A Large Deviation Principle for a Large Star-Shaped Loss Network with Links of Capacity One

Ben Graham, S. Meleard

1997, v.3, №4, 475-492

ABSTRACT

We consider the empirical measure process of a network of $n$ links of capacity $C=1$, with Poisson call arrivals. Each call involves $K$ uniformly chosen links; if each of these links is free the call holds them for an exponential time and then releases them simultaneously, else it is lost. We consider the limit of large $n$. In a previous paper we proved exponential tightness, and gave a large deviation upper bound in terms of a rate function expressed in a variational form related to exponential martingales. Here we give an explicit non-variational form for this rate function and prove it provides a large deviation lower bound, hence proving the full LDP. The main difficulty comes from the presence of the function $e^x-x-1$ in the rate function, instead of the function $x^2/2$ in more classical Gaussian situations, and from the unboundedness of the rate function.

Keywords: networks,propagation of chaos,large deviation principle