Let τn 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 𝔼τn and the Laplace transform 𝔼e–sτn is derived in closed form using a martingale introduced in Kella and Whitt. For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given Asmussen and Kella 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.
