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Modeling Buffer Occupancy in bittide Systems

Sanjay Lall, Tammo Spalink

TL;DR

This paper addresses steady-state buffer occupancy in bittide-based logically synchronous systems.It develops a fluid abstract-frame model and proves convergence of buffer occupancy to a steady state, deriving an explicit expression $\beta^{ss} = \lambda + Y L S^\mathsf{T} H v + k^{-1} B^\mathsf{T} Q^\# (H - I) v$ that ties occupancy to network topology, latencies, and gains.The results express $\beta^{ss}$ in terms of graph-related operators $Q$, $S$, $B$, the latency matrix $L$, and the controller gain $k$, enabling practical buffer sizing and latency minimization for large-scale distributed systems.Limitations arise from the fluid approximation and neglect of quantization and discrete-time effects; future work includes incorporating sampling, quantization, dynamic topology, and fault scenarios.

Abstract

The bittide mechanism enables logically synchronous computation across distributed systems by leveraging the continuous frame transmission inherent to wired networks such as Ethernet. Instead of relying on a global clock, bittide uses a decentralized control system to adjust local clock frequencies, ensuring all nodes operate with a consistent notion of time by utilizing elastic buffers at each node to absorb frequency variations. This paper presents an analysis of the steady-state occupancy of these elastic buffers, a critical factor influencing system latency. Using a fluid model of the bittide system, we prove that buffer occupancy converges and derive an explicit formula for the steady-state value in terms of system parameters, including network topology, physical latencies, and controller gains. This analysis provides valuable insights for optimizing buffer sizes and minimizing latency in bittide-based distributed systems.

Modeling Buffer Occupancy in bittide Systems

TL;DR

This paper addresses steady-state buffer occupancy in bittide-based logically synchronous systems.It develops a fluid abstract-frame model and proves convergence of buffer occupancy to a steady state, deriving an explicit expression $\beta^{ss} = \lambda + Y L S^\mathsf{T} H v + k^{-1} B^\mathsf{T} Q^\# (H - I) v$ that ties occupancy to network topology, latencies, and gains.The results express $\beta^{ss}$ in terms of graph-related operators $Q$, $S$, $B$, the latency matrix $L$, and the controller gain $k$, enabling practical buffer sizing and latency minimization for large-scale distributed systems.Limitations arise from the fluid approximation and neglect of quantization and discrete-time effects; future work includes incorporating sampling, quantization, dynamic topology, and fault scenarios.

Abstract

The bittide mechanism enables logically synchronous computation across distributed systems by leveraging the continuous frame transmission inherent to wired networks such as Ethernet. Instead of relying on a global clock, bittide uses a decentralized control system to adjust local clock frequencies, ensuring all nodes operate with a consistent notion of time by utilizing elastic buffers at each node to absorb frequency variations. This paper presents an analysis of the steady-state occupancy of these elastic buffers, a critical factor influencing system latency. Using a fluid model of the bittide system, we prove that buffer occupancy converges and derive an explicit formula for the steady-state value in terms of system parameters, including network topology, physical latencies, and controller gains. This analysis provides valuable insights for optimizing buffer sizes and minimizing latency in bittide-based distributed systems.
Paper Structure (16 sections, 7 theorems, 50 equations, 3 figures)

This paper contains 16 sections, 7 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Modeling
  4. The bittide mechanism
  5. Abstract frame model
  6. Fluid model
  7. Limiting behavior
  8. Relative equilibrium.
  9. Relative dynamics.
  10. Stability.
  11. Limiting buffer occupancy
  12. The zero-latency case.
  13. Steady initial conditions
  14. Related work
  15. Conclusions
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose the parameters $\mathcal{P}$ are admissible. Then there exists $\omega^\text{ss},\theta^\text{ss}\in\mathbb{R}^n$ such that satisfies eqn:model. Such $\omega^\text{ss}$ is unique, given by and $\theta^\text{ss}$ is not unique in general but must satisfy

Figures (3)

  • Figure 1: Graph showing edge and node numbering
  • Figure 2: Per-node frequencies
  • Figure 3: Per-edge buffer occupancies

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Proof 1
  • Theorem 2
  • Corollary 1
  • Proof 2
  • Theorem 3
  • Proof 3
  • Definition 3
  • ...and 6 more