Delay reduction via Lagrange multipliers in stochastic network optimization

In this paper, we consider the problem of reducing network delay in stochastic network utility optimization problems. We start by studying the recently proposed quadratic Lyapunov function based algorithms (QLA, also known as the MaxWeight algorithm). We show that for every stochastic problem, there is a corresponding deterministic problem, whose dual optimal solution “exponentially attracts” the network backlog process under QLA. In particular, the probability that the backlog vector under QLA deviates from the attractor is exponentially decreasing in their Euclidean distance. This is the first such result for the class of algorithms built upon quadratic Lyapunov functions. The result quantifies the “network gravity” role of Lagrange Multipliers in network scheduling. It not only helps to explain how QLA achieves the desired performance but also suggests that one can roughly “subtract out” a Lagrange multiplier from the system induced by QLA.