# Technical Document Extraction: Roofline Model (Llama 33B, A40)
## 1. Header Information
* **Title:** Roofline Model (Llama 33B, A40)
* **Subject:** Performance analysis of the Llama 33B model running on an NVIDIA A40 GPU.
## 2. Chart Specifications
* **Type:** Log-Log Roofline Plot.
* **X-Axis (Horizontal):** Operational Intensity (FLOP/Byte).
* **Scale:** Logarithmic, ranging from approximately 0.6 to 10,000 (10k).
* **Major Markers:** 1, 10, 100, 1k (1,000), 10k (10,000).
* **Y-Axis (Vertical):** Performance (FLOP/s).
* **Scale:** Logarithmic, ranging from 10G to over 100T.
* **Major Markers:** 10G, 100G, 1T, 10T, 100T.
## 3. Legend and Thresholds
The legend is located in the bottom-right quadrant of the chart area.
| Legend Item | Color/Style | Description |
| :--- | :--- | :--- |
| **696GB/s** | Blue Dashed Line (Diagonal) | Memory Bandwidth Limit. Slopes upward from left to right. |
| **149.7 TFLOP/s** | Red Dashed Line (Horizontal) | Peak Compute Performance Limit (Roof). |
| **qkv mlp init** | Blue 'x' | Data points for QKV/MLP initialization. |
| **qkv mlp ar** | Orange 'x' | Data points for QKV/MLP auto-regressive phase. |
| **up/gate/down init** | Green 'x' | Data points for Up/Gate/Down projection initialization. |
| **up/gate/down ar** | Red 'x' | Data points for Up/Gate/Down projection auto-regressive phase. |
| **qk/pv init** | Purple 'x' | Data points for QK/PV initialization. |
| **qk/pv ar** | Brown 'x' | Data points for QK/PV auto-regressive phase. |
*Note: A vertical green dashed line intersects the "elbow" where the bandwidth limit meets the compute limit, occurring at an operational intensity of approximately 215 FLOP/Byte.*
## 4. Component Analysis and Trends
### Memory-Bound Region (Left of the Green Vertical Line)
In this region, performance is limited by memory bandwidth (the diagonal blue line).
* **Trend:** Data points follow the upward slope of the 696GB/s line. As operational intensity increases, performance increases linearly on the log-log scale.
* **Series `qk/pv ar` (Brown):** Clustered at the lowest operational intensity (~1 FLOP/Byte) with performance between 60G and 600G FLOP/s.
* **Series `up/gate/down ar` (Red) & `qkv mlp ar` (Orange):** These follow the diagonal line closely between 2 and 40 FLOP/Byte. Performance ranges from ~1T to ~20T FLOP/s.
* **Series `qk/pv init` (Purple):** Clustered between 40 and 150 FLOP/Byte. These points are slightly below the theoretical bandwidth limit, ranging from ~7T to ~70T FLOP/s.
### Compute-Bound Region (Right of the Green Vertical Line)
In this region, performance is limited by the GPU's peak compute capability (the horizontal red line).
* **Trend:** Data points flatten out and move horizontally, plateauing near the 149.7 TFLOP/s limit.
* **Series `up/gate/down init` (Green):** Located at high operational intensities (approx. 250 to 5,000 FLOP/Byte). These points sit very close to the 149.7 TFLOP/s roof.
* **Series `qkv mlp init` (Blue):** Located at high operational intensities (approx. 400 to 4,000 FLOP/Byte). These points also sit near the peak compute roof, showing high efficiency for initialization tasks.
## 5. Summary of Data Observations
1. **Auto-regressive (ar) tasks** (Brown, Red, Orange) are predominantly **memory-bound**, characterized by low operational intensity and performance that scales with memory bandwidth.
2. **Initialization (init) tasks** (Purple, Green, Blue) have higher operational intensity. While `qk/pv init` is still transitioning, `up/gate/down init` and `qkv mlp init` are clearly **compute-bound**, reaching the hardware's maximum TFLOP/s capacity.
3. The "Ridge Point" or "Elbow" of the machine is at **~215 FLOP/Byte**. Any operation with intensity lower than this cannot reach peak TFLOP/s on an A40.
