$\alpha $-stable limits for multiple channel queues in heavy traffic

Zbigniew Michna¹

¹ Department of Mathematics Wrocław University of Economics Komandorska 118/120 53-345 Wrocław, Poland

Applicationes Mathematicae, Tome 30 (2003) no. 1, pp. 55-68

Résumé

We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law ($1\alpha 2$). We show that this sequence is not tight in the Skorokhod $J_1$ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod $M_1$ topology to an $\alpha $-stable Lévy motion. This example leads to a weaker notion of weak convergence. As an application, we present limit theorems for multiple channel queues in heavy traffic. The convergence of the queue length process to a linear combination of $\alpha $-stable Lévy motions is derived.

Détail

DOI : 10.4064/am30-1-4

Keywords: consider sequence renewal processes constructed sequence random variables belonging domain attraction stable law alpha sequence tight skorokhod topology convergence functionals derived using structure sample paths renewal process derive convergence skorokhod topology alpha stable motion example leads weaker notion weak convergence application present limit theorems multiple channel queues heavy traffic convergence queue length process linear combination alpha stable motions derived