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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%                          %%
-%%       CHAPTER 06         %%
+%%       CHAPTER 03         %%
 %%                          %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \chapter{Energy Optimization of Heterogeneous Platforms}
 \label{ch3}
 
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}
+\newcommand{\Eind}{\Xsub{E}{ind}}
+\newcommand{\Enorm}{\Xsub{E}{Norm}}
+\newcommand{\Eoriginal}{\Xsub{E}{Original}}
+\newcommand{\Ereduced}{\Xsub{E}{Reduced}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
+\newcommand{\Fnew}{\Xsub{F}{new}}
+\newcommand{\Ileak}{\Xsub{I}{leak}}
+\newcommand{\Kdesign}{\Xsub{K}{design}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}
+\newcommand{\Ntrans}{\Xsub{N}{trans}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}
+\newcommand{\Pnorm}{\Xsub{P}{Norm}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
+\newcommand{\Tnew}{\Xsub{T}{New}}
+\newcommand{\Told}{\Xsub{T}{Old}}
+\newcommand{\Pmax}[1][]{\Xsub{P}{max}_{\fxheight{#1}}}
+\newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
+
+
+\renewcommand{\algorithmicdo}{\textbf{do}}
+\renewcommand{\algorithmicwhile}{\textbf{while}}
+\renewcommand{\algorithmicrequire}{\textbf{Require:}}
+\renewcommand{\algorithmicensure}{\textbf{Ensure:}}
+\renewcommand{\algorithmicend}{\textbf{end}}
+\renewcommand{\algorithmicif}{\textbf{if}}
+\renewcommand{\algorithmicthen}{\textbf{then}}
+
+\section{Introduction}
+\label{ch3:intro}
+
+
+  Computing platforms are consuming more and more energy due to the increasing
+  number of nodes composing them.  In a heterogeneous computing platform composed 
+  of multiple computing   nodes,  nodes may differ in the computing power from 
+  each others.  Accordingly, the fast nodes have to wait for the slow ones to finish 
+  their works. The resulting waiting times are called  idle times which are increased 
+  proportionally to the increase in the heterogeneity between the computing nodes. 
+  This leads to a big waste in the computing power and thus the energy consumed by  fast nodes.   
+  To minimize the operating costs of these  platforms many techniques have been used. 
+  Dynamic voltage and   frequency  scaling (DVFS) is one of them. It reduces the frequency 
+  of a CPU to lower its energy consumption.  However, lowering the frequency of a CPU may 
+  increase the execution time of an application running on that processor.  Therefore,
+  the frequency that gives the best trade-off between the energy consumption and
+  the performance of an application must be selected.
+
+  In this chapter, two new online frequency selecting algorithms for heterogeneous local
+  clusters (heterogeneous CPUs)  and grid platforms are presented.  
+  They select the frequencies that try to give the best
+  trade-off between energy saving and performance degradation, for each node
+  computing the synchronous message passing  application with iterations. These algorithms have a small
+  overhead and work without training or profiling. They use new energy models
+  for  message passing  synchronous applications with iterations running on both the heterogeneous
+  local cluster and the grid platform. The first proposed algorithm for a heterogeneous local 
+  cluster was evaluated on the SimGrid simulator while running the class C of the NAS parallel
+  benchmarks. The experiments conducted  over 8 heterogeneous nodes show that it reduces on 
+  average the energy consumption by  29.8\%  while limiting the performance degradation to 3.8\%.  
+  The second proposed algorithm for a grid platform was evaluated on the Grid5000 testbed 
+  platform  while running the class D of the NAS parallel benchmarks.
+  The experiments were run on 16 nodes, distributed on three clusters, and show that the algorithm reduces  
+  on average the energy consumption by 30\% while  the performance  is on average only degraded
+   by 3.2\%.  
+  Finally, both algorithms were compared to the EDP method. The comparison 
+  results show that they outperform the latter in the energy reduction and performance trade-off. 
+
+
+This chapter is organized as follows: Section~\ref{ch3:relwork} presents some
+related works from other authors.  Section~\ref{ch3:1} presents the performance and energy 
+models of synchronous message passing programs running over a heterogeneous local cluster.
+It also describes the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified. 
+Section~\ref{ch3:2} presents the simulation results of applying the algorithm on the NAS parallel 
+benchmarks class C and executing them on a heterogeneous local cluster. It shows the results of running
+three different power scenarios and comparing them. Moreover, it also shows the
+comparison results between the proposed method and an existing method.
+Section~\ref{ch3:3} shows the energy and performance models in addition to the frequencies 
+selecting algorithm of synchronous message passing programs running over a grid platform.
+Section~\ref{ch3:4} presents the results of applying the algorithm on the 
+NAS parallel benchmarks (class D) and executing them on the Grid'5000 testbed. 
+The algorithm is also evaluated over multi-core architectures and over three different power scenarios. Moreover, Section~\ref{ch3:4}, shows the comparison results between the proposed method and the EDP method.
+Finally, in Section~\ref{ch3:concl}  the chapter ends with a summary.
+
+\section{Related works}
+\label{ch3:relwork}
+
+The process of selecting the appropriate frequency for a
+processor to satisfy some objectives, while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{ref64,ref67}.
+Others used offline methods that may need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{ref58,ref91}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \dots{}
+
+In this chapter, we are interested in reducing the energy consumption when running a message passing
+synchronous applications with iterations over a heterogeneous platform.  Some
+works have already been done for such platforms which can be classified into
+two types of heterogeneous platforms:
+\begin{itemize}
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+\end{itemize}
+
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{ref68}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{ref69}, Kai Ma et al. developed a scheduling algorithm that distributes  
+workloads proportional to
+the computing power of the nodes which could be a GPU or a CPU. All the tasks
+must be completed at the same time.  In~\cite{ref70},
+Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
+DVFS operations gave better energy and performance efficiency than other clusters only
+composed of CPUs.
+
+The work presented in this chapter concerns the second type of platform, with
+heterogeneous CPUs.  Many methods were conceived to reduce the energy
+consumption of this type of platform.  Naveen et
+al.~\cite{ref71} developed a method that
+minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
+the sum of slack times that happen during synchronous communications) by
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{ref72} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than 10\%.
+In~\cite{ref73}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{ref74} and
+\cite{ref75}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+on heuristic.  Chen et
+al.~\cite{ref76} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting the given time constraint.  This approach had considerable
+overhead.  In contrast to the above described works, the work of this chapter presents the
+following contributions:
+\begin{enumerate}
+\item two new energy and two performance models for message passing 
+  synchronous applications with iterations running over a heterogeneous local cluster and a grid platform. 
+  All the models take into account the communications and the slack times. The models can predict the
+  energy  consumption and the execution time of the application.
+
+\item two new online frequencies selecting algorithms for a heterogeneous
+  local cluster and a grid platform. The algorithms have a very small overhead and do not need any
+  training or profiling. They use a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing synchronous application with iterations.
+\end{enumerate}
+
+\section[The energy optimization of a heterogeneous cluster]{The energy optimization of parallel  applications with iterations running over local heterogeneous 
+clusters}
+\label{ch3:1}
+
+\subsection{The execution time of message passing distributed 
+  applications with iterations on a heterogeneous local cluster}
+\label{ch3:1:1}
+In this section, we are interested in reducing the energy consumption of message
+passing distributed synchronous applications with iterations running over heterogeneous local clusters. 
+In this work, a heterogeneous local cluster is defined as a collection of
+heterogeneous computing nodes interconnected via a high speed homogeneous
+network. Therefore, the nodes may have different characteristics such as computing
+power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
+have the same network bandwidth and latency.
+
+\begin{figure}[h!]
+  \centering
+  \includegraphics[scale=0.8]{fig/ch3/commtasks}
+  \caption{Parallel tasks on a heterogeneous platform}
+  \label{fig:task-heter}
+\end{figure}
+
+The overall execution time of a distributed synchronous application with iterations 
+over a heterogeneous local cluster consists of the sum of the computation time and
+the communication time for every iteration on a node. However, due to the
+heterogeneous computation power of the computing nodes, slack times may occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:task-heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
+
+Reducing the frequency of a processor by applying DVFS operation can be expressed by the scaling
+factor S which is the ratio between  the maximum frequency and the new frequency of a CPU
+as in (\ref{eq:s}).
