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BMM算子优化分析_Release/BMM算子优化分析_v0.92.html
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<tr><td>$T_{bd}$</td><td>batch 边界固定开销</td><td>fixpipe 启动握手（详见下文物理成因）</td></tr>
<tr><td>$BW_{pc}$</td><td>单核 GM 带宽份额</td><td>$W_{GM}/C$</td></tr></table>
<p> *端到端时延模型*：对每核执行过程建模。稳态流水（unitflag 交叠 batch 间 drain/startup）加上 batch 边界固定开销 $T_{bd}$（每边界一次）：</p>
<div class="math"> $$T_{iter} = \underbrace{b_{core} \cdot n_K \cdot \max(T_{load},\, T_{comp})}_{\text{稳态流水}} + \underbrace{T_{comp} + T_{write}}_{\text{末 batch drain}} + \underbrace{b_{core} \cdot T_{bd}}_{\text{边界开销}}$$</div>
<div class="math"> $$T_{mb} = \underbrace{b_{core} \cdot n_K \cdot \max(T_{load},\, b_0 T_{comp})}_{\text{稳态流水}} + \underbrace{b_0(T_{comp} + T_{write})}_{\text{末合并 batch drain}} + \underbrace{\frac{b_{core}}{b_0} \cdot T_{bd}}_{\text{边界开销}}$$</div>
<div class="math"> $$
T_{iter} = \underbrace{b_{core} \cdot n_K \cdot \max(T_{load},\, T_{comp})}_{\text{稳态流水}} + \underbrace{T_{comp} + T_{write}}_{\text{末 batch drain}} + \underbrace{b_{core} \cdot T_{bd}}_{\text{边界开销}}
$$</div>
<div class="math"> $$
T_{mb} = \underbrace{b_{core} \cdot n_K \cdot \max(T_{load},\, b_0 T_{comp})}_{\text{稳态流水}} + \underbrace{b_0(T_{comp} + T_{write})}_{\text{末合并 batch drain}} + \underbrace{\frac{b_{core}}{b_0} \cdot T_{bd}}_{\text{边界开销}}
$$</div>
<p> 其中 MergeBatch 合并后 $k_{L1}^m = \min(K,\; k_{L1}/b_0)$——L1 绑定（$k_{L1}&lt;K$）时 $k_{L1}^m = k_{L1}/b_0$；K 截断（$k_{L1}=K$）时 $k_{L1}^m = K$ 不减半。以 L1 绑定为例。</p>
<p> 条件 5 保证合并后仍访存 Bound（$T_{load} &gt; b_0 T_{comp}$），两式相减：</p>
<div class="math"> $$\Delta = T_{mb} - T_{iter} = \underbrace{(b_0-1)(T_{comp} + T_{write})}_{\text{drain 惩罚}} - \underbrace{b_{core}\Big(1-\frac{1}{b_0}\Big) T_{bd}}_{\text{边界节省}}$$</div>
<div class="math"> $$
\Delta = T_{mb} - T_{iter} = \underbrace{(b_0-1)(T_{comp} + T_{write})}_{\text{drain 惩罚}} - \underbrace{b_{core}\Big(1-\frac{1}{b_0}\Big) T_{bd}}_{\text{边界节省}}
$$</div>
<p> <b>分界条件</b>（MergeBatch 优于 IterBatch 当且仅当 $\Delta &lt; 0$）：</p>
<div class="math"> $$b_{core} > \frac{b_0 \cdot \text{penalty}}{T_{bd}},\qquad
\text{penalty} = \begin{cases} (b_0-1)(T_{comp} + T_{write}) & \text{L1 绑定（}k_{L1} \lt K\text{）} \\ (b_0^2-1)T_{comp} + (b_0-1)T_{write} & \text{K 截断（}k_{L1} = K\text{）} \end{cases}$$</div>
<div class="math"> $$
b_{core} > \frac{b_0 \cdot \text{penalty}}{T_{bd}},\qquad
\text{penalty} = \begin{cases} (b_0-1)(T_{comp} + T_{write}) & \text{L1 绑定（}k_{L1} \lt K\text{）} \\ (b_0^2-1)T_{comp} + (b_0-1)T_{write} & \text{K 截断（}k_{L1} = K\text{）} \end{cases}
$$</div>
<p> penalty 两种情形的含义：drain 暴露 = 末（合并）batch 最后一个 K 分块的计算时延 + 输出写回时延——这部分没有下一块搬移可交叠，是流水线的 drain 尾部。</p>
<ul class="tight">
<li><b>L1 绑定</b>（$k_{L1} &lt; K$）：L1 容量限制了 K 分块粒度，MergeBatch 合并后 tile 更大 → $k_{L1}$ 减半 → 每分块计算量 $b_0$ 倍 → drain 惩罚 $= (b_0-1)(T_{comp}+T_{write})$；</li>