A Loosely Self-stabilizing Protocol for Randomized Congestion Control with Logarithmic Memory

Abstract

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 $v \in V$ sends a message with probability $p(v) \in [0,1)$ and the server has a capacity of $\sigma \in \mathbb {N}$, i.e., it can receive at most $\sigma$ 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 $\sum p(v) = \sigma$ and $p(v) = p(w)$ for all $v,w \in V$.

We propose a loosely self-stabilizing protocol with a slightly relaxed legitimate state. Our protocol lets the system converge from any initial state to a state where $\sum p(v) \in \left[ \sigma \pm \epsilon \right]$ and $|p(v)-p(w)| \in O(\frac{1}{n})$. This property is then maintained for $\varOmega (n^{\mathfrak {c}})$ 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 $O(W + \log n)$ bits of memory where W is length of node identifiers, 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) in certain cases.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center On-The-Fly Computing (GZ: SFB 901/3) under the project number 160364472.