+The execution time of a compute bound sequential program is linearly
+proportional to the frequency scaling factor $S$.  On the other hand, message
+passing distributed applications consist of two parts: computation and
+communication.  The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the communications~\cite{ref53}.  The
+communication time for a task is the summation of periods of time that begin
+with an MPI call for sending or receiving a message until the message is
+synchronously sent or received.
+
+Since in a heterogeneous cluster the nodes may have different characteristics,
+especially different frequency gears, when applying DVFS operations on these
+nodes, they may get different scaling factors represented by a scaling vector:
+$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
+be able to predict the execution time of message passing synchronous 
+applications with iterations running over a heterogeneous local cluster, for different vectors of
+scaling factors, the communication time and the computation time for all the
+tasks must be measured during the first iteration before applying any DVFS
+operation. Then the execution time for one iteration of the application with any
+vector of scaling factors can be predicted using (\ref{eq:perf_heter}).
+\begin{equation}
+  \label{eq:perf_heter}
+  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \min_{i=1,2,\dots,N} (\Tcm[i])
+\end{equation}
+
+where $\TcpOld[i]$ is the computation time of processor $i$ during the first
+iteration.  The model computes the maximum computation time with
+scaling factor from each node added to the communication time of the slowest
+node. It means only the communication time without any slack time is taken into
+account.  Therefore, the execution time of the  application with iterations is equal to
+the execution time of one iteration as in (\ref{eq:perf_heter}) multiplied by the
+number of iterations of that application.
+
+This prediction model is improved from the model that predicts the execution time
+of message passing distributed applications for homogeneous
+architectures presented in Chapter \ref{ch2} Section \ref{ch2:3}.  The execution time prediction model is
+used in the method that optimizes both the energy consumption and the performance
+of parallel application with iterations, which is presented in the following sections.
+
+\subsection{Energy model for heterogeneous local cluster}
+\label{ch3:1:2}
+In Chapter \ref{ch2}, the dynamic and the static  energy consumption of a 
+processor is computed according to Equations \ref{eq:Edyn_new} and \ref{eq:Estatic_new} respectively. Then, the total energy consumption of a processor is the sum of these two metrics.  
+Therefore, the overall energy consumption for the parallel tasks over  a parallel cluster 
+is the  summation of the energies consumed by all the processors. 
+
+In the considered heterogeneous platform, each processor $i$ may have
+different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
+respectively.  Therefore, even if the distributed message passing  
+application with iterations is load balanced, the computation time of each CPU $i$ noted
+$\Tcp[i]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce the slack times.  The communication time of a processor $i$ is noted as
+$\Tcm[i]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:task-heter}.  Therefore, all the nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in
+(\ref{eq:Edyn_new}), the static energy is computed as the sum of the execution time
+of one iteration as in \ref{eq:perf_heter}  multiplied by the static power of each processor.  
+The overall energy consumption of a message passing distributed application executed over a
+heterogeneous cluster during one iteration is the summation of the dynamic and
+static energies for all the processors.  It is computed as follows:
+\begin{equation}
+  \label{eq:energy-heter}
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + 
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
+  { \min_{i=1,2,\dots,N} (\Tcm[i]) ))}
+\end{equation}
+
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the application
+and thus, increase the consumed static energy because the execution time is
+increased~\cite{ref78}. The overall energy consumption
+for an  application with iterations can be measured by measuring the energy
+consumption for one iteration as in (\ref{eq:energy-heter}) multiplied by the number
+of iterations of that application.
+
+\subsection{Optimization of both energy consumption and performance}
+\label{ch3:1:3}
+Using the lowest frequency for each processor does not necessarily give the most
+energy efficient execution of an application. Indeed, even though the dynamic
+power is reduced while scaling down the frequency of a processor, its
+computation power is proportionally decreased. Hence, the execution time might
+be drastically increased and during that time, dynamic and static powers are
+being consumed.  Therefore, it might cancel any gains achieved by scaling down
+the frequency of all nodes to the minimum and the overall energy consumption of
+the application might not be the optimal one.  It is not trivial to select the
+appropriate frequency scaling factor for each processor while considering the
+characteristics of each processor (computation power, range of frequencies,
+dynamic and static powers) and the task it is executing (computation/communication
+ratio).  In  Chapter~\ref{ch2}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous cluster executing a message passing
+ synchronous application with iterations while giving the best trade-off between the
+energy consumption and the performance for such applications.  
+In this section, this optimization method is improved while considering a heterogeneous clusters.
+
+As described before, the  relation between the energy consumption and the execution time for an
+application is complex and nonlinear. Thus, to find the trade-off relation between the energy consumption computed in Equation \ref{eq:energy-heter} and the performance with Equation \ref{eq:perf_heter}  for the  message passing applications with iterations, first we need to normalize both terms as follows:
+
+
+\begin{equation}
+  \label{eq:enorm-heter}
+  \Enorm = \frac{\Ereduced}{\Eoriginal} 
+    = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
+\end{equation}
+
+
+
+\begin{equation}
+  \label{eq:pnorm-heter}
+  \Pnorm = \frac{\Told}{\Tnew}
+          = \frac{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
+            { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \min_{i=1,2,\dots,N} (\Tcm[i])}
+\end{equation}
+
+
+\begin{figure}[!t]
+  \centering
+    \includegraphics[width=.7\textwidth]{fig/ch3/heter}
+  \caption{The energy and performance relation in heterogeneous cluster}
+  \label{fig:rel-heter}
+\end{figure}
+
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm-heter}) and the performance curve
+(\ref{eq:pnorm-heter}) over all available sets of scaling factors for the processors of the heterogeneous 
+cluster.  This represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:rel-heter}. Then the objective function has the following form:
+\begin{equation}
+  \label{eq:max-heter}
+  \MaxDist =                                                                                                                                                                                                               
+  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,F_i}
+      (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )
+\end{equation}
+where $N$ is the number of nodes and $F_i$ is the number of available frequencies
+for the node $i$.  Then, the set of scaling factors that maximizes the objective function 
+(\ref{eq:max-heter}) should be selected. 
+
+\subsection[The scaling algorithm for heterogeneous cluster]{The scaling factors selection algorithm for heterogeneous cluster }
+\label{ch3:1:4}
+
+
+\begin{algorithm}[h!]
+  \begin{algorithmic}[1]
+    % \footnotesize
+    \Require ~
+    %\begin{description}
+    \item[{$\Tcp[i]$}] array of all computation times for all nodes during one iteration and with highest frequency.
+    \item[{$\Tcm[i]$}] array of all communication times for all nodes during one iteration and with highest frequency.
+    \item[{$\Fmax[i]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[i]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[i]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[i]$}] array of the differences between two successive frequencies for all nodes.
+    %\end{description}
+    \Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors
+
+    \State $\Scp[i] \gets \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]} $
+    \State $F_{i} \gets  \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\cdots,N}$
+    \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
+    \If{(not the first frequency)}
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
+    \EndIf
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \State $\Sopt[i] \gets 1,~i=1,\dots,N. $
+    \State $\Dist \gets 0 $
+    \While {(all nodes not reach their  minimum  frequency)}
+        \If{(not the last freq. \textbf{and} not the slowest node)}
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$
+        \EndIf
+       \State $\Tnew \gets  \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \min_{i=1,2,\dots,N} (\Tcm[i])$
+%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Ereduced \gets \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + 
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
+  { \min_{i=1,2,\dots,N} (\Tcm[i]) ))} $
+       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
+      \If{$(\Pnorm - \Enorm > \Dist)$}
+        \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $
+        \State $\Dist \gets \Pnorm - \Enorm$
+      \EndIf
+    \EndWhile
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$
+  \end{algorithmic}
+  \caption{Scaling factors selection algorithm for heterogeneous cluster}
+  \label{HSA}
+\end{algorithm}
+
+
+
+\begin{algorithm}[h!]
+  \begin{algorithmic}[1]
+    % \footnotesize
+    \For {$k=1$ to \textit{some iterations}}
+      \State Computations section.
+      \State Communications section.
+      \If {$(k=1)$}
+        \State Gather all times of computation and communication from each node.
+        \State Call Algorithm \ref{HSA}.
+        \State Compute the new frequencies from the returned optimal scaling factors.
+        \State Set the new frequencies to nodes.
