Large Deviations Behavior of Counting Processes and Their Inverses

P. W. Glynn and W. Whitt

Queueing Systems:  Theory and Application, Vol. 17, 107-128 (1994)

We show, under regularity conditions, that a counting process satisfies a large deviations principle in Ropf or the Gärtner-Ellis condition (convergence of the normalized logarithmic moment generating functions) if and only if its inverse process does. We show, again under regularity conditions, that embedded regenerative structure is sufficient for the counting process or its inverse process to have exponential asymptotics, and thus satisfy the Gärtner-Ellis condition. These results help characterize the small-tail asymptotic behavior of steady-state distributions in queueing models, e.g., the waiting time, workload and queue length.