More corrections.

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index 933f0a13684300a70af85fc5ab84e5b60bcd544f..5a598e954c1df7142e0d31be1f5c5457799b30ad 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -200,18 +200,25 @@ the number of FLOPS executed by the processor which may increase the execution
 time of the application running over that processor.  Therefore, researchers use
 different optimization strategies to select the frequency that gives the best
 trade-off between the energy reduction and performance degradation ratio. In
 time of the application running over that processor.  Therefore, researchers use
 different optimization strategies to select the frequency that gives the best
 trade-off between the energy reduction and performance degradation ratio. In

-\cite{Our_first_paper} and \cite{pdsec2015} , a frequency selecting algorithm was proposed to reduce
-the energy consumption of message passing iterative applications running over
-homogeneous  and heterogeneous clusters respectively.  
-The results of the experiments showed significant energy
-consumption reductions. All the experimental results were conducted over the
-SimGrid simulator \cite{SimGrid}, which offers easy tools to create homogeneous and heterogeneous platforms and runs message passing parallel applications over them. In this paper, a new frequency selecting algorithm,
-adapted to  grid platforms composed of heterogeneous clusters, is presented. It is applied to the NAS parallel benchmarks and evaluated over a real testbed, 
-the Grid'5000 platform \cite{grid5000}. It selects  for a grid platform running a message passing iterative
-application the vector of
-frequencies  that simultaneously tries to offer the maximum energy reduction and
-minimum performance degradation ratios. The algorithm has a very small overhead,
-works online and does not need any training or profiling.
+\cite{Our_first_paper} and \cite{pdsec2015} , a frequency selecting algorithm
+was proposed to reduce the energy consumption of message passing iterative
+applications running over homogeneous and heterogeneous clusters respectively.
+The results of the experiments showed significant energy consumption
+reductions. All the experimental results were conducted over the SimGrid
+simulator \cite{SimGrid}, which offers easy tools to create homogeneous and
+heterogeneous platforms and runs message passing parallel applications over
+them. %
+\AG{[\dots], which offers easy tools to describe homogeneous and heterogeneous
+  platforms, and to simulate the execution of message passing parallel
+  applications over them.}%
+In this paper, a new frequency selecting algorithm, adapted to grid platforms
+composed of heterogeneous clusters, is presented. It is applied to the NAS
+parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
+\cite{grid5000}. It selects for a grid platform running a message passing
+iterative application the vector of frequencies that simultaneously tries to
+offer the maximum energy reduction and minimum performance degradation
+ratios. The algorithm has a very small overhead, works online and does not need
+any training or profiling.

 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some


@@ -266,7 +273,7 @@ heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
 goal was to maximize the energy efficiency of the platform during computation by
 maximizing the number of FLOPS per watt generated.
 In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
 goal was to maximize the energy efficiency of the platform during computation by
 maximizing the number of FLOPS per watt generated.
 In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et

-al. developed a scheduling algorithm that distributes workloads proportional to
+al. developed a scheduling algorithm that distributes workload proportional to

 the computing power of the nodes which could be a GPU or a CPU. All the tasks
 must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
 Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
 the computing power of the nodes which could be a GPU or a CPU. All the tasks
 must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
 Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables


@@ -305,7 +312,7 @@ following contributions :
 
 \item a new online frequency selecting algorithm for heterogeneous grid
   platforms. The algorithm has a very small overhead and does not need any
 
 \item a new online frequency selecting algorithm for heterogeneous grid
   platforms. The algorithm has a very small overhead and does not need any

-  training or profiling. It uses a new optimization function which
+  training nor profiling. It uses a new optimization function which

   simultaneously maximizes the performance and minimizes the energy consumption
   of a message passing iterative synchronous application.
 
   simultaneously maximizes the performance and minimizes the energy consumption
   of a message passing iterative synchronous application.
 


@@ -325,13 +332,6 @@ heterogeneous grid platforms. A heterogeneous grid platform could be defined as
 heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
 and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
 
 heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
 and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
 

-\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 grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. However, due to the
 The overall execution time of a distributed iterative synchronous application 
 over a heterogeneous grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. However, due to the


@@ -341,6 +341,13 @@ 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 has the highest computation time and no slack time.
 
 overall execution time of the program is the execution time of the slowest task 
 which has 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}
+

 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
 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


@@ -374,19 +381,20 @@ scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 vector of scaling factors can be predicted using (\ref{eq:perf}).
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 vector of scaling factors can be predicted using (\ref{eq:perf}).

+%

 \begin{equation}
   \label{eq:perf}
   \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
   +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}
 \begin{equation}
   \label{eq:perf}
   \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
   +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}

-
+%

 where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 

-first  iteration. the execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors 
- and the slowest communication time without slack time during one iteration. 
+first  iteration.  The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
+and the slowest communication time without slack time during one iteration.

