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Retrial queues with constant retrial times

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Abstract

We consider the M/D/1 retrial queueing system with constant retrial times, which makes up a natural abstraction for optical fibre delay line buffers. Drawing on a time-discretisation approach and on an equivalence with polling systems, we find explicit expressions for the distribution of the number of retrials, and the probability generating function of the number of customers in orbit. While the state space of the queueing system at hand is complicated, the results are strikingly simple. The number of retrials follows a geometric distribution, while the orbit size decomposes into two independent random variables: the system content of the M/D/1 queue at departure times and the orbit size of the M/D/1 retrial queue when the server is idle. We finally obtain explicit expressions for the retrial rate after a departure and for the distribution of the time until the nth retrial after a departure.

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Acknowledgements

The author would like to thank Rudesindo Núñez Queija and Jan-Pieter Dorsman for fruitful discussions and for pointing to the simulation results of P. Bongers which motivated the present investigation.

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  1. Department of Telecommunication and Information Processing, Ghent University, St-Pietersnieuwstraat 41, 9000, Gent, Belgium

    Dieter Fiems

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Correspondence to Dieter Fiems.

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Fiems, D. Retrial queues with constant retrial times. Queueing Syst 103, 347–365 (2023). https://doi.org/10.1007/s11134-022-09866-4

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