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\begin{document}

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

\author{Raphaël Couturier \and
        Arnaud Giersch \and
        Abderrahmane Sider
}

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              Fax: +123-45-678910\\
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%             \emph{Present address:} of F. Author  %  if needed
           \and
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              second address
}

\maketitle


\begin{abstract}

Most of the  time, asynchronous load balancing algorithms  have extensively been
studied in a theoretical point  of view. The Bertsekas and Tsitsiklis'
algorithm~\cite[section~7.4]{bertsekas+tsitsiklis.1997.parallel}
is certainly  the most well known  algorithm for which the  convergence proof is
given. From a  practical point of view, when  a node wants to balance  a part of
its  load to some  of its  neighbors, the  strategy is  not described.   In this
paper, we propose a strategy  called \texttt{best effort} which tries to balance
the load of a node to all  its less loaded neighbors while ensuring that all the
nodes  concerned by  the load  balancing  phase have  the same  amount of  load.
Moreover,  asynchronous  iterative  algorithms  in which  an  asynchronous  load
balancing  algorithm is  implemented most  of the  time can  dissociate messages
concerning load transfers and message  concerning load information.  In order to
increase  the  converge of  a  load balancing  algorithm,  we  propose a  simple
heuristic called \texttt{virtual load} which allows a node that receives an load
information message  to integrate the  load that it  will receive later  in its
load (virtually) and consequently sends a (real) part of its load to some of its
neighbors.  In order to  validate our  approaches, we  have defined  a simulator
based on SimGrid which allowed us to conduct many experiments.


\end{abstract}

\section{Introduction}

Load  balancing algorithms  are  extensively used  in  parallel and  distributed
applications in  order to  reduce the  execution times. They  can be  applied in
different scientific  fields from high  performance computation to  micro sensor
networks.   They are  iterative by  nature.  In  literature many  kinds  of load
balancing  algorithms  have been  studied.   They  can  be classified  according
different  criteria:   centralized  or  decentralized,  in   static  or  dynamic
environment,  with  homogeneous  or  heterogeneous load,  using  synchronous  or
asynchronous iterations, with  a static topology or a  dynamic one which evolves
during time.  In  this work, we focus on  asynchronous load balancing algorithms
where computer nodes  are considered homogeneous and with  homogeneous load with
no external  load. In  this context, Bertsekas  and Tsitsiklis have  proposed an
algorithm which is definitively a reference  for many works. In their work, they
proved that under classical  hypotheses of asynchronous iterative algorithms and
a  special  constraint   avoiding  \texttt{ping-pong}  effect,  an  asynchronous
iterative algorithm  converge to  the uniform load  distribution. This  work has
been extended by many authors. For example,
DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous} propose a version working
with integer load. {\bf Rajouter des choses ici}.

Although  the Bertsekas  and Tsitsiklis'  algorithm describes  the  condition to
ensure the convergence,  there is no indication or  strategy to really implement
the load distribution. In other word, a node  can send a part of its load to one
or   many  of   its  neighbors   while  all   the  convergence   conditions  are
followed. Consequently,  we propose a  new strategy called  \texttt{best effort}
that tries to balance the load of  a node to all its less loaded neighbors while
ensuring that all the nodes concerned  by the load balancing phase have the same
amount of  load.  Moreover, when real asynchronous  applications are considered,
using  asynchronous   load  balancing   algorithms  can  reduce   the  execution
times. Most of the times, it is simpler to distinguish load information messages
from  data  migration  messages.  Formers  ones  allows  a  node to  inform  its
neighbors of its  current load. These messages are very small,  they can be sent
quite often.  For example, if an  computing iteration takes  a significant times
(ranging from seconds to minutes), it is possible to send a new load information
message at each  neighbor at each iteration. Latter  messages contains data that
migrates from one node to another one. Depending on the application, it may have
sense or not  that nodes 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 nore 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 \texttt{virtual load} mecanism.



So, in  this work, we propose a  new strategy for improving  the distribution of
the  load  and  a  simple  but  efficient trick  that  also  improves  the  load
balacing. Moreover, we have conducted  many simulations with simgrid in order to
validate 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{BT algo} describes  the Bertsekas
and Tsitsiklis'  asynchronous load balancing  algorithm. Moreover, we  present a
possible  problem  in  the  convergence  conditions.   Section~\ref{Best-effort}
presents the best effort strategy which  provides an efficient way to reduce the
execution  times. In Section~\ref{Virtual  load}, the  virtual load  mecanism is
proposed. Simulations allowed to show that both our approaches are valid using a
quite realistic  model detailed in  Section~\ref{Simulations}. Finally we  give a
conclusion and some perspectives to this work.




\section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
\label{BT algo}

Comment on the problem in the convergence condition.

\section{Best effort strategy}
\label{Best-effort}



\section{Virtual load}
\label{Virtual load}

\section{Simulations}
\label{Simulations}

\subsection{Simulation model}

\subsection{Validation of our approaches}


\section{Conclusion and perspectives}


\bibliographystyle{spmpsci}
\bibliography{biblio}

\end{document}

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% LocalWords:  Raphaël Couturier Arnaud Giersch Abderrahmane Sider
% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD
