X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/61214208b8b94c75a87f2d1902473298d8ad6185..e5b137c77dc66ab0403d1b711352d7df14235075:/loba-besteffort/loba-besteffort.tex?ds=inline

diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex
index b0885c1..b95f6d5 100644
--- a/loba-besteffort/loba-besteffort.tex
+++ b/loba-besteffort/loba-besteffort.tex
@@ -56,9 +56,7 @@
 \begin{abstract}
   Most of the time, asynchronous load balancing algorithms are extensively 
   studied from a theoretical point of view. The Bertsekas and Tsitsiklis'
-  algorithm~\cite
-  %[section~7.4]
-  {bertsekas+tsitsiklis.1997.parallel} is undeniably the best known algorithm for which the asymptotic convergence proof is given. 
+  algorithm~\cite{bertsekas+tsitsiklis.1997.parallel} is undeniably the best known algorithm for which the asymptotic convergence proof is given. 
   From a
   practical point of view, when a node needs to balance a part of its load to
   some of its neighbors, the algorithm's description is unfortunately too succinct, and no details are given on what is really sent and how the load balancing decisions are taken. In this paper, we
@@ -68,8 +66,7 @@
   and their variations \cite{bcvc07:bc}, both load transfer and load information messages are dissociated. 
   To speedup the convergence time of the load balancing process, we propose {\it a clairvoyant virtual load} heuristic. This heuristic allows a node receiving a load
   information message to integrate the future virtual load (if any) in its load's list, even if the load has not been received yet. This leads to have predictive snapshots of nodes' loads at each iteration of the load balancing process.  Consequently, the notified node sends a real part of its load to some of
-  its neighbors taking into account the virtual load it will receive in the subsequent time-steps. Based on the SimGrid simulator, some series of test-bed scenarios are considered and many QoS metrics are evaluated to show the usefulness of the proposed algorithm.  %In order to validate our approaches, we have defined a
- % simulator based on SimGrid which allowed us to conduct many experiments.
+  its neighbors taking into account the virtual load it will receive in the subsequent time-steps. Based on the SimGrid simulator, some series of test-bed scenarios are considered and several QoS metrics are evaluated to show the usefulness of the proposed algorithm.
 \end{abstract}
 
 % \begin{keywords}
@@ -91,7 +88,7 @@ polynomial time complexity.
 
 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. 
@@ -99,21 +96,14 @@ 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
@@ -142,7 +132,7 @@ The main contributions and novelties of our work are summarized in the following
 \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}
 
@@ -174,13 +164,12 @@ that characterize a set of negotiation rules for users' jobs. For instance, Izak
 
 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. 
 
-Several LB techniques, based on artificial intelligence, have also been proposed in the literature:  genetic algorithm (GA) \cite{subrata2007artificial}, honey bee behavior \cite{krishna2013honey, kwok2004new},  tabu search  \cite{subrata2007artificial} and fuzzy logic \cite{salimi2014task}. The main strength of these techniques comes from their ability to seek in large search spaces, which arises in many combinatorial optimization problems. For instance, the works in~\cite{cao2005grid, shen2014achieving} have been proposed to tackle the load balancing problem using the multiagent approach where each agent is responsible for load balancing for a subset of nodes in the network. The agent objective is to minimize jobs' response time and host idle time dynamically. In~\cite{GrosuC05}, the authors formulate the load balancing problem as a non-cooperative game among users. They use the Nash equilibrium as the solution of this game to optimize the response time of all jobs in the entire system. The proposed scheme guarantees the optimal task allocation for each user with low time complexity. A game theoretic approach to tackle the static load balancing problem is also investigated in~\cite{PenmatsaC11}. To provide fairness to all users in  the system, the load balancing problem is formulated as a non-cooperative game among the users to minimize the response time of the submitted users' jobs. As in~\cite{GrosuC05}, the authors use the concept of Nash equilibrium as the solution of a non-cooperative game. Simulation results show that the proposed scheme offers near optimal solutions compared to other existing techniques in terms of fairness.
+Several LB techniques, based on artificial intelligence, have also been proposed in the literature:  genetic algorithm (GA) \cite{subrata2007artificial}, honey bee behavior \cite{krishna2013honey, kwok2004new},  tabu search  \cite{subrata2007artificial} and fuzzy logic \cite{salimi2014task}. The main strength of these techniques comes from their ability to seek in large search spaces, which arises in many combinatorial optimization problems. For instance, the works in~\cite{cao2005grid, shen2014achieving} have been proposed to tackle the load balancing problem using the multi-agent approach where each agent is responsible for load balancing for a subset of nodes in the network. The agent objective is to minimize jobs' response time and host idle time dynamically. In~\cite{GrosuC05}, the authors formulate the load balancing problem as a non-cooperative game among users. They use the Nash equilibrium as the solution of this game to optimize the response time of all jobs in the entire system. The proposed scheme guarantees the optimal task allocation for each user with low time complexity. A game theoretic approach to tackle the static load balancing problem is also investigated in~\cite{PenmatsaC11}. To provide fairness to all users in  the system, the load balancing problem is formulated as a non-cooperative game among the users to minimize the response time of the submitted users' jobs. As in~\cite{GrosuC05}, the authors use the concept of Nash equilibrium as the solution of a non-cooperative game. Simulation results show that the proposed scheme offers near optimal solutions compared to other existing techniques in terms of fairness.
 
