% can be deleted if we need space, we can just say we are interested in this
% paper in homogeneous clusters}
-\subsection{The performance of parallel tasks on heterogeneous cluster}
+\subsection{The execution time of message passing distributed iterative applications on a heterogeneous platform}
+In this paper, we are interested in reducing the energy consumption of message passing distributed iterative synchronous applications running over heterogeneous platforms. We define a heterogeneous platform as a collection of heterogeneous computing nodes interconnected via a high speed homogeneous network. Therefore, each node has different characteristics such as computing power (FLOPS), energy consumption, cpu's frequency range, ... but they all have the same network bandwidth and latency.
-The heterogeneous cluster is a collection of non identical computing nodes. Each node in
-a cluster is connected via a high speed network. The communication capabilities between nodes
-are identical or different. In this work we are interested in identical communications. While each
-node has different processing capabilities such as CPU speeds and memory. Tasks executed
-on this model can be either synchronous or asynchronous. In this paper we are consider execution of
-the synchronous tasks on distributed heterogeneous platform. These tasks can exchange
-the data via synchronous message passing.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{fig/commtasks}
- \caption{Parallel tasks on heterogeneous platform}
+ \caption{Parallel tasks on a heterogeneous platform}
\label{fig:heter}
\end{figure}
- Therefore, the execution time of a task consists of the computation time and
- the communication time. Due to heterogeneous computations can lead to slack times while the tasks
- wait at the synchronization barrier for other tasks to finish their jobs (see Figure~(\ref{fig:heter})).
- In this case the fastest tasks have to wait at the synchronization barrier for the slowest ones to begin
- the next task. Therefore, the overall execution time of the program is the execution time of the slowest
- task as in EQ~(\ref{eq:T1}).
-\begin{equation}
- \label{eq:T1}
- \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
-\end{equation}
- where $T_i$ is the execution time of the task $i$ and all the tasks are executed concurrently on different processors. DVFS is a process that is allowed in modern processors to reduce the dynamic
-power by scaling down the voltage and frequency. Then any DVFS operation used to reduce energy of the processor has direct affect on the execution time of the MPI program. The reduction process of the frequency can be expressed by the scaling factor S which is the ratio between the maximum and the new frequency as in EQ (\ref{eq:s}).
+ The overall execution time of a distributed iterative synchronous application over a heterogeneous platform consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur when fast nodes have to
+ wait, during synchronous communications, for the slower nodes to finish their computations (see Figure~(\ref{fig:heter})).
+ Therefore, the overall execution time of the program is the execution time of the slowest
+ task which have the highest computation time and no slack time.
+
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
+of a CPU by scaling down its voltage and frequency. Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound. The frequency reduction process can be expressed by the scaling factor S which is the ratio between the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
- The execution time of a parallel program is linearly proportional to the frequency scaling factor $S$.
- However, in most MPI applications the processes exchange data. During these communications the
- processors involved remain idle until the communications are finished. For that reason, any change in
- the frequency has no impact on the time of communication~\cite{17}. The communication time for a task is the summation of periods
- of time that begin with an MPI call for sending or receiving a message till the message is synchronously sent or received.
- Each node has different DVFS features such as frequency values and the number of available frequencies
- (Pstates) for each node. By contrast there are different frequency scaling factors for each node $S_1, S_2,..., S_N$. To be able to predict the execution time of MPI program, the communication time and the computation time for the slowest
- task must be measured before scaling. These times are used to predict the execution time for any MPI program running on heterogeneous cluster as a function
- of the new scaling factors as in EQ (\ref{eq:perf}). The model is computes the maximum production of computation time
- with scaling factor from each node added to the minimum communication time of the slowest node, it means only the
- communication time without slack times, because in MPI the slack times is measured with communication times.
+ The execution time of a compute bound sequential program is linearly proportional to the frequency scaling factor $S$.
+ On the other hand, message passing distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but the communication time is not affected by the scaling factor because the processors involved remain idle during the communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving a message till the message is synchronously sent or received.
+
+Since in a heterogeneous platform, each node has different characteristics, especially different frequency gears, when applying DVFS operations on these nodes, they may get different scaling factors represented by a scaling vector: $(S_1, S_2,..., S_N)$ where $S_i$ is the scaling factor of processor $i$. To be able to predict the execution time of message passing synchronous iterative applications running over a heterogeneous platform, for different vectors of scaling factors, the communication time and the computation time for all the
+ tasks must be measured during the first iteration before applying any DVFS operation. Then the execution time for one iteration of the application with any vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+
+
+
\begin{multline}
\label{eq:perf}
- \textit T_\textit{new} = \\
- {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +
- \min_{i=1,2,\dots,N} Tcm Old_{i}
+ \textit T_\textit{new} =
+ {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) + TcmOld_{j}
\end{multline}
-This prediction modal is developed from our model for predicting the execution time of parallel task on homogeneous architecture~\cite{45}. The execution time predicting model is useful to used in our method for optimizing both energy and performance of iterative methods as in the coming sections.
+where $TcpOld_i$ is the computation time of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
+ The model computes the maximum computation time
+ with scaling factor from each node added to the communication time of the slowest node, it means only the
+ communication time without any slack time.
+
+This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
\subsection{Energy model for heterogeneous platform}
Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
two power metrics: the static and the dynamic power. While the first one is
-consumed as long as the computing unit is on, the latter is only consumed during
+consumed as long as the computing unit is turned on, the latter is only consumed during
computation times. The dynamic power $P_{d}$ is related to the switching
activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
operational frequency $F$, as shown in EQ(\ref{eq:pd}).
