\usepackage[english]{babel}
\usepackage{algpseudocode}
\usepackage{graphicx}
+\usepackage{algorithm}
\usepackage{subfig}
\usepackage{amsmath}
\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
\newcommand{\Tmax}{\Xsub{T}{max}}
\newcommand{\Tnew}{\Xsub{T}{New}}
-\newcommand{\Told}{\Xsub{T}{Old}}
+\newcommand{\Told}{\Xsub{T}{Old}}
+\begin{document}
-\begin{document}
-
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
-
-\author{%
+\title{Energy Consumption Reduction in a Heterogeneous Architecture Using DVFS}
+
+\author{%
\IEEEauthorblockN{%
Jean-Claude Charr,
Raphaël Couturier,
Ahmed Fanfakh and
Arnaud Giersch
- }
+the normalized performance equation, as follows:
+ }
\IEEEauthorblockA{%
FEMTO-ST Institute\\
University of Franche-Comté\\
\section{Introduction}
\label{sec.intro}
-
+Modern processors continue to increased in a performance.
+The CPUs constructors are competing to achieve maximum number
+of floating point operations per second (FLOPS).
+Thus, the energy consumption and the heat dissipation are increased
+drastically according to this increase. Because the number of FLOPS
+is linearly related to the power consumption of a CPU~\cite{51}.
+As an example of the more power hungry cluster, Tianhe-2 became in
+the top of the Top500 list in June 2014 \cite{43}. It has more than
+3 millions of cores and consumed more than 17.8 megawatts.
+Moreover, according to the U.S. annual energy outlook 2014 \cite{60},
+the price of energy for 1 megawatt-hour was approximately equal to \$70.
+Therefore, we can consider the price of the energy consumption for the
+Tianhe-2 platform is approximately more than \$10 millions for
+one year. For this reason, the heterogeneous clusters must be offer more
+energy efficiency due to the increase in the energy cost and the environment
+influences. Therefore, a green computing clusters with maximum number of
+FLOPS per watt are required nowadays. For example, the GSIC center of Tokyo,
+became the top of the Green500 list in June 2014 \cite{59}. This platform
+has more than four thousand of MFLOPS per watt. Dynamic voltage and frequency
+scaling (DVFS) is a process used widely to reduce the energy consumption of
+the processor. In a heterogeneous clusters enabled DVFS, many researchers
+used DVFS in a different ways. DVFS can be minimized the energy consumption
+but it leads to a disadvantage due to increase in performance degradation.
+Therefore, researchers used different optimization strategies to overcame
+this problem. The best tradeoff relation between the energy reduction and
+performance degradation ratio is became a key challenges in a heterogeneous
+platforms. In this paper we are propose a heterogeneous scaling algorithm
+that selects the optimal vector of the frequency scaling factors for distributed
+iterative application, producing maximum energy reduction against minimum
+performance degradation ratio simultaneously. The algorithm has very small
+overhead, works online and not needs for any training or profiling.
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+related works from other authors. Section~\ref{sec.exe} describes how the
+execution time of MPI programs can be predicted. It also presents an energy
+model for heterogeneous platforms. Section~\ref{sec.compet} presents
+the energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed heterogeneous scaling algorithm.
+Section~\ref{sec.expe} presents the results of running the NAS benchmarks on
+the proposed heterogeneous platform. It also shows the comparison of three
+different power scenarios and it verifies the precision of the proposed algorithm.
+Finally, we conclude in Section~\ref{sec.concl} with a summary and some future works.
\section{Related works}
\label{sec.relwork}
-
-
-
+Energy reduction process for a high performance clusters recently performed using
+dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled
+in a modern processors to scaled down both of the voltage and the frequency of
+the CPU while it is in the computing mode to reduce the energy consumption. DVFS is
+also allowed in the graphical processors GPUs, to achieved the same goal. Applying
+DVFS has a dramatical side effect if it is applied to minimum levels to gain more
+energy reduction, producing a high percentage of performance degradations for the
+parallel applications. Many researchers used different strategies to solve this
+nonlinear problem for example in~\cite{19,42}, their methods add big overheads to
+the algorithm to select the suitable frequency. In this paper we present a method
+to find the optimal set of frequency scaling factors for a heterogeneous cluster to
+simultaneously optimize both the energy and the execution time without adding a big
+overhead. This work is developed from our previous work of a homogeneous cluster~\cite{45}.
