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Finding the Conjugate of Markov Fluid Processes

Published online by Cambridge University Press:  27 July 2009

Michel Mandjes  and
Ad Ridder
Michel Mandjes
Affiliation:
Department of Econometrics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
Ad Ridder
Affiliation:
Department of Econometrics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
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Abstract

This paper addresses characteristics of finite-buffer Markov-modulated fluid processes, particularly those related to their deviant behavior. Our aim in this paper is to find rough asymptotics for the probability of a loss cycle. Apart from that, we derive some properties of the fluid process in case of the buffer contents reaching a high level (a process we call the conjugate of the original process). Our main goal is to obtain practicable methods to find the rate matrix of this conjugate process. For this purpose we use large deviations techniques, but we consider the governing eigensystem, as well, and we discuss the relation between these two approaches. We extend the analysis to the multiple source case. Finally, we use the obtained results in simulation. We examine variance reduction by importance sampling in a multiple source example. The new statistical law of the fluid process is based on the conjugate rate matrices.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 2 , April 1995 , pp. 297 - 315
Copyright
Copyright © Cambridge University Press 1995

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References

1.Anantharam, V. (1988). How large delays build up in a GI/G/1 queue. Queueing Systems 5: 345–368.CrossRefGoogle Scholar
2.Anick, D., Mitra, D. & Sondhi, M.M. (1982). Stochastic theory of data-handling system with multiple sources. Bell System Technical Journal 61: 1871–1894.Google Scholar
3.Asmussen, S. (1985). Conjugate processes and the simulation of ruin probabilities. Stochastic Processes and Their Applications 20: 213–229.Google Scholar
4.Bucklew, J.A. (1990). Large deviation techniques in decision, simulation, and estimation. New York: Wiley.Google Scholar
5.Chang, C.-S., Heidelberger, P., Juneja, S. & Shahabuddin, P. (1993). Effective bandwidth and fast simulation of ATM intree networks. In Proceedings of the Performance '93 Conference.Google Scholar
6.Cottrell, M., Fort, J. -C, & Malgouyres, G. (1983). Large deviations and rare events in the study of stochastic algorithms. IEEE Transactions on Automatic Control 28: 907–920.CrossRefGoogle Scholar
7.Courcoubetis, C., Kesidis, G., Ridder, A., Walrand, J.& Weber, R. (1993). Admission control and routing in ATM networks using inferences from measured buffered occupancy. IEEE Transactions on Communications (to appear).Google Scholar
8.de Veciana, G., Olivier, C., & Walrand, J. (1993). Large deviations of birth death Markov fluids. Probability in the Engineering and Informational Sciences 7: 237–255.CrossRefGoogle Scholar
9.Dembo, A. & Zeitouni, O. (1993). Large deviations techniques and applications. Boston: Jones and Bartlett.Google Scholar
10.Donsker, M.D. & Varadhan, S.R.S. (1975). Asymptotic evaluation of certain Markov process expectations for large time, I. Communications on Pure and Applied Mathematics 28: 1–47.CrossRefGoogle Scholar
11.Ellis, R.S. (1985). Entropy, large deviations, and statistical mechanics. Berlin: Springer-Verlag.Google Scholar
12.Elwalid, A.I. & Mitra, D. (1991). Analysis and design of rate-based congestion control of high speed networks, I: Stochastic fluid models, access regulation. Queueing Systems 9: 29–64.Google Scholar
13.Elwalid, A.I. & Mitra, D. (1993). Effective bandwidth of general Markovian traffic sources and admission control of high speed networks. IEEE/ACM Transactions on Networking 1: 329–343.Google Scholar
14.Feller, W. (1971). An introduction to probability theory and its applications, Vol. 2, 2nd ed.New York: Wiley.Google Scholar
15.Gibbens, R.J. & Hunt, P.J. (1991). Effective bandwidths for the multi-type UAS channel. Queueing Systems 9: 17–28.CrossRefGoogle Scholar
16.Glynn, P.W. & Iglehart, D.L. (1989). Importance sampling for stochastic simulations. Management Science 35: 1367–1392.Google Scholar
17.Kelly, F.P. (1991). Effective bandwidths at multi-class queues. Queueing Systems 9: 5–16.CrossRefGoogle Scholar
18.Kesidis, G. & Walrand, J. (1993). Relative entropy between Markov transition rate matrices IEEE Transactions on Information Theory 39: 1056–1057.CrossRefGoogle Scholar
19.Kesidis, G., Walrand, J., & Chang, C.S. (1993). Effective bandwidths for multiclass Markov fluids and other ATM sources. IEEE/ACM Transactions on Networking 1: 424–428.CrossRefGoogle Scholar
20.Knessl, C. & Morrison, J.A. (1991). Heavy traffic analysis of a data-handling system with many sources. SIAM Journal of Applied Mathematics 51: 187–213.CrossRefGoogle Scholar
21.Kobayashi, H. & Ren, Q. (1993). A diffusion approximation analysis of an ATM statistical multiplexer with multiple types of traffic, part I: Equilibrium state solutions. Proceedings of the 1993 IEEE International Conference on Communications 2: 1047–1053.Google Scholar
22.Kosten, L. (1984). Stochastic theory of data-handling systems with groups of multiple sources. In Rudin, H. & Bux, W. (eds.), Performance of computer-communication systems.Amsterdam: Elsevier, pp. 321–331.Google Scholar
23.Mandjes, M. (1993). Fast simulation of Markov fluid models in conjunction with large deviations. Research Memorandum 1993–58, Department of Econometrics, Free University, Amsterdam.Google Scholar
24.Mitra, D. (1988). Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Advances in Applied Probability 20: 646–676.Google Scholar
25.Parekh, S. & Walrand, J. (1989). A quick simulation method for excessive backlogs in networks of queues. IEEE Transactions on Automatic Control 34: 54–66.Google Scholar
26.Ridder, A. (1993). Fast simulation of Markov fluid models. Research Memorandum 1993–1921, Department of Econometrics, Free University, Amsterdam.Google Scholar
27.Ridder, A. & Walrand, J. (1992). Some large deviations results in Markov fluid models. Probability in the Engineering and Informational Sciences 6: 543–560.Google Scholar
28.Roberts, J.W. (1992). Performance evaluation and design of multi-service networks. Final report of the COST 224 Project, Commission of the European Communities, Luxembourg.Google Scholar
29.Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Annals of Statistics 4: 673–684.Google Scholar
30.Stern, T.E. & Elwalid, A.I. (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Advances in Applied Probability 23: 105–139.CrossRefGoogle Scholar
31.Walrand, J. (1988). An introduction to queueing networks. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
32.Weiss, A. (1986). A new technique of analyzing large traffic systems. Advances in Applied Probability 18: 506–532.CrossRefGoogle Scholar
33.Whitt, W. (1993). Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommunication Systems 2: 71–107.Google Scholar