\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{mathptmx}
+\usepackage{amsmath}
\usepackage{courier}
\usepackage{graphicx}
+\usepackage{url}
+\usepackage[ruled,lined]{algorithm2e}
+
+\newcommand{\abs}[1]{\lvert#1\rvert} % \abs{x} -> |x|
+
+\newenvironment{algodata}{%
+ \begin{tabular}[t]{@{}l@{:~}l@{}}}{%
+ \end{tabular}}
+
+\newcommand{\FIXMEmargin}[1]{%
+ \marginpar{\textbf{[FIXME]} {\footnotesize #1}}}
+\newcommand{\FIXME}[2][]{%
+ \ifx #2\relax\relax \FIXMEmargin{#1}%
+ \else \textbf{$\triangleright$\FIXMEmargin{#1}~#2}\fi}
+
+\newcommand{\VAR}[1]{\textit{#1}}
\begin{document}
-\title{Best effort strategy and virtual load for asynchronous iterative load balancing}
+\title{Best effort strategy and virtual load
+ for asynchronous iterative load balancing}
\author{Raphaël Couturier \and
- Arnaud Giersch \and
- Abderrahmane Sider
+ Arnaud Giersch
}
-\institute{F. Author \at
- first address \\
- Tel.: +123-45-678910\\
- Fax: +123-45-678910\\
- \email{fauthor@example.com} % \\
-% \emph{Present address:} of F. Author % if needed
- \and
- S. Author \at
- second address
+\institute{R. Couturier \and A. Giersch \at
+ FEMTO-ST, University of Franche-Comté, Belfort, France \\
+ % Tel.: +123-45-678910\\
+ % Fax: +123-45-678910\\
+ \email{%
+ raphael.couturier@femto-st.fr,
+ arnaud.giersch@femto-st.fr}
}
\maketitle
\begin{abstract}
Most of the time, asynchronous load balancing algorithms have extensively been
-studied in a theoretical point of view. The Bertsekas and Tsitsiklis' algorithm
+studied in a theoretical point of view. The Bertsekas and Tsitsiklis'
+algorithm~\cite[section~7.4]{bertsekas+tsitsiklis.1997.parallel}
is certainly the most well known algorithm for which the convergence proof is
given. From a practical point of view, when a node wants to balance a part of
its load to some of its neighbors, the strategy is not described. In this
-paper, we propose a strategy called \texttt{best effort} which tries to balance
+paper, we propose a strategy called \emph{best effort} which tries to balance
the load of a node to all its less loaded neighbors while ensuring that all the
nodes concerned by the load balancing phase have the same amount of load.
Moreover, asynchronous iterative algorithms in which an asynchronous load
balancing algorithm is implemented most of the time can dissociate messages
concerning load transfers and message concerning load information. In order to
increase the converge of a load balancing algorithm, we propose a simple
-heuristic called \texttt{virtual load} which allows a node that receives an load
-information message to integrate the load that it will receive latter in its
+heuristic called \emph{virtual load} which allows a node that receives a load
+information message to integrate the load that it will receive later in its
load (virtually) and consequently sends a (real) part of its load to some of its
neighbors. In order to validate our approaches, we have defined a simulator
based on SimGrid which allowed us to conduct many experiments.
\end{abstract}
-
+\section{Introduction}
Load balancing algorithms are extensively used in parallel and distributed
applications in order to reduce the execution times. They can be applied in
no external load. In this context, Bertsekas and Tsitsiklis have proposed an
algorithm which is definitively a reference for many works. In their work, they
proved that under classical hypotheses of asynchronous iterative algorithms and
-a special constraint avoiding \texttt{ping-pong} effect, an asynchronous
+a special constraint avoiding \emph{ping-pong} effect, an asynchronous
iterative algorithm converge to the uniform load distribution. This work has
-been extended by many authors. For example, DASUD propose a version working with
-integer load.
+been extended by many authors. For example, Cortés et al., with
+DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous}, propose a
+version working with integer load. This work was later generalized by
+the same authors in \cite{cedo+cortes+ripoll+al.2007.convergence}.