+Therefore we are interested to present some works that concerned the heterogeneous clusters
+enabled DVFS. In general, the heterogeneous cluster works fall into two categorizes:
+GPUs-CPUs heterogeneous clusters and CPUs-CPUs heterogeneous clusters. In GPUs-CPUs
+heterogeneous clusters some parallel tasks executed on a GPUs and the others executed
+on a CPUs. As an example of this works, Luley et al.~\cite{51}, proposed a heterogeneous
+cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal is to determined the
+energy efficiency as a function of performance per watt, the best tradeoff is done when the
+performance per watt function is maximized. In the work of Kia Ma et al.~\cite{49},
+They developed a scheduling algorithm to distributed different workloads proportional
+to the computing power of the node to be executed on a CPU or a GPU, emphasize all tasks
+must be finished in the same time.
+Recently, Rong et al.~\cite{50}, Their study explain that a heterogeneous clusters enabled
+DVFS using GPUs and CPUs gave better energy and performance efficiency than other clusters
+composed of only CPUs. The CPUs-CPUs heterogeneous clusters consist of number of computing
+nodes all of the type CPU. Our work in this paper can be classified to this type of the
+clusters. As an example of this works see Naveen et al.~\cite{52} work, They developed a
+policy to dynamically assigned the frequency to a heterogeneous cluster. The goal is to
+minimizing a fixed metric of $energy*delay^2$. Where our proposed method is automatically
+optimized the relation between the energy and the delay of the iterative applications.
+Other works such as Lizhe et al.~\cite{53}, their algorithm divided the executed tasks into
+two types: the critical and non critical tasks. The algorithm scaled down the frequency of
+the non critical tasks as function to the amount of the slack and communication times that
+have with maximum of performance degradation percentage of 10\%. In our method there is no
+fixed bounds for performance degradation percentage and the bound is dynamically computed
+according to the energy and the performance tradeoff relation of the executed application.
+There are some approaches used a heterogeneous cluster composed from two different types
+of Intel and AMD processors such as~\cite{54} and \cite{55}, they predicated both the energy
+and the performance for each frequency gear, then the algorithm selected the best gear that gave
+the best tradeoff. In contrast our algorithm works over a heterogeneous platform composed of
+four different types of processors. Others approaches such as \cite{56} and \cite{57}, they
+are selected the best frequencies for a specified heterogeneous clusters offline using some
+heuristic methods. While our proposed algorithm works online during the execution time of
+iterative application. Greedy dynamic approach used by Chen et al.~\cite{58}, minimized
+the power consumption of a heterogeneous severs with time/space complexity, this approach
+had considerable overhead. In our proposed scaling algorithm has very small overhead and
+it is works without any previous analysis for the application time complexity.
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
% can be deleted if we need space, we can just say we are interested in this
% paper in homogeneous clusters}
-\subsection{The execution time of message passing distributed iterative applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed
+ iterative applications on a heterogeneous platform}
In this paper, we are interested in reducing the energy consumption of message
passing distributed iterative synchronous applications running over
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.
-
-\begin{figure}[t]
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous platform 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 might occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~(\ref{fig:heter}).
+Therefore, the overall execution time of the program is the execution time of the slowest
+task which have the highest computation time and no slack time.
+
+ \begin{figure}[t]
\centering
\includegraphics[scale=0.6]{fig/commtasks}
\caption{Parallel tasks on a heterogeneous platform}
\label{fig:heter}
\end{figure}
- The overall execution time of a distributed iterative synchronous application over a heterogeneous platform 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 might occur when fast nodes have to
- wait, during synchronous communications, for the slower nodes to finish their computations (see Figure~(\ref{fig:heter})).
- Therefore, the overall execution time of the program is the execution time of the slowest
- task which have the highest computation time and no slack time.
