For queueing models that can be analyzed as (embedded) Markov chains, many results are presented in terms of the probability generating function (PGF) of the stationary queue length distribution. Queueing models that belong to this category are bulk service queues, M/G/l and G/M/l-type queues, and discrete or discrete-time queues. The determination of the PGF typically requires a fixed number of complex-valued zeros on and within the unit circle of some analytic function. Rouche's theorem can be used to prove the existence of these zeros and fulfills as such a prominent role in queueing theory. For most queueing models the analytic function of interest is of the type z' A(z), where A(z) is the PGF of a discrete random variable. The standard application of Rouche's theorem requires that A(z) has a radius of convergence strictly larger than one. However, in some applications this is not true. In this note we present an elementary proof of the existence of the zeros for z' A (z) that includes functions A(z) with a radius of convergence of one. The proof is based on applying the classical argument principle to a truncation of the series A(z).