\end{equation}
where $T$ is the execution time of the program, $T_{cp}$ is the computation
time and $T_{cp} \leq T$. $T_{cp}$ may be equal to $T$ if there is no
-communication, no slack time and no synchronization.
+communication and no slack time.
The main objective of DVFS operation is to
reduce the overall energy consumption~\cite{37}. The operational frequency $F$
voltage with respect to various frequency values in~\cite{3}. The reduction
process of the frequency can be expressed by the scaling factor $S$ which is the
ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-The value of the scaling factor $S$ is greater than 1 when changing the
-frequency of the CPU to any new frequency value~(\emph{P-state}) in the
-governor. The CPU governor is an interface driver supplied by the operating
+The CPU governors are power schemes supplied by the operating
system's kernel to lower a core's frequency. we can calculate the new frequency
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
\label{eq:fnew}
F_\textit{new} = S^{-1} . F_\textit{max}
\end{equation}
-By substituting this equation in EQ(\ref{eq:pd}) results the following equation for dynamic
+Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic
power consumption:
\begin{multline}
\label{eq:pdnew}
- {P}_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
- = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{d} \cdot S^{-3}
+ {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
+ = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduce by a factor of $S^{-3}$ when
-reducing the frequency by a factor of $S$~\cite{3}. The dynamic energy is the energy consumed by a CPU when its in the computation mode.
-So, the dynamic energy is the dynamic power multiply by the time of computations. While the time of computation is decreased by a factor of $S$. Therefore the
-the dynamic energy is decreased by a factor of $S^{-2}$ as follow:
+where $ {P}_\textit{dNew}$ and $P_{dOld}$ are the dynamic power consumed with the new frequency and the maximum frequency respectively.
+
+According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when
+reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$. The new dynamic energy is the dynamic power multiplied by the new time of computation and is given by the following equation:
\begin{equation}
\label{eq:Edyn}
- E_\textit{d} = P_{d} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{d} \cdot Tcp
+ E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot T_{cp}
\end{equation}
-The static power is related to leakage power consumption, its mean the CPU continue consumes energy
-whereas in idle state. Therefore, we are make an assumption that the static power is constant as in~\cite{3,46}.
-The static energy is the static power multiply by the execution time of the program. Moreover, the CPU consumes static
-energy in all times of the program such as computation, communication and slacks times. According to the execution time model in EQ(\ref{eq:perf}),
-the execution time of the program is the summation of the computation and the communication times. The computation time is related
-to frequency scaling factor linearly, while this scaling factor not affecting on the time of communication~\cite{17}, then the static energy
-of individual processor is as follow:
+The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies.
+The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}),
+the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related
+to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy
+of a processor after scaling its frequency is computed as follows:
\begin{equation}
\label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S + Tcm)
+ E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S + T_{cm})
\end{equation}
-In heterogeneous architecture there is a number of different processors $P_1, P_2,...,P_N$, where $N$ is the number of nodes . Moreover, each processor perhaps has different frequency scaling factor, so there are a set of frequency scaling factors for such platform $S_1,S_2,...,S_N$. According to these different
-scaling factors producing different computation time, $Tcp_1,Tcp_2,...,Tcp_N$, because these times linearly related to these scaling factors. In MPI program the communication times is measured with slacks times. The slack times also has linear relation with the scaling factors. So, there are different mesured communication times $Tcm_1,Tcm_2,...,Tcm_N$ even if its identical communications, e.g. see figure(\ref{fig:heter}). The energy modal of an heterogeneous architecture represents the summation of all dynamic and static energies from each processors, each processor has its dynamic and static powers, for example, in the hole architecture their are: $Pd_1,Pd_2,...,Pd_N$ and $Ps_1,Ps_2,...,Ps_N$. The dynamic energy is computes as in EQ(\ref{eq:Edyn}) with regarding to the frequency scaling factor and the dynamic power of each node. While the static energy is computes using EQ(\ref{eq:perf}) multiplied by the static power of each processor. So, the energy modal of an heterogeneous platform has the following form:
+
+In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing distributed application executed over a heterogeneous platform is the summation of all dynamic and static energies for each processor. It is computed as follows:
\begin{multline}
\label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +\\
- {}\sum_{i=1}^{N} {(Ps_i \cdot (\max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +}
- {}\min_{i=1,2,\dots,N} {Tcm_{i}))}
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot T_{cpi})} +\\
+ {}\sum_{i=1}^{N} {(P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +}
+ {}\min_{i=1,2,\dots,N} {T_{cmi}))}
\end{multline}
-These set of frequency scaling factors $S_i$ reduce quadratically the dynamic power which may cause degradation in performance and thus, the
-increase of the static energy because the execution time is increased~\cite{36}.
+Reducing the the frequencies of the processors according to the vector of scaling factors $(S_1, S_2,..., S_N)$ may degrade the performance of the application and thus,
+increase the static energy because the execution time is increased~\cite{36}.
\section{Optimization of both energy consumption and performance}
\label{sec.compet}
\begin{figure}
\centering
\subfloat[Homogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/homo.eps}\label{fig:r1}}%
+ \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
\qquad%
\subfloat[Heterogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/heter.eps}\label{fig:r2}}
+ \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
\label{fig:rel}
\caption{The energy and performance relation}
\end{figure}
\begin{figure}
\centering
\subfloat[Balanced nodes type scenario]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_eq.eps}\label{fig:avg_eq}}%
+ \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
\quad%
\subfloat[Imbalanced nodes type scenario]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_neq.eps}\label{fig:avg_neq}}
+ \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
\label{fig:avg}
\caption{The average of energy and performance for all Nas benchmarks running with difference number of nodes}
\end{figure}
