Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-25T05:54:58.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov-renewal fluid queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Guy Latouche  and
Tetsuya Takine
Guy Latouche*
Affiliation:
Université Libre de Bruxelles
Tetsuya Takine*
Affiliation:
Kyoto University
*
∗ Postal address: Université Libre de Bruxelles, Département d'Informatique, Boulevard du Triomphe, CP 212, 1050 Bruxelles, Belgium. Email address: guy.latouche@ulb.ac.be
∗∗ Postal address: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan. Email address: takine@amp.i.kyoto-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

Keywords

Fluid queuesemi-Markov environmentMarkov-renewal analysis

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K37: Processes in random environments
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 746 - 757
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrawal, R., Makowski, A. M., and Nain, P. (1999). On a reduced load equivalence for fluid queues under subexponentiality. Queueing Systems 33, 5–41.10.1023/A:1019111809660Google Scholar
Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Commun. Statist. Stoch. Models 11, 21–49.10.1080/15326349508807330Google Scholar
Boxma, O. J., and Dumas, V. (1998). Fluid queues with long-tailed activity period distributions. Comput. Commun. 21, 1509–1529.10.1016/S0140-3664(98)00219-9Google Scholar
Da Silva Soares, A., and Latouche, G. (2002). Further results on the similarity between fluid queues and QBDs. In Matrix-Analytic Methods: Theory and Applications, eds Latouche, G. and Taylor, P., World Scientific, River Edge, NJ, pp. 89–106.10.1142/9789812777164_0005Google Scholar
Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queueing: Models and Applications in Science and Engineering, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 321–338.Google Scholar
Latouche, G., and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, PA.10.1137/1.9780898719734Google Scholar
Miyazawa, M., and Takada, H. (2002). A matrix exponential form for hitting probabilities and its application to a Markov modulated fluid queue with downward jumps. J. Appl. Prob. 39, 604–618.10.1239/jap/1034082131Google Scholar
Ramaswami, V. (1996). Matrix analytic methods: a tutorial overview with some extensions and new results. In Matrix-Analytic Methods in Stochastic Models, eds Chakravarthy, S. R. and Alfa, A. S., Marcel Dekker, New York, pp. 261–295.10.1201/b17050-15Google Scholar
Ramaswami, V. (1999). Matrix analytic methods for stochastic fluid flows. In Teletraffic Engineering in a Competitive World, eds Smith, D. and Hey, P., Elsevier, Amsterdam, pp. 1019–1030.Google Scholar
Resnick, S., and Samorodnitsky, G. (2001). Steady-state distribution of the buffer content for M/G/∞ input fluid queues. Bernoulli 7, 191–210.10.2307/3318735Google Scholar
Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413.10.1214/aoap/1177005065Google Scholar
Sengupta, B. (1989). Markov processes whose steady-state distribution is matrix-exponential with an application to the GI/G/1 queue. Adv. Appl. Prob. 21, 159–180.10.2307/1427202Google Scholar
Takada, H. (2001). Markov modulated fluid queues with batch fluid arrivals. J. Operat. Res. Soc. Japan 44, 344–365.10.15807/jorsj.44.344Google Scholar
Takine, T., and Hasegawa, T. (1994). The workload in the MAP/G/1 queue with state-dependent services: its application to a queue with preemptive resume priority. Commun. Statist. Stoch. Models 10, 183–204.10.1080/15326349408807292Google Scholar