From f3dd4a9b098e411495f36b94efe02ad7bab52053 Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Tue, 7 Nov 2017 11:29:28 +0100 Subject: [PATCH 1/1] [sharelatex-git-integration Best effort strategy and virtual load for asynchronous iterative load balancing 2017/11/07 11:29:27] --- loba-besteffort/loba-besteffort.tex | 84 ++++++++++++++++------------- 1 file changed, 47 insertions(+), 37 deletions(-) diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex index c387ad6..9f26237 100644 --- a/loba-besteffort/loba-besteffort.tex +++ b/loba-besteffort/loba-besteffort.tex @@ -40,18 +40,16 @@ asynchronous iterative load balancing} \author{Raphaël Couturier} -\ead{raphael.couturier@femto-st.fr} +\ead{raphael.couturier@univ-fcomte.fr} \author{Arnaud Giersch\corref{cor}} -\ead{arnaud.giersch@femto-st.fr} +\ead{arnaud.giersch@univ-fcomte.fr} + +\author{Mourad Hakem} +\ead{mourad.hakem@univ-fcomte.fr} \address{% - Institut FEMTO-ST (UMR 6174), - Université de Franche-Comté (UFC), - Centre National de la Recherche Scientifique (CNRS), - École Nationale Supérieure de Mécanique et des Microtechniques (ENSMM), - Université de Technologie de Belfort Montbéliard (UTBM)\\ - 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France} + FEMTO-ST Institute, Univ Bourgogne Franche-Comté, Belfort, France} \cortext[cor]{Corresponding author.} @@ -65,14 +63,13 @@ propose a strategy called \besteffort{} 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 + asynchronous iterative algorithms, in which an asynchronous load balancing + algorithm is implemented, can dissociate, most of the time, 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 \emph{virtual load} which allows a node that receives a 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 + called \emph{virtual load}. This heuristic allows a node that receives a load + information message to integrate this information, even if the load has not been received yet. Consequently the node sends a (real) part of its load to some of + its neighbors taking into account the virtual load it will receive soon. In order to validate our approaches, we have defined a simulator based on SimGrid which allowed us to conduct many experiments. \end{abstract} @@ -87,19 +84,19 @@ 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.\FIXME{really?} +networks. In a distributed context (i.e. without centralization), 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 +where computing 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 +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 \emph{ping-pong} effect, an asynchronous -iterative algorithm converge to the uniform load distribution. This work has +iterative algorithm converges to the uniform load distribution. This work has been extended by many authors. For example, Cortés et al., with DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous}, propose a version working with integer load. This work was later generalized by @@ -117,11 +114,10 @@ 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. Former ones allow a node to inform its -neighbors of its current load. These messages are very small, they can be sent -quite often. For example, if a computing iteration takes a significant times +neighbors of its current load. These messages are very small, they can be sent +often and very quickly. For example, if a computing iteration takes a significant times (ranging from seconds to minutes), it is possible to send a new load information -message to each neighbor at each iteration. Latter messages contain data that -migrates from one node to another one. Depending on the application, it may have +message to each 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 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 more longer that to time to transfer a load information message. So, @@ -130,17 +126,21 @@ 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 +{\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. Moreover, we have conducted many simulations with SimGrid in order to +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} 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. +present a possible problem in the convergence conditions. In Section~\ref{sec.related.works}, related works are presented. 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 @@ -168,7 +168,7 @@ each of its neighbors $j \in V(i)$ represented by $x_j^i(t)$. According to asynchronism and communication delays, this estimate may be outdated. We also consider that the load is described by a continuous variable. -When a processor send a part of its load to one or some of its neighbors, the +When a processor sends a part of its load to one or 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 load received by processor $j$ from processor $i$ at time $t$. Then @@ -198,6 +198,8 @@ chain which 3 processors). Now consider we have the following values at time $t$ 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} + In this case, processor $2$ can either sends load to processor $1$ or processor $3$. If it sends load to processor $1$ it will not satisfy condition \eqref{eq.ping-pong} because after the sending it will be less loaded that @@ -213,6 +215,13 @@ 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. + +\section{Related works} +\label{sec.related.works} +{\bf A FAIRE} + + + \section{Best effort strategy} \label{sec.besteffort} @@ -233,21 +242,20 @@ he proceeds as following. \item First, the neighbors are sorted in non-decreasing order of their known loads $x^i_j(t)$. -\item Then, this sorted list is traversed in order to find its largest - prefix such as the load of each selected neighbor is lesser than: +\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 processor's own load, and + \item the load of processor $i$, and \item the mean of the loads of the selected neighbors and of the processor's load. \end{itemize} - Let's call $S_i(t)$ the set of the selected neighbors, and - $\bar{x}(t)$ the mean of the loads of the selected neighbors and of - the processor load: + Let $S_i(t)$ be the set of the selected neighbors, and + $\bar{x}(t)$ be the mean of the loads of the selected neighbors plus the load of processor $i$: \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*} - The following properties hold: + The following properties hold: {\bf RAPH : la suite tombe du ciel :-)} \begin{equation*} \begin{cases} S_i(t) \subset V(i) \\ @@ -284,8 +292,8 @@ In order to reduce this effect, we add the ability to level the amount to send. The idea, here, is to make smaller steps toward the equilibrium, such that a potentially wrong decision has a lower impact. -Concretely, once $s_{ij}$ has been evaluated as before, it is simply divided by -some configurable factor. That's what we named the ``parameter $k$'' in +Roughtly speaking, once $s_{ij}$ has been evaluated as previously explained, it is simply divided by +a given factor. 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}]{} @@ -326,10 +334,10 @@ require more time to be transferred. The concept of \emph{virtual load} allows a node that received a load information message to integrate the load that it will receive later in its load -(virtually) and consequently send a (real) part of its load to some of its +(virtually). Consequently the considered node can send a (real) part of its load to some of its neighbors. In fact, a node that receives a load information message knows that later it will receive the corresponding load balancing message containing the -corresponding data. So if this node detects it is too loaded compared to some +corresponding data. So, if this node detects it is too loaded compared to some of its neighbors and if it has enough load (real load), then it can send more load to some of its neighbors without waiting the reception of the load balancing message. @@ -339,6 +347,8 @@ information of the load they will receive, so they can take it into account. \FIXME{Est ce qu'on donne l'algo avec virtual load?} +With integer load, we adapt this algorithm by .... {\bf RAPH a faire} + \FIXME{describe integer mode} \section{Simulations} -- 2.39.5