Stability for Fluid Queues: Characteristic Inequalities

Govorun, Maria; Latouche, Guy; Remiche, Marie-Ange

doi:10.1080/15326349.2013.750533

Cited by 15 publications

(28 citation statements)

References 15 publications

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“…Taking the limit as ε → 0 in (31) and replacing W by (40) leads to (6). Thus, lim ε→0 Ψ +− (ε) = Ψ, and (33) is proved.…”

Section: Remark 33mentioning

confidence: 83%

“…where |C where τ + = inf{t > 0 : Y (t) > 0}, i ∈ S − and j ∈ S + , it satisfies a Riccati equation similar to (6). The present article focuses on the perturbation analysis of Ψ only, as the analysis forΨ is similar.…”

Section: Assumption 11mentioning

confidence: 99%

“…When the mean stationary drift of the fluid model is negative, from Govorun et al [6], we have, for x > 0,…”

Section: By (55) We Havementioning

confidence: 99%

“…From Rogers [14] and Govorun et al [6], we have sp(K) ∈ {z ∈ C : Re(z) < 0} and sp(−U ) ∈ {z ∈ C : Re(z) ≥ 0}. Thus, K and −U have no common eigenvalue and, by Lancaser and Tismenetsky [10, page 414], (14) has a unique solution, so that ∂ X F (ε, X )| ε=0,X =Ψ(0) is a nonsingular operator.…”

Section: Perturbation Of the Infinitesimal Generatormentioning

confidence: 99%

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Perturbation analysis of Markov modulated fluid models

Dendievel

Latouche

2017

Stochastic Models

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We consider perturbations of positive recurrent Markov modulated fluid models. In addition to the infinitesimal generator of the phases, we also perturb the rate matrix, and analyze the effect of those perturbations on the matrix of first return probabilities to the initial level. Our main contribution is the construction of a substitute for the matrix of first return probabilities, which enables us to analyze the effect of the perturbation under consideration.

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“…Taking the limit as ε → 0 in (31) and replacing W by (40) leads to (6). Thus, lim ε→0 Ψ +− (ε) = Ψ, and (33) is proved.…”

Section: Remark 33mentioning

confidence: 83%

Section: Assumption 11mentioning

confidence: 99%

“…When the mean stationary drift of the fluid model is negative, from Govorun et al [6], we have, for x > 0,…”

Section: By (55) We Havementioning

confidence: 99%

Section: Perturbation Of the Infinitesimal Generatormentioning

confidence: 99%

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Perturbation analysis of Markov modulated fluid models

Dendievel

Latouche

2017

Stochastic Models

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“…Proof It is well established [4, Section 5] that the first return matrix G is stochastic for a recurrent QBD and substochastic for a transient one. An analogous characterization holds for a fluid queue [11,Theorem 4.5]: the matrix U is a generator for a recurrent queue and a sub-generator for a transient queue. Hence we want to show that ρ(G C ) = ρ(G D ) = ρ(W ) equals 1 if and only if U has a 0 eigenvalue.…”

Section: Mean Number Of Visits and Recurrencementioning

confidence: 99%

Doubling algorithms for stationary distributions of fluid queues: A probabilistic interpretation

Bean

Nguyen

Poloni

2018

Performance Evaluation

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Fluid queues are mathematical models frequently used in stochastic modelling. Their stationary distributions involve a key matrix recording the conditional probabilities of returning to an initial level from above, often known in the literature as the matrix Ψ. Here, we present a probabilistic interpretation of the family of algorithms known as doubling, which are currently the most effective algorithms for computing the return probability matrix Ψ.To this end, we first revisit the links described in [18,10] between fluid queues and Quasi-Birth-Death processes; in particular, we give new probabilistic interpretations for these connections. We generalize this framework to give a probabilistic meaning for the initial step of doubling algorithms, and include also an interpretation for the iterative step of these algorithms. Our work is the first probabilistic interpretation available for doubling algorithms.

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