Exact buffer overflow calculations for queues via martingales

Let τ_n be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean double struck E sugneτ_n and the Laplace transform double-struck E signe^-Sτn is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/ 1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.