物理融合系统：负荷建模与设计优化 - 图文(2)

from both a practical and a theoretical perspective. For example, from a physician’s perspective, the absence or tendency of losing the self-similarity signifies a high likelihood of congestive heart failure. On the other hand, from a mathematical standpoint, the fractal pattern exhibited in Figure 1a is not differentiable when considered as a time function describing a physical process; this means that the integer order derivatives should be replaced by derivatives of fractional orders when analyzing the system behavior.

无论从实践角度还是理论角度，图1中显示出的波形的尖锐变化都具有很重要的含义。例如从医师的角度来看，自相似性的缺失或者有消失的趋势都表示出充血性心力衰竭的很大可能性。另一方面，从数学角度出发，当图1a中表现出的分形图案被看做一个描述物理过程的时间函数时是不可微的；这意味着当分析系统行为时，整数阶导数应当替换成分数阶导数。

Figure 1b shows the time dependency of the mean, variance, and kurtosis of the R-R intervals in Figure 1a. By way of context, the kurtosis (i.e., the ratio between the fourth-order moment and the standard deviation of a probability distribution) captures the frequency of rare events. The existence of a nonstationary behavior can be observed from the moving average graphs of the mean and variance of R-R intervals in Figure 1b.This implies that, at every point in time, the physical process can be characterized by some local fractal exponent. In addition to the intrinsic nonstationarity of data, a nonzero kurtosis proves that a Gaussian approach does not fit the data well, and thus a higher-order moment analysis is needed to properly characterize this type of behavior.

图1b显示了图1a中R间期的平均值、方差和峰值的时间依赖性。通过上下文联系，可以知道峰值（即介于四阶矩函数和概率分布的标准偏差之间的比例）采集的是稀有事件的发生频率。非平稳特性的存在可以从图1b中R间期的平均值和方差的移动平均线图表中观测得知。这意味着在每一个时间点，物理过程都可以以一些局部的分形指数为特征。除了数据固有的非稳定之外，非零的峰值也证明了高斯方法并不适用于这些数据，因此为了描绘这种类型的性态特性就需要用到高阶力矩解析。

Figure 2. Cumulative concentration per cubic centimeter of cloud condensation nuclei (CCN) collected during flights also exhibits a self-similar signature. (Data sets shown here courtesy of National Center for Atmospheric Research, which allowed access to the Ice in Clouds Experiment; http ://data.eol.ucar.edu/codiac/projs?ICE-L.)

Many other physical processes also exhibit a similar behavior over long periods of time. For instance, Figure 2 shows another example of a self-similar process: it displays the cumulative concentration of cloud condensation nuclei (CCN) collected via a CCN spectrometer. From a practical standpoint, the CCN measurements are used to assess the impact of industrial pollution on climate change. Indeed,atmospheric

measurements show that a higher concentration of CCN determine that clouds reflect more solar radiation and therefore contribute to more abnormal temperature fluctuations on the earth’s surface. Beyond the intrinsic variability originating in the spatial location of the ice droplets within the cloud, however, the data in Figure 2 shows that there is also self-similarity in the time domain.

许多其他的物理过程经过一段较长的时间也会显示出相似的特性。例如，图2中演示了另一种具有自相似性的过程实例：它显示了通过云凝结核的光谱仪的收集到的云凝结核的累积浓度。从实践的角度来说，云凝结核的测量是用来评定工业污染对气候改变带了的影响的。事实上。大气监测显示更高浓度的云凝结核决定了云层会反射更多的日光照射，并因此造成地球表面更多异常的温度波动。但是越过起源于云层冰滴空间位置的内在变异性，图2中的数据还显示出时域中也存在着自相似性。

Although these geophysical processes, as well as many others such as daily average wind speeds, display self-similar behavior, their societal impact is quite different when we consider various time scales. More precisely, while the study of climate change and the impact of the human footprint on Earth’s atmosphere has a larger time scale (years or decades), the daily average wind speeds have an immediate impact with a time scale of days and possibly minutes. Indeed, the information about wind speeds, precipitation formation, and cloud movement has an immediate and enormous economic impact on air, road, and rail traffic. Despite these differences in characteristics, a CPS must be able to collect and communicate all this data to make various predictions.

