Convergence of Dual Algorithm with Arbitrary Communication Delays

We study the issue of convergence of user rates and resource prices under a family of rate controlschemes called dual algorithms with arbitrary communication delays. We first consider a case where asingle resource is shared by many users. Then, we study a general network shared by heterogeneoususers and derive sufficient conditions for convergence. We show that in the case of a single user utilizinga single resource, our condition is also necessary. Using our results we derive a sufficient conditionfor convergence with a family of popular utility and resource price functions. We present numericalexamples to validate our analysis.