Computing platforms are consuming more and more energy due to the increasing
- number of nodes composing them. In the heterogeneous computing platform composed
- of multiple computing nodes, each node is different in the computing power from
- the others. Accordingly, the fast nodes have to waits to the slow ones to finish
- their works. The resulting waiting times is called the idle times that are increased
+ number of nodes composing them. In a heterogeneous computing platform composed
+ of multiple computing nodes, nodes may differ in the computing power from
+ each others. Accordingly, the fast nodes have to wait for the slow ones to finish
+ their works. The resulting waiting times are called idle times which are increased
proportionally to the increase in the heterogeneity between the computing nodes.
- This leads to a big waste in the computing power and thus the energy consumed by the fast nodes.
+ This leads to a big waste in the computing power and thus the energy consumed by fast nodes.
To minimize the operating costs of these platforms many techniques have been used.
Dynamic voltage and frequency scaling (DVFS) is one of them. It reduces the frequency
of a CPU to lower its energy consumption. However, lowering the frequency of a CPU may
the performance of an application must be selected.
In this chapter, two new online frequency selecting algorithms for heterogeneous local
- cluster (heterogeneous CPUs) and grid platform are presented.
- They select the frequencies that tray to give the best
+ clusters (heterogeneous CPUs) and grid platforms are presented.
+ They select the frequencies that try to give the best
trade-off between energy saving and performance degradation, for each node
computing the synchronous message passing iterative application. These algorithms have a small
overhead and work without training or profiling. They use new energy models
for message passing iterative synchronous applications running on both the heterogeneous
- local cluster and grid platform. The first proposed algorithm for a heterogeneous local
- cluster is evaluated on the SimGrid simulator while running the NAS parallel
- benchmarks class C. The experiments conducted over 8 heterogeneous nodes show that it reduces on
- average the energy consumption by 29.8\% while limiting the performance degradation by 3.8\%.
- The second proposed algorithm for a grid platform is evaluated on the Grid5000 testbed
- platform while running the NAS parallel benchmarks class D.
- Its experiments on 16 nodes, distributed on three clusters, show that it reduces on average the
- energy consumption by 30\% while the performance is on average only degraded
+ local cluster and the grid platform. The first proposed algorithm for a heterogeneous local
+ cluster was evaluated on the SimGrid simulator while running the class C of the NAS parallel
+ benchmarks. The experiments conducted over 8 heterogeneous nodes show that it reduces on
+ average the energy consumption by 29.8\% while limiting the performance degradation to 3.8\%.
+ The second proposed algorithm for a grid platform was evaluated on the Grid5000 testbed
+ platform while running the class D of the NAS parallel benchmarks.
+ The experiments were run on 16 nodes, distributed on three clusters, and show that it reduces
+ on average the energy consumption by 30\% while the performance is on average only degraded
by 3.2\%.
- Finally, both the two algorithms are compared to an existing methods, the comparison
- results show that they outperform the latter in term of energy and performance trade-off.
+ Finally, both algorithms were compared to the EDP method. The comparison
+ results show that they outperform the latter in the energy reduction and performance trade-off.
This chapter is organized as follows: Section~\ref{ch3:relwork} presents some
related works from other authors. Section~\ref{ch3:1} presents the performance and energy
models of synchronous message passing programs running over a heterogeneous local cluster.
-It also describes the proposed frequencies selecting algorithm then the precision of the proposed algorithm is verified.
+It also describes the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified.
Section~\ref{ch3:2} presents the simulation results of applying the algorithm on the NAS parallel
benchmarks class C and executing them on a heterogeneous local cluster. It shows the results of running
three different power scenarios and comparing them. Moreover, it also shows the
Section~\ref{ch3:3} shows the energy and performance models in addition to the frequencies
selecting algorithm of synchronous message passing programs running over a grid platform.
Section~\ref{ch3:4} presents the results of applying the algorithm on the
-NAS parallel benchmarks class D and executing them on the Grid'5000 testbed.
-It also evaluates the algorithm over multi-cores per node architectures and over three different power scenarios. Moreover, it shows the comparison results between the proposed method and an existing method.
+NAS parallel benchmarks (class D) and executing them on the Grid'5000 testbed.
+The algorithm is also evaluated over multi-core architectures and over three different power scenarios. Moreover, section~\ref{ch3:4}, shows the comparison results between the proposed method and the EDP method.
Finally, in Section~\ref{ch3:concl} the chapter ends with a summary.
\section{Related works}
\label{ch3:relwork}
-DVFS is a technique used in modern processors to scale down both the voltage and
-the frequency of the CPU while computing, in order to reduce the energy
-consumption of the processor. DVFS is also allowed in GPUs to achieve the same
-goal. Reducing the frequency of a processor lowers its number of FLOPS and may
-degrade the performance of the application running on that processor, especially
-if it is compute bound. Therefore selecting the appropriate frequency for a
+As same as in CPUs, DVFS is also allowed in GPUs to reduce their energy consumption.
+The process of selecting the appropriate frequency for a
processor to satisfy some objectives, while taking into account all the
constraints, is not a trivial operation. Many researchers used different
strategies to tackle this problem. Some of them developed online methods that
sequential, parallel or distributed architecture, homogeneous or heterogeneous
platform, synchronous or asynchronous application, \dots{}
-In this chapter, we are interested in reducing energy for message passing
-iterative synchronous applications running over heterogeneous platforms. Some
-works have already been done for such platforms and they can be classified into
+In this chapter, we are interested in reducing the energy consumption when running a message passing
+iterative synchronous applications over a heterogeneous platform. Some
+works have already been done for such platforms which can be classified into
two types of heterogeneous platforms:
\begin{itemize}
\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
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{ref70},
Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
-DVFS gave better energy and performance efficiency than other clusters only
+DVFS operations gave better energy and performance efficiency than other clusters only
composed of CPUs.
The work presented in this chapter concerns the second type of platform, with
In~\cite{ref74} and
\cite{ref75}, the best
frequencies for a specified heterogeneous cluster are selected offline using
-some heuristic. Chen et
+on heuristic. Chen et
al.~\cite{ref76} used a greedy dynamic
programming approach to minimize the power consumption of heterogeneous servers
-while respecting given time constraints. This approach had considerable
+while respecting the given time constraint. This approach had considerable
overhead. In contrast to the above described works, the work of this chapter presents the
following contributions:
\begin{enumerate}
\item two new energy and two performance models for message passing iterative
- synchronous applications running over a heterogeneous local cluster and grid platform.
- All the models take into account communication and slack times. The models can predict the
- required energy and the execution time of the application.
+ synchronous applications running over a heterogeneous local cluster and a grid platform.
+ All the models take into account the communications and the slack times. The models can predict the
+ energy consumption and the execution time of the application.
-\item two new online frequencies selecting algorithms for heterogeneous
- local cluster and grid platform. The algorithms have a very small overhead and do not need any
+\item two new online frequencies selecting algorithms for a heterogeneous
+ local cluster and a grid platform. The algorithms have a very small overhead and do not need any
training or profiling. They use a new optimization function which
simultaneously maximizes the performance and minimizes the energy consumption
of a message passing iterative synchronous application.
\end{enumerate}
-\section[The energy optimization of heterogeneous cluster]{The energy optimization of parallel iterative applications running over local heterogeneous
-cluster}
+\section[The energy optimization of a heterogeneous cluster]{The energy optimization of parallel iterative applications running over local heterogeneous
+clusters}
\label{ch3:1}
\subsection{The execution time of message passing distributed iterative
applications on a heterogeneous local cluster}
\label{ch3:1:1}
In this section, we are interested in reducing the energy consumption of message
-passing distributed iterative synchronous applications running over
-heterogeneous local cluster. A heterogeneous local cluster is defined as a collection of
+passing distributed iterative synchronous applications running over heterogeneous local clusters.
+In this work, a heterogeneous local cluster is defined as a collection of
heterogeneous computing nodes interconnected via a high speed homogeneous
-network. Therefore, each node has different characteristics such as computing
+network. Therefore, the nodes may have different characteristics such as computing
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.
overall execution time of the program is the execution time of the slowest task
which has the highest computation time and no slack time.
-The frequency reduction process by applying DVFS operation can be expressed by the scaling
-factor S which is the ratio between the maximum and the new frequency of a CPU
+Reducing the frequency of a processor by applying DVFS operation can be expressed by the scaling
+factor S which is the ratio between the maximum frequency and the new frequency of a CPU
as in (\ref{eq:s}).
The execution time of a compute bound sequential program is linearly
proportional to the frequency scaling factor $S$. On the other hand, message
with an MPI call for sending or receiving a message until the message is
synchronously sent or received.
-Since in a heterogeneous cluster each node has different characteristics,
+Since in a heterogeneous cluster the nodes may have different characteristics,
especially different frequency gears, when applying DVFS operations on these
nodes, they may get different scaling factors represented by a scaling vector:
$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
the execution time of one iteration as in (\ref{eq:perf_heter}) multiplied by the
number of iterations of that application.
-This prediction model is developed from the model to predict the execution time
+This prediction model is improved from the model that predicts the execution time
of message passing distributed applications for homogeneous
architectures presented in chapter \ref{ch2} section \ref{ch2:3}. The execution time prediction model is
-used in the method to optimize both the energy consumption and the performance
+used in the method that optimizes both the energy consumption and the performance
of iterative methods, which is presented in the following sections.
\subsection{Energy model for heterogeneous local cluster}
\label{ch3:1:2}
-In the chapter \ref{ch2}, the dynamic and the static energy consumption of the individual
-processor is computed in \ref{eq:Edyn_new} and \ref{eq:Estatic_new} respectively. Then,
-the total energy consumption of the individual processor is the sum of these two metrics.
-Therefore, the overall energy consumption for the parallel tasks over parallel cluster
-is the summation of the individual energies consumed for all processors.
+In chapter \ref{ch2}, the dynamic and the static energy consumption of a
+processor is computed according to Equations \ref{eq:Edyn_new} and \ref{eq:Estatic_new} respectively. Then, the total energy consumption of a processor is the sum of these two metrics.
+Therefore, the overall energy consumption for the parallel tasks over a parallel cluster
+is the summation of the energies consumed by all the processors.
In the considered heterogeneous platform, each processor $i$ may have
different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
application is load balanced, the computation time of each CPU $i$ noted
$\Tcp[i]$ may be different and different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
-and reduce slack times. The communication time of a processor $i$ is noted as
+and reduce the slack times. The communication time of a processor $i$ is noted as
$\Tcm[i]$ and could contain slack times when communicating with slower nodes,
-see Figure~\ref{fig:task-heter}. Therefore, all nodes do not have equal
+see Figure~\ref{fig:task-heter}. Therefore, all the nodes do not have equal
communication times. While the dynamic energy is computed according to the
frequency scaling factor and the dynamic power of each node as in
(\ref{eq:Edyn_new}), the static energy is computed as the sum of the execution time
of one iteration as in \ref{eq:perf_heter} multiplied by the static power of each processor.
The overall energy consumption of a message passing distributed application executed over a
-heterogeneous cluster during one iteration is the summation of all dynamic and
-static energies for each processor. It is computed as follows:
+heterogeneous cluster during one iteration is the summation of all the dynamic and
+static energies for all the processors. It is computed as follows:
\begin{equation}
\label{eq:energy-heter}
E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot \Tcp[i])} +
Reducing the frequencies of the processors according to the vector of scaling
factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the application
-and thus, increase the static energy because the execution time is
+and thus, increase the consumed static energy because the execution time is
increased~\cite{ref78}. The overall energy consumption
-for the iterative application can be measured by measuring the energy
+for an iterative application can be measured by measuring the energy
consumption for one iteration as in (\ref{eq:energy-heter}) multiplied by the number
of iterations of that application.
the application might not be the optimal one. It is not trivial to select the
appropriate frequency scaling factor for each processor while considering the
characteristics of each processor (computation power, range of frequencies,
-dynamic and static powers) and the task executed (computation/communication
-ratio). The aim being to reduce the overall energy consumption and to avoid
-increasing significantly the execution time. In last chapter
-~\ref{ch2}, we proposed a method that selects the optimal
+dynamic and static powers) and the task it is executing (computation/communication
+ratio). In chapter~\ref{ch2}, we proposed a method that selects the optimal
frequency scaling factor for a homogeneous cluster executing a message passing
iterative synchronous application while giving the best trade-off between the
-energy consumption and the performance for such applications. In this section, we
-are interested in heterogeneous clusters as described above. Due to the
-heterogeneity of the processors, a vector of scaling factors should be selected
-and it must give the best trade-off between energy consumption and performance.
+energy consumption and the performance for such applications.
+In this section, this optimization method is improved while considering a heterogeneous clusters.
As described before, the relation between the energy consumption and the execution time for an
-application is complex and nonlinear. Thus, to find the trade-off relation between the energy consumption in \ref{eq:energy-heter} and the performance in \ref{eq:perf_heter} of the iterative message passing applications, first we need to normalized both of them as follows:
+application is complex and nonlinear. Thus, to find the trade-off relation between the energy consumption computed in Equation \ref{eq:energy-heter} and the performance with Equation \ref{eq:perf_heter} for the iterative message passing applications, first we need to normalize both term as follows:
\begin{equation}
{ \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \min_{i=1,2,\dots,N} (\Tcm[i])}
\end{equation}
-Therefore, the vector of frequency scaling factors $S_1,S_2,\dots,S_N$ of the heterogeneous
-cluster reduce both the energy and the execution time simultaneously.
\begin{figure}[!t]
\centering
Then, the objective function can be modeled in order to find the maximum
distance between the energy curve (\ref{eq:enorm-heter}) and the performance curve
-(\ref{eq:pnorm-heter}) over all available sets of scaling factors of the heterogeneous
-computing cluster. This represents the minimum energy consumption with minimum execution time (maximum
+(\ref{eq:pnorm-heter}) over all available sets of scaling factors for the processors of the heterogeneous
+cluster. This represents the minimum energy consumption with minimum execution time (maximum
performance) at the same time, see Figure~\ref{fig:rel-heter}. Then the objective function has the following form:
\begin{equation}
\label{eq:max-heter}
- \MaxDist =
- \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
+ \MaxDist =
+ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,F_i}
(\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
\overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )
\end{equation}
-where $N$ is the number of nodes and $F$ is the number of available frequencies
-for each node. Then, the optimal set of scaling factors that satisfies
-(\ref{eq:max-heter}) can be selected.
