[sharelatex-git-integration Best effort strategy and virtual load for asynchronous...

[loba-papers.git] / loba-besteffort / loba-besteffort.tex

diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex
index b95f6d5ee7dc9378b47c208a43c1cd71c37f7536..5983308c6f5b65a59ee7fa7d9ad528612b780607 100644 (file)
--- a/loba-besteffort/loba-besteffort.tex
+++ b/loba-besteffort/loba-besteffort.tex
@@ -34,7 +34,7 @@
 
 \begin{frontmatter}
 
 
 \begin{frontmatter}
 

-\journal{Parallel Computing}
+\journal{Journal of Computational Science}

 
 \title{Best effort strategy and virtual load for\\
   asynchronous iterative load balancing}
 
 \title{Best effort strategy and virtual load for\\
   asynchronous iterative load balancing}


@@ -59,14 +59,14 @@
   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
   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
-  propose a new strategy called \besteffort{} which aims to balance the load
+  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 made. In this paper, we
+  propose a new strategy called \besteffort{} which aims at balancing the load

   of a node to all its less loaded neighbors while ensuring that all involved nodes by the load balancing phase have the same amount of load. Moreover, since 
   of a node to all its less loaded neighbors while ensuring that all involved nodes by the load balancing phase have the same amount of load. Moreover, since 

-  asynchronous iterative algorithms are less sensitive to communications delays 
+  asynchronous iterative algorithms are less sensitive to communication delays 

   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
   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 several QoS metrics are evaluated to show the usefulness of the proposed algorithm.
+  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}
 \end{abstract}
 
 % \begin{keywords}


@@ -108,9 +108,9 @@ 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
 (ranging from seconds to minutes), it is possible to send a new load information
 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
 (ranging from seconds to minutes), it is possible to send a new load information

-message to each involved neighbor at each iteration. Then, the load is sent, but the reception may take time when the amount of load is huge and when communication links are slow.  Depending on the application, it may make sense or not for the nodes to 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 longer than the  time to transfer a load information message. So,
+message to each involved neighbor at each iteration. The load is then sent, but the reception may take time when the amount of load is huge and when communication links are slow.  Depending on the application, it may or may not make sense for the nodes to try to balance  a part of their load  at each computing
+iteration. But the time needed to transfer a load message from one node to another is
+often much longer than the time needed to transfer a load information message. So,

 when a node is notified 
 %receives the information  
 that later it will receive a data message,
 when a node is notified 
 %receives the information  
 that later it will receive a data message,


@@ -132,7 +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 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 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. 
+\item The SimGrid simulator, which is known to handle realistic models of computation and communication in different types of platforms was used. 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}
 
 
 \end{itemize}
 


@@ -150,7 +150,7 @@ In Section~\ref{sec.simulations}, a comprehensive set of numerical results that
 \label{sec.related.works}
 In this section, the relevant techniques proposed in the literature to tackle the problem of load balancing in a general context of distributed systems are reviewed. 
 
 \label{sec.related.works}
 In this section, the relevant techniques proposed in the literature to tackle the problem of load balancing in a general context of distributed systems are reviewed. 
 

-As pointed above, the most interesting approach to this issue has been proposed by Bertsekas  and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. This algorithm which is outlined in Section~\ref{sec.bt-algo} for the sake of comparison, 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 converging 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.
+As pointed above, the most interesting approach to this issue has been proposed by Bertsekas  and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. This algorithm, which is outlined in Section~\ref{sec.bt-algo} for the sake of comparison, 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 converging 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 the links between the network's resources may be intermittent.

 
 Cybenko~\cite{Cybenko89} proposes a {\it diffusion} approach for hypercube multiprocessor networks. 
 The author targets both static and dynamic random models of work distribution. 
 
 Cybenko~\cite{Cybenko89} proposes a {\it diffusion} approach for hypercube multiprocessor networks. 
 The author targets both static and dynamic random models of work distribution. 


@@ -158,8 +158,8 @@ The convergence proof is derived based on the {\it eigenstructure} of the
 iteration matrices that arise in load balancing of equal amount of
 computational works. A static load balancing for  both synchronous and asynchronous ring networks is addressed in~\cite{GehrkePR99}. The authors assume that at any time step, one token at the most (units of load) can be transmitted along any edge of the ring and no tokens are created during the balancing phase. They show that for every initial token distribution, the proposed algorithm converges to the stable equilibrium with tighter linear bounds of time step-complexity.
 
 iteration matrices that arise in load balancing of equal amount of
 computational works. A static load balancing for  both synchronous and asynchronous ring networks is addressed in~\cite{GehrkePR99}. The authors assume that at any time step, one token at the most (units of load) can be transmitted along any edge of the ring and no tokens are created during the balancing phase. They show that for every initial token distribution, the proposed algorithm converges to the stable equilibrium with tighter linear bounds of time step-complexity.
 

