Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse

We consider a queueing network in which there are constraints on which queues may be served simultaneously; such networks may be used to model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time. We consider a family of scheduling policies, related to the maximum-weight policy of Tassiulas and Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936–1948], for single-hop and multihop networks. We specify a fluid model and show that fluid-scaled performance processes can be approximated by fluid model solutions. We study the behavior of fluid model solutions under critical load, and characterize invariant states as those states which solve a certain network-wide optimization problem. We use fluid model results to prove multiplicative state space collapse. A notable feature of our results is that they do not assume complete resource pooling.

💡 Research Summary

The paper studies a broad class of constrained queueing networks that arise in input‑queued switches, wireless systems, and other settings where only certain subsets of queues can be served simultaneously. The authors focus on a family of scheduling policies that generalize the classic Maximum‑Weight (MW) rule of Tassiulas and Ephremides. In this family each queue i is assigned a weight f_i(Q_i(t)) that is a (possibly non‑linear) function of its current backlog Q_i(t); at each decision epoch the scheduler selects a feasible schedule s ∈ S (the set of admissible simultaneous service patterns) that maximizes the total weighted sum Σ_i s_i f_i(Q_i(t)). This formulation covers single‑hop and multi‑hop networks, linear and non‑linear weight functions, and arbitrary service‑constraint graphs.

The authors first develop a fluid‑scale model. By scaling time and queue lengths by a factor r and letting r → ∞, they define fluid‑scaled processes (\bar Q^r(t)=\frac{1}{r}Q(rt)). Using compactness arguments in the Skorokhod space and continuity of the scheduling map, they prove that any sequence of fluid‑scaled processes has a subsequence converging weakly to a deterministic fluid trajectory that satisfies a set of differential inclusions derived from the original stochastic dynamics. This establishes that the fluid model faithfully approximates the stochastic network under the considered policies.

Next, the paper examines the behavior of fluid solutions under critical loading, i.e., when the arrival rate vector λ lies exactly on the boundary of the network’s capacity region. They show that fluid trajectories converge to an invariant set 𝕀, which can be characterized as the set of optimal solutions to a network‑wide convex optimization problem: \