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). We show that for every stochastic problem, there is a corresponding deterministic problem, whose dual optimal solution ldquoexponentially attractsrdquo 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 suggests that one can roughly ldquosubtract outrdquo a Lagrange multiplier from the system induced by QLA. We thus develop a family of Fast Quadratic Lyapunov based Algorithms (FQLA) that achieve an [O(1/V ),O(log²(V ))] performance-delay tradeoff. These results highlight the ldquonetwork gravityrdquo role of Lagrange Multipliers in network scheduling. This role can be viewed as the counterpart of the ldquoshadow pricerdquo role of Lagrange Multipliers in flow regulation for classic flow-based network problems.