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A second-order Markov-modulated fluid queue with linear service rate

Part of: Special processes Operations research and management science Markov processes

Published online by Cambridge University Press:  14 July 2016

Landy Rabehasaina  and
Bruno Sericola
Landy Rabehasaina
Affiliation:
IRISA-INRIA, Rennes
Bruno Sericola*
Affiliation:
IRISA-INRIA, Rennes
*
∗∗ Postal address: IRISA-INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: [email protected]
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Abstract

We consider an infinite-capacity second-order fluid queue governed by a continuous-time Markov chain and with linear service rate. The variability of the traffic is modeled by a Brownian motion and a local variance function modulated by the Markov chain and proportional to the fluid level in the queue. The behavior of this second-order fluid-flow model is described by a linear stochastic differential equation, satisfied by the transient queue level. We study the transient level's convergence in distribution under weak assumptions and we obtain an expression for the stationary queue level. For the first-order case, we give a simple expression of all its moments as well as of its Laplace transform. For the second-order model we compute its first two moments.

Keywords

Fluid queueMarkov chainsecond-order modelrisk theory

MSC classification

Primary: 60K25: Queueing theory 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 90B05: Inventory, storage, reservoirs
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 758 - 777
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

∗

Current address: LMC-IMAG, 51, Rue des Mathématiques, BP53, 38043 Grenoble Cedex 9, France. Email address: [email protected]

References

Asmussen, S., and Kella, O. (1996). Rate modulation in dams and ruin problem. J. Appl. Prob. 33, 523–535.10.2307/3215076Google Scholar
Chen, D., Hong, Y., and Trivedi, K. S. (2002). Second-order stochastic fluid models with fluid-dependent flow rates. Performance Evaluation 49, 341–358.10.1016/S0166-5316(02)00113-XGoogle Scholar
Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Karandikar, R. L., and Kulkarni, V. G. (1995). Second-order fluid flow models: reflected Brownian motion in a random environment. Operat. Res. 43, 77–88.10.1287/opre.43.1.77Google Scholar
Kella, O., and Stadje, W. (2002). Exact results for a fluid model with state-dependent flow rates. Prob. Eng. Inf. Sci. 16, 389–402.10.1017/S0269964802164011Google Scholar
Kella, O., and Stadje, W. (2002). Markov modulated linear fluid networks with Markov additive input. J. Appl. Prob. 39, 413–420.10.1239/jap/1025131438Google Scholar
Kella, O., and Whitt, W. (1999). Linear stochastic fluid networks. J. Appl. Prob. 36, 244–260.10.1239/jap/1032374245Google Scholar
Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queuing, Models and Applications in Science and Engineering, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 321–338.Google Scholar
Rabehasaina, L., and Sericola, B. (2003). Stability of second order fluid flow models in a stationary ergodic environment. Ann. Appl. Prob. 13, 1449–1473.Google Scholar
Rabehasaina, L., and Sericola, B. (2003). Transient analysis of a Markov modulated fluid queue with linear service rate. In Proc. 10th Conf. Anal. Stoch. Modelling Tech. Appl. (ASMTA'03, Nottingham, June 2003), ed. Al-Dabass, D., SCS European Publishing House, pp. 234–239.Google Scholar
Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.10.1007/978-3-662-06400-9Google Scholar
Sigman, K., and Ryan, R. (2000). Continuous-time stochastic recursions and duality. Adv. Appl. Prob. 32, 426–445.10.1239/aap/1013540172Google Scholar
Yao, D. D., Zhang, Q., and Zhou, X. Y. (2003). A regime-switching model for European option pricing. Working paper, Columbia University. Available at http://www.columbia.edu/∼yao/.Google Scholar