From 7851a05d8c696e72f32cc2b36170ff31a213bfc1 Mon Sep 17 00:00:00 2001
From: jean-claude <jean-claude.charr@univ-fcomte.fr>
Date: Wed, 23 Sep 2015 15:37:16 +0200
Subject: [PATCH] merged

---
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 mpi-energy2-extension/Heter_paper.tex | 351 +++++++++++++++++++-------
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diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index 79c3511..f5917fa 100644
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -9,6 +9,7 @@
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}
+\usepackage{multirow}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage[autolanguage,np]{numprint}
@@ -106,8 +107,9 @@
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
-heterogeneous grid platforms. A heterogeneous grid platform is defined as a collection of
-heterogeneous computing clusters interconnected via a long distance network (the internet network). Each computing cluster in the grid composed from homogeneous nodes, where are connected together via high speed homogeneous network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
+heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
+heterogeneous computing clusters interconnected via a long distance network which has 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
@@ -164,7 +166,7 @@ vector of scaling factors can be predicted using (\ref{eq:perf}).
   +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}
 
-where $N$ is the number of the clusters in the grid, $M$ is the number of the nodes in
+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 model computes the maximum computation time with scaling factor 
@@ -180,6 +182,7 @@ of message passing distributed applications for homogeneous and heterogeneous cl
 used in the method to optimize both the energy consumption and the performance
 of iterative methods, which is presented in the following sections.
 
+
 \subsection{Energy model for heterogeneous platform}
 
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
@@ -349,6 +352,7 @@ maximum frequency for all  nodes:
 Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
 computed as in ().
 
+\textcolor{red}{A reference is missing}
 \begin{equation}
   \label{eq:eorginal}
     \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 
@@ -404,7 +408,7 @@ values for each node (static and dynamic powers). However, the most important
 energy reduction gain can be achieved when the energy curve has a convex form as shown 
 in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
 
-\section{The scaling factors selection algorithm for heterogeneous grid platforms }
+\section{The scaling factors selection algorithm for  grids }
 \label{sec.optim}
 
 \begin{algorithm}
@@ -475,10 +479,12 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 
 \subsection{The algorithm details}
 
-In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
-scaling factors vector that gives the best trade-off between minimizing the
+\textcolor{red}{Delete the subsection if there's only one.}
+
+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
-synchronous iterative application executed on a heterogeneous grid platform. It works
+synchronous iterative application executed on a  grid. 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
@@ -496,13 +502,13 @@ scaling algorithm is called in the iterative MPI program.
   \label{fig:st_freq}
 \end{figure}
 
-The nodes in a heterogeneous grid have different computing powers, thus
+Nodes from distinct clusters in a grid 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.  The algorithm takes into account this
-problem and tries to reduce these slack times when selecting the frequency
-scaling factors vector. At first, it selects initial frequency scaling factors
+problem and tries to reduce these slack times when selecting the vector of the frequency
+scaling factors. 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
@@ -523,25 +529,24 @@ follows:
 \end{equation}
 If the computed initial frequency for a node is not available in the gears of
 that node, it is replaced by the nearest available frequency.  In
-Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power 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 highlighted 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 scaling factors higher
+optimal vector of frequencies because selecting higher frequencies
 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 or reaching to the lower bound. The lower bound is used to  stop 
-the algorithm search process when the new computed distance between the energy and performance is less than zero.
-The new negative distance is  mean that the performance degradation ratio is higher than energy saving ratio.
-Therefore, the algorithm must stop the iterations before reaching to the end of the search space, the minimum frequencies, 
-because the all the coming new distances are negative values.
-The algorithm iterates on all remaining frequencies, from the higher
-bound until all nodes reach their minimum frequencies or to the lower bound, 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
+frequencies until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 
+
+Therefore, the algorithm iterates on all 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
 according to the 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
@@ -549,48 +554,67 @@ factors.  The optimal set of frequency scaling factors is the set that gives the
 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 platform and a
-heterogeneous grid platform respectively while increasing the scaling factors. It can
-be noticed that in a homogeneous platform the search for the optimal scaling
+consumed energy for an application running on a homogeneous cluster and a
+ grid platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster the search for the optimal scaling
 factor should start from the maximum frequency because the performance and the
 consumed energy decrease from the beginning of the plot. On the other hand, in
-the heterogeneous grid platform the performance is maintained at the beginning of the
+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,
 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 the
-scaling factors are varying which results in bigger energy savings. 
+maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 
 
 
 \section{Experimental results}
 \label{sec.expe}
+While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid.org}, 
+in this paper real experiments were conducted over the grid'5000 platform. 
 