+      \EndIf
+    \EndFor
+  \end{algorithmic}
+  \caption{DVFS algorithm of heterogeneous platform}
+  \label{dvfs-heter}
+\end{algorithm}
+
+
+
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous application with iterations executed on a heterogeneous local cluster. It works
+online during the execution time of the  message passing program with iterations.  It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max-heter}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs-heter} shows where and when the proposed
+scaling algorithm is called in the  MPI program with iterations.
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.75]{fig/ch3/start_freq}
+  \caption{Selecting the initial frequencies in heterogeneous cluster}
+  \label{fig:st_freq-cluster}
+\end{figure}
+
+The nodes in a heterogeneous cluster may have different computing powers. 
+ The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of the fast nodes and the slow ones. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+  \label{eq:Scp}
+  \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computed scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+  \label{eq:Fint}
+  F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
+\end{equation}
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency.  In
+Figure~\ref{fig:st_freq-cluster}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq-cluster}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption.  Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to  Equation (\ref{eq:perf_heter}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear.  The new overall
+energy consumption and execution time are computed according to the new scaling
+factors.  The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max-heter}).
+
+Figure~\ref{fig:rel-heter} illustrates the normalized performance and
+consumed energy for an application running on a
+heterogeneous cluster  while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster, as in the figure \ref{fig:rel} (a), 
+the search for the optimal scaling
+factor should start from the maximum frequency because the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand, in
+the heterogeneous cluster the performance is maintained at the beginning of the
+plot even if the frequencies of the faster nodes decrease until the computing
+power of scaled down nodes are lower than the slowest node. In other words,
+until they reach the higher bound. It can also be noticed that the higher the
+difference between the faster nodes and the slower nodes is, the bigger the
+maximum distance between the energy curve and the performance curve is which results in bigger energy savings. 
+
+\subsection{The evaluation of the proposed algorithm}
+\label{ch3:1:5}
+The accuracy of the proposed algorithm mainly depends on the execution time
+prediction model defined in (\ref{eq:perf_heter}) and the energy model computed by Equation 
+(\ref{eq:energy-heter}).  The energy model is also significantly dependent on the
+execution time model because the static energy is linearly related to the
+execution time and the dynamic energy is related to the computation time. So,
+all the works presented in this chapter are based on the execution time model. To
+verify this model, the predicted execution time was compared to the real
+execution time over SimGrid/SMPI simulator,
+v3.10~\cite{ref66}, for all the NAS
+parallel benchmarks NPB v3.3 \cite{ref65}, running class B on
+8 or 9 nodes. The comparison showed that the proposed execution time model is
+very accurate, the maximum normalized difference between the predicted execution
+time and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since the proposed algorithm is not an exact method, it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+cluster composed of four different types of nodes having the characteristics
+presented in Table~\ref{table:platform-cluster}, it takes on average  0.04 \textit{ms} for 4
+nodes and 0.15 \textit{ms} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F_i \cdot N)$, where $F_i$ is the
+maximum number of available frequencies in the node $i$, and $N$ is the number of computing
+nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
+frequency scaling factors that gives the results of the next sections.
+
+\begin{table}[h!]
+  \caption{Heterogeneous nodes characteristics}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Node   & Simulated & Max   & Min   & Diff. & Dynamic    & Static    \\
+    type   & GFLOPS    & Freq. & Freq. & Freq. & power      & power     \\
+           &           & GHz   & GHz   & GHz   &            &           \\
+    \hline
+    1      & 40        & 2.50  & 1.20  & 0.100 & 20 W & 4 W \\
+    \hline
+    2      & 50        & 2.66  & 1.60  & 0.133 & 25 W & 5 W \\
+    \hline
+    3      & 60        & 2.90  & 1.20  & 0.100 & 30 W & 6 W \\
+    \hline
+    4      & 70        & 3.40  & 1.60  & 0.133 & 35 W & 7 W \\
+    \hline
+  \end{tabular}
+  \label{table:platform-cluster}
+\end{table}
+
+\section{Experimental results over a heterogeneous  local cluster}
+\label{ch3:2}
+To evaluate the efficiency and the overall energy consumption reduction of
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3 which 
+is composed of synchronous message passing applications. The 
+experiments were executed on the simulator SimGrid/SMPI which offers easy tools
+to create a heterogeneous local cluster and run message passing applications over it.
+The heterogeneous cluster that was used in the experiments, had one core per
+node because just one process was executed per node.  The heterogeneous cluster
+was composed of four types of nodes. Each type of nodes had different
+characteristics such as the maximum CPU frequency, the number of available
+frequencies and the computational power, see Table~\ref{table:platform-cluster}. The
+characteristics of these different types of nodes are inspired from the
+specifications of real Intel processors.  The heterogeneous cluster had up to
+144 nodes and had nodes from the four types in equal proportions, for example if
+a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
+constructors of CPUs do not specify the dynamic and the static power of their
+CPUs, for each type of node they were chosen proportionally to their computing
+powers (FLOPS).  The dynamic power corresponds to 80\% of the overall power consumption while executing at 
+the higher frequency and the
+remaining 20\% is the static power. The same assumption was made in Chapter \ref{ch2} and
+\cite{ref3}.  Finally, These nodes were connected via an Ethernet network with 1 \textit{Gbit/s} bandwidth.
+
+
+\subsection{The experimental results of the scaling algorithm }
+\label{ch3:2:1}
+
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+MG, FT, BT, LU and SP). The benchmarks were executed with class C while being 
+run on different number of nodes, ranging from 8 to 128 or 144 nodes depending 
+on the benchmark being executed. 
+Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on 1,
+2, 4, 8, 16, 32, 64, or 128 nodes.  The other benchmarks such as BT and SP had
+to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.
+
+
+  
+ \begin{table}[h!]
+  \caption{Running NAS benchmarks on 8 and 9 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      &  36.11    &  3263.49      & 31.25    & 7.12          & 24.13    \\
+    \hline
+    MG      &   8.99    &   953.39      & 33.78    & 6.41          & 27.37    \\
+    \hline
+    EP      &  40.39    &  5652.81      & 27.04    & 0.49          & 26.55    \\
+    \hline
+    LU      & 218.79    & 36149.77      & 28.23    & 0.01          & 28.22    \\
+    \hline
+    BT      & 166.89    & 23207.42      & 32.32    & 7.89          & 24.43    \\
+    \hline
+    SP      & 104.73    & 18414.62      & 24.73    & 2.78          & 21.95    \\
+    \hline
+    FT      &  51.10    &  4913.26      & 31.02    & 2.54          & 28.48    \\
+    \hline
+  \end{tabular}
+  \label{table:res_8n}
+ \end{table}
+
+  \medskip
+ \begin{table}[h!]
+  \caption{Running NAS benchmarks on 16 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      &  31.74    &  4373.90      & 26.29    & 9.57          & 16.72    \\
+    \hline
+    MG      &   5.71    &  1076.19      & 32.49    & 6.05          & 26.44    \\
+    \hline
+    EP      &  20.11    &  5638.49      & 26.85    & 0.56          & 26.29    \\
+    \hline
+    LU      & 144.13    & 42529.06      & 28.80    & 6.56          & 22.24    \\
+    \hline
+    BT      &  97.29    & 22813.86      & 34.95    & 5.80          & 29.15    \\
+    \hline
+    SP      &  66.49    & 20821.67      & 22.49    & 3.82          & 18.67    \\
+    \hline
+    FT      &  37.01    &  5505.60      & 31.59    & 6.48          & 25.11    \\
+    \hline
+  \end{tabular}
+  \label{table:res_16n}
+ \end{table}
+
+  \medskip
+ \begin{table}[h!]
+  \caption{Running NAS benchmarks on 32 and 36 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      & 32.35     &  6704.21      & 16.15    & 5.30          & 10.85    \\
+    \hline
+    MG      &  4.30     &  1355.58      & 28.93    & 8.85          & 20.08    \\
+    \hline
+    EP      &  9.96     &  5519.68      & 26.98    & 0.02          & 26.96    \\
+    \hline
+    LU      & 99.93     & 67463.43      & 23.60    & 2.45          & 21.15    \\
+    \hline
+    BT      & 48.61     & 23796.97      & 34.62    & 5.83          & 28.79    \\
+    \hline
+    SP      & 46.01     & 27007.43      & 22.72    & 3.45          & 19.27    \\
+    \hline
+    FT      & 28.06     &  7142.69      & 23.09    & 2.90          & 20.19    \\
+    \hline
+  \end{tabular}
+  \label{table:res_32n}
+ \end{table}
+
+  \medskip
+ \begin{table}[h!]
