\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}}
-
-\begin{document}
+\newcommand{\Told}{\Xsub{T}{Old}}
+\begin{document}
\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
-
\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 performance of parallel tasks on heterogeneous cluster}
+\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
+heterogeneous platforms. We define a heterogeneous platform as a collection of
+heterogeneous computing nodes interconnected via a high speed homogeneous
+network. Therefore, each node has different characteristics such as computing
+power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
+have the same network bandwidth and latency.
-The heterogeneous cluster is a collection of non identical computing nodes. Each node in
-a cluster is connected via a high speed network. The communication capabilities between nodes
-are identical or different. In this work we are interested in identical communications. While each
-node has different processing capabilities such as CPU speeds and memory. Tasks executed
-on this model can be either synchronous or asynchronous. In this paper we are consider execution of
-the synchronous tasks on distributed heterogeneous platform. These tasks can exchange
-the data via synchronous message passing.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{fig/commtasks}
- \caption{Parallel tasks on heterogeneous platform}
+ \caption{Parallel tasks on a heterogeneous platform}
\label{fig:heter}
\end{figure}
- Therefore, the execution time of a task consists of the computation time and
- the communication time. Due to heterogeneous computations can lead to slack times while the tasks
- wait at the synchronization barrier for other tasks to finish their jobs (see Figure~(\ref{fig:heter})).
- In this case the fastest tasks have to wait at the synchronization barrier for the slowest ones to begin
- the next task. Therefore, the overall execution time of the program is the execution time of the slowest
- task as in EQ~(\ref{eq:T1}).
-\begin{equation}
- \label{eq:T1}
- \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
-\end{equation}
- where $T_i$ is the execution time of the task $i$ and all the tasks are executed concurrently on different processors. DVFS is a process that is allowed in modern processors to reduce the dynamic
-power by scaling down the voltage and frequency. Then any DVFS operation used to reduce energy of the processor has direct affect on the execution time of the MPI program. The reduction process of the frequency can be expressed by the scaling factor S which is the ratio between the maximum and the new frequency as in EQ (\ref{eq:s}).
+ 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}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
- The execution time of a parallel program is linearly proportional to the frequency scaling factor $S$.
- However, in most MPI applications the processes exchange data. During these communications the
- processors involved remain idle until the communications are finished. For that reason, any change in
- the frequency has no impact on the time of communication~\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.
- Each node has different DVFS features such as frequency values and the number of available frequencies
- (Pstates) for each node. By contrast there are different frequency scaling factors for each node $S_1, S_2,..., S_N$. To be able to predict the execution time of MPI program, the communication time and the computation time for the slowest
- task must be measured before scaling. These times are used to predict the execution time for any MPI program running on heterogeneous cluster as a function
- of the new scaling factors as in EQ (\ref{eq:perf}). The model is computes the maximum production of computation time
- with scaling factor from each node added to the minimum communication time of the slowest node, it means only the
- communication time without slack times, because in MPI the slack times is measured with communication times.
-\begin{multline}
+ 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
+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 iterative
+applications running over a heterogeneous platform, 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 EQ (\ref{eq:perf}).
+\begin{equation}
\label{eq:perf}
- \textit T_\textit{new} = \\
- {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +
- \min_{i=1,2,\dots,N} Tcm Old_{i}
-\end{multline}
-This prediction modal is developed from our model for predicting the execution time of parallel task on homogeneous architecture~\cite{45}. The execution time predicting model is useful to used in our method for optimizing both energy and performance of iterative methods as in the coming sections.
+ \textit T_\textit{new} =
+ \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 $MinTcm$ 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}
+\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 on, the latter is only consumed during
+consumed as long as the computing unit is turned on, the latter is only consumed during
computation times. The dynamic power $P_{d}$ is related to the switching
activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
operational frequency $F$, as shown in EQ(\ref{eq:pd}).
to execute a given program can be computed as:
\begin{equation}
\label{eq:eind}
- E_\textit{ind} = P_\textit{d} \cdot T_{cp} + P_\textit{s} \cdot T
+ E_\textit{ind} = P_\textit{d} \cdot Tcp + P_\textit{s} \cdot T
\end{equation}
where $T$ is the execution time of the program, $T_{cp}$ is the computation
time and $T_{cp} \leq T$. $T_{cp}$ may be equal to $T$ if there is no
-communication, no slack time and no synchronization.
+communication and no slack time.
