Scheduling of a Generalized Switch: Heavy Traffic Regime

Abstract

We consider a generalized switch model, which is a natural model of scheduling multiple data flows over a shared time-varying wireless environment. It also includes as special cases the input-queued cross-bar switch model, and a discrete time version of a parallel server queueing system.

Input flows, n = 1,..., N, are served in discrete time by a switch. Switch state follows a finite discrete time Markov chain. In each state m, the switch chooses a scheduling decision k from a finite set K(m), which has the associated service rate vector µ ^m₁ (k),...,µ ^m_n (k).

We study the Max Weight discipline which always chooses a decision

$$ k \in \arg \mathop {\max }\limits_k \sum\limits_n {{\gamma _n}\mu _n^m(k){Q_n}} $$

where Qn’ s are the queue lengths, and γn ’s, are arbitrary positive parameters. It has been shown in previous work, that Max Weight discipline is optimal in terms of system stability, i.e. it stabilizes queues if it is feasible to do all.

We show that Max Weight also has striking optimality and “self-organizing” properties in the heavy traffic limit regime. Namely, under a non-restrictive additional conditions, Max Weight minimizes system equivalent workload X = ∑v ^*_n Q_n, where v ^*= ( ^*₁ ,..., ^*_N )is some fixed vector with positive components; moreover, in the limit, vector (γ₁ Q₁,..., γ_N Q_N) is always proportional to v ^*. These properties of Max Weight discipline can be utilized in applications to optimize various system performance criteria.