+  \caption{Running NAS benchmarks on 64 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      & 46.65     &  17521.83     &  8.13    &  1.68         &  6.45    \\
+    \hline
+    MG      &  3.27     &   1534.70     & 29.27    & 14.35         & 14.92    \\
+    \hline
+    EP      &  5.05     &   5471.11     & 27.12    &  3.11         & 24.01    \\
+    \hline
+    LU      & 73.92     & 101339.16     & 21.96    &  3.67         & 18.29    \\
+    \hline
+    BT      & 39.99     &  27166.71     & 32.02    & 12.28         & 19.74    \\
+    \hline
+    SP      & 52.00     &  49099.28     & 24.84    &  0.03         & 24.81    \\
+    \hline
+    FT      & 25.97     &  10416.82     & 20.15    &  4.87         & 15.28    \\
+    \hline
+  \end{tabular}
+  \label{table:res_64n}
+ \end{table}
+
+  \medskip \begin{table}[h!]
+  \caption{Running NAS benchmarks on 128 and 144 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      & 56.92     &  41163.36     &  4.00    &  1.10         &  2.90    \\
+    \hline
+    MG      &  3.55     &   2843.33     & 18.77    & 10.38         &  8.39    \\
+    \hline
+    EP      &  2.67     &   5669.66     & 27.09    &  0.03         & 27.06    \\
+    \hline
+    LU      & 51.23     & 144471.90     & 16.67    &  2.36         & 14.31    \\
+    \hline
+    BT      & 37.96     &  44243.82     & 23.18    &  1.28         & 21.90    \\
+    \hline
+    SP      & 64.53     & 115409.71     & 26.72    &  0.05         & 26.67    \\
+    \hline
+    FT      & 25.51     &  18808.72     & 12.85    &  2.84         & 10.01    \\
+    \hline
+  \end{tabular}
+  \label{table:res_128n}
+\end{table}
+
+\begin{figure}[h!]
+  \centering
+  \centering
+    \includegraphics[width=.7\textwidth]{fig/ch3/energy}\\~ ~ ~ ~ ~(a) \\ 
+    
+    \includegraphics[width=.7\textwidth]{fig/ch3/per_deg}\\~ ~ ~ ~ ~(b) 
+  \caption{NAS benchmarks running with a different number of nodes  (a) the  energy saving and 
+      (b) the performance degradation }
+      \label{fig:res}
+\end{figure}
+
+The overall energy consumption was computed for each instance according to the
+energy consumption model (\ref{eq:energy-heter}), with and without applying the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.  The results are presented in Tables 
+\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
+values obtained from many experiments for energy savings and performance degradation.
+The tables show the experimental results for running the NAS parallel benchmarks
+on different numbers of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to 34\%) and tries to
+limit the performance degradation.  They also show that the energy saving
+percentage decreases when the number of computing nodes increases.  This
+reduction is due to the increase of the communication times compared to the
+execution times when the benchmarks are run over a higher number of nodes.
+Indeed, the benchmarks with the same class, C, are executed on different numbers
+of nodes, so the computation required for each iteration is divided by the
+number of computing nodes.  On the other hand, more communications are required
+when increasing the number of nodes so the static energy increases linearly
+according to the communication time and the dynamic power is less relevant in
+the overall energy consumption.  Therefore, reducing the frequency with
+Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It
+can also be noticed that for the benchmarks EP and SP that contain little or no
+communications, the energy savings are not significantly affected by the high
+number of nodes.  No experiments were conducted using bigger classes than D,
+because they require a lot of memory (more than 64 \textit{GB}) when being executed
+by the simulator on one machine.  The maximum distance between the normalized
+energy curve and the normalized performance for each instance is also shown in
+the result tables. It decreases in the same way as the energy saving percentage.
+The tables also show that the performance degradation percentage is not
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
+
+Figure~\ref{fig:res} (a) and (b) present the energy saving and
+performance degradation respectively for all the benchmarks according to the
+number of used nodes. As shown in the first plot, the energy saving percentages
+of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
+increase. While for the EP and SP benchmarks, the energy saving percentage is
+not affected by the increase of the number of computing nodes, because in these
+benchmarks there are little or no communications. Finally, the energy saving of
+the CG benchmark significantly decreases when the number of nodes increase
+because this benchmark has more communications than the others. The second plot
+shows that the performance degradation percentages of most of the benchmarks
+decrease when they run on a big number of nodes because they spend more time
+communicating than computing, thus, scaling down the frequencies of some nodes
+has less effect on the performance.
+
+\subsection{The results for different power consumption scenarios}
+\label{ch3:2:2}
+
+The results of the previous section were obtained while using processors that
+consume during computation an overall power which is 80\% composed of
+dynamic power and  20\% of static power. In this section, these ratios
+are changed and two new power scenarios are considered in order to evaluate how
+the proposed algorithm adapts itself according to the static and dynamic power
+values.  The two new power scenarios are the following:
+
+\begin{itemize}
+\item 70\% of dynamic power and 30\% of static power
+\item 90\% of dynamic power and 10\% of static power
+\end{itemize}
+
+The NAS parallel benchmarks were executed again over processors that follow the
+new power scenarios.  The class C of each benchmark was run over 8 or 9 nodes
+and the results are presented in Tables~\ref{table:res_s1} and
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+70\%-30\% scenario is smaller for all benchmarks compared to the
+energy saving of the 90\%-10\% scenario.  Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the 70\%-30\% scenario. On the other hand,
+the performance degradation percentage is smaller in the 70\%-30\%
+scenario compared to the 90\%-10\% scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy.  Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases.  Therefore, the proposed algorithm does not
+significantly scale down  the frequencies of the nodes in order to
+limit the increase of the execution time and thus limiting the effect of the
+consumed static energy.
+
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:powers-heter} (a). It shows the average of the performance degradation,
+the energy saving and the distances for all the NAS benchmarks running  class C on 
+8 or 9 nodes.  The comparison shows that the energy saving ratio is proportional
+to the dynamic power ratio: it is increased when applying the
+90\%-10\% scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the 70\%-30\% scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings.  Moreover, the average of the
+performance degradation is decreased when using a higher ratio for the static power
+(e.g.  70\%-30\% scenario and 80\%-20\% scenario). Since the proposed 
+algorithm optimizes the energy consumption when
+using a higher ratio for the dynamic power, the algorithm selects bigger frequency
+scaling factors that results in more energy saving but degrade the performance, for
+example see Figure~\ref{fig:powers-heter} (b). The opposite happens when using a
+higher ratio for the static power, the algorithm proportionally selects smaller
+scaling values which results in less energy saving but also less performance
+degradation.
