An estimate for the loss probability in a queueing system of the $MAP/G/m/0$ type in the case of light traffic

Authors:
D. Baum and I. N. Kovalenko

Translated by:
S. V. Kvasko

Journal:
Theor. Probability and Math. Statist. **71** (2005), 17-24

MSC (2000):
Primary 60K25

DOI:
https://doi.org/10.1090/S0094-9000-05-00644-7

Published electronically:
December 28, 2005

MathSciNet review:
2144317

Full-text PDF Free Access

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References |
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Additional Information

Abstract: We consider a queueing system with losses and with a general distribution of the service time. It is assumed that the input is of the $MAP$ type and the phase process assumes values in a general measurable space. The asymptotic behavior of the loss probability is studied for the case where the mean service time tends to zero. In particular, we find conditions under which the loss probability is asymptotically invariant with respect to the shape of the service time.

References
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*The fundamental period of the queue with Markov-modulated arrivals*, Probability, Statistics, and Mathematics (J. W. Anderson, K. B. Athreya, and D. L. Iglehart, eds.), In Honor of Professor Samuel Karlin, Academic Press, New York, 1989, pp. 187–200. MR **1031285** (91f:60173)
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- T. Erhardsson,
*On the number of lost customers in stationary loss systems in the light traffic case*, KTH, INTERNET, Stockholm (2002), 1–19.
- V. Klimenok, C. S. Kim, D. Orlovsky, and A. Dudin,
*Lack of invariant property of Erlang loss model in case of the MAP input*, QUESTA (to appear).
- Yu. V. Malinkovskiĭ,
*Invariance of the stationary distribution of the states of modified Jackson and Gordon–Newell networks*, Avtomat. i Telemekh. **59** (1998), no. 9, 29–36; English transl. in Automat. Remote Control **59** (1999), no. 9, 1226–1231. MR **1680017**
- Yu. V. Malinkovskiĭ and O. V. Yakubovich,
*Invariance in closed networks with bypasses*, Mathematical Methods for Investigations for Telecommunication Networks, Proceedings of the 13-th Belorussian Winter School–Seminar on the Theory of Queues, (International Conference BWWQT-97), Minsk, February 3–5, 1997, Belorussian State University, Minsk, 1997, pp. 118–119. (Russian)
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*Invariance of Markov queueing networks with bypasses of nodes and immediate service*, Mathematical Methods for Investigations of Queueing Systems and Networks, Proceedings of the 14-th Belorussian Winter School-Seminar on the Theory of Queues (International Conference BWWQT-98), Minsk, January 27–29, 1998, Belorussian State University, Minsk, 1998, pp. 121–122. (Russian)
- A. V. Krylenko,
*Invariance of queueing networks with several types of nodes and customers, with immediate service, and bypasses of nodes*, Mathematical Methods for Investigations of Queueing Systems and Networks, Proceedings of the 14-th Belorussian Winter School-Seminar on the Theory of Queues (International Conference BWWQT-98), Minsk, January 27–29, 1998, Belorussian State University, Minsk, 1998, pp. 112–115. (Russian)

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Additional Information

**D. Baum**

Affiliation:
LG Stochastische Modellierung und Rechnernetze, FB4-Abteilung Informatik, Universität Trier, D-54286 Trier, Germany

Email:
baum@info04.uni-trier.de

**I. N. Kovalenko**

Affiliation:
Glushkov Institute for Cybernetics, Kyiv, Ukraine

Email:
kovigo@yandex.ru

Received by editor(s):
March 29, 2004

Published electronically:
December 28, 2005

Additional Notes:
The second author is partially supported by the DLR Foundation (project ADLON, Trier University, Germany).

Article copyright:
© Copyright 2005
American Mathematical Society