Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T05:21:47.115Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Generalized Drift Skorokhod Problem in One Dimension

Part of: Stochastic processes Operations research and management science Markov processes

Published online by Cambridge University Press:  30 January 2018

Josh Reed ,
Amy Ward  and
Dongyuan Zhan
Josh Reed*
Affiliation:
New York University
Amy Ward*
Affiliation:
University of Southern California
Dongyuan Zhan*
Affiliation:
University of Southern California
*
∗ Postal address: Stern School of Business, New York University, New York, NY 10012, USA. Email address: [email protected]
∗∗ Postal address: Marshall School of Business, University of Southern California, Los Angeles, CA 90089-0809, USA.
∗∗ Postal address: Marshall School of Business, University of Southern California, Los Angeles, CA 90089-0809, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

Keywords

Skorokhod mapreflected Ornstein–Uhlenbeck processabandonmentqueueing

MSC classification

Primary: 90B22: Queues and service
Secondary: 90B15: Network models, stochastic 60G17: Sample path properties 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 16 - 28
Copyright
Copyright © Applied Probability Trust 2013 

References

Alili, L., Patie, P. and Pedersen, J. L. (2010). Representations of the first hitting time density of an Ornstein-Uhlenbeck process. Preprint.Google Scholar
Anantharam, V. and Konstantopoulos, T. (2011). Integral representation of Skorokhod reflection. Proc. Amer. Math. Soc. 139, 2227–2237.CrossRefGoogle Scholar
Burdzy, K., Kang, W. and Ramanan, K. (2009). The Skorokhod problem in a time-dependent interval. Stoch. Process. Appl. 119, 428–452.CrossRefGoogle Scholar
Chaleyat-Maurel, M., El Karoui, N. and Marchal, B. (1980). Reflexion discontinue et systèmes stochastiques. Ann. Prob. 8, 1049–1067.CrossRefGoogle Scholar
Cox, J. T. and Rosler, U. (1984). A duality relation for entrance and exit laws for Markov processes. Stoch. Process. Appl. 16, 141–156.Google Scholar
Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stoch. Stoch. Reports 35, 31–62.CrossRefGoogle Scholar
Fralix, B. H. (2012). On the time-dependent moments of Markovian queues with reneging. Queueing Systems 20pp.Google Scholar
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.CrossRefGoogle Scholar
Kruk, Ł., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0,a]. Ann. Prob. 35, 1740–1768.Google Scholar
Kruk, Ł., Lehoczky, J., Ramanan, K. and Shreve, S. (2008). Double Skorokhod map and reneging real-time queues. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4), Institute of Mathematical Statistics, Beachwood, OH, pp. 169–193.CrossRefGoogle Scholar
Linetsky, V. (2004). Computing hitting time densities for OU and CIR processes: applications to mean-reverting models. J. Comput. Finance 7, 1–22.CrossRefGoogle Scholar
Pitman, J. and Yor, M. (1981). Bessel processes and infinitely divisible laws. In Stochastic Integrals (Lecture Notes Math. 851), Springer, Berlin, pp. 285–370.CrossRefGoogle Scholar
Ramanan, K. (2006). Reflected diffusions via the extended Skorokhod map. Electron. J. Prob. 36, 934–992.Google Scholar
Sigman, K. and Ryan, R. (2000). Continuous-time monotone stochastic recursions and duality. Adv. Appl. Prob. 32, 426–445.CrossRefGoogle Scholar
Skorokhod, A. V. (1961). Stochastic equations for diffusions in a bounded region. Theoret. Prob. Appl. 6, 264–274.CrossRefGoogle Scholar
Srikant, R. and Whitt, W. (1996). Simulation run lengths to estimate blocking probabilities. ACM Trans. Model. Comput. Simul. 6, 7–52.CrossRefGoogle Scholar
Tanaka, H. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math J. 9, 163–177.CrossRefGoogle Scholar
Ward, A. R. and Glynn, P. W. (2003). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43, 103–128.Google Scholar
Ward, A. R. and Glynn, P. W. (2003). Properties of the reflected Ornstein–Uhlenbeck process. Queueing Systems 44, 109–123.CrossRefGoogle Scholar
Ward, A. R. and Glynn, P. W. (2005). A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Systems 50, 371–400.Google Scholar
Whitt, W. (1999). Improving service by informing customers about anticipated delays. Manag. Sci. 45, 192–207.CrossRefGoogle Scholar
Zhang, T. S. (1994). On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary. Stoch. Process. Appl. 50, 135–147.Google Scholar