made all the tables, figures, sections and equations references homogeneous

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index 3e1bb461175032175abd0d22b3fd90e0406061d8..49ccbf7815edf10c632198eb5b75a9a3295cd51b 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -432,7 +432,7 @@ communication and no slack time.
 The main objective of DVFS operation is to reduce the overall energy
 consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
 frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
 The main objective of DVFS operation is to reduce the overall energy
 consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
 frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot

-F$ with some constant $\beta$.~This equation is used to study the change of the
+F$ with some constant $\beta$. This equation is used to study the change of the

 dynamic voltage with respect to various frequency values
 in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
 frequency can be expressed by the scaling factor $S$ which is the ratio between
 dynamic voltage with respect to various frequency values
 in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
 frequency can be expressed by the scaling factor $S$ which is the ratio between


@@ -580,7 +580,7 @@ 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 the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+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
 maximum energy reduction with minimum execution time reduction.
 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.


@@ -653,8 +653,8 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \If{(not the first frequency)}
           \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
     \EndIf
     \If{(not the first frequency)}
           \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
     \EndIf

-    \State $\Told \gets $ computed as in equations (\ref{eq:told}).
-    \State $\Eoriginal \gets $ computed as in equations (\ref{eq:eorginal}) .
+    \State $\Told \gets $ computed as in Equation \ref{eq:told}.
+    \State $\Eoriginal \gets $ computed as in Equation \ref{eq:eorginal}.

     \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
     \State $\Dist \gets 0 $
     \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
     \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
     \State $\Dist \gets 0 $
     \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}


@@ -662,8 +662,8 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
         \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
         \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
         \EndIf
         \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
         \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
         \EndIf

-       \State $\Tnew \gets $ computed as  in equations (\ref{eq:perf}). 
-       \State $\Ereduced \gets $ computed as  in equations (\ref{eq:energy}). 
+       \State $\Tnew \gets $ computed as  in Equation \ref{eq:perf}. 
+       \State $\Ereduced \gets $ computed as  in Equation \ref{eq:energy}. 

        \State $\Pnorm \gets \frac{\Told}{\Tnew}$,  $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
         \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
        \State $\Pnorm \gets \frac{\Told}{\Tnew}$,  $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
         \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $


@@ -763,11 +763,11 @@ 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 the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+according to Equation~\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
 while it lowers the frequency of all other nodes by one gear.  The new overall
 energy consumption and execution time are computed according to the new scaling
 factors.  The optimal set of frequency scaling factors is the set that gives the

-highest distance according to the objective function (\ref{eq:max}).
+highest distance according to the objective function~\ref{eq:max}.

 
 Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
 consumed energy for an application running on a homogeneous cluster and a
 
 Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
 consumed energy for an application running on a homogeneous cluster and a


@@ -802,10 +802,10 @@ Since grid'5000 is dedicated to  testing, contrary to production grids it allows
 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
 \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 
 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
 \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}). 
+dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 

 
 
 
 

-The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
+The dynamic power $\Pd[j]$ is computed as in Equation~\ref{eq:pdyn}

 \begin{equation}
   \label{eq:pdyn}
     \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
 \begin{equation}
   \label{eq:pdyn}
     \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])


@@ -819,14 +819,14 @@ 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 in Figure~\ref{fig:grid5000}.

 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 
 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 
 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 

-frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
-the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using equation (\ref{eq:pdyn}) for all the nodes in the 
-selected clusters and are presented in table  \ref{table:grid5000}.
+frequency ranges and local network features: the bandwidth and the latency.  Table~\ref{table:grid5000} shows 
+the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using Equation~\ref{eq:pdyn} for all the nodes in the 
+selected clusters and are presented in Table~\ref{table:grid5000}.

 
 
 \begin{figure}[!t]
 
 
 \begin{figure}[!t]


@@ -903,7 +903,7 @@ is very low due to the higher communication times which reduce the effect of DVF
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
 is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
 is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.

-Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
+Table~\ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 

 
 \begin{table}[h]
 
 
 \begin{table}[h]
 


@@ -938,18 +938,17 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
 
 
 The NAS parallel benchmarks are executed over these two platforms
 
 
 The NAS parallel benchmarks are executed over these two platforms

- with different number of nodes, as in Table \ref{tab:sc}. 
+ with different number of nodes, as in Table~\ref{tab:sc}. 

 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 
 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 

-using the equation of the reduced energy consumption, equation 
-(\ref{eq:energy}). This model uses the measured dynamic power showed in Table \ref{table:grid5000}
+using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}

 
 and the static 
 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 
 
 and the static 
 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 

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


@@ -965,8 +964,8 @@ However, the  execution times and the energy consumptions of EP and MG benchmark
 
 
 The energy saving percentage is computed as the ratio between the reduced 
 
 
 The energy saving percentage is computed as the ratio between the reduced 

-energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
-equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}. 
+energy consumption, Equation~\ref{eq:energy}, and the original energy consumption,
+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   


@@ -1030,7 +1029,7 @@ The rest of the benchmarks showed different performance degradation percentages,
 when the communication times increase and vice versa.
 
 Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The tradeoff distance percentage can be 
 when the communication times increase and vice versa.
 
 Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The tradeoff distance percentage can be 

-computed as in equation \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
+computed as in Equation~\ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 

 tradeoff, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
 tradeoff comparing to the two sites scenario  because the former has high speed local communications 
 which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 
 tradeoff, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
 tradeoff comparing to the two sites scenario  because the former has high speed local communications 
 which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 


@@ -1045,16 +1044,17 @@ which increase the computations to communications ratio  and the latter uses lon
 \label{sec.res-mc}
 
 The  clusters of grid'5000 have different number of cores embedded in their nodes
 \label{sec.res-mc}
 
 The  clusters of grid'5000 have different number of cores embedded in their nodes

-as shown in Table \ref{table:grid5000}. In 
-this section, the proposed scaling algorithm is evaluated over the  grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in the section \ref{sec.res}.
+as shown in Table~\ref{table:grid5000}. In 
+this section, the proposed scaling algorithm is evaluated over the  grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in  Section~\ref{sec.res}.

 The one site scenario uses  32 cores from multi-cores nodes instead of 32 distinct nodes. For example if 
 the participating number of cores from a certain cluster is equal to 14, 
 in the multi-core scenario the selected nodes is equal to  4 nodes while using 
 3 or 4 cores from each node. The platforms with one  
 The one site scenario uses  32 cores from multi-cores nodes instead of 32 distinct nodes. For example if 
 the participating number of cores from a certain cluster is equal to 14, 
 in the multi-core scenario the selected nodes is equal to  4 nodes while using 
 3 or 4 cores from each node. The platforms with one  

-core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 
+core per node and  multi-cores nodes are  shown in Table~\ref{table:sen-mc}. 

 The energy consumptions and execution times of running  class D of the NAS parallel 
 benchmarks over these two different scenarios are presented 
 The energy consumptions and execution times of running  class D of the NAS parallel 
 benchmarks over these two different scenarios are presented 

-in  figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+

 
 \begin{table}[]
 \centering
 
 \begin{table}[]
 \centering


@@ -1099,7 +1099,7 @@ consumption and the execution times of the EP and MG benchmarks do not change si
 scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
   
   
 scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
   
   

-The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in figure \ref{fig:eng-s-mc}. 
+The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}. 

 The figure shows that  the energy saving percentages in the one 
 core and the multi-cores scenarios
 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
 The figure shows that  the energy saving percentages in the one 
 core and the multi-cores scenarios
 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.


@@ -1107,10 +1107,10 @@ 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 is 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 is lower because  the computations to communications ratio is smaller than the ratio 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 tradeoff distance percentages of the NAS benchmarks over the two scenarios are presented 
-in figure \ref{fig:dist-mc}. These  tradeoff distance 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  tradeoff distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
+The tradeoff distances percentages of the NAS benchmarks over the two scenarios are presented 
+in ~Figure~\ref{fig:dist-mc}. These  tradeoff 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  tradeoff distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).

 
 
 
 
 
 


@@ -1132,7 +1132,7 @@ in figure \ref{fig:dist-mc}. These  tradeoff distance between energy consumption
 \subsection{Experiments with different static power scenarios}
 \label{sec.pow_sen}
 
 \subsection{Experiments with different static power scenarios}
 \label{sec.pow_sen}
 

-In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
+In Section~\ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 

 
 The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
 In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
 
 The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
 In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.


@@ -1163,18 +1163,18 @@ 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 tradeoff 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 tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 

-are presented in figure \ref{fig:dist}. 
+are presented in Figure~\ref{fig:dist}. 

 It shows that the best  tradeoff
 distance percentage is obtained with  the  10\% static power scenario  and this percentage 
 is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
 It shows that the best  tradeoff
 distance percentage is obtained with  the  10\% static power scenario  and this percentage 
 is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.


@@ -1193,16 +1193,16 @@ In this section, the proposed frequencies selecting algorithm is compared to a m
 This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 
 architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
 
 This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 
 architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
 

-To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  equation \ref{eq:energy} and
-execution time model, equation \ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
-Moreover, both algorithms start the search space from the upper bound computed as in equation   \ref{eq:Fint}.
+To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy} and
+execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.

 Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
 and selects the vector of frequencies that minimize the EDP product.
 
 Both algorithms were applied to class D of the NAS benchmarks over 16 nodes.
 Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
 and selects the vector of frequencies that minimize the EDP product.
 
 Both algorithms were applied to class D of the NAS benchmarks over 16 nodes.

-The participating computing nodes are distributed  according to the two scenarios described in  section \ref{sec.res}. 
+The participating computing nodes are distributed  according to the two scenarios described in  Section~\ref{sec.res}. 

 The experimental results, the energy saving, performance degradation and tradeoff distance percentages, are 
 The experimental results, the energy saving, performance degradation and tradeoff distance percentages, are 

-presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.

 
 
 \begin{figure*}[t]
 
 
 \begin{figure*}[t]