 \subsection{Grid'5000 architature and power consumption}
 \label{sec.grid5000}
-The grid'5000 is a large-scale testbed found in France \cite{grid5000}.
-The grid infrastructure consist of ten sites distributed over all France 
-metropolitan regions. Each site in the grid'5000 composed from number of heterogeneous 
-computing clusters, while each cluster includes a collection of homogeneous nodes.
-In general, the grid'5000 had one thousand of heterogeneous nodes and eight thousand of cores. 
-All the sites are connected together via special long distance network called RENATER,
-which is the French National Telecommunication Network for Technology. Whereas inside each site 
-the clusters and their nodes are connected throw high speed local area networks. 
-There are different types of local networks used such as Ethernet and Infiniband netwoks,  
-which allowed different gigabits bandwidth and latencies.  On the other hand, the nodes inside each cluster 
-are homogeneous, while they are different from the nodes of the other clusters. Therefore, there are
-a wide diversity of processors in grid'5000, that mainly had different processors families 
-such as  Intel Xeon and AMD Opteron families. 
-
-In this paper we are interested  to run NAS parallel v3.3 \cite{NAS.Parallel.Benchmarks} over grid'5000.
-We are used seven benchmarks, CG, MG, EP, LU, BT, SP and FT. These benchmarks used seven different types of classes.
-These classes are S, W, A, B, C, D, E, where S represents the smaller problem size that used by benchmark and
-E is represents the biggest class. In this work, the class D is used for all benchmarks in all the experiments that will 
-be showed in the coming sections.
-Moreover, the NAS parallel benchmarks have different computations and communications ratios, then it is interested 
-to study their energy consumption and their performance on real testbed such as grid'5000.
-In this work, the NAS benchmarks are executed over two sites, Lyon and Nancy sites, of grid'5000.
-These two sites had seven different types of computing clusters as in figure (\ref{fig:grid5000}).
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via 	a special long distance network called RENATER,
+which is the French National Telecommunication Network for Technology.
+Each site of the grid is composed of few heterogeneous 
+computing clusters and each cluster contains many homogeneous nodes. In total,
+ grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
+the clusters and their nodes are connected via  high speed local area networks. 
+Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
+
+Since grid'5000 is dedicated for 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 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}. To just 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 $P\max[jx]$. The difference between the two measured power consumption represents the 
+dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
+
+\textcolor{red}{why maximum and minimum, change peak in the equation and the figure}
+
+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} (P\max[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
+
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values, 
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum 
+measured value in peak powers vector and the minimum measured value in the idle powers vector.
+
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and 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  np[\%]{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}).
+
+Four clusters from the two sites were selected in the experiments: one cluster from 
+Lyon's site, Taurus cluster, 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 details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
+selected clusters and are presented in table  \ref{table:grid5000}.
+
+
+
 
 \begin{figure}[!t]
   \centering
@@ -599,12 +623,22 @@ These two sites had seven different types of computing clusters as in figure (\r
   \label{fig:grid5000}
 \end{figure}
 
-Four clusters from the two sites are selected in the experiments, one cluster from 
-Lyon site, Taurus cluster, and three clusters from Nancy site where are Graphene, 
-Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while their nodes are
-different from the nodes of other clusters in many aspects such as: computing power, power consumption, available 
-frequencies ranges and the network features, the bandwidth and the latency. The Table \ref{table:grid5000} shows 
-the details characteristics of these four clusters.
+
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
+
+
+
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
+  \caption{The power consumption by one core from Taurus cluster}
+  \label{fig:power_cons}
+\end{figure}
+
+
 
   
 \begin{table}[!t]
@@ -637,48 +671,179 @@ the details characteristics of these four clusters.
   \label{table:grid5000}
 \end{table} 
 
-The grid'5000 testbed provided some monitoring and measurements features to captured 
-the power consumption values for each node in any cluster of Lyon and Nancy sites. 
-The power consumed for each node from the selected four clusters is measured. 
-While the power consumed by any computing node is a collection of the powers  consumed by 
-hard drive, main-board, memory and node's computing cores, for more detail refer to
-\cite{Energy_measurement}. Therefore, the dynamic power consumed
-by one core is not allowed to measured alone. To overcome this problem, firstly, 
-we measured the power consumed by one node when there is no computation, when 
-the CPU is in the idle state. The second step, we run EP benchmark, there is no communications 
-in this benchmarks, over one core with maximum frequency of the desired node and 
-capturing the power consumed by a node, this representing the peak power of the node with one core.  
-The difference between the peak power  and the idle power representing the 
-dynamic power consumption of that core with maximum frequency, for example see figure(\ref{fig:power_cons}). 
-The $\Ppeak[jx]$ is the peak power value in time $x$ with maximum frequency for one core of node $j$, 
-and $\Pidle[jy]$ is the idle power value in time $y$ for the one  core of the node $j$ . 
-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} (\Ppeak[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
-\end{equation}
 