+\FIXME{Rajouter des choses ici. Lesquelles ?}
+
+Although the Bertsekas and Tsitsiklis' algorithm describes the condition to
+ensure the convergence, there is no indication or strategy to really implement
+the load distribution. In other word, a node can send a part of its load to one
+or many of its neighbors while all the convergence conditions are
+followed. Consequently, we propose a new strategy called \emph{best effort}
+that tries to balance the load of a node to all its less loaded neighbors while
+ensuring that all the nodes concerned by the load balancing phase have the same
+amount of load. Moreover, when real asynchronous applications are considered,
+using asynchronous load balancing algorithms can reduce the execution
+times. Most of the times, it is simpler to distinguish load information messages
+from data migration messages. Former ones allows a node to inform its
+neighbors of its current load. These messages are very small, they can be sent
+quite often. For example, if an computing iteration takes a significant times
+(ranging from seconds to minutes), it is possible to send a new load information
+message at each neighbor at each iteration. Latter messages contains data that
+migrates from one node to another one. Depending on the application, it may have
+sense or not that nodes try to balance a part of their load at each computing
+iteration. But the time to transfer a load message from a node to another one is
+often much more longer that to time to transfer a load information message. So,
+when a node receives the information that later it will receive a data message,
+it can take this information into account and it can consider that its new load
+is larger. Consequently, it can send a part of it real load to some of its
+neighbors if required. We call this trick the \emph{virtual load} mechanism.
+
+
+
+So, in this work, we propose a new strategy for improving the distribution of
+the load and a simple but efficient trick that also improves the load
+balancing. Moreover, we have conducted many simulations with SimGrid in order to
+validate our improvements are really efficient. Our simulations consider that in
+order to send a message, a latency delays the sending and according to the
+network performance and the message size, the time of the reception of the
+message also varies.
+
+In the following of this paper, Section~\ref{BT algo} describes the Bertsekas
+and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we present a
+possible problem in the convergence conditions. Section~\ref{Best-effort}
+presents the best effort strategy which provides an efficient way to reduce the
+execution times. This strategy will be compared with other ones, presented in
+Section~\ref{Other}. In Section~\ref{Virtual load}, the virtual load mechanism
+is proposed. Simulations allowed to show that both our approaches are valid
+using a quite realistic model detailed in Section~\ref{Simulations}. Finally we
+give a conclusion and some perspectives to this work.
+
+
+
+\section{Bertsekas and Tsitsiklis' asynchronous load balancing algorithm}
+\label{BT algo}
+
+In order prove the convergence of asynchronous iterative load balancing
+Bertsekas and Tsitsiklis proposed a model
+in~\cite{bertsekas+tsitsiklis.1997.parallel}. Here we recall some notations.
+Consider that $N={1,...,n}$ processors are connected through a network.
+Communication links are represented by a connected undirected graph $G=(N,V)$
+where $V$ is the set of links connecting different processors. In this work, we
+consider that processors are homogeneous for sake of simplicity. It is quite
+easy to tackle the heterogeneous case~\cite{ElsMonPre02}. Load of processor $i$
+at time $t$ is represented by $x_i(t)\geq 0$. Let $V(i)$ be the set of
+neighbors of processor $i$. Each processor $i$ has an estimate of the load of
+each of its neighbors $j \in V(i)$ represented by $x_j^i(t)$. According to
+asynchronism and communication delays, this estimate may be outdated. We also
+consider that the load is described by a continuous variable.
+
+When a processor send a part of its load to one or some of its neighbors, the
+transfer takes time to be completed. Let $s_{ij}(t)$ be the amount of load that
+processor $i$ has transferred to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
+amount of load received by processor $j$ from processor $i$ at time $t$. Then
+the amount of load of processor $i$ at time $t+1$ is given by:
+\begin{equation}
+x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
+\label{eq:ping-pong}
+\end{equation}
+
+
+Some conditions are required to ensure the convergence. One of them can be
+called the \emph{ping-pong} condition which specifies that:
+\begin{equation}
+x_i(t)-\sum _{k\in V(i)} s_{ik}(t) \geq x_j^i(t)+s_{ij}(t)
+\end{equation}
+for any processor $i$ and any $j \in V(i)$ such that $x_i(t)>x_j^i(t)$. This
+condition aims at avoiding a processor to send a part of its load and being
+less loaded after that.