-In order to achieve the load balancing of cloud data centers, a LB technique based on Bayes theorem and Clustering is proposed in~\cite{zhao2016heuristic}. The main idea of this approach is that, the Bayes theorem is combined with the clustering process to obtain the optimal clustering set of physical target hosts leading to the overall load balancing equilibrium.  Bidding is a market-technique for task scheduling and load balancing in distributed systems
-that characterize a set of negotiation rules for users' jobs. For instance, Izakian et al~\cite{IzakianAL10} formulate a double auction mechanism for tasks-resources matching in grid computing environments where resources are considered as provider agents and users as consumer ones. Each entity participates in the network independently and makes autonomous decisions. A provider agent determines its bid price based on its current workload and each consumer agent defines its bid value based on two main parameters: average remaining time and remaining resources for bidding. Based on JADE simulator, the proposed algorithm exhibits better performances in terms of successful execution rates, resource utilization rates and fair profit allocation.
+In order to achieve the load balancing of cloud data centers, a LB technique based on Bayes theorem and Clustering is proposed in~\cite{zhao2016heuristic}. The main idea of this approach is that the Bayes theorem is combined with the clustering process to obtain the optimal clustering set of physical target hosts leading to the overall load balancing equilibrium.  Bidding is a market-technique for task scheduling and load balancing in distributed systems
+that characterize a set of negotiation rules for users' jobs. For instance, Izakian et al~\cite{IzakianAL10} formulate a double auction mechanism for tasks-resources matching in grid computing environments where resources are considered as provider agents and users as consumer ones. Each entity participates in the network independently and makes autonomous decisions. A provider agent determines its bid price based on its current workload and each consumer agent defines its bid value based on two main parameters: the average remaining time and the remaining resources for bidding. Based on JADE simulator, the proposed algorithm exhibits better performances in terms of successful execution rates, resource utilization rates and fair profit allocation.

 
 
 Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
 
 
 Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed


@@ -167,9 +167,9 @@ approaches combine a bidding approach for task selection and a consensus procedu
 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.
 
 
 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. 
+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~\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 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.
+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 proposed to tackle the load balancing problem using the multi-agent approach where each agent is responsible for the load balancing of a subset of nodes in the network. The objective of the agent is to minimize the jobs' response time and the 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 an 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.

 
 
 
 
 
 


@@ -177,16 +177,16 @@ Several LB techniques, based on artificial intelligence, have also been proposed
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
 \label{sec.bt-algo}
 
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
 \label{sec.bt-algo}
 

-In this section, we present a brief description of Bertsekas and Tsitsiklis' algorithm~\cite{bertsekas+tsitsiklis.1997.parallel} using its original notations. 
+In this section, a brief description of Bertsekas and Tsitsiklis' algorithm~\cite{bertsekas+tsitsiklis.1997.parallel} is given, using its original notations. 

 A network is modeled as a connected undirected graph $G=(N,A)$, where $N$ is a set 
 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 $V(i)$.
  
 A network is modeled as a connected undirected graph $G=(N,A)$, where $N$ is a set 
 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 $V(i)$.
  

-Load of processor $i$
+The load of processor $i$

 at  time $t$  is  represented  by $x_i(t)\geq  0$.   
 Each processor $i$ has an estimate  of the load of
 at  time $t$  is  represented  by $x_i(t)\geq  0$.   
 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 
+each  of its  neighbors $j  \in V(i)$  denoted by  $x_j^i(t)$. This estimate 

 may be outdated due to %.  According to
 asynchronism and communication  delays. 
  
 may be outdated due to %.  According to
 asynchronism and communication  delays. 
  


@@ -194,7 +194,7 @@ asynchronism and communication  delays.
 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
 processor $i$ has transferred to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
 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
 processor $i$ has transferred to processor $j$ at time $t$ and let $r_{ij}(t)$ be the

-amount of  loads received by  $j$  from  $i$ at  time $t$. Then
+amount of  load received by  $j$  from  $i$ at  time $t$. Then

 the amount of load of processor $i$ at time $t+1$ is given by:
 
 \begin{equation}
 the amount of load of processor $i$ at time $t+1$ is given by:
 
 \begin{equation}


@@ -222,7 +222,7 @@ other back and forth, without reaching equilibrium.
 
