Optimal Sampling of Overflow Paths in Jackson Networks

Blanchet, José

doi:10.1287/moor.2013.0586

Cited by 10 publications

(22 citation statements)

References 33 publications

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“…Remark 1. The previous result can be used to efficiently estimate conditional expectations involving Ornstein-Uhlenbeck processes, conditioned on {T N < T 0 } when N is large, in a way that is analogous to the methods described in Blanchet (2013). This then leads to algorithms that have linear running time uniformly as N ↑ ∞.…”

Section: Ornstein-uhlenbeck Process Given Rare First Passage Time Eventsmentioning

confidence: 92%

A weak convergence criterion for constructing changes of measure

Blanchet

Ruf

2015

Stochastic Models

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Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes. * We thank Richard Davis, Paul Embrechts, and Thomas Mikosch for putting together a very interesting Oberwolfach seminar, where this project started. We are grateful to Kay Giesecke, Peter Glynn, Jan Kallsen, Ioannis Karatzas, and Philip Protter for stimulating conversations on topics related to the theme of this project, and to Philippe Charmoy, Zhenyu Cui, Roseline Falafala, and Nicolas Perkowski for their comments on an earlier version of this note. †

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Section: Ornstein-uhlenbeck Process Given Rare First Passage Time Eventsmentioning

confidence: 92%

A weak convergence criterion for constructing changes of measure

Blanchet

Ruf

2015

Stochastic Models

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“…In particular, P (Q j (∞) = m) = ρ m j (1 − ρ j ) for m ≥ 0. The next proposition follows directly from standard properties of the geometric distribution (see Proposition 3 in [4]).…”

Section: Jackson Network: Notation and Propertiesmentioning

confidence: 95%

“…So, we must enhance the large deviations approximations in order to provide a more precise estimate for p V n . Developing such an estimate is the aim of the following proposition which follows as a direct consequence of Proposition 2 and the analysis in Section 5 in [4]; see also Section 4 in this paper for a sketch of the proof. Proposition 1.…”

Section: Jackson Network: Notation and Propertiesmentioning

confidence: 99%

“…We believe that our results shed light into the type of performance that can be expected when applying particle algorithms beyond the setting of Jackson networks. This feature should be emphasized, specially given the fact that a linear time algorithm for computing overflows in Jackson networks has been developed very recently (see [4]). Contrary to particle methods, which are versatile and that can in principle be applied in great generality, the algorithm in [4] takes advantage of certain properties of Jackson networks which are not shared by all classes of systems.…”

Section: Introductionmentioning

confidence: 99%

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

2011

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We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in [8] that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffices to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n 2β V +1 ) function evaluations suffice to achieve a given relative precision, where βV is the number of bottleneck stations in the subset of stations under consideration in the network. This is the first rigorous analysis that favorably compares splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n d ) variables.

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“…In both articles, the so-called subsolution method is used, which is also briefly discussed in this article. Improved results are given in Reference [2], where the author focuses on optimal simulation algorithms for overflow probabilities during a busy period. Instead of using exponential twisting forward in time, the author proposes a method that goes backwards in time.…”

Section: Introductionmentioning

confidence: 99%

Estimating Large Delay Probabilities in Two Correlated Queues

Cahen

Mandjes²,

Zwart

2018

ACM Trans. Model. Comput. Simul.

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This article focuses on evaluating the probability that both components of a two-dimensional stochastic process will ever, but not necessarily at the same time, exceed some large level u. An important application is in determining the probability of large delays occurring in two correlated queues. Since exact analysis of this probability seems prohibitive, we focus on deriving asymptotics and on developing efficient simulations techniques. Large deviations theory is used to characterise logarithmic asymptotics. The second part of this article focuses on efficient simulation techniques. Using "nearest-neighbour random walk" as an example, we first show that a "naive" implementation of importance sampling, based on the decay rate, is not asymptotically efficient. A different approach, which we call partitioned importance sampling, is developed and shown to be asymptotically efficient. The results are illustrated through various simulation experiments.

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Optimal Sampling of Overflow Paths in Jackson Networks

Cited by 10 publications

References 33 publications

A weak convergence criterion for constructing changes of measure

A weak convergence criterion for constructing changes of measure

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

Estimating Large Delay Probabilities in Two Correlated Queues

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