+
+Nevertheless, we think that this condition may lead to deadlocks in some
+cases. For example, if we consider only three processors and that processor $1$
+is linked to processor $2$ which is also linked to processor $3$ (i.e. a simple
+chain which 3 processors). Now consider we have the following values at time $t$:
+\begin{eqnarray*}
+x_1(t)=10 \\
+x_2(t)=100 \\
+x_3(t)=99.99\\
+ x_3^2(t)=99.99\\
+\end{eqnarray*}
+In this case, processor $2$ can either sends load to processor $1$ or processor
+$3$. If it sends load to processor $1$ it will not satisfy condition
+(\ref{eq:ping-pong}) because after the sending it will be less loaded that
+$x_3^2(t)$. So we consider that the \emph{ping-pong} condition is probably to
+strong. Currently, we did not try to make another convergence proof without this
+condition or with a weaker condition.
+
+Nevertheless, we conjecture that such a weaker condition exists. In fact, we
+have never seen any scenario that is not leading to convergence, even with
+load-balancing strategies that are not exactly fulfilling these two conditions.
+
+It may be the subject of future work to express weaker conditions, and to prove
+that they are sufficient to ensure the convergence of the load-balancing
+algorithm.
+
+\section{Best effort strategy}
+\label{Best-effort}
+
+In this section we describe a new load-balancing strategy that we call
+\emph{best effort}. First, we explain the general idea behind this strategy,
+and then we describe some variants of this basic strategy.
+
+\subsection{Basic strategy}
+
+The general idea behind the \emph{best effort} strategy is that each processor,
+that detects it has more load than some of its neighbors, sends some load to the
+most of its less loaded neighbors, doing its best to reach the equilibrium
+between those neighbors and himself.
+
+More precisely, when a processor $i$ is in its load-balancing phase,
+he proceeds as following.
+\begin{enumerate}
+\item First, the neighbors are sorted in non-decreasing order of their
+ known loads $x^i_j(t)$.
+
+\item Then, this sorted list is traversed in order to find its largest
+ prefix such as the load of each selected neighbor is lesser than:
+ \begin{itemize}
+ \item the processor's own load, and
+ \item the mean of the loads of the selected neighbors and of the
+ processor's load.
+ \end{itemize}
+ Let's call $S_i(t)$ the set of the selected neighbors, and
+ $\bar{x}(t)$ the mean of the loads of the selected neighbors and of
+ the processor load:
+ \begin{equation*}
+ \bar{x}(t) = \frac{1}{\abs{S_i(t)} + 1}
+ \left( x_i(t) + \sum_{j\in S_i(t)} x^i_j(t) \right)
+ \end{equation*}
+ The following properties hold:
+ \begin{equation*}
+ \begin{cases}
+ S_i(t) \subset V(i) \\
+ x^i_j(t) < x_i(t) & \forall j \in S_i(t) \\
+ x^i_j(t) < \bar{x} & \forall j \in S_i(t) \\
+ x^i_j(t) \leq x^i_k(t) & \forall j \in S_i(t), \forall k \in V(i) \setminus S_i(t) \\
+ \bar{x} \leq x_i(t)
+ \end{cases}
+ \end{equation*}
+
+\item Once this selection is completed, processor $i$ sends to each of
+ the selected neighbor $j\in S_i(t)$ an amount of load $s_{ij}(t) =
+ \bar{x} - x^i_j(t)$.
