[Paper Review] Heavy-Traffic Insensitive Bounds for Weighted Proportionally Fair Bandwidth Sharing Policies

Weina Wang, Siva Theja Maguluri+2 more|arXiv (Cornell University)|Aug 6, 2018

Network Traffic and Congestion Control30 references3 citations

TL;DR

This paper proposes a Lyapunov-drift-based method to derive explicit, heavy-traffic insensitive bounds on the weighted sum of expected flow counts in bandwidth-sharing networks under weighted proportionally fair policies. It establishes multiplicative state-space collapse in steady state, yielding bounds linear in the number of critically loaded links and independent of phase-type file size distributions, while also implying interchange of limits for diffusion approximations.

ABSTRACT

We consider a connection-level model proposed by Massoulié and Roberts for bandwidth sharing among file transfer flows in a communication network. We study weighted proportionally fair sharing policies and establish explicit-form bounds on the weighted sum of the expected numbers of flows on different routes in heavy traffic. The bounds are linear in the number of critically loaded links in the network, and they hold for a class of phase-type file-size distributions; i.e., the bounds are heavy-traffic insensitive to the distributions in this class. Our approach is Lyapunov-drift based, which is different from the widely used diffusion approximation approach. A key technique we develop is to construct a novel inner product in the state space, which then allows us to obtain a multiplicative type of state-space collapse in steady state. Furthermore, this state-space collapse result implies the interchange of limits as a by-product for the diffusion approximation of the equal-weight case under phase-type file-size distributions, demonstrating the heavy-traffic insensitivity of the stationary distribution.

Motivation & Objective

• To analyze the steady-state performance of weighted proportionally fair bandwidth sharing policies in heavy-traffic networks.
• To derive explicit, tight bounds on the weighted sum of expected flow counts on different routes.
• To establish heavy-traffic insensitivity of these bounds with respect to phase-type file size distributions.
• To demonstrate the interchange of limits for diffusion approximations via state-space collapse.
• To show the applicability of the drift method in complex bandwidth-sharing models where diffusion approximations fail to yield explicit results.

Proposed method

• A novel inner product is constructed in the state space to enable multiplicative-type state-space collapse in steady state.
• The Lyapunov-drift method is applied directly to analyze the steady-state behavior of the system, avoiding reliance on diffusion approximations.
• Tight tail bounds on the expected number of flows are derived using the state-space collapse result.
• Prokhorov’s theorem is used to establish weak convergence of the scaled steady-state process by proving tightness.
• The convergence of subsequences is analyzed by showing that the perpendicular component of the state vector vanishes in the limit.
• The interchange of limits is established by combining state-space collapse with convergence of the diffusion-scaled process to a limiting diffusion.

Experimental results

Research questions

• RQ1 Can explicit, closed-form bounds be derived for the weighted sum of expected flow counts in weighted proportionally fair bandwidth sharing under heavy traffic?
• RQ2 Are these bounds insensitive to the specific form of file size distributions, provided they are phase-type?
• RQ3 Can the drift method be used to establish state-space collapse and performance bounds in systems where diffusion approximations are intractable?
• RQ4 Does the state-space collapse result imply the interchange of limits for diffusion approximations in the equal-weight case?
• RQ5 Can the Lyapunov-drift approach yield stronger results than the standard diffusion approximation framework in this setting?

Key findings

• The paper establishes explicit upper and lower bounds on the weighted sum of expected flow counts that are linear in the number of critically loaded links.
• These bounds are heavy-traffic insensitive, meaning their dominant terms do not depend on the specific phase-type distribution of file sizes.
• A multiplicative-type state-space collapse is proven in steady state, showing that the perpendicular component of the state vector vanishes as the system approaches heavy traffic.
• The state-space collapse result implies the interchange of limits for the diffusion approximation of the equal-weight case, validating the use of diffusion limits.
• The Lyapunov-drift method successfully yields explicit bounds in settings where diffusion approximation fails to produce closed-form results.
• The convergence of the scaled steady-state process to the limiting diffusion is established via tightness and subsequence analysis, confirming the validity of the approximation.