ARP Reference

This page highlights the core ARP model and the most‑used closed‑forms.

Model

dG/dt = α·|I(t)| − μ·G(t)

Closed‑form Solution

G(t) = e^{−μt} [ G(0) + α ∫₀ᵗ e^{μs} |I(s)| ds ]

Special Cases

• Constant input |I| = I₀: G(t) = G∞ + (G(0) − G∞) e^{−μt}, with G∞ = (α/μ)·I₀.
• Piecewise‑constant input: G relaxes exponentially to the current G∞ on each interval.
• Stationary ergodic input: limₜ→∞ G(t) = (α/μ)·E[|I|].

For full derivations and Laplace/Z‑domain views, consult the Markdown in the source repository.

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