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Exact buffer overflow calculations for queues via martingales. (English) Zbl 1035.90018

Summary: Let \(\tau_n\) 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 \(\mathbb E\tau_n\) and the Laplace transform \(\mathbb Ee^{-s\tau_n}\) is derived in closed form using a martingale introduced in O. Kella and W. Whitt [J. Appl. Probab. 29, 396–403 (1992; Zbl 0761.60065)]. For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in the first author and O. Kella [Adv. appl. Probab. 32, 376–393 (2000; Zbl 0961.60081)] 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.

MSC:

90B22 Queues and service in operations research
60J27 Continuous-time Markov processes on discrete state spaces
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60G48 Generalizations of martingales
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