A Loosely Self-stabilizing Protocol for Randomized Congestion Control with Logarithmic Memory
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We consider congestion control in peer-to-peer distributed systems. The problem can be reduced to the following scenario: Consider a set V of n peers (called clients in this paper) that want to send messages to a fixed common peer (called server in this paper). We assume that each client v ∈ V sends a message with probability p(v) ∈ [0,1) and the server has a capacity of σ∈N, i.e., it can recieve at most σ messages per round and excess messages are dropped. The server can modify these probabilities when clients send messages. Ideally, we wish to converge to a state with ∑ p(v) = σ and p(v) = p(w) for all v,w ∈ V. We propose a loosely self-stabilizing protocol with a slightly relaxed legimate state. Our protocol lets the system converge from any initial state to a state where ∑ p(v) ∈[σ±ϵ] and |p(v)-p(w)| ∈ O(1/n). This property is then maintained for Ω(n^c) rounds in expectation. In particular, the initial client probabilities and server variables are not necessarily well-defined, i.e., they may have arbitrary values. Our protocol uses only O(W + n) bits of memory where W is length of node identifers, making it very lightweight. Finally we state a lower bound on the convergence time an see that our protocol performs asymptotically optimal (up to some polylogarithmic factor).
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