The main objective of DVFS operation is to
reduce the overall energy consumption~\cite{37}. The operational frequency $F$
voltage with respect to various frequency values in~\cite{3}. The reduction
process of the frequency can be expressed by the scaling factor $S$ which is the
ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-The value of the scaling factor $S$ is greater than 1 when changing the
-frequency of the CPU to any new frequency value~(\emph{P-state}) in the
-governor. The CPU governor is an interface driver supplied by the operating
+The CPU governors are power schemes supplied by the operating
system's kernel to lower a core's frequency. we can calculate the new frequency
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
\label{eq:fnew}
- F_\textit{new} = S^{-1} . F_\textit{max}
+ F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
-By substituting this equation in EQ(\ref{eq:pd}) results the following equation for dynamic
+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{d} = \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_{d} \cdot S^{-3}
+ {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_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduce by a factor of $S^{-3}$ when
-reducing the frequency by a factor of $S$~\cite{3}. The dynamic energy is the energy consumed by a CPU when its in the computation mode.
-So, the dynamic energy is the dynamic power multiply by the time of computations. While the time of computation is decreased by a factor of $S$. Therefore the
-the dynamic energy is decreased by a factor of $S^{-2}$ as follow:
+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:
\begin{equation}
\label{eq:Edyn}
- E_\textit{d} = P_{d} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{d} \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 leakage power consumption, its mean the CPU continue consumes energy
-whereas in idle state. Therefore, we are make an assumption that the static power is constant as in~\cite{3,46}.
-The static energy is the static power multiply by the execution time of the program. Moreover, the CPU consumes static
-energy in all times of the program such as computation, communication and slacks times. 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 related
-to frequency scaling factor linearly, while this scaling factor not affecting on the time of communication~\cite{17}, then the static energy
-of individual processor is as follow:
-
+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 heterogeneous architecture there is a number of different processors $P_1, P_2,...,P_N$, where $N$ is the number of nodes . Moreover, each processor perhaps has different frequency scaling factor, so there are a set of frequency scaling factors for such platform $S_1,S_2,...,S_N$. According to these different
-scaling factors producing different computation time, $Tcp_1,Tcp_2,...,Tcp_N$, because these times linearly related to these scaling factors. In MPI program the communication times is measured with slacks times. The slack times also has linear relation with the scaling factors. So, there are different mesured communication times $Tcm_1,Tcm_2,...,Tcm_N$ even if its identical communications, e.g. see figure(\ref{fig:heter}). The energy modal of an heterogeneous architecture represents the summation of all dynamic and static energies from each processors, each processor has its dynamic and static powers, for example, in the hole architecture their are: $Pd_1,Pd_2,...,Pd_N$ and $Ps_1,Ps_2,...,Ps_N$. The dynamic energy is computes as in EQ(\ref{eq:Edyn}) with regarding to the frequency scaling factor and the dynamic power of each node. While the static energy is computes using EQ(\ref{eq:perf}) multiplied by the static power of each processor. So, the energy modal of an heterogeneous platform has the following form:
+
+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)} +\\
- {}\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}))}
+ 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}) +
+ {MinTcm))}
\end{multline}
-
-These set of frequency scaling factors $S_i$ reduce quadratically the dynamic power which may cause degradation in performance and thus, the
-increase of the static energy because the execution time is increased~\cite{36}.
+
+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 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 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}
-Applying DVFS to lower level not surly reducing the energy consumption to minimum level. Also, a big scaling for the frequency produces high performance degradation percent. Moreover, by considering the drastically increase in execution time of parallel program, the static energy is related to this time
-and it also increased by the same ratio. Thus, the opportunity for gaining more energy reduction is restricted. For that choosing frequency scaling factors is very important process to taking into account both energy and performance. In our previous work~\cite{45}, we are proposed a method that selects the optimal frequency scaling factor for an homogeneous cluster, depending on the trade-off relation between the energy and performance. In this work we have an heterogeneous cluster, at each node there is different scaling factors, so our goal is to selects the optimal set of frequency scaling factors, $Sopt_1,Sopt_2,...,Sopt_N$, that gives the best trade-off between energy consumption and performance. The relation between the energy and the execution time is complex and nonlinear, Thus, unlike the relation between the performance 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 calculating the ratio between the new execution time (the scaled execution time) and the old one as follow:
+
+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:
\begin{multline}
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
- = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
+ {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}
{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
\end{multline}
-By the same way, we are normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency:
+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}} \\
- = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
+ {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
\sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
+ \sum_{i=1}^{N} {(Ps_i@+eYd162 \cdot T_{Old})}}
\end{multline}
-Where $T_{New}$ and $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). The second problem
-is that the optimization operation for both energy and performance is not in the same direction.
