Nonclassical stochastic flows and continuous products

Tsirelson, Boris

doi:10.1214/154957804100000042

Cited by 50 publications

(92 citation statements)

References 36 publications

(58 reference statements)

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“…(24) In this section we identify the distribution of observed idle (and busy) periods (see Propositions 5 and 6, below). In both cases we shall appeal to the result of Lemma 2, below, a short proof of which is provided.…”

Section: The Intervals (G(n) D(n)) Are Called Idle Periods While Thmentioning

confidence: 99%

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Analysis of stochastic fluid queues driven by local-time processes

Konstantopoulos¹,

Kyprianou

Sirvio

et al. 2008

Adv. Appl. Probab.

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We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

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Section: The Intervals (G(n) D(n)) Are Called Idle Periods While Thmentioning

confidence: 99%

“…We are interested in the distribution of the observed busy period, as defined in (24). On the conditioning event {Q 0 > 0}, we have, by our enumeration convention,…”

Section: Observed Busy Periodsmentioning

confidence: 99%

Analysis of stochastic fluid queues driven by local-time processes

Konstantopoulos¹,

Kyprianou

Sirvio

et al. 2008

Adv. Appl. Probab.

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“…Two examples found by Warren [8], [9] are 'the noise of splitting' and 'the noise of stickiness'; see also [10] and [7,Sect. 4].…”

Section: 5mentioning

confidence: 99%

Probabilistic approach

Dynkin

2004

Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations

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“…Several non-exclusive approaches to this problem have been proposed and pursued in various settings, out of which we emphasize W. Arveson's operator algebraic approach using representations of spectral C * -algebras [5], R.T. Powers' analytic approach using CP -flows [11], and B. Tsirelson's probabilistic approach using random sets [14] (see also [10]) and nonclassical stochastic flows [15].…”

Section: Introductionmentioning

confidence: 99%

The enveloping E-semigroup of a product system

Floricel

2015

Journal of Mathematical Analysis and Applications

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It is shown that the enveloping von Neumann algebra of the spectral C * -algebra of a product system admits an E-semigroup whose conjugacy class is a complete isomorphism invariant for the product system. This semigroup, called the enveloping E-semigroup, displays a Wold-type decomposition that reduces the classification problem of product systems to classification of the constituent semigroups of this decomposition.

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Nonclassical stochastic flows and continuous products

Cited by 50 publications

References 36 publications

Analysis of stochastic fluid queues driven by local-time processes

Analysis of stochastic fluid queues driven by local-time processes

Probabilistic approach

The enveloping E-semigroup of a product system

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