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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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
- M. Krämer, Computational method for Markov chains occurring in queueing theory, Messung, Modellierung und Bewertung von Rechensystemen (V. Herzog and M. Paterok, eds.), Informatik-Fachberichte 154, Berlin, 1987, pp. 164–175.
- D. M. Lucantoni, An algorithmic analysis of a communication model with retransmission of flawed messages, Research Notes in Mathematics, vol. 81, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1983. MR 693753
- M. F. Neuts, The fundamental period of the queue with Markov-modulated arrivals, Probability, statistics, and mathematics, Academic Press, Boston, MA, 1989, pp. 187–200. MR 1031285
- Marcel F. Neuts, Matrix-geometric solutions in stochastic models, Johns Hopkins Series in the Mathematical Sciences, vol. 2, Johns Hopkins University Press, Baltimore, Md., 1981. An algorithmic approach. MR 618123
- Ī. Ī. Ēžov and A. V. Skorohod, Markov processes which are homogeneous in the second component. I, Teor. Verojatnost. i Primenen. 14 (1969), 3–14 (Russian, with English summary). MR 0247666
- I. N. Kovalenko, J. B. Atkinson, and K. V. Mykhalevych, Three cases of light-traffic insensitivity of the loss probability in a $GI/G/m/0$ loss system to the shape of the service time distribution, Queueing Syst. 45 (2003), no. 3, 245–271. MR 2024180, DOI https://doi.org/10.1023/A%3A1027328719269
- 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. 9 (1998), 29–36 (Russian, with Russian summary); English transl., Automat. Remote Control 59 (1998), no. 9, 1226–1231 (1999). 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)
- Yu. V. Malinkovskiĭ and O. V. Yakubovich, 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)
References
- M. Krämer, Computational method for Markov chains occurring in queueing theory, Messung, Modellierung und Bewertung von Rechensystemen (V. Herzog and M. Paterok, eds.), Informatik-Fachberichte 154, Berlin, 1987, pp. 164–175.
- D. M. Lucantoni, An Algorithmic Analysis of a Communication Model with Retransmission of Flawed Messages, Pitman, London, 1983. MR 0693753 (84h:60166)
- M. F. Neuts, 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)
- M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins, Baltimore, 1981. MR 0618123 (82j:60177)
- I. I. Ezhov and A. V. Skorokhod, Markov processes which are homogeneous in the second component. I, Teor. Veroyatnost. i Primenen. XIV (1969), no. 1, 3–14; II, no. 4, 679–692; English transl. in Theory Probab. Appl. XIV (1970), no. 1, 1–13; (1970), no. 4, 652–667. MR 0247666 (40:929); MR 0267640 (42:2542)
- I. N. Kovalenko, J. B. Atkinson, and K. V. Mykhalevych, Three cases of light-traffic insensitivity of the loss probability in a $GI/G/m/0$ loss system to the shape of the service time distribution, Queueing Systems 45 (2003), 245–271. MR 2024180 (2004k:60255)
- 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)
- Yu. V. Malinkovskiĭ and O. V. Yakubovich, 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