 \medskip
 Nevertheless,  we  think that  this  condition may  lead  to  deadlocks in  some
 
 \medskip
 Nevertheless,  we  think that  this  condition may  lead  to  deadlocks in  some

-cases. For example, consider a linear chain graph network of only three processors in which processor $1$
+cases. For example, let us consider a linear chain graph network of only three processors in which processor $1$

 is linked to processor $2$ which is  also linked to processor $3$, but in which processors $1$ and $3$ are not neighbors. 
 %(i.e. a simple chain which 3 processors). 
 
 is linked to processor $2$ which is  also linked to processor $3$, but in which processors $1$ and $3$ are not neighbors. 
 %(i.e. a simple chain which 3 processors). 
 


@@ -235,15 +235,15 @@ 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}
 
 \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 a part of its load to processor $1$ or processor
-$3$.  If it sends 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 to processor
+$3$.  If it sends its load 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.
 
 \smallskip  
 \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.
 
 \smallskip  

-Despite this, we conjecture that a weaker condition may exist since we
-have never seen any scenario that is not leading to convergence, even with
+In spite of this, a weaker condition can be conjectured to exist since
+there does not seem to be any scenario that does not lead to convergence, even with

 load-balancing strategies that are not exactly fulfilling the authors' own conditions. %se two conditions.
 
 %It may be the subject of future work to express weaker conditions, and to prove
 load-balancing strategies that are not exactly fulfilling the authors' own conditions. %se two conditions.
 
 %It may be the subject of future work to express weaker conditions, and to prove


@@ -257,12 +257,12 @@ questions arise when dealing with realistic models of
 computation and communication. As reported above, the 
 algorithm's description is too succinct and no details are 
 given on what is really sent and how the load balancing decisions 
 computation and communication. As reported above, the 
 algorithm's description is too succinct and no details are 
 given on what is really sent and how the load balancing decisions 

-are taken. To our knowledge, the only first attempt for a possible 
+are made. To our knowledge, the only first attempt for a possible 

 implementation of this algorithm is investigated in~\cite{bahi+giersch+makhoul.2008.scalable} under the same conditions. Thus, in order to assess the performances
 implementation of this algorithm is investigated in~\cite{bahi+giersch+makhoul.2008.scalable} under the same conditions. Thus, in order to assess the performances

-of the new \besteffort{}, we naturally chose to compare it to this previous
-work.  More precisely, we will use the algorithm~2 from
-\cite{bahi+giersch+makhoul.2008.scalable} and, throughout the paper, we will
-reference it under the original name {\it Bertsekas and Tsitsiklis} for the sake of convenience and readability. 
+of the new \besteffort{}, it seemed natural to compare it with this previous
+work.  More precisely, algorithm~2 from
+\cite{bahi+giersch+makhoul.2008.scalable} will be used and, throughout the paper, will be
+referenced under the original name {\it Bertsekas and Tsitsiklis} for the sake of convenience and readability. 

 
 \smallskip 
 Here is an outline of the main principle of the borrowed algorithm.  When a given node  $i$ has to take
 
 \smallskip 
 Here is an outline of the main principle of the borrowed algorithm.  When a given node  $i$ has to take


@@ -271,21 +271,21 @@ order of their loads.  Then, it computes the difference between its own load, an
 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
 $1/(|V(i)|+1)$ of the load difference%, with $n$ being the number of neighbors
 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
 $1/(|V(i)|+1)$ of the load difference%, with $n$ being the number of neighbors

-. This process is iterated as long as the node is more loaded than the considered
+. This process is iterated as long as a node is more loaded than its considered

 neighbors.
 
 
 \section{Best effort strategy}
 \label{sec.besteffort}
 
 neighbors.
 
 
 \section{Best effort strategy}
 \label{sec.besteffort}
 

-In this section, we describe a new load-balancing strategy that we call
-\besteffort{}.  First, we explain the general idea behind this strategy,
-and then we present some variants of this basic strategy.
+In this section, a new load-balancing strategy that is called
+\besteffort{} is described.  First, the general idea behind this strategy is given,
+and then some variants of this basic strategy are presented.

