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The Instability of all Backoff Protocols

Leslie Ann Goldberg, John Lapinskas

TL;DR

This paper proves Aldous's conjecture from 1987 that there is no backoff protocol that is stable for any positive arrival rate, and proves that binary exponential backoff is unstable for any positive $\lambda".

Abstract

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.

The Instability of all Backoff Protocols

TL;DR

This paper proves Aldous's conjecture from 1987 that there is no backoff protocol that is stable for any positive arrival rate, and proves that binary exponential backoff is unstable for any positive $\lambda".

Abstract

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 . If a message has had collisions before a time step then, with probability , it sends during that time step, whereas with probability it is silent (waiting for later). The most famous backoff protocol is binary exponential backoff, where . 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 , Aldous proved that binary exponential backoff is unstable for any positive . He conjectured that the same is true for any backoff protocol. We prove this conjecture.
Paper Structure (46 sections, 118 theorems, 294 equations, 4 figures)

This paper contains 46 sections, 118 theorems, 294 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Ideas from previous work
  3. A problematic send sequence
  4. Proof sketch
  5. Infinitely-many strongly exposed bins
  6. Finitely-many strongly exposed bins
  7. Key properties of the high-level state transition algorithm
  8. A brief description of our high-level state transition algorithm and its properties
  9. Putting it all together (for the case with finitely-many strongly exposed bins)
  10. Related work
  11. Organisation of the paper (with commentary)
  12. Preliminaries
  13. General notation
  14. Probability distributions and domination
  15. Tail bounds
  16. ...and 31 more sections

Key Result

Theorem 1.2

Consider any send sequence $\overline{p}$ and any birth rate $\lambda >0$. Then the backoff process with send sequence $\overline{p}$ and birth rate $\lambda$ is not positive recurrent.

Figures (4)

  • Figure 1: The high-level state $\Psi(t)$ is the interface between $V$ and $E$ at time $t$.
  • Figure 2: Our high-level state transition algorithm.
  • Figure 3: Illustration of Definitions \ref{['def:volume-process-ext']} and \ref{['def:volume-process']}.
  • Figure 4: High-level state transitions under the backoff-bounding rule from Definition \ref{['def:backoff-bounding-rule']}.

Theorems & Definitions (321)

  • Conjecture 1.1: Aldous's Conjecture
  • Theorem 1.2
  • Definition 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.8
  • proof
  • ...and 311 more