-In other words, the normalized energy and the normalized execution time curves are not at the same direction.
-While the main goal is to optimize the energy and execution time in the same time. According to the
-equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency scaling factors $S_1,S_2,...,S_N$ 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 see~\cite{19,42}, their
-methods add big overheads to the algorithm to select the suitable frequency.
-In this paper we are present a method to find the optimal set of frequency scaling factors to optimize both energy and execution time simultaneously
-without adding a big overhead. 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, the normalized performance, as follows:
-
+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
+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:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
= \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}}}
+ { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm}
\end{multline}
Then, we can model our objective function as finding the maximum distance
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 (better
+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
function has the following form:
-\begin{multline}
+\begin{equation}
\label{eq:max}
Max Dist =
\max_{i=1,\dots F, j=1,\dots,N}
(\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
\overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
-\end{multline}
+\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 energy values stored in a data file.
-Moreover, this function works in optimal way when the energy curve has a convex
-form over the available frequency scaling factors as shown in~\cite{15,3,19}.
+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 proposed an heterogeneous scaling algorithm, (figure~\ref{HSA}), that selects the optimal set of scaling factors from each node.
-The algorithm is numerates the suitable range of available scaling factors for each node in the heterogeneous cluster, returns a set of optimal frequency scaling factors for each node. Using heterogeneous cluster is produces different workloads for each node. Therefore, the fastest nodes waiting at the barrier for the slowest nodes to finish there work as in figure (\ref{fig:heter}). Our algorithm takes into account these imbalanced workloads when is starts to search for selecting the best scaling factors. So, the algorithm is selecting 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 test the first frequencies of the fastest nodes until it converge their frequencies to the frequency of the slowest node. If the algorithm is starts test changing the frequency of the slowest nodes from beginning, we are loosing performance and then not selecting the best tradeoff (the distance). This case will be similar to the homogeneous cluster when all nodes scales their frequencies together from the beginning. In this case there is a small distance between energy and performance curves, for example see the figure(\ref{fig:r1}). Then the algorithm searching for optimal frequency scaling factor from the selected frequencies until the last available ones.
-\begin{figure}[t]
- \centering
- \includegraphics[scale=0.5]{fig/start_freq}
- \caption{Selecting the initial frequencies}
- \label{fig:st_freq}
-\end{figure}
+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.
-To compute the initial frequencies, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ for each node. Each one of these factors represent a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
+
+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 $F_{i} \gets \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
\State Round the computed initial frequencies $F_i$ to the closest one available in each node.
\If{(not the first frequency)}
- \State $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
+ \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
\EndIf
- \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
+ \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
\State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
\State $Dist \gets 0$
- \State $Sopt_{i} \gets 1,~i=1,...,N. $
+ \State $Sopt_{i} \gets 1,~i=1,\dots,N. $
\While {(all nodes not reach their minimum frequency)}
\If{(not the last freq. \textbf{and} not the slowest node)}
- \State $F_i \gets F_i - Fdiff_i,~i=1,...,N.$
- \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$
+ \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 $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
+ \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $
\State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} + $ \hspace*{43 mm}
$\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
\State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
\State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
\If{$(\Pnorm - \Enorm > \Dist)$}
- \State $Sopt_{i} \gets S_{i},~i=1,...,N. $
+ \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
\State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndWhile
- \State Return $Sopt_1,Sopt_2, \dots ,Sopt_N$
+ \State Return $Sopt_1,Sopt_2,\dots,Sopt_N$
\end{algorithmic}
\caption{Heterogeneous scaling algorithm}
\label{HSA}
-\end{figure}
-When the initial frequencies are computed the algorithm numerates all available scaling factors starting from these frequencies until all nodes reach their
-minimum frequencies. At each iteration the algorithm remains the frequency of the slowest node without change and scaling the frequency of the other nodes. This is gives better performance and energy tradeoff.
-The proposed algorithm works online during the execution time of the MPI
-program. Its returns a set of optimal frequency scaling factors $Sopt_i$ 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 an heterogeneous cluster composed of four different types of nodes having the characteristics presented in
-table~(\ref{table:platform}), it takes \np[ms]{0.04} on average for 4 nodes and
-\np[ms]{0.1} on average for 128 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 for all the NAS benchmark on class C to selects the best set of frequency scaling factors. Its called just once during the execution of the program. The DVFS figure~(\ref{dvfs}) shows where and when the algorithm is
-called in the 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}
-The experiments of this work are executed on the simulator Simgrid/SMPI v3.10. We 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 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 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 running the MPI programs, for example if we execute the program on 8 node, there are 2 nodes from each type participating in the computing. 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 performance/GFlops, 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 from the hole power consumption, the same assumption is made in \cite{45,3}. These nodes are connected via an ethernet network with 1 Gbit/s bandwidth.
