物理融合系统：负荷建模与设计优化 - 图文(3)

phenomena in nature and implies that CPS workloads cannot be described accurately using average values based on characteristic space-time scales.

从统计物理学的角度看，1/fβ类型波动的存在表明了CPS负载实际上是由包含着数据和控制信息的长包与由控制标志位组成的短包混合而成的。这与一些自然界的临界现象类似，暗示了CPS负载不能用基于典型时空尺度的平均值准确描述。

CPS workloads can also exhibit nonstationary behavior. Indeed, CPS workloads more often exhibit a heterogeneous set of scaling exponents, rather than a homogeneous or monofractal set. Such a nonstationary physical process can be understood by recalling the time dependency of the fluctuations in the R-R intervals (see Figure 1). The existence of various heterogeneous scaling exponents indicates that some CPS workloads may exhibit multifractal properties.

CPS负载当然也能显示出非稳定特性。事实上，CPS负载的标度指数更多地表现为异构的集合，而不是均匀的或具有单分形特征的集合。我们可以通过回想R间期（见图1）内波形的起伏所表现出的时间依赖性来理解这样一个非稳定性的物理过程。各种异构的标度指数的存在表明了一些CPS负载或许具有多重分形特征。

Simply speaking, the multifractal approach extends the concept of self-similarity to a distribution (instead of a single value) of space-time scaling exponents. Such a multifractal perspective is equivalent to stating that the multifractal spectrum encompasses the most significant short-range interactions among CPS components, which determine the overall macroscopic behavior and scaling phenomena and which are reflected in the CPS workload via certain weights and scaling exponents. For instance, Figure 3b shows the multifractal spectrum of four communication traces collected from a local area network (LAN) established between several vehicles communicating via wireless links.(For detailed information about the measurement instrumentation, refer to Mahajan et al.)We can see that although trace 1 exhibits a monofractal behavior (i.e., a narrow spike), the remaining three traces exhibit a multifractal behavior (a wide

bell-like shape).

简单地说，多重分形的方法将自相似性的概念延伸至关于时空标度指数的广义函数，替代了一个单一值。这种多重分形的观点相当于说明了多重分形谱在所有CPS组件中具有最重要的短程交互作用，这种影响决定了整个宏观行为和标度现象，并且体现在通过一定的重量和标度指数的CPS负载上。例如，图3b显示的四种通信轨迹的多重分形谱，是从建立在几个通过无线电线路通讯的媒介之间的局域网收集得到的。（更多关于仪表测量的详细的信息参考Mahajan等等）我们可以看到虽然轨迹1显示出单分形特性（也就是一道狭窄的突起），但是剩下的三道轨迹都显示了多重分形特性（一个扁的钟形图样）。

Figure 4. Cyber physical systems’ operation from physical processes to workloads (a).In (a), the processes might involve volcanic activity monitoring, precipitation formation, or traffic conditions, for example, each of which then undergoes data measurements compression and communication to data centers for further analysis. The feedback

control that enforces maintaining the quality-of-service (QoS) reference via statistical physics approaches (b). In (b), distributed controllers can dynamically estimate the workload and, based on specific QoS metrics (e.g., latency), decide on prioritizing data transmission or allocating more efficiently the communication bandwidth.

Statistical physics approaches to CPS workload modeling 运用统计物理学方法进行CPS负载建模

A natural question to ask is, How can space-time self-similarity that propagates through a networked infrastructure be captured into a mathematical description of workloads (or communication flows)?For decades, the science of systems design tacitly assumed that workloads can be modeled by linear time-invariant equations. However, due to the multifractal behavior of CPS workloads, we argue that this situation has to change. Moreover, the major developments in statistical physics (e.g., master equation, path integrals, or renormalization group theory) developed specifically for processes characterized by strong fluctuations, pseudo-periodicity, and long-range memory, for instance, should become essential tools for future CPS design. We argue, also, that it is not only necessary to estimate the correlation structure observed in CPS traffic traces, but also to incorporate such characteristics into system-dynamical state equations.

