In this paper, we focus on asynchronous load balancing of non negative real numbers of {\it divisible loads}
in homogeneous distributed systems. Loads can be divided in arbitrary {\it fine-grain} parallel parts size
-that can be processed independently of each other. This model of divisible loads arises in
+that can be processed independently of each other~\cite{Bharadwaj1996, Drozdowski1998, Casanova2008}. This model of divisible loads arises in
a wide range of real-world applications. Common examples, among many, include signal processing,
feature extraction and edge detection in image processing, records search in huge databases,
average consensus in WSN, pattern search in Big data and so on.
In the literature, the problem of load balancing has been formulated and studied in various ways. The first pioneering work is due to Bertsekas and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. Under some specific hypothesis and {\it ping-pong} awareness conditions (see section~\ref{sec.bt-algo} for more details), an asymptotic convergence proof is derived.
-%%RAPH Attention cette partie n'apparait plus
-\begin{comment}
-This algorithm has been borrowed and adapted in many works. For instance, in~\cite{CortesRCSL02} a static load balancing (called DASUD) for non negative integer number of divisible loads in arbitrary networks topologies is investigated. The term {\it "static"} stems from the fact that no loads are added or consumed during the load balancing process. The theoretical correctness proofs of the convergence property are given. Some generalizations of the same authors' own work for partially asynchronous discrete load balancing model are presented in~\cite{cedo+cortes+ripoll+al.2007.convergence}. The authors prove that the algorithm's convergence is finite and bounded by the straightforward network's diameter of the global equilibrium threshold in the network. In~\cite{bahi+giersch+makhoul.2008.scalable}, a fault tolerant communication version is addressed to deal with average consensus in wireless sensor networks. The objective is to have all nodes converged to the average of their initial measurements based only on nodes' local information. A slight adaptation is also considered in~\cite{BahiCG10} for dynamic networks with bounded delays asynchronous diffusion. The dynamical aspect stands at the communication level as links between the network's resources may be intermittent.
-\end{comment}
Although Bertsekas and Tsitsiklis describe the necessary conditions to
ensure the algorithm's convergence, there is no indication nor any strategy to really implement
-the load distribution. %In other word, a node can send some amount of its load to one or many of its neighbors while all the convergence conditions are followed.
+the load distribution.
Consequently, we propose a new strategy called \besteffort{}
that tries to balance the load of a node to all its less loaded neighbors while
ensuring that all the nodes involved in the load balancing phase have the same
-amount of load. Moreover, %when real-world asynchronous applications are considered,
-%using asynchronous load balancing algorithms can reduce the execution
-%times.
-most of the time, it is simpler to dissociate load information messages
+amount of load. Moreover, most of the time, it is simpler to dissociate load information messages
from data migration messages. Former ones allow a node to inform its
neighbors about its current load. These messages are in fact very small and can often be sent
very quickly. For example, if a computing iteration takes a significant time
\item Unlike earlier works, we use a new concept of virtual loads transfer which allows nodes to predict the future loads they will receive in the subsequent iterations.
This leads to a noticeable speedup of the global convergence time of the load balancing process.
-\item We use SimGrid simulator which is known to be able to characterize and modelize realistic models of computation and communication in different types of platforms. We show that taking into account both loads transfers' costs and network contention is essential and has a real impact on the quality of the load balancing performances.
+\item We use SimGrid simulator which is known to be able to characterize and model realistic models of computation and communication in different types of platforms. We show that taking into account both loads transfers' costs and network contention is essential and has a real impact on the quality of the load balancing performances.
\end{itemize}
Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
approaches combine a bidding approach for task selection and a consensus procedure scheme for
-decentralized conflict resolution. The developed algorithms are proven to converge to a conflict-free assignment in
-both single and multiple task assignment problems. An online stochastic dual gradient LB algorithm, which is called DGLB, is proposed in~\cite{chen2017dglb}. The authors deal with both workload and energy management for cloud networks consisting of multiple geo-distributed mapping nodes and data Centers. To enable online distributed implementation, tasks are decomposed both across time and space by leveraging a dual decomposition approach. Experimental results corroborate the merits of the proposed algorithm.
