A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle

References

• Antunes N, Fricker C, Robert P, Tibi D. Analysis of loss networks with routing. Ann. Appl. Prob. (2006) 16(4):2007–2026Crossref, Google Scholar
• Asmussen S. Extreme value theory for queues via cycle maxima. Extremes (1998) 1(2):137–168Crossref, Google Scholar
• Asmussen S. Applied Probability and Queues (2003) 2nd ed.(Springer, New York) Google Scholar
• Bassamboo A, Harrison JM, Zeevi A. Dynamic routing and admission control in high-volume service systems: Asymptotic analysis via multi-scale fluid limits. Queueing Systems (2005) 51(3–4):249–285Crossref, Google Scholar
• Bassamboo A, Harrison JM, Zeevi A. Dynamic and control of a large call center: Asymptotic analysis of an LP-based method. Oper. Res. (2006) 54(3):419–435Link, Google Scholar
• Baxendale P. Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. (2005) 15(1):700–738Crossref, Google Scholar
• Billingsley P. Convergence of Probability Measures (1999) 2nd ed.(John Wiley & Sons, New York) Crossref, Google Scholar
• Bolthausen E. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete (1980) 54(1):59–73Crossref, Google Scholar
• Borkovec M, Klüppelberg C. Extremal behavior of diffusion models in finance. Extremes (1998) 1(1):47–80Crossref, Google Scholar
• Coffman EG, Puhalskii AA, Reiman MI. Polling systems with zero switchover times: A heavy-traffic averaging principle. Ann. Appl. Probab. (1995) 5(3):681–719Crossref, Google Scholar
• Davis RA. Maximum and minimum of one-dimensional diffusions. Stochastic Processes and Their Appl. (1982) 13(1):1–9Crossref, Google Scholar
• Ethier SN, Kurtz TG. Markov Processes: Characterization and Convergence (1985) (John Wiley & Sons, New York) Google Scholar
• Glynn PW, Whitt W. Limit theorems for cumulative processes. Stoch. Proc. Appl. (1993) 47(2):299–314Crossref, Google Scholar
• Gurvich I, Whitt W. Queue-and-idleness-ratio controls in many-server service systems. Math. Oper. Res. (2009) 34(2):363–396Link, Google Scholar
• Gurvich I, Whitt W. Scheduling flexible servers with convex delay costs in many-server service systems. Manufacturing Service Oper. Management (2009) 11(2):237–253Link, Google Scholar
• Gurvich I, Whitt W. Service-level differentiation in many-server service systems via queue-ratio routing. Oper. Res. (2010) 58(2):316–328Link, Google Scholar
• He Q. Differentiability of the matrices R and G in the matrix analytic method. Stochastic Models (1993) 11(1):123–132Google Scholar
• Hunt PJ, Kurtz TG. Large loss networks. Stochastic Processes and Their Appl. (1994) 53(2):363–378Crossref, Google Scholar
• Jacod J, Shiryaev AN. Limit Theorems for Stoschastic Processes (1987) (Springer, Berlin) Crossref, Google Scholar
• Kamae T, Krengel U, O'Brien GL. Stochastic inequalities on partially ordered spaces. Ann. Prob. (1977) 5(6):899–912Crossref, Google Scholar
• Kang W, Ramanan K. Fluid limits of many-server queues with reneging. Ann. Appl. Prob. (2010) 20:2204–2260Crossref, Google Scholar
• Karr AF. Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete (1975) 33(1):41–48Crossref, Google Scholar
• Kaspi H, Ramanan K. Law of large numbers limits for many-server queues. Ann. Appl. Prob. (2011) 21(1):33–114Crossref, Google Scholar
• Khalil HK. Nonlinear Systems (2002) (Prentice Hall, Englewood Cliffs, NJ) Google Scholar
• Khasminskii RZ, Yin G. On averaging principles: An asymptotic expansion approach. SIAM J. Math. Anal. (2004) 35(6):1534–1560Crossref, Google Scholar
