An Upper Bound of Large Deviations for a Generalized Star-Shaped Loss Network

#### Ben Graham, S. Meleard

1997, v.3, №2, 199-223

ABSTRACT

We consider a network with $n$ links of capacity $C$. Any call involves $K$ uniformly chosen links and releases them simultaneously. The processes of interest are the occupancies of each link, which as a system are neither Markovian nor in mean-field interaction. We let $n$ go to infinity while the offered load per link remains constant. We proved in earlier papers propagation of chaos on path-space in variation norm, and a Gaussian fluctuation result for the flow of marginals. Here we prove for this flow an exponential tightness result that involves an auxiliary process pertaining to simultaneous release. We then obtain a large deviation upper bound for closed sets. In order to derive from this a useful upper bound, we need a precise asymptotic control of the auxiliary process, which we obtain only when $C=1$. This allows us to derive an explicit rate of exponential decay of the probabilities of exiting a tube around the limit path, which implies a.s. convergence of the empirical measures.

Keywords: networks,interaction graphs,propagation of chaos,large deviations