 
 
@@ -193,31 +182,14 @@ A network is modeled as a connected undirected graph $G=(N,A)$, where $N$ is a s
 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
@@ -263,9 +235,8 @@ is linked to processor $2$ which is  also linked to processor $3$, but in which
 \end{align*}
 %{\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
-$3$.  If it sends loads to processor $1$, it will not satisfy condition
+Owing to the algorithm's specifications, processor $2$ can either send a part of its load to processor $1$ or processor
+$3$.  If it sends 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
 strong. %Currently, we did not try to make another convergence proof without this condition or with a weaker condition.
@@ -391,36 +362,11 @@ Section~\ref{sec.results}.  The amount of data to send is then $s_{ij}(t) =
 
 
 
-%\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 naÃ¯ve implementation of Bertsekas' load balancing algorithm.  {\bf : RAPH j'ai renommÃ© 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
@@ -437,30 +383,12 @@ neighbors when needed. By and large, this allows a node on the one hand, to pred
 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}
 
-In order to test and validate our approache, we wrote a simulator
+In order to test and validate our approach, we wrote a simulator
 using the SimGrid
 framework~\cite{simgrid.web,casanova+giersch+legrand+al.2014.simgrid}.  This
 simulator, which consists of about 2,700 lines of C++, allows to run
@@ -635,38 +563,16 @@ To summarize the different load balancing strategies, we have:
 \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
@@ -701,23 +607,6 @@ two different types of applications:
 %\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}
@@ -731,30 +620,16 @@ settings. With these constraints in mind, the following metrics are defined:
 \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}
 
@@ -765,22 +640,7 @@ settings. With these constraints in mind, the following metrics are defined:
 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}
 
@@ -815,16 +675,12 @@ The results in Figure~\ref{fig.results1} are when the load to balance is
 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
@@ -832,7 +688,6 @@ because the algorithm did not reach the convergence state in the
 allocated time.
 
 
-%\FIXME{annoncer le plan de la suite}
 
 \subsubsection{The \besteffort{} and  \makhoul{} strategies without virtual load}
 
@@ -877,7 +732,7 @@ load fairly when needed and is better for sparse connected applications.
 \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,
@@ -892,12 +747,12 @@ that for small CCR values,  high communication costs play quite a significant ro
 communication becomes less important as the CCR values increase, since larger CCR values result in smaller communication times. The impact of CCR values were also tested on the performance of each algorithm in terms of idle times. From Figures~\ref{fig.results1} and ~\ref{fig.resultsN} virtual load scheme can be seen to achieve really good  average idle times, which is quite close to both its own simple version and its direct competitor {\it Bertsekas and Tsitsiklis} algorithm.  As expected, for coarse grain applications (CCR =10/1), idle times are close to 0 since processors are inactive most of the time compared to fine grain applications. 
 
