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Volume and duration of losses in finite buffer fluid queues

Part of: Basic methods Markov processes

Published online by Cambridge University Press:  30 March 2016

Fabrice Guillemin  and
Bruno Sericola
Fabrice Guillemin*
Affiliation:
Orange Labs
Bruno Sericola*
Affiliation:
INRIA
*
∗ Postal address: Orange Labs, 2 Avenue Pierre Marzin, 22300 Lannion, France. Email address: fabrice.guillemin@orange.com
∗∗ Postal address: INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: bruno.sericola@inria.fr
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Abstract

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We study congestion periods in a finite fluid buffer when the net input rate depends upon a recurrent Markov process; congestion occurs when the buffer content is equal to the buffer capacity. Similarly to O'Reilly and Palmowski (2013), we consider the duration of congestion periods as well as the associated volume of lost information. While these quantities are characterized by their Laplace transforms in that paper, we presently derive their distributions in a typical stationary busy period of the buffer. Our goal is to compute the exact expression of the loss probability in the system, which is usually approximated by the probability that the occupancy of the infinite buffer is greater than the buffer capacity under consideration. Moreover, by using general results of the theory of Markovian arrival processes, we show that the duration of congestion and the volume of lost information have phase-type distributions.

Keywords

Fluid queueMarkov chaincongestion

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 80M35: Asymptotic analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 826 - 840
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Commun. Statist. Stoch. Models 11 21-49.Google Scholar
Bean, N. G. and O'Reilly, M. M. (2008). Performance measures of a multi-layer Markovian fluid model. Ann. Operat. Res. 160 99-120.Google Scholar
Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2005). Algorithms for return probabilities for stochastic fluid flows. Stoch. Models 21 149-184.CrossRefGoogle Scholar
Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2005). Hitting probabilities and hitting times for stochastic fluid flows. Stoch. Process. Appl. 115 1530-1556.Google Scholar
Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2008). Algorithms for the Laplace–Stieltjes transforms of first return times for stochastic fluid flows. Method. Comput. Appl. Prob. 10 381-408.CrossRefGoogle Scholar
Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2009). Hitting probabilities and hitting times for stochastic fluid flows: the bounded model. Prob. Eng. Inf. Sci. 23 121-147.Google Scholar
Soares, A. Da Silva and Latouche, G. (2006). Matrix-analytic methods for fluid queues with finite buffers. Performance Evaluation 63 295-314.CrossRefGoogle Scholar
Guillemin, F. and Sericola, B. (2012). On the fluid queue driven by an ergodic birth and death process. In Telecommunications Networks – Current Status and Future Trends, In Tech, Rijeka, pp. 379–404.Google Scholar
Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16 764-779.Google Scholar
O'Reilly, M. M. and Palmowski, Z. (2013). Loss rates for stochastic fluid models. Performance Evaluation 70 593-606.Google Scholar
Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Adv. Appl. Prob. 4 390-413.Google Scholar
Rubino, G. and Sericola, B. (1989). Accumulated reward over the n first operational periods in fault-tolerant computing systems. Res. Rep. 1028, INRIA.Google Scholar
Sericola, B. (2013). Markov Chains: Theory, Algorithms and Applications. Iste-Wiley, London.Google Scholar
Sericola, B. and Remiche, M.-A. (2011). Maximum level and hitting probabilities in stochastic fluid flows using matrix differential Riccati equations. Method. Comput. Appl. Prob. 13 307-328.CrossRefGoogle Scholar