Abstract
We consider a multihop switched network operating under a max-weight 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 independent and identically distributed arrivals, an additive SSC result over an exponential time scale.
Highlights
The subject of this paper is a new line of analysis of the maximum-weight (MW) scheduling policy for singlehop and multihop networks
We consider a multihop switched network operating under a max-weight 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
The main ingredient is a purely deterministic qualitative property of the queue dynamics: the trajectory followed by the queue vector under an MW policy tracks the trajectory of an associated deterministic fluid model within a constant multiple of the cumulative fluctuation of the arrival processes
Summary
The subject of this paper is a new line of analysis of the maximum-weight (MW) scheduling policy for singlehop and multihop networks. The main ingredient is a purely deterministic qualitative property of the queue dynamics: the trajectory followed by the queue vector under an MW policy tracks the trajectory of an associated deterministic fluid model within a constant multiple of the cumulative fluctuation of the arrival processes. The fluid model underlies a general technique for dealing with discrete-time networks: approximate the queue lengths by fluid solutions and analyze the fluid model This approach has proved useful in the study of the MW dynamics, leading to results on stability (Dai and Prabhakar 2000, Andrews et al 2004), SSC (Stolyar 2004; Dai and Lin 2005; Shah and Wischik 2006, 2012), and delay stability under heavy-tailed arrivals (Markakis et al 2016, 2018). A key ingredient behind such results is an understanding of the accuracy with which fluid solutions approximate the original queue length processes; this paper contributes to this understanding
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