Buffer size for routing limited-rate adversarial traffic

We consider he slight variation of the adversarial queuing theory model in which an adversary injects packets with routes into the network subject to the following constraint: For any link e, the total number of packets injected in any time window [t, t’) and whose route contains e is at most ρ(t’−t)+σ, where ρ and σ are non-negative parameters. Informally, ρ bounds the long-term rate of injections and σ bounds the “burstiness” of injection: σ = 0 means that the injection is as smooth as it can be. It is known that greedy scheduling of the packets (under which a link is not idle if there is any packet ready to be sent over it) may result in Ω(n) buffer size even on an n-node line network and very smooth injections (σ = 0). In this paper, we propose a simple non-greedy scheduling policy and show that, in a tree where all packets are destined at the root, no buffer needs to be larger than σ+2ρ to ensure that no overflows occur, which is optimal in our model. The rule of our algorithm is to forward a packet only if its next buffer is completely empty. The policy is centralized: in a single step, a long “train” of packets may progress together. We show that, in some sense, central coordination is required for our algorithm, and even for the more sophisticated “downhill” algorithm in which each node forwards a packet only if its next buffer is less occupied than its current one. This is shown by presenting an injection pattern with σ = 0 for the n-node line that results in Ω(n) packets in a buffer if local control is used.