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Buffer Centering for bittide Synchronization via Frame Rotation

Sanjay Lall, Tammo Spalink

TL;DR

Frame rotation addresses buffer centering in bittide by rotating buffer occupancy around the midpoint $\beta^{\text{off}}$ using a directed spanning-tree-guided pulsed feedback strategy. The method combines an initial proportional control to align node frequencies with a subsequent tree-ordered pulsed correction that centers every elastic buffer edge; mathematical analysis shows that all buffers converge to the equilibrium and nodes reach a common frequency, encapsulated by $\omega^{\mathrm{ss}} = W \omega^{\mathrm{u}}$ and $\beta^{\mathrm{ss}} = -k^{-1} B^\mathsf{T} Q^\ddag \omega^{\mathrm{u}}$. The authors prove that with a consistent tree-based ordering, tree-edge occupancies can be driven to zero without disrupting previously centered buffers, and extend this to all buffers via range arguments. The approach offers a robust, scalable decentralization for maintaining data-flow balance in bittide networks, with opportunities to adapt to dynamic topology and to develop fully decentralized tree construction.

Abstract

Maintaining consistent time in distributed systems is a fundamental challenge. The bittide system addresses this by providing logical synchronization through a decentralized control mechanism that observes local buffer occupancies and controls the frequency of an oscillator at each node. A critical aspect of bittide's stability and performance is ensuring that these elastic buffers operate around a desired equilibrium point, preventing data loss due to overflow or underflow. This paper introduces a novel method for centering buffer occupancies in a bittide network using a technique we term frame rotation. We propose a control strategy utilizing a directed spanning tree of the network graph. By adjusting the frequencies of nodes in a specific order dictated by this tree, and employing a pulsed feedback controller that targets the buffer occupancy of edges within the spanning tree, we prove that all elastic buffers in the network can be driven to their desired equilibrium. This ordered adjustment approach ensures that prior centering efforts are not disrupted, providing a robust mechanism for managing buffer occupancy in bittide synchronized systems.

Buffer Centering for bittide Synchronization via Frame Rotation

TL;DR

Frame rotation addresses buffer centering in bittide by rotating buffer occupancy around the midpoint using a directed spanning-tree-guided pulsed feedback strategy. The method combines an initial proportional control to align node frequencies with a subsequent tree-ordered pulsed correction that centers every elastic buffer edge; mathematical analysis shows that all buffers converge to the equilibrium and nodes reach a common frequency, encapsulated by and . The authors prove that with a consistent tree-based ordering, tree-edge occupancies can be driven to zero without disrupting previously centered buffers, and extend this to all buffers via range arguments. The approach offers a robust, scalable decentralization for maintaining data-flow balance in bittide networks, with opportunities to adapt to dynamic topology and to develop fully decentralized tree construction.

Abstract

Maintaining consistent time in distributed systems is a fundamental challenge. The bittide system addresses this by providing logical synchronization through a decentralized control mechanism that observes local buffer occupancies and controls the frequency of an oscillator at each node. A critical aspect of bittide's stability and performance is ensuring that these elastic buffers operate around a desired equilibrium point, preventing data loss due to overflow or underflow. This paper introduces a novel method for centering buffer occupancies in a bittide network using a technique we term frame rotation. We propose a control strategy utilizing a directed spanning tree of the network graph. By adjusting the frequencies of nodes in a specific order dictated by this tree, and employing a pulsed feedback controller that targets the buffer occupancy of edges within the spanning tree, we prove that all elastic buffers in the network can be driven to their desired equilibrium. This ordered adjustment approach ensures that prior centering efforts are not disrupted, providing a robust mechanism for managing buffer occupancy in bittide synchronized systems.

Paper Structure

This paper contains 10 sections, 9 theorems, 35 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Model
  4. Properties of the Laplacian.
  5. Behavior with proportional control.
  6. Control of Buffer Occupancy
  7. Example: triangular network
  8. Example: mesh
  9. Formulation of the controller
  10. Conclusions

Key Result

Proposition 1

Under Assumption asm:feas, the dynamics eqn:model2 are equivalent to the following together with the boundary condition $\tilde{\theta}(t^0)=0$.

Figures (6)

  • Figure 1: Graph used for example in §\ref{['s:example']}
  • Figure 2: Frequency behavior as a function of time for the system in Figure \ref{['fig:three']}.
  • Figure 3: Relative buffer occupancies for the system in Figure \ref{['fig:three']}.
  • Figure 4: Relative buffer occupancies for the system in Figure \ref{['fig:three']} after centering by both nodes 2 and node 1.
  • Figure 5: Graph with directed spanning tree
  • ...and 1 more figures

Theorems & Definitions (18)

  • Proposition 1
  • Proposition 2
  • Proof 1
  • Definition 1
  • Lemma 1
  • Proof 2
  • Lemma 2
  • Proof 3
  • Lemma 3
  • Proof 4
  • ...and 8 more