We consider a symmetric network composed of $N$ links, each with capacity $C$. Calls arrive according to a Poisson process, and each call concerns $L$ distinct links chosen uniformly at random. If each of these links has free capacity, the call is held for an exponential time; otherwise it is lost. The semiexplicit stationary distribution for this process is similar to a Gibbs measure: it involves a normalizing factor, the partition function, which is very difficult to evaluate.We let $N$ go to infinity and keep fixed the rate of call attempts concerning any link. We use asymptotic combinatorics and recent techniques involving the law of large numbers to obtain the asymptotic equivalent for the logarithm of the partition function and then the large deviation principle for the empirical measure of the occupancies of the links. We give an explicit formula for the rate function and examine its properties.
"Large deviations at equilibrium for a large star-shaped loss network." Ann. Appl. Probab. 10 (1) 104 - 122, February 2000. https://doi.org/10.1214/aoap/1019737666