An Optimal Analytical Solution for Processor Speed Control with Thermal Constraints

Ravishankar Rao, S. Vrudhula, C. Chakrabarti, N. Chang
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引用次数: 51

Abstract

As semiconductor manufacturing technology scales to smaller device sizes, the power consumption of clocked digital ICs begins to increase. Dynamic voltage and frequency scaling (DVFS) is a well-known technique for conserving energy. Recently, it has also been used to control the CPU temperature as part of dynamic thermal management (DTM) techniques. Most works in these areas assume that the optimum speed profile (for either minimizing energy or maximizing performance) is a constant profile. However, in the presence of thermal constraints, we show that the optimal profile is in general, a time-varying function. We formulate the problem of maximizing the average throughput of a processor over a given time period, subject to thermal and speed constraints, as a problem in the calculus of variations. The variational approach provides a powerful framework for precisely specifying and solving the speed control problem, and allows us to obtain an exact analytical solution. The solution methodology is very general, and works for any convex power model, and simple lumped RC thermal models. The resulting speed profiles were found to consist of up to three segments, of which one of them is a decreasing function of time, and the others are constant. We analyze the effect of different parameters like the initial temperature, thermal capacitance and the maximum rated speed on the nature and the cost of the optimum solution. We also propose a two-speed solution that approximates the optimal speed curve. This solution was found to achieve a performance close to that of the optimum, and is also easier to implement in real processors
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热约束下处理器速度控制的最优解析解
随着半导体制造技术向更小的器件尺寸扩展,时钟数字集成电路的功耗开始增加。动态电压频率标度(DVFS)是一种众所周知的节能技术。最近,它也被用于控制CPU温度作为动态热管理(DTM)技术的一部分。这些领域的大多数工作都假设最佳速度轮廓(对于最小化能量或最大化性能)是恒定的轮廓。然而,在存在热约束的情况下,我们表明最优剖面通常是一个时变函数。我们将在给定时间内,受热和速度限制,使处理器的平均吞吐量最大化的问题表述为变分法中的一个问题。变分方法为精确指定和求解速度控制问题提供了一个强有力的框架,并使我们能够获得精确的解析解。求解方法非常通用,适用于任何凸幂模型和简单的集总RC热模型。结果发现,所得的速度曲线最多由三个部分组成,其中一个是时间的递减函数,而其他部分是恒定的。分析了初始温度、热容和最大额定转速等参数对最优解性质和成本的影响。我们还提出了一个近似于最优速度曲线的双速解决方案。该解决方案的性能接近于最佳性能,并且在实际处理器中更容易实现
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