Optimal queue-size scaling in switched networks

We consider a switched (queuing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. In the main result of this paper, we provide a new class of online scheduling policies that achieve optimal queue-size scaling for a class of switched networks including input-queued switches. In particular, it establishes the validity of a conjecture (documented in Shah, Tsitsiklis and Zhong [Queueing Syst. 68 (2011) 375-384]) about optimal queue-size scaling for input-queued switches.

💡 Research Summary

The paper studies a broad class of single‑hop switched networks—systems in which a finite set of service schedules S ⊂ {0,1}^N determines which queues can be served simultaneously. Each of the N queues receives unit‑size packets according to independent Poisson processes with rates λ_i, and the scheduling decision at each discrete time slot must be made online, i.e., based only on past arrivals and past service decisions. The authors first formalize the admissible region C as the set of arrival rate vectors that can be supported by convex combinations of schedules in S, and define the load ρ(λ) as the optimal value of a static linear program (PRIMAL(λ)). Under the monotonicity assumption on S, C coincides with the convex hull of S, and ρ(λ) can be expressed as the maximum over a small number J of linear constraints: ρ(λ)=max_j (∑i R{ji} λ_i / C_j).

The central contribution is a two‑stage online scheduling mechanism that achieves the optimal scaling of the average total queue length with respect to both the number of queues N and the distance to capacity 1−ρ. The first stage introduces an “error‑bounded” discretization of a continuous‑time bandwidth allocation process. It maintains a virtual continuous‑time queueing system and monitors the gap between the virtual and the actual discrete‑time queues; whenever the gap exceeds a pre‑specified bound, the scheduler injects a corrective service schedule. This ensures that the discrete system tracks the ideal continuous system within a bounded error.

The second stage implements the Store‑and‑Forward Allocation (SFA) policy, originally studied in the context of insensitive queueing networks. SFA has the remarkable property that its stationary distribution is product‑form and does not depend on the detailed service‑time distribution (insensitivity). In the equivalent product‑form network each class behaves like an M/M/1 queue with effective load ρ/(1−ρ). By emulating this network within the switched system, the authors obtain explicit bounds on the stationary moments of the queue lengths.

Combining the two stages yields a rigorous bound on the average total queue size: \