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On cycle maxima, first passage problems and extreme value theory for queues. (English) Zbl 0762.60086
For a single server with a Markov-modulated arrival stream of different classes of customers and with class-dependent phase-type service times a detailed study of the maximum virtual waiting time during a busy cycle is given, including numerical illustrations. The results are applicable to the study of the maximum virtual waiting time up to time \(t\), as \(t\) tends to infinity. It is pointed out that actual waiting times can be treated in a similar way. The paper also contains new material on steady state solutions of queues with a Markovian arrival process, some basic formulas in extreme value theory for semi-regenerative processes and a simple proof of Takacs’ formula for the distribution of \(\overline V(C)\) in the \(M/G/1\) case.

60K25 Queueing theory (aspects of probability theory)
60G70 Extreme value theory; extremal stochastic processes
90B22 Queues and service in operations research
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