+decentralized conflict resolution. The developed algorithms are proven to converge to a conflict-free assignment in both single and multiple task assignment problem. An online stochastic dual gradient LB algorithm, which is called DGLB, is proposed in~\cite{chen2017dglb}. The authors deal with both workload and energy management for cloud networks consisting of multiple geo-distributed mapping nodes and data Centers. To enable online distributed implementation, tasks are decomposed both across time and space by leveraging a dual decomposition approach. Experimental results corroborate the merits of the proposed algorithm.
In~\cite{tripathi2017non} a LB algorithm based on game theory is proposed for distributed data centers. The authors formulate the LB problem as a non-cooperative game among front-end proxy servers and characterize the structure of Nash equilibrium. Based on the obtained Nash equilibrium structure, they derive a LB algorithm to compute the Nash equilibrium. They show through simulations that the proposed algorithm ensures fairness among the users and a good average latency across all client regions. A hybrid task scheduling and load balancing dependent and independent tasks for master-slaves platforms are addressed in~In~\cite{liu2017dems}. To minimize the response time of the submitted jobs, the proposed algorithm which is called DeMS is split into three stages: i) communication overhead reduction between masters and slaves, ii) task migration to keep the workload balanced iii) and precedence task graphs partitioning.
of processors and $A$ is a set of communication links. The processors are
labeled $i = 1,...,n$, and a link between processors $i$ and
$j$ is denoted by $(i, j)\in A$. The set of processor $i$'s neighbors is denoted by $V(i)$.
-%In this work, we consider that
-%Processors are considered to be homogeneous for the sake of simplicity. It is easily extendable to the case of heterogeneous platforms by scaling the processor's load by its computing power~\cite{ElsMonPre02}.
-%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 $G=(N,A)$
-%where $A$ 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 $j \in V(i)$ denoted by $x_j^i(t)$ and this estimate
may be outdated due to %. According to
asynchronism and communication delays.
-%, this estimate may be outdated.
-%We also
-%consider that the load is described by a continuous variable.
-
-%Since we deal with large {\it fine grain} parallelism of divisible loads,
-%the processor's load is represented by a continuous variable for notational
-%convenience.
+
\medskip
When a processor sends a part of its load to one or to some of its neighbors, the
transfer takes time to be completed. Let $s_{ij}(t)$ be the amount of load that
%{\bf RAPH, pourquoi il y a $x_3^2$?. Sinon il faudra reformuler la suite, c'est mal dit}
Owing to the algorithm's specifications, processor $2$ can either send
-loads to processor $1$ or processor
+a load to processor $1$ or processor
$3$. If it sends loads to processor $1$, it will not satisfy condition
\eqref{eq.ping-pong} because after that sending it will be less loaded than
$x_3^2(t)$. So we consider that the \emph{ping-pong} condition is probably too
-%\section{Other strategies}
-%\label{sec.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 \besteffort{}, 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 naïve implementation of Bertsekas' load balancing algorithm. {\bf : RAPH j'ai renommé MAKHOUL en naive, il faut valider !!!! LE SOUCI, il faudrait refaire les figures}
-
-%Here is an outline of the \makhoul{} 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{sec.virtual-load}
In this section, we present the new concept of \emph{virtual load} which aims to improve the global convergence time. For this end, both load transfer messages and load information messages are dissociated.
-%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 some amount of its load to one (or more) of its neighbors
+More precisely, a node wanting to send some amount of its load to one (or more) of its neighbors
can first send a load information message about the load it will send, and
later it can send the load message containing data to be transferred.
Load information messages are in fact short
into account the information about the predictive loads not
received yet.
-% repetition !
-%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 it into account.
-
-%\FIXME{Est ce qu'on donne l'algo avec virtual load?}
-
-%With integer load, this algorithm has been adapted by rounding the load value. In fact, we consider that the total amount of load is big enough and that it can be split with integer numbers.
-
-
-%\FIXME{describe integer mode}
-
\section{Implementation with SimGrid and simulations}
\label{sec.simulations}
\item[\textbf{variants:}] with, or without virtual loads
%\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 reaching the global equilibrium threshold.
-%the load was nearly balanced among the participating nodes.
