Notes on ‘Discrete-time queueing models for ATM switching

The principle of the procedure to go from (4) to (5) is explained in the paper. The function S(z) given by (4) is a probability generating function (of the system contents of an output buffer during an arbitrary slot). Probability generating functions (pgf’s) have the property that they are analytic inside the unit disc, i.e. for {z : |z|6 1}. Among other things, this means that a pgf cannot have poles in this area. Now, if there is a value z = z∗ for which the denominator of (4) is zero and which lies inside the unit disc, the function S(z) would have a pole in z∗, unless of course, the nominator of S(z) also vanishes for z = z∗. So, because of the fact that S(z) has no poles inside the unit disc, all the zeros of the denominator zc−A(z) for which |z|6 1 are also zeros of the nominator