Chattering and Congestion Collapse in an Overload Switching Control

References

• Asmussen, S. (2003). Applied probability and queues. Springer. MR1978607Google Scholar
• Bell, S. L. and Williams, R.J. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy. The Annals of Applied Probability, 11 (3) 608–649. MR1865018Google Scholar
• Billah, K. Y. and Scanlan, R. H. (1991). Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, American Journal of Physics, 59 (2), 118–124.Google Scholar
• Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems, 30, (1–2), 89–140. MR1663763Google Scholar
• Bramson, M. (1994). Instability of FIFO queueing networks. The Annals of Applied Probability 4 (2), 414–431. MR1272733Google Scholar
• Bramson, M. (2008). Stability of queueing networks. Springer. MR2445100Google Scholar
• Bramson, M. and Williams, R. J. (2000). On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process Proceedings of the 39th IEEE Conference on Decisions and Control, 516–521.Google Scholar
• Chase, C., Serrano, J., Ramadge, P. J. (1993). Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. IEEE Transactions on Automatic Control 38 (1), 70–83. MR1201496Google Scholar
• Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. The Annals of Applied Probability 5 (1) 49–77. MR1325041Google Scholar
• Durrett, R. (1991). Probability: theory and examples. Wadsworth and Brooks/Cole, Pacific Grove, California. MR1068527Google Scholar
• Erramilli, A. and Forys, L. J. (1991). Oscillations and chaos in a flow model of a switching system. IEEE Journal on Selected Areas in Communications, 9 (2), 171–178.Google Scholar
• Filippov, A. F. (1988) Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, the Netherlands. MR1028776Google Scholar
• Garnett, O., Mandelbaum, A. and Reiman, M. (2002). Designing a call center with impatient customers, Manufacturing & Service Operations Management, 4, (3), 208–227.Link, Google Scholar
• Gurvich, I. and Whitt, W. (2009). Scheduling Flexible Servers with Convex Delay Costs in Many-Server Service Systems. Manufacturing & Service Operations Management, 11 (2), 237–253.Link, Google Scholar
• Gurvich, I. and Whitt, W. (2009). Queue-and-Idleness-Ratio Controls in Many-Server Service Systems. Mathematics of Operations Research 34 (2), 363–396. MR2554064Link, Google Scholar
• Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operations Research 29 (3) 567–588. MR0629195Link, Google Scholar
• Harrison, J. M. (1988). Brownian models of queueing networks with heterogeneous customer populations Stochastic differential systems, stochastic control theory and applications, 147–186. MR0934722Google Scholar
• Harrison, J.M. (1998). Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies. Annals of Applied Probability 8 (3) 822–848. MR1627791Google Scholar
• Harrison, J. M. and Taskar, M. I. (1983). Instantaneous control of Brownian motion. Mathematics of Operations research, 8 (3) 439–453. MR0716123Link, Google Scholar
• Harrison, J. M. and Williams, R. J. (2005). Workload reduction of a generalized Brownian network. The Annals of Applied Probability, 15 (4), 2255–2295. MR2187295Google Scholar
• Khalil, H. K. (2002). Nonlinear Systems. Prentice Hall, New Jersey.Google Scholar
• Kontoyiannis I. and S.P. Meyn. (2003). Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes. The Annals of Applied Probability, Vol. 13, 304–362. MR1952001Google Scholar
• Liberzon, D. (2003). Switching in Systems and Control. Birkhäuser, Boston. MR1987806Google Scholar
• Liu, Y. and Whitt, W. (2011). Nearly Periodic Behavior in the The Overloaded G/D/S + GI Queue. Stochastic Systems, 1 (2), 340–410. MR2949544Link, Google Scholar
• Lu S.H., Kumar P.R. (1991). Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Automat. Control. 36(12), 1406–1416.Google Scholar