+
+\begin{table}[!t]
+  \caption{The results of the 70\%-30\% power scenario}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{6}{r|}}
+    \hline
+    Program & Energy        & Energy   & Performance   & Distance \\
+    name    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      &  4144.21      & 22.42    & 7.72          & 14.70    \\
+    \hline
+    MG      &  1133.23      & 24.50    & 5.34          & 19.16    \\
+   \hline
+    EP      &  6170.30      & 16.19    & 0.02          & 16.17    \\
+   \hline
+    LU      & 39477.28      & 20.43    & 0.07          & 20.36    \\
+    \hline
+    BT      & 26169.55      & 25.34    & 6.62          & 18.71    \\
+   \hline
+    SP      & 19620.09      & 19.32    & 3.66          & 15.66    \\
+   \hline
+    FT      &  6094.07      & 23.17    & 0.36          & 22.81    \\
+    \hline
+  \end{tabular}
+  \label{table:res_s1}
+\end{table}
+
+\begin{table}[!t]
+  \caption{The results of the 90\%-10\% power scenario}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{6}{r|}}
+    \hline
+    Program & Energy          & Energy      & Performance   & Distance \\
+    name    & consumption/J   & saving\%    & degradation\% &          \\
+    \hline
+    CG      &  2812.38        & 36.36       &  6.80         & 29.56    \\
+    \hline
+    MG      &   825.43        & 38.35       &  6.41         & 31.94    \\
+    \hline
+    EP      &  5281.62        & 35.02       &  2.68         & 32.34    \\
+    \hline
+    LU      & 31611.28        & 39.15       &  3.51         & 35.64    \\
+    \hline
+    BT      & 21296.46        & 36.70       &  6.60         & 30.10    \\
+    \hline
+    SP      & 15183.42        & 35.19       & 11.76         & 23.43    \\
+    \hline
+    FT      &  3856.54        & 40.80       &  5.67         & 35.13    \\
+    \hline
+  \end{tabular}
+  \label{table:res_s2}
+\end{table}
+
+\begin{table}[!t]
+  \caption{Comparing the MaxDist algorithm to the EDP method}
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Program & \multicolumn{2}{c|}{Energy saving \%}
+            & \multicolumn{2}{c|}{Perf.  degradation \%}
+            & \multicolumn{2}{c|}{Distance} \\
+    \cline{2-7}
+    name    & EDP   & MaxDist & EDP  & MaxDist & EDP   & MaxDist \\
+    \hline
+    CG      & 27.58 & 31.25   & 5.82 & 7.12    & 21.76 & 24.13   \\
+    \hline
+    MG      & 29.49 & 33.78   & 3.74 & 6.41    & 25.75 & 27.37   \\
+    \hline
+    LU      & 19.55 & 28.33   & 0.00 & 0.01    & 19.55 & 28.22   \\
+    \hline
+    EP      & 28.40 & 27.04   & 4.29 & 0.49    & 24.11 & 26.55   \\
+    \hline
+    BT      & 27.68 & 32.32   & 6.45 & 7.87    & 21.23 & 24.43   \\
+    \hline
+    SP      & 20.52 & 24.73   & 5.21 & 2.78    & 15.31 & 21.95   \\
+    \hline
+    FT      & 27.03 & 31.02   & 2.75 & 2.54    & 24.28 & 28.48   \\
+    \hline
+  \end{tabular}
+  \label{table:compare_EDP}
+\end{table}
+
+\begin{figure}[h!]
+  \centering
+ 
+    \includegraphics[width=.75\textwidth]{fig/ch3/sen_comp}\\~ ~ ~ ~ ~ (a)\\
+
+    \includegraphics[width=.75\textwidth]{fig/ch3/three_scenarios}\\~ ~ ~ ~ ~ (b)
+ 
+  \caption{(a) Comparison the results of the three power scenarios and
+    (b) Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes}
+     \label{fig:powers-heter}
+\end{figure}
+
+\begin{figure}[h!]
+  \centering
+   \includegraphics[scale=0.85]{fig/ch3/compare_EDP.pdf}
+  \caption{Trade-off comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\end{figure}
+
+
+\subsection{Comparison between  the proposed scaling algorithm and the EDP method}
+\label{ch3:2:3}
+In this section, the scaling factors selection algorithm, called MaxDist, is
+compared to  \cite{ref67},  EDP method.  They developed a
+green governor that regularly applies an online frequency selecting algorithm to
+reduce the energy consumed by a multi-core architecture without degrading much
+its performance. The algorithm selects the frequencies that minimize the energy
+and delay product, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$,  using
+the predicted overall energy consumption and execution time delay for each
+frequency.  To fairly compare both algorithms, the same energy and execution
+time models, Equations (\ref{eq:energy-heter}) and (\ref{eq:perf_heter}), were used for both
+algorithms to predict the energy consumption and the execution times. 
+Spiliopoulos et al. algorithm was adapted to start the search from the initial
+frequencies computed using  Equation (\ref{eq:Fint}). The resulting algorithm
+is an exhaustive search algorithm that minimizes the EDP and has the initial
+frequencies values as an upper bound.
+
+Both algorithms were applied to the parallel NAS benchmarks to compare their
+efficiency. Table~\ref{table:compare_EDP} presents  the
+execution times and the energy consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in 29.76\% energy
+saving while their algorithm saves just 25.75\%. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about 4\%.
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+the   energy reduction  to  performance trade-off, see  Figure~\ref{fig:compare_EDP},
+because it maximizes the distance  between the energy saving and the performance
+degradation values while giving the same weight for both metrics.
+
+\section[The energy optimization of grid]{The energy optimization of parallel  applications with iterations  running over grids}
+\label{ch3:3}
+
+\subsection{The energy and performance models of grid platform}
+\label{ch3:3:1}
+In this section, we are interested in reducing the energy consumption of message
+passing   applications with synchronous iterations running over
+heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
+heterogeneous computing clusters interconnected via a long distance network which has a lower bandwidth 
+and a higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via a high speed network. However, nodes from distinct  clusters may have different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
+
+Since in a heterogeneous grid each cluster has different characteristics,
+ when applying DVFS operations on the nodes 
+of these clusters, they may get different scaling factors represented by a scaling vector:
+$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$. To
+be able to predict the execution time of message passing  
+applications with synchronous iterations running over a heterogeneous grid, for different vectors of
+scaling factors, the communication time and the computation time for all the
+tasks must be measured during the first iteration before applying any DVFS
+operation. Then the execution time for one iteration of the application with any
+vector of scaling factors can be predicted using Equation (\ref{eq:perf-grid}).
+%
+\begin{equation}
+  \label{eq:perf-grid}
+  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots,M_i}}  (\Tcm[hj])
+\end{equation}
+%
+where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
+ cluster $i$, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
+and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
+first  iteration.  The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
+and the slowest communication time without slack time during one iteration.
+The latter is equal to the  communication time of the slowest node in the slowest cluster $h$.
+It means that only the communication time without any slack time is taken into account.
+Therefore, the execution time of the parallel  application with iterations is equal to
+the execution time of one iteration as in Equation (\ref{eq:perf-grid}) multiplied by the
+number of iterations of that application.
+
+
+In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
+different dynamic and static powers from the nodes of the other clusters, 
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Therefore, even if the distributed 
+message passing  application with iterations is load balanced, the computation time of each CPU $j$ 
+in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce slack times.  The communication time of a processor $j$ in cluster $i$ is noted as
+$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:task-heter}.  Therefore, all nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in Equation 
+(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
+of one iteration multiplied by the static power of each processor.  The overall
+energy consumption of a message passing distributed application executed over a
+heterogeneous grid platform during one iteration is the summation of all dynamic and
+static energies for $M_i$ processors in $N$ clusters.  It is computed as follows:
+\begin{equation}
+  \label{eq:energy-grid}
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
+ \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot 
+  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
+\end{equation}
+
+
+To optimize both of the energy consumption model computed by \ref{eq:energy-grid} and the performance model computed by \ref{eq:perf-grid},
+they must be normalized  as in Equation \ref{eq:enorm-heter} and Equation \ref{eq:pnorm-heter}  respectively.
+While the original energy consumption is the consumed energy with the 
+maximum frequency for all the  nodes computed as follows:
+
+\begin{equation}
+  \label{eq:eorginal-grid}
+    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot  \Tcp[ij])  + 
+     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)       
+\end{equation}
+
+By the same way, the old execution time with the maximum frequency for all the nodes is computed as follows:
+
+\begin{equation}
+  \label{eq:told-grid}
+   \Told =  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]}) 
+  +\mathop{\min_{j=1,\dots,M_i}}  (\Tcm[hj])          
+\end{equation}
+Therefore, the objective function can be modelled in order to find the maximum
+distance between the normalized energy curve  and the normalized performance curve
+over all possible sets of scaling factors as follows:
+
+ \begin{equation}
+  \label{eq:max-grid}
+  \MaxDist =
+\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}}_{k=1,\dots,F_j}
+      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
+\end{equation}
+
+where $N$ is the number of clusters, $M_i$ is the number of nodes in each cluster and
+$F_j$ is the number of available frequencies for the  node $j$.  Then, the optimal set 
+of scaling factors that satisfies (\ref{eq:max-grid}) can be selected. 
+
+\subsection{The scaling factors selection algorithm for a grid architecture}
+\label{ch3:3:2}
+
+\begin{algorithm}
+\setstretch{1}
+  \begin{algorithmic}[1]
+    % \footnotesize
+    
+    \Require ~
+
+    \item [{$N$}] number of clusters in the grid. 
+    \item [{$M_i$}] number of nodes in each cluster.
+    \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
+    \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
+    \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[ij]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
+
+    \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$,  a vector of scaling factors that gives the optimal trade-off between energy consumption and execution time
+
+    \State $\Scp[ij] \gets \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]} $
+    \State $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
+    \State Round the computed initial frequencies $F_i$ to the closest  available frequency for each node.
