State Dependent AIMD Algorithms and Consensus Problems

The standard linear AIMD algorithm employed in TCP (transmission control protocol) describes probing strategies that evolve in cycles, each cycle having two phases. The first phase of the cycles is instantaneous. It occurs when capacity is reached, users are notified and each responds by down-scaling its utilization-rate (abruptly) by a multiplicative factor. This phase is called the Multiplicative Decrease (MD) phase. During the second phase of a cycle, each user increases his utilization-rate linearly until congestion is reached again, at which point the first phase of the next cycle is entered. The second phase is called the Additive Increase (AI) phase. Denote the share of the collective resource allocated to user i at time t by xi(t) and let x(t) = [x1(t), . . . , xn(t)] . The capacity constraint requires that ∑n i=1 xi(t) < C, with C as the total capacity of the resource available to the entire system. The k cycle begins at a time t(k) at which the global utilization of the resource reaches capacity. The instantaneous decrease of the utilization-rate of user i during the MD phase of the k cycle is expressed by: