Price based protocols for fair resource allocation: convergence time analysis and extension to Leontief utilities

We analyze several distributed, continuous time protocols for a fair allocation of bandwidths to flows in a network (or resources to agents). Our protocols converge to an allocation which is a logarithmic approximation, simultaneously, to all canonical social welfare functions (i.e. functions which are symmetric, concave, and non-decreasing). These protocols can be started in an arbitrary state. While a similar protocol was known before, it only applied to the simple bandwidth allocation problem, and its stability and convergence time was not understood. In contrast, our protocols also apply to the more general case of Leontief utilities, where each user may place a different requirement on each resource. Further, we prove that our protocols converge in polynomial time. The best convergence time we prove is O(n log ^nc_max^a_max/^c_min^a_min), where n is the number of agents in the network, c_max and c_min are the maximum and minimum capacity of the links, and a_max, a_min are the largest and smallest Leontief coefficients, respectively. This time is achieved by a simple MIMD (multiplicative increase, multiplicative decrease) protocol which had not been studied before in this setting. We also identify combinatorial properties of these protocols that may be useful in proving stronger convergence bounds. The final allocations by our protocols are supported by usage-sensitive dual prices which are fair in the sense that they shield light users of a resource from the impact of heavy users. Thus our protocols can also be thought of as efficient distributed schemes for computing fair prices.