 
 \subsection{Basic strategy}
 
 \subsection{Basic strategy}

-The description of our algorithm will be given from the point of view a processor~$i$.
+The description of our algorithm will be given from the point of view of a processor~$i$.

 The principle of the \besteffort{} strategy is that each processor
 The principle of the \besteffort{} strategy is that each processor

-detecting itself to be more loaded than some of its neighbors, sends some load to its less loaded neighbors, doing its best to reach the equilibrium
+detecting itself to be more loaded than some of its neighbors, sends part of its load to its less loaded neighbors, doing its best to reach the equilibrium

 between the involved neighbors and itself.
 
 More precisely, %when a processor $i$ is in its load-balancing phase,
 between the involved neighbors and itself.
 
 More precisely, %when a processor $i$ is in its load-balancing phase,


@@ -340,7 +340,7 @@ at each iteration of the load balancing process, processor~$i$
 
 With the aforementioned basic strategy, each node does its best to reach the
 equilibrium with its neighbors. However, one question should be outlined here:
 
 With the aforementioned basic strategy, each node does its best to reach the
 equilibrium with its neighbors. However, one question should be outlined here:

-how can we handle the case where two (or more) node initiators might concurrently send 
+how to handle the case where two (or more) node initiators might concurrently send 

  some loads to the same least loaded neighbor? Indeed,  
 %since each node may take the same kind of decision at the same time, 
 there is a risk that a node will receive loads from
  some loads to the same least loaded neighbor? Indeed,  
 %since each node may take the same kind of decision at the same time, 
 there is a risk that a node will receive loads from


@@ -348,24 +348,21 @@ several of its neighbors, and then might temporary go off the equilibrium state.
 This is particularly true with strongly connected applications.
 
 
 This is particularly true with strongly connected applications.
 
 

-
-In order to reduce this effect, we add the ability to level the amount of loads to send.
+In order to reduce this effect, the ability to level the amount of load to send is added.

 The idea, here, is to make as few steps as possible toward the equilibrium, such that a
 potentially unsuitable decision pointed above has a lower impact on the local equilibrium.
 The idea, here, is to make as few steps as possible toward the equilibrium, such that a
 potentially unsuitable decision pointed above has a lower impact on the local equilibrium.

-Roughly speaking, once $s_{ij}$ is estimated as previously explained, it is simply weighted by
-a given prescribed threshold parameter which we call 
-%.  This parameter is called 
-$k$ in
-Section~\ref{sec.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{sec.results}]{}
+A weighting system parameter $k$ is introduced to orchestrate the right balance between the topology structure and the computation to communication ratios (CCR) values of the deployed application. Indeed, to speedup the convergence time of the load balancing process, one is faced with a difficult trade-off to choose an appropriate amount of load to send between node neighbors upon load imbalance detection. On the one hand, if $k$ is small, faster convergence times are expected for sparsely connected applications and large CCR values. On the other hand, for strongly connected applications and small CCR values, a large value of $k$ will enable us to better balance the load locally and therefore minimize the number of iterations toward the global equilibrium. In the experiments section (Section~\ref{sec.results}), it can be observed that choosing $k$ in $\{1, 2, 4\}$ leads to good results for the considered CCR values and the targeted topology structures.
+So the amount of data to send is then $s_{ij}(t) = (\bar{x} - x^i_j(t))/k$.
+
+
+

 
 
 
 \section{Virtual load}
 \label{sec.virtual-load}
 
 
 
 
 \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 this section, the new concept of \emph{virtual load} is presented. It aims at improving the global convergence time. For that purpose, both load transfer messages and load information messages are dissociated. 

 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.
 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.


@@ -379,18 +376,17 @@ require more time to be transferred.
 The  concept  of  \emph{virtual load}  allows  a  node receiving a  load
 information message to integrate (virtually) the future load it will receive later in its load's list
  even if the load has not been received yet. Consequently, the notified node can send  a (real)  part of  its load  to some  of its
 The  concept  of  \emph{virtual load}  allows  a  node receiving a  load
 information message to integrate (virtually) the future load it will receive later in its load's list
  even if the load has not been received yet. Consequently, the notified node can send  a (real)  part of  its load  to some  of its

-neighbors when needed. By and large, this allows a node on the one hand, to predict the load it will receive in the subsequent time steps, and on the other hand, to take suitable decisions when detecting load imbalance in its closed neighborhoods. Doing so, we expect faster convergence time since nodes can take 
-into account the information about the predictive loads not 
-received yet.
+neighbors when needed. By and large, this allows a node on the one hand, to predict the load it will receive in the subsequent time steps, and on the other hand, to make suitable decisions when detecting load imbalance in its closed neighborhoods. Doing so, faster convergence times are expected since nodes can take 
+into account the information about the predictive loads even if these have not yet been received.