 The latter is equal to the  communication time of the slowest node in the slowest cluster $h$.
 The latter is equal to the  communication time of the slowest node in the slowest cluster $h$.

-It means only the communication time without any slack time is taken into account.  
+It means\AG[]{It means that\dots} only the communication time without any slack time is taken into account.

 Therefore, the execution time of the iterative application is equal to
 the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
 number of iterations of that application.
 Therefore, the execution time of the iterative application is equal to
 the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
 number of iterations of that application.


@@ -549,29 +557,31 @@ are not measured using the same metric.  To solve this problem, the execution
 time is normalized 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:
 time is normalized 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{equation}
   \label{eq:pnorm}
   \Pnorm = \frac{\Tnew}{\Told}                 
 \end{equation}
 \begin{equation}
   \label{eq:pnorm}
   \Pnorm = \frac{\Tnew}{\Told}                 
 \end{equation}

-
-
-Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told})
+%
+where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
+%

 \begin{equation}
   \label{eq:told}
    \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
 \end{equation}
 \begin{equation}
   \label{eq:told}
    \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
 \end{equation}

+%

 In the same way, the energy is normalized 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, the energy is normalized by computing the ratio between the
 consumed energy while scaling down the frequency and the consumed energy with
 maximum frequency for all  nodes:

+%

 \begin{equation}
   \label{eq:enorm}
   \Enorm = \frac{\Ereduced}{\Eoriginal} 
 \end{equation}
 \begin{equation}
   \label{eq:enorm}
   \Enorm = \frac{\Ereduced}{\Eoriginal} 
 \end{equation}

-
-Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
+%
+where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 

 computed as in (\ref{eq:eorginal}).
 computed as in (\ref{eq:eorginal}).

-
-
+%

 \begin{equation}
   \label{eq:eorginal}
     \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 
 \begin{equation}
   \label{eq:eorginal}
     \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 


@@ -580,9 +590,9 @@ computed as in (\ref{eq:eorginal}).
 
 While the main goal is to optimize the energy and execution time at the same
 time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
 
 While the main goal is to optimize the energy and execution time at the same
 time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 

-According to Equations~\ref{eq:pnorm} and \ref{eq:enorm}, 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
+According to (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduces both the energy
+and the execution time,  but the main objective is to produce

 maximum energy reduction with minimum execution time reduction.
 
 This problem can be solved by making the optimization process for energy and
 maximum energy reduction with minimum execution time reduction.
 
 This problem can be solved by making the optimization process for energy and


@@ -609,7 +619,7 @@ Then, the objective function can be modeled in order to find the maximum
 distance between the energy curve (\ref{eq:enorm}) and the performance curve
 (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 represents the minimum energy consumption with minimum execution time (maximum
 distance between the energy curve (\ref{eq:enorm}) and the performance curve
 (\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
+performance) at the same time, see Figure~\ref{fig:r1} and

 Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}
 Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}


@@ -696,7 +706,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 \end{algorithm}
 
 
 \end{algorithm}
 
 

-In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, 
+In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 

 is presented. It selects the vector of the frequency
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing
 is presented. It selects the vector of the frequency
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing


@@ -763,7 +773,9 @@ Therefore, the algorithm iterates on all remaining frequencies, from the higher
 bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
 energy consumption and performance and selects the optimal vector of the frequency scaling
 factors. At each iteration the algorithm determines the slowest node
 bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
 energy consumption and performance and selects the optimal vector of the frequency scaling
 factors. At each iteration the algorithm determines the slowest node

-according to Equation~\ref{eq:perf} and keeps its frequency unchanged,
+according to Equation~\ref{eq:perf}
+%\AG[]{Be consistent: remove word ``Equation'' and add parentheses around equation number, here and all along the rest of the text.}
+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
 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


@@ -778,6 +790,7 @@ consumed energy decrease from the beginning of the plot. On the other hand, in
 the  grid platform the performance is maintained at the beginning of the
 plot even if the frequencies of the faster nodes decrease until the computing
 power of scaled down nodes are lower than the slowest node. In other words,
 the  grid platform the performance is maintained at the beginning of the
 plot even if the frequencies of the faster nodes decrease until the computing
 power of scaled down nodes are lower than the slowest node. In other words,

+\AG[]{That's not a sentence.}

 until they reach the higher bound. It can also be noticed that the higher the
 difference between the faster nodes and the slower nodes is, the bigger the
 maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 
 until they reach the higher bound. It can also be noticed that the higher the
 difference between the faster nodes and the slower nodes is, the bigger the
 maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 


