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\begin{document}

\begin{frontmatter}

\journal{Parallel Computing}

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

\author{Raphaël Couturier}
\ead{raphael.couturier@univ-fcomte.fr}

\author{Arnaud Giersch\corref{cor}}
\ead{arnaud.giersch@univ-fcomte.fr}

\author{Mourad Hakem}
\ead{mourad.hakem@univ-fcomte.fr}

\address{%
  FEMTO-ST Institute, Univ Bourgogne Franche-Comté, Belfort, France}

\cortext[cor]{Corresponding author.}

\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{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
  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 
  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.
\end{abstract}

% \begin{keywords}
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\end{frontmatter}

\section{Introduction}

Load  balancing algorithms  are  widely used  in  parallel and  distributed
applications to achieve high performances in terms of response time, throughput and resources usage. They play an important role and arise in various fields ranging from parallel and distributed 
computing systems to wireless sensor networks (WSN). 
The objective of load balancing is to orchestrate the distribution of the global load so that 
the load difference between the computational resources of the network is 
minimized as much as possible. Unfortunately, this problem is known to be {\bf NP-hard} in its 
general form and heuristics are required to achieve sub-optimal solutions but in 
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~\cite{Bharadwaj1996, Drozdowski1998, Casanova2008}. This model of divisible loads arises in 
a wide range of real-world applications. Common examples, among many, include signal processing, 
feature extraction and edge detection in image processing, records search in huge databases, 
average consensus in WSN, pattern search in Big data and so on. 


In the literature, the problem of load balancing has been formulated and studied in various ways. The first pioneering work is due to Bertsekas  and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. Under some specific hypothesis and {\it ping-pong} awareness conditions (see section~\ref{sec.bt-algo} for more details), an asymptotic convergence proof is derived. 


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. 
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, 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
(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,
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 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. 

\end{itemize}



The reminder of the paper is organized as follows. Section~\ref{sec.related.works} offers a review of the relevant approaches in the literature. Section~\ref{sec.bt-algo} describes the
Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. 
Section~\ref{sec.besteffort} presents the best effort strategy which provides
efficient local loads equilibrium. 
In Section~\ref{sec.virtual-load}, the clairvoyant virtual load scheme is proposed to speedup the convergence time of the load balancing process.
In Section~\ref{sec.simulations}, a comprehensive set of numerical results that exhibit the usefulness of our proposal when dealing with realistic models of computation and communication is provided. Finally, some concluding remarks are made in Section~\ref{conclusions-remarks}.


\section{Related works}
\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.

Cybenko~\cite{Cybenko89} proposes 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, 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.


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 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.




\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. 
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 $i$
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 
may be outdated due to %.  According to
asynchronism and communication  delays. 
 
\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
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
the amount of load of processor $i$ at time $t+1$ is given by:

\begin{equation}
x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
\label{eq.ping-pong}
\end{equation}


%Some  conditions are  required to  ensure the  convergence. One  of them  can be
%called the \emph{ping-pong} condition which specifies that:
\medskip
The asymptotic convergence is derived based on the {\it ping-pong} awareness condition which specifies that:

\begin{equation}
x_i(t)-\sum _{k\in V(i)} s_{ik}(t) \geq x_j^i(t)+s_{ij}(t)
\end{equation}

for any  processor $i$ and any  $j \in V(i)$ such  that $x_i(t)>x_j^i(t)$.  
%This condition aims  at avoiding a processor  to send a  part of its load  and being
%less loaded after that.

\medskip
This condition prohibits the possibility that two nodes keep sending loads to each
other back and forth, without reaching equilibrium. 

\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 $1$
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). 

\noindent Now consider that we have the following load values at time~$t$:
\begin{align*}
  x_1(t)   &= 10    \\
  x_2(t)   &= 100   \\
  x_3(t)   &= 99.99 \\
  x_3^2(t) &= 99.99 
\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
\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
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
%that they are sufficient to ensure the convergence of the load-balancing
%algorithm.

\smallskip

Even though this approach is interesting, several practical
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 
are taken. 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
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. 

\smallskip 
Here is an outline of the main principle of the borrowed algorithm.  When a given node  $i$ has to take
a load balancing decision, it starts by sorting its neighbors by non-increasing
order of their loads.  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
$1/(|V(i)|+1)$ 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
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.

