76 research outputs found
Fast simulation of the leaky bucket algorithm
We use fast simulation methods, based on importance sampling, to efficiently estimate cell loss probability in queueing models of the Leaky Bucket algorithm. One of these models was introduced by Berger (1991), in which the rare event of a cell loss is related to the rare event of an empty finite buffer in an "overloaded" queue. In particular, we propose a heuristic change of measure for importance sampling to efficiently estimate the probability of the rare empty-buffer event in an asymptotically unstable GI/GI/1/k queue. This change of measure is, in a way, "dual" to that proposed by Parekh and Walrand (1989) to estimate the probability of a rare buffer overflow event. We present empirical results to demonstrate the effectiveness of our fast simulation method. Since we have not yet obtained a mathematical proof, we can only conjecture that our heuristic is asymptotically optimal, as k/spl rarr//spl infin/
Sample-path Large Deviations in Credit Risk
The event of large losses plays an important role in credit risk. As these
large losses are typically rare, and portfolios usually consist of a large
number of positions, large deviation theory is the natural tool to analyze the
tail asymptotics of the probabilities involved. We first derive a sample-path
large deviation principle (LDP) for the portfolio's loss process, which enables
the computation of the logarithmic decay rate of the probabilities of interest.
In addition, we derive exact asymptotic results for a number of specific
rare-event probabilities, such as the probability of the loss process exceeding
some given function
Large Deviations without Principle: Join the Shortest Queue
We develop a methodology for studying "large deviations type" questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a larg class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this sytem may exhibit unexpected behavior
On convergence to stationarity of fractional Brownian storage
With denoting the running maximum of a
fractional Brownian motion with negative drift, this paper studies
the rate of convergence of to . We
define two metrics that measure the distance between the (complementary)
distribution functions and . Our
main result states that both metrics roughly decay as , where is the decay rate corresponding to the tail
distribution of the busy period in an fBm-driven queue, which was computed
recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs
extensively rely on application of the well-known large deviations theorem for
Gaussian processes. We also show that the identified relation between the decay
of the convergence metrics and busy-period asymptotics holds in other settings
as well, most notably when G\"artner--Ellis-type conditions are fulfilled.Comment: Published in at http://dx.doi.org/10.1214/08-AAP578 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotic analysis of Levy-driven tandem queues
We analyze tail asymptotics of a two-node tandem queue with spectrally-positive L\'evy input. A first focus lies on tail probabilities of the type , for and large, and denoting the steady-state workload in the th queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize to the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed L\'evy input. It is also indicated how the results can be extended to tandem queues with more than two nodes
Sojourn time asymptotics in processor sharing queues
This paper addresses the sojourn time asymptotics for a GI/GI/β’ queue operating under the
Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the
logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize
distribution has a light tail. Whereas upper bounds on the decay rate can be derived under
fairly general conditions, the establishment of the corresponding lower bounds requires that the
service process satisfies a samplepath large-deviation principle. We show that the class of
allowed service processes includes the case where the service rate is modulated by a Markov
process. Finally, we extend our results to a similar system operation under the Discriminatory
Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations
techniques
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