+
+ From the above equations, and notably from the definition of
+ $\bar{x}$, it can easily be verified that:
+ \begin{equation*}
+ \begin{cases}
+ x_i(t) - \sum_{j\in S_i(t)} s_{ij}(t) = \bar{x} \\
+ x^i_j(t) + s_{ij}(t) = \bar{x} & \forall j \in S_i(t)
+ \end{cases}
+ \end{equation*}
+\end{enumerate}
+
+\subsection{Leveling the amount to send}
+
+With the aforementioned basic strategy, each node does its best to reach the
+equilibrium with its neighbors. Since each node may be taking the same kind of
+decision at the same moment, there is the risk that a node receives load from
+several of its neighbors, and then is temporary going off the equilibrium state.
+This is particularly true with strongly connected applications.
+
+In order to reduce this effect, we add the ability to level the amount to send.
+The idea, here, is to make smaller steps toward the equilibrium, such that a
+potentially wrong decision has a lower impact.
+
+Concretely, once $s_{ij}$ has been evaluated as before, it is simply divided by
+some configurable factor. That's what we named the ``parameter $k$'' in
+Section~\ref{Results}. The amount of data to send is then $s_{ij}(t) = (\bar{x}
+- x^i_j(t))/k$.
+\FIXME[check that it's still named $k$ in Sec.~\ref{Results}]{}
+
+\section{Other strategies}
+\label{Other}
+
+Another load balancing strategy, working under the same conditions, was
+previously developed by Bahi, Giersch, and Makhoul in
+\cite{bahi+giersch+makhoul.2008.scalable}. In order to assess the performances
+of the new \emph{best effort}, we naturally chose to compare it to this anterior
+work. More precisely, we will use the algorithm~2 from
+\cite{bahi+giersch+makhoul.2008.scalable} and, in the following, we will
+reference it under the name of Makhoul's.
+
+Here is an outline of the Makhoul's algorithm. When a given node needs to take
+a load balancing decision, it starts by sorting its neighbors by increasing
+order of their load. Then, it computes the difference between its own load, and
+the load of each of its neighbors. Finally, taking the neighbors following the
+order defined before, the amount of load to send $s_{ij}$ is computed as
+$1/(N+1)$ of the load difference, with $N$ being the number of neighbors. This
+process continues as long as the node is more loaded than the considered
+neighbor.
+
+
+\section{Virtual load}
+\label{Virtual load}
+
+In this section, we present the concept of \texttt{virtual load}. In order to
+use this concept, load balancing messages must be sent using two different kinds
+of messages: load information messages and load balancing messages. More
+precisely, a node wanting to send a part of its load to one of its neighbors,
+can first send a load information message containing the load it will send and
+then it can send the load balancing message containing data to be transferred.
+Load information message are really short, consequently they will be received
+very quickly. In opposition, load balancing messages are often bigger and thus
+require more time to be transferred.
+
+The concept of \texttt{virtual load} allows a node that received a load
+information message to integrate the load that it will receive later in its load
+(virtually) and consequently send a (real) part of its load to some of its
+neighbors. In fact, a node that receives a load information message knows that
+later it will receive the corresponding load balancing message containing the
+corresponding data. So if this node detects it is too loaded compared to some
+of its neighbors and if it has enough load (real load), then it can send more
+load to some of its neighbors without waiting the reception of the load
+balancing message.
+
+Doing this, we can expect a faster convergence since nodes have a faster
+information of the load they will receive, so they can take in into account.
+
+\FIXME{Est ce qu'on donne l'algo avec virtual load?}
+
+\FIXME{describe integer mode}
+
+\section{Simulations}
+\label{Simulations}
+
+In order to test and validate our approaches, we wrote a simulator
+using the SimGrid
+framework~\cite{casanova+legrand+quinson.2008.simgrid}. This
+simulator, which consists of about 2,700 lines of C++, allows to run
+the different load-balancing strategies under various parameters, such
+as the initial distribution of load, the interconnection topology, the
+characteristics of the running platform, etc. Then several metrics
+are issued that permit to compare the strategies.
+
+The simulation model is detailed in the next section (\ref{Sim
+ model}), and the experimental contexts are described in
+section~\ref{Contexts}. Then the results of the simulations are
+presented in section~\ref{Results}.