 \smallskip 
-Taken as a whole, the results illustrated in Figures~\ref{fig.results1} and ~\ref{fig.resultsN} clearly show that our proposal outperforms the Betsekas and Tsistlikis algorithm. 
+Taken as a whole, the results illustrated in Figures~\ref{fig.results1} and ~\ref{fig.resultsN} clearly show that our proposal outperforms the Bertsekas and Tsitsiklis algorithm. 
 These results indicate that local load balancing decisions have a significant impact on the global 
-convergence time achieved by the compared strategies. This is because, upon load imbalance detection, assigning  an amount of load in an unfair way between neighbors will severely increase the total number of iterations required by the algorithm before reaching the final stable distributions. The reason of the poorer performance of {\it Bertsekas and tsistsilikis} algorithm  can be explained by the inconvenience of the iterative load balance policy adopted for load distribution between neighbors. Neighbors are selected in such a way that the {\it ping-pong} condition holds. Doing so, loads are not really assigned to processor neighbors which would allow them to be fairly balanced.  
+convergence time achieved by the compared strategies. This is because, upon load imbalance detection, assigning  an amount of load in an unfair way between neighbors will severely increase the total number of iterations required by the algorithm before reaching the final stable distributions. The reason of the poorer performance of {\it Bertsekas and Tsitsiklis} algorithm  can be explained by the inconvenience of the iterative load balance policy adopted for load distribution between neighbors. Neighbors are selected in such a way that the {\it ping-pong} condition holds. Doing so, loads are not really assigned to processor neighbors which would allow them to be fairly balanced.  
 
 \smallskip 
-Unlike the {\it Betsekas and Tsistlikis} algorithm, our approach is not really sensitive when dealing with realistic models of computation and communication. This is due to two main features: i) the use of "virtual load" transfers which allows nodes to predict the load they receive in the subsequent iterations steps, ii) and the greedy neighbors selection adopted by our algorithm at each time step in the load balancing process. The involved neighbors are selected in such a way that the load difference between the computational resources is minimized as much as possible. 
+Unlike the {\it Bertsekas and Tsitsiklis} algorithm, our approach is not really sensitive when dealing with realistic models of computation and communication. This is due to two main features: i) the use of "virtual load" transfers which allows nodes to predict the load they receive in the subsequent iterations steps, ii) and the greedy neighbors selection adopted by our algorithm at each time step in the load balancing process. The involved neighbors are selected in such a way that the load difference between the computational resources is minimized as much as possible. 
 
 \smallskip 
 Comparing the results of the extended version (with virtual load) to the results of the simple one, it can be observed in Figs.~\ref{fig.results1} and ~\ref{fig.resultsN} that the improved version gives the best performances. It always improves both convergence and idle times significantly in all figures. This is because, with virtual load transfers, the algorithm seeks greedily to ensure a certain degree of load balancing for processors by taking into account the information about the predictive loads not received yet. Consequently, this leads to optimizing the final convergence time of the load balancing process. Similarly, the extended version achieves much better results than the simple one when considering larger platforms, as shown in Figs.~\ref{fig.results1} and ~\ref{fig.resultsN}.
@@ -937,7 +792,7 @@ the virtual  load, most  of the time,  processors converge  to what is  called t
 obtain 10 9 8 7 6 6 7 8 9 10 instead of 8 8 8 8 8 8 8 8 8 8).
 
 \smallskip 
-To summarize  the simulation results led us to show that, with a few exceptions (without virtual load), our proposal is superior to the {\it Bertsekas and Tsiltsikis} algorithm in all the tested scenarios. The illustrated results indicate that network size, CCR values and initial load distribution have a significant impact on the algorithm's performances. Thus, this experimental study corroborates the usefulness of our algorithm, and confirms that when dealing with realistic model platforms, both  {\it best effort} strategy and {\it virtual load} transfers play an important role on the achieved idle and convergence times.
+To summarize  the simulation results led us to show that, with a few exceptions (without virtual load), our proposal is superior to the {\it Bertsekas and Tsitsiklis} algorithm in all the tested scenarios. The illustrated results indicate that network size, CCR values and initial load distribution have a significant impact on the algorithm's performances. Thus, this experimental study corroborates the usefulness of our algorithm, and confirms that when dealing with realistic model platforms, both  {\it best effort} strategy and {\it virtual load} transfers play an important role on the achieved idle and convergence times.
 
 
 
@@ -958,21 +813,8 @@ ANR-11-LABX-01-01).  We also thank the supercomputer facilities of the MÃ©socent
  
 \bibliographystyle{elsarticle-num}
 \bibliography{biblio}
-%\FIXME{find and add more references}
+
 
 \end{document}
 
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-% LocalWords:  RaphaÃ«l Couturier Arnaud Giersch Franche ij Bertsekas Tsitsiklis
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+
