Maximum Level and Hitting Probabilities in Stochastic Fluid Flows Using Matrix Differential Riccati Equations

Séricola, Bruno; Remiche, Marie-Ange

doi:10.1007/s11009-009-9149-z

Cited by 7 publications

(13 citation statements)

References 21 publications

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“…For this model, they derive a system of first-order non-homogeneous linear differential equations for the mean passage time. Sericola and Remiche [22] propose a method to analyse the maximum level and the hitting probabilities in a Markov driven fluid queue for various initial condition scenarios, allowing for both finite and infinite buffers. Their analysis leads to matrix differential Ricatti equations for which there is a unique solution.…”

Section: ϕ(T)mentioning

confidence: 99%

“…Our analysis proceeds as follows: We use results from [22] for the analysis of the maximum in a busy period. Furthermore, we show that the busy period maximum has an exponential tail and the maximum grows logarithmically.…”

Section: ϕ(T)mentioning

confidence: 99%

“…In Sericola and Remiche [22] the distribution of the maximum level reached in a busy period is derived using matrix exponential forms. The resulting equations are rewritten such that they can be transformed into matrix differential Riccati equations.…”

Section: Maximum Over Busy Cyclesmentioning

confidence: 99%

“…The joint distribution of the maximum in a busy period Ψ i,j (x) is calculated by solving a matrix differential Riccati equation [22]. Function Ψ i,j (x) can be expressed in terms of the matrix exponential form of matrix Q:…”

Section: Maximum Over Busy Cyclesmentioning

confidence: 99%

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A spectral theory approach for extreme value analysis in a tandem of fluid queues

Bosman

Núñez-Queija

2014

Queueing Syst

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We consider a model for to evaluate performance of streaming media over an unreliable network. Our model consists of a tandem of two fluid queues. The first fluid queue is a Markov Modulated fluid queue that models the network congestion, and the second queue represents the play-out buffer. For this model the distribution of the total amount of fluid in the congestion and play-out buffer corresponds to the distribution of the maximum attained level of the first buffer. We show that, under proper scaling and when we let time go to infinity, the distribution of the total amount of fluid converges to a Gumbel extreme value distribution. From this result, we derive a simple closed-form expression for the initial play-out buffer level that provides a probabilistic guarantee for undisturbed play-out. Motivation and literatureOver the past few years, the tremendous popularity of smart phone end devices and services (like Youtube) has boosted the demand for streaming media applications offered via the Internet. One of the key requirements for the success

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Section: ϕ(T)mentioning

confidence: 99%

Section: ϕ(T)mentioning

confidence: 99%

Section: Maximum Over Busy Cyclesmentioning

confidence: 99%

Section: Maximum Over Busy Cyclesmentioning

confidence: 99%

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A spectral theory approach for extreme value analysis in a tandem of fluid queues

Bosman

Núñez-Queija

2014

Queueing Syst

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“…Our approach is based on the analysis of level crossings during a vacation+service cycle. The analysis of level crossings in stochastic fluid models is introduced by the matrix analytic method community [8,9,14]. We apply several existing level crossing results from those works, but the analysis of fluid vacation model requires the introduction and analysis of previously not considered level crossing measures.…”

Section: Introductionmentioning

confidence: 99%

Exhaustive fluid vacation model with positive fluid rate during service

Horvth

Telek

2015

Performance Evaluation

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Using an elegant transform domain iterative operator approach exhaustive fluid vacation models with strictly negative fluid rate during service has been analyzed, recently. Unfortunately, the potential presence of positive fluid rate during service (when the fluid input rate is larger than the fluid service rate) inhibits the use of all previously applied methodologies and makes the extension of the analysis approaches of discrete vacation and polling models toward fluid vacation and polling models rather difficult.Based on the level crossing analysis of Markov fluid models the paper introduces an analysis approach which is applicable for the stationary analysis of the fluid level distribution and its moments. In the course of the analysis compact new matrix exponential expressions are obtained for the distribution of fluid level during a busy period starting from a given positive fluid level and by exploiting the relations of upward and downward measures of fluid processes several matrix transformations are applied to avoid Kronecker expansion matrix multiplications. Finally, the obtained fluid level distribution is related with previous results of fluid vacation models.

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On the overflow time of a fluid model

Löpker

2016

Math Meth Oper Res

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Maximum Level and Hitting Probabilities in Stochastic Fluid Flows Using Matrix Differential Riccati Equations

Cited by 7 publications

References 21 publications

A spectral theory approach for extreme value analysis in a tandem of fluid queues

A spectral theory approach for extreme value analysis in a tandem of fluid queues

Exhaustive fluid vacation model with positive fluid rate during service

On the overflow time of a fluid model

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