-where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
-$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values, 
-$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
-Therefore, the dynamic power of one core is computed as the difference between the maximum 
-measured value in peak powers vector and the minimum measured value in the idle powers vector.
-We are computed the dynamic powers, using the equation (\ref{eq:pdyn}), for all nodes in the 
-selected clusters, which is recorded in table  \ref{table:grid5000}.
-On the other side, the static power consumption by one core is embedded with whole idle power consumption of the node.
-Indeed, the static power is represents as ratio from  dynamic power. So, we supposed  
-the static power consumption represented  as \np[\%]{20} of dynamic power consumption of the core, 
-the same assumption was made in \cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy}.
-\begin{figure}[!t]
+
+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}
+In this section, the scaling factor selection algorithm \ref{HSA}, is applied 
+to NAS parallel benchmarks. Seven benchmarks, CG, MG, EP, LU, BT, SP and FT, of the class D
+are executed over grid'5000 computing clusters. As mentioned previously, the experiments 
+of this paper obtained from a collection of many clusters distributed in two sites, Lyon and Nancy sites, 
+of grid'5000. Four different clusters are selected from these two sites to generate two 
+different scenarios. Each of these two scenarios used three clusters. The first scenario,
+is composed from three clusters that located in two sites, Lyon and Nancy sites. One of these three
+clusters is from Lyon site, Taurus cluster and the other two clusters are form Nancy site, 
+Graphene and Griffon clusters. The second scenario, is composed from three clusters that are 
+located in one site, Nancy site. These cluster are Graphite, Graphene and griffon. The main reason 
+behind using these two scenarios is because the first one is executing the NAS parllel benchmarks over 
+two sites that are connected via long distance network, then the computations to communications ratio 
+is very low due to the increase in communication times, while in the second scenario, all of the three clusters are 
+located in one site and they are connected via high speed local area networks, where the computations 
+to communications ratio is higher. Therefore, it is very interested to know the performance behaviour 
+and the energy consumption of NAS parallel benchmarks using the proposed method, when they run 
+over these two different platform scenarios. Moreover, The NAS parallel benchmarks are executed over 
+16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
+are different, this  depends on the available number of nodes in each cluster. 
+Table \ref{tab:sc} shows the details of these two scenarios and the number of nodes 
+used from each cluster.
+
+\begin{table}[h]
+
+\caption{The different clusters scenarios}
+\centering
+\begin{tabular}{|*{3}{c|}}
+\hline
+\multirow{2}{*}{Scenario name}        & \multicolumn{2}{c|} {The participating clusters} \\ \cline{2-3} 
+                                      & Cluster name           & No. of  nodes of each cluster     \\ 
+\hline
+\multirow{3}{*}{Two sites / 16 nodes} & Taurus                 & 5                      \\ \cline{2-3} 
+                                      & Graphene               & 5                      \\ \cline{2-3} 
+                                      & Griffon                & 6                      \\ 
+\hline
+\multirow{3}{*}{Tow sites / 32 nodes} & Taurus                 & 10                     \\ \cline{2-3} 
+                                      & Graphene               & 10                     \\ \cline{2-3} 
+                                      & Griffon                & 12                     \\ 
+\hline
+\multirow{3}{*}{One site / 16 nodes}  & Graphite               & 4                      \\ \cline{2-3} 
+                                      & Graphene               & 6                      \\ \cline{2-3} 
+                                      & Griffon                & 6                      \\ 
+\hline
+\multirow{3}{*}{One site / 32 nodes}  & Graphite               & 4                      \\ \cline{2-3} 
+                                      & Graphene               & 12                     \\ \cline{2-3} 
+                                      & Griffon                & 12                       \\ 
+\hline
+\end{tabular}
+ \label{tab:sc}
+\end{table}
+
+\begin{figure}
   \centering
-  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
-  \caption{The power consumption by one core from Taurus cluster}
-  \label{fig:power_cons}
+  \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}
+  \caption{The energy consumptions of NAS benchmarks over different scenarios }
+  \label{fig:eng_sen}
 \end{figure}
 
