Optimal Protocols for 2-Party Contention Resolution

Abstract

Contention Resolution is a fundamental symmetry-breaking problem in which n devices must acquire temporary and exclusive access to some shared resource, without the assistance of a mediating authority. For example, the n devices may be sensors that each need to transmit a single packet of data over a broadcast channel. In each time step, devices can (probabilistically) choose to acquire the resource or remain idle; if exactly one device attempts to acquire it, it succeeds, and if two or more devices make an attempt, none succeeds. The complexity of the problem depends heavily on what types of collision detection are available. In this paper we consider acknowledgement-based protocols, in which devices only learn whether their own attempt succeeded or failed; they receive no other feedback from the environment whatsoever, i.e., whether other devices attempted to acquire the resource, succeeded, or failed.

Nearly all work on the Contention Resolution problem evaluated the performance of algorithms asymptotically, as $n\rightarrow \infty$. In this work we focus on the simplest case of $n=2$ devices, but look for precisely optimal algorithms. We design provably optimal algorithms under three natural cost metrics: minimizing the expected average of the waiting times (avg), the expected waiting time until the first device acquires the resource (min), and the expected time until the last device acquires the resource (max). We first prove that the optimal algorithms for $n=2$ are periodic in a certain sense, and therefore have finite descriptions, then we design optimal algorithms under all three objectives.

• avg. The optimal contention resolution algorithm under the avg objective has expected cost $\sqrt{3/2} + 3/2 \approx 2.72474$.

• min. The optimal contention resolution algorithm under the min objective has expected cost 2. (This result can be proved in an ad hoc fashion, and may be considered folklore.)

• max. The optimal contention resolution algorithm under the max objective has expected cost $1/\gamma \approx 3.33641$, where $\gamma \approx 0.299723$ is the smallest root of $3x^3 - 12x^2 + 10x -2$ (We may also express $\gamma$ in radical form: $\gamma = -\frac{1}{6} \left( 1-i \sqrt{3}\right) \root 3 \of {13+i \sqrt{47}}+\frac{4}{3}-\frac{1+i \sqrt{3}}{\root 3 \of {13+i \sqrt{47}}}$.).