# UvA/CWI Seminar: Mariska Heemskerk (UvA)

Queueing networks often face highly variable (i.e. overdispersed) arrival streams, to the point where the standard assumption of Poissonian arrivals is not statistically valid anymore.

Abstract:

Queueing networks often face highly variable (i.e. overdispersed) arrival streams, to the point where the standard assumption of Poissonian arrivals is not statistically valid anymore.

Taking an overdispersed arrival process as a leading example with the objective to approximate its tail distribution, we study the probability \xi_n(u):={\mathbb P}\left(C_n\geqslant u n \right) for a composition of Lévy processes C_n. Here C_n:=A(\psi_n B(\varphi_n)) consists of Lévy processes A(\cdot) and B(\cdot), and scaling variables \varphi_n and \psi_n. The latter are non-negative sequences such that \varphi_n \psi_n =n and \varphi_n\to\infty as n\to\infty.  Two timescale regimes can arise: a `fast' regime in case \varphi_n is superlinear (hence psi_n \to 0), and a 'slow' regime, in the setting where \varphi_n is sublinear (in which case psi_n must be sublinear too).

We provide the exact asymptotics of xi_n(u) as n\to\infty for both regimes, relying on change-of-measure techniques in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms. When using the asymptotic expression as an approximation for the pre-limit probability, the retrieved refinements should thus improve the accuracy of the approximation. We show that under mediocre timescale separation this is indeed the case.