Let τ{sub}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 Eτ{sub}n and the Laplace transform Ee{sup}(-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/i, 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.
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