 
-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/time_scenarios.eps}
+  \caption{The execution times of NAS benchmarks over different scenarios }
+  \label{fig:time_sen}
+\end{figure}
+
+The NAS parallel benchmarks are executed over these two platform
+scenarios with different number of nodes, as in Table \ref{tab:sc}. 
+The overall energy consumption of all benchmark, class D, with 
+applying 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 and static 
+power values that showed in Table \ref{table:grid5000}. The execution
+time is measured for all benchmarks over these different scenarios.  
+The energy consumptions  and the execution times for all benchmarks are 
+demonstrated in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+In general, the energy consumptions of NAS benchmarks over one site scenario 
+for  16 and 32 nodes are less than those executed over the two sites 
+scenarios. This because in the two sites scenario the communication times 
+are higher, due to long distance communications between the two distributed sites. 
+This leading to more static energy consumption which is linearly related to the 
+increased in the communication time. The execution times of these benchmarks 
+over one sites for 16 and 32 nodes are less  comparing to the two sites 
+scenario according to the increase in communications times.
+
+The EP and MG benchmarks, where there are no or small communications, showed 
+that their execution times and the energy consumptions are not effected 
+significantly in both scenarios and when the number of nodes is increase, 
+while the other benchmarks showed the inverse, because they have more communications 
+that proportionally increase the communication times if there are slow 
+communications or using more number of nodes or both of them.
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/eng_s.eps}
+  \caption{The energy saving of NAS benchmarks over different scenarios }
+  \label{fig:eng_s}
+\end{figure}
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/per_d.eps}
+  \caption{The performance degradation of NAS benchmarks over different scenarios }
+  \label{fig:per_d}
+\end{figure}
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/dist.eps}
+  \caption{The tradeoff distance of NAS benchmarks over different scenarios }
+  \label{fig:dist}
+\end{figure}
+
+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}. 
+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. This is because
+the computations to communications ratio in  one site scenario is higher 
+than the ratio of the two sites scenarios, due to the increase in the communication
+times. Moreover, the frequencies selecting algorithm selects smaller frequencies, bigger
+scaling factors, when the computations times are higher than communication times, 
+producing  smaller energy consumption, because the dynamic energy consumption 
+is decreased linearly with computation times that decreased exponentially with 
+scaling factors. On the other side, the increase in the number of computing nodes can be 
+increase the communication times and thus producing less energy saving depending on the 
+benchmarks being executed. The benchmarks CG, MG, BT and FT show more 
+energy saving percentage in one site scenario when executed  over 16 nodes comparing to 32 nodes. While 
+the benchmarks  LU and SP showed the inverse, because there computations to 
+communications ratio is not effected to the increase in local site communications. 
+While all benchmarks are effected by the long distance communications in the two sites 
+scenarios, except  EP benchmarks. In EP benchmark  there is no communications 
+in their iterations, then it is independent from the effect of local and long 
+distance communications. Therefore, the energy saving percentage of this benchmarks is 
+depend on differences between the computing powers of the computing nodes, for example 
+in the one site scenario, the graphite cluster is selected but in the two sits scenario 
+this cluster is replaced with Taurus cluster that be more powerful in computing power. 
+Therefore, the energy saving of EP benchmarks are bigger in the two site scenario due 
+to increase in the differences between the computing powers of the nodes. This means, the higher 
+differences between the nodes' computing powers make the proposed frequencies selecting  
+algorithm  to selects smaller frequencies in the nodes of the higher computing power, 
+producing less energy consumption and thus more energy saving.
+The best energy saving percentage was for one site scenario with 16 nodes, on average it
+saves the energy consumption up to 30\%. 
+
+Figure \ref{fig:per_d}, presents the performance degradation percentages for all benchmarks.
+It shows that the performance degradation percentages  of the one site scenario with
+32 nodes, on average equal to 10\%, is higher than the performance degradation of one 16 nodes, 
+which on average equal to 3\%. This because selecting smaller frequencies in the one site scenarios,
+when the computations grater than the communications , increase the number of the critical nodes 
+when the number of nodes increased. The inverse happens in the tow sites scenario,
+this due to the lower computations to communications ratio that decreased with highest 
+communications. Therefore, the number of the critical nodes are decreased. The average performance 
+degradation for the two sites scenario with 16 nodes is equal to 8\% and for 32 nodes is equal to 4\%.
+The EP benchmarks is gives the bigger performance degradation ratio, because there is no 
+communications and no slack times in this benchmarks that is always their performance effected 
+by selecting big or small frequencies.
+The tradeoff between these scenarios can be computed as in the trade-off function \ref{eq:max}.
+Figure \ref{fig:dist}, presents the tradeoff distance for all benchmarks  over all 
+platform scenarios.  The one site scenario with 16 and 32 nodes had the best tradeoff distance 
+compared to the two sites scenarios, because the increase in the communications as mentioned before.
+The one site scenario with 16 nodes is the best scenario in term of energy and performance tradeoff,
+which on average is up 26\%.  Then, the tradeoff distance is related linearly to the  energy saving  
+percentage. Finally, the best energy and performance tradeoff depends on the increase in all of:
+1) the computations to communications ratio, 2) the differences in computing powers 
+between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.
 
 \subsection{The experimental results of multi-cores clusters}
 \label{sec.res}
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2.39.5