+
+\subsection{Simulation model}
+\label{Sim model}
+
+In the simulation model the processors exchange messages which are of
+two kinds. First, there are \emph{control messages} which only carry
+information that is exchanged between the processors, such as the
+current load, or the virtual load transfers if this option is
+selected. These messages are rather small, and their size is
+constant. Then, there are \emph{data messages} that carry the real
+load transferred between the processors. The size of a data message
+is a function of the amount of load that it carries, and it can be
+pretty large. In order to receive the messages, each processor has
+two receiving channels, one for each kind of messages. Finally, when
+a message is sent or received, this is done by using the non-blocking
+primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
+ and \texttt{MSG\_task\_irecv()}.}.
+
+During the simulation, each processor concurrently runs three threads:
+a \emph{receiving thread}, a \emph{computing thread}, and a
+\emph{load-balancing thread}, which we will briefly describe now.
+
+For the sake of simplicity, a few details were voluntary omitted from
+these descriptions. For an exhaustive presentation, we refer to the
+actual source code that was used for the experiments%
+\footnote{As mentioned before, our simulator relies on the SimGrid
+ framework~\cite{casanova+legrand+quinson.2008.simgrid}. For the
+ experiments, we used a pre-release of SimGrid 3.7 (Git commit
+ 67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
+ \url{https://gforge.inria.fr/scm/?group_id=12})}, and which is
+available at
+\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.
+
+\subsubsection{Receiving thread}
+
+The receiving thread is in charge of waiting for messages to come, either on the
+control channel, or on the data channel. Its behavior is sketched by
+Algorithm~\ref{algo.recv}. When a message is received, it is pushed in a buffer
+of received message, to be later consumed by one of the other threads. There
+are two such buffers, one for the control messages, and one for the data
+messages. The buffers are implemented with a lock-free FIFO
+\cite{sutter.2008.writing} to avoid contention between the threads.
+
+\begin{algorithm}
+ \caption{Receiving thread}
+ \label{algo.recv}
+ \KwData{
+ \begin{algodata}
+ \VAR{ctrl\_chan}, \VAR{data\_chan}
+ & communication channels (control and data) \\
+ \VAR{ctrl\_fifo}, \VAR{data\_fifo}
+ & buffers of received messages (control and data) \\
+ \end{algodata}}
+ \While{true}{%
+ wait for a message to be available on either \VAR{ctrl\_chan},
+ or \VAR{data\_chan}\;
+ \If{a message is available on \VAR{ctrl\_chan}}{%
+ get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
+ }
+ \If{a message is available on \VAR{data\_chan}}{%
+ get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
+ }
+ }
+\end{algorithm}
+
+\subsubsection{Computing thread}
+
+The computing thread is in charge of the real load management. As exposed in
+Algorithm~\ref{algo.comp}, it iteratively runs the following operations:
+\begin{itemize}
+\item if some load was received from the neighbors, get it;
+\item if there is some load to send to the neighbors, send it;
+\item run some computation, whose duration is function of the current
+ load of the processor.
+\end{itemize}
+Practically, after the computation, the computing thread waits for a
+small amount of time if the iterations are looping too fast (for
+example, when the current load is near zero).
+
+\begin{algorithm}
+ \caption{Computing thread}
+ \label{algo.comp}
+ \KwData{
+ \begin{algodata}
+ \VAR{data\_fifo} & buffer of received data messages \\
+ \VAR{real\_load} & current load \\
+ \end{algodata}}
+ \While{true}{%
+ \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
+ wait until a message is pushed into \VAR{data\_fifo}\;
+ }
+ \While{\VAR{data\_fifo} is not empty}{%
+ pop a message from \VAR{data\_fifo}\;
+ get the load embedded in the message, and add it to \VAR{real\_load}\;
+ }
+ \ForEach{neighbor $n$}{%
+ \If{there is some amount of load $a$ to send to $n$}{%
+ send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
+ }
+ }
+ \If{$\VAR{real\_load} > 0.0$}{
+ simulate some computation, whose duration is function of \VAR{real\_load}\;
+ ensure that the main loop does not iterate too fast\;
+ }
+ }
+\end{algorithm}
+
+\subsubsection{Load-balancing thread}
+
+The load-balancing thread is in charge of running the load-balancing algorithm,
+and exchange the control messages. As shown in Algorithm~\ref{algo.lb}, it
+iteratively runs the following operations:
+\begin{itemize}
+\item get the control messages that were received from the neighbors;
+\item run the load-balancing algorithm;
+\item send control messages to the neighbors, to inform them of the
+ processor's current load, and possibly of virtual load transfers;
+\item wait a minimum (configurable) amount of time, to avoid to
+ iterate too fast.
