import Mathlib
import ClassicalAnalysisAPI

/-! # Heavy Traffic Diffusion Limits

This module provides the mathematical foundation for analyzing session-typed
systems under high load. Heavy traffic theory studies systems operating near
their capacity limits, where fluctuations become important and can be
characterized by diffusion processes.

## Physical Intuition

Consider a highway during rush hour. When traffic is light, cars flow smoothly
and individual variations average out quickly. But when traffic approaches the
road's capacity, small fluctuations can trigger traffic jams that persist for
long periods. Heavy traffic theory captures this regime: when arrival rates
approach service capacity, the system's behavior becomes dominated by random
fluctuations rather than average behavior.

The mathematical trick is *scaling*: we zoom out in time (by a factor of n)
while zooming in on fluctuations (by a factor of √n). In this limit, the
discrete random walk of queue lengths converges to a continuous Brownian motion.

## Canonical Mathematical Formulation

Consider a queue with arrival rate λ and service rate μ. Define the scaled
queue length process:

  Q̂ₙ(t) = [Q(nt) - n(μ - λ)t] / √n

where Q(t) is the queue length at time t. The **Heavy Traffic Theorem** states
that as ρ = λ/μ → 1 (load approaches capacity):

  Q̂ₙ → σB     (in distribution)

where B is standard Brownian motion and σ² = λ + μ is the variance parameter.

The key scaling relationships are:
  - Time scales as n (longer observation window)
  - Fluctuations scale as √n (central limit theorem scaling)
  - Variance scales as σ² · n (variance grows linearly in time)

## Application to Telltale

In session-typed systems, heavy traffic analysis applies when:

  message_arrival_rate ≈ message_processing_rate

This occurs in high-throughput scenarios where the system is operating near
its capacity. The heavy traffic perspective provides:

1. **Buffer sizing**: Expected buffer occupancy scales as √n times the variance
   parameter, guiding memory allocation.

2. **Latency distribution**: Waiting times follow a reflected Brownian motion,
   giving precise tail distributions for SLA compliance.

3. **Bottleneck identification**: The variance parameter σ² identifies which
   edges contribute most to queueing delay.

For Coherence, heavy traffic analysis reveals that buffer bounds depend on:
  - The "slack" (μ - λ) between processing and arrival rates
  - The variability (σ) of message interarrival times
  - The observation timescale (n)

## Key Properties

- **Centering**: The `centered x μ = x - μ` operation removes the deterministic
  drift, isolating the random fluctuations.

- **Diffusion scaling**: The `diffusionScale n = √n` captures the central limit
  theorem scaling for fluctuations.

- **Variance preservation**: Under diffusion scaling, variance transforms as
  (σ · √n)² = σ² · n, matching the expected growth rate.

## References

- Kingman, J.F.C. (1961). The single server queue in heavy traffic.
- Iglehart, D.L. and Whitt, W. (1970). Multiple channel queues in heavy traffic.
- Williams, R.J. (1998). Diffusion approximations for open multiclass queueing
  networks.
- See also `Runtime/Proofs/WeightedMeasure.lean` for buffer bound calculations.
-/

/-
The Problem. This module needs a concise statement of its theorem-family boundary and reusable assumptions.
Solution Structure. Introduce the core model/contracts first, then derive canonical lemmas and API wrappers from those contracts.
-/

/-! ## Core Development -/

namespace Classical
namespace HeavyTrafficDiffusion

/-- Center a value around its mean.

    In the Telltale setting:
    - x is an observed buffer length or message count
    - μ is the expected value under steady state

    Centering isolates the fluctuation component for diffusion analysis. -/
def centered (x μ : Real) : Real :=
  x - μ

/-- The diffusion scaling factor √n.

    This is the characteristic scaling of the central limit theorem: sums of
    n independent random variables have fluctuations of order √n. For protocols,
    after n steps, buffer deviations from the mean are typically of order √n. -/
def diffusionScale [EntropyAPI.AnalysisModel] (n : Nat) : Real :=
  EntropyAPI.fluctuationScale n

/-- The diffusion scale is positive for positive n.

    This ensures we can divide by the scale factor in normalization. -/
theorem diffusion_scale_pos [EntropyAPI.AnalysisLaws] {n : Nat} (hn : 0 < n) :
    0 < diffusionScale n := by
  simpa [diffusionScale] using EntropyAPI.fluctuation_scale_pos hn

/-- Centering commutes with scaling.

    The normalized centered value equals the difference of normalized values.
    This allows us to compute scaled fluctuations either by centering then
    scaling, or by scaling then taking differences. -/
theorem centered_div_scale [EntropyAPI.AnalysisLaws] {n : Nat} (hn : 0 < n) (x μ : Real) :
    centered x μ / diffusionScale n = x / diffusionScale n - μ / diffusionScale n := by
  unfold centered
  have hscale : diffusionScale n ≠ 0 := by
    exact ne_of_gt (diffusion_scale_pos hn)
  field_simp [hscale]

/-- Variance under diffusion scaling.

    If a random variable has standard deviation σ, then after multiplying by
    √n, its variance becomes σ² · n. This is the fundamental variance growth
    law for random walks: variance accumulates linearly in time.

    For protocols: if per-step buffer fluctuation has variance σ², then after
    n steps the total buffer deviation has variance σ² · n. -/
theorem variance_scaling [EntropyAPI.AnalysisLaws] (σ : Real) (n : Nat) :
    (σ * diffusionScale n) ^ 2 = σ ^ 2 * n := by
  have hsq : (diffusionScale n) ^ 2 = n := by
    simpa [diffusionScale] using EntropyAPI.fluctuation_scale_sq n
  calc
    (σ * diffusionScale n) ^ 2 = σ ^ 2 * (diffusionScale n) ^ 2 := by ring
    _ = σ ^ 2 * n := by
      rw [hsq]

end HeavyTrafficDiffusion
end Classical