+where $N$ is the number of nodes and $F_i$ is the number of available frequencies
+for the node $i$. Then, the set of scaling factors that maximizes the objective function
+(\ref{eq:max-heter}) should be selected.
\subsection[The scaling algorithm for heterogeneous cluster]{The scaling factors selection algorithm for heterogeneous cluster }
\label{ch3:1:4}
\label{fig:st_freq-cluster}
\end{figure}
-The nodes in a heterogeneous cluster 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:task-heter}. These
-periods are called idle or slack times. The algorithm takes into account this
+The nodes in a heterogeneous cluster may have different computing powers.
+ 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
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
+between the computation times of the fast nodes and the slow ones. The value of the initial
frequency scaling factor for each node is inversely proportional to its
computation time that was gathered from the first iteration. These initial
frequency scaling factors are computed as a ratio between the computation time
\end{equation}
Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
algorithm computes the initial frequencies for all nodes as a ratio between the
-maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+maximum frequency of node $i$ and the computed scaling factor $\Scp[i]$ as
follows:
\begin{equation}
\label{eq:Fint}
bound until all nodes reach their minimum frequencies, to compute their overall
energy consumption and performance, and select the optimal frequency scaling
factors vector. At each iteration the algorithm determines the slowest node
-according to the equation (\ref{eq:perf_heter}) and keeps its frequency unchanged,
+according to Equation (\ref{eq:perf_heter}) and keeps its frequency unchanged,
while it lowers the frequency of all other nodes by one gear. The new overall
energy consumption and execution time are computed according to the new scaling
factors. The optimal set of frequency scaling factors is the set that gives the
highest distance according to the objective function (\ref{eq:max-heter}).
-Figure~\ref{fig:rel-heter} illustrate the normalized performance and
+Figure~\ref{fig:rel-heter} illustrates the normalized performance and
consumed energy for an application running on a
heterogeneous cluster while increasing the scaling factors. It can
be noticed that in a homogeneous cluster, as in the figure \ref{fig:rel} (a),
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.
-Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high.
-Indeed, the dynamic energy saved by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.
+maximum distance between the energy curve and the performance curve is which results in bigger energy savings.
\subsection{The evaluation of the proposed algorithm}
\label{ch3:1:5}
-The precision of the proposed algorithm mainly depends on the execution time
-prediction model defined in (\ref{eq:perf_heter}) and the energy model computed by
+The accuracy of the proposed algorithm mainly depends on the execution time
+prediction model defined in (\ref{eq:perf_heter}) and the energy model computed by Equation
(\ref{eq:energy-heter}). The energy model is also significantly dependent on the
execution time model because the static energy is linearly related to the
execution time and the dynamic energy is related to the computation time. So,
v3.10~\cite{ref66}, for all the NAS
parallel benchmarks NPB v3.3 \cite{ref65}, running class B on
8 or 9 nodes. The comparison showed that the proposed execution time model is
-very precise, the maximum normalized difference between the predicted execution
+very accurate, the maximum normalized difference between the predicted execution
time and the real execution time is equal to 0.03 for all the NAS benchmarks.
Since the proposed algorithm is not an exact method, it does not test all the
cluster composed of four different types of nodes having the characteristics
presented in Table~\ref{table:platform-cluster}, it takes on average 0.04 \textit{ms} for 4
nodes and 0.15 \textit{ms} on average for 144 nodes to compute the best scaling
-factors vector. The algorithm complexity is $O(F\cdot N)$, where $F$ is the
-maximum number of available frequencies, and $N$ is the number of computing
+factors vector. The algorithm complexity is $O(F_i \cdot N)$, where $F_i$ is the
+maximum number of available frequencies in the node $i$, and $N$ is the number of computing
nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
frequency scaling factors that gives the results of the next sections.
\label{table:platform-cluster}
\end{table}
-\section{Experimental results over heterogeneous local cluster}
+\section{Experimental results over a heterogeneous local cluster}
\label{ch3:2}
To evaluate the efficiency and the overall energy consumption reduction of
Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3 which
a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
constructors of CPUs do not specify the dynamic and the static power of their
CPUs, for each type of node they were chosen proportionally to its computing
-power (FLOPS). In the initial heterogeneous cluster, while computing with
-highest frequency, each node consumed an amount of power proportional to its
-computing power (which corresponds to 80\% of its dynamic power and the
-remaining 20\% to the static power), the same assumption was made in chapter \ref{ch2} and
-\cite{ref3}. Finally, These
-nodes were connected via an Ethernet network with 1 \textit{Gbit/s} bandwidth.
+powers (FLOPS). The dynamic power corresponds to 80\% of the overall power consumption while executing at
+the higher frequency and the
+remaining 20\% is the static power. The same assumption was made in chapter \ref{ch2} and
+\cite{ref3}. Finally, These nodes were connected via an Ethernet network with 1 \textit{Gbit/s} bandwidth.
\subsection{The experimental results of the scaling algorithm }
instance. The results are presented in Tables
\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
-values from many experiments for energy savings and performance degradation.
+values obtained from many experiments for energy savings and performance degradation.
The tables show the experimental results for running the NAS parallel benchmarks
on different numbers of nodes. The experiments show that the algorithm
significantly reduces the energy consumption (up to 34\%) and tries to
can also be noticed that for the benchmarks EP and SP that contain little or no
communications, the energy savings are not significantly affected by the high
number of nodes. No experiments were conducted using bigger classes than D,
-because they require a lot of memory (more than 64 \textit{CB}) when being executed
+because they require a lot of memory (more than 64 \textit{GB}) when being executed
by the simulator on one machine. The maximum distance between the normalized
energy curve and the normalized performance for each instance is also shown in
-the result tables. It decrease in the same way as the energy saving percentage.
+the result tables. It decreases in the same way as the energy saving percentage.
The tables also show that the performance degradation percentage is not
significantly increased when the number of computing nodes is increased because
the computation times are small when compared to the communication times.
The results of the previous section were obtained while using processors that
consume during computation an overall power which is 80\% composed of
-dynamic power and of 20\% of static power. In this section, these ratios
+dynamic power and 20\% of static power. In this section, these ratios
are changed and two new power scenarios are considered in order to evaluate how
the proposed algorithm adapts itself according to the static and dynamic power
values. The two new power scenarios are the following:
relevant in the overall consumed energy. Indeed, the static energy is related
to the execution time and if the performance is degraded the amount of consumed
static energy directly increases. Therefore, the proposed algorithm does not
-really significantly scale down much the frequencies of the nodes in order to
+significantly scale down the frequencies of the nodes in order to
limit the increase of the execution time and thus limiting the effect of the
consumed static energy.
Both new power scenarios are compared to the old one in
Figure~\ref{fig:powers-heter} (a). It shows the average of the performance degradation,
-the energy saving and the distances for all NAS benchmarks of class C running on
+the energy saving and the distances for all the NAS benchmarks running class C on
8 or 9 nodes. The comparison shows that the energy saving ratio is proportional
to the dynamic power ratio: it is increased when applying the
90\%-10\% scenario because at maximum frequency the dynamic energy
performance degradation is decreased when using a higher ratio for static power
(e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed
algorithm optimizes the energy consumption when
-using a higher ratio for dynamic power the algorithm selects bigger frequency
-scaling factors that result in more energy saving but less performance, for
+using a higher ratio for the dynamic power, the algorithm selects bigger frequency
+scaling factors that results in more energy saving but degrade the performance, for
example see Figure~\ref{fig:powers-heter} (b). The opposite happens when using a
-higher ratio for static power, the algorithm proportionally selects smaller
+higher ratio for the static power, the algorithm proportionally selects smaller
scaling values which result in less energy saving but also less performance
degradation.
\end{table}
\begin{table}[!t]
- \caption{Comparing the proposed algorithm}
+ \caption{Comparing the MaxDist algorithm to the EDP method}
\centering
\begin{tabular}{|*{7}{r|}}
\hline
\end{figure}
-\subsection{The comparison of the proposed scaling algorithm }
+\subsection{Comparison between the proposed scaling algorithm and the EDP method}
\label{ch3:2:3}
In this section, the scaling factors selection algorithm, called MaxDist, is
-compared to Spiliopoulos et al. algorithm
-\cite{ref67}, called EDP. They developed a
+compared to \cite{ref67}, EDP method. They developed a
green governor that regularly applies an online frequency selecting algorithm to
-reduce the energy consumed by a multicore architecture without degrading much
+reduce the energy consumed by a multi-core architecture without degrading much
its performance. The algorithm selects the frequencies that minimize the energy
-and delay products, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ using
+and delay product, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$, using
the predicted overall energy consumption and execution time delay for each
frequency. To fairly compare both algorithms, the same energy and execution
-time models, equations (\ref{eq:energy-heter}) and (\ref{eq:perf_heter}), were used for both
-algorithms to predict the energy consumption and the execution times. Also
+time models, Equations (\ref{eq:energy-heter}) and (\ref{eq:perf_heter}), were used for both
+algorithms to predict the energy consumption and the execution times.
Spiliopoulos et al. algorithm was adapted to start the search from the initial
-frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm
+frequencies computed using Equation (\ref{eq:Fint}). The resulting algorithm
is an exhaustive search algorithm that minimizes the EDP and has the initial
frequencies values as an upper bound.
Both algorithms were applied to the parallel NAS benchmarks to compare their
-efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
+efficiency. Table~\ref{table:compare_EDP} presents the
execution times and the energy consumption for both versions of the NAS
benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
nodes. The results show that our algorithm provides better energy savings than
Spiliopoulos et al. algorithm, on average it results in 29.76\% energy
-saving while their algorithm returns just 25.75\%. The average of
+saving while their algorithm saves just 25.75\%. The average of
performance degradation percentage is approximately the same for both
algorithms, about 4\%.
For all benchmarks, our algorithm outperforms Spiliopoulos et al. algorithm in
-terms of energy and performance trade-off, see Figure~\ref{fig:compare_EDP},
+the energy reduction to performance trade-off, see Figure~\ref{fig:compare_EDP},
because it maximizes the distance between the energy saving and the performance
degradation values while giving the same weight for both metrics.
-\section[The energy optimization of grid]{The energy optimization of parallel iterative applications running over grid}
+\section[The energy optimization of grid]{The energy optimization of parallel iterative applications running over grids}
\label{ch3:3}
\subsection{The energy and performance models of grid platform}
\label{ch3:3:1}
In this section, we are interested in reducing the energy consumption of message
-passing distributed iterative synchronous applications running over
+passing iterative synchronous applications running over
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 a 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 a lower bandwidth
+and a 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 a high speed network. However, nodes from distinct clusters may have different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
Since in a heterogeneous grid each cluster has different characteristics,
-especially different frequency gears, when applying DVFS operations on the nodes
+ when applying DVFS operations on the nodes
of these clusters, they may get different scaling factors represented by a scaling vector:
$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
be able to predict the execution time of message passing synchronous iterative
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-grid}).
+vector of scaling factors can be predicted using Equation (\ref{eq:perf-grid}).
%
\begin{equation}
\label{eq:perf-grid}
- \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots,M}} (\Tcm[hj])
+ \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots,M_i}} (\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 $i$
+where $N$ is the number of clusters in the grid, $M_i$ is the number of nodes in
+ cluster $i$, $\TcpOld[ij]$ 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.
The latter is equal to the communication time of the slowest node in the slowest cluster $h$.
It means that 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-grid}) multiplied by the
+the execution time of one iteration as in Equation (\ref{eq:perf-grid}) multiplied by the
number of iterations of that application.
$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
see Figure~\ref{fig:task-heter}. Therefore, all nodes do not have equal
communication times. While the dynamic energy is computed according to the
-frequency scaling factor and the dynamic power of each node as in
+frequency scaling factor and the dynamic power of each node as in Equation
(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
of one iteration multiplied by the static power of each processor. The overall
energy consumption of a message passing distributed application executed over a
heterogeneous grid platform during one iteration is the summation of all dynamic and
-static energies for $M$ processors in $N$ clusters. It is computed as follows:
+static energies for $M_i$ processors in $N$ clusters. It is computed as follows:
\begin{equation}
\label{eq:energy-grid}
- E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
- \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot
- (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
+ \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot
+ (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
\end{equation}
-To optimize both of the energy model \ref{eq:energy-grid} and the performance model\ref{eq:perf-grid},
-they must normalizes respectively as in \ref{eq:enorm-heter} and \ref{eq:pnorm-heter}.
-While the original energy consumption is the consumed energy with
-maximum frequency for all nodes computes as follows:
+To optimize both of the energy consumption model computed by \ref{eq:energy-grid} and the performance model computed by \ref{eq:perf-grid},
+they must be normalized as in \ref{eq:enorm-heter} and \ref{eq:pnorm-heter} Equations respectively.
+While the original energy consumption is the consumed energy with the
+maximum frequency for all the nodes computed as follows:
\begin{equation}
\label{eq:eorginal-grid}
- \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot \Tcp[ij]) +
- \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)
+ \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot \Tcp[ij]) +
+ \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)
\end{equation}
-By the same way, the old execution time with maximum frequency for all nodes computes as follows:
+By the same way, the old execution time with the maximum frequency for all the nodes is computed as follows:
\begin{equation}
\label{eq:told-grid}
- \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])
+ \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]})
+ +\mathop{\min_{j=1,\dots,M_i}} (\Tcm[hj])
\end{equation}
-Therefore, the objective function can be modeled in order to find the maximum
+Therefore, the objective function can be modelled in order to find the maximum
distance between the normalized energy curve and the normalized performance curve
-over all available sets of scaling factors as follows:
+over all possible sets of scaling factors as follows:
\begin{equation}
\label{eq:max-grid}
\MaxDist =
-\mathop{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+\mathop{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}}_{k=1,\dots,F_j}
(\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
\overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
\end{equation}
-where $N$ is the number of clusters, $M$ is the number of nodes in each cluster and
-$F$ is the number of available frequencies for each node. Then, the optimal set
+where $N$ is the number of clusters, $M_i$ is the number of nodes in each cluster and
+$F_j$ is the number of available frequencies for the node $j$. Then, the optimal set
of scaling factors that satisfies (\ref{eq:max-grid}) can be selected.
