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

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations Special 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: [email protected] 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: [email protected] 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: [email protected]
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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 

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