Mathematical Derivations¶
This document contains all mathematical formulas and proofs used in A-LEMS.
📐 Core Energy Equations¶
RAPL Counter Reading¶
\[E_{domain}(t) = \text{value}_t \quad [\mu J]\]
Energy Difference¶
\[\Delta E_{domain} = E_{domain}(t_2) - E_{domain}(t_1)\]
Average Power¶
\[P_{avg} = \frac{\Delta E_{domain}}{\Delta t} = \frac{E_{domain}(t_2) - E_{domain}(t_1)}{t_2 - t_1}\]
🧮 Workload Isolation¶
Total Energy Decomposition¶
\[E_{total} = E_{workload} + E_{idle} + E_{noise}\]
Baseline Subtraction¶
\[E_{workload} = E_{total} - P_{idle} \cdot \Delta t\]
With Confidence Interval¶
\[E_{workload} = E_{total} - (\mu_{idle} - 2\sigma_{idle}) \cdot \Delta t\]
🔍 Orchestration Tax Derivation¶
Step 1: Workload Energy¶
\[E_{workload} = E_{pkg} - P_{idle,pkg}\Delta t\]
Step 2: Reasoning Energy¶
\[E_{reasoning} = E_{core} - P_{idle,core}\Delta t\]
Step 3: Tax Definition¶
\[E_{tax} = E_{workload} - E_{reasoning}\]
Step 4: Substitution¶
\[
\begin{aligned}
E_{tax} &= (E_{pkg} - P_{idle,pkg}\Delta t) - (E_{core} - P_{idle,core}\Delta t) \\
&= (E_{pkg} - E_{core}) - (P_{idle,pkg} - P_{idle,core})\Delta t
\end{aligned}
\]
Step 5: Package Decomposition¶
\[E_{pkg} = E_{core} + E_{uncore}\]
\[P_{idle,pkg} = P_{idle,core} + P_{idle,uncore}\]
Step 6: Final Form¶
\[
\begin{aligned}
E_{tax} &= E_{uncore} - P_{idle,uncore}\Delta t \\
&= \text{(uncore energy)} - \text{(idle uncore energy)}
\end{aligned}
\]
🔄 C-State Energy Model¶
C-State Power Levels¶
\[P_{Cx} = P_{static} + P_{dynamic}(f,V) \cdot \text{activity}_x\]
Residency Calculation¶
\[t_{Cx} = \sum_{i=1}^{n} \Delta t_i \cdot \mathbb{1}_{state=Cx}\]
Energy Saved¶
\[E_{saved} = \sum_{x} (P_{C0} - P_{Cx}) \cdot t_{Cx}\]
🌡️ Thermal Model¶
Newton's Law of Cooling with Power Input¶
\[\frac{dT}{dt} = \frac{1}{C_{th}} \left(P(t) - \frac{T(t) - T_{amb}}{R_{th}}\right)\]
Where: - \(C_{th}\) = thermal capacitance (J/K) - \(R_{th}\) = thermal resistance (K/W) - \(T_{amb}\) = ambient temperature
Discrete Time Solution¶
\[T_{i+1} = T_i + \frac{\Delta t}{C_{th}} \left(P_i - \frac{T_i - T_{amb}}{R_{th}}\right)\]
Thermal Gradient¶
\[\nabla T = \frac{T_{max} - T_{start}}{t_{rise}}\]
📈 Statistical Formulas¶
Sample Mean¶
\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\]
Sample Standard Deviation¶
\[s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2}\]
Standard Error¶
\[SE = \frac{s}{\sqrt{n}}\]
Confidence Interval (95%)¶
\[CI = \bar{x} \pm t_{n-1, 0.975} \cdot \frac{s}{\sqrt{n}}\]
For large n (\(n \geq 30\)):
\[CI \approx \bar{x} \pm 1.96 \cdot \frac{s}{\sqrt{n}}\]
🎯 Sample Size Calculation¶
For Desired Effect Size \(\delta\)¶
\[n = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2 \sigma^2}{\delta^2}\]
Where: - \(\alpha\) = significance level (typically 0.05) - \(1-\beta\) = desired power (typically 0.8) - \(\sigma\) = estimated standard deviation
Common Values¶
| Power | \(\alpha = 0.05\) | \(\alpha = 0.01\) |
|---|---|---|
| 0.80 | \(z = 2.80\) | \(z = 3.42\) |
| 0.90 | \(z = 3.24\) | \(z = 3.86\) |
| 0.95 | \(z = 3.60\) | \(z = 4.22\) |
📊 Efficiency Metrics¶
Energy per Instruction¶
\[EPI = \frac{E_{workload}}{\text{instructions}}\]
Energy per Token¶
\[EPT = \frac{E_{workload}}{\text{total\_tokens}}\]
Instructions per Token¶
\[IPT = \frac{\text{instructions}}{\text{total\_tokens}}\]
Energy Efficiency Ratio¶
\[EER = \frac{\text{linear\_energy}}{\text{agentic\_energy}}\]
🧪 Error Analysis¶
Measurement Uncertainty¶
\[\epsilon_{RAPL} = \pm 1 \mu J\]
\[\epsilon_t = \pm 1 \text{ ns}\]
Combined Relative Error¶
\[\frac{\Delta P}{P} = \sqrt{\left(\frac{\epsilon_{RAPL}}{E}\right)^2 + \left(\frac{\epsilon_t}{\Delta t}\right)^2}\]
Error Propagation in Tax Calculation¶
\[
\begin{aligned}
\sigma_{tax}^2 &= \sigma_{workload}^2 + \sigma_{reasoning}^2 \\
&= \left(\frac{\partial tax}{\partial workload}\right)^2 \sigma_{workload}^2 + \left(\frac{\partial tax}{\partial reasoning}\right)^2 \sigma_{reasoning}^2 \\
&= \sigma_{workload}^2 + \sigma_{reasoning}^2
\end{aligned}
\]
🔬 Advanced Metrics¶
Thermal-Energy Coupling Coefficient¶
\[\kappa = \frac{\Delta P_{leak}}{\Delta T} \quad [W/K]\]
C-State Transition Latency¶
\[L_{Cx \to C0} = t_{wake} - t_{request}\]
Orchestration Overhead Index¶
\[OOI = \frac{E_{agentic}}{E_{linear}} \cdot \left(1 + \frac{t_{planning} + t_{synthesis}}{t_{execution}}\right)\]
📚 References¶
- Papoulis, A. (1991). "Probability, Random Variables and Stochastic Processes"
- Taylor, J. R. (1997). "An Introduction to Error Analysis"
- Hennessy, J. L., & Patterson, D. A. (2017). "Computer Architecture: A Quantitative Approach"