18 research outputs found
Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks
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 DeanDup09 that a
subsolution to the Isaacs equation guarantees that a subexponential number of
function evaluations (in n) suffice to estimate such overflow probabilities
within a given relative accuracy. Our analysis here shows that in fact
O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative
precision, where {\beta} is the number of bottleneck stations in the network.
This is the first rigorous analysis that allows to favorably compare 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.Comment: 23 page
A new, analysis-based, change of measure for tandem queues
In this paper, we introduce a simple analytical approximation for the overflow probability of a two-node tandem queue. From this, we derive a change of measure, which turns out to have good performance in almost the entire parameter space. The form of our new change of measure sheds an interesting new light on earlier changes of measure for the same problem, because here the transition zone from one measure to another - that they all have - arises naturally.\u
Efficient Rare-event Simulation for Perpetuities
We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) +
... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable
Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition
with associated root theta_{ast} in (0,infty) and that the tails of the B_j's
are appropriately behaved so that D is regularly varying with index
theta_{ast}. We illustrate by means of an example that the natural
state-independent importance sampling estimator obtained by exponentially
tilting the Y_j's according to theta_{ast} fails to provide an efficient
estimator (in the sense of appropriately controlling the relative mean squared
error as the tail probability of interest gets smaller). Then, we construct
estimators based on state-dependent importance sampling that are rigorously
shown to be efficient
Moderate deviations for recursive stochastic algorithms
We prove a moderate deviation principle for the continuous time interpolation
of discrete time recursive stochastic processes. The methods of proof are
somewhat different from the corresponding large deviation result, and in
particular the proof of the upper bound is more complicated. The results can be
applied to the design of accelerated Monte Carlo algorithms for certain
problems, where schemes based on moderate deviations are easier to construct
and in certain situations provide performance comparable to those based on
large deviations.Comment: Submitted to Stochastic System
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Rare Events in Stochastic Systems: Modeling, Simulation Design and Algorithm Analysis
This dissertation explores a few topics in the study of rare events in stochastic systems, with a particular emphasis on the simulation aspect. This line of research has been receiving a substantial amount of interest in recent years, mainly motivated by scientific and industrial applications in which system performance is frequently measured in terms of events with very small probabilities.The topics mainly break down into the following themes: Algorithm Analysis: Chapters 2, 3, 4 and 5. Simulation Design: Chapters 3, 4 and 5. Modeling: Chapter 5. The titles of the main chapters are detailed as follows: Chapter 2: Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks Chapter 3: Splitting for Heavy-tailed Systems: An Exploration with Two Algorithms Chapter 4: State Dependent Importance Sampling with Cross Entropy for Heavy-tailed Systems Chapter 5: Stochastic Insurance-Reinsurance Networks: Modeling, Analysis and Efficient Monte Carl