  We investigate the connections between max-weight approaches and dual
subgradient methods for convex optimisation. We find that strong connections
exist and we establish a clean, unifying theoretical framework that includes
both max-weight and dual subgradient approaches as special cases. Our analysis
uses only elementary methods, and is not asymptotic in nature. It also allows
us to establish an explicit and direct connection between discrete queue
occupancies and Lagrange multipliers.