\subsection{Basic strategy}
The description of our algorithm will be given from the point of view a processor~$i$.
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
between the involved neighbors and itself.

More precisely, %when a processor $i$ is in its load-balancing phase,
at each iteration of the load balancing process, processor~$i$ 
 proceeds as follows.
\begin{enumerate}
\item First, the neighbors are sorted in non-decreasing order of their
  known loads $x^i_j(t)$.

\item Then, this sorted list is used to find its largest
  prefix such as the load of each selected neighbor is smaller than:
  \begin{itemize}
  \item the load of processor $i$, and
  \item the mean of the loads of the selected neighbors and processor i.
  \end{itemize}
  Let $S_i(t)$ be the set of the selected neighbors, and
  $\bar{x}(t)$ be the mean of the loads between the selected neighbors and processor $i$ which is given as follows:
  \begin{equation*}
    \bar{x}(t) = \frac{1}{\abs{S_i(t)} + 1}
      \left( x_i(t) + \sum_{j\in S_i(t)} x^i_j(t) \right)
  \end{equation*}
  so that the following properties hold: %{\bf RAPH : la suite tombe du ciel :-)}
  \begin{equation*}
    \begin{cases}
      S_i(t) \subset V(i) \\
      x^i_j(t) < x_i(t) & \forall j \in S_i(t) \\
      x^i_j(t) < \bar{x} & \forall j \in S_i(t) \\
      x^i_j(t) \leq x^i_k(t) & \forall j \in S_i(t), \forall k \in V(i) \setminus S_i(t) \\
      \bar{x} \leq x_i(t)
    \end{cases}
  \end{equation*}

\item Once this selection is done, processor $i$ sends to each selected neighbor $j\in S_i(t)$ an amount of load $s_{ij}(t) =
  \bar{x} - x^i_j(t)$.

  %From the above equations, and notably from the definition of $\bar{x}$, it can easily be verified that:
  
  \smallskip
  In this way we obtain: 
  
  \begin{equation*}
    \begin{cases}
      x_i(t) - \sum_{j\in S_i(t)} s_{ij}(t) = \bar{x} \\
      x^i_j(t) + s_{ij}(t) = \bar{x} & \forall j \in S_i(t)
    \end{cases}
  \end{equation*}
\end{enumerate}



\subsection{Leveling the amount of load to move}

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 
 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
several of its neighbors, and then might temporary go off the equilibrium state.
This is particularly true with strongly connected applications.



In order to reduce this effect, we add the ability to level the amount of loads to send.
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}]{}



\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. 
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
%, consequently they 
and will be received soon.
%very quickly.  
In contrast, load  transfer messages are often larger ones and thus
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
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.



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

In order to test and validate our approache, 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
the different load-balancing strategies under various parameters, such
as the initial distribution of load, the interconnection topology, the
characteristics of the running platform, etc.  Then several metrics
were considered to assess and compare the behavior of the different 
%are issued that permit to compare the 
strategies.

The simulation model is detailed in the next section (\ref{sec.model}), and the
experimental contexts are described in section~\ref{sec.exp-context}.  Then the
results of the simulations are presented in section~\ref{sec.results}.

\subsection{Simulation model}
\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
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
load transferred between processors.  The size of a data message
is a function of the amount of load that it carries, and it can be
pretty large.  In order to receive the messages, each processor has
two receiving channels, one for each type of messages.  Finally, when
a message is sent or received, this is done by using non-blocking
primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
  and \texttt{MSG\_task\_irecv()}.}.

During the simulation, each processor concurrently runs three threads:
a \emph{receiving thread}, a \emph{computing thread}, and a
\emph{load-balancing thread}, which we will briefly describe hereafter.

For the sake of simplicity, a few details were voluntary omitted from
these descriptions.  For an exhaustive presentation, we refer to the
actual source code that was used for the experiments%
\footnote{As mentioned before, our simulator relies on the SimGrid
  framework~\cite{casanova+giersch+legrand+al.2014.simgrid}.  For the
  experiments, we used a pre-release of SimGrid 3.7 (Git commit
  67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
  \url{https://github.com/simgrid/simgrid})}, and which is
available at
\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.