-\subsection{The scaling factors selection algorithm for a grid }
+\subsection{The scaling factors selection algorithm for a grid architecture}
\label{ch3:3:2}
\begin{algorithm}
\Require ~
\item [{$N$}] number of clusters in the grid.
- \item [{$M$}] number of nodes in each cluster.
+ \item [{$M_i$}] number of nodes in each cluster.
\item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
\item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
\item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
\Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$, a vector of scaling factors that gives the optimal trade-off between energy consumption and execution time
- \State $\Scp[ij] \gets \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]} $
+ \State $\Scp[ij] \gets \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]} $
\State $F_{ij} \gets \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
\State Round the computed initial frequencies $F_i$ to the closest available frequency for each node.
\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 \mathop{\max\limits_{i=1,2,\dots,N}}\limits_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij]) $
- \State $\Eoriginal \gets \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot \Tcp[ij]) +
- \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told) $
+ \State $\Told \gets \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\Tcp[ij]})
+ +\mathop{\min\limits_{j=1,\dots,M_i}} (\Tcm[hj]) $
+ \State $\Eoriginal \gets \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot \Tcp[ij]) +
+ \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told) $
\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 frequency \textbf{or} $\Pnorm - \Enorm < 0 $)}
\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 \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}({\TcpOld[ij]}
- \cdot S_{ij}) +\mathop{\min\limits_{j=1,\dots,M}} (\Tcm[hj]) $.
- \State $\Ereduced \gets \sum\limits_{i=1}^{N} \sum\limits_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij]
- \cdot \Tcp[ij])} + \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M} (\Ps[ij] \cdot
- (\mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij})
- +\mathop{\min\limits_{j=1,\dots M}} (\Tcm[hj]) ))$
+ \State $\Tnew \gets \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\TcpOld[ij]}
+ \cdot S_{ij}) +\mathop{\min\limits_{j=1,\dots,M_i}} (\Tcm[hj]) $
+ \State $\Ereduced \gets \sum\limits_{i=1}^{N} \sum\limits_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij]
+ \cdot \Tcp[ij])} + \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} (\Ps[ij] \cdot
+ (\mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij})
+ +\mathop{\min\limits_{j=1,\dots,M_i}} (\Tcm[hj]) ))$
\State $\Pnorm \gets \frac{\Told}{\Tnew}$
\State $\Enorm \gets \frac{\Ereduced}{\Eoriginal}$
\If{$(\Pnorm - \Enorm > \Dist)$}
- \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
+ \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i$
\State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndWhile
In this section, the scaling factors selection algorithm for a grid, Algorithm~\ref{HSA-grid},
-is presented. It selects the vector of the frequency
+is presented. It selects the vector of 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 grid that satisfies the objective function
-(\ref{eq:max-grid}).
-It has the same principles and specifications of the frequencies selection algorithm of the heterogeneous
-local cluster \ref{HSA}.
+synchronous iterative application executed on a grid.
+It is similar to the frequency selection algorithm for heterogeneous
+local clusters presented in section \ref{ch3:1:4}.
The value of the initial frequency scaling factor for each node is inversely proportional to its
-computation time that was gathered from the first iteration. These initial
-frequency scaling factors are computed as a ratio between the computation time
+computation time that was gathered in the first iteration. The initial
+frequency scaling factor for a node $i$ is computed as a ratio between the computation time
of the slowest node and the computation time of the node $i$ as follows:
\begin{equation}
\label{eq:Scp-grid}
- \Scp[ij] = \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+ \Scp[ij] = \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]}
\end{equation}
Using the initial frequency scaling factors computed in (\ref{eq:Scp-grid}), the
algorithm computes the initial frequencies for all nodes as a ratio between the
-maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
-follows:
+maximum frequency of the node and its computed scaling factor, as follows:
\begin{equation}
\label{eq:Fint-grid}
- F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+ F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M_i}
\end{equation}
Figure \ref{fig:st_freq-grid} shows the selected initial frequencies for a grid composed of three clusters.
In contrast to algorithm \ref{HSA}, algorithm \ref{HSA-grid} replaces the computed initial frequency for a node by the nearest available frequency if not available in the gears of
that node.
-The frequency scaling algorithm of the grid stops its iteration if it reaches to lower bound, which is the computed distance between the energy and performance at this frequency if it is less than zero.
+The frequency scaling algorithm of the grid stops its iteration if it reaches the lower bound, which is the frequency that gives a negative distance between the energy and performance.
A negative distance means that the performance degradation ratio is higher than the energy saving ratio as in figure \ref{fig:rel-grid}.
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
+Therefore, the algorithm iterates on all the 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. The DVFS algorithm~\ref{dvfs-heter} is also used to call the algorithm \ref{HSA-grid} in the MPI
-program executed over grid platform.
+energy consumption and execution time. Then, it selects the vector of frequency scaling
+factors that give the maximum distance (MaxDist). Algorithm~\ref{dvfs-heter} is also used to call the Algorithm \ref{HSA-grid} in the MPI program executed over the grid platform.
-\section{Experimental results over Grid5000 platform}
+\section{Experimental results over the Grid5000 platform}
\label{ch3:4}
In this section, real experiments were conducted over the Grid'5000 platform.
-Grid'5000~\cite{ref21} is a large-scale testbed that consists of ten sites distributed all over metropolitan France and Luxembourg. These sites are: Grenoble, Lille, Luxembourg, Lyon, Nancy, Reims, Rennes , Sophia, Toulouse, Bordeaux. Figure \ref{fig:grid5000-dis} shows the geographical distribution of grid'5000 sites over France and Luxembourg. All the sites are connected together via a special long distance network called RENATER, which is abbreviation of the French
+Grid'5000~\cite{ref21} is a large-scale testbed that consists of ten sites distributed all over metropolitan France and Luxembourg. These sites are: Grenoble, Lille, Luxembourg, Lyon, Nancy, Reims, Rennes , Sophia, Toulouse and Bordeaux. Figure \ref{fig:grid5000-dis} shows the geographical distribution of grid'5000 sites over France and Luxembourg. All the sites are connected together via a special long distance network called RENATER, which is the abbreviation of the French
National Telecommunication Network for Technology. Each site in the grid is
composed of a 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.
-Grid'5000 is dedicated as a test-bed for grid computing and thus users can book the required nodes from different sites.
-It also gives the opportunity to the users to deploy their configured image of the operating system over the reserved nodes.
-Indeed, many software tools are available for users in order to control and manage the reservation and deployment processes from their local machines. For example, OAR \cite{ref22} is a batch scheduler that is used to manage the heterogeneous resources of the grid'5000.
+Grid'5000 is dedicated for research experiments and users can book nodes from different sites to conduct their experiments.
+It also gives the opportunity to the users to deploy their customized operating system over the reserved nodes.
+Indeed, many software tools are available for users in order to control and manage the reservation and deployment processes remotely. For example, OAR \cite{ref22} is a batch scheduler that is used to manage the heterogeneous resources of the grid'5000.
\begin{figure}[h!]
\centering
\end{figure}
- 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, \dots{} For more details refer to \cite{ref79}.
+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. For more details refer to \cite{ref79}.
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}.
Therefore, the dynamic power of one core is computed as the difference between the maximum
measured value in maximum 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 same as in sections \ref{ch3:2} and \ref{ch2:6} 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 it was assumed, as in sections \ref{ch3:2} and \ref{ch2:6}, that the static power represents a ratio of the dynamic power, the value of the static power is assumed to be equal to 20\% of the 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 shown on 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 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
+Griffon and Graphite. Each one of these clusters composed of homogeneous nodes, 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:grid5000-1} 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:grid5000-1}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.8]{fig/ch3/power_consumption.pdf}
- \caption{The power consumption by one core from the Taurus cluster}
+ \caption{The power consumed by one core from the Taurus cluster}
\label{fig:power_cons}
\end{figure}
The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{ref65} 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, class D was used for all benchmarks in all the experiments presented in the next sections.
-
+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 the next sections, the class D was used for all the benchmarks in all the experiments.
\begin{table}[!t]
- \caption{CPUs characteristics of the selected clusters}
+ \caption{The characteristics of the CPUs in the selected clusters}
% title of Table
\centering
\begin{tabular}{|*{7}{c|}}
\end{tabular}
\label{table:grid5000-1}
\end{table}
+CPUs
-
-\subsection{The experimental results of the scaling algorithm of Grid}
+\subsection{The experimental results of the scaling algorithm on a Grid}
\label{ch3:4:1}
-In this section, the results of applying the scaling factors selection algorithm \ref{HSA}
-to NAS parallel benchmarks are presented.
-
+In this section, the results of applying the scaling factors selection algorithm
+to the NAS parallel benchmarks are presented.
As mentioned previously, the experiments
were conducted over two sites of Grid'5000, Lyon and Nancy sites.
Two scenarios were considered while selecting the clusters from these two sites :
\begin{itemize}
\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected
via a long distance network.
-\item In the second scenario nodes from three clusters located in one site, Nancy site, were selected.
+\item In the second scenario nodes from three clusters located in one site, Nancy's site, were selected.
\end{itemize}
The main reason
for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the
scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio
-is very low due to the higher communication times which reduce the effect of DVFS operations.
+is very low due to the higher communication times which reduces the effect of the DVFS operations.
The NAS parallel benchmarks are executed over
16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster
\begin{table}[h]
-\caption{The different clusters scenarios}
+\caption{The different grid scenarios}
\centering
\begin{tabular}{|*{4}{c|}}
\hline
& Graphene & Nancy & 5 \\ \cline{2-4}
& Griffon & Nancy & 6 \\
\hline
-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus & Lyon & 10 \\ \cline{2-4}
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus & Lyon & 10 \\ \cline{2-4}
& Graphene & Nancy & 10 \\ \cline{2-4}
& Griffon &Nancy & 12 \\
\hline
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
-using the equation of the reduced energy consumption, Equation~\ref{eq:energy-grid}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000-1}
-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.
+using Equation~\ref{eq:energy-grid}.
The energy consumptions and the execution times for all the benchmarks are
presented in Figures~\ref{fig:exp-time-energy} (a) and (b) 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.
-The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption.
+For the majority of the benchmarks, the energy consumed while executing the NAS benchmarks over one site on 16 and 32 nodes is lower than the energy consumed while using two sites.
+The long distance communications between the two distributed sites increase the idle times, which lead to more static energy consumption.
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 have their execution times 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).
+over one site with 16 and 32 nodes are also lower than those of the two sites
+scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times approximately halved when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).
-However, the execution times and the energy consumptions of EP and MG
+However, the execution times and the energy consumptions of the 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 communication times of the rest of the benchmarks increases when
-using long distance communications between two sites or increasing the number of
+other hand, the communication times of the rest of the benchmarks increase when
+using long distance communications between two sites or when increasing the number of
computing nodes.
The energy saving percentage is computed as the ratio between the reduced
energy consumption, Equation~\ref{eq:energy-grid}, and the original energy consumption,
-Equation~\ref{eq:eorginal-grid}, for all benchmarks as in Figure~\ref{fig:eng_s}.
-This figure shows that the energy saving percentages of one site scenario for
+Equation~\ref{eq:eorginal-grid}, for all the benchmarks as in Figure~\ref{fig:eng_s}.
+This figure shows that the energy saving percentages of the 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 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
-benchmarks being executed. The results of 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, LU and SP consume more energy with 16 nodes than 32 in one site because their computations to communications ratio is not affected by the increase of the number of local communications.
-\begin{figure}[!t]
+benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more
+energy saving percentage in the one site scenario when executed over 16 nodes than on 32 nodes. LU and SP consume more energy with 16 nodes than with 32 node on one site because their computations to communications ratio is not affected by the increase of the number of local communications.
+\begin{figure}[!h]
\centering
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/eng_con_scenarios.eps}\\~~~~~~~~~(a)\\
\includegraphics[width=.7\textwidth]{fig/ch3/time_scenarios.eps}\\~~~~~~~~~(b)
\caption{ (a) energy consumption and (b) execution time of NAS Benchmarks over different scenarios}
- \label{fig:exp-time-energy}
-
+ \label{fig:exp-time-energy}
\end{figure}
-
-The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites
-scenario, except for the EP benchmark which has no communication. Therefore, the energy saving percentage of this benchmark is
-dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example
-in the one site scenario, the graphite cluster is selected but in the two sites scenario
-this cluster is replaced with the Taurus cluster which is more powerful.
-Therefore, the energy savings of the EP benchmark are bigger in the two sites scenario due
-to the higher maximum difference between the computing powers of the nodes.
-
-In fact, high differences between the nodes' computing powers make the proposed frequencies selecting
-algorithm select smaller frequencies for the powerful nodes which
-produces less energy consumption and thus more energy saving.
-The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
-
-
-
-\begin{figure*}[t]
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/eng_s.eps}
\caption{The energy reduction while executing the NAS benchmarks over different scenarios}
\label{fig:eng_s}
\end{figure*}
-\begin{figure*}[t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/per_d.eps}
\caption{The performance degradation of the NAS benchmarks over different scenarios}
\label{fig:per_d}
\end{figure*}
-\begin{figure*}[t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/dist.eps}
\caption{The trade-off distance between the energy reduction and the performance of the NAS benchmarks
\label{fig:dist-grid}
\end{figure*}
+The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites
+scenario, except for the EP benchmark which has no communication. Therefore, the energy saving percentage of this benchmark is
+dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example
+in the one site scenario, the graphite cluster is selected but in the two sites scenario
+this cluster is replaced with the Taurus cluster which is more powerful.