+    \If{(not the first frequency)}
+          \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
+    \EndIf
+    \State $\Told \gets  \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\Tcp[ij]}) 
+  +\mathop{\min\limits_{j=1,\dots,M_i}}  (\Tcm[hj])   $ 
+    \State $\Eoriginal \gets \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot  \Tcp[ij])  + 
+     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)  $ 
+    \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
+    \State $\Dist \gets 0 $
+    \While {(all nodes have not reached their  minimum   frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
+        \If{(not the last freq. \textbf{and} not the slowest node)}
+        \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
+        \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
+        \EndIf
+       \State $\Tnew \gets \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\TcpOld[ij]}
+        \cdot S_{ij})  +\mathop{\min\limits_{j=1,\dots,M_i}}  (\Tcm[hj]) $
+       \State $\Ereduced \gets  \sum\limits_{i=1}^{N} \sum\limits_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] 
+        \cdot \Tcp[ij])} +  \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} (\Ps[ij] \cdot  
+        (\mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 
+         +\mathop{\min\limits_{j=1,\dots,M_i}} (\Tcm[hj]) ))$ 
+       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+  
+       \State $\Enorm \gets \frac{\Ereduced}{\Eoriginal}$
+      \If{$(\Pnorm - \Enorm > \Dist)$}
+        \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i$
+        \State $\Dist \gets \Pnorm - \Enorm$
+      \EndIf
+    \EndWhile
+    \State  Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
+  \end{algorithmic}
+  \caption{Scaling factors selection algorithm for grid}
+  \label{HSA-grid}
+\end{algorithm}
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.7]{fig/ch3/init_freq}
+  \caption{Selecting the initial frequencies in the grid architecture}
+  \label{fig:st_freq-grid}
+\end{figure}
+
+\begin{figure}[!t]
+  \centering
+    \includegraphics[width=.7\textwidth]{fig/ch3/heter2}
+  \caption{The energy and performance relation in grid}
+  \label{fig:rel-grid}
+\end{figure}
+
+
+In this section, the scaling factors selection algorithm for a grid, Algorithm~\ref{HSA-grid}, 
+is presented. It selects the vector of  frequency
+scaling factors  that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+  application with synchronous iterations executed on a grid.
+It is similar to the frequency selection algorithm for heterogeneous 
+local clusters presented in Section \ref{ch3:1:4}. 
+
+The value of the initial frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered in the first iteration. The initial
+frequency scaling factor for a node $i$ is computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+  \label{eq:Scp-grid}
+  \Scp[ij] =  \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp-grid}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of the node  and its computed scaling factor, as follows:
+\begin{equation}
+  \label{eq:Fint-grid}
+  F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M_i}
+\end{equation}
+Figure \ref{fig:st_freq-grid} shows the selected initial frequencies for a grid composed of three clusters.
+In contrast to algorithm \ref{HSA}, algorithm \ref{HSA-grid} replaces the computed initial frequency for a node by the nearest available frequency if not available in the gears of
+that node.
+The frequency scaling algorithm of the grid stops its iteration if it reaches the lower bound, which is the frequency that gives a negative distance between the energy and performance.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio as in figure \ref{fig:rel-grid}.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 
+Therefore, the algorithm iterates on all the remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
+energy consumption and execution time. Then, it selects the  vector of  frequency scaling
+factors that give the maximum distance (MaxDist).  Algorithm~\ref{dvfs-heter} is also used to call the Algorithm \ref{HSA-grid} in the MPI program executed over the grid platform. 
+
+\section{Experimental results over the Grid5000 platform}
+\label{ch3:4}
+
+In this section,  real experiments were conducted over the Grid'5000 platform. 
+Grid'5000~\cite{ref21} is a large-scale testbed that consists of ten sites distributed all over  metropolitan France and Luxembourg.  These sites are: Grenoble, Lille, Luxembourg, Lyon, Nancy, Reims, Rennes , Sophia, Toulouse and Bordeaux. Figure \ref{fig:grid5000-dis} shows the geographical distribution of grid'5000 sites over France and Luxembourg. All the sites are connected together via a  special long distance network called RENATER, which is the abbreviation of the French
+National Telecommunication Network for Technology. Each site in the grid is
+composed of a few heterogeneous computing clusters and each cluster contains
+many homogeneous nodes. In total, Grid'5000 has about one thousand heterogeneous nodes and eight thousand cores. In each site, the clusters and their nodes
+are connected via high speed local area networks. Two types of local networks
+are used, Ethernet or Infiniband networks, which have different characteristics
+in terms of bandwidth and latency.
+Grid'5000 is dedicated for research experiments and  users can book nodes from different sites to conduct their experiments. 
+It also gives the opportunity to the users to deploy their customized operating system over the reserved nodes.
+Indeed, many software tools are available for  users in order to control and manage the reservation and deployment processes remotely. For example, OAR \cite{ref22} is a  batch scheduler that is used to manage the heterogeneous resources of the grid'5000.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=1]{fig/ch3/grid5000.pdf}
+\caption{Grid5000's sites distribution in France and Luxembourg}
+\label{fig:grid5000-dis}
+\end{figure}
+
+
+Moreover, the Grid'5000 testbed provides at some sites a power measurement tool to capture 
+the power consumption for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant.  For more details refer to \cite{ref79}. 
+In order to correctly measure the CPU power of one core in a node $j$, 
+ firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
+dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 
+
+
+The dynamic power $\Pd[j]$ is computed as in Equation~\ref{eq:pdyn}
+\begin{equation}
+  \label{eq:pdyn}
+    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
+
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured maximum power values, 
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum 
+measured value in maximum powers vector and the minimum measured value in the idle powers vector.
+
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in Grid'5000 there is no way to measure precisely the consumed static power and it was assumed, as in Sections  \ref{ch3:2} and \ref{ch2:6}, that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed to be equal to 20\% of the dynamic power consumption of the core.
+
+In the experiments presented in the following sections, two sites of Grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as shown in Figure~\ref{fig:grid5000}.
+
+Four clusters from the two sites were selected in the experiments: one cluster from 
+Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 
+Griffon and Graphite. Each one of these clusters composed of homogeneous nodes, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
+frequency ranges and local network features: the bandwidth and the latency.  Table~\ref{table:grid5000-1} shows 
+the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using Equation~\ref{eq:pdyn} for all the nodes in the 
+selected clusters and are presented in Table~\ref{table:grid5000-1}.
+
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=1.4]{fig/ch3/grid5000-2}
+  \caption{The selected two sites of Grid'5000}
+  \label{fig:grid5000}
+\end{figure}
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.8]{fig/ch3/power_consumption.pdf}
+  \caption{The power consumed by one core from the Taurus cluster}
+  \label{fig:power_cons}
+\end{figure}
+
+
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{ref65} and evaluated over Grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In the next sections,  the class D was used for all the benchmarks in all the experiments. 
+
+  
+\begin{table}[!t]
+  \caption{The characteristics of the CPUs in the selected clusters}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{c|}}
+    \hline
+                &             & Max   & Min   & Diff. &                 &               \\
+    Cluster     & CPU         & Freq. & Freq. & Freq. & Cores           & Dynamic power \\
+    Name        & model       & GHz   & GHz   & GHz   & per CPU         & of one core   \\
+    \hline
+                & Intel       &       &       &         &           &              \\
+    Taurus      & Xeon        & 2.3   & 1.2   & 0.1     & 6         & \np[W]{35}    \\
+                & E5-2630     &       &       &         &           &            \\         
+    \hline
+                & Intel       &       &       &         &           &             \\
+    Graphene    & Xeon        & 2.53  & 1.2   & 0.133   & 4         & \np[W]{23}  \\
+                & X3440       &       &       &         &           &             \\    
+    \hline
+                & Intel       &       &       &         &           &            \\
+    Griffon     & Xeon        & 2.5   & 2     & 0.5     & 4         & \np[W]{46}  \\
+                & L5420       &       &       &         &           &            \\  
+    \hline
+                & Intel       &       &       &         &           &            \\
+     Graphite   & Xeon        & 2     & 1.2   & 0.1     & 8         & \np[W]{35} \\
+                & E5-2650     &       &       &         &           &            \\  
+    \hline
+  \end{tabular}
+  \label{table:grid5000-1}
+\end{table} 
+
+
+\subsection{The experimental results of the scaling algorithm on a Grid}
+\label{ch3:4:1}
+In this section, the results of applying  the scaling factors selection algorithm  
+to the NAS parallel benchmarks are presented. 
