Abstract
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 \(\mathbb{E}\)τ n and the Laplace transform \(\mathbb{E}\)e-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.
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Asmussen, S., Jobmann, M. & Schwefel, HP. Exact Buffer Overflow Calculations for Queues via Martingales. Queueing Systems 42, 63–90 (2002). https://doi.org/10.1023/A:1019994728099
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DOI: https://doi.org/10.1023/A:1019994728099