 
 
 
 \section{Implementation with SimGrid and simulations}
 \label{sec.simulations}
 
 
 
 
 \section{Implementation with SimGrid and simulations}
 \label{sec.simulations}
 

-In order to test and validate our approach, we wrote a simulator
+In order to test and validate our approach, a simulator

 using the SimGrid
 using the SimGrid

-framework~\cite{simgrid.web,casanova+giersch+legrand+al.2014.simgrid}.  This
+framework~\cite{simgrid.web,casanova+giersch+legrand+al.2014.simgrid} was written.  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
 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


@@ -407,7 +403,7 @@ results of the simulations are presented in section~\ref{sec.results}.
 \label{sec.model}
 
 In the simulation model the processors exchange messages which are of
 \label{sec.model}
 
 In the simulation model the processors exchange messages which are of

-two types.  First, there are \emph{control messages} which carry only the information exchanged between processors, such as the
+two types.  First, there are \emph{control messages} which only carry the information exchanged between processors, such as the

 current load, or the virtual load transfers if this option is
 considered.  These messages are rather small, and their size is
 constant.  Then, there are \emph{data messages} that carry the real
 current load, or the virtual load transfers if this option is
 considered.  These messages are rather small, and their size is
 constant.  Then, there are \emph{data messages} that carry the real


@@ -421,10 +417,10 @@ primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
 
 During the simulation, each processor concurrently runs three threads:
 a \emph{receiving thread}, a \emph{computing thread}, and a
 
 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 hereafter.
+\emph{load-balancing thread}, which will be briefly described hereafter.

 
 For the sake of simplicity, a few details were voluntary omitted from
 
 For the sake of simplicity, a few details were voluntary omitted from

-these descriptions.  For an exhaustive presentation, we refer to the
+these descriptions.  For an exhaustive presentation, interested readers are referred to the

 actual source code that was used for the experiments%
 \footnote{As mentioned before, our simulator relies on the SimGrid
   framework~\cite{casanova+giersch+legrand+al.2014.simgrid}.  For the
 actual source code that was used for the experiments%
 \footnote{As mentioned before, our simulator relies on the SimGrid
   framework~\cite{casanova+giersch+legrand+al.2014.simgrid}.  For the


@@ -550,11 +546,11 @@ In order to assess the performances of our algorithm, simulations with various p
 \subsubsection{Load balancing strategies}
 
 Several load balancing strategies were compared.  Experiments with
 \subsubsection{Load balancing strategies}
 
 Several load balancing strategies were compared.  Experiments with

-the \besteffort{}, and with the \makhoul{} strategies have been performed.  First the \emph{best
+the \besteffort{}, and with the \makhoul{} strategies have been compared.  First the \emph{best

   effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$.  Then,
 each strategy was run in its two variants: with, and without the management of
 \emph{virtual load}.  Finally, each configuration with \emph{real},
   effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$.  Then,
 each strategy was run in its two variants: with, and without the management of
 \emph{virtual load}.  Finally, each configuration with \emph{real},

-and with \emph{integer} load values is considered.
+and with \emph{integer} load values was considered.

 
 To summarize the different load balancing strategies, we have:
 \begin{description}
 
 To summarize the different load balancing strategies, we have:
 \begin{description}


@@ -566,7 +562,7 @@ To summarize the different load balancing strategies, we have:
 
 \subsubsection{End of the simulation}
 
 
 \subsubsection{End of the simulation}
 

-The simulations were run until reaching the global equilibrium threshold. 
+The simulations were run until the global equilibrium threshold was reached. 

   
 More precisely, the simulation stops when each node holds
 an amount of load at least inferior to 1\% of the load average.
   
 More precisely, the simulation stops when each node holds
 an amount of load at least inferior to 1\% of the load average.


@@ -574,7 +570,7 @@ an amount of load at least inferior to 1\% of the load average.
 \subsubsection{Platform}
 
 
 \subsubsection{Platform}
 
 

-In order to make our experiments, an heterogeneous grid platform description were created by taking a subset of the
+In order to make our experiments, an heterogeneous grid platform description was 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
 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


@@ -593,13 +589,13 @@ nodes: 16, 64, 256, and 1024 nodes is built in a similar way.
 
