Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

Bean, Nigel; O’Reilly, Małgorzata M.; Taylor, Peter G.

doi:10.1007/s11009-008-9077-3

Cited by 46 publications

(59 citation statements)

References 18 publications

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“…Note, as expected, the very significant difference for the probability densities ψ(t) 12 between the bounded and the unbounded FIGURE 5. Probability density ψ(t) 12 for the unbounded model in Example 1. …”

Section: Example 1: Letsupporting

confidence: 78%

“…We construct two different bounded processes ( M(t), ϕ(t)) with parameters The results are given in Figures 7 and 8. Note again the significant difference for the probability densities ψ(t) 12 between the bounded and the unbounded models, even though the unbounded process has a downward drift. The probability densities for the bounded models (a) and (b) vary in the expected way as well.…”

Section: The Unbounded Process (M(t) ϕ(T))mentioning

confidence: 98%

“…However, as we wish to numerically invert the Laplace-Stieltjes transforms to determine a probability density function, this is not a problem, as we FIGURE 7. Probability density ψ(t) 12 for the unbounded model in Example 2. 12 for the bounded models in Example 2: (a) dashed line, (b) solid line.…”

Section: The Unbounded Process (M(t) ϕ(T))mentioning

confidence: 99%

“…In [10], we derived results for several such measures of the fluid flow model without an upper bound. In [11,12] we described and analyzed several algorithms that can be used to calculate measures for the model in [10]. As we will see, these results are also useful for the calculation of the measures of the model introduced here.…”

Section: Introductionmentioning

confidence: 99%

“…The probability matrixP 12 , which appears in this expression, is the contribution to the Laplace-Stieltjes transform from the paths in which no time elapses.…”

mentioning

confidence: 99%

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Hitting Probabilities and Hitting Times for Stochastic Fluid Flows: The Bounded Model

Bean

O’Reilly²,

Taylor

2008

Prob. Eng. Inf. Sci.

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We consider a Markovian stochastic fluid flow model in which the fluid level has a lower bound zero and a positive upper bound. The behavior of the process at the boundaries is modeled by parameters that are different than those in the interior and allow for modeling a range of desired behaviors at the boundaries. We illustrate this with examples. We establish formulas for several time-dependent performance measures of significance to a number of applied probability models. These results are achieved with techniques applied within the fluid flow model directly. This leads to useful physical interpretations, which are presented.

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Section: Example 1: Letsupporting

confidence: 78%

Section: The Unbounded Process (M(t) ϕ(T))mentioning

confidence: 98%

Section: The Unbounded Process (M(t) ϕ(T))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The probability matrixP 12 , which appears in this expression, is the contribution to the Laplace-Stieltjes transform from the paths in which no time elapses.…”

mentioning

confidence: 99%

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Hitting Probabilities and Hitting Times for Stochastic Fluid Flows: The Bounded Model

Bean

O’Reilly²,

Taylor

2008

Prob. Eng. Inf. Sci.

Self Cite

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Busy Period, Congestion Analysis and Loss Probability in Fluid Queues

2021

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Stochastic fluid flow models and in particular those driven by Markov chains have been intensively studied in the last two decades. Not only they have been proven to be efficient tools to mimic Internet traffic flow at a macroscopic level but they are useful tools in many areas of applications such as manufacturing systems or in actuarial sciences to cite but a few. This chapter proposes to focus on such a model in the context of performance analysis of a potentially congested system. The latter is modeled by means of a finite-capacity system whose content is described by a Markov driven stable fluid flow. We step-by-step describe a methodology to compute exactly the loss probability of the system. Our approach is based on the computation of hitting probabilities jointly with the peak level reached during a busy period, both in the infinite and finite buffer case. Accordingly we end up with differential Riccati equations that can be solved numerically. Moreover we are able to characterize the complete distribution of both the duration of congestion and of the total information lost during such a busy period.

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Two-Dimensional Fluid Queues with Temporary Assistance

Latouche

Nguyen

Palmowski

2012

Matrix-Analytic Methods in Stochastic Models

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We consider a two-dimensional stochastic fluid model with N ON-OFF inputs and temporary assistance, which is an extension of the same model with N = 1 in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures.We derive the marginal probability distribution of Buffer 1, and bounds for that of Buffer 2. Furthermore, restricting Buffer 1 to a finite size, we determine its marginal probability distribution in the specific case of N = 1, thus providing numerical comparisons to the corresponding results in Mahabhashyam et al. (2008) where Buffer 1 is assumed to be infinite.

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Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

Cited by 46 publications

References 18 publications

Hitting Probabilities and Hitting Times for Stochastic Fluid Flows: The Bounded Model

Hitting Probabilities and Hitting Times for Stochastic Fluid Flows: The Bounded Model

Busy Period, Congestion Analysis and Loss Probability in Fluid Queues

Two-Dimensional Fluid Queues with Temporary Assistance

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