+\end{itemize}
+
+\begin{algorithm}
+ \caption{Load-balancing}
+ \label{algo.lb}
+ \While{true}{%
+ \While{\VAR{ctrl\_fifo} is not empty}{%
+ pop a message from \VAR{ctrl\_fifo}\;
+ identify the sender of the message,
+ and update the current knowledge of its load\;
+ }
+ run the load-balancing algorithm to make the decision about load transfers\;
+ \ForEach{neighbor $n$}{%
+ send a control messages to $n$\;
+ }
+ ensure that the main loop does not iterate too fast\;
+ }
+\end{algorithm}
+
+\paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
+par ex, donner l'idée générale de l'implémentation. l'idée générale est déja décrite en section~\ref{Virtual load}}
+
+\subsection{Experimental contexts}
+\label{Contexts}
+
+In order to assess the performances of our algorithms, we ran our
+simulator with various parameters, and extracted several metrics, that
+we will describe in this section.
+
+\subsubsection{Load balancing strategies}
+
+Several load balancing strategies were compared. We ran the experiments with
+the \emph{Best effort}, and with the \emph{Makhoul} strategies. \emph{Best
+ effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$. Secondly,
+each strategy was run in its two variants: with, and without the management of
+\emph{virtual load}. Finally, we tested each configuration with \emph{real},
+and with \emph{integer} load.
+
+To summarize the different load balancing strategies, we have:
+\begin{description}
+\item[\textbf{strategies:}] \emph{Makhoul}, or \emph{Best effort} with $k\in
+ \{1,2,4\}$
+\item[\textbf{variants:}] with, or without virtual load
+\item[\textbf{domain:}] real load, or integer load
+\end{description}
+%
+This gives us as many as $4\times 2\times 2 = 16$ different strategies.
+
+\subsubsection{End of the simulation}
+
+The simulations were run until the load was nearly balanced among the
+participating nodes. More precisely the simulation stops when each node holds
+an amount of load at less than 1\% of the load average, during an arbitrary
+number of computing iterations (2000 in our case).
+
+Note that this convergence detection was implemented in a centralized manner.
+This is easy to do within the simulator, but it's obviously not realistic. In a
+real application we would have chosen a decentralized convergence detection
+algorithm, like the one described by Bahi, Contassot-Vivier, Couturier, and
+Vernier in \cite{10.1109/TPDS.2005.2}.
+
+\subsubsection{Platforms}
+
+In order to show the behavior of the different strategies in different
+settings, we simulated the executions on two sorts of platforms. These two
+sorts of platforms differ by their underlaid network topology. On the one hand,
+we have homogeneous platforms, modeled as a cluster. On the other hand, we have
+heterogeneous platforms, modeled as the interconnection of a number of clusters.
+
+The clusters were modeled by a fixed number of computing nodes interconnected
+through a backbone link. Each computing node has a computing power of
+1~GFlop/s, and is connected to the backbone by a network link whose bandwidth is
+of 125~MB/s, with a latency of 50~$\mu$s. The backbone has a network bandwidth
+of 2.25~GB/s, with a latency of 500~$\mu$s.
+
+The heterogeneous platform descriptions were created by taking a subset of the
+Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
+ Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
+\texttt{g5k.xml} distributed with SimGrid. Note that the heterogeneity of the
+platform here only comes from the network topology. Indeed, since our
+algorithms currently do not handle heterogeneous computing resources, the
+processor speeds were normalized, and we arbitrarily chose to fix them to
+1~GFlop/s.
