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Analysis of stochastic fluid queues driven by local-time processes

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Takis Konstantopoulos ,
Andreas E. Kyprianou ,
Paavo Salminen  and
Marina Sirviö
Takis Konstantopoulos*
Affiliation:
Heriot-Watt University
Andreas E. Kyprianou*
Affiliation:
University of Bath
Paavo Salminen*
Affiliation:
Åbo Akademi University
Marina Sirviö*
Affiliation:
Åbo Akademi University
*
∗ Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.
∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
∗∗∗ Postal address: Department of Mathematics, Åbo Akademi University, Turku, FIN-20500, Finland.
∗∗∗ Postal address: Department of Mathematics, Åbo Akademi University, Turku, FIN-20500, Finland.
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Abstract

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We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

Keywords

Local timefluid queueLévy processSkorokhod reflectionperformance analysisPalm calculusinspection paradox

MSC classification

Primary: 60G10: Stationary processes 60G55: Point processes 60G51: Processes with independent increments; L'evy processes 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1072 - 1103
Copyright
Copyright © Applied Probability Trust 2008 

References

Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory. Palm Martingale Calculus and Stochastic Recurrences (Appl. Math. 26), 2nd edn. Springer, Berlin.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
Blumenthal, R. M. and Getoor, R. (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Distributed Computing Industry Weekly Newsletter, Vol. XXII, Issue 8, June 30, 2008. Available at http://www.dcia.info/news/newsletter_2008-06-30.htm.Google Scholar
Fristedt, B. E. (1974). Sample functions of stochastic processes with stationary independent increments. In Adv. Prob. 3, eds Ney, P. and Port, S., Dekker, New York, pp. 241–396.Google Scholar
Harrison, J. M. (2003). A broader view of Brownian networks. Ann. Appl. Prob. 13, 1119–1150.CrossRefGoogle Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, London.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.CrossRefGoogle Scholar
Kella, O. (1997). Stochastic storage networks: stationarity and the feedforward case. J. Appl. Prob. 34, 498–507.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992). A tandem fluid network with Lévy input. In Queueing and Related Models, eds Basawa, I. and Bhat, U. N., Oxford University Press, pp. 112–128.Google Scholar
Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.CrossRefGoogle Scholar
Konstantopoulos, T. and Last, G. (2000). On the dynamics and performance of stochastic fluid systems. J. Appl. Prob. 37, 652–667.CrossRefGoogle Scholar
Konstantopoulos, T., Zazanis, M. and de Veciana, G. (1997). Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues. Stoch. Process Appl. 65, 139–146.CrossRefGoogle Scholar
Kozlova, M. and Salminen, P. (2004). Diffusion local time storage. Stoch. Process Appl. 114, 211–229.CrossRefGoogle Scholar
Kyprianou, A. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. and Palmowski, Z. (2004). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 28). Springer, Berlin, pp. 16–29.Google Scholar
Mannersalo, P., Norros, I. and Salminen, P. (2004). A storage process with local time input. Queueing Systems 46, 557–577.Google Scholar
Pitman, J. (1986). Stationary excursions. In Séminaire de Probabilités XXI (Lecture Notes Math. 1247). Springer, Berlin, pp. 289–302.Google Scholar
Salminen, P. (1993). On the distribution of diffusion local time. Statist. Prob. Lett. 18, 219–225.CrossRefGoogle Scholar
Sirviö, M. (2006). On an inverse subordinator storage. Helsinki University of Technology, Inst. Math. Rep. Series A501.Google Scholar
Tsirelson, B. (2004). Nonclassical stochastic flows and continuous products. Prob. Surveys 1, 173–298.Google Scholar
Zolotarev, V. M. (1964). The first-passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–664.CrossRefGoogle Scholar