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On the excursions of reflected local-time processes and stochastic fluid queues

Part of: Operations research and management science Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Takis Konstantopoulos ,
Andreas E. Kyprianou  and
Paavo Salminen
Takis Konstantopoulos
Affiliation:
Uppsala University, Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden. Email address: takis@math.uu.se
Andreas E. Kyprianou
Affiliation:
University of Bath, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk
Paavo Salminen
Affiliation:
Åbo Akademi, Department of Mathematics, Åbo Akademi University, Turku, FIN-20500, Finland. Email address: phsalmin@abo.fi
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Abstract

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In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

Keywords

Lévy processlocal timeSkorokhod reflectionstationary process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G10: Stationary processes
Secondary: 90B15: Network models, stochastic
Type
Part 2. Lévy Processes
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 79 - 98
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Baccelli, F. and Brémaud, P., (2003). Elements of Queueing Theory, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[2] Bertoin, J., (1996). Lévy Processes. Cambridge University Press.Google Scholar
[3] Bertoin, J., (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.CrossRefGoogle Scholar
[4] Bingham, N. H., (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
[5] Blumenthal, R. M. and Getoor, R. K., (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
[6] Borodin, A. N. and Salminen, P., (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
[7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., (1954). Tables of Integral Transforms, Vol. II. McGraw-Hill, New York.Google Scholar
[8] Fristedt, B. E., (1974). Sample functions of stochastic processes with stationary, independent increments. In Advances in Probability and Related Topics, Vol. 3, Dekker, New York, pp. 241–396.Google Scholar
[9] Hubalek, F. and Kyprianou, A. E., (2011). Old and new examples of scale functions for spectrally negative Lévy processes. In Sixth Seminar on Stochastic Analysis, Random Fields and Applications, eds Dalang, R., Dozzi, M. and Russo, F., Birkhäuser, Basel.Google Scholar
[10] Itö, K., and McKean, H. P. Jr., (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
[11] Kallenberg, O., (1983). Random Measures, 3rd edn. Academic Press, London.Google Scholar
[12] Kallenberg, O., (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[13] Konstantopoulos, T. and Last, G., (2000). On the dynamics and performance of stochastic fluid systems. J. Appl. Prob. 37, 652–667.Google Scholar
[14] Konstantopoulos, T., Zazanis, M. and De Veciana, G., (1995). 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
[15] Konstantopoulos, T., Kyprianou, A. E., Salminen, P. and Sirviö, M., (2008). Analysis of stochastic fluid queues driven by local-time processes. Adv. Appl. Prob. 40, 1072–1103.Google Scholar
[16] Kozlova, M. and Salminen, P., (2004). Diffusion local time storage. Stoch. Process. Appl. 114, 211–229.Google Scholar
[17] Kyprianou, A. E., (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[18] Kyprianou, A. E. and Palmowski, Z., (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 16–29.Google Scholar
[19] Mannersalo, P., Norros, I. and Salminen, P., (2004). A storage process with local time input. Queueing Systems 46, 557–577.CrossRefGoogle Scholar
[20] Mecke, J., (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
[21] Pitman, J., (1987). Stationary excursions. In Séminaire de Probabilités XXI (Lecture Notes Math. 1247), Springer, Berlin, pp. 289–302.Google Scholar
[22] Salminen, P., (1993). On the distribution of supremum of diffusion local time. Statist. Prob. Lett. 18, 219–225.CrossRefGoogle Scholar
[23] Sirviö, M., (2006). On an inverse subordinator storage. Inst. Math. Rep. Ser. A501, Helsinki University of Technology.Google Scholar
[24] Tsirelson, B., (2004). Non-classical stochastic flows and continuous products. Prob. Surveys 1, 173–298.Google Scholar
[25] 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, 653–661.Google Scholar