The Reflection Map with Discontinuities

References

• Abate J., Whitt W. The Fourier-series method for inverting transforms of probability distributions. Queueing Sys. (1992) 10:5–88Crossref, Google Scholar
• Beneš V. E.General Stochastic Processes in the Theory of Queues (1963) (Addison-Wesley, Reading, MA) Google Scholar
• Berger A. W., Whitt W. The Brownian approximation for rate-control throttles and the G/G/1/C queue. J. Disc. Event Dynamic Sys. (1992) 2:7–60Crossref, Google Scholar
• Billingsley P.Convergence of Probability Measures (1968) (Wiley, New York) Google Scholar
• Chen H., Mandelbaum A. Discrete flow networks: Bottleneck analysis and fluid approximation. Math. Oper. Res. (1991a) 16:408–446Link, Google Scholar
• Chen H., Mandelbaum A. Discrete flow networks: Diffusion approximations and bottlenecks. Ann. Probab. (1991b) 19:1463–1519Crossref, Google Scholar
• Chen H., Mandelbaum A., Davis M. H. A., Elliot R. J. Leontief systems, RBV's and RBM's. Proc. Imperial College Workshop on Applied Stochastic Processes (1991c) (Gordon and Breach, London) Google Scholar
• Chen H., Whitt W. Diffusion approximations for open queueing networks with service interruptions. Queueing Sys. (1993) 13:335–359Crossref, Google Scholar
• Chen H., Yao D. D.Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization (2001) (Springer, New York) Crossref, Google Scholar
• Cottle R. W., Pang J.-S., Stone R. E.The Linear Complementarity Problem (1992) (Academic, New York) Google Scholar
• Csörgő M., Rëvész P.Strong Approximations in Probability and Statistics (1981) (Academic, New York) Google Scholar
• Dupuis P., Ishii H. On when the solution to the Skorohod problem is Lipschitz continuous with applications. Stochastics (1991) 35:31–62Google Scholar
• Dupuis P., Ramanan K. Convex duality and the Skorohod problem—I. Probab. Theory Rel. Fields (1999a) 115:153–195Crossref, Google Scholar
• Dupuis P., Ramanan K. Convex duality and the Skorohod problem—II. Probab. Theory Rel. Fields (1999b) 115:197–236Crossref, Google Scholar
• Harrison J. M.Brownian Motion and Stochastic Flow Systems (1985) (Wiley, New York) Google Scholar
• Harrison J. M., Reiman M. I. Reflected Brownian motion in an orthant. Ann. Probab. (1981) 9:302–308Crossref, Google Scholar
• Harrison J. M., Williams R. J. A multiclass closed queueing network with unconventional heavy traffic behavior. Ann. Appl. Probab. (1996) 6:1–47Crossref, Google Scholar
• Iglehart D. L., Whitt W. Multiple channel queues in heavy traffic—II. Adv. Appl. Probab. (1970a) 2:150–177Crossref, Google Scholar
• Iglehart D. L., Whitt W. Multiple channel queues in heavy traffic—II. Adv. Appl. Probab. (1970b) 2:355–369Crossref, Google Scholar
• Jacod J., Shiryaev A. N.Limit Theorems for Stochastic Processes (1987) (Springer-Verlag, New York) Crossref, Google Scholar
• Kella O. Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Probab. (1993) 3:682–695Crossref, Google Scholar
• Kella O. Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Probab. (1996) 6:186–199Crossref, Google Scholar
• Kella O., Whitt W. Diffusion approximations for queues with server vacations. Adv. Appl. Probab. (1990) 22:706–729Crossref, Google Scholar
• Kella O., Whitt W., Basawa I. V., Bhat U. N. A tandem fluid network with Lévy input. Queues and Related Models (1992) (Oxford University Press, Oxford) 112–128Google Scholar
• Kella O., Whitt W. Stability and structural properties of stochastic storage networks. J. Appl. Probab. (1996) 33:1169–1180Crossref, Google Scholar
• Konstantopoulos T. The Skorohod reflection problem for functions with discontinuities (contractive case). (1999) (University of Texas, Austin) Google Scholar
• Konstantopoulos T., Lin S.-J. Fractional Brownian motion as limits of stochastic traffic models. Proceedings of the 34th Allerton Conference on Communication, Control and Computing (1996) University of Illinois:913–922Google Scholar
• Konstantopoulos T., Lin S.-J. Macroscopic models for long-range dependent network traffic. Queueing Sys. (1998) 28:215–243Crossref, Google Scholar
• Kushner H.Heavy Traffic Analysis of Controlled and Uncontrolled Queueing and Communication Networks (2001) (Springer, New York) Crossref, Google Scholar
• Lindvall T. Weak convergence of probability measures and random functions in the function space D[0, ∞). J. Appl. Probab. (1973) 10:109–121Crossref, Google Scholar
• Lions P.-L., Sznitman A.-S. Stochastic differential equations with reflecting boundary conditions. Coomun. Pure Appl. Math. (1984) 37:511–553Crossref, Google Scholar
• Mandelbaum A., Massey W. A. Strong approximations for time-dependent queues. Math. Oper. Res. (1995) 20:33–64Link, Google Scholar
• Park K., Willinger W.Self-Similar Network Traffic and Performance Evaluation (2000) (Wiley, New York) Crossref, Google Scholar
• Pollard D.Convergence of Stochastic Processes (1984) (Springer-Verlag, New York) Crossref, Google Scholar
• Puhalskii A. A., Whitt W. Functional large deviation principles for first-passage-time processes. Ann. Appl. Probab. (1997) 7:362–381Crossref, Google Scholar
• Puhalskii A. A., Whitt W. Functional large deviation principles for waiting and departure processes. Probab. Engrg. Info. Sci. (1998) 12:479–507Crossref, Google Scholar
• Reiman M. I. Open queueing networks in heavy traffic. Math. Oper. Res. (1984) 9:441–458Link, Google Scholar
• Resnick S., van den Berg E. Weak convergence of high-speed network traffic models. J. Appl. Probab. (2000) 37:575–597Crossref, Google Scholar
• Skorohod A. V. Limit theorems for stochastic processes. Theory Probab. Appl. (1956) 1:261–290Crossref, Google Scholar
• Skorohod A. V. Stochastic differential equations for a bounded region. Theory Probab. Appl. (1961) 6:264–274Crossref, Google Scholar
• Tanaka H. Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. (1979) 9:163–177Crossref, Google Scholar
• Whitt W. Preservation of rates of convergence under mappings. Zeit. Wahrscheinlichskeitstheorie verw. Gebiete (1974) 29:39–44Crossref, Google Scholar
• Whitt W. Some useful functions for functional limit theorems. Math. Oper. Res. (1980) 5:67–85Link, Google Scholar
• Whitt W. Limits for cumulative input processes to queues. Probab. Engrg. Info. Sci. (2000a) 14:123–150Crossref, Google Scholar
• Whitt W. An overview of Brownian and non-Brownian FCLTs for the single server queue. Queueing Sys. (2000b) 36:39–70Crossref, Google Scholar
• Whitt W.Stochastic-Process Limits (2001) (Springer, New York) . In preparation. http://www.research.att.com/∼wowGoogle Scholar
• Williams R. J. Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Rel. Fields (1987) 75:459–485Crossref, Google Scholar
• Williams R. J., Kelly F. P., Williams R. J. Semimartingale reflecting Brownian motions in the orthant. Stochastic Networks (1995) (Springer-Verlag, New York) 125–137Crossref, Google Scholar