-
-Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
-of a CPU by scaling down its voltage and frequency. Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound. The frequency reduction process can be expressed by the scaling factor S which is the ratio between the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency. Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor might increase, especially if the program is
+compute bound. The frequency reduction process can be expressed by the scaling
+factor S which is the ratio between the maximum and the new frequency of a CPU
+as in EQ (\ref{eq:s}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
- 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{17}. 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 till the message is synchronously sent or received.
+ 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{17}. 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
+ till the message is synchronously sent or received.
Since in a heterogeneous platform, each node has different characteristics,
especially different frequency gears, when applying DVFS operations on these
\begin{equation}
\label{eq:perf}
\textit T_\textit{new} =
- \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) + MinTcm
+ \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) + MinTcm
\end{equation}
-where $TcpOld_i$ is the computation time of processor $i$ during the first iteration and $MinT_{c}m$ is the communication time of the slowest processor from 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. Therefore, we can consider the execution time of the iterative application is the execution time of one iteration as in EQ(\ref{eq:perf}) multiply by the number of iterations of the application.
-
-This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
+where $TcpOld_i$ is the computation time of processor $i$ during the first
+iteration and $MinTcm$ is the communication time of the slowest processor from
+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.
+Therefore, we can consider the execution time of the iterative application is
+equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied
+by the number of iterations of that application.
+
+This prediction model is based on our model for predicting the execution time of
+message passing distributed applications for homogeneous architectures~\cite{45}.
+The execution time prediction model is used in our method for optimizing both
+energy consumption and performance of iterative methods, which is presented in the
+following sections.
\subsection{Energy model for heterogeneous platform}
-
Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
two power metrics: the static and the dynamic power. While the first one is
consumed as long as the computing unit is turned on, the latter is only consumed during
\label{eq:fnew}
F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic
-power consumption:
+Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following
+equation for dynamic power consumption:
\begin{multline}
\label{eq:pdnew}
{P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
- {} = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+ {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
-where $ {P}_\textit{dNew}$ and $P_{dOld}$ are the dynamic power consumed with the new frequency and the maximum frequency respectively.
+where $ {P}_\textit{dNew}$ and $P_{dOld}$ are the dynamic power consumed with the
+new frequency and the maximum frequency respectively.
According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when
-reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$. The new dynamic energy is the dynamic power multiplied by the new time of computation and is given by the following equation:
+reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional
+to the frequency of a CPU, the computation time is increased proportionally to $S$.
+The new dynamic energy is the dynamic power multiplied by the new time of computation
+and is given by the following equation:
\begin{equation}
\label{eq:Edyn}
- E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot Tcp
+ E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \cdot Tcp
\end{equation}
-The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies.
-The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}),
-the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related
-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy
-of a processor after scaling its frequency is computed as follows:
+The static power is related to the power leakage of the CPU and is consumed during computation
+and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant
+during idle and computation periods, and for all its available frequencies.
+The static energy is the static power multiplied by the execution time of the program.
+According to the execution time model in EQ(\ref{eq:perf}), the execution time of the program
+is the summation of the computation and the communication times. The computation time is linearly related
+to the frequency scaling factor, while this scaling factor does not affect the communication time.
+The static energy of a processor after scaling its frequency is computed as follows:
\begin{equation}
\label{eq:Estatic}
E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S + Tcm)
\end{equation}
-In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency scaling factors might 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 $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig: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 EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing distributed application executed over a heterogeneous platform during one iteration is the summation of all dynamic and static energies for each processor. It is computed as follows:
+In the considered heterogeneous platform, each processor $i$ might have different dynamic and
+static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed
+message passing iterative application is load balanced, the computation time of each CPU $i$
+noted $Tcp_{i}$ might be different and different frequency scaling factors might 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
+if it is communicating with slower nodes, see figure(\ref{fig: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 EQ(\ref{eq:Edyn}), the static energy is
+computed as the sum of the execution time of each processor multiplied by its static power.
+The overall energy consumption of a message passing distributed application executed over a
+heterogeneous platform during one iteration is the summation of all dynamic and static energies
+for each processor. It is computed as follows:
\begin{multline}
\label{eq:energy}
E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot Tcp_i)} + {} \\
scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
application and thus, increase the static energy because the execution time is
increased~\cite{36}. We can measure the overall energy consumption for the iterative
-application by measuring the energy consumption from one iteration as in EQ(\ref{eq:energy}) multiply by
-the number of iterations of the iterative application.
