Efficiency of oblivious versus nonoblivious schedulers for optimistic, rate-based flow control

Two important performance parameters of distributed, rate-based flow control algorithms are their locality and convergence complexity. The former is characterized by the amount of global knowledge that is available to their scheduling mechanisms, while the latter is defined as the number of update operations performed on rates of individual sessions until max-min fairness is reached. Optimistic algorithms allow any session to intermediately receive a rate larger than its max-min fair rate; bottleneck algorithms finalize the rate of a session only if it is restricted by a certain, highly congested link of the network. In this work, we present a comprehensive collection of lower and upper bounds on convergence complexity, under varying degrees of locality, for optimistic, bottleneck, rate-based flow control algorithms. Say that an algorithm is oblivious if its scheduling mechanism uses no information of either the session rates or the network topology. We present a novel, combinatorial construction of a capacitated network, which we use to establish a fundamental lower bound of dn/4 + n/2 on the convergence complexity of any oblivious algorithm, where n is the number of sessions laid out on a network, and d, the session dependency, is a measure of topological dependencies among sessions. Moreover, we devise a novel simulation proof to establish that, perhaps surprisingly, the lower bound of dn/4 + n/2 on convergence complexity still holds for any partially oblivious algorithm, in which the scheduling mechanism is allowed to use information about session rates, but is otherwise unaware of network topology. On the positive side, we prove that the lower bounds for oblivious and partially oblivious algorithms are both tight. We do so by presenting optimal oblivious algorithms, which converge after dn/2 + n/2 update operations are performed in the worst case. To complete the picture, we show that linear convergence complexity can indeed be achieved if information about both session rates and network topology is available to schedulers. We present a counterexample, nonoblivious algorithm, which converges within an optimal number of n update operations. Our results imply a surprising convergence complexity collapse of oblivious and partially oblivious algorithms, and a convergence complexity separation between (partially) oblivious and nonoblivious algorithms for optimistic, bottleneck rate-based flow control.