 The distributed processes of the application were then logically organized along
 three possible typologies: a line, a torus or an hypercube.  Tests were divided into two groups on the basis of the initial distribution of the global load: i) some tests were performed with the total load initially on only one node, ii) and other tests were performed for which the load was initially randomly distributed across all the
 
 The distributed processes of the application were then logically organized along
 three possible typologies: a line, a torus or an hypercube.  Tests were divided into two groups on the basis of the initial distribution of the global load: i) some tests were performed with the total load initially on only one node, ii) and other tests were performed for which the load was initially randomly distributed across all the

-participating nodes of the platform.  The total amount of loads was fixed to a number of load
+participating nodes of the platform.  The total amount of load was fixed to a number of load

 units equal to 1,000 times the number of node.  The average load is then of 1,000
 load units.
 
 For all the previous configurations, the
 computation and communication costs of a load unit are defined.  They were chosen so as to
 units equal to 1,000 times the number of node.  The average load is then of 1,000
 load units.
 
 For all the previous configurations, the
 computation and communication costs of a load unit are defined.  They were chosen so as to

-have two different computation to communication ratios (CCR), and hence characterize 
+have two different CCR, and hence characterize 

 two different types of applications:
 \begin{itemize}
 \item mainly communicating, with a CCR of $1/10$;
 two different types of applications:
 \begin{itemize}
 \item mainly communicating, with a CCR of $1/10$;


@@ -612,8 +608,8 @@ two different types of applications:
 \label{sec.metrics}
 
 In order to evaluate and compare the different load balancing strategies, several metrics were considered. Our goal, when choosing these metrics, is to have
 \label{sec.metrics}
 
 In order to evaluate and compare the different load balancing strategies, several metrics were considered. Our goal, when choosing these metrics, is 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, the goal is to 
+something tending to a constant value, i.e. to have a measure which does not
+change once the convergence state is reached.  Moreover, the goal is to 

 have some normalized values, in order to be able to compare them across different
 settings. With these constraints in mind, the following metrics are defined:
 %
 have some normalized values, in order to be able to compare them across different
 settings. With these constraints in mind, the following metrics are defined:
 %


@@ -674,9 +670,9 @@ The main results for our simulations on grid platforms are presented in Figures~
 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. 
 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
+In both figures, the CCR is $10/1$ on the left

 column, and $1/10$ on the right column.  
 column, and $1/10$ on the right column.  

-On each Figure, ~\ref{fig.results1} and~\ref{fig.resultsN}, the results
+In 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.
 
 are given for the process topology being, from top to bottom, a line, a torus or
 an hypercube.
 


@@ -684,7 +680,7 @@ Finally, the vertical bars show the measured times for the evaluated metrics. Th
 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
 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

-because the algorithm did not reach the convergence state in the
+because the algorithm did not manage to reach the convergence state in the

 allocated time.
 
 
 allocated time.
 
 


@@ -698,7 +694,7 @@ Each algorithm is evaluated in terms of achieved idle time and convergence time.
 
 Before looking  at the different variations,  we will first show  that the simple
 \besteffort{}  strategy  is valuable,  and  may be  as  good  as the  \makhoul{}
 
 Before looking  at the different variations,  we will first show  that the simple
 \besteffort{}  strategy  is valuable,  and  may be  as  good  as the  \makhoul{}

-strategy.  On  Figures~\ref{fig.results1} and~\ref{fig.resultsN},
+strategy.  In  Figures~\ref{fig.results1} and~\ref{fig.resultsN},

 these strategies are respectively labeled ``b'' and ``a''.
 
 We  can  see  that  the  relative  performance of  these  strategies  is  mainly
 these strategies are respectively labeled ``b'' and ``a''.
 
 We  can  see  that  the  relative  performance of  these  strategies  is  mainly


@@ -711,7 +707,7 @@ to load  balance the load efficiently.
 
 In contrast, for the hypercube topology, the \besteffort{}' performances are lower than
 the \makhoul{} strategy. In this case, the \makhoul{} strategy, which
 
 In contrast, for the hypercube topology, the \besteffort{}' performances are lower than
 the \makhoul{} strategy. In this case, the \makhoul{} strategy, which

-tries to give more load to few neighbors, reaches the equilibrium faster.
+tries to give more load to a small number of neighbors, reaches the equilibrium faster.