@@ -799,7 +812,7 @@ the clusters and their nodes are connected via  high speed local area networks.
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 
 Since Grid'5000 is dedicated to  testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the Grid'5000 testbed provides at some sites a power measurement tool to capture 
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 
 Since Grid'5000 is dedicated to  testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the Grid'5000 testbed provides at some sites a power measurement tool to capture 

-the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
+the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, \dots{} For more details refer to

 \cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$, 
  firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 
 \cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$, 
  firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 


@@ -819,7 +832,7 @@ measured value in maximum powers vector and the minimum measured value in the id
 
 On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in Grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
 
 
 On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in Grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
 

-In the experiments presented in the following sections, two sites of Grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in Figure~\ref{fig:grid5000}.
+In the experiments presented in the following sections, two sites of Grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as shown on Figure~\ref{fig:grid5000}.

 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 
 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 


@@ -838,6 +851,8 @@ selected clusters and are presented in Table~\ref{table:grid5000}.
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}

+  \AG{I don't understand the labels on the horizontal axis: 10:30:37, 10:30:38,
+    etc.}

   \caption{The power consumption by one core from the Taurus cluster}
   \label{fig:power_cons}
 \end{figure}
   \caption{The power consumption by one core from the Taurus cluster}
   \label{fig:power_cons}
 \end{figure}


@@ -947,7 +962,7 @@ power is assumed to be equal to 20\% of the dynamic power. The execution
 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 
 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 

-presented in  Plots~\ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+presented in Figures~\ref{fig:eng_sen} and \ref{fig:time_sen} respectively.

 
 For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
 for  16 and 32 nodes is lower than the energy consumed while using two sites. 
 
 For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
 for  16 and 32 nodes is lower than the energy consumed while using two sites. 


@@ -955,10 +970,14 @@ The long distance communications between the two distributed sites increase the
 
 The execution times of these benchmarks 
 over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 
 
 The execution times of these benchmarks 
 over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 

-scenario. Moreover, most of the benchmarks running over the one site scenario their execution times  are approximately divided by two  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).  
+scenario. Moreover, most of the benchmarks running over the one site scenario their execution times  are approximately divided by two  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).\AG{Parse error (cannot understand the previous sentence).}

 
 

-However, the  execution times and the energy consumptions of EP and MG benchmarks, which have no or small communications, are not significantly affected 
- in both scenarios. Even when the number of nodes is doubled. On the other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing nodes.
+However, the execution times and the energy consumptions of EP and MG
+benchmarks, which have no or small communications, are not significantly
+affected in both scenarios, even when the number of nodes is doubled.  On the
+other hand, the communications\AG[]{the communication time?} of the rest of the benchmarks increases when
+using long distance communications between two sites or increasing the number of
+computing nodes.

 
 
 
 
 
 


@@ -968,7 +987,7 @@ Equation~\ref{eq:eorginal}, for all benchmarks as in Figure~\ref{fig:eng_s}.
 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   
 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   

-than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computation times are bigger than the communication times which

 results in  a lower energy consumption. Indeed, the dynamic  consumed power
 is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 
 increase the communication times and thus produces less energy saving depending on the 
 results in  a lower energy consumption. Indeed, the dynamic  consumed power
 is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 
 increase the communication times and thus produces less energy saving depending on the 


@@ -1117,7 +1136,7 @@ The energy consumption is reduced at the same rate in the two scenarios when com
 
 
 The performance degradation percentages of the NAS benchmarks are presented in
 
 
 The performance degradation percentages of the NAS benchmarks are presented in

-Figure\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 
+Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 

 
 The trade-off distances percentages of the NAS benchmarks over the two scenarios are presented 
 in ~Figure~\ref{fig:dist-mc}. These  trade-off distances between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  trade-off distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
 
 The trade-off distances percentages of the NAS benchmarks over the two scenarios are presented 
 in ~Figure~\ref{fig:dist-mc}. These  trade-off distances between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  trade-off distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).


@@ -1162,14 +1181,14 @@ In these experiments, class D of the NAS parallel benchmarks are executed over t
 \end{figure}
 
 The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
 \end{figure}
 
 The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 

-in Figure\ref{fig:eng_sen}. This figure shows that the  10\% of static power scenario 
+in Figure~\ref{fig:eng_sen}. This figure shows that the  10\% of static power scenario 

 gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
 scenarios. The small value of the static power consumption makes the proposed 
 scaling algorithm  select smaller frequencies for the CPUs. 
 These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives            less overall energy consumption. 
 The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
 
 gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
 scenarios. The small value of the static power consumption makes the proposed 
 scaling algorithm  select smaller frequencies for the CPUs. 
 These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives            less overall energy consumption. 
 The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
 

-The performance degradation percentages are presented in Figure\ref{fig:per-pow}.
+The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.

 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
 the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios 
 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
 the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios