Moderate deviations of many--server queues, idempotent processes and quasipotentials

Abstract

A Large Deviation Principle (LDP) is established for the stationary distribution of the number of customers in a many--server queue in heavy traffic for a moderate deviation scaling akin to the Halfin--Whitt regime. The interarrival and service times are assumed generally distributed. The deviation function is given by a quasipotential. It is related to the longterm idempotent distribution of the large deviation limit of the number-in-the-system process. New results on the trajectorial LDP for that process are also obtained. Proofs rely on the characterisation of large deviation relatively compact sequences as exponentially tight ones and use methods of weak convergence and idempotent processes. Another contribution of the paper concerns bounds on the higher--order moments of counting renewal processes.