Framing the Problem

DAGs describe flows of jobs, assets, and dependencies. In practice, errors, retries, and external failures distort the DAG, introducing apparent cycles. Monitoring today is event-driven but not predictive — lacking a formal basis for quantifying disorder and anticipating collapse.

Entropy as a Field on DAGs

  • Each edge or stage has associated metrics: throughput, backlog, error rate.
  • We define entropy as a measure of unpredictability in those distributions.
  • More important is the rate of change of entropy (dS/dt): fast rises signal systemic instability.

This makes entropy a field defined across the DAG, like a potential that can be differentiated to predict flow failures.

Markov Chains for Flow Dynamics

Flows can be modeled as Markov chains:

  • States = stages of the DAG, plus failure/retry states.
  • Transitions = probabilities of job movement between states.
  • Transition matrices adapt over time from empirical metrics.
  • Entropy then characterizes uncertainty in the transition matrix.

This enables simulation of “most likely future paths” and early-warning signals for rare but costly transitions (e.g., cascading retries).

Bayesian Belief Updates

Bayesian inference layers on top of Markov dynamics:

  • Prior = baseline flow performance distribution.
  • Evidence = observed metrics (backlog, latency, errors).
  • Posterior = updated belief about DAG health, SLA compliance, and risk.

This allows continuous learning: flows become self-aware of their reliability, with confidence intervals for risk prediction.

Analogy to Ampère–Maxwell Law

The Ampère–Maxwell law describes how currents and changing electric fields generate magnetic fields:

∇ × B = μ₀ ( J + ε₀ dE/dt )

By analogy:

  • Flow current (J) = throughput of jobs across the DAG.
  • Electric field (E) = accumulated backlog or pressure in the system.
  • Magnetic field (B) = governance or operational response field (alerts, tickets, interventions).
  • Displacement current (dE/dt) = rate of change of backlog/entropy, which drives governance activity even without immediate throughput.

This reframes governance as a field response proportional not only to actual flow but also to changes in systemic stress. Just as in physics, ignoring displacement terms underestimates risk.

Implementation Sketch

  1. Data Plane: Ingest job transitions, metrics, logs.
  2. Markov Substrate: Build adaptive transition matrices from observed data.
  3. Entropy Fields: Compute entropy and its derivative per stage/edge.
  4. Bayesian Inference: Continuously update DAG health beliefs.
  5. Governance Field: Encode SLO/SLA response as the Maxwell-like closure law.

Research Implications

  • Unifies stochastic models (Markov), inferential models (Bayesian), and field-theory analogies (Maxwell) into one framework.
  • Provides a scientific foundation for predictive observability, going beyond dashboards into principled AI-driven governance.
  • Offers a bridge between SRE practice and theoretical physics-inspired research.

Next Steps

  • Formalize the entropy field equations for DAG substrates.
  • Test predictive accuracy of dS/dt against incident data.
  • Extend the Ampère–Maxwell analogy into full Maxwell-like equations for observability, potentially linking with information-theoretic conservation laws.

This technical note is designed to guide deeper research collaboration and experimental validation of entropy-augmented DAG observability.