+Therefore, the energy savings of the EP benchmark are bigger in the two sites scenario due
+to the higher maximum difference between the computing powers of the nodes.
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting
+algorithm select smaller frequencies for the powerful nodes which
+produces less energy consumption and thus more energy saving.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
-Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
+Figure \ref{fig:per_d} presents the performance degradation percentages for all the benchmarks over the two scenarios.
The performance degradation percentage for the benchmarks running on two sites with
-16 or 32 nodes is on average equal to 8.3\% or 4.7\% respectively.
-For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
-16 or 32 nodes is on average equal to 3.2\% or 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
+16 and 32 nodes is on average equal to 8.3\% and 4.7\% respectively.
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are high with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
+16 and 32 nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
nodes when the communications occur in high speed network does not decrease the computations to
communication ratio.
The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to
the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its
performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
-The rest of the benchmarks showed different performance degradation percentages, which decrease
+The rest of the benchmarks showed different performance degradation percentages which decrease
when the communication times increase and vice versa.
Figure \ref{fig:dist-grid} presents the distance percentage between the energy saving and the performance degradation for each benchmark over both scenarios. The trade-off distance percentage can be
-\subsection{The experimental results over multi-cores clusters}
+\subsection{The experimental results over multi-core clusters}
\label{ch3:4:2}
The clusters of Grid'5000 have different number of cores embedded in their nodes
as shown in Table~\ref{table:grid5000-1}. In
-this section, the proposed scaling algorithm of the grid is evaluated over the Grid'5000 platform while using multi-cores nodes selected according to the one site scenario described in Section
+this section, the proposed scaling algorithm is evaluated over the Grid'5000 platform while using multi-core nodes selected according to the one site scenario described in Section
~\ref{ch3:4:1}.
-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}.
-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.
-
-\begin{table}[]
+\begin{table}[!h]
\centering
-\caption{The multicores scenarios}
+\caption{The multi-core scenarios}
\begin{tabular}{|*{4}{c|}}
\hline
Scenario name & Cluster name & Nodes per cluster &
\multirow{3}{*}{One core per node} & Graphite & 4 & 1 \\ \cline{2-4}
& Graphene & 14 & 1 \\ \cline{2-4}
& Griffon & 14 & 1 \\ \hline
-\multirow{3}{*}{Multi-cores per node} & Graphite & 1 & 4 \\ \cline{2-4}
+\multirow{3}{*}{Multi-core per node} & Graphite & 1 & 4 \\ \cline{2-4}
& Graphene & 4 & 3 or 4 \\ \cline{2-4}
& Griffon & 4 & 3 or 4 \\ \hline
\end{tabular}
+
\label{table:sen-mc}
\end{table}
-
-
-
-
-\begin{figure}[!t]
+\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/time.eps}
- \caption{The execution times of running NAS benchmarks over one core and multicores scenarios}
+ \caption{The execution times of NAS benchmarks running over the one core and the multi-core scenarios}
\label{fig:time-mc}
\end{figure}
-\begin{figure}[!t]
+\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/eng_con.eps}
- \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-cores per node architectures}
+ \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-core per node architectures}
\label{fig:eng-cons-mc}
\end{figure}
-The execution times for most of the NAS benchmarks are higher over the multi-cores per node scenario
-than over single core per node scenario. Indeed,
- the communication times are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
+The one site scenario uses 32 cores from multi-core 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 4 nodes are selected and 3 or 4 cores from each node are used. The platforms with one
+core per node and multi-core 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 platforms are presented
+in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+The execution times for most of the NAS benchmarks are higher over the multi-core per node scenario
+than over the single core per node scenario. Indeed,
+ the communication times are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and might become a bottleneck.
Moreover, the energy consumptions of the NAS benchmarks are lower over the
- one core scenario than over the multi-cores scenario because
+ one core scenario than over the multi-core scenario because
the first scenario had less execution time than the latter which results in less static energy being consumed.
The computations to communications ratios of the NAS benchmarks are higher over
-the one site one core scenario when compared to the ratio of the multi-cores scenario.
+the one site one core scenario when compared to the ratio of the multi-core scenario.
More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy
consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
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.
-
-
-\begin{figure*}[!t]
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/eng_s_mc.eps}
- \caption{The energy saving of running NAS benchmarks over one core and multicores scenarios}
+ \caption{The energy saving of running NAS benchmarks over one core and multi-core scenarios}
\label{fig:eng-s-mc}
\end{figure*}
-\begin{figure*}[!t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/per_d_mc.eps}
- \caption{The performance degradation of running NAS benchmarks over one core and multicores scenarios}
+ \caption{The performance degradation of running NAS benchmarks over one core and multi-core scenarios}
\label{fig:per-d-mc}
\end{figure*}
-\begin{figure*}[!t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/dist_mc.eps}
- \caption{The trade-off distance of running NAS benchmarks over one core and multicores scenarios}
+ \caption{The trade-off distance of running NAS benchmarks over one core and multi-core scenarios}
\label{fig:dist-mc}
\end{figure*}
-
-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
+The energy saving percentages of all the NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}.
+It shows that the energy saving percentages in the one
+core and the multi-core scenarios
are approximately equivalent, on average they are equal to 25.9\% and 25.1\% respectively.
The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS.
-
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 executed over the one core per node scenario (on average equal to 10.6\%) than over the multi-core scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-core 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 both 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-core scenario (17.6\%) than over the one core per node scenario (15.3\%).
\subsection{Experiments with different static power scenarios}
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 experiments have been executed with these two new static power scenarios over the one site one core per node scenario.
-In these experiments, class D of the NAS parallel benchmarks are executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment.
+In these experiments, the class D of the NAS parallel benchmarks were executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, were used in this experiment.
-\begin{figure}[!t]
+\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/eng_pow.eps}
\caption{The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios}
\label{fig:eng-pow}
\end{figure}
-\begin{figure}[!t]
+\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/per_pow.eps}
\caption{The performance degradation percentages for the NAS benchmarks over the three power scenarios}
\label{fig:per-pow}
\end{figure}
-\begin{figure}[!t]
+
+\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/dist_pow.eps}
\caption{The trade-off distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios}
\end{figure}
-\begin{figure}
+\begin{figure}[!h]
\centering
\includegraphics[scale=0.7]{fig/ch3/three_scenarios2.pdf}
\caption{Comparing the selected frequency scaling factors for the MG benchmark over the three static power scenarios}
-\subsection{Comparison of the proposed frequencies selecting algorithm }
+\subsection{Comparison between the proposed frequencies selecting algorithm and the EDP method}
\label{ch3:4:4}
Finding the frequencies that give the best trade-off between the energy consumption and the performance for a parallel
application is not a trivial task. Many algorithms have been proposed to tackle this problem.
In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers \cite{ref80,ref81,ref82}.
-This objective function was also used by Spiliopoulos et al. algorithm \cite{ref67} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to the multi-cores
+This objective function was also used by Spiliopoulos et al. algorithm \cite{ref67} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to the multi-core
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-grid} and
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.
+Both algorithms were applied to the class D of the NAS benchmarks running over 16 nodes.
The participating computing nodes are distributed according to the two scenarios described in Section~\ref{ch3:4:1}.
The experimental results, the energy saving, performance degradation and trade-off distance percentages, are
presented in Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
-
-
-
-\begin{figure*}[!t]
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/edp_eng}
\caption{The energy reduction induced by the Maxdist method and the EDP method}
\label{fig:edp-eng}
\end{figure*}
-\begin{figure*}[!t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/edp_per}
\caption{The performance degradation induced by the Maxdist method and the EDP method}
\label{fig:edp-perf}
\end{figure*}
-\begin{figure*}[!t]
+
+\begin{figure*}[!h]
\centering
\includegraphics[width=.7\textwidth]{fig/ch3/edp_dist}
\caption{The trade-off distance between the energy consumption reduction and the performance for the Maxdist method and the EDP method}
As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios.
-The proposed algorithm gives better results than EDP because it
+The proposed algorithm gives better results than the EDP method because it
maximizes the energy saving and the performance at the same time.
Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
The high positive values of the trade-off distance percentage mean that the energy saving percentage is much higher than the performance degradation percentage.
-The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and
-$O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the
-maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average $0.01 ms$ to compute the best frequencies while EDP is on average ten times slower over the same architecture.
+The complexity of both algoriths, Maxdist and EDP, are of order $O(N \cdot M_i \cdot F_j)$ and
+$O(N \cdot M_i \cdot F_j^2)$ respectively, where $N$ is the number of the clusters, $M_i$ is the number of nodes and $F_j$ is the
+maximum number of available frequencies of node $j$. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average $0.01$ $ms$ to compute the best frequencies while the EDP method is on average ten times slower over the same architecture.
\section{Conclusion}
\label{ch3:concl}
In this chapter, two new online frequency scaling factors selecting algorithms have been presented. They select the best possible vectors of frequency scaling factors that give the
-maximum distance (optimal trade-off) between the predicted energy and the
+maximum distance (optimal trade-off) between the predicted energy and the
predicted performance curves for a heterogeneous cluster and grid. Both algorithms use a
-new energy models for measuring and predicting the energy of distributed
+new energy models for measuring and predicting the energy consumption of message passing
iterative applications running over a heterogeneous local cluster and a grid platform.
-Firstly, the proposed scaling factors selection algorithm for a heterogeneous local cluster is applied to NAS parallel benchmarks class C and simulated by SimGrid.
-The results of the experiments showed that the algorithm on average reduces by 29.8\% the energy
-consumption of NAS benchmarks executed over 8 nodes while limiting the degradation of the performance by 3.8\%. The algorithm also selects different scaling factors according to
+Firstly, the proposed scaling factors selection algorithm for a heterogeneous local cluster is applied to the class C of NAS parallel benchmarks and simulated by SimGrid.
+The results of the simulations showed that the algorithm on average reduces by 29.8\% the energy
+consumption of the NAS benchmarks executed over 8 nodes while limiting the degradation of the performance by 3.8\%. The algorithm also selects different scaling factors according to
the percentage of the computing and communication times, and according to the
values of the static and dynamic powers of the CPUs.
-Secondly, the proposed scaling factors selection algorithm for a grid is applied to NAS parallel benchmarks class D and executed over Grid5000 testbed platform.
-The experiments on 16 nodes, distributed over three clusters, showed that the algorithm on average reduces by 30\% the energy consumption
-for all the NAS benchmarks while on average only degrading by 3.2\% the performance.
-The algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or use multi-cores per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the
+Secondly, the proposed scaling factors selection algorithm for a grid is applied to the class D of the NAS parallel benchmarks and executed over the Grid5000 testbed platform.
+The experiments executed on 16 nodes distributed over three clusters, showed that the algorithm on average reduces by 30\% the energy consumption
+of all the NAS benchmarks while on average only degrading by 3.2\% their performance.
+The algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or use multi-core per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the
computations and communication times ratios, and the values of the static and measured dynamic powers of the CPUs. Thus, the simulation and the real results are comparable in term of energy saving and performance degradation percentages.
-Finally, both the proposed algorithms were compared to another method that uses
+Finally, both algorithms were compared to a method that uses
the well known energy and delay product as an objective function. The comparison results showed
that the proposed algorithms outperform the latter by selecting vectors of frequencies that give a better
trade-off results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
-%% CHAPTER 05 %%
+%% CHAPTER 04 %%
%% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Energy Optimization of Asynchronous Iterative Applications}
+\newcommand{\Tnorm}{\Xsub{T}{Norm}}
+\newcommand{\Ltcm}[1][]{\Xsub{L}{tcm}_{\fxheight{#1}}}
+\newcommand{\Etcm}[1][]{\Xsub{E}{tcm}_{\fxheight{#1}}}
+\newcommand{\Niter}[1][]{\Xsub{N}{iter}_{\fxheight{#1}}}
+
+\chapter{Energy Optimization of Asynchronous Applications}
\label{ch4}
+\section{Introduction}
+\label{ch4:1}
+
+A grid is composed of heterogeneous clusters: CPUs from distinct clusters might have different computing power, energy consumption or frequency range.
+Running synchronous parallel applications on grids results in long slack times where the fast nodes have to wait for the slower ones to finish their computations before synchronously exchanging data with them. Therefore, it is widely accepted that asynchronous parallel methods are more suitable than synchronous ones for such architectures because there is no slack time and the asynchronous communications are overlapped by computations. However, they usually execute more iterations than the synchronous ones and thus consume more energy.
+In order to make the asynchronous method a good alternative to the synchronous one, it should not be just competitive in performance but also in energy consumption.
+To reduce the energy consumption of a CPU executing the asynchronous iterative method, the Dynamic voltage and frequency scaling (DVFS) technique can be used. Modern operating systems automatically adjust the frequency of the processor according to their needs using DVFS operations. However, the user can scale down the frequency of the CPU using the on-demand governor \cite{ref96}. It lowers the frequency of a CPU to reduce its energy
+consumption, but it also decreases its computing power and thus it might increase the
+execution time of an application running on that processor. Therefore, the frequency that gives the best trade-off between energy consumption and performance must be selected. For parallel asynchronous methods running over a grid, a different frequency might be selected for each CPU in the grid depending on its characteristics.
+In chapters \ref{ch2} and \ref{ch3}, three frequencies selecting algorithms were proposed
+to reduce the energy consumption of synchronous message passing iterative applications running over homogeneous and heterogeneous platforms respectively. In this chapter, a new frequency selecting algorithm for asynchronous iterative message passing applications running over grids is presented. An adaptation for hybrid methods, with synchronous and asynchronous communications, is also proposed.
+The algorithm and its adaptation select the vector of frequencies which simultaneously offers a maximum energy reduction and minimum performance degradation ratio. The algorithm has a very small overhead and works online without needing any training nor any profiling.