 
 For the torus  topology, for which the  number of links is between  the line and
 the hypercube, the \makhoul{} strategy  is slightly better but the difference is
 
 For the torus  topology, for which the  number of links is between  the line and
 the hypercube, the \makhoul{} strategy  is slightly better but the difference is


@@ -721,7 +717,7 @@ random initial distribution when communications are slow.
 
 Generally speaking,  the  number  of   interconnection  is   very  important.  Indeed, the  more
 numerous the interconnection links are, the  faster the \makhoul{} strategy is because
 
 Generally speaking,  the  number  of   interconnection  is   very  important.  Indeed, the  more
 numerous the interconnection links are, the  faster the \makhoul{} strategy is because

-it distributes quickly significant amount of loads, even if the distribution may be unfair, between
+it quickly distributes significant amount of load, even if the distribution may be unfair, between

 all neighbors.  However,  the \besteffort{} strategy  distributes the
 load fairly when needed and is better for sparse connected applications.
 
 all neighbors.  However,  the \besteffort{} strategy  distributes the
 load fairly when needed and is better for sparse connected applications.
 


@@ -731,25 +727,25 @@ load fairly when needed and is better for sparse connected applications.
 
 \subsubsection{With virtual load}
 
 
 \subsubsection{With virtual load}
 

-The impact of virtual load scheme is most of the time really significant compared to
+The impact of virtual load schemes is, most of the time, really significant compared to

 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
 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,
+than the simple version. This  can be explained  by the  fact that, with a virtual load,

 processors  will exchange all  the load  they need  to exchange  as soon  as the
 virtual load has been balanced  between all the processors. As a consequence, they
 processors  will exchange all  the load  they need  to exchange  as soon  as the
 virtual load has been balanced  between all the processors. As a consequence, they

-cannot  compute  at  the  beginning.  This is  especially  noticeable  when  the
+cannot  compute  from  the  beginning.  This is  especially  noticeable  when  the

 communication are slow (on the left part of Figure ~\ref{fig.results1}).
 
 \smallskip 
 When the load to balance is initially randomly distributed over all nodes, we can see from Figure \ref{fig.resultsN} that the effect of virtual load is not significant for the line topology structure. However, for both torus and hypercube structures with CCR = 1/10 (on the left of the figure), the performance of virtual load transfers is significantly better. This is explained by the fact 
 that for small CCR values,  high communication costs play quite a significant role. Moreover, the impact of
 communication are slow (on the left part of Figure ~\ref{fig.results1}).
 
 \smallskip 
 When the load to balance is initially randomly distributed over all nodes, we can see from Figure \ref{fig.resultsN} that the effect of virtual load is not significant for the line topology structure. However, for both torus and hypercube structures with CCR = 1/10 (on the left of the figure), the performance of virtual load transfers is significantly better. This is explained by the fact 
 that for small CCR values,  high communication costs play quite a significant role. Moreover, the impact of

-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. 
+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 schemes 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 Bertsekas and Tsitsiklis algorithm. 
 These results indicate that local load balancing decisions have a significant impact on the global 
 
 \smallskip 
 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 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.  
+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 the {\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. Therefore, loads are not really assigned to neighboring processors which would allow them to be fairly balanced.  

 
 \smallskip 
 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 
 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. 


@@ -758,10 +754,10 @@ Unlike the {\it Bertsekas and Tsitsiklis} algorithm, our approach is not really
 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}.
 
 \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}.
 
 \smallskip 

-We also find in Figs.~\ref{fig.results1} and ~\ref{fig.resultsN} that the performance difference between the improved version of our proposal and its simple version (without virtual load) increases when the CCR increases. This interesting result comes from the fact that larger CCR values reveal that we are dealing with intensive computations applications in grid platforms. Thus, in order to reduce the convergence time of the load balancing for such applications, it is important to take suitable decisions upon local load imbalance detection. That is why we added {\it virtual load} transfers scheme to the {\it best effort} strategy to perfectly balance the load of processors at each step of the load balancing process.
+We also find in Figs.~\ref{fig.results1} and ~\ref{fig.resultsN} that the performance difference between the improved version of our proposal and its simple version (without virtual load) increases when the CCR increases. This interesting result comes from the fact that larger CCR values reveal that we are dealing with intensive computations applications in grid platforms. Thus, in order to reduce the convergence time of the load balancing for such applications, it is important to make suitable decisions upon local load imbalance detection. That is why we added {\it virtual load} transfers scheme to the {\it best effort} strategy to perfectly balance the load of processors at each step of the load balancing process.