+
+Then we derived each sort of platform with four different number of computing
+nodes: 16, 64, 256, and 1024 nodes.
+
+\subsubsection{Configurations}
+
+The distributed processes of the application were then logically organized along
+three possible topologies: a line, a torus or an hypercube. We ran tests where
+the total load was initially on an only node (at one end for the line topology),
+and other tests where the load was initially randomly distributed across all the
+participating nodes. The total amount of load was fixed to a number of load
+units equal to 1000 times the number of node. The average load is then of 1000
+load units.
+
+For each of the preceding configuration, we finally had to choose the
+computation and communication costs of a load unit. We chose them, such as to
+have three different computation over communication cost ratios, and hence model
+three different kinds of applications:
+\begin{itemize}
+\item mainly communicating, with a computation/communication cost ratio of $1/10$;
+\item mainly computing, with a computation/communication cost ratio of $10/1$ ;
+\item balanced, with a computation/communication cost ratio of $1/1$.
+\end{itemize}
+
+To summarize the various configurations, we have:
+\begin{description}
+\item[\textbf{platforms:}] homogeneous (cluster), or heterogeneous (subset of
+ Grid'5000)
+\item[\textbf{platform sizes:}] platforms with 16, 64, 256, or 1024 nodes
+\item[\textbf{process topologies:}] line, torus, or hypercube
+\item[\textbf{initial load distribution:}] initially on a only node, or
+ initially randomly distributed over all nodes
+\item[\textbf{computation/communication ratio:}] $10/1$, $1/1$, or $1/10$
+\end{description}
+%
+This gives us as many as $2\times 4\times 3\times 2\times 3 = 144$ different
+configurations.
+%
+Combined with the various load balancing strategies, we had $16\times 144 =
+2304$ distinct settings to evaluate. In fact, as it will be shown later, we
+didn't run all the strategies, nor all the configurations for the bigger
+platforms with 1024 nodes, since to simulations would have run for a too long
+time.
+
+Anyway, all these the experiments represent more than 240 hours of computing
+time.
+
+\subsubsection{Metrics}
+
+In order to evaluate and compare the different load balancing strategies we had
+to define several metrics. Our goal, when choosing these metrics, was to have
+something tending to a constant value, i.e. to have a measure which is not
+changing anymore once the convergence state is reached. Moreover, we wanted to
+have some normalized value, in order to be able to compare them across different
+settings.
+
+With these constraints in mind, we defined the following metrics:
+%
+\begin{description}
+\item[\textbf{average idle time:}] that's the total time spent, when the nodes
+ don't hold any share of load, and thus have nothing to compute. This total
+ time is divided by the number of participating nodes, such as to have a number
+ that can be compared between simulations of different sizes.
+
+ This metric is expected to give an idea of the ability of the strategy to
+ diffuse the load quickly. A smaller value is better.
+
+\item[\textbf{average convergence date:}] that's the average of the dates when
+ all nodes reached the convergence state. The dates are measured as a number
+ of (simulated) seconds since the beginning of the simulation.
+
+\item[\textbf{maximum convergence date:}] that's the date when the last node
+ reached the convergence state.
+
+ These two dates give an idea of the time needed by the strategy to reach the
+ equilibrium state. A smaller value is better.
+
+\item[\textbf{data transfer amount:}] that's the sum of the amount of all data
+ transfers during the simulation. This sum is then normalized by dividing it
+ by the total amount of data present in the system.
+
+ This metric is expected to give an idea of the efficiency of the strategy in
+ terms of data movements, i.e. its ability to reach the equilibrium with fewer
+ transfers. Again, a smaller value is better.
+
+\end{description}
+
+
+\subsection{Experimental results}
+\label{Results}
+
+In this section, the results for the different simulations will be presented,
+and we'll try to explain our observations.
+
+\subsubsection{Cluster vs grid platforms}
+
+As mentioned earlier, we simulated the different algorithms on two kinds of
+physical platforms: clusters and grids. A first observation that we can make,
+is that the graphs we draw from the data have a similar aspect for the two kinds
+of platforms. The only noticeable difference is that the algorithms need a bit
+more time to achieve the convergence on the grid platforms, than on clusters.