+application by measuring the energy consumption for one iteration as in EQ(\ref{eq:energy})
+multiplied by the number of iterations of that application.
\section{Optimization of both energy consumption and performance}
\label{sec.compet}
-Using the lowest frequency for each processor does not necessarily gives 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 and thus the execution time might be drastically increased during which 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 executed (computation/communication ratio) in order to reduce the overall energy consumption and not significantly increase the execution time. In our previous work~\cite{45}, we proposed a method that selects the optimal
-frequency scaling factor for a homogeneous cluster executing a message passing iterative synchronous application while giving the best trade-off
- between the energy consumption and the performance for such applications. In this work we are interested in
-heterogeneous clusters as described above. Due to the heterogeneity of the processors, not one but a vector of scaling factors should be selected and it must give the best trade-off between energy consumption and performance.
-
-The relation between the energy consumption and the execution
-time for an application is complex and nonlinear, Thus, unlike the relation between the execution time
+Using the lowest frequency for each processor does not necessarily gives 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 and thus the execution time might be drastically increased during which 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 executed
+(computation/communication ratio) in order to reduce the overall energy consumption and not
+significantly increase the execution time. In our previous work~\cite{45}, we proposed a method
+that selects the optimal frequency scaling factor for a homogeneous cluster executing a message
+passing iterative synchronous application while giving the best trade-off between the energy
+consumption and the performance for such applications. In this work we are interested in
+heterogeneous clusters as described above. Due to the heterogeneity of the processors, not
+one but a vector of scaling factors should be selected and it must give the best trade-off
+between energy consumption and performance.
+
+The relation between the energy consumption and the execution time for an application is
+complex and nonlinear, Thus, unlike the relation between the execution time
and the scaling factor, the relation of the energy with the frequency scaling
factors is nonlinear, for more details refer to~\cite{17}. Moreover, they are
not measured using the same metric. To solve this problem, we normalize the
-execution time by computing the ratio between the new execution time (after scaling down the frequencies of some processors) and the initial one (with maximum frequency for all nodes,) as follows:
+execution time by computing the ratio between the new execution time (after
+scaling down the frequencies of some processors) and the initial one (with maximum
+frequency for all nodes,) as follows:
\begin{multline}
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
\end{multline}
-In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
+In the same way, we normalize the energy by computing the ratio between the consumed energy
+while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
While the main
-goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According
+goal is to optimize the energy and execution time at the same time, the normalized
+energy and execution time curves are not in the same direction. According
to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the vector of frequency
scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
time simultaneously. But the main objective is to produce maximum energy
reduction with minimum execution time reduction.
-Many researchers used different strategies to solve this nonlinear problem for example
-in~\cite{19,42}, their methods add big overheads to the algorithm to select the
-suitable frequency. In this paper we present a method to find the optimal
-set of frequency scaling factors to simultaneously optimize both energy and execution time
- without adding a big overhead. \textbf{put the last two phrases in the related work section}
-Our solution for this problem is to make the optimization process for energy and execution time follow the same
-direction. Therefore, we inverse the equation of the normalized execution time which gives
-the normalized performance equation, as follows:
+Our solution for this problem is to make the optimization process for energy and
+execution time follow the same direction. Therefore, we inverse the equation of the
+normalized execution time which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
between the energy curve EQ~(\ref{eq:enorm}) and the performance
curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
represents the minimum energy consumption with minimum execution time (maximum
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) . Then our objective
+performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then our objective
function has the following form:
\begin{equation}
\label{eq:max}
\overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of nodes and $F$ is the number of available frequencies for each nodes.
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}). Our objective function can
-work with any energy model or any power values for each node (static and dynamic powers).
-However, the most energy reduction gain can be achieved when the energy curve has a convex form as shown in~\cite{15,3,19}.
+Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).
+Our objective function can work with any energy model or any power values for each node
+(static and dynamic powers). However, the most energy reduction gain can be achieved when
+the energy curve has a convex form as shown in~\cite{15,3,19}.
