  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
\emph{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 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. We thus
develop a family of \emph{Fast Quadratic Lyapunov based Algorithms} (FQLA) that
achieve an $[O(1/V), O(\log^2(V))]$ performance-delay tradeoff for problems
with a discrete set of action options, and achieve a square-root tradeoff for
continuous problems. This is similar to the optimal performance-delay tradeoffs
achieved in prior work by Neely (2007) via drift-steering methods, and shows
that QLA algorithms can also be used to approach such performance.
  These results highlight the "network gravity" role of Lagrange Multipliers in
network scheduling. This role can be viewed as the counterpart of the "shadow
price" role of Lagrange Multipliers in flow regulation for classic flow-based
network problems.
