Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T03:33:42.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundary

Part of: Probabilistic methods, simulation and stochastic differential equations Special processes Stochastic processes

Published online by Cambridge University Press:  12 July 2019

Jose Blanchet ,
Jing Dong  and
Zhipeng Liu
Jose Blanchet*
Affiliation:
Stanford University
Jing Dong*
Affiliation:
Columbia University
Zhipeng Liu*
Affiliation:
Columbia University
*
*Postal address: Stanford University, 475 Via Ortega, Stanford, CA 94305, USA. Email address: jose.blanchet@stanford.edu Support from NSF grant DMS-132055 and NSF grant CMMI-1538217 is gratefully acknowledged.
**Postal address: Decision, Risk, and Operations Division, Columbia University, 3022 Broadway, New York, NY 10027, USA. Email address: jing.dong@gsb.columbia.edu Support from NSF grant DMS-1720433 is gratefully acknowledged.
***Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 20017, USA. Email address: zl2337@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the first algorithm that samples maxn≥0{Sn − nα}, where Sn is a mean zero random walk, and nα with $\alpha \in ({1 \over 2},1)$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean.

Keywords

Exact simulationMonte Carloqueueing theoryrandom walk

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60K25: Queueing theory 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 116 - 138
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn, Springer, New York.Google Scholar
Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Probab. 25 (6), 3209–3250.CrossRefGoogle Scholar
Blanchet, J. and Dong, J. (2014). Perfect sampling for infinite server and loss systems. Adv. Appl. Prob. 47 (3), 761–786.CrossRefGoogle Scholar
Blanchet, J. and Wallwater, A. (2015). Exact sampling for the steady-state waiting times of a heavy-tailed single server queue. ACM Trans. Model. Comput. Simul. 25 (4), art. 26.Google Scholar
Blanchet, J., Dong, J. and Pei, Y. (2018). Perfect sampling of GI/GI/c queues. Queueing Systems 90 (1–2), 1–33.CrossRefGoogle Scholar
Ensor, K. and Glynn, P. (2000). Simulating the maximum of a random walk. J. Statist. Plann. Inference 85, 127–135.CrossRefGoogle Scholar
Ganesh, A., O’Connell, N. and Wischik, D. (2004). Big Queues (Lecture Notes Math. 1838). Springer, Berlin.CrossRefGoogle Scholar
Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000 (Lecture Notes Statist. 128), pp. 218–234. Springer, New York.CrossRefGoogle Scholar
Propp, J. and Wilson, D. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar