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ON CONVERGENCE TO STATIONARITY OF FRACTIONAL BROWNIAN STORAGE
, 2009
"... With M(t): = sup s∈[0,t] A(s) − s denoting the running maximum of a fractional Brownian motion A(·) with negative drift, this paper studies the rate of convergence of P(M(t)> x) to P(M> x). We define two metrics that measure the distance between the (complementary) distribution functions P(M( ..."
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With M(t): = sup s∈[0,t] A(s) − s denoting the running maximum of a fractional Brownian motion A(·) with negative drift, this paper studies the rate of convergence of P(M(t)> x) to P(M> x). We define two metrics that measure the distance between the (complementary) distribution functions P(M(t)> ·) and P(M> ·). Our main result states that both metrics roughly decay as exp(−ϑt 2−2H), where ϑ is the decay rate corresponding to the tail distribution of the busy period in an fBmdriven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269–1293]. The proofs extensively rely on application of the wellknown large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busyperiod asymptotics holds in other settings as well, most notably when Gärtner–Ellistype conditions are fulfilled.
Bridge MonteCarlo: a novel approach to rare events of Gaussian processes
 In Proc. 5th Workshop on Simulation
, 2005
"... In this work we describe a new technique, alternative to Importance Sampling (IS), for the MonteCarlo estimation of rare events of Gaussian processes which we call Bridge MonteCarlo (BMC). This topic is relevant in teletraffic engineering where queueing systems can be fed by longrange dependent s ..."
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In this work we describe a new technique, alternative to Importance Sampling (IS), for the MonteCarlo estimation of rare events of Gaussian processes which we call Bridge MonteCarlo (BMC). This topic is relevant in teletraffic engineering where queueing systems can be fed by longrange dependent stochastic processes usually modelled through fraction Brownian motion. We show that the proposed approach has clear advantages over the widespread singletwist IS and at the same time has the same computational complexity. 1
Efficient identification of uncongested links for topological downscaling of Internetlike networks
, 2007
"... In [33, 34] two methods have been presented to scale down the topology of the Internet, while preserving important performance metrics. In partcular, based on the observation that only the congested links along the path of each flow introduce sizable queueing delays and dependencies among flows two ..."
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In [33, 34] two methods have been presented to scale down the topology of the Internet, while preserving important performance metrics. In partcular, based on the observation that only the congested links along the path of each flow introduce sizable queueing delays and dependencies among flows two methods have been proposed that can infer the performance of the larger Internet by creating and observing a suitably scaleddown replica, consisting of the congested links only. It has been demonstrated that these techniques can be used in practice to greatly simplify and expedite performance prediction. While a main requirement for topology downscaling is that uncongested links are known in advance, the question of whether one can identify them, in an efficient and scalable way, has not been addressed yet. However, this is quite important, as it is directly related to the practicability of topological downscaling. In this paper we provide simple rules that can be used to identify uncongested links. In particular, we first identify scenarios under which one can easily deduce whether a link is uncongested by inspecting the network topology. Then, we identify scenarios in which this is not possible and show how one can efficiently use known results, based on the large deviations theory, to approximate the queue length distribution. While our main motivation in this paper is to complement the work on topology downscaling, our approach is quite general and can be used beyond this context, e.g. for traffic engineering and capacity planning.
EFFICIENT SIMULATION FOR TAIL PROBABILITIES OF GAUSSIAN RANDOM FIELDS
"... We are interested in computing tail probabilities for the maxima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite number of distinct point and fields with finite KarhunenLoève expansions. For the first case we propose an importance sampling ..."
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We are interested in computing tail probabilities for the maxima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite number of distinct point and fields with finite KarhunenLoève expansions. For the first case we propose an importance sampling estimator which yields asymptotically zero relative error. Moreover, it yields a procedure for sampling the field conditional on it having an excursion above a high level with a complexity that is uniformly bounded as the level increases. In the second case we propose an estimator which is asymptotically optimal. These results serve as a first step analysis of rareevent simulation for Gaussian random fields. 1