Volume and duration of losses in finite buffer fluid queues. (English) Zbl 1326.60128

Guillemin, Fabrice; Sericola, Bruno

J. Appl. Probab. 52, No. 3, 826-840 (2015).

Summary: 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 [M. M. O’Reilly and Z. Palmowski, “Loss rates for stochastic fluid models”, Perform. Eval. 70, No. 9, 593–606 (2013; doi:10.1016/j.peva.2013.05.005)], 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.

MSC:

60K25	Queueing theory (aspects of probability theory)
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

fluid queue; Markov chain; congestion

References:

[1]	Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Commun. Statist. Stoch. Models 11, 21-49. · Zbl 0817.60086
[2]	Bean, N. G. and O’Reilly, M. M. (2008). Performance measures of a multi-layer Markovian fluid model. Ann. Operat. Res. 160, 99-120. · Zbl 1145.60048
[3]	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. · Zbl 1064.60162
[4]	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. · Zbl 1074.60078
[5]	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. · Zbl 1184.60027
[6]	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. · Zbl 1162.76046
[7]	Da Silva Soares, A. and Latouche, G. (2006). Matrix-analytic methods for fluid queues with finite buffers. Performance Evaluation 63, 295-314.
[8]	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.
[9]	Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764-779. · Zbl 0422.60043
[10]	O’Reilly, M. M. and Palmowski, Z. (2013). Loss rates for stochastic fluid models. Performance Evaluation 70, 593-606.
[11]	Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Adv. Appl. Prob. 4, 390-413. · Zbl 0806.60052
[12]	Rubino, G. and Sericola, B. (1989). Accumulated reward over the n first operational periods in fault-tolerant computing systems. Res. Rep. 1028, INRIA.
[13]	Sericola, B. (2013). Markov Chains: Theory, Algorithms and Applications . Iste-Wiley, London. · Zbl 1277.60002
[14]	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. · Zbl 1216.60062