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The N/G/1 finite capacity queue. (English) Zbl 0673.60102

Summary: We study a single server queue with finite capacity where the arrival process is M. F. Neuts’ [J. Appl. Probab. 16, 764-774 (1979; Zbl 0422.60043)] versatile Markovian point process (the N-process). Many arrival processes are special cases of this N-process, such as the Markov modulated Poisson process, the renewal process of phase-type and others. The service times are generally distributed. We obtain recursive formulas for the joint distribution of the length of the busy period and the number of customers served during such a period.
The queue length distribution, both at departure instants and at an arbitrary time instant are derived. The Laplace-Stieltjes transform of the virtual waiting time distribution is also obtained. This result generalizes S. S. Lavenberg’s formula [Management Sci., Theory 21, 501-506 (1975; Zbl 0302.60058)] for the M/G/1 finite capacity queue to the present model.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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