In this paper we prove Aldous's conjecture from 1987 that there is no backoff protocol that is stable for any positive arrival rate. The setting is a communication channel for coordinating requests for a shared resource. Each user who wants to access the resource makes a request by sending a message to the channel. The users don't have any way to communicate with each other, except by sending messages to the channel. The operation of the channel proceeds in discrete time steps. If exactly one message is sent to the channel during a time step then this message succeeds (and leaves the system). If multiple messages are sent during a time step then these messages collide. Each of the users that sent these messages therefore waits a random amount of time before re-sending. A backoff protocol is a randomised algorithm for determining how long to wait -- the waiting time is a function of how many collisions a message has had. Specifically, a backoff protocol is described by a send sequence $\overline{p} = (p_0,p_1,p_2,\ldots)$. If a message has had $k$ collisions before a time step then, with probability $p_k$, it sends during that time step, whereas with probability $1-p_k$ it is silent (waiting for later). The most famous backoff protocol is binary exponential backoff, where $p_k = 2^{-k}$. Under Kelly's model, in which the number of new messages that arrive in the system at each time step is given by a Poisson random variable with mean $λ$, Aldous proved that binary exponential backoff is unstable for any positive $λ$. He conjectured that the same is true for any backoff protocol. We prove this conjecture.

翻译：本文证明了 Aldous 于 1987 年提出的猜想：不存在对任意正到达率均保持稳定的退避协议。研究场景是一个用于协调共享资源请求的通信信道。每个希望访问资源的用户通过向信道发送消息来发出请求。用户之间除了向信道发送消息外，没有任何其他通信方式。信道的运行以离散时间步进行。若在一个时间步内恰好有一条消息发送至信道，则该消息成功（并离开系统）。若在一个时间步内有多条消息发送，则这些消息发生碰撞。因此，每个发送了碰撞消息的用户在重新发送前需等待一段随机时间。退避协议是一种用于确定等待时长的随机化算法——等待时间是消息已发生碰撞次数的函数。具体而言，一个退避协议由发送序列 $\overline{p} = (p_0,p_1,p_2,\ldots)$ 描述。若一条消息在某个时间步前已发生 $k$ 次碰撞，则在该时间步内，它以概率 $p_k$ 发送消息，而以概率 $1-p_k$ 保持静默（等待后续时机）。最著名的退避协议是二进制指数退避，其中 $p_k = 2^{-k}$。在 Kelly 模型中，每个时间步到达系统的新消息数量服从均值为 $λ$ 的泊松随机变量，Aldous 证明了二进制指数退避对任意正 $λ$ 均不稳定。他猜想这一结论对所有退避协议均成立。本文证明了该猜想。