• Kontoyiannis I, Meyn SP. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. (2003) 13(1):304–362Crossref, Google Scholar
• Kurtz TG. Averaging for martingale problems and stochastic approximations. Appl. Stochastic Anal., Proc. US-French Workshop (1992) 177(Springer, Berlin) 186–209Lecture Notes in Control Inform. Sci.Crossref, Google Scholar
• Latouche G, Ramaswami V. Introduction to Matrix Analystic Methods in Stochastic Modeling (1999) (Siam and ASA, Philadelphia) Crossref, Google Scholar
• Leadbetter MR, Lindgren G, Rootzén H. Extremes and Related Properties of Random Sequences and Processes (1983) (Springer-Verlag, New York) Crossref, Google Scholar
• Lindvall T. Lectures on the Coupling Method (1992) (John Wiley & Sons, New York) Google Scholar
• Massey WA, Whitt W. Uniform acceleration expansions for Markov chains with time-varying rates. Ann. Appl. Prob. (1998) 8(4):1130–1155Crossref, Google Scholar
• Meyn SP, Tweedie RL. Markov Chains and Stochastic Stability (1993) (Springer, New York) Crossref, Google Scholar
• Müller A, Stoyan D. Comparison Methods for Stochastic Models with Risks (2002) (John Wiley & Sons, New York) Google Scholar
• Pang G, Talreja R, Whitt W. Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surveys (2007) 4:193–267Crossref, Google Scholar
• Perry O, Whitt W. Responding to unexpected overloads in large-scale service systems. Management Sci. (2009) 55(8):1353–1367Link, Google Scholar
• Perry O, Whitt W. A fluid approximation for service systems responding to unexpected overloads. Oper. Res. (2011) 59(5):1159–1170Link, Google Scholar
• Perry O, Whitt W. An ODE for an overloaded X model involving a stochastic averaging principle. Stochastic Systems (2011) 1(1):17–66Link, Google Scholar
• Perry O, Whitt W. Diffusion approximation for an overload X model via a stochastic averaging principle. (2012) . Working paper, Columbia University. Available at: http://www.columbia.edu/∼ww2040/allpapers.htmlGoogle Scholar
• Resnick SI. Extreme Values, Regular Variation and Point Processes (1987) (Springer-Verlag, New York) Crossref, Google Scholar
• Roberts GO, Rosenthal JS. Quantative bounds for convergence rates of continuous time Markov processes. Electronic J. Appl. Probab. (1996) 1(9):1–21Google Scholar
• Rootzén H. Maxima and exceedances of stationary Markov chains. Adv. Appl. Probab. (1988) 20(2):371–390Crossref, Google Scholar
• Silvestrov DS, Teugels JL. Limit theorems for extremes with random sample size. Adv. Appl. Prob. (1998) 30(3):777–806Crossref, Google Scholar
• Whitt W. Continuity of generalized semi-Markov processes. Math. Oper. Res. (1980) 5(4):494–501Link, Google Scholar
• Whitt W. Comparing counting processes and queues. Adv. Appl. Prob. (1981) 13(1):207–220Crossref, Google Scholar
• Whitt W. Departures from a queue with many busy servers. Math. Oper. Res. (1984) 9(4):534–544Link, Google Scholar
• Whitt W. The pointwise stationary approximation for Mt/Mt/s queues is asymptotically correct as the rates increase. Management Sci. (1991) 37(3):307–314Link, Google Scholar
• Whitt W. Stochastic-Process Limits (2002) (Springer, New York) Crossref, Google Scholar
• Whitt W. Efficiency-driven heavy-traffic approximations for many-server queues with abandonments. Management Sci. (2004) 50(10):1449–1461Link, Google Scholar
• Zachary S, Ziedins J. A refinement of the Hunt-Kurtz theory of large loss networks, with an application to virtual partitioning. Ann. Appl. Prob. (2002) 12(1):1–22Google Scholar