Price differentiation in the kelly mechanism

Resource allocation [7, 4] and pricing [13, 1, 12] for network resources, e.g., bandwidth, have been studied extensively during the last decade. Among various proposed pricing and allocation mechanisms, the Kelly mechanism [7] stands out as a simple and scalable solution. Kelly et al [7, 6] first showed that it can be used as a congestion pricing mechanism to achieve proportional fairness. However, Johari and Tsisiklis [4] found that the resource allocation from the Kelly mechanism might induce an efficiency loss up to 25% of the social optimality. In this paper, we generalize the Kelly mechanism by designing a build-in price differentiation and show that the efficiency gap can be closed. In particular, we analyze the resource competition game under the generalized mechanism and show that any price differentiation induces a unique Nash equilibrium. We further reveal the relationship between the price differentiation and its resulting resource allocations and derive the optimal condition under which the social welfare reaches maximum. Our generalization extends the flexibility of the Kelly mechanism in two very different ways. First, it allows autonomous resource owners to use price differentiations so as to achieve individual objectives. Second, it provides a method for altruistic resource owners to make tradeoffs between user fairness (in terms of price differentiation) and system efficiency (in terms of social welfare).