  The dominant pole approximation (DPA) is a classical analytic method to
obtain from a generating function asymptotic estimates for its underlying
coefficients. We apply DPA to a discrete queue in a critical many-sources
regime, in order to obtain tail asymptotics for the stationary queue length. As
it turns out, this regime leads to a clustering of the poles of the generating
function, which renders the classical DPA useless, since the dominant pole is
not sufficiently dominant. To resolve this, we design a new DPA method, which
might also find application in other areas of mathematics, like combinatorics,
particularly when Gaussian scalings related to the central limit theorem are
involved.