\subsubsection{Receiving thread}

The receiving thread is in charge of waiting for incoming messages, either on the
control channel, or on the data channel.  Its behavior is sketched by
Algorithm~\ref{algo.recv}.  When a message is received, it is pushed in a buffer
of received messages, to be later consumed by one of the other threads.  There
are two such buffers, one for the control messages, and one for the data
messages.
The buffers are implemented with first-in, first-out queues (FIFO).

\begin{algorithm}
  \caption{Receiving thread}
  \label{algo.recv}
  \KwData{
    \begin{algodata}
      \VAR{ctrl\_chan}, \VAR{data\_chan}
      & communication channels (control and data) \\
      \VAR{ctrl\_fifo}, \VAR{data\_fifo}
      & buffers of received messages (control and data) \\
    \end{algodata}}
  \While{true}{%
    wait for a message to be available on either \VAR{ctrl\_chan},
    or \VAR{data\_chan}\;
    \If{a message is available on \VAR{ctrl\_chan}}{%
      get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
    }
    \If{a message is available on \VAR{data\_chan}}{%
      get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
    }
  }
\end{algorithm}

\subsubsection{Computing thread}

The computing thread is in charge of the real load management.  As outlined in
Algorithm~\ref{algo.comp}, it iteratively runs the following operations:
\begin{itemize}
\item if some load was received from the neighbors, get it;
\item if there is some load to send to the neighbors, send it;
\item run some computations, whose duration is a function of the processor's current
  load.
\end{itemize}
Practically, after the computation, the computing thread waits for a
small amount of time if the iterations are looping too fast (for
example, when the current load is near zero).

\begin{algorithm}
  \caption{Computing thread}
  \label{algo.comp}
  \KwData{
    \begin{algodata}
      \VAR{data\_fifo} & buffer of received data messages \\
      \VAR{real\_load} & current load \\
    \end{algodata}}
  \While{true}{%
    \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
      wait until a message is pushed into \VAR{data\_fifo}\;
    }
    \While{\VAR{data\_fifo} is not empty}{%
      pop a message from \VAR{data\_fifo}\;
      get the load embedded in the message, and add it to \VAR{real\_load}\;
    }
    \ForEach{neighbor $n$}{%
      \If{there is some amount of load $a$ to send to $n$}{%
        send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
      }
    }
    \If{$\VAR{real\_load} > 0.0$}{
      simulate some computation, whose duration is function of \VAR{real\_load}\;
      ensure that the main loop does not iterate too fast\;
    }
  }
\end{algorithm}

\subsubsection{Load-balancing thread}

The load-balancing thread is in charge of running the load-balancing algorithm,
and exchanging the control messages.  As shown in Algorithm~\ref{algo.lb}, it
iteratively runs the following operations:
\begin{itemize}
\item get the control messages that were received from the neighbors;
\item run the load-balancing algorithm;
\item send control messages to the neighbors, to inform them about the
  processor's current load, and possibly the future virtual load transfers;
\item wait a minimum (configurable) amount of time, to avoid iterating too fast.
\end{itemize}

\begin{algorithm}
  \caption{Load-balancing}
  \label{algo.lb}
  \While{true}{%
    \While{\VAR{ctrl\_fifo} is not empty}{%
      pop a message from \VAR{ctrl\_fifo}\;
      identify the sender of the message,
      and update the current knowledge of its load\;
    }
    run the load-balancing algorithm to make the decision about load transfers\;
    \ForEach{neighbor $n$}{%
      send a control messages to $n$\;
    }
    ensure that the main loop does not iterate too fast\;
  }
\end{algorithm}

%\paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
%  par ex, donner l'idée générale de l'implémentation.  l'idée générale est déja
%  décrite en section~\ref{sec.virtual-load}}

\subsection{Experimental contexts}
\label{sec.exp-context}

In order to assess the performances of our algorithm, simulations with various parameters have been achieved out, and several metrics are described in this section.

\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
  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.

To summarize the different load balancing strategies, we have:
\begin{description}
\item[\textbf{strategies:}] \makhoul{}, or \besteffort{} with $k\in
  \{1,2,4\}$
\item[\textbf{variants:}] with, or without virtual loads
%\item[\textbf{domain:}] real load, or integer load
\end{description}

\subsubsection{End of the simulation}

The simulations were run until reaching the global equilibrium threshold. 
  