-\section{The heterogeneous scaling algorithm }
+\section{The scaling factors selection algorithm for heterogeneous platforms }
\label{sec.optim}
-In this section we are proposed a heterogeneous scaling algorithm,
-(figure~\ref{HSA}), that selects the optimal vector of the frequency scaling factors from each
-node. The algorithm is numerates the suitable range of available frequency scaling
-factors for each node in a heterogeneous cluster, returns a vector of optimal
-frequency scaling factors for all node define as $Sopt_1,Sopt_2,\dots,Sopt_N$. Using heterogeneous cluster
-has different computing powers is produces different workloads for each node. Therefore, the fastest nodes waiting at the
-synchronous barrier for the slowest nodes to finish there work as in figure
-(\ref{fig:heter}). Our algorithm is takes into account these imbalanced workloads
-when is starts to search for selecting the best vector of the frequency scaling factors. So, the
-algorithm is selects the initial frequencies values for each node proportional
-to the times of computations that gathered from the first iteration. As an
-example in figure (\ref{fig:st_freq}), the algorithm don't tests the first
-frequencies of the computing nodes until it is converge their frequencies to the
-frequency of the slowest node. If the algorithm is starts to test changing the
-frequency of the slowest node from the first gear, we are loosing the performance and
-then the best trade-off relation (the maximum distance) be not reachable. This case will be similar
-to a homogeneous cluster when all nodes scales their frequencies together from
-the first gear. Therefore, there is a small distance between the energy and
-the performance curves in a homogeneous cluster compare to heterogeneous one, for example see the figure(\ref{fig:r1}). Then the
-algorithm starts to search for the optimal vector of the frequency scaling factors from the selected initial
-frequencies until all node reach their minimum frequencies.
-\begin{figure}[t]
- \centering
- \includegraphics[scale=0.5]{fig/start_freq}
- \caption{Selecting the initial frequencies}
- \label{fig:st_freq}
-\end{figure}
-
-
-To compute the initial frequencies in each node, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ of the node $i$. Each one of these factors is represents a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
+In this section we propose algorithm~(\ref{HSA}) which 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 iterative application executed on a heterogeneous
+platform. It works online during the execution time of the iterative message passing program.
+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 EQ(\ref{eq:max}). The program apply 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}) shows where and when the proposed scaling algorithm is called
+in the iterative MPI program.
+
+The nodes in a heterogeneous platform have different computing powers, thus while executing message
+passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their
+computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}).
+These periods are called idle or slack times.
+Our 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
+fast and slow nodes. 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}
-Depending on the initial computation scaling factors EQ(\ref{eq:Scp}), the algorithm computes the initial frequencies for all nodes as a ratio between the
-maximum frequency of node $i$ and the computation scaling factor $Scp_i$ as follow:
+Using the initial frequency scaling factors computed in EQ(\ref{eq:Scp}), the algorithm computes
+the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$
+and the computation scaling factor $Scp_i$ as follows:
\begin{equation}
\label{eq:Fint}
F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
-\begin{figure}[tp]
+If the computed initial frequency for a node is not available in the gears of that node, the computed
+initial frequency is replaced by the nearest available frequency. In figure (\ref{fig:st_freq}),
+the nodes are sorted by their computing powers 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 colored in blue in figure (\ref{fig:st_freq}). This set of frequencies can be considered
+as a higher bound for the search space of the optimal vector of frequencies because selecting frequency
+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 left 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 EQ(\ref{eq:perf}) 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 EQ(\ref{eq:max}).
+
+The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an
+application running on a homogeneous platform and a heterogeneous platform respectively while increasing the
+scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor
+should be started from the maximum frequency because the performance and the consumed energy is decreased since
+the beginning of the plot. On the other hand, in the heterogeneous platform the performance is maintained at
+the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes
+have computing powers 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 while varying the scaling factors
+which results in bigger energy savings.
+\begin{figure}[t]
+ \centering
+ \includegraphics[scale=0.5]{fig/start_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
+\end{figure}
+
+
+
+
+\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\Require ~
\item[$Ps_i$] array of the static powers for all nodes.
