On the application of Rouché's theorem in queueing theory

The determination of the PGF of the queue length distribution often requires the zeros on and within the unit circle of functions of the type zs-A(z)zs-A(z), where A(z)A(z) is a PGF. We present an elementary proof of the existence of these zeros including series A(z)A(z) with a radius of convergence of one.