-often much more longer that to time to transfer a load information message. So,
-when a node receives the information that later it will receive a data message,
-it can take this information into account and it can consider that its new load
-is larger. Consequently, it can send a part of it real load to some of its
-neighbors if required. We call this trick the \emph{virtual load} mechanism.
-
-So, in this work, we propose a new strategy to improve the distribution of the
-load and a simple but efficient trick that also improves the load
-balancing. Moreover, we have conducted many simulations with SimGrid in order to
-validate that our improvements are really efficient. Our simulations consider
-that in order to send a message, a latency delays the sending and according to
-the network performance and the message size, the time of the reception of the
-message also varies.
-
-In the following of this paper, Section~\ref{sec.bt-algo} describes the
-Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we
-present a possible problem in the convergence conditions.
-Section~\ref{sec.besteffort} presents the best effort strategy which provides an
-efficient way to reduce the execution times. This strategy will be compared
-with other ones, presented in Section~\ref{sec.other}. In
-Section~\ref{sec.virtual-load}, the virtual load mechanism is proposed.
-Simulations allowed to show that both our approaches are valid using a quite
-realistic model detailed in Section~\ref{sec.simulations}. Finally we give a
-conclusion and some perspectives to this work.
+often much more longer that the time to transfer a load information message. So,
+when a node is notified
+%receives the information
+that later it will receive a data message,
+it can take this information into account in its load's queue list for preventive purposes.
+%and it can consider that its new load is larger.
+Consequently, it can send a part of its predictive
+%real
+load to some of its
+neighbors if required. We call this trick the \emph{clairvoyant virtual load} transfer mechanism.
+
+\medskip
+The main contributions and novelties of our work are summarized in the following section.
+
+\section{Our contributions}
+\label{contributions}
+\begin{itemize}
+\item We propose a {\it best effort strategy} which proceeds greedily to achieve efficient local neighborhoods equilibrium. Upon local load imbalance detection, a {\it significant amount} of load is moved from a highly loaded node (initiator) to less loaded neighbors.
+
+\item Unlike earlier works, we use a new concept of virtual loads transfers 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 improve the straightforward network's diameter bound of the global equilibrium threshold in the network. % not sure, it depends on the remaining time before the paper submission ...
+\end{itemize}
+
+
+%{\bf The contributions of this paper are the following:}
+%\begin{itemize}
+%\item We propose a new strategy to improve the distribution of the
+%load and a simple but efficient trick that also improves the load
+%balancing.
+%\item we have conducted many simulations with SimGrid in order to
+%validate that our improvements are really efficient. Our simulations consider
+%that in order to send a message, a latency delays the sending and according to
+%the network performance and the message size, the time of the reception of the
+%message also varies.
+%\end{itemize}
+
+The reminder of the paper is organized as follows.
+In Section~\ref{sec.related.works}, we review the relevant approaches in the literature. Section~\ref{sec.bt-algo} describes the
+Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. %Moreover, we present a possible problem in the convergence conditions.
+Section~\ref{sec.besteffort} presents the best effort strategy which provides
+efficient local loads equilibrium. %This strategy will be compared with the one presented in Section~\ref{sec.other}.
+In Section~\ref{sec.virtual-load}, the clairvoyant virtual load scheme is proposed to speedup the convergence time of the load balancing process.
+We provide in Section~\ref{sec.simulations}, a comprehensive set of numerical results that exhibit the usefulness of our proposals when we deal with realistic models of computation and communication. Finally, we give some concluding remarks in Section~\ref{conclusions-remarks}.
+
+
+\section{Related works}
+\label{sec.related.works}
+In this section, we fairly review the relevant techniques proposed in the literature to tackle the problem of load balancing in a general context of distributed systems.
+
+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 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.
+
+Cybenko~\cite{Cybenko89} propose a {\it diffusion} approach for hypercube multiprocessor networks.
+The author targets both static and dynamic random models of work distribution.
+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, at most one token (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.
+
+
+Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
+approaches combine bidding approach for task selection and 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 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 for computing the Nash equilibrium. They show through simulations that the proposed algorithm ensures fairness among the users and good average latency across all client regions. A hybrid task scheduling and load balancing dependent and independent tasks for master-slaves platforms is addressed in~In~\cite{liu2017dems}. To minimize the response time of the submitted jobs, the proposed algorithm which is called DeMS is splitted in 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{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. Simulations results show that the proposed scheme perform near optimal solutions compared to other existing techniques in terms of fairness.
+
