Properties of the Reflected Ornstein–Uhlenbeck Process
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.
J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, I: Starting at the origin, Adv. in Appl. Probab. 19 (1987) 560–598.
J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, II: Non-zero initial conditions, Adv. in Appl. Probab. 19 (1987) 599–631.
S. Asmussen, Extreme value theory for queues via cycle maxima, Extremes 1(2) (1998) 137–168.
A. Berger and W. Whitt, Maximum values in queueing processes, Probab. Engrg. Inform. Sci. 9 (1995) 375–409.
A. Borovkov, Asymptotic Methods in Queueing Theory (Wiley, New York, 1984).
S. Browne and W. Whitt, Piecewise-linear diffusion processes, in: Advances in Queueing: Theory, Methods, and Open Problems, ed. J. Dshalalow (CRC Press, Boca Raton, FL, 1995) pp. 463–480.
E. Coffman, A. Puhalskii, M. Reiman and P. Wright, Processor-shared buffers with reneging, Performance Evaluation 19 (1994) 25–46.
P.W. Glynn and D. Iglehart, Trading securities using trailing stops, Managm. Sci. 41 (1995) 1096– 1106.
J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer, New York, 1991).
S. Karlin and H. Taylor, A Second Course in Stochastic Processes (Academic Press, New York, 1976).
J. Keilson, A limit theorem for passage times in ergodic regenerative processes, Ann. Math. Statist. 37 (1966) 866–870.
T. Lindvall, On coupling of diffusion processes, J. Appl. Probab. 20 (1983) 82–93.
T. Lindvall, Lectures on the coupling method, in: Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics (Wiley, New York, 1992).
P. Lions and A. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984) 511–537.
H. Rootzen, Maxima and exceedances of stationary Markov chains, Adv. in Appl. Probab. 20 (1989) 371–390.
S. Ross, Stochastic Processes (Wiley, New York, 1996).
R. Serfozo, Extreme values of birth and death processes and queues, Stochastic Process. Appl. 27 (1988) 291–306.
R. Srikant and W. Whitt, Simulation run lengths to estimate blocking probabilities, ACMTrans.Modeling Comput. Simulation 6 (1996) 7–52.
A. Ward and P. Glynn, A diffusion approximation for a GI/G/1 queue with reneging, Working paper (2002).
A. Ward and P. Glynn, A diffusion approximation for a Markovian queue with reneging, under review (2002).
- Properties of the Reflected Ornstein–Uhlenbeck Process
Volume 44, Issue 2 , pp 109-123
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- diffusion approximation
- Ornstein–Uhlenbeck process
- reflecting diffusion
- transient moment
- level crossing time
- maximum process
- Industry Sectors