+
+
+This chapter is organized as follows: Section~\ref{ch4:2} presents some
+related works from other authors. models for predicting the performance and the energy consumption
+ of both synchronous and asynchronous message passing programs
+running over a grid are explained in Section~\ref{ch4:3}.
+It also presents the objective function that maximizes the reduction of energy consumption while minimizing
+the degradation of the program's performance, used to select the frequencies.
+Section~\ref{ch4:5} details the proposed frequencies selecting algorithm.
+Section~\ref{ch4:6} presents the iterative multi-splitting application which is a hybrid method and was used as a benchmark to evaluate the efficiency of the proposed algorithm.
+Section~\ref{ch4:7} presents the simulation results of applying the algorithm on the multi-splitting application
+and executing it on different grid scenarios. It also shows the results of running
+three different power scenarios and comparing them. Moreover, in the last subsection, the proposed algorithm is compared to the energy and delay product (EDP) method. Section \ref{ch4:8} shows the real experiment results of applying the proposed algorithm over Grid'5000 platform and the results with the EDP method . Finally, the chapter ends with a summary in section
+\ref{ch4:9}.
+
+
+
+
+\section{Related works}
+\label{ch4:2}
+
+
+A message passing application is in general composed of two types of sections, which are the computations and the communications sections. The communications can be done synchronously or asynchronously. In a synchronous message passing application, when a process synchronously sends a message to another node, it is blocked until the latter receives the message. During that time, there is no computation on both nodes and that period is called slack time.
+On the contrary, in an asynchronous message passing application, the asynchronous communications are overlapped by computations, thus, there is no slack time.
+Many techniques have been used to reduce the energy consumption of message passing applications,
+such as scheduling, heuristics and DVFS. For example, different scheduling techniques, to switch off the idle nodes to save their energy consumption, were presented in \cite{ref83,ref84,ref85} and \cite{ref86}. In \cite{ref87}
+and \cite{ref88}, an heuristic to manage the workloads between the computing resources of the cluster and reduce their energy, was published.
+However, the dynamic voltage and frequency scaling (DVFS) is the most popular technique to reduce the energy consumption of computing processors.
+
+As shown in the related works of chapter \ref{ch2}, most of the works in this field targeted the synchronous message passing applications because they are more common than the asynchronous ones and easier to work on. Some researchers tried to reduce slack times in synchronous applications running over homogeneous clusters. These slack times can happen on such architectures if the distributed workloads over the computing nodes are imbalanced.
+Other works focused on reducing the energy consumption of synchronous applications running over heterogeneous architectures such as heterogeneous clusters or grids. When executing synchronous message passing applications on these architectures, slack times are generated when fast nodes have to communicate with slower ones. Indeed, the fast nodes have to wait for the slower ones to finish their computations to be able to communicate with them. In this case, some energy was saved as in the work of chapter \ref{ch3} and its related works by reducing the frequencies of the fast nodes with DVFS operations while minimizing the slack times.
+
+Whereas, no work has been conducted to optimize the energy consumption of asynchronous message passing applications. Some works use asynchronous communications when applying DVFS operations on synchronous applications. For example, Hsu et al. \cite{ref92} proposed an online adaptive algorithm that divides the synchronous message passing application into several time periods and selects the suitable frequency for each one. The algorithm asynchronously applies the new computed frequencies to overlap the multiple DVFS switching times with computation. Similarly to this work, Zhu et al. \cite{ref93} studied the difference between applying synchronously or asynchronously the frequency changing algorithm during the execution time of the program. The results of the proposed asynchronous scheduler were more energy efficient than synchronous one. In \cite{ref94}, Vishnu et al. presented an energy efficient asynchronous agent that reduces the slack times in a parallel program to reduce the energy consumption. They used asynchronous communications in the proposed algorithm, which calls the DVFS algorithm many times during the execution time of the program. The three previous presented works were applied on applications running over homogeneous platforms.
+
+In \cite{ref95}, the energy consumption of an asynchronous iterative linear solver running over a heterogeneous platform, is evaluated. The results showed that the asynchronous version of the application had less execution time than the synchronous one. Therefore, according to their energy model the asynchronous method consumes less energy.
+However, in their model they do not consider that during synchronous communications only static power which is significantly lower than dynamic power, is consumed.
+
+This chapter presents the following contributions:
+\begin{enumerate}
+\item new model to predict the energy consumption and the execution time
+ of asynchronous iterative message passing applications running over a grid platform.
+
+\item a new online algorithm that selects a vector of frequencies which gives the best trade-off between energy consumption and performance for asynchronous iterative message passing applications running over a grid platform. The algorithm has a very small overhead
+ and does not need any training or profiling. The new algorithm can be applied synchronously and asynchronously on an iterative message passing application.
+
+\end{enumerate}
+
+
+
+
+\section{The performance and the energy consumption measurement models}
+\label{ch4:3}
+
+\subsection{The execution time of iterative asynchronous message passing applications}
+\label{ch4:3:1}
+In this chapter, we are interested in running asynchronous iterative message
+ passing distributed applications over a grid while reducing the energy consumption of the
+ CPUs during the execution.
+ Figure \ref{fig:heter} is an example of a grid with four different clusters. Inside each cluster, all the nodes are homogeneous, have the same specifications, but are different from the nodes of the other clusters.
+ To reduce the energy consumption of these applications while running on a grid,
+ the heterogeneity of the clusters' nodes, such as nodes' computing powers (FLOPS), energy consumptions and
+ CPU's frequency ranges, must be taken into account. To reduce the complexity of the experiments and focus on the heterogeneity of the nodes, the local networks of all the clusters are assumed to be identical, with the same latency and bandwidth. The networks connecting the clusters are also assumed to be homogeneous but they are slower than the local networks.
+
+ \begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.9]{fig/ch4/GRID}
+ \caption{A grid platform composed of heterogeneous clusters}
+ \label{fig:heter}
+\end{figure}
+
+
+An iterative application consists of a block of instructions that is repeatedly executed until convergence. A distributed iterative application with interdependent tasks requires, at each iteration, exchanging data between nodes to compute the distributed tasks. The communications between the nodes can be done synchronously or asynchronously. In the synchronous model, each node has to wait to receive data from all its neighbors to compute its iteration, see figures \ref{fig:ch1:15} and \ref{fig:ch1:16}.
+Since the tasks are synchronized, all the nodes execute the same number of iterations.
+Then, The overall execution time of an iterative synchronous message passing application with balanced tasks, running on the grid described above, is equal to the execution time of the slowest node in the slowest cluster running a task as presented in \ref{eq:perf_heter}.
+
+
+Whereas, in the asynchronous model, the fast nodes do not have to wait for the slower nodes to finish their computations to exchange data, see Figure \ref{fig:ch1:17}. Therefore, there are no idle times between successive iterations, the node executes the computations with the last received data from its neighbors and the communications are overlapped by computations. Since there are no synchronizations between nodes, all nodes do not have the same number of iterations.
+The difference in the number of executed iterations between the nodes depends on the heterogeneity of the computing powers of the nodes. The execution time of an asynchronous iterative message passing application is not equal to the execution time of the slowest node like in the synchronous mode because each node executes a different number of iterations. Moreover, the overall execution time is directly dependent on the method used to detect the global convergence of the asynchronous iterative application. The global convergence detection method might be synchronous or asynchronous and centralized or distributed.
+
+In a grid, the nodes in each cluster have different characteristics, especially different frequency gears.
+Therefore, when applying DVFS operations on these nodes, they may get different scaling factors represented
+by a scaling vector: $(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the
+scaling factor of processor $j$ in the cluster $i$.
+To be able to predict the execution time of asynchronous iterative message passing applications running
+over a grid, for different vectors of scaling factors, the communication times and the computation times for all the tasks must be measured during the first iteration before applying any DVFS operation. Then, the execution time of one iteration of an asynchronous iterative message passing application,
+running on a grid after applying a vector of scaling factors, is equal to the execution time of the synchronous application but without its communication times. The communication times are overlapped by computations and the execution time can be evaluated for all the application as the average of the execution time of all the parallel tasks. This is presented in Equation \ref{eq:asyn_time}.
+
+\begin{equation}
+ \label{eq:asyn_time}
+ \Tnew = \frac{\sum_{i=1}^{N} \sum_{j=1}^{M_i}({\TcpOld[ij]} \cdot S_{ij})} {N \cdot M_i }
+\end{equation}
+
+
+In this work, a hybrid (synchronous/asynchronous) message passing application \cite{ref99} is being used. It is composed of two loops:
+\begin{enumerate}
+\item In the inner loop, at each iteration, the nodes in a cluster synchronously exchange data between them. There is no communication between nodes from different clusters.
+\item In the outer loop, at each iteration, the nodes from different clusters asynchronously exchange their data between them because the network interconnecting the clusters has a high latency.
+\end{enumerate}
+
+Therefore, the execution time of one outer iteration of such a hybrid application can be evaluated by computing the average of the execution time of the slowest node in each cluster. The overall execution time of the asynchronous iterative applications can be evaluated as follows:
+
+\begin{equation}
+ \label{eq:asyn_perf}
+ \Tnew = \frac{\sum_{i=1}^{N} (\max_{j=1,\dots, M_i} ({\TcpOld[ij]} \cdot S_{ij}) +
+ \min_{j=1,\dots,M_i} ({\Ltcm[ij]}))}{N}
+\end{equation}
+
+In Equation (\ref{eq:asyn_perf}), the communication times $\Ltcm[ij]$ are only the communications between the local nodes because the communications between the clusters are asynchronous and overlapped by computations.
+
+
+\subsection{The energy model and tradeoff optimization}
+\label{ch3:3:3}
+
+The energy consumption of an asynchronous application running over a heterogeneous grid is the summation of
+the dynamic and static power of each node multiplied by the computation time of that node as in Equation (\ref{eq:asyn_energy1}). The computation time of each node is equal to the overall execution time of
+the node because the asynchronous communications are overlapped by computations.
+\begin{equation}
+ \label{eq:asyn_energy1}
+ E = \sum_{i=1}^{N} \sum_{j=1}^{M_i} {(S_{ij}^{-2} \cdot \Tcp[ij] \cdot (\Pd[ij]+\Ps[ij]) )}
+\end{equation}
+
+
+It is common for distributed algorithms running over grids to have asynchronous external communications between clusters and synchronous ones between the nodes of the same cluster. In this hybrid communication scheme, the dynamic energy consumption can be computed in the same way as for the synchronous application with Equation (\ref{eq:Edyn_new}).
+However, since the nodes of different clusters are not synchronized and do not have the same execution time as in the synchronous application, the static energy consumption is different between them. The cluster execution time is equal to the execution time of the slowest task in that cluster. The energy
+consumption of the asynchronous iterative message passing application running on a
+heterogeneous grid platform during one iteration can be computed as follows:
+
+ \begin{equation}
+ \label{eq:asyn_energy}
+ E = \sum_{i=1}^{N} \sum_{j=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} + \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot
+ ( \mathop{\max_{j=1,\dots,M_i}} ({\Tcp[ij]} \cdot S_{ij}) + \mathop{\min_{j=1,\dots,M_i}} ({\Ltcm[ij]})))
+\end{equation}
+Where $\Ltcm[ij]$ is the local communication time of the cluster $i$ of node $j$.
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_{11}, S_{12},\dots, S_{NM_i})$ may degrade the performance of the application
+and thus, increase the static energy consumed because the execution time is
+increased~\cite{ref78}. The overall
+energy consumption for the asynchronous application can be computed by multiplying the energy consumption
+from one iteration of each cluster by the number of the iterations of that cluster, $\Niter[i]$,
+as in Equation (\ref{eq:asyn_energy_it}).
+
+
+ \begin{multline}
+ \label{eq:asyn_energy_it}
+ E = \sum_{i=1}^{N} (\sum_{j=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])}) \cdot \Niter[i] + \sum_{i=1}^{N} (\sum_{j=1}^{M_i} (\Ps[ij] \cdot \\
+ ( \mathop{\max_{j=1,\dots,M_i}} ({\Tcp[ij]} \cdot S_{ij}) + \mathop{\min_{j=1,\dots,M_i}} ({\Ltcm[ij]})))) \cdot \Niter[i]
+\end{multline}
+
+In order to optimize the energy consumption and the performance of the asynchronous iterative applications at the same time, the maximum distance between the two metrics can be computed as in the previous chapters.
+However, both the energy model and performance must be normalized as in the Equations \ref{eq:enorm-heter} and
+\ref{eq:pnorm-heter} respectively.
+Hence, $\Tnew$ should be computed as in Equation~\ref{eq:asyn_perf} and $\Told$ computed as follows:
+\begin{equation}
+ \label{eq:asyn_told}
+ \Told = \frac{\sum_{i=1}^{N} (\max_{j=1,\dots, M_i} ({\TcpOld[ij]}) +
+ \min_{j=1,\dots,M_i} ({\Ltcm[ij]}))}{N}
+\end{equation}
+
+The original energy consumption of asynchronous applications, $\Eoriginal$ is computed as in (\ref{eq:asyn_energy_original}).
+
+
+
+\begin{equation}
+ \label{eq:asyn_energy_original}
+ E_{original} = \sum_{i=1}^{N} \sum_{j=1}^{M_i} {( \Pd[ij] \cdot \TcpOld[ij])} + \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot
+ ( \mathop{\max_{j=1,\dots,M_i}} ({\TcpOld[ij]} ) + \mathop{\min_{j=1,\dots,M_i}} ({\Ltcm[ij]})))
+\end{equation}
+
+
+Then, the objective function can be modeled as the maximum
+distance between the normalized energy curve and the normalized
+performance curve over all available sets of scaling factors and is computed as in the
+objective function \ref{eq:max-grid}.
+
+
+\section[The scaling algorithm of asynchronous applications]{The scaling factors selection algorithm of asynchronous applications over grid}
+\label{ch4:5}
+The frequency selection algorithm~(\ref{HSA-asyn}) works online during the first iteration of asynchronous iterative message passing program running over a grid. The algorithm selects
+ the set of frequency scaling factors $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$ which maximizes
+ the distance, the tradeoff function (\ref{eq:max-grid}), between the predicted normalized energy consumption
+ and the normalized performance of the program. The algorithm is called just once in the iterative program and
+ it uses information gathered from the first iteration to approximate the vector of frequency scaling factors that gives the best tradeoff.