+Nevertheless their relative performances remain generally identical.
+
+This suggests that the relative performances of the different strategies are not
+influenced by the characteristics of the physical platform. The differences in
+the convergence times can be explained by the fact that on the grid platforms,
+distant sites are interconnected by links of smaller bandwith.
+
+Therefore, in the following, we'll only discuss the results for the grid
+platforms. The different results are presented on the
+figures~\ref{fig.results1} and~\ref{fig.resultsN}.
+
+\FIXME{explain how to read the graphs}
+ratio 1:1 not given here
+
+\begin{figure*}[p]
+ \centering
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-line}%
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-line}
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-torus}%
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-torus}
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-hcube}%
+ \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-hcube}
+ \caption{Real mode, initially on an only mode, comp/comm ratio = 1/10 (left), or 10/1 (right).}
+ \label{fig.results1}
+\end{figure*}
+
+\begin{figure*}[p]
+ \centering
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-line}%
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-line}
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-torus}%
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-torus}
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-hcube}%
+ \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-hcube}
+ \caption{Real mode, random initial distribution, comp/comm ratio = 1/10 (left), or 10/1 (right).}
+ \label{fig.resultsN}
+\end{figure*}
+
+\subsubsection{Main results}
+
+On fig.~\ref{fig.results1}, \dots
+
+\subsubsection{With the virtual load extension}
+
+\subsubsection{The $k$ parameter}
+
+\subsubsection{With an initial random repartition, and larger platforms}
+
+\subsubsection{With integer load}
+
+\FIXME{what about the amount of data?}
+
+\begin{itshape}
+\FIXME{remove that part}
+Dans cet ordre:
+...
+- comparer be/makhoul -> be tient la route
+ -> en réel uniquement
+- valider l'extension virtual load -> c'est 'achement bien
+- proposer le -k -> ça peut aider dans certains cas
+- conclure avec la version entière -> on n'a pas l'effet d'escalier !
+Q: comment inclure les types/tailles de platesformes ?
+Q: comment faire des moyennes ?
+Q: comment introduire les distrib 1/N ?
+...
+
+On constate quoi (vérifier avec les chiffres)?
+\begin{itemize}
+\item cluster ou grid, entier ou réel, ne font pas de grosses différences
+
+\item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
+
+\item makhoul? se fait battre sur les grosses plateformes
+
+\item taille de plateforme?
+
+\item ratio comp/comm?
+
+\item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
+
+\item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
+
+\item répartition initiale de la charge ?
+
+\item integer mode sur topo. line n'a jamais fini en plain? vérifier si ce n'est
+ pas à cause de l'effet d'escalier que bk est capable de gommer.
+
+\end{itemize}
+
+% On veut montrer quoi ? :
+
+% 1) best plus rapide que les autres (simple, makhoul)
+% 2) avantage virtual load
+
+% Est ce qu'on peut trouver des contre exemple?
+% Topologies variées
+
+
+% Simulation avec temps définies assez long et on mesure la qualité avec : volume de calcul effectué, volume de données échangées
+% Mais aussi simulation avec temps court qui montre que seul best converge
+
+% Expés avec ratio calcul/comm rapide et lent
+
+% Quelques expés avec charge initiale aléatoire plutot que sur le premier proc
+
+% Cadre processeurs homogènes
+
+% Topologies statiques
+
+% On ne tient pas compte de la vitesse des liens donc on la considère homogène
+
+% Prendre un réseau hétérogène et rendre processeur homogène
+
+% Taille : 10 100 très gros
+\end{itshape}
+
+\section{Conclusion and perspectives}
+
+\FIXME{conclude!}
+\begin{acknowledgements}
+ Computations have been performed on the supercomputer facilities of
+ the Mésocentre de calcul de Franche-Comté.
+\end{acknowledgements}
+\FIXME{find and add more references}
+\bibliographystyle{spmpsci}
+\bibliography{biblio}
\end{document}
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