## Stochastic Systems

### The morphing of fluid queues into Markov-modulated Brownian motion

#### Abstract

Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. We follow a Markov-renewal approach and we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Thus, we provide a new approach for obtaining the stationary distribution of a reflected MMBM without time-reversal or solving partial differential equations. Our results open the way to the analysis of more complex Markov-modulated processes.

Key matrices in the limiting stationary distribution are shown to be solutions of a matrix-quadratic equation, and we describe how this equation can be efficiently solved.

#### Article information

Source
Stoch. Syst., Volume 5, Number 1 (2015), 62-86.

Dates
First available in Project Euclid: 23 December 2015

https://projecteuclid.org/euclid.ssy/1450879281

Digital Object Identifier
doi:10.1214/13-SSY133

Mathematical Reviews number (MathSciNet)
MR3442389

Zentralblatt MATH identifier
1336.60151

#### Citation

Latouche, Guy; Nguyen, Giang T. The morphing of fluid queues into Markov-modulated Brownian motion. Stoch. Syst. 5 (2015), no. 1, 62--86. doi:10.1214/13-SSY133. https://projecteuclid.org/euclid.ssy/1450879281

#### References

• [1] Ahn, S. and Ramaswami, V., Transient analysis of fluid flow models via stochastic coupling to a queue. Stochastic Models, 20:71–101, 2004.
• [2] Ahn, S. and Ramaswami, V., An approach to the stationary and transient analysis on the Markov modulated fluid model. Private communication, 2012.
• [3] Akar, N. and Sohraby, K., An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. Comm. Statist. Stochastic Models, 13:381–416, 1997.
• [4] Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory. Springer-Verlag, New York, 1997.
• [5] Asmussen, S., Stationary distributions for fluid flow models with or without Brownian noise. Communications in Statistics: Stochastic Models, 11(1):21–49, 1995.
• [6] Asmussen, S. and Kella, O., A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab., 32:376–393, 2000.
• [7] Bean, N. G. and O’Reilly, M. M., The stochastic fluid-fluid model: A stochastic fluid model driven by an uncountable-state process, which is a stochastic fluid model itself. Stoch. Proc. Appl., 124:1741–1772, 2014.
• [8] Billingsley, P., Convergence of Probability Measures. John Wiley & Sons, N.Y., second edition, 1999.
• [9] Bini, D. A. and Gemignani, L., Solving quadratic matrix equations and factoring polynomials: New fixed point iterations based on Schur complements of Toeplitz matrices. Numer. Linear Algebra Appl., 12:181–189, 2005.
• [10] Bini, D. A., Gemignani, L., and Meini, B., Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl., 343–344:21–61, 2002.
• [11] Bini, D. A., Iannazzo, B., and Meini, B., Numerical Solution of Algebraic Riccati Equations. Fundamentals of Algorithms. SIAM, Philadelphia PA, 2012.
• [12] Bini, D. A., Latouche, G., and Meini, B., Solving matrix polynomial equations arising in queueing problems. Linear Algebra Appl., 340:225–244, 2002.
• [13] Bini, D. A., Latouche, G., and Meini, B., Numerical Methods for Structured Markov Chains. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2005.
• [14] Breuer, L., First passage times for Markov additive processes with positive jumps of Phase-Type. Journal of Applied Probability, 45:778–799, 2008.
• [15] da Silva Soares, A. and Latouche, G., Fluid queues with level dependent evolution. European J. Oper. Res., 196:1041–1048, 2009.
• [16] D’Auria, B., Ivanovs, J., Kella, O., and Mandjes, M., Two-sided reflection of Markov-modulated Brownian motion. Stoch. Models, 28(2):316–332, 2012.
• [17] D’Auria, B. and Kella, O., Markov modulation of a two-sided reflected Brownian motion with application to fluid queues. Stoch. Proc. Appl., 122(4):1566–1581, 2012.
• [18] Enikeeva, E., Kalashnikov, V., and Rusaityte, D., Continuity estimates for ruin probabilities. Scand. Actuarial J., 2001(1):18–39, 2001.
• [19] Freedman, D., Approximating Countable Markov Chains. Holden–Day, San Francisco, CA, 1972.
• [20] Govorun, M., Latouche, G., and Remiche, M.-A., Stability for fluid queues: Characteristic inequalities. Stoch. Models, 29(1):64–88, 2013.
• [21] Higham, N. J., Functions of Matrices: Theory and Computation. SIAM, 2008.
• [22] Ivanovs, J., Markov-modulated Brownian motion with two reflecting barriers. Journal of Applied Probability, 47:1034–1047, 2010.
• [23] Karandikar, R. L. and Kulkarni, V., Second-order fluid flow models: Reflected Brownian motion in a random environment. Oper. Res, 43:77–88, 1995.
• [24] Kato, T., Perturbation Theory for Linear Operators. Springer-Verlag, New York, 2nd edition, 1976.
• [25] Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia PA, 1999.
• [26] Loynes, R. M., Stationary waiting-time distribution for single-server queues. Ann. Math. Statist., 33:1323–1339, 1962.
• [27] Neuts, M. F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD, 1981.
• [28] Ramaswami, V., Matrix analytic methods for stochastic fluid flows. In D. Smith and P. Hey, editors, Teletraffic Engineering in a Competitive World (Proceedings of the 16th International Teletraffic Congress), pages 1019–1030. Elsevier Science B.V., Edinburgh, UK, 1999.
• [29] Ramaswami, V., A fluid introduction to Brownian motion and stochastic integration. In G. Latouche, V. Ramaswami, J. Sethuraman, K. Sigman, M. Squillante, and D. Yao, editors, Matrix-Analytic Methods in Stochastic Models, volume 27 of Springer Proceedings in Mathematics & Statistics, chapter 10, pages 209–225. Springer Science, New York, NY, 2013.
• [30] Rogers, L. C. G., Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Probab., 4:390–413, 1994.
• [31] Seneta, E., Non-Negative Matrices and Markov Chains. Springer-Verlag, New York, second edition, 1981.
• [32] Stenflo, O., Ergodic theorems for Markov chains represented by iterated function systems. Bulletin of the Polish Academy of Sciences: Mathematics, 49:427–443, 2001.
• [33] Whitt, W., Weak convergence of probability measures on the function space ${C}[0,\infty)$. Annals of Mathematical Statistics, 41(2):939–944, 1970.