Two-Dimensional Fluid Queues with Temporary Assistance

Latouche, Guy; Nguyen, Giang T.; Palmowski, Zbigniew

doi:10.1007/978-1-4614-4909-6_9

Cited by 5 publications

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Latouche and Nguyen [38,39] are two recent papers that follow a regenerative approach similar to the one developed here. Two-dimensional fluid A few authors have considered systems where the component X is two-dimensional: Bean and O'Reilly [10,11], Foss and Miyazawa [29] and Latouche et al [40] among others. The area of two-or higher-dimensional fluids is still wide open, with many exciting unanswered questions.…”

Section: Finite Buffersmentioning

confidence: 99%

Analysis of fluid flow models

Latouche¹,

Nguyen²

2018

Preprint

Self Cite

View full text Add to dashboard Cite

Markov-modulated fluids have a long history. They form a simple class of Markov additive processes, and were initially developed in the 1950s as models for dams and reservoirs, before gaining much popularity in the 1980s as models for buffers in telecommunication systems, when they became known as fluid queues. More recent applications are in risk theory and in environmental studies.In telecommunication systems modelling, the attention focuses on determining the stationary distribution of the buffer content. Early ODE resolution techniques have progressively given way to approaches grounded in the analysis of the physical evolution of the system, and one only needs now to solve a Riccati equation in order to obtain several quantities of interest. To the early algorithms proposed in the Applied Probability literature, numerical analysts have added new algorithms, improved in terms of convergence speed, numerical accuracy, and domain of applicability.We give here a high-level presentation of the matrix-analytic approach to the analysis of fluid queues, briefly address computational issues, and conclude by indicating how this has been extended to more general processes.

show abstract

Section: Finite Buffersmentioning

confidence: 99%

Analysis of fluid flow models

Latouche¹,

Nguyen²

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

A Discontinuous Galerkin Method for Approximating the Stationary Distribution of Stochastic Fluid-Fluid Processes

Bean¹,

Nguyen²,

O’Reilly³

et al. 2022

Methodol Comput Appl Probab

View full text Add to dashboard Cite

The stochastic fluid-fluid model (SFFM) is a Markov process $$\{(X_t,Y_t,\varphi _t),t\ge 0\}$$ { ( X t , Y t , φ t ) , t ≥ 0 } , where $$\{\varphi _t,{t\ge 0}\}$$ { φ t , t ≥ 0 } is a continuous-time Markov chain, the first fluid, $$\{X_t,t\ge 0\}$$ { X t , t ≥ 0 } , is a classical stochastic fluid process driven by $$\{\varphi _t,t\ge 0\}$$ { φ t , t ≥ 0 } , and the second fluid, $$\{Y_t,t\ge 0\}$$ { Y t , t ≥ 0 } , is driven by the pair $$\{(X_t,\varphi _t),t\ge 0\}$$ { ( X t , φ t ) , t ≥ 0 } . Operator-analytic expressions for the stationary distribution of the SFFM, in terms of the infinitesimal generator of the process $$\{(X_t,\varphi _t),t\ge 0\}$$ { ( X t , φ t ) , t ≥ 0 } , are known. However, these operator-analytic expressions do not lend themselves to direct computation. In this paper the discontinuous Galerkin (DG) method is used to construct approximations to these operators, in the form of finite dimensional matrices, to enable computation. The DG approximations are used to construct approximations to the stationary distribution of the SFFM, and results are verified by simulation. The numerics demonstrate that the DG scheme can have a superior rate of convergence compared to other methods.

show abstract