这里就有一个很自然的问题要问了，网络化基础结构传播的时空自相似性要如何归类成对工作负荷或通讯流量的数学描述？很长时间以来，设计系统的科学家们都心照不宣的假定可以通过线性定常微分方程来对工作负载建模。然而，由于CPS负载的多重分形特性，我们认为这种想法需要改变了。而且，统计物理学的主要成就（例如主方程、路径积分或者重整化群）将会成为未来CPS设计的基本工具，而这些理论是为了特定的以例如大幅波动、伪周期性和远程记忆为特征物理过程开发的。我们也认为不仅需要判断从CPS通信量痕迹中观测得出的序列结构，还需要在动态系统状态方程中加入那些特性。

To discuss the mathematical underpinnings of CPS workload modeling, we first define some parameters. As Figure 4a shows, various

types of sensors monitor diverse physical processes—for example, volcanic activity, heart rate, or CCN concentration—and communicate their measurements to specialized data centers for further analysis. For instance, a bio-implantable SoC can monitor the heart rate by constructing a time series based on its electrical activity as Figure 1 shows. The collected data can be digitized for local actuation (such as in the case of pacemakers), but, if required, it can also be packetized and communicated for further analysis to various data decision centers generating the CPS workload (see Figure 4a).Similarly, airplanes in flight can sense the movement of clouds or collect pollution measures (e.g., CO2 or CCN concentration) and communicate it to data centers for weather prediction and climate change analysis.

要讨论CPS负载建模的数学基础，我们首先要规定一些参数。就像图4中显示的，各种类型的传感器检测不同的物理过程，例如火山活动、心率、或者云凝结核浓度，然后将他们的测量数据传送到指定的信息交换中心进行进一步的分析。例如，可植入式生命检测器可以想图1中显示的通过创建一组基于其电活动的时间序列来监控心率。收集到的数据可直接由本地驱动器数字化（例如使用起搏器），但如果有其他要求，它们也可以被分装打包传送给生成CPS负载的各种数据决策中心进行进一步分析（见图4a）。相似的，航行中的飞机可以检测云层运动或者收集污染数据（例如二氧化碳或云凝结核的浓度），然后传送给数据处理中心进行气象预测和气候变化分析。

The CPS workloads typically consist of many types of data (e.g., volcanic activity or traffic conditions) transmitted over the same network. Let us denote by a(t) the stochastic process characterizing the CPS workloads (e.g., communication volume or packet delays).Because of the inherent fractal nature of many physical processes, the CPS workload can also exhibit a complex self-similar behavior. To capture this complex behavior, we define by g (y, t) a distribution function of the scaling exponents y that characterizes the CPS workload a(t).

CPS的工作负载由在同一网络上传输的数据构成（例如火山活动或交通状况数据）。让我们用a(t)来表示一个具有CPS负载（例如通信业务量或封包延迟）特征的随机过程。由于物理过程固有的多充分形性质，CPS负载也可能表现有复杂的自相似行为。为了掌握这种复杂的行为，我们定义一个分布函数g(y,t),其中y代表标度指数，来描述工作负载a(t)

的特征。

On the basis of these definitions, we can define a master equation governing the evolution of the stochastic process a(t) as follows:

基于这些定义，我们可以将一个随机过程的发展用下列主方程表示：

In this equation, P(a, t) denotes the probability of finding the system at time t in a particular state a. For instance, the atmospheric measurements done during commercial flights can be aggregated with road traffic information from cars into various heterogeneous workloads and transmitted via satellite or intermediate nodes to data centers. In this case, the stochastic process a(t) represents the amount of information communicated at a particular time.

在这个方程中，P(a,t)表示在特定状态a下，t时间内查找系统的概率。举个例子，通过民航飞机进行的大气测量可以与无论是汽车的还是异构工作负载的运输信息合计在一起，并通过卫星或中间节点传送给数据中心。这种情况下，随机过程a(t)代表了在特定时间下通讯信息的总量。

To capture the fractal features of the stochastic process a(t), the first term in Equation 1 represents the time-based dynamics of the CPS workload as a power law function rather than an exponential one. The second term is meant to describe how the power law exponent evolves as a function of the intrinsic interactions among the CPS components. More precisely, it relates the probability of the stochastic process to attain value a at time t as a weighted sum (i.e., via the g(y, t) distribution) of the previous realizations (i.e., the scaling term a/y).The reason behind introducing the g(y, t)distribution is that, in many practical situations, the fractality of the stochastic process a(t), if it exists, will be affected by a series of factors (e.g., video packets of variable length because of variations in the input stream).

为了获得随机过程a(t)的多重分形特性，方程的第一项代表了CPS负载的基于时间的

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