\item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
\end{description}
- \Ensure $Sopt_1, \dots, Sopt_N$ is a set of optimal scaling factors
+ \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 $F_{i} \gets \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
\end{algorithmic}
\caption{Heterogeneous scaling algorithm}
\label{HSA}
-\end{figure}
-When the initial frequencies are computed, the algorithm numerates all available
-frequency scaling factors starting from the initial frequencies until all nodes reach their
-minimum frequencies. At each iteration the algorithm determine the slowest node according to EQ(\ref{eq:perf}).
-It is remains the frequency of the slowest node without change, while it is scale down the frequency of the other
-nodes. This is improved the execution time degradation and energy saving in the same time.
-The proposed algorithm works online during the execution time of the iterative MPI program. It is
-returns a vector of optimal frequency scaling factors depending on the
-objective function EQ(\ref{eq:max}). The program changes the new frequencies of
-the CPUs according to the computed scaling factors. This algorithm 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}), it is
-takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.15} on average for 144
-nodes. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the
-number of iterations and $N$ is the number of computing nodes. The algorithm
-needs on average from 12 to 20 iterations to selects the best vector of frequency scaling factors that give the results of the next section. It is called just once during the execution of the program. The DVFS algorithm in figure~(\ref{dvfs}) shows where
-and when the proposed scaling algorithm is called in the iterative MPI program.
-\begin{figure}[tp]
+\end{algorithm}
+
+\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\For {$k=1$ to \textit{some iterations}}
\end{algorithmic}
\caption{DVFS algorithm}
\label{dvfs}
-\end{figure}
+\end{algorithm}
\section{Experimental results}
\label{sec.expe}
+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 \cite{44}. The experiments were executed
+on the simulator SimGrid/SMPI v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers
+easy tools to create a heterogeneous platform and run message passing applications over it. The
+heterogeneous platform that was used in the experiments, had one core per node because just one
+process was executed per node. The heterogeneous platform 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}). The characteristics
+of these different types of nodes are inspired from the specifications of real Intel processors.
+The heterogeneous platform 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 its computing power (FLOPS). In the initial heterogeneous platform, while computing
+with highest frequency, each node consumed power proportional to its computing power which 80\% of it was
+dynamic power and the rest was 20\% for the static power, the same assumption was made in \cite{45,3}.
+Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
+
-The experiments of this work are executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We are configure the
-simulator to use a heterogeneous cluster with one core per node. The proposed
-heterogeneous cluster has four different types of nodes. Each node in the cluster
-has different characteristics such as the maximum frequency speed, the number of
-available frequencies and dynamic and static powers values, see table
-(\ref{table:platform}). These different types of processing nodes are simulate some
-real Intel processors. The maximum number of nodes that supported by the cluster
-is 144 nodes according to characteristics of some MPI programs of the NAS
-benchmarks that used. We are use the same number from each type of nodes when we
-run the iterative MPI programs, for example if we are execute the program on 8 node, there
-are 2 nodes from each type participating in the computation. The dynamic and
-static power values is different from one type to other. Each node has a dynamic
-and static power values proportional to their computing power (FLOPS), for more
-details see the Intel data sheets in \cite{47}. Each node has a percentage of
-80\% for dynamic power and 20\% for static power of the total power
-consumption of a CPU, the same assumption is made in \cite{45,3}. These nodes are
-connected via an ethernet network with 1 Gbit/s bandwidth.
\begin{table}[htb]
\caption{Heterogeneous nodes characteristics}
% title of Table
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Node & Similar & Max & Min & Diff. & Dynamic & Static \\
- type & to & Freq. GHz & Freq. GHz & Freq GHz & power & power \\
+ Node &Simulated & Max & Min & Diff. & Dynamic & Static \\
+ type &GFLOPS & Freq. & Freq. & Freq. & power & power \\
+ & & GHz & GHz &GHz & & \\
\hline
- 1 & core-i3 & 2.5 & 1.2 & 0.1 & 20~w &4~w \\
- & 2100T & & & & & \\
+ 1 &40 & 2.5 & 1.2 & 0.1 & 20~w &4~w \\
+ & & & & & & \\
\hline
- 2 & Xeon & 2.66 & 1.6 & 0.133 & 25~w &5~w \\
- & 7542 & & & & & \\
+ 2 &50 & 2.66 & 1.6 & 0.133 & 25~w &5~w \\
+ & & & & & & \\
\hline
- 3 & core-i5 & 2.9 & 1.2 & 0.1 & 30~w &6~w \\
- & 3470s & & & & & \\
+ 3 &60 & 2.9 & 1.2 & 0.1 & 30~w &6~w \\
+ & & & & & & \\
\hline
- 4 & core-i7 & 3.4 & 1.6 & 0.133 & 35~w &7~w \\
- & 2600s & & & & & \\
+ 4 &70 & 3.4 & 1.6 & 0.133 & 35~w &7~w \\
+ & & & & & & \\
\hline
\end{tabular}
\label{table:platform}
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.3
-\cite{44}, which were run with three classes (A, B and C).