+ According to the returned vector of scaling factors, the DVFS algorithm (\ref{dvfs-heter}) computes the new frequency
+ for each node in the grid. It also shows where and when the proposed scaling algorithm is called in
+ the iterative message passing program.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.65]{fig/ch4/init_freq}
+ \caption{Selecting the initial frequencies in a grid composed of four clusters}
+ \label{fig:st_freq}
+\end{figure}
+
+In contrast to the scaling factors selection algorithm of synchronous applications running on the grid
+(algorithm \ref{HSA-grid}), this algorithm computed the initial frequencies depending on the Equations
+\ref{eq:Scp-grid} and \ref{eq:Fint-grid}. Figure~\ref{fig:st_freq} shows the selected initial frequencies of the grid composed of four different types of clusters that are presented in the Figure \ref{fig:heter}.
+The only difference between the two algorithms is the energy and performance models that are used. Furthermore, this algorithm scales down all frequencies of nodes at each iteration, while other algorithm don't scaled down the frequency of the slowest node. However, the performance of asynchronous application does not depend on the performance of the slower nodes, while it depends on the performance of all nodes.
+
+\begin{algorithm}
+ \begin{algorithmic}[1]
+ % \footnotesize
+ \Require ~
+ \item [{$N$}] number of clusters in the grid.
+ \item [{$M$}] number of nodes in each cluster.
+ \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
+ \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
+ \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
+ \item[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
+ \item[{$\Ps[ij]$}] array of the static powers for all nodes.
+ \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
+
+ \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$, a vector of scaling factors that gives the optimal tradeoff between energy consumption and execution time
+
+ \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
+ \State $F_{ij} \gets \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
+ \State Round the computed initial frequencies $F_i$ to the closest available frequency for each node.
+ \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 \frac{\sum_{i=1}^{N} (\max\limits_{j=1,\dots, M_i} ({\TcpOld[ij]}) +
+ \min\limits_{j=1,\dots,M_i} ({\Ltcm[ij]}))}{N} $
+ \vspace*{0.2 cm}
+ \State $\Eoriginal \gets \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} {( \Pd[ij] \cdot \TcpOld[ij])} + \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} (\Ps[ij] \cdot
+ (\mathop{\max\limits_{j=1,\dots,M_i}} ({\TcpOld[ij]} ) + \mathop{\min\limits_{j=1,\dots,M_i}} ({\Ltcm[ij]})))$
+ \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 frequency \textbf{or} $\Pnorm - \Enorm < 0 $)}
+ \If{(not the last frequency)}
+ \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 \frac{\sum\limits_{i=1}^{N} (\max\limits_{j=1,\dots, M_i} ({\TcpOld[ij]} \cdot S_{ij}) + \min\limits_{j=1,\dots,M_i} ({\Ltcm[ij]}))}{N} $
+ \vspace*{0.2 cm}
+ \State $\Ereduced \gets \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} + \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M_i} (\Ps[ij] \cdot
+ ( \mathop{\max\limits_{j=1,\dots,M_i}} ({\Tcp[ij]} \cdot S_{ij}) + \mathop{\min\limits_{j=1,\dots,M_i}} ({\Ltcm[ij]}))) $
+ \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+ \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
+ \If{$(\Pnorm - \Enorm > \Dist)$}
+ \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
+ \State $\Dist \gets \Pnorm - \Enorm$
+ \EndIf
+ \EndWhile
+ \State Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
+ \end{algorithmic}
+ \caption{Scaling factors selection algorithm of asynchronous applications over grid}
+ \label{HSA-asyn}
+\end{algorithm}
+
+
+
+
+\section{The iterative multi-splitting method }
+ \label{ch4:6}
+
+ Multi-splitting algorithms have been initially studied to
+ solve linear systems of equations in parallel
+ \cite{ref97}. Thereafter, they were used to design
+ non linear iterative algorithms and asynchronous iterative
+ algorithms~\cite{ref98}. The principle of multi-splitting
+ algorithms lies in splitting the system of equations, then solving
+ each sub-system using a direct or an iterative method and then
+ combining the results in order to build a global solution. Since a
+ multi-splitting method is iterative, it requires executing several iterations
+ in order to reach global convergence.
+
+ In this chapter, we have used an asynchronous iterative multisplitting method
+ to solve a 3D Poisson problem as described in~\cite{ref99}. The
+ problem is divided into small 3D sub-problems and each one is solved by a
+ parallel GMRES method. For more information about multi-splitting
+ algorithms, interested readers are invited to
+ consult the previous references.
+
+
+\section{The experimental results over SimGrid}
+ \label{ch4:7}
+In this section, the heterogeneous scaling algorithm (HSA), Algorithm~(\ref{HSA-asyn}), is applied to the parallel iterative
+multi-splitting method. The performance of this algorithm is evaluated by
+ executing the iterative multi-splitting method on the Simgrid/SMPI simulator v3.10
+\cite{ref66}. This simulator offers flexible tools to create a
+grid architecture and run the iterative application over it. The grid used in these
+experiments has four different types of nodes. Two types of nodes have different computing powers, frequency ranges, static and dynamic powers. Table \ref{table:platform}
+presents the characteristics of the four types of nodes. The specifications of the simulated nodes are similar to real Intel processors.
+Many grid configurations have been used in the experiments where the number of clusters and the number of nodes per cluster are equal to 4 or 8.
+For the grids composed of 8 clusters, two clusters of each type of nodes were used. The number of nodes per cluster is the same for all the clusters in a given grid.
+
+
+\begin{table}[!t]
+ \caption{The characteristics of the four types of nodes}
+ % title of Table
+ \centering
+ \begin{tabular}{|*{7}{r|}}
+ \hline
+ node& Simulated & Max & Min & Diff. & Dynamic & Static \\
+ type & GFLOPS & Freq. & Freq. & Freq. & power & power \\
+ & of one node & GHz & GHz & GHz & & \\
+ \hline
+ A & 40 & 2.50 & 1.20 & 0.100 & \np[W]{20} & \np[W]{4} \\
+ \hline
+ B & 50 & 2.66 & 1.60 & 0.133 & \np[W]{25} & \np[W]{5} \\
+ \hline
+ C & 60 & 2.90 & 1.20 & 0.100 & \np[W]{30} & \np[W]{6} \\
+ \hline
+ D & 70 & 3.40 & 1.60 & 0.133 & \np[W]{35} & \np[W]{7} \\
+ \hline
+ \end{tabular}
+ \label{table:platform}
+\end{table}
+
+
+ The CPUs' constructors do not specify the amount of static and dynamic powers their CPUs consume.
+The maximum power consumption for each node's CPU was chosen to be proportional to its computing power (FLOPS). The dynamic power was assumed to represent \np[\%]{80} of the overall power consumption and the rest (\np[\%]{20}) is the static power. Similar assumptions were made in last two chapters and \cite{ref47}.
+The clusters of the grid are connected via a long distance Ethernet network with
+\np[Gbit/s]{1} bandwidth, while inside each cluster the nodes are connected via a high-speed \np[Gbit/s]{10} bandwidth local Ethernet network. The local networks have ten times less latency than the network connecting the clusters.
+
+\subsection{The energy consumption and the execution time of the multi-splitting application}
+ \label{ch4:7:1}
+ The multi-splitting (MS) method solves a three dimensional problem of size $N=N_x \cdot N_y \cdot N_z$. The problem is divided into equal sub-problems which are distributed to the computing nodes of the grid and then solved.
+ The experiments were conducted on problems of size $N=400^3$
+ or $N=500^3$ that require more than $12$ and $24$ Gigabyte of memory, respectively.
+ Table \ref{table:comp} presents the different experiment scenarios with different numbers of clusters, nodes per cluster and problem sizes. A name, consisting in the values of these parameters was given to each scenario.
+
+ \begin{table}[!t]
+ \caption{The different experiment scenarios}
+ % title of Table
+ \centering
+ \begin{tabular}{|*{7}{r|}}
+ \hline
+ Platform & Clusters & Number of nodes &Vector & Total number of \\
+ scenario & number & in cluster &size & nodes in grid \\
+ \hline
+ Grid.4*4.400 & 4 & 4 &$400^3$ &16 \\
+ \hline
+ Grid.4*8.400 & 4 & 8 &$400^3$ &32 \\
+ \hline
+ Grid.8*4.400 & 8 & 4 &$400^3$ &32 \\
+ \hline
+ Grid.8*8.400 & 8 & 8 &$400^3$ &64 \\
+ \hline
+ Grid.4*4.500 & 4 & 4 &$500^3$ &16 \\
+ \hline
+ Grid.4*8.500 & 4 & 8 &$500^3$ &32 \\
+ \hline
+ Grid.8*4.500 & 8 & 4 &$500^3$ &32 \\
+ \hline
+ Grid.8*8.500 & 8 & 8 &$500^3$ &64 \\
+ \hline
+ \end{tabular}
+ \label{table:comp}
+\end{table}
+
+This section focuses on the execution time and the energy consumed by
+the MS application while running over the grid platform without using DVFS operations.
+The energy consumption of the synchronous and asynchronous MS was
+measured using the energy Equations \ref{eq:energy-grid} and \ref{eq:asyn_energy} respectively.
+Figures \ref{fig:eng_time_ms} (a) and (b) show the energy consumption
+and the execution time, respectively, of the multi-splitting application running over a heterogeneous grid
+with different numbers of clusters and nodes per cluster.
+The synchronous and the asynchronous versions of the MS application were executed over each scenario in Table \ref{table:comp}.
+As shown in Figure \ref{fig:eng_time_ms} (a), the asynchronous MS consumes more
+energy than the synchronous one. Indeed, the asynchronous application overlaps the asynchronous communications with computations and thus it executes more iterations than the synchronous one and has no slack times. More computations result in more dynamic energy consumption by the CPU in the asynchronous MS and since the dynamic power is chosen to be four times higher than the static power, the asynchronous MS method consumes more overall energy than the synchronous one. However, the execution times of the experiments, presented in Figure \ref{fig:eng_time_ms} (b), show that the execution times of the
+asynchronous MS are smaller than the execution times of the synchronous one. Indeed, in the
+asynchronous application the fast nodes do not have to wait for the slower ones to exchange data. So there are no slack times and more iterations are executed by fast nodes which accelerates the convergence to the final solution.
+
+\begin{figure}[!t]
+ \centering
+ \centering
+ \includegraphics[width=.80\textwidth]{fig/ch4/energy_ms.eps}\\~~~~~~(a)\\
+ \includegraphics[width=.82\textwidth]{fig/ch4/time_ms.eps}\\~~~~~~~~(b)
+ \caption{(a) energy consumption and (b) execution time of multi-splitting application without applying the HSA algorithm}
+ \label{fig:eng_time_ms}
+\end{figure}
+
+
+The synchronous and asynchronous MS scale well. The execution times of both methods decrease linearly with the increase of the
+number of computing nodes in the grid, whereas the energy consumption is approximately
+the same when the number of computing nodes increases. Therefore, the energy consumption
+of this application is not directly related to the number of computing nodes.
+
+\subsection{The results of the scaling factor selection algorithm}
+ \label{ch4:7:2}
+ The scaling factor selection algorithm~\ref{HSA-asyn} was applied to both
+ synchronous and asynchronous MS applications which were
+ executed over the 8 possible scenarios presented in table~\ref{table:comp}.
+ The DVFS algorithm \ref{dvfs} needs to send and receive some information before
+ calling the scaling factor selection algorithm algorithm~\ref{HSA-asyn}. The communications of the DVFS algorithm
+ can be applied synchronously or asynchronously which results in four different versions of the application: synchronous or asynchronous MS with synchronous or asynchronous DVFS communications. Figures \ref{fig:eng_time_dvfs} (a) and (b) present the energy consumption and the execution time for the four different versions of the application running on all the scenarios in Table \ref{table:comp}.
+
+
+ \begin{figure}[!t]
+ \centering
+ \centering
+ \includegraphics[width=.82\textwidth]{fig/ch4/energy_dvfs.eps}\\~~~~~~~(a)\\
+ \includegraphics[width=.80\textwidth]{fig/ch4/time_dvfs.eps}\\~~~~~~~~(b)
+ \caption{(a) energy consumption and (b) execution time of different versions of the multi-splitting application after applying the HSA algorithm}
+ \label{fig:eng_time_dvfs}
+\end{figure}
+ Figure \ref{fig:eng_time_dvfs} (a) shows that the energy
+ consumption of all four versions of the method, running over the 8 grid scenarios described in Table \ref{table:comp}, are not affected by the increase in the number of computing nodes. MS without applying DVFS operations had the same behavior. On the other hand, Figure \ref{fig:eng_time_dvfs} (b) shows that the execution time of the MS application with DVFS operations
+ decreases in inverse proportion to the number of nodes. Moreover, it can be noticed that the asynchronous MS with synchronous DVFS consumes less energy when compared to the other versions of the method. Two reasons explain this energy consumption reduction:
+ \begin{enumerate}
+ \item The asynchronous MS with synchronous DVFS version uses synchronous DVFS communications which allow it to apply the new computed frequencies at the begining of the second iteration. Thus, reducing the consumption of dynamic energy by the application from the second iteration until the end of the application. Whereas in
+ asynchronous DVFS versions where the DVFS communications are asynchronous, the new frequencies cannot be computed at the end of the first iteration and consequently cannot be applied at the begining of the second iteration.
+ Indeed, since the performance information gathered during the first iteration is not sent synchronously at the end of the first iteration, fast nodes might execute many iterations before receiving the performance information, computing the new frequencies based on this information and applying the new computed frequencies. Therefore, many iterations might be computed by CPUs running on their highest frequency and consuming more dynamic energy than scaled down processors.
