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On a property of a refusals stream

  • Vyacheslav M. Abramov (a1)

Abstract

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

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Postal address: Institute of Clinical Epidemiology, The Chaim Sheba Medical Center, Tel Hashomer, Ramat-Gan 52621, Israel. Present address: 26/7 Rambam St., Petach Tiqwa 49542, Israel.

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Research was supported by the Nate H. and Beatrice G. Sherman Fellowship at the Technion — Israel Institute of Technology.

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References

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Abramov, V. M. (1984) Some properties of lost customers. Izv. Akad. Nauk SSSR 3, 148150. (In Russian.)
Abramov, V. M. (1991a) Investigation of a Queueing System with Service Depending on Queue Length. Donish, Dushanbe. (In Russian.)
Abramov, V. M. (1991b) Asymptotic properties of lost customers for one queueing system with refusals. Kibernetika 2, 123124. (In Russian.)
Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.
Cooper, R. B. and Tilt, B. (1976) On the relationship between the distribution of the maximal queue length in the M/G/1 queue and the mean busy period in the M/M/G/1/n queue. J. Appl. Prob. 13, 195199.
Karlin, S. and Taylor, H. G. (1975) A First Course in Stochastic Processes. 2nd edn. Academic Press, New York.
Subkhankulov, M. A. (1976) Tauberian Theorems with Remainder. Nauka, Moscow. (In Russian.)
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.
Tomko, J. (1967) One limit theorem in the queueing problem as input stream increases infinitely. Studia Sci. Math. Hungarica 2, 447454. (In Russian.)

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On a property of a refusals stream

  • Vyacheslav M. Abramov (a1)

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