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A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa  and
Hiroyuki Takada
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
Hiroyuki Takada*
Affiliation:
Tokyo University of Science
*
∗ Postal address: Department of Information Sciences, Tokyo University of Science, Noda, Chiba 278-8510, Japan.
∗ Postal address: Department of Information Sciences, Tokyo University of Science, Noda, Chiba 278-8510, Japan.
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Abstract

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

Keywords

Matrix exponentialfluid queueMarkov modulatedbatch fluidGI/M/1-type queuestationary distributionhitting probability

MSC classification

Primary: 60K25: Queueing theory 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J75: Jump processes 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 604 - 618
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Aalto, S. (1998). Characterization of the output rate process for a Markovian storage model. J. Appl. Prob. 35, 184–199.Google Scholar
[2]. Anick, D., Mitra, D., and Sondhi, M. M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.CrossRefGoogle Scholar
[3]. Asmussen, S. (1991). Ladder heights and the Markov-modulated M/G/1 queue. Stoch. Process. Appl. 37, 313–326.Google Scholar
[4]. Asmussen, S. (1994). Busy period analysis, rare events and transient behavior in fluid flow models. J. Appl. Math. Stoch. Anal. 7, 269–299.Google Scholar
[5]. Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11, 21–49.CrossRefGoogle Scholar
[6]. Asmussen, S., and O’Cinneide, C. A. (1998). Representations for matrix-geometric and matrix-exponential steady-state distributions to many-server queues. Stoch. Models 14, 369–387.Google Scholar
[7]. Chao, X., Miyazawa, M., and Pinedo, M. (1999). Queueing Networks: Customers, Signals and Product Form Solutions. John Wiley, Chichester.Google Scholar
[8]. Elwalid, A. I., and Stern, T. E. (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23, 105–139.Google Scholar
[9]. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[10]. Hasegawa, T., and Takine, T. (1994). The workload in the MAP/G/1 queue with state-dependent services: its application to a queue with preemptive resume priority. Stoch. Models 10, 183–204.Google Scholar
[11]. Jain, G., and Sigman, K. (1996). A Pollaczek–Khinchine formulation for M/G/1 queues with disasters. J. Appl. Prob. 33, 1191–1200.Google Scholar
[12]. Mitra, D. (1988). Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Prob. 20, 646–676.CrossRefGoogle Scholar
[13]. Miyazawa, M. (1994). Palm calculus for a process with a stationary random measure and its applications to fluid queues. Queueing Systems 17, 183–211.CrossRefGoogle Scholar
[14]. Miyazawa, M. (1994). Time-dependent rate conservation laws for a process defined with a stationary marked point process and their applications. J. Appl. Prob. 31, 114–129.Google Scholar
[15]. Prabhu, N. U. (1980). Stochastic Storage Processes. Springer, New York.Google Scholar
[16]. Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413.CrossRefGoogle Scholar
[17]. Seneta, E. (1981). Non-Negative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar
[18]. Sengupta, B. (1989). Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue. Adv. Appl. Prob. 21, 159–180.CrossRefGoogle Scholar
[19]. Takada, H. (2001). Markov modulated fluid queues with batch fluid arrivals. J. Operat. Res. Soc. Japan 44, 344–365.Google Scholar