+
+\item As shown in Figure \ref{fig:eng_time_ms} (b), the execution time of the asynchronous MS version is lower than the execution time of the synchronous MS version because there is no idle time in the asynchronous version and the communications are overlapped by computations. Since the consumption of static energy is proportional to the execution time, the asynchronous MS version consumes less static energy than the synchronous version.
+
+ \end{enumerate}
+
+ \begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.7]{fig/ch4/energy_saving.eps}
+ \caption{The energy saving percentages after applying the HSA algorithm to the different versions and scenarios}
+ \label{fig:energy_saving}
+\end{figure}
+
+
+ The energy saving percentage is the ratio between the reduced energy consumption after applying the HSA algorithm and the original energy consumption of synchronous MS without DVFS.
+ Whereas, the performance degradation percentage is the ratio between the original execution time of the synchronous MS without DVFS and the new execution time after applying the HSA algorithm.
+Therefore, in this section, the synchronous MS method without DVFS serves as a reference for comparison with the other methods for the following terms: energy saving, performance degradation and the distance between the two previous terms.
+
+ In Figure \ref{fig:energy_saving}, the energy saving is computed for the four versions of the MS method which
+ are the synchronous or asynchronous MS that apply synchronously or asynchronously the HSA algorithm.
+ The fifth version is the asynchronous MS without any DVFS operations. Figure \ref{fig:energy_saving} shows that some versions have positive or negative energy saving percentages which means that the corresponding version respectively consumes less or more energy than the reference method.
+As in Figure \ref{fig:eng_time_dvfs} (a) and for the same reasons presented above, the asynchronous MS with synchronous DVFS version gives the best energy saving percentage when compared to the other versions.
+
+
+ \begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.7]{fig/ch4/perf_degra.eps}
+ \caption{The results of the performance degradation}
+ \label{fig:perf_degr}
+\end{figure}
+
+ \begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.7]{fig/ch4/dist.eps}
+ \caption{The results of the tradeoff distance}
+ \label{fig:dist}
+\end{figure}
+
+Figure \ref{fig:perf_degr} shows that some versions have negative performance
+degradation percentages which means that the new execution time of a given version of the application is less than the execution time of the synchronous MS without DVFS.
+ Therefore, the version with the smallest negative performance degradation percentage has actually the best speed up when compared to the other versions. The version that gives the best execution time is the
+ asynchronous MS without DVFS which on average outperforms the synchronous MS without DVFS version by
+ $16.9\%$. While the worst case is the synchronous MS with synchronous DVFS where the performance is on average degraded by $2.9\%$ when compared to the reference method.
+
+
+ The energy consumption and performance tradeoff between these five versions is presented in Figure \ref{fig:dist}.
+ These distance values are computed as the differences between the energy saving
+ and the performance degradation percentages as in the optimization function
+ (\ref{eq:max-grid}). Thus, the best MS version is the one that has the maximum distance between the energy saving and performance degradation. The distance can be negative if the energy saving percentage is less than the performance degradation percentage.
+ The asynchronous MS applying synchronously the HSA algorithm gives the best distance which is on average equal to $27.72\%$.
+ This version saves on average up to $22\%$ of energy and on average speeds up the application by $5.72\%$. This overall improvement is due to combining asynchronous computing and the synchronous application of the HSA algorithm.
+
+
+The two platform scenarios, Grid 4*8 and Grid 8*4, use the same
+number of computing nodes but give different trade-off results.
+The versions applying the HSA algorithm and running over the Grid 4*8 platform, give higher distance percentages than those running on the Grid 8*4 platform. In the Grid 8*4 platform scenario more clusters are used than in the Grid 4*8 platform and thus the global system is divided into 8 small subsystems instead of 4. Indeed, each subsystem is assigned to a cluster and synchronously solved by the nodes of that cluster. Dividing the global system into smaller subsystems, increases the number of outer iterations required for the global convergence of the system because for the multi-splitting system the more the system is decomposed the higher the spectral radius is. For example, the asynchronous MS, applying synchronously the HSA algorithm, requires on average 135 outer iterations when running over the Grid 4*8 platform and 148 outer iterations when running over the Grid 8*4 platform. The increase in the number of executed iterations over the Grid 8*4 platform justifies the increase in energy consumption by applications running over that platform.
+
+
+\subsection{Comparing the number of iterations executed by the different MS versions}
+ \label{ch4:7:3}
+
+ The heterogeneity in the computing power of the nodes in the grid has a direct
+ effect on the number of iterations executed by the nodes of each cluster when running an asynchronous iterative message passing method. The fast nodes execute more iterations than the slower ones because the iterations are not synchronized.
+ On the other hand, in the synchronous versions, all the nodes in all the clusters have the same number of iterations and have to wait for the slowest node to finish its iteration before starting the next iteration because the iterations are synchronized.
+
+ When the fast nodes asynchronously execute more iterations than the slower ones, they consume more energy without significantly improving the global convergence of the system. Reducing the frequency of the fast nodes will decrease the number of iterations executed by them. If all the nodes, the fast and the slow ones, execute close numbers of iterations, the asynchronous application will consume less energy and its performance will not be significantly affected.
+ Therefore, applying the HSA algorithm over asynchronous applications is very promising. In this section, the number of iterations executed by the asynchronous MS method, while solving a 3D problem of size $400^3$ with and without applying the HSA algorithm, is evaluated. In Table \ref{table:sd}, the standard deviation of the number of iterations executed by the asynchronous application over all the grid platform scenarios, is presented.
+
+
+\begin{table}[h]
+\centering
+\caption{The standard deviation of the numbers of iterations for different asynchronous MS versions running over different grid platforms}
+\label{table:sd}
+\begin{tabular}{|l|l|l|l|}
+\hline
+\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Grid\\platform \end{tabular}}
+& \multicolumn{3}{c|}{Standard deviation} \\ \cline{2-4}
+& \begin{tabular}[c]{@{}l@{}}Asyn. MS without \\ HSA\end{tabular}
+& \begin{tabular}[c]{@{}l@{}}Asyn. MS with \\ Asyn. HSA\end{tabular}
+& \begin{tabular}[c]{@{}l@{}}Asyn. MS with \\ Syn. HSA\end{tabular} \\ \hline
+Grid.4*4.400 & 60.43 & 13.86 & 1.12 \\ \hline
+Grid.4*8.400 & 58.06 & 27.43 & 1.22 \\ \hline
+Grid.8*4.400 & 50.97 & 20.76 & 1.15 \\ \hline
+Grid.8*8.400 & 52.46 & 48.40 & 2.38 \\ \hline
+\end{tabular}
+\end{table}
+
+A small standard deviation value means that there is a very small difference between
+ the numbers of iterations executed by the nodes which means fast nodes did not uselessly execute more iterations than the slower ones and the application does not waste a lot of energy. As shown in Table \ref{table:sd},
+ the asynchronous MS that applies synchronously the HSA algorithm has the best standard deviation value when compared to the other versions. Two reasons explain the advantage of this method:
+ \begin{enumerate}
+\item The applied HSA algorithm selects new frequencies that reduce the computation power of the fast nodes while maintaining the computation power of the slower nodes. Therefore, it tries to balance as much as possible the computation powers of the heterogeneous nodes.
+
+\item Applying synchronously the HSA algorithm scales down the frequencies of the CPUs at the end of the first iteration of the application. Therefore the computation power of all the nodes is balanced as much as possible since the beginning of the application. On the other hand, applying asynchronously the HSA algorithm onto the asynchronous MS application only changes the frequencies of the nodes after executing many iterations. Therefore, before the frequencies are scaled down, the fast nodes have enough time to execute many more iterations than the slower ones and consequently increase the overall energy consumption of the application.
+
+ \end{enumerate}
+
+Finally, the asynchronous MS version that does not apply the HSA algorithm gives the worst standard deviation values because there is a big difference between the numbers of iterations executed by the heterogeneous nodes. Therefore, this version consumes more energy than the other versions and thus saves less energy as shown in Figure \ref{fig:eng_time_dvfs} (a).
+
+
+\subsection{Comparing different power scenarios}
+ \label{ch4:7:4}
+
+ In the previous sections, all the results were obtained by assuming that the dynamic and the static powers are respectively equal to 80\% and 20\% of the total power consumed by a CPU during computation at its highest frequency. The goal
+ of this section is to evaluate the proposed frequency scaling factors selection algorithm when
+ these two power ratios are changed. Two new power scenarios are proposed in this section:
+ \begin{enumerate}
+\item The dynamic and the static power are respectively equal to 90\% and 10\% of the total power consumed by a CPU during computation at its highest frequency.
+ \item The dynamic and the static power are respectively equal to 70\% and 30\% of the total power consumed by a CPU during computation at its highest frequency.
+ \end{enumerate}
+ The asynchronous MS method solving a 3D problem of size $400^3$ was executed over two
+ platform scenarios, the Grid 4*4 and Grid 8*4. Two versions of the asynchronous MS method, with synchronous or asynchronous application of the HSA algorithm, were evaluated on each platform scenario.
+ The energy saving, performance degradation and distance percentages for both versions over both platform
+ scenarios and with the three power scenarios are presented in Figures \ref{fig:three_power_syn} and \ref{fig:three_power_asyn}.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/three_powers_syn.eps}
+\caption{The results of the three power scenarios: Synchronous application of the HSA algorithm}
+\label{fig:three_power_syn}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/three_powers_Asyn.eps}
+\caption{The results of the three power scenarios: Asynchronous application of the HSA algorithm}
+\label{fig:three_power_asyn}
+\end{figure}
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=.7]{fig/ch4/three_scenarios.pdf}
+ \caption{Comparison of the selected frequency scaling factors by the HSA algorithm for the three power scenarios}
+ \label{fig:three_scenarios}
+\end{figure}
+
+
+ The displayed results are the average of the percentages obtained from multiple runs.
+ Both figures show that the \np[\%]{90}-\np[\%]{10} power scenario gives the biggest energy saving percentages.
+ The high dynamic power ratio pushes the HSA algorithm to select bigger scaling factors
+ which decreases exponentially the dynamic energy consumption. Figure \ref{fig:three_scenarios} shows that the HSA algorithm selects in the \np[\%]{90}-\np[\%]{10} power scenario higher frequency scaling factors than in the other power scenarios for the same application. Moreover, the \np[\%]{90}-\np[\%]{10} power scenario has the smallest static power consumption per CPU which reduces the effect of the performance degradation, due to scaling down the frequencies of the CPUs, on the total energy consumption of the application. Finally, the \np[\%]{90}-\np[\%]{10} power scenario gives higher distance percentages than the other two scenarios which means the difference between the energy reduction and the performance degradation percentages is the highest for this scenario. From these observations, it can be concluded that in a platform with CPUs that consume low static power and high dynamic power, a lot of energy consumption can be reduced by applying the HSA algorithm but the performance degradation might be significant.
+
+The energy saving percentages are the smallest with the \np[\%]{70}-\np[\%]{30} power scenario. The high static power consumption in this scenario force the HSA algorithm to select small scaling factors in order not to significantly decrease the performance of the application. Indeed, scaling down more the frequency of the CPUs will significantly increase the total execution time and consequently increase the static energy consumption which will outweigh the reduction of dynamic energy consumption. Finally, since the dynamic power consumption ratio is relatively small in this power scenario less dynamic energy reduction can be gained in lowering the frequencies of the CPUs than in the other power scenarios. On the other hand, the \np[\%]{70}-\np[\%]{30} power scenario's main advantage is that its performance suffers the least from the application of the HSA algorithm. From these observations, it can be concluded that in a high static power model just a small percentage of energy can be saved by applying the HSA algorithm.
+
+The asynchronous application of the HSA algorithm on average
+improves the performance of the application more than the synchronous
+application of the HSA algorithm. This difference can be explained by the fact that applying the HSA algorithm synchronously scales down the frequencies of the CPUs after the first iteration, while applying the HSA algorithm asynchronously scales them down after many iterations, depending on the heterogeneity of the platform.
+However, for the same reasons as above, the synchronous application of the HSA algorithm reduces the energy consumption more than the asynchronous one even though, the method applying the first has a bigger execution time than the one applying the latter.
+
+\subsection{Comparing the HSA algorithm to the energy and delay product method}
+\label{ch4:7:5}
+
+Many methods have been proposed to optimize the trade-off between the energy consumption and the performance of message passing applications. A well known optimization model used to solve this
+ problem is the energy and delay product, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$.
+In \cite{ref100,ref60,ref55}, the researchers used equal weights for the energy and delay factors.
+However, others added some weights to the factors in order to direct the optimization towards more energy saving or less performance degradation. For example, in ~\cite{ref71} they used the product $\mathit{EDP}=\mathit{energy}\times \mathit{delay}^2$ which favour performance over energy consumption reduction.
+
+In this work, the proposed scaling factors selection algorithm optimizes both the energy consumption and the performance at the same time and gives the same weight to both factors as in Equation \ref{eq:max-grid}. In this section, to evaluate the performance of the HSA algorithm, it is compared to the algorithm proposed by Spiliopoulos et al. \cite{ref67}. The latter is an online method that selects for each processor the frequency that minimizes the energy and delay product in order to reduce the energy consumption of a parallel application running over a homogeneous multi-cores platform. It gives the same weight to both metrics and predicts both the energy consumption and the execution time for each frequency gear as in the HSA algorithm.
+To fairly compare the HSA algorithm with the algorithm of Spiliopoulos et al., the same energy models, Equation (\ref{eq:energy-grid}) or (\ref{eq:asyn_energy}), and execution time models, Equation (\ref{eq:perf-grid}) or (\ref{eq:asyn_perf}), are used to predict the energy consumptions and the execution times.
