×

Stability analysis of second-order fluid flow models in a stationary ergodic environment. (English) Zbl 1036.60088

Summary: We study the stability of a fluid queue with an infinite-capacity buffer. The input and service rates are governed by a stochastic process, called the environment process, and are allowed to depend on the fluid level in the buffer. The variability of the traffic is modeled by a Brownian motion and a local variance function, which also depends on the fluid level in the buffer. The behavior of this second-order fluid flow model is described by a reflected stochastic differential equation, and, under stationarity and ergodicity assumptions on the environment process, we obtain stability conditions for this general fluid queue.

MSC:

60K25 Queueing theory (aspects of probability theory)
60G35 Signal detection and filtering (aspects of stochastic processes)
60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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.
[2] Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Comm. Statist. Stochastic Models 11 21–49. · Zbl 0817.60086 · doi:10.1080/15326349508807330
[3] Atar, R., Budhiraja, A. and Dupuis, P. (2001). On positive recurrence of constrained diffusion processes. Ann. Probab. 29 979–1000. · Zbl 1018.60081 · doi:10.1214/aop/1008956699
[4] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory . Springer, New York. · Zbl 0801.60081
[5] El Karoui, N. and Chaleyat-Maurel, M. (1978). Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur \(\nbR\), cas continu. Astérisque 52 –53 117–144.
[6] Karandikar, R. L. and Kulkarni, V. G. (1995). Second-order fluid flow models: Reflected Brownian motion in a random environment. Oper. Res. 1 77–88. · Zbl 0821.60087 · doi:10.1287/opre.43.1.77
[7] Karatzas, I. and Shreve, S. E. (1997). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0638.60065
[8] Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queuing. Models and Applications in Science and Engineering (J. D. Dshalalow, ed.) 321–338. CRC Press, Boca Raton, FL. · Zbl 0871.60079
[9] Lam, C. H. and Lee, T. T. (1997). Fluid flow models with state-dependent service rate. Comm. Statist. Stochastic Models 13 547–576. · Zbl 0896.60064 · doi:10.1080/15326349708807439
[10] Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrivals and service times. Math. Proc. Cambridge Philos. Soc. 58 497–520. · Zbl 0203.22303 · doi:10.1017/S0305004100036781
[11] Mitra, D. (1988). Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. in Appl. Probab. 20 646–676. · Zbl 0656.60079 · doi:10.2307/1427040
[12] Prabhu, N. U. (1997). Stochastic Storage Processes : Queues , Insurance Risk , Dams , and Data Communication , 2nd ed. Springer, New York. · Zbl 0453.60094
[13] Rogers, L. C. G. and Williams, D. (1987). Diffusions , Markov Processes , and Martingales : Itô Calculus 2 . Wiley, New York. · Zbl 0627.60001
[14] Sericola, B. and Tuffin, B. (1999). A fluid queue driven by a Markovian queue. Queueing Syst. Theory Appl. 31 253–264. · Zbl 0934.90023 · doi:10.1023/A:1019114415595
[15] Sigman, K. and Ryan, R. (2000). Continuous-time stochastic recursions and duality. Adv. in Appl. Probab. 32 426–445. · Zbl 0971.60040 · doi:10.1239/aap/1013540172
[16] Skorokhod, A. V. (1961). Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl . 6 264–274. · Zbl 0215.53501
[17] Stern, T. E. and Elwalid, A. I. (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Adv. in Appl. Probab. 23 105–139. · Zbl 0716.60114 · doi:10.2307/1427514
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.