-In this experiments we are interested to run the class C, the biggest class compared to A and B, on different number of
-nodes, from 4 to 128 or 144 nodes according to the type of the iterative MPI program. Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values are used under the same assumption used by \cite{45,3}. We are used a percentage of 80\% for dynamic power and 20\% for static of the total power consumption of a CPU. The heterogeneous nodes in table (\ref{table:platform}) have different simulated computing power (FLOPS), ranked from the node of type 1 with smaller computing power (FLOPS) to the highest computing power (FLOPS) for node of type 4. Therefore, the power values are used proportionally increased from nodes of type 1 to nodes of type 4 that with highest computing power. Then, we are used an assumption that the power consumption is increased linearly when the computing power (FLOPS) of the processor is increased, see \cite{48}.
+
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP)
+and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in
+this paper, only the results of the biggest class, C, are presented while being run on different number
+of nodes, ranging from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the
+benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes.
+The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+
+
\begin{table}[htb]
\caption{Running NAS benchmarks on 4 nodes }
\end{tabular}
\label{table:res_128n}
\end{table}
+The overall energy consumption was computed for each instance according to the energy
+consumption model EQ(\ref{eq:energy}), 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_4n}, \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 from many experiments for energy savings and performance degradation.
+
+The tables show the experimental results for running the NAS parallel benchmarks on different
+number of nodes. The experiments show that the algorithm reduce significantly the energy
+consumption (up to 35\%) and tries to limit the performance degradation. They also show that
+the energy saving percentage is decreased when the number of the computing nodes is increased.
+This reduction is due to the increase of the communication times compared to the execution times
+when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C,
+are executed on different number 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 is increased 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}) have less effect 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 with the high number of nodes. No experiments were conducted using bigger
+classes such as D, because they require a lot of memory(more than 64GB) 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 is decreased 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.
-The results of applying the proposed scaling algorithm to the NAS benchmarks is demonstrated in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). These tables are show the experimental results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased when the number of the computing nodes is increased. Also the distance is decreased by the same direction of the energy saving. This because when we are run the iterative MPI programs on a big number of nodes the communications times is increased, so the static energy is increased linearly to these times. The tables also show that the performance degradation percent still approximately the same ratio or decreased in a very small present when the number of computing nodes is increased. This is gives a good prove that the proposed algorithm keeping the performance degradation as mush as possible is the same.
+
\begin{figure}
\centering
- \subfloat[CG, MG, LU and FT benchmarks]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
+ \subfloat[Energy saving]{%
+ \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
\quad%
- \subfloat[BT and SP benchmarks]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
+ \subfloat[Performance degradation ]{%
+ \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
\label{fig:avg}
- \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
+ \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}
-In the NAS benchmarks there are some programs executed on different number of
-nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8,
-\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
-power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
-saving, performance degradation and distances for all results of NAS
-benchmarks. The average of values of these three objectives are plotted to the number of
-nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}). In CG, MG, LU, and
-FT benchmarks the average of energy saving is decreased when the number of nodes
-is increased because the communication times is increased as mentioned
-before. Thus, the average of distances (our objective function) is decreased
-linearly with energy saving while keeping the average of performance degradation approximately is
-the same. In BT and SP benchmarks, the average of the energy saving is not decreased
-significantly compare to other benchmarks when the number of nodes is
-increased. Nevertheless, the average of performance degradation approximately
-still the same ratio. This difference is depends on the characteristics of the
-benchmarks such as the computation to communication ratio that has.