+
+The EDP objective function can be equal to zero when the predicted delay is equal to zero. Moreover, this product is equal to zero before applying any DVFS operation. To eliminate the zero values, the EDP function must take the following form:
+
+
+\begin{equation}
+ \label{eq:EDP}
+ EDP = E_{Norm} \times (1+ D_{Norm})
+\end{equation}
+where $E_{Norm}$ is the normalized energy consumption which is computed as in Equation (\ref{eq:enorm})
+and $D_{Norm}$ is the normalized delay of the execution time which is computed as follows:
+\begin{equation}
+ \label{eq:Dnorm}
+ D_{Norm}= 1 -P_{Norm}= 1- (\frac{T_{old}}{T_{new}})
+\end{equation}
+Where $P_{Norm}$ is computed as in Equation (\ref{eq:pnorm}). Furthermore, the EDP algorithm starts the search process from the initial frequencies that are computed as in Equation (\ref{eq:Fint}). It stops the search process when it reaches the minimum available frequency for each processor. The EDP algorithm was applied to the synchronous and asynchronous MS algorithm solving a 3D problem of size $400^3$. Two platform scenarios, Grid 4*4 and Grid 4*8, were chosen for this experiment. The EDP method was applied synchronously and asynchronously to the MS application as for the HSA algorithm. The comparison results of the EDP and HSA algorithms are presented in the Figures \ref{fig:compare_syndvfs_synms}, \ref{fig:compare_asyndvfs_asynms},\ref{fig:compare_asyndvfs_synms} and \ref{fig:compare_asyndvfs_asynms}. Each of these figures presents the energy saving, performance degradation and distance percentages for one version of the MS algorithm. The results shown in these figures are also the average of the results obtained from running each version of the MS method over the two platform scenarios described above.
+
+
+
+
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/compare_syndvfs_synms.eps}
+ \caption{Synchronous application of the frequency scaling selection method on the synchronous MS version}
+ \label{fig:compare_syndvfs_synms}
+\end{figure}
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/compare_syndvfs_asynms.eps}
+ \caption{Synchronous application of the frequency scaling selection method on the asynchronous MS version}
+ \label{fig:compare_syndvfs_asynms}
+\end{figure}
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/compare_asyndvfs_synms.eps}
+ \caption{Asynchronous application of the frequency scaling selection method on the synchronous MS version}
+ \label{fig:compare_asyndvfs_synms}
+\end{figure}
+\begin{figure}[!h]
+ \centering
+ \includegraphics[width=.7\textwidth]{fig/ch4/compare_asyndvfs_asynms.eps}
+ \caption{Asynchronous application of the frequency scaling selection method on the asynchronous MS version}
+ \label{fig:compare_asyndvfs_asynms}
+\end{figure}
+
+
+
+
+All the figures show that the proposed HSA algorithm outperforms the EDP algorithm
+in terms of energy saving and performance degradation. EDP gave for some scenarios negative trade-off values which mean that the performance degradation percentages are higher than
+the energy saving percentages, while the HSA algorithm gives positive trade-off values over all scenarios.
+The frequency scaling factors selected by the EDP are most of the time higher than those selected by the HSA algorithm as shown in Figure \ref{fig:three_methods}.
+The results confirm that higher frequency scaling factors do not always give more energy saving, especially when the overall execution time is drastically increased. Therefore, the HSA method that computes the maximum distance between the energy saving and the performance degradation is an effective method to optimize these two metrics at the same time.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[scale=0.6]{fig/ch4/compare_scales.eps}
+ \caption{Comparison of the selected frequency scaling factors by the two algorithms
+ over the Grid 4*4 platform scenario}
+ \label{fig:three_methods}
+\end{figure}
+
+
+
+
+\section{The Experimental Results over Grid'5000}
+\label{ch4:8}
+The performance of algorithm ~(\ref{HSA-asyn}) was evaluated by
+ executing the iterative multi-splitting method on the Grid'5000 textbed \cite{ref21}.
+ This testbed is a large-scale platform that consists of ten sites distributed
+all over metropolitan France and Luxembourg. Moreover, some sites are equipped with power measurement tools that capture the power consumption for each node on those sites. Same method for computing the dynamic power consumption described in section \ref{ch3:4} is used.
+Table \ref{table:grid5000} presents the characteristics of the selected clusters which are located on four different sites.
+\begin{table}[!t]
+ \caption{CPUs characteristics of the selected clusters}
+ % title of Table
+ \centering
+ \begin{tabular}{|*{7}{c|}}
+ \hline
+ Cluster & CPU & Max Freq. & Min Freq. & Diff. Freq. & Site & Dynamic power \\
+ Name & model & GHz & GHz & GHz & & of one core \\
+ \hline
+ Taurus & Intel & 2.3 & 1.2 & 0.1 & Lyon & \np[W]{35} \\
+ & E5-2630 & & & & & \\
+ \hline
+ Graphene & Intel & 2.53 & 1.2 & 0.133 & Nancy & \np[W]{23} \\
+ & X3440 & & & & & \\
+ \hline
+ Parapide & Inte & 2.93 & 1.6 & 0.133 & Rennes & \np[W]{23} \\
+ & X5570 & & & & & \\
+ \hline
+ StRemi & AMD & 1.7 & 0.8 & 0.2 & Reims & \np[W]{6} \\
+ &6164 HE & & & & & \\
+ \hline
+ \end{tabular}
+ \label{table:grid5000}
+\end{table}
+The dynamic power of each core with maximum frequency is computed as the difference between the measured power of the core, only when it is computing at maximum frequency, and the measured power of that core when it is idle as in \ref{eq:pdyn}. The CPUs' constructors do not specify the amount of static power their CPUs consume. Therefore, the static power consumption is assumed to be equal to \np[\%]{20} of the dynamic power consumption.
+The experiments were conducted on problems of size $N=400^3$ and $N=500^3$ over 4 distributed clusters described in Table \ref{table:grid5000}. Each cluster is composed of 8 homogeneous nodes.
+
+
+Algorithm~\ref{HSA-asyn} was applied synchronously and asynchronously to both synchronous and asynchronous MS applications.
+Figures \ref{fig:time-compare} and \ref{fig:energy-compare} show the energy consumption and the execution time of the multi-splitting application with and without the application of the HSA algorithm respectively.
+The asynchronous MS consumes more energy than the synchronous one.
+Also, it can be noticed that both the asynchronous and synchronous MS with synchronous application of the HSA algorithm consume less energy than the other versions of the application. Synchronously applying the HSA algorithm allows them to scale down the CPUs' frequencies at the beginning of the second iteration. Thus, the consumption of dynamic energy by the application is reduced from the second iteration until the end of the application. On the contrary, with the asynchronous application of the HSA algorithm, the new frequencies cannot be computed at the end of the first iteration and consequently cannot be applied at the beginning of the second iteration. Indeed, since the performance information gathered during the first iteration is not sent synchronously at the end of the first iteration, fast nodes might execute many iterations before receiving the performance information, computing the new frequencies based on this information and applying the new computed frequencies. Therefore, many iterations might be computed by CPUs running on their highest frequency and consuming more dynamic energy than the scaled down processors.
+Moreover, the execution time of the asynchronous MS version is lower than the execution time of the synchronous MS version because there is no idle time in the asynchronous version and the communications are overlapped by computations. Since the consumption of static energy is proportional to the execution time, the asynchronous MS version consumes less static energy than the synchronous version.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[width=.8\textwidth]{fig/ch4/time-compare.eps}
+ \caption{ Comparing the execution time}
+ \label{fig:time-compare}
+ \end{figure}
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[width=.8\textwidth]{fig/ch4/energy-compare.eps}
+ \caption{ Comparing the energy consumption}
+ \label{fig:energy-compare}
+ \end{figure}
+
+
+\begin{table}[]
+\centering
+\begin{tabular}{|l|l|l|l|l|}
+\hline
+Size & Method &\begin{tabular}[c]{@{}l@{}}Energy\\ saving \%\end{tabular} & \begin{tabular}[c]{@{}l@{}}Perf. \\ degra.\%\end{tabular} & Distance \\ \hline
+\multirow{4}{*}{400} & Sync MS with Sync DVFS & 23.16 & 4.12 & 19.04 \\ \cline{2-5}
+ & Sync MS with Async DVFS & 18.36 & 2.59 & 15.77 \\ \cline{2-5}
+ & Async MS with Sync DVFS & 26.93 & -21.48 & 48.41 \\ \cline{2-5}
+ & Async MS with Async DVFS & 14.9 & -26.41 & 41.31 \\ \hline
+\multirow{4}{*}{500} & Sync MS with Sync DVFS & 24.57 & 3.15 & 21.42 \\ \cline{2-5}
+ & Sync MS with Async DVFS & 19.97 & 0.60 & 19.37 \\ \cline{2-5}
+ & Async MS with Sync DVFS & 20.69 & -10.95 & 31.64 \\ \cline{2-5}
+ & Async MS with Async DVFS & 9.06 & -18.22 & 27.28 \\ \hline
+\end{tabular}
+\caption{The experimental results of HSA algorithm}
+\label{table:exper}
+\end{table}
+
+Table \ref{table:exper} shows that there are positive and negative performance
+degradation percentages. A negative value means that the new execution time of a given version of the application is less than the execution time of the synchronous MS without DVFS.
+ Therefore, the version with the smallest negative performance degradation percentage has actually the best speed up when compared to the other versions.
+ The energy consumption and performance tradeoffs between these four versions can be computed as in the optimization Function
+ (\ref{eq:max-grid}). The asynchronous MS applying synchronously the HSA algorithm gives the best distance which is equal to $48.41\%$.
+ This version saves up to $26.93\%$ of energy and even reduces the execution time of the application by
+ $21.48\%$. This overall improvement is due to combining asynchronous computing and the synchronous application of the HSA algorithm.
+
+
+
+
+Finally, this section shows that the obtained results over Grid'5000 are comparable to
+simulation results of section \ref{ch4:7:2}, where the asynchronous MS applying synchronously the HSA algorithm is the best version in both of them. Moreover, results of Grid'5000 are better
+than simulation ones because its computing clusters are more heterogeneous in term of the computing power and network characteristics. For example, the StRemi cluster has smaller computing power compared to others three clusters of Grid'5000 platform.
+As a result, The increase in the idle times forces the proposed algorithm to select a big scaling factors and thus more energy saving.
+
+
+
+
+\subsection{Comparing the HSA algorithm to the energy and delay product method}
+\label{res-comp}
+
+The EDP algorithm, described in section \ref{ch4:7:5}, was applied synchronously and asynchronously to both the synchronous and asynchronous MS application of size $N=400^3$. The experiments were conducted over 4 distributed clusters, described in Table \ref{table:grid5000}, and 8 homogeneous nodes were used from each cluster.
+Table \ref{table:comapre} presents the results of energy saving, performance degradation and distance percentages when applying the EDP method on four different MS versions.
+Figure \ref{fig:compare} compares the distance percentages, computed as the difference between energy saving and performance degradation percentages, of the EDP and HSA
+algorithms. This comparison shows that the proposed HSA algorithm gives better energy reduction and performance trade-off than the EDP method. The results of EDP method over Grid'5000 are better than those for EDP obtained by the simulation according to the increase in the heterogeneity between the computing clusters of Grid'5000 as mentioned before.
+
+\begin{table}
+\centering
+\caption{The EDP algorithm results over the Grid'5000}
+\label{table:comapre}
+\begin{tabular}{|l|l|l|l|}
+\hline
+Method name & Energy saving \% & Perf. degra.\% & Distance \% \\ \hline
+Sync MS with Sync DVFS & 21.83 & 12.78 & 9.05 \\ \hline
+Sync MS with Async DVFS & 18.26 & 7.68 & 10.58 \\ \hline
+Async MS with Sync DVFS & 24.95 & -12.24 & 37.19 \\ \hline
+Async MS with Async DVFS & 10.32 & -17.04 & 27.36 \\ \hline
+\end{tabular}
+\end{table}
+
+\begin{figure}[!h]
+ \centering
+ \includegraphics[scale=0.65]{fig/ch4/compare.eps}
+ \caption{Comparing the trade-off percentages of HSA and EDP methods over the Grid'5000}
+ \label{fig:compare}
+\end{figure}
+
+
+
+\section{Conclusions}
+ \label{ch4:9}
+
+This chapter presents a new online frequency selection algorithm for asynchronous iterative
+applications running over a grid. It selects the best vector of frequencies that maximizes
+the distance between the predicted energy consumption and the predicted execution time.
+The algorithm uses new
+energy and performance models to predict the energy consumption and the execution time of asynchronous or hybrid message passing iterative applications running over grids.
+The proposed algorithm was evaluated twice over the SimGrid simulator and Grid'5000 testbed while running a multi-splitting (MS) application that solves 3D problems.
+The experiments were executed over different
+ grid scenarios composed of different numbers of clusters and different numbers of nodes per cluster.
+ The HSA algorithm was applied synchronously and asynchronously on a synchronous and an asynchronous version of the MS application. Both the simulation and real experiment results show that applying synchronous HSA algorithm on an asynchronous MS application gives the best tradeoff between energy consumption reduction and performance compared to other scenarios.
+In the simulation results, this scenario saves on average the energy consumption by 22\% and reduces the execution time of the application by 5.72\%. This version optimizes both of the dynamic energy consumption by applying synchronously the HSA algorithm at the end of the first iteration and the static energy consumption by using asynchronous communications between nodes from different clusters which are overlapped by computations. The HSA algorithm was also evaluated over three power scenarios. As expected, the algorithm selects different vectors of frequencies for each power scenario. The highest energy consumption reduction was achieved in the power scenario with the highest dynamic power and the lowest performance degradation was obtained in the power scenario with the highest static power.
+The proposed algorithm was compared to another method that
+uses the well known energy and delay product as an objective function.
+The comparison results showed that the proposed algorithm outperforms the latter
+by selecting a vector of frequencies that gives a better trade-off between the energy
+consumption reduction and the performance.
+
+The experiments conducted over Grid'5000 were showed that applying the synchronous HSA algorithm on an asynchronous MS application saves the energy consumption by 26.93\% and reduces the execution time of the application by 21.48\%. On the other hand, these results are better than simulation ones, according to the increase in the heterogeneity level between the clusters of Grid'5000 compared to the simulated grid platform.
\ No newline at end of file
