The Morphing of Fluid Queues into Markov-Modulated Brownian Motion
Published Online:20 Apr 2015https://doi.org/10.1287/13-SSY133
References
- , Transient analysis of fluid flow models via stochastic coupling to a queue. Stochastic Models, 20:71–101, 2004. MR2036297Google Scholar
- , An approach to the stationary and transient analysis on the Markov modulated fluid model. Private communication, 2012.Google Scholar
- , An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. Comm. Statist. Stochastic Models, 13:381–416, 1997. MR1457654Google Scholar
- , Modular Functions and Dirichlet Series in Number Theory. Springer-Verlag, New York, 1997.Google Scholar
- , Stationary distributions for fluid flow models with or without Brownian noise. Communications in Statistics: Stochastic Models, 11(1):21–49, 1995. MR1316767Google Scholar
- , A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab., 32:376–393, 2000. MR1778570Google Scholar
- , 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. MR3170224Google Scholar
- , Convergence of Probability Measures. John Wiley & Sons, N.Y., second edition, 1999. MR1700749Google Scholar
- , 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. MR2125629Google Scholar
- , Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl., 343–344:21–61, 2002. MR1878936Google Scholar
- , Numerical Solution of Algebraic Riccati Equations. Fundamentals of Algorithms. SIAM, Philadelphia PA, 2012. MR2896454Google Scholar
- , Solving matrix polynomial equations arising in queueing problems. Linear Algebra Appl., 340:225–244, 2002. MR1869430Google Scholar
- , Numerical Methods for Structured Markov Chains. Numerical, Mathematics and Scientific Computation. Oxford University Press, Oxford, 2005. MR2132031Google Scholar
- , First passage times for Markov additive processes with positive jumps of Phase-Type. Journal of Applied Probability, 45:778–799, 2008. MR2455184Google Scholar
- , Fluid queues with level dependent evolution. European J. Oper. Res., 196:1041–1048, 2009. MR2498464Google Scholar
- , Two-sided reflection of Markov-modulated Brownian motion. Stoch. Models, 28(2):316–332, 2012. MR2926589Google Scholar
- , Markov modulation of a two-sided reflected Brownian motion with application to fluid queues. Stoch. Proc. Appl., 122(4):1566–1581, 2012. MR2914763Google Scholar
- , Continuity estimates for ruin probabilities. Scand. Actuarial J., 2001(1):18–39, 2001. MR1834970Google Scholar
- , Approximating Countable Markov Chains. Holden–Day, San Francisco, CA, 1972. MR0428472Google Scholar
- , Stability for fluid queues: Characteristic inequalities. Stoch. Models, 29(1):64–88, 2013. MR3018924Google Scholar
- , Functions of Matrices: Theory and Computation. SIAM, 2008. MR2396439Google Scholar
- , Markov-modulated Brownian motion with two reflecting barriers. Journal of Applied Probability, 47:1034–1047, 2010. MR2752892Google Scholar
- , Second-order fluid flow models: Reflected Brownian motion in a random environment. Oper. Res, 43:77–88, 1995.Link, Google Scholar
- , Perturbation Theory for Linear Operators. Springer-Verlag, New York, 2nd edition, 1976. MR0407617Google Scholar
- , Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia PA, 1999. MR1674122Google Scholar
- , Stationary waiting-time distribution for single-server queues. Ann. Math. Statist., 33:1323–1339, 1962. MR0154342Google Scholar
- , Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD, 1981. MR0618123Google Scholar
- , 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.Google Scholar
- , 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. MR3067683Google Scholar
- , Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Probab., 4:390–413, 1994. MR1272732Google Scholar
- , Non-Negative Matrices and Markov Chains. Springer-Verlag, New York, second edition, 1981.Google Scholar
- , Ergodic theorems for Markov chains represented by iterated function systems. Bulletin of the Polish Academy of Sciences: Mathematics, 49:427–443, 2001. MR1824155Google Scholar
- , Weak convergence of probability measures on the function space C[0, ∞). Annals of Mathematical Statistics, 41(2):939–944, 1970. MR0261646Google Scholar