+Plots (\ref{fig:energy} and \ref{fig:per_deg}) 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 are decreased linearly when the the
+number of nodes is increased. 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 no
+communications. Finally, the energy saving of the GC benchmark is significantly decreased when the number
+of nodes is increased because this benchmark has more communications than the others. The second plot
+shows that the performance degradation percentages of most of the benchmarks are decreased 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 have less effect on the performance.
+
+
+
\subsection{The results for different power consumption scenarios}
-The results of the previous section are obtained using a percentage of 80\% for
-dynamic power and 20\% for static power of the total power consumption of a CPU. In this
-section we are change these ratio by using two others power scenarios. Because is
-interested to measure the ability of the proposed algorithm when these power ratios are changed.
-In fact, we are used two different scenarios for dynamic and static power ratios in addition to the previous
-scenario in section (\ref{sec.res}). Therefore, we have three different
-scenarios for three different dynamic and static power ratios refer to these as:
-70\%-20\%, 80\%-20\% and 90\%-10\% scenario respectively. The results of these scenarios
-running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
-(\ref{table:res_s1} and \ref{table:res_s2}).
+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\% dynamic power and 30\% static power
+\item 90\% dynamic power and 10\% static power
+\end{itemize}
+
+The NAS parallel benchmarks were executed again over processors that follow the 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 less 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 less 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 total consumed
+static energy is directly increased. Therefore, the proposed algorithm do not scales down much 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 .
+
+The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of
+the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running 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 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 is decreased when the 70\%-30\% scenario is used because the dynamic energy is less relevant in
+the overall consumed energy and lowering the frequency do not returns big energy savings.
+Moreover, the average of the performance degradation is decreased when using a higher ratio for 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 dynamic power the algorithm selects bigger frequency scaling factors that result in
+more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens
+when using a higher ratio for static power, the algorithm proportionally selects smaller scaling values which
+results in less energy saving but less performance degradation.
+
\begin{table}[htb]
\caption{The results of 70\%-30\% powers scenario}
\subfloat[Comparison the average of the results on 8 nodes]{%
\includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
\quad%
- \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
+ \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
\includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
\label{fig:comp}
\caption{The comparison of the three power scenarios}
-\end{figure}
+\end{figure}
+
-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of these results, the energy saving ratio is increased when using a higher percentage for dynamic power (e.g. 90\%-10\% scenario), due to increase in dynamic energy. While the average of energy saving is decreased in 70\%-30\% scenario. Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. The average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for static power, the algorithm proportionally selects a smaller scaling values, less energy saving versus less performance degradation. This is because the
-algorithm is optimizes the static energy consumption that is always related to the execution time.
\subsection{The verifications of the proposed method}
\label{sec.verif}
-The precision of the proposed algorithm mainly depends on the execution time prediction model EQ(\ref{eq:perf}) and the energy model EQ(\ref{eq:energy}). The energy model is significantly depends on the execution time model, that the static energy is related linearly. So, our work is depends mainly on execution time model. To verifying thid model, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered offline from the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts the real execution time by maximum normalized error equal to 0.03 for all the NAS benchmarks. The second verification that we are made is for the proposed scaling algorithm to prove its ability to selects the best vector of the frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all available scaling factors of the other nodes, all possible solutions. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the expanded algorithms and the proposed algorithm are identical. While the proposed algorithm is runs by 10 times faster on average compare to the expanded algorithm.
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
+EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
+The energy model is also significantly dependent on the execution time model because the static energy is
+linearly related the execution time and the dynamic energy is related to the computation time. So, all of
+the work presented in this paper is based on the execution time model. To verify this model, the predicted
+execution time was compared to the real execution time over Simgrid for all the NAS parallel benchmarks
+running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise,
+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 and do not test all the possible solutions (vectors of scaling factors)
+in the search space and 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}), it takes on average \np[ms]{0.04} for 4 nodes and \np[ms]{0.15} on average for 144 nodes
+to compute the best scaling factors vector. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number
+of iterations 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 section (\ref{sec.res}).
\section{Conclusion}
\label{sec.concl}