 
 \smallskip 
 
 \smallskip 

-Finally, it is worthwhile noting from Figures~\ref{fig.results1} and ~\ref{fig.resultsN}, that the algorithm's convergence time increases together with the size of the network. We also see that the idle time increases together with the size of the network when a load is initially on a single node (Figure~\ref{fig.results1}),
+Finally, it is worth noting from Figures~\ref{fig.results1} and ~\ref{fig.resultsN}, that the algorithm's convergence time increases together with the size of the network. We also see that the idle time increases together with the size of the network when a load is initially on a single node (Figure~\ref{fig.results1}),

 as expected. In addition, it is interesting to note that when the number of nodes increases, there is no substantial difference in the increase of the convergence time, compared to the simple version without virtual load. This is explained by the fact that the increase in the convergence time is already absorbed by the virtual load transfers between processors being in line with the network's size. 
 
 
 as expected. In addition, it is interesting to note that when the number of nodes increases, there is no substantial difference in the increase of the convergence time, compared to the simple version without virtual load. This is explained by the fact that the increase in the convergence time is already absorbed by the virtual load transfers between processors being in line with the network's size. 
 
 


@@ -784,15 +780,15 @@ in the same conditions.
 
 \subsubsection{With non negative integer load values}
 In addition to the first tests devoted to the case of non negative real load values, further experiments were also carried with integer load values to assess the performance of our proposal.
 
 \subsubsection{With non negative integer load values}
 In addition to the first tests devoted to the case of non negative real load values, further experiments were also carried with integer load values to assess the performance of our proposal.

-As expected, the  obtained results globally have the same behavior, that is why we decided not to show similar figures.   The most
-interesting  result, from  our point  of view,  is that  the virtual  mode allows
+As expected, the  obtained results globally have the same behavior, that is why similar figures do not appear in this paper.   The most
+interesting  result is that  the virtual  mode allows

 processors in a line topology to converge to the uniform load balancing state. Without
 the virtual  load, most  of the time,  processors converge  to what is  called the
 ``stairway effect'', that  is to say that  there is only a difference of at most one unit load between any pairs of neighbor nodes, i.e. the load difference between each processor and its neighbors is within one unit load (for example with 10 processors, we
 obtain 10 9 8 7 6 6 7 8 9 10 instead of 8 8 8 8 8 8 8 8 8 8).
 
 \smallskip 
 processors in a line topology to converge to the uniform load balancing state. Without
 the virtual  load, most  of the time,  processors converge  to what is  called the
 ``stairway effect'', that  is to say that  there is only a difference of at most one unit load between any pairs of neighbor nodes, i.e. the load difference between each processor and its neighbors is within one unit load (for example with 10 processors, we
 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 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.
+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.

 
 
 
 
 
 


@@ -800,11 +796,11 @@ To summarize  the simulation results led us to show that, with a few exceptions
 \label{conclusions-remarks}
 
 In this paper, a new asynchronous load balancing algorithm for non negative real numbers
 \label{conclusions-remarks}
 
 In this paper, a new asynchronous load balancing algorithm for non negative real numbers

-of divisible loads in distributed systems was presented. The proposed algorithm which is called {\it best effort strategy} 
+of divisible loads in distributed systems was presented. The proposed algorithm, which is called {\it best effort strategy},

 seeks greedily for loads imbalance detection and tries to achieve efficient local load equilibrium  
 between neighbors. Our proposal is based on {\it a clairvoyant virtual loads' transfer} scheme 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. 
 seeks greedily for loads imbalance detection and tries to achieve efficient local load equilibrium  
 between neighbors. Our proposal is based on {\it a clairvoyant virtual loads' transfer} scheme 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. 

-Based on SimGrid simulator, we have demonstrated that, when dealing with realistic models of computation and communication, our algorithm exhibits better performances than its direct competitor from the literature. This makes it a viable choice for load balancing of both non negative real and integer divisible loads in distributed computing systems. % un peu gonflé peut être pour la dernière phrase.
+Based on SimGrid simulator, it was shown that, when dealing with realistic models of computation and communication, our algorithm exhibits better performances than its direct competitor from the literature. This makes it a viable choice for load balancing of both non negative real and integer divisible loads in distributed computing systems. % un peu gonflé peut être pour la dernière phrase.

 
 \section*{Acknowledgments}
 
 
 \section*{Acknowledgments}