More precisely, the simulation stops when each node holds
an amount of load at least inferior to 1\% of the load average.

\subsubsection{Platform}


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
  Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
\texttt{g5k.xml} distributed with SimGrid.  Note that the heterogeneity of the
platform here only comes from the network topology.  Indeed, 
processors are  considered to be homogeneous for the sake of simplicity.
However, this situation is easily extendable to the case of heterogeneous platforms 
by scaling the processor's load by its computing power~\cite{ElsMonPre02}.
%since our
%algorithms currently do not handle heterogeneous computing resources, 
 The
processor speeds were normalized, and we arbitrarily chose to fix them at
1~GFlop/s. Each type of platform with four different numbers of computing
nodes: 16, 64, 256, and 1024 nodes is built in a similar way.

\subsubsection{Configurations}

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
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 
two different types of applications:
\begin{itemize}
\item mainly communicating, with a CCR of $1/10$;
\item mainly computing, with a CCR of $10/1$.
%\item balanced, with a computation/communication cost ratio of $1/1$.
\end{itemize}


\subsubsection{Metrics}
\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 
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:
%
\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. 
   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.  A smaller value is better.



\end{description}


\subsection{Experimental results}
\label{sec.results}

In this section, the results for the different simulations are presented,
and our observations are explained.



\subsubsection{Main results}

\begin{figure*}[p]
  \centering
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-line}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-line}
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-torus}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-torus}
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-hcube}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-hcube}
  \caption{Real mode, initially on an only mode, CCR = $10/1$ (left), or $1/10$ (right). For each bar, from bottom to top starting at $t=0$, the first part represents the average idle
time, the second part represents the average convergence time, and then the third part represents the maximum convergence time.}
  \label{fig.results1}
\end{figure*}

\begin{figure*}[p]
  \centering
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-line}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-line}
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-torus}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-torus}
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-hcube}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-hcube}
  \caption{Real mode, random initial distribution, CCR = $10/1$ (left), or $1/10$ (right).}
  \label{fig.resultsN}
\end{figure*}

The main results for our simulations on grid platforms are presented in Figures~\ref{fig.results1} and~\ref{fig.resultsN}.
%
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.  
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. 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
because the algorithm did not reach the convergence state in the
allocated time.



\subsubsection{The \besteffort{} and  \makhoul{} strategies without virtual load}

The {\it simple} ({\it plain}) version of each strategy is defined as the load balancing 
algorithm without virtual load's transfers. For each strategy, we compare the simple 
version (without virtual load) and the improved one (with virtual load). 
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{}
strategy.  On  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
influenced by  the application topology structure.  It  is for the line  topology that the
difference is the  most important.  In this case,  the \besteffort{} strategy is
really faster than the \makhoul{} strategy.  This can  be explained by the
fact that the \besteffort{} strategy tries to distribute the load fairly between
all the nodes  and is in a good agreement with the line topology since  it is easy 
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
tries to give more load to few 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
more nuanced when the initial load is  only on one node. The only case where the
\makhoul{} strategy is really faster than the \besteffort{} strategy is with the
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
it distributes quickly significant amount of loads, 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.





\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. 
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,
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
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. 

\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. 
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.  

\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. 

\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.

\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}),
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. 



\subsubsection{The $k$ parameter}
\label{results-k}

As  explained  previously when  the  communication  are  slow the  \besteffort{}
strategy is efficient. This is due to the fact that it tries to balance the load
fairly and consequently  a significant amount of the  load is transferred between
processors.  In this case, it is possible to reduce the convergence time by
using  the leveler  parameter  (parameter  $k$).  The  advantage  of using  this
solution is particularly true when the initial load is randomly distributed
on  the nodes with  torus and  hypercube topologies  and slow  communication. When
a virtual load  scheme is used,  the effect of  this parameter is  also perceptible
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.
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
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 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.



\section{Conclusion}
\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} 
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.

\section*{Acknowledgments}

This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
ANR-11-LABX-01-01).  We also thank the supercomputer facilities of the Mésocentre de calcul de Franche-Comté.
 
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\end{document}