• Matveev, A. S., Savkin, A. V. (2000). Qualitative theory of hybrid dynamical systems. Birkhäuser, Boston.Google Scholar
• Pang, G., Talreja, R., Whitt, W., (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probability Surveys, 4, 193–267. MR2368951Google Scholar
• Perry, O., W. Whitt. (2009). Responding to unexpected overloads in large-scale service systems. Management Sci., 55 (8), 1353–1367.Link, Google Scholar
• Perry, O., W. Whitt. (2011a). A fluid approximation for service systems responding to unexpected overloads. Oper. Res., 59 (5), 1159–1170. MR2864331Link, Google Scholar
• Perry, O., W. Whitt. (2011b). An ODE for an overloaded X model involving a stochastic averaging principle. Stochastic Systems, 1 (1), 17–66. MR2948918Link, Google Scholar
• Perry, O., W. Whitt. (2013). A fluid limit for an overloaded X model via an averaging principle. Math, Oper. Res., 38 (2), 294–349. MR3062009Link, Google Scholar
• Perry, O., W. Whitt. (2015a). Achieving rapid recovery in an overload control for large-scale service systems. Forthcoming in INFORMS Journal on Computing. Available at: http://www.columbia.edu/~ww2040/allpapers.html MR3382939Google Scholar
• Perry, O., W. Whitt. (2015b). Online Supplement to Achieving rapid recovery in an overload control for large-scale service systems. Forthcoming in INFORMS Journal on Computing. Available at: http://www.columbia.edu/~ww2040/allpapers.html MR3382939Google Scholar
• Reiman, M. I. (1984). Some diffusion approximations with state space collapse in Modelling and performance evaluation methodology. 207–240, Springer. MR0893658Google Scholar
• Robert, P. (2003). Stochastic networks and queues, Springer-Verlag. MR1996883Google Scholar
• Rudin, W. (1991). Functional Analysis, McGraw-Hill, Inc., New York. MR1157815Google Scholar
• Rybko A.N., Stolyar A.L. (1992). Ergodicity of stochastic processes describing the operations of open queueing networks. Problems Inform. Transmission 28, 3–26 (in Russian). MR1189331Google Scholar
• Sastry, S.S. and Desoer, C. A. (1981). Jump behavior of circuits and systems. IEEE Transactions on Circuits and Systems 28 (12) 1109–1124. MR0643025Google Scholar
• Shah, D., D. Wischik. (2011). Fluid models of congestion collapse in overloaded switched networks. Queueing Systems 69 121–143. MR2836736Google Scholar
• Schaft, V.D. and Schumacher, H. (2000). An introduction to hybrid dynamical systems. Springer Lecture Notes in Control and Information Sciences, Vol. 251. Springer-Verlag, London. MR1734638Google Scholar
• Shakkottai, S, R. Srikant, A. L. Stolyar. (2004). Pathwise optimality of the exponential scheduling rule for wireless channels. Adv. Appl. Prob. 36 1021–1045. MR2119854Google Scholar
• Stewart, D.E. (2000). Rigid-body dynamics with friction and impact. SIAM review 42 (1) 3–39. MR1738097Google Scholar
• Stolyar, A. L. (2004). Maxweight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic. Ann. Appl. Prob. 14 (1) 1–53. MR2023015Google Scholar
• Teschl, G. (2009). Ordinary Differential Equations and Dynamical Systems, Universität Wien. Available online: http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf MR2961944Google Scholar
• Tezcan, T. and Dai, J. G. (2010). Dynamic control of N-systems with many servers: Asymptotic optimality of a static priority policy in heavy traffic. Operations Research, 58 (1) 94–110. MR2668929Link, Google Scholar
• Whitt, W. (1971). Weak Convergence Theorems for Priority Queues: Preemptive-Resume Discipline. Journal of Applied Probability 8 (1) 74–94. MR0307389Google Scholar
• Whitt, W. (1981). Comparing counting processes and queues. Adv. Appl. Prob. 13 (1) 207–220. MR0595895Google Scholar
• Whitt, W. (2002). Stochastic-Process Limits, New York, Springer. MR1876437Google Scholar
• Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing systems 30 (1–2) 27–88. MR1663759Google Scholar