Exploring the impact of pipeline on energy consumption
Summary
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Analysis: Single Module
Background
引入k级pipeline以后(使用k-1个Reg.):
Some Symbol
| Symbol | Description | Notes |
|---|---|---|
| $P_{M,Static}$ | M的静态功率 | |
| $W_{M,Dynamic}$ | M的动态功耗 | 完成一次运算消耗的能量，暂不考虑随着频率变化而改变 |
| $T_{M}$ | M的时延 | |
| $P_{R,Static}$ | Reg.的静态功率 | 引入一级Pipeline所需的Reg |
| $W_{R,Dynamic}$ | Reg.的动态功耗 | 完成一次运算消耗的能量，暂不考虑随着频率变化而改变 |
| $T_{R}$ | Reg.的时延 | |
| $PPW$ | 能耗比(Performance Per Watt) | 完成一次运算所需能量 |
时延
$
T_{k} = \frac{T_{M}}{k} +T_{R}
$
静态功率
$
\begin{aligned}
P_{k,Static} & = (k-1) \cdot P_{R,Static} + P_{M1,Static} + P_{M2,Static} +... + P_{Mk,Static} \\\\
& = (k-1) \cdot P_{R,Static} + P_{M,Static}
\end{aligned}
$
动态功率
$
\begin{aligned}
W_{k,Dynamic} & = (k-1) \cdot W_{R,Dynamic} + W_{M1,Dynamic} + W_{M2,Dynamic} +... + W_{Mk,Dynamic} \\\\
& = (k-1) \cdot W_{R,Dynamic} + W_{M,Dynamic}
\end{aligned}
$
能耗比
$
\begin{aligned}
PPW &= \frac{1}{P_{k,Static} \cdot T_{k} + W_{k,Dynamic} } \\\\
& = \frac{1}{[(k-1) \cdot P_{R,Static} + P_{M,Static}]\cdot(\frac{T_{M}}{k} +T_{R})+ (k-1) \cdot W_{R,Dynamic} + W_{M,Dynamic}} \\\\
& = \frac{1}{k \cdot (P_{R,Static}T_{R}+W_{R,Dynamic}) + \frac{1}{k} \cdot (P_{M,Static}T_{M}-P_{R,Static}T_{M}) - P_{R,Static}T_{R} +P_{R,Static}T_{M} +P_{M,Static}T_{R} - W_{R,Dynamic} + W_{M,Dynamic}} \\\\
& \le \frac{1}{ 2\sqrt{(P_{R,Static}T_{R}+W_{R,Dynamic}) \cdot (P_{M,Static}T_{M}-P_{R,Static}T_{M})} - P_{R,Static}T_{R} +P_{R,Static}T_{M} +P_{M,Static}T_{R} - W_{R,Dynamic} + W_{M,Dynamic}} \\\\
& when \ and \ only \ when \ \ k= \sqrt{\frac{P_{M,Static}T_{M}-P_{R,Static}T_{M}}{P_{R,Static}T_{R}+W_{R,Dynamic}}} , \ we \ have \ PPW_{max}
\end{aligned}
$
Analysis: Multi-States Module
Some Symbol
| Symbol | Description | Notes |
|---|---|---|
| $P_{C_{mn},Static}$ | $C_{mn}$的静态功率 | |
| $W_{C_{mn},Dynamic}$ | $C_{mn}$的动态功耗 | 单个Cell完成一次运算消耗的能量，暂不考虑随着频率变化而改变 |
| $T_{C_{mn}}$ | $C_{mn}$的时延 | |
| $P_{R,Static}$ | Reg.的静态功率 | 引入一级Pipeline所需的Reg |
| $W_{R,Dynamic}$ | Reg.的动态功耗 | 完成一次运算消耗的能量，暂不考虑随着频率变化而改变 |
| $T_{R}$ | Reg.的时延 | |
| $PPW$ | 整个模块的能耗比(Performance Per Watt) | 整个模块的完成一次运算所需能量 |
时延
整个网络共享最大时延
$
T_{k} = \frac{T_{M}}{k} +T_{R}
$
静态功率
$
\begin{aligned}
P_{k,Static} & = (k-1) \cdot P_{R,Static} + P_{M1,Static} + P_{M2,Static} +... + P_{Mk,Static} \\\\
& = (k-1) \cdot P_{R,Static} + P_{M,Static}
\end{aligned}
$
动态功率
$
\begin{aligned}
W_{k,Dynamic} & = (k-1) \cdot W_{R,Dynamic} + W_{M1,Dynamic} + W_{M2,Dynamic} +... + W_{Mk,Dynamic} \\\\
& = (k-1) \cdot W_{R,Dynamic} + W_{M,Dynamic}
\end{aligned}
$
能耗比
$
\begin{aligned}
PPW &= \frac{1}{\sum_{m=1}^{M} \sum_{n=1}^{N} (P_{m,n,k,Static} \cdot T_{m,n,k} + W_{m,n,k,Dynamic}) }
\end{aligned}
$
Outcome
To be continue..