+
More precisely, the simulation stops when each node holds
an amount of load at least inferior to 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 is obviously not realistic. In a
-%real application we would have chosen a decentralized convergence detection
-%algorithm, like the one described in \cite{ccl09:ij}.
\subsubsection{Platform}
-%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 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 are 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.
In order to make our experiments, an heterogeneous grid platform description were created by taking a subset of the
Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
%\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 cost 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, $16\times 144 =
-% 2,304$ distinct settings have been evaluated. In fact, as it will be shown later, only configurations with a maximum number of 1,024 nodes are considered in order to limit the time of experiments.
-
\subsubsection{Metrics}
\label{sec.metrics}
\begin{description}
\item[\it{average idle time:}] that is the total time spent, when the nodes
do not 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.
+ A smaller value is better.
\item[\it{average convergence time:}] that is the average of the times when
all nodes reached the final balanced load distribution. Times are measured as a number
of (simulated) seconds from the beginning of the simulation.
\item[\it{maximum convergence time:}] that is the time when the last node
- reached the final stable equilibrium.
- %These two dates give an idea of the time needed by the strategy to reach the
- %equilibrium state.
- A smaller value is better.
+ reached the final stable equilibrium. A smaller value is better.
-% \item[\textbf{data transfer amount:}] that is 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}
In this section, the results for the different simulations are presented,
and our observations are explained.
-% \subsubsection{Cluster versus grid platforms}
-
-% As mentioned earlier, different algorithms have been simulated on two kinds of
-% physical platforms: clusters and grids. A first observation,
-% 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 similar.
-
-% 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 bandwidth.
-% Therefore, in the following, we only discuss the results for the grid
-% platforms.
\subsubsection{Main results}
initially on only one node, while the results in Figure~\ref{fig.resultsN} are
when the load to balance is initially randomly distributed over all nodes.
On both figures, the CCR is $10/1$ on the left
-column, and $1/10$ on the right column. %With a computation/communication cost
-%ratio of $1/1$ the results are just between these two extrema, and definitely
-%don not give additional information, so we chose not to show them here.
+column, and $1/10$ on the right column.
On each Figure, ~\ref{fig.results1} and~\ref{fig.resultsN}, the results
are given for the process topology being, from top to bottom, a line, a torus or
an hypercube.
-Finally, the vertical bars show the measured times for the evaluated metrics
-%each of the algorithms
-. These measured times are, starting at $t=0$ and from bottom to top, the average idle
+Finally, the vertical bars show the measured times for the evaluated metrics. These measured times are, starting at $t=0$ and from bottom to top, the average idle
time, the average convergence time, and the maximum convergence time (see
Section~\ref{sec.metrics}). The measurements are repeated for the different
platform sizes. Some bars are missing, especially for large platforms. This is
allocated time.
-%\FIXME{annoncer le plan de la suite}
\subsubsection{The \besteffort{} and \makhoul{} strategies without virtual load}
\subsubsection{With virtual load}
The impact of virtual load scheme is most of the time really significant compared to
-the simple version of the algorithm with the same configuration. %Sometimes it has no effect but, based on our observations, it has never a negative effect on the load balancing we tested.
+the simple version of the algorithm with the same configuration.
For instance, as can be seen from Figure~\ref{fig.results1}, when the load is initially on one node, it can be
noticed that the average idle times are generally longer with the virtual load
than the simple version. This can be explained by the fact that, with virtual load,
\bibliographystyle{elsarticle-num}
\bibliography{biblio}
-%\FIXME{find and add more references}
+
\end{document}
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% fill-column: 80
-%%% ispell-local-dictionary: "american"
-%%% End:
-
-% LocalWords: Raphaël Couturier Arnaud Giersch Franche ij Bertsekas Tsitsiklis
-% LocalWords: SimGrid DASUD Comté asynchronism ji ik isend irecv Cortés et al
-% LocalWords: chan ctrl fifo Makhoul GFlop xml pre FEMTO Makhoul's fca bdee
-% LocalWords: cdde Contassot Vivier underlaid du de Maréchal Juin cedex calcul
-% LocalWords: biblio Institut UMR Université UFC Centre Scientifique CNRS des
-% LocalWords: École Nationale Supérieure Mécanique Microtechniques ENSMM UTBM
-% LocalWords: Technologie Bahi
+
