# Technical Document Extraction: Roofline Model (Llama 13B, A6000) ## 1. Document Header * **Title:** Roofline Model (Llama 13B, A6000) * **Language:** English ## 2. Chart Specifications This is a **Roofline Model** chart, a standard visualization used to represent the performance limits of a computing system (NVIDIA A6000 GPU) running a specific workload (Llama 13B model). ### Axis Definitions * **Y-Axis (Vertical):** Performance (FLOP/s) * **Scale:** Logarithmic (Base 10) * **Range:** 10G to ~300T * **Major Markers:** 10G, 100G, 1T, 10T, 100T * **X-Axis (Horizontal):** Operational Intensity (FLOP/Byte) * **Scale:** Logarithmic (Base 10) * **Range:** ~0.6 to 10k * **Major Markers:** 1, 10, 1k, 10k ### Legend and Thresholds The legend is located in the bottom-right quadrant of the plot area. | Legend Item | Color/Style | Description | Value/Threshold | | :--- | :--- | :--- | :--- | | **768GB/s** | Blue Dashed Line | Memory Bandwidth Limit (Slope) | 768 GB/s | | **181 TFLOP/s** | Red Dashed Line | Peak Compute Performance (Ceiling) | 181 TFLOP/s | | **qkv mlp init** | Blue 'x' | Data points for QKV/MLP initialization | High Intensity/High Perf | | **qkv mlp ar** | Orange 'x' | Data points for QKV/MLP auto-regressive | Mid Intensity/Mid Perf | | **up/gate/down init** | Green 'x' | Data points for Up/Gate/Down initialization | High Intensity/High Perf | | **up/gate/down ar** | Red 'x' | Data points for Up/Gate/Down auto-regressive | Low Intensity/Low Perf | | **qk/pv init** | Purple 'x' | Data points for QK/PV initialization | Mid Intensity/Mid Perf | | **qk/pv ar** | Brown 'x' | Data points for QK/PV auto-regressive | Low Intensity/Low Perf | --- ## 3. Component Analysis ### The "Roofline" Structure 1. **Memory-Bound Region (The Slope):** Represented by the blue dashed line starting from the bottom left. It follows the formula $Performance = Bandwidth \times Intensity$. Any data point sitting on or near this line is limited by how fast data can be moved from memory (768 GB/s). 2. **Compute-Bound Region (The Ceiling):** Represented by the horizontal red dashed line at the top. It represents the hardware's maximum theoretical throughput (181 TFLOP/s). 3. **Ridge Point:** The intersection of the two lines occurs at an Operational Intensity of approximately **235 FLOP/Byte** (indicated by a vertical green dashed line). ### Data Series Trends and Distribution * **Initialization (init) Phases:** * **Trend:** These points (Blue, Green, Purple 'x') cluster toward the right side of the graph (High Operational Intensity). * **Observation:** Most "init" points for `qkv mlp` and `up/gate/down` are located on the horizontal "ceiling," meaning they are compute-bound and utilizing the GPU's maximum TFLOP/s. * **Auto-regressive (ar) Phases:** * **Trend:** These points (Orange, Red, Brown 'x') cluster toward the left side of the graph (Low Operational Intensity). * **Observation:** These points follow the diagonal blue dashed line. This indicates that the auto-regressive decoding phase of the Llama 13B model is strictly memory-bandwidth bound, operating significantly below the peak TFLOP/s of the A6000. --- ## 4. Data Point Extraction (Approximate Values) | Category | Operational Intensity (FLOP/Byte) | Performance (FLOP/s) | Bottleneck | | :--- | :--- | :--- | :--- | | **up/gate/down ar** | ~1 to 10 | 100G to 5T | Memory (768GB/s) | | **qk/pv ar** | ~1 (Vertical stack) | 40G to 800G | Memory (768GB/s) | | **qkv mlp ar** | ~2 to 40 | 1T to 20T | Memory (768GB/s) | | **qk/pv init** | ~40 to 150 | 10T to 80T | Transition/Memory | | **qkv mlp init** | ~800 to 3k | ~150T to 181T | Compute (181 TFLOP/s) | | **up/gate/down init** | ~200 to 4k | ~100T to 181T | Compute (181 TFLOP/s) | ## 5. Summary of Findings The chart demonstrates that for a Llama 13B model on an A6000 GPU: 1. **Initialization** is highly efficient and hits the hardware's compute ceiling (181 TFLOP/s). 2. **Auto-regressive decoding** is inefficient in terms of raw compute utilization because it is bottlenecked by the 768 GB/s memory bandwidth. 3. The **qk/pv ar** (Brown 'x') operations are the least efficient, clustered at the lowest operational intensity (~1 FLOP/Byte).