+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\% was 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.
+
+
\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.
-\cite{44}, which were run with three classes (A, B and C).
-In this experiments we are focus on running of 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 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 used under the same assumption used by \cite{45,3}. We used a percentage of 80\% for dynamic power from all power consumption of the CPU and 20\% for static power. The heterogeneous nodes in table (\ref{table:platform}) have different simulated performance, ranked from the node of type 1 with smaller performance/GFlops to the highest performance/GFlops for node 4. Therefore, the power values used proportionally increased from node 1 to node 4 that with highest performance. Then we used an assumption that the power consumption increases linearly with the performance/GFlops of the processor 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 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 show the 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. While the distance is decreased by the same direction. This because when we are run the MPI programs on a big number of nodes the communications is biggest than the computations, so the static energy is increased linearly to these times. The tables also show that performance degradation percent still approximately the same or decreased a little when the number of computing nodes is increased. This gives good a prove that the proposed algorithm keeping as mush as possible the performance degradation.
+
\begin{figure}
\centering
- \subfloat[Balanced nodes type scenario]{%
- \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[Imbalanced nodes type scenario]{%
- \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, ...) of nodes. The other benchmarks such as BT and SP executed on 2 to a power of (1, 2, 4, 9, ...) of nodes. We are take the average of energy saving, performance degradation and distances for all results of NAS benchmarks. The average 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 due to the increasing in the communication times as mentioned before. Thus, the average of distances (our objective function) is decreased linearly with energy saving while keeping the average of performance degradation the same. In BT and SP benchmarks, the average of 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.
+
+ \textbf{ The energy saving and performance degradation of all benchmarks are plotted to the number of
+nodes as in plots (\ref{fig:energy} and \ref{fig:per_deg}). As shown in the plots, the energy saving percentage of the benchmarks MG, LU, BT and FT is decreased linearly when the the number of nodes increased. While in EP benchmark the energy saving percentage is approximately the same percentage when the number of computing nodes is increased, because in this benchmark there is no communications. In the SP benchmark the energy saving percentage is decreased when it runs on a small number of nodes, while this percentage is increased when it runs on a big number of nodes. The energy saving of the GC benchmarks is significantly decreased when the number of nodes is increased, because this benchmark has more communications compared to other benchmarks. The performance degradation percentage of the benchmarks CG, EP, LU and BT is decreased when they run on a big number of nodes. While in MG benchmark has a higher percentage of performance degradation when it runs on a big number of nodes. The inverse happen in SP benchmark has smaller performance degradation percentage when it runs on a big number of nodes.}
+
+
+\subsection{The results for different power consumption scenarios}
+
+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}). \textbf{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, because this scenario uses higher percentage of dynamic dynamic power that is quadratically related to scaling factors. While the performance degradation percentage is less in 70\%-30\% scenario compared to 90\%-10\% scenario, because the first scenario used higher percentage for static power consumption that is linearly related to scaling factors and thus the execution time. }
+
+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.
-\subsection{The results for different powers scenarios}
-The results of the previous section are obtained using a percentage of 80\% for dynamic power and 20\% for static power of total power consumption. In this section we are change these ratio by using two others scenarios. Because is interested to measure the ability of the proposed algorithm to changes it behaviour when these power ratios are changed. In fact, we are use 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 as: 70\%-20\%, 80\%-20\% and 90\%-10\% scenario. The results of these scenarios running NAS benchmarks class C on 8 or 9 nodes are place in the tables (\ref{table:res_s1} and \ref{table:res_s2}).
\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 the results, the energy saving ratio is increased when using the a higher percentage for dynamic power (e.g. 90\%-10\% scenario). 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 also keeps as much as possible the static energy consumption that is always related to 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 and the energy model. The energy model is significantly depends on the execution time model EQ(\ref{eq:perf}), that the energy static related linearly. So, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered offline for 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 0.03 of all NAS benchmarks. The second verification that we are make is for the scaling algorithm to prove its ability to selects the best set of 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 scaling factors available of the other nodes. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the two algorithms is identical. While the proposed algorithm is runs faster, 10 times faster on average than 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}
%%% ispell-local-dictionary: "american"
%%% End:
-% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
+% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
+% LocalWords: de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
+% LocalWords: Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
