State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

Abstract

We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185--201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted $\alpha$-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] [$\alpha \in (0,\infty )$]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.

Published in at http://dx.doi.org/10.1214/08-AAP591 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Structural properties of proportional fairness: stability and insensitivity
• Fluid model for a network operating under a fair bandwidth-sharing policy
• Fluid limits for networks with bandwidth sharing and general document size distributions
• An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries
• Network stability under max--min fair bandwidth sharing

Cited by in corpus (28)

• Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse
• Optimal queue-size scaling in switched networks
• Proportional fairness and its relationship with multi-class queueing networks
• Transform Methods for Heavy-Traffic Analysis
• An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries
• Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems
• Qualitative properties of $\alpha$-fair policies in bandwidth-sharing networks
• Load balancing system under Join the Shortest Queue: Many-Server-Heavy-Traffic Asymptotics
• Existence and uniqueness of reflecting diffusions in cusps
• Network stability under max--min fair bandwidth sharing
• Heavy traffic on a controlled motorway
• A multi-dimensional SRBM: Geometric views of its product form stationary distribution
• Markov selection for constrained martingale problems
• A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions
• Heavy-Traffic Insensitive Bounds for Weighted Proportionally Fair Bandwidth Sharing Policies
• Localization for constrained martingale problems and optimal conditions for uniqueness of reflecting diffusions in 2-dimensional domains
• Control Policies Approaching HGI Performance in Heavy Traffic for Resource Sharing Networks
• Decomposable stationary distribution of a multidimensional SRBM
• Multidimensional Sticky Brownian Motions: Tail Behaviour of the Joint Stationary Distribution
• Continuity and monotonicity of solutions to a greedy maximization problem
• Heavy Traffic Approximation of Equilibria in Resource Sharing Games
• Asymptotically Tight Steady-State Queue Length Bounds Implied By Drift Conditions
• Dynamics of the supermarket model
• Stability of MaxWeight-(α,g)
• Store-Forward and its implications for Proportional Scheduling
• Fluid Limits for Bandwidth-Sharing Networks with Rate Constraints
• Obliquely reflecting Brownian motion in nonpolyhedral, piecewise smooth cones, with an example of application to diffusion approximation of bandwidth sharing queues
• Network iso-elasticity and weighted $\alpha$-fairness