Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T04:49:32.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary behavior and product-form stationary distributions of jump diffusions in the orthant with state-dependent reflections

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Francisco J. Piera ,
Ravi R. Mazumdar  and
Fabrice M. Guillemin
Francisco J. Piera*
Affiliation:
University of Chile
Ravi R. Mazumdar*
Affiliation:
University of Waterloo
Fabrice M. Guillemin*
Affiliation:
France Telecom R&D
*
∗ Postal address: Department of EE, University of Chile, Av. Tupper 2007, Santiago, 8370451, Chile. Email address: [email protected]
∗∗ Postal address: Department of ECE, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email address: [email protected]
∗∗∗ Postal address: France Telecom R&D, 2 Avenue Pierre Marzin, 22300 Lannion, France. Email address: [email protected]
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.

In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝn. We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

Keywords

Diffusionjumpsreflection mapregulator processlocal timesemimartingalestationary distributionproduct form

MSC classification

Primary: 60J60: Diffusion processes 60J75: Jump processes
Secondary: 60J50: Boundary theory 60J55: Local time and additive functionals 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 529 - 547
Copyright
Copyright © Applied Probability Trust 2008 

References

Bardhan, I. (1995). Further applications of a general rate conservation law. Stoch. Process. Appl. 60, 113–130.Google Scholar
Gamarnik, D. and Zeevi, A. (2006). Validity of heavy traffic steady state approximations in open queueing networks. Ann. Appl. Prob. 16, 56–90.CrossRefGoogle Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion in an orthant. Ann. Prob. 9, 302–308.Google Scholar
Harrison, J. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115.Google Scholar
Kella, O. and Whitt, W. (1990). Diffusion approximations for queues with server vacations. Adv. Appl. Prob. 22, 706–729.Google Scholar
Konstantopoulos, T., Last, G. and Lin, S.-J. (2004). On a class of Lévy stochastic networks. Queueing Systems 46, 409–437.Google Scholar
Kushner, H. J. (2001). Heavy Traffic Analysis of Controlled Queueing and Communication Networks (Appl. Math. 47). Springer, New York.Google Scholar
Mazumdar, R. R. and Guillemin, F. M. (1996). Forward equations for reflected diffusions with Jumps. Appl. Math. Optimization 33, 81–102.Google Scholar
Piera, F., Mazumdar, R. and Guillemin, F. (2005). On product-form stationary distributions for reflected diffusions with Jumps in the positive orthant. Adv. Appl. Prob. 37, 212–228.Google Scholar
Piera, F., Mazumdar, R. and Guillemin, F. (2006). On the local times and boundary properties of reflected diffusions with Jumps in the positive orthant. Markov Process. Relat. Fields 12, 561–582.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach (Appl. Math. 21). Springer, New York.Google Scholar
Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.Google Scholar
Reiman, M. I. and Williams, R. J. (1988). A boundary property of semimartingale reflecting Brownian motions. Prob. Theory Relat. Fields 77, 87–97.CrossRefGoogle Scholar
Rong, S. (2000). Reflecting Stochastic Differential Equations with Jumps and Applications (Research Notes Math. 408). Chapman & Hall/CRC, London.Google Scholar
Shen, X., Chen, H., Dai, J. G. and Dai, W. (2002). The finite element method for computing the stationary distribution of an SRBM in a hypercube with applications to finite buffer queueing networks. Queueing Systems Theory Appl. 42, 33–62.CrossRefGoogle Scholar
Whitt, W. (2001). The reflection map with discontinuities. Math. Operat. Res. 26, 447–484.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks (IMA Vol. Math. Appl. 71), Springer, New York, pp. 125–137.Google Scholar
Williams, R. J. (1998). Reflecting diffusions and queueing networks. In Proceedings of the International Congress of Mathematicians (Berlin, 1998), Vol. III, pp. 321–330 (electronic).Google Scholar