Roots, symmetry and contour integrals in queueing systems

Many queueing systems are analysed using the probability-generating-
function (pgf) technique. This approach often leads to expressions in
terms of the (complex) roots of a certain equation. In this paper, we
show that it is not necessary to compute the roots in order to evaluate
these expressions. We focus on a certain class of pgfs with a rational form
and represent it explicitly using symmetric functions of the roots. These
functions can be computed using contour integrals.
We also study when the mean of the random variable corresponding
to the considered pgf is an additive function of the roots. In this case,
it may be found using one contour integral, which is more reliable than
the root-finding approach. We give a necessary and sufficient condition
for an additive mean. For example, the mean is an additive function
when the numerator of the pgf has a polynomial-like structure of a certain
degree, which means that the pgf can be represented in a special product
form. We also give a necessary and sufficient condition for the mean to
be independent of the roots.