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

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. * A. Sharifnassab and S. J. Golestani are with the

Jumping Fluid Models and Delay Stability of Max-Weight Dynamics under Heavy-Tailed Traffic

“…For simplicity and ease of presentation, we restrict ourselves to single-hop networks. However, our results are easily generalized to multi-hop networks of the type considered in [18].…”

“…The literature provides a few, somewhat different but equivalent, definitions of the fluid model [16,12], which typically involve differential equations with boundary conditions. Here, we adopt an equivalent but somewhat simpler definition, 3 from [18].…”

“…Given some λ 0 and q(0) 0, there always exists a unique (and nonnegative) fluid trajectory q(•) corresponding to λ and initialized at q(0) [12,18]. 4.2.…”

“…As in [18], we assume throughout the paper that for any µ ∈ M, the set M also contains all vectors that result from setting some entries of µ to zero. This assumption is naturally valid in most contexts.…”

Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

“…where the second and third inequalities follow from the definitions of ρ min and θ * in (14) and (60), respectively. Using also the fact d p i , B ρ (p i )\B r (p i ) ≥ r, we obtain…”