+As mentioned previously, the experiments 
+were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 
+Two scenarios were considered while selecting the clusters from these two sites :
+\begin{itemize}
+\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
+ via a long distance network.
+\item In the second scenario nodes from three clusters located in one site, Nancy's site, were selected.  
+\end{itemize}
+
+The main reason 
+for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
+scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
+is very low due to the higher communication times which reduces the effect of the DVFS operations.
+
+The NAS parallel benchmarks are executed over 
+16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
+is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
+Table~\ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
+
+\begin{table}[h]
+
+\caption{The different grid scenarios}
+\centering
+\begin{tabular}{|*{4}{c|}}
+\hline
+\multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
+                                      & Cluster & Site           & Nodes per cluster     \\ 
+\hline
+\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
+                                      & Griffon       & Nancy        & 6                      \\ 
+\hline
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
+                                      & Griffon     &Nancy           & 12                     \\ 
+\hline
+\multirow{3}{*}{One site / 16 nodes}  & Graphite    & Nancy            & 4                      \\ \cline{2-4} 
+                                      & Graphene     & Nancy           & 6                      \\ \cline{2-4} 
+                                      & Griffon         & Nancy        & 6                      \\ 
+\hline
+\multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
+                                      & Graphene      & Nancy          & 14                     \\ \cline{2-4} 
+                                      & Griffon          & Nancy       & 14                       \\ 
+\hline
+\end{tabular}
+ \label{tab:sc}
+\end{table}
+
+
+The NAS parallel benchmarks are executed over these two platforms
+with different number of nodes, as in Table~\ref{tab:sc}. 
+The overall energy consumption of all the benchmarks solving the class D instance and
+using the proposed frequency selection algorithm is measured 
+using Equation~\ref{eq:energy-grid}.
+
+The energy consumptions  and the execution times for all the benchmarks are 
+presented in Figures~\ref{fig:exp-time-energy} (a) and (b) respectively.
+For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site on 16 and 32 nodes is lower than the energy consumed while using two sites. 
+The long distance communications between the two distributed sites increase the idle times, which lead to more static energy consumption. 
+
+The execution times of these benchmarks 
+over one site with 16 and 32 nodes are also lower than those of the  two sites 
+scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times  approximately halved when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).
+
+However, the execution times and the energy consumptions of the EP and MG
+benchmarks, which have no or small communications, are not significantly
+affected in both scenarios, even when the number of nodes is doubled.  On the
+other hand, the communication times of the rest of the benchmarks increase when
+using long distance communications between two sites or when increasing the number of
+computing nodes.
+
+
+The energy saving percentage is computed as the ratio between the reduced 
+energy consumption, Equation~\ref{eq:energy-grid}, and the original energy consumption,
+Equation~\ref{eq:eorginal-grid}, for all the benchmarks as in Figure~\ref{fig:eng_s}. 
+This figure shows that the energy saving percentages of the one site scenario for
+16 and 32 nodes are bigger than those of the two sites scenario which is due
+to the higher  computations to communications ratio in the first scenario   
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computation times are bigger than the communication times which
+results in  a lower energy consumption. Indeed, the dynamic  consumed power
+is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 
+increase the communication times and thus produces less energy saving depending on the 
+benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
+energy saving percentage in the one site scenario when executed over 16 nodes than on 32 nodes.  LU and SP consume more energy with 16 nodes than with 32 nodes  on one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
+\begin{figure}[!h]
+  \centering
+  \centering
+   \includegraphics[width=.7\textwidth]{fig/ch3/eng_con_scenarios.eps}\\~~~~~~~~~(a)\\
+   \includegraphics[width=.7\textwidth]{fig/ch3/time_scenarios.eps}\\~~~~~~~~~(b)
+  \caption{ (a) energy consumption and (b) execution time of NAS  Benchmarks over different scenarios}
+  \label{fig:exp-time-energy} 
+\end{figure}
+
+\begin{figure*}[!h]
+\centering
+\includegraphics[width=.7\textwidth]{fig/ch3/eng_s.eps}
+\caption{The energy reduction percentages while executing the NAS benchmarks over different scenarios}
+\label{fig:eng_s}
+\end{figure*}
+
+\begin{figure*}[!h]
+\centering
+\includegraphics[width=.7\textwidth]{fig/ch3/per_d.eps}
+\caption{The performance degradation percentages of the NAS benchmarks over different scenarios}
+\label{fig:per_d}
+\end{figure*}
+
+\begin{figure*}[!h]
+\centering
+\includegraphics[width=.7\textwidth]{fig/ch3/dist.eps}
+\caption{The trade-off distance percentages between the energy reduction and the performance of the NAS benchmarks  
+      over different scenarios}
+\label{fig:dist-grid}
+\end{figure*}
+
+The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
+scenario, except for the   EP benchmark which has  no communication. Therefore, the energy saving percentage of this benchmark is 
+dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
+in the one site scenario, the graphite cluster is selected but in the two sites scenario 
+this cluster is replaced with the Taurus cluster which is more powerful. 
+Therefore, the energy savings of the EP benchmark are bigger in the two sites scenario due 
+to the higher maximum difference between the computing powers of the nodes. 
+
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  
+algorithm  select smaller frequencies for the powerful nodes which 
+produces less energy consumption and thus more energy saving.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
+
+Figure \ref{fig:per_d} presents the performance degradation percentages for all the benchmarks over the two scenarios.
+The performance degradation percentage for the benchmarks running on two sites  with
+16 and 32  nodes is on average equal to 8.3\% and 4.7\% respectively. 
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are high with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
+16 and 32  nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
+nodes when the communications occur in high speed network does not decrease the computations to 
+communication ratio. 
+
+The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
+performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
+The rest of the benchmarks showed different performance degradation percentages which decrease
+when the communication times increase and vice versa.
+
+Figure \ref{fig:dist-grid} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The trade-off distance percentage can be 
+computed as in Equation~\ref{eq:max-grid}. The one site scenario with 16 nodes gives the best energy and performance 
+trade-off, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
+trade-off comparing to the two sites scenario  because the former has high speed local communications 
+which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 
+
+ Finally, the best energy and performance trade-off depends on all of the following:
+1) the computations to communications ratio when there are  communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed  static and dynamic powers of the nodes.
+
+
+
+
+\subsection{The experimental results over multi-core clusters}
+\label{ch3:4:2}
+
+The  clusters of Grid'5000 have different number of cores embedded in their nodes
+as shown in Table~\ref{table:grid5000-1}. In 
+this section, the proposed scaling algorithm  is evaluated over the Grid'5000 platform  while using multi-core nodes selected according to the one site scenario described in  Section
+~\ref{ch3:4:1}.
+
+\begin{table}[!h]
+\centering
+\caption{The multi-core scenarios}
+\begin{tabular}{|*{4}{c|}}
+\hline
+Scenario name                          & Cluster name & Nodes per cluster & 
+                                       Cores per node  \\ \hline
+\multirow{3}{*}{One core per node}    & Graphite     & 4               & 1                   \\  \cline{2-4}
+                                       & Graphene     & 14              & 1                   \\  \cline{2-4}
+                                       & Griffon      & 14              & 1                   \\ \hline
+\multirow{3}{*}{Multi-core per node}  & Graphite     & 1               &  4              \\  \cline{2-4}
+                                       & Graphene     & 4               & 3 or 4              \\  \cline{2-4}
+                                       & Griffon      & 4               & 3 or 4                   \\ \hline
+\end{tabular}
+
+\label{table:sen-mc}
+\end{table}
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch3/time.eps}
+ \caption{The execution times of the NAS benchmarks running over the one core and the multi-core scenarios}
+  \label{fig:time-mc}
+\end{figure}
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch3/eng_con.eps}
+ \caption{The energy consumptions and execution times of the NAS benchmarks over one core and multi-core per node architectures}
+\label{fig:eng-cons-mc}
+\end{figure}
+
+The one site scenario uses  32 cores from multi-core nodes instead of 32 distinct nodes. For example if 
+the participating number of cores from a certain cluster is equal to 14, 
+in the multi-core  4 nodes are selected and 3 or 4 cores  from each node are used. The platforms with one  
+core per node and  multi-core nodes are  shown in Table~\ref{table:sen-mc}. 
