Fluid Model for a Data Network with α -Fair Bandwidth Sharing and General Document Size Distributions: Two Examples of Stability

Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

Stochastic Systems ◽

10.1287/stsy.2019.0058 ◽

2020 ◽

Vol 10 (3) ◽

pp. 251-273

Author(s):

Yingjia Fu ◽

Ruth J. Williams

Keyword(s):

Lyapunov Function ◽

Fluid Model ◽

Model Solution ◽

Size Distributions ◽

File Size ◽

Bandwidth Sharing ◽

Initial State ◽

Model Solutions ◽

Slight Generalization ◽

Subcritical Fluid

This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth-sharing policy. Here we consider fair bandwidth-sharing policies that are a slight generalization of the [Formula: see text]-fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in time and space that they are strong solutions of a partial differential equation and assuming that no fluid level on any route touches zero before all route levels reach zero. The aim of the current paper is to prove stability of the subcritical fluid model without these strong assumptions. Starting with a slight generalization of the Lyapunov function proposed by Paganini et al., assuming that each component of the initial state of a measure-valued fluid model solution, as well as the file size distributions, have no atoms and have finite first moments, we prove absolute continuity in time of the composition of the Lyapunov function with any subcritical fluid model solution and describe the associated density. We use this to prove that the Lyapunov function composed with such a subcritical fluid model solution converges to zero as time goes to infinity. This implies that each component of the measure-valued fluid model solution converges vaguely on [Formula: see text] to the zero measure as time goes to infinity. Under the further assumption that the file size distributions have finite pth moments for some p > 1 and that each component of the initial state of the fluid model solution has finite pth moment, it is proved that the fluid model solution reaches the measure with all components equal to the zero measure in finite time and that the time to reach this zero state has a uniform bound for all fluid model solutions having a uniform bound on the initial total mass and the pth moment of each component of the initial state. In contrast to the analysis of Paganini et al., we do not need their strong smoothness assumptions on fluid model solutions and we rigorously treat the realistic, but singular situation, where the fluid level on some routes becomes zero, whereas other route levels remain positive.

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Fluid model for a network operating under a fair bandwidth-sharing policy

The Annals of Applied Probability ◽

10.1214/105051604000000224 ◽

2004 ◽

Vol 14 (3) ◽

pp. 1055-1083 ◽

Cited By ~ 70

Author(s):

F. P. Kelly ◽

R. J. Williams

Keyword(s):

Fluid Model ◽

Bandwidth Sharing

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Fluid limits for networks with bandwidth sharing and general document size distributions

The Annals of Applied Probability ◽

10.1214/08-aap541 ◽

2009 ◽

Vol 19 (1) ◽

pp. 243-280 ◽

Cited By ~ 32

Author(s):

H. Christian Gromoll ◽

Ruth J. Williams

Keyword(s):

Size Distributions ◽

Bandwidth Sharing ◽

Fluid Limits

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Numerical Simulation of Heterogeneous Bubbly Flow in a Bubble Column

Volume 2: Fora ◽

10.1115/fedsm2006-98178 ◽

2006 ◽

Author(s):

Munenori Maekawa ◽

Naoki Shimada ◽

Kouji Kinoshita ◽

Akira Sou ◽

Akio Tomiyama

Keyword(s):

Size Distribution ◽

Bubble Size ◽

Bubble Column ◽

Fluid Model ◽

Bubbly Flow ◽

Bubble Size Distribution ◽

Bubbly Flows ◽

Bubble Coalescence ◽

Size Distributions ◽

Bubble Sizes

Numerical methods for predicting heterogeneous bubbly flows are indispensable for the design of a Fisher-Tropsh reactor for GTL (Gas To Liquid). It is necessary to take into account bubble size distribution determined by bubble coalescence and breakup for the accurate prediction of heterogeneous bubbly flows. Hence we implemented several bubble coalescence and breakup models into the (N+2) field model, which is a hybrid combination of an interface tracking method and a multi-fluid model. Void and bubble size distributions in an open rectangular bubble column were measured and compared with predicted ones. As a result, the following conclusions were obtained: (1) Void and bubble size distributions were not affected by inlet bubble sizes because the bubble size distribution reaches an equilibrium state at which the birth rate is equal to the death rate, and (2) the combination of Luo’s bubble breakup model and a coalescence model consisting of Prince & Blanch’s model and Wang’s wake entrainment model gave good predictions.

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