Fluctuation Bounds for the Max-Weight Policy, with Applications to State Space Collapse

TL;DR

This paper establishes bounds on queue length deviations under Max-Weight scheduling in multi-hop networks and applies these to analyze state space collapse over various time scales.

Contribution

It provides the first bounds on queue deviations under Max-Weight and characterizes the time scales for state space collapse in different arrival scenarios.

Findings

Bounded queue length deviations proportional to arrival process deviations

Matching upper and lower bounds for state space collapse time scales

State space collapse occurs over exponential time scales for i.i.d. arrivals

Abstract

We consider a multi-hop switched network operating under a Max-Weight (MW) scheduling policy, and show that the distance between the queue length process and a fluid solution remains bounded by a constant multiple of the deviation of the cumulative arrival process from its average. We then exploit this result to prove matching upper and lower bounds for the time scale over which additive state space collapse (SSC) takes place. This implies, as two special cases, an additive SSC result in diffusion scaling under non-Markovian arrivals and, for the case of i.i.d. arrivals, an additive SSC result over an exponential time scale.