尽管这些地球物理学过程和很多其他的数据诸如每日平均风速一样表现出自相似的特性，但是当我们考虑到不同的时间尺度时它们的社会影响是完全不一样的。更准确的说，相较于气候变化的研究和人类活动对大气层的影响需要一个更长的时间尺度（例如一年或十年），每日平均风速在以一天或更可能是一分钟为时间尺度时具有更接近的效果。事实上，关于风速、降水形成、云运动的信息会对航空、公路和铁路交通产生直接并巨大的经济影响。除了这些特性上的不同，物理融合系统必须能够收集并传达所有的这些数据以做出各种预测。

Figure 3. Power spectrum of short-range communication in a local area network (LAN) established via wireless links between moving vehicles (a). Multifractal spectrum of the transaction events (i.e., sent and received packets) in a LAN in which connectivity is established via wireless links and access points in an urban environment (b).

Modeling CPS workloads CPS工作负荷的建模

From the previous discussion, we can see that the physical processes relevant to a CPS might exhibit a systematic relationship at different scales in space and time. This intrinsic property can also be thought of as one of the main causes for observing self-similarity in CPS workloads. A powerful approach for investigating the existence of self-similarity is to move the investigation from the time domain to the frequency domain and analyze the power spectrum of CPS workloads.

通过前面的讨论，我们可以看到与CPS有关的物理过程应该在不同的时间和空间尺度下显示出必然联系。这个内在性能也可以作为我们观察CPS工作负荷自相似性的主要目标之一。研究自相似性存在的一个有效方法是将调查报告的时域换成频域，然后分析CPS负载的功率谱。

From a mathematical point of view, the power spectrum

characterizes the contribution of each frequency to the overall signal. If the power spectrum of a CPS workload follows a flat horizontal line on a logarithmic scale, then it means that its associated stochastic process does not display any correlations because each frequency plays an equally important role. The lack of correlation would be similar to having a white-noise type of behavior that appears as a flat line when represented on a log-log scale.

从数学的角度出发，功率谱描绘出每个频率对整体信号贡献。如果CPS负载的功率谱遵循对数刻度尺的平缓水平线走向，那么就意味着与它有关的随机过程没有表现出任何相关性，因为其中的每一个频率都扮演了相同重要的角色。相关性的缺失与一种白噪声类型的特性相似，这种特性在重对数图尺上描绘图形时表现为平缓的线条。

In contrast, if the power spectrum diverges for high frequencies, then the CPS workload is said to display long-term memory effects. In this case, the CPS workload exhibits a 1/fβ type of scaling, where f and β are the frequency and power law coefficient respectively. In fact, the existence of 1/fβ scaling is also referred to as the lack of any characteristic scale because the workload appears to behave similarly across all frequency scales. For example, Figure 3a shows the power spectrum of the communication throughput in a heterogeneous network with a frequency exponent of approximately 1.8; this type of behavior confirms the existence of self-similarity in these workloads.

与此相反，如果功率谱向高频率偏离，CPS负载可以说是显示了长期记忆效应。这种情况下，CPS负载显示出一种1/fβ类型的扩展，这里的f和β各自代表了频率和幂律系数。事实上，1/fβ类型扩展的存在被认为是毫无特征尺度的表现，因为这时的工作负载似乎在所有的频率范围内都有相似的表现。例如，图3a中显示异构网络中通信吞吐量的功率谱的频率指数近似于1.8；这种类型的表现证实了这些工作负载中自相似特性的存在。

From a statistical physics perspective, the existence of 1/fβ type of fluctuations indicates that the CPS workloads are actually a mix of long packets containing data and control information and short packets consisting of control flags. This is similar to many other critical

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