+The energy consumptions and execution times of running  class D of the NAS parallel 
+benchmarks over these two different platforms are presented 
+in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+The execution times for most of  the NAS  benchmarks are higher over the multi-core per node scenario 
+than over the  single core per node  scenario. Indeed,  
+ the communication times  are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and might become a bottleneck.    
+Moreover, the energy consumptions of the NAS benchmarks are lower over the 
+ one core scenario  than over the multi-core scenario because 
+the first scenario had less execution time than the latter which results in less static energy being consumed.
+The computations to communications ratios of the NAS benchmarks are higher over 
+the one site one core scenario  when compared to the ratio of the multi-core scenario. 
+More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy 
+consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
+scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
+\begin{figure*}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch3/eng_s_mc.eps}
+  \caption{The energy saving percentages of running NAS benchmarks over one core and multi-core scenarios}
+  \label{fig:eng-s-mc}
+\end{figure*}   
+
+\begin{figure*}[!h]
+ \centering
+\includegraphics[width=.7\textwidth]{fig/ch3/per_d_mc.eps}
+  \caption{The performance degradation percentages of running NAS benchmarks over one core and multi-core scenarios}
+  \label{fig:per-d-mc}
+\end{figure*}
+
+\begin{figure*}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch3/dist_mc.eps}
+  \caption{The trade-off distance percentages of running NAS benchmarks over one core and multi-core scenarios}
+  \label{fig:dist-mc}
+\end{figure*}
+The energy saving percentages of all the NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}. 
+It shows that  the energy saving percentages in the one 
+core and the multi-core scenarios
+are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
+The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS. 
+
+The performance degradation percentages of the NAS benchmarks are presented in
+Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks executed over the one core per node scenario  (on average equal to 10.6\%)  than over the  multi-core scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-core scenario are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 
+
+The trade-off distances percentages of the NAS benchmarks over both scenarios are presented 
+in~Figure~\ref{fig:dist-mc}. These  trade-off distances between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  trade-off distance percentages are on average   bigger over the multi-core scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
+
+
+\subsection{Experiments with different static power scenarios}
+\label{ch3:4:3}
+
+In Section~\ref{ch3:4}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
+
+The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
+In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
+The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.
+In these experiments, the class D of the NAS parallel benchmarks were executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, were used in this experiment. 
+
+
+
+
+\begin{figure}[!h]
+  \centering
+  \includegraphics[width=.7\textwidth]{fig/ch3/eng_pow.eps}
+  \caption{The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios}
+  \label{fig:eng-pow}
+\end{figure}
+\begin{figure}[!h]
+  \centering
+   \includegraphics[width=.7\textwidth]{fig/ch3/per_pow.eps}
+  \caption{The performance degradation percentages for the NAS benchmarks over the three power scenarios}
+  \label{fig:per-pow}
+\end{figure}
+
+\begin{figure}[!h]
+  \centering
+  \includegraphics[width=.7\textwidth]{fig/ch3/dist_pow.eps}
+  \caption{The trade-off distance percentages between the energy reduction and the performance of the NAS benchmarks over the three power scenarios}
+  \label{fig:dist-pow}
+\end{figure}
+
+
+\begin{figure}[!h]
+  \centering
+  \includegraphics[scale=0.7]{fig/ch3/three_scenarios2.pdf}
+  \caption{Comparing the selected frequency scaling factors for the MG benchmark over the three static power scenarios}
+  \label{fig:fre-pow}
+\end{figure}
+
+The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
+in Figure~\ref{fig:eng-pow}. This figure shows that the  10\% of static power scenario 
+gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
+scenarios. The small value of the static power consumption makes the proposed 
+scaling algorithm  select smaller frequencies for the CPUs. 
+These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives 		less overall energy consumption. 
+The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
+
+The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.
+The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
+had  selected big frequencies for the CPUs. While, 
+the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios 
+are presented in Figure~\ref{fig:dist-pow}. 
+It shows that the best  trade-off
+distance percentage is obtained with  the  10\% static power scenario  and this percentage 
+is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
+
+In the EP benchmark, the energy saving, performance degradation and trade-off 
+distance percentages for these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.
+
+
+ 
+\subsection{Comparison between the proposed frequencies selecting algorithm and the EDP method}
+\label{ch3:4:4}
+
+Finding the frequencies that give the best trade-off between the energy consumption and the performance for a parallel 
+application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
+In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{ref80,ref81,ref82}. 
+This objective function  was also used by Spiliopoulos et al. algorithm \cite{ref67} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-core 
+architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
+
+To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy-grid} and
+execution time model, Equation~\ref{eq:perf-grid}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
+Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
+and selects the vector of frequencies that minimize the EDP product.
+Both algorithms were applied to the class D of the NAS benchmarks running over 16 nodes.
+The participating computing nodes are distributed  according to the two scenarios described in  Section~\ref{ch3:4:1}. 
+The experimental results, the energy saving, performance degradation and trade-off distance percentages, are 
+presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+\begin{figure*}[!h]
+  \centering
+  \includegraphics[width=.7\textwidth]{fig/ch3/edp_eng}
+\caption{The energy reduction percentages induced by the Maxdist method and the EDP method}
+\label{fig:edp-eng}
+\end{figure*}
+
+\begin{figure*}[!h]
+  \centering
+  \includegraphics[width=.7\textwidth]{fig/ch3/edp_per}
+\caption{The performance degradation percentages induced by the Maxdist method and the EDP method}
+\label{fig:edp-perf}
+\end{figure*}
+
+\begin{figure*}[!h]
+  \centering
+   \includegraphics[width=.7\textwidth]{fig/ch3/edp_dist}
+\caption{The trade-off distance percentages between the energy consumption reduction and the performance for the Maxdist method and the  EDP method}
+\label{fig:edp-dist}
+\end{figure*}
+
+
+As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
+The proposed algorithm gives better results than the EDP method because it 
+maximizes the energy saving and the performance at the same time. 
+Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
+Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
+These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
+The high positive values of the trade-off distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
+The  complexity of both algorithms, Maxdist and EDP, are of order $O(N \cdot M_i \cdot F_j)$ and 
+$O(N \cdot M_i \cdot F_j^2)$ respectively, where $N$ is the number of the clusters, $M_i$ is the number of nodes and $F_j$ is the 
+maximum number of available frequencies of node $j$. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average  $0.01$ $ms$  to compute the best frequencies while the EDP method is on average ten times slower over the same architecture.  
+
+
+\section{Conclusion}
+\label{ch3:concl}
+In this chapter,  two new online frequency scaling factors selecting algorithms have been presented. They select the best possible vectors of frequency scaling factors that give the
+maximum distance (optimal trade-off) between the predicted energy   and the
+predicted performance curves for a heterogeneous cluster and grid. Both  algorithms use a
+new energy models for measuring and predicting the energy consumption of message passing 
+ applications with iterations running over a heterogeneous local cluster and a grid platform. 
+Firstly, the  proposed scaling factors selection algorithm for a heterogeneous local cluster  is applied to  the class C of the NAS parallel benchmarks and  simulated by SimGrid.
+The results of the simulations showed  that the algorithm on average reduces by 29.8\% the energy 
+consumption of the NAS benchmarks executed over 8 nodes while limiting the degradation of the performance by 3.8\%.  The algorithm also selects different scaling factors according to
+the percentage of the computing and communication times, and according to the
+values of the static and dynamic powers of the CPUs. 
+Secondly, the proposed scaling factors selection algorithm for a grid is applied to the class D of the  NAS parallel benchmarks  and  executed over the Grid5000 testbed platform.
+The experiments executed on 16 nodes distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption
+of all the NAS benchmarks   while  on average only degrading by 3.2\%   their performance. 
+The  algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or  use multi-core per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the 
+computations and communication times ratios, and  the values of the static and measured dynamic powers of the CPUs. Thus, the simulation and the real results are comparable in term of energy saving and performance degradation percentages.
+Finally, both algorithms were compared to a method that uses
+the well known energy and delay product as an objective function. The comparison results showed 
+that the proposed algorithms outperform the latter by selecting vectors of frequencies that give a better 
+trade-off results. 
