Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

Blanchet, José; Leder, Kevin; Shi, Yixi

doi:10.1287/11-ssy026

Cited by 4 publications

(3 citation statements)

References 20 publications

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“…Nevertheless, the fact is that weakly efficient estimators can only guarantee that a subexponential number of replications (as a function of N) suffice to estimate overflow probabilities of interest within a prescribed relative accuracy. On the other hand it is possible to setup an associated linear system of equations for computing overflow probabilities in Jackson networks which requires O ¡ N d ¢ variables where d is the dimension of the network (see for instance [4]). Although this implies that polynomial time algorithms are available for computing overflow probabilities in Jackson networks, it is clear that the computational burden can be substantial even for networks of moderate size.…”

Section: Introductionmentioning

confidence: 99%

Optimal Sampling of Overflow Paths in Jackson Networks

Blanchet

2013

Mathematics of OR

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We consider the problems of computing overflow probabilities at level N in any subset of stations in a Jackson network and of simulating sample paths conditional on overflow. We construct algorithms that take O (N ) function evaluations to estimate such overflow probabilities within a prescribed relative accuracy and to simulate paths conditional on overflow at level N . The algorithms that we present are optimal in the sense that the best possible performance that can be expected for conditional sampling involves Ω (N ) running time. As we explain in our development, our techniques have the potential to be applicable to more general classes of networks.

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Section: Introductionmentioning

confidence: 99%

Optimal Sampling of Overflow Paths in Jackson Networks

Blanchet

2013

Mathematics of OR

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“…Dean and Dupuis (2009) show how one can use a large deviation principle to ensure that the splitting algorithm is stable and efficiently estimates the rare event of interest. There have been several works (e.g., Dupuis 2009, 2011;Blanchet et al 2011) that have looked at rare event simulations using particle methods, but to our knowledge, our work is the first to study rare event simulations for SRBMs using any method.…”

Section: Introductionmentioning

confidence: 99%

Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions

2021

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We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.

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“…Glasserman, Heidelberger, Shahabuddin and Zajic (1999) analyse the performance of multilevel splitting techniques for rare event estimation and give, under certain conditions, the optimal degree of splitting as the probability of the event goes to 0. Multilevel splitting methods have had many applications, such as the estimation of network reliability (Botev, L'Ecuyer, Rubino, Simard and Tuffin 2013) and of rare events in Jackson networks (Blanchet, Leder and Shi 2011). Multilevel splitting techniques for rare event simulation with finite time constraints are analysed in (Jiang and Fu 2017).…”

Section: Introductionmentioning

confidence: 99%

Randomized Dimension Reduction for Monte Carlo Simulations

Kahale

2020

Management Science

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We present a new unbiased algorithm that estimates the expected value of f (U ) via Monte Carlo simulation, where U is a vector of d independent random variables, and f is a function of d variables. We assume that f does not depend equally on all its arguments. Under certain conditions we prove that, for the same computational cost, the variance of our estimator is lower than the variance of the standard Monte Carlo estimator by a factor of order d. Our method can be used to obtain a low-variance unbiased estimator for the expectation of a function of the state of a Markov chain at a given time-step. We study applications to volatility forecasting and time-varying queues. Numerical experiments show that our algorithm dramatically improves upon the standard Monte Carlo method for large values of d, and is highly resilient to discontinuities.

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Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

Cited by 4 publications

References 20 publications

Optimal Sampling of Overflow Paths in Jackson Networks

Optimal Sampling of Overflow Paths in Jackson Networks

Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions

Randomized Dimension Reduction for Monte Carlo Simulations

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