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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.

##### MSC:
 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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