1 \documentclass[preprint]{elsarticle}
3 \usepackage[utf8]{inputenc}
4 \usepackage[T1]{fontenc}
6 %\usepackage{newtxtext}
7 %\usepackage[cmintegrals]{newtxmath}
8 \usepackage{mathptmx,helvet,courier}
13 \usepackage[ruled,lined]{algorithm2e}
15 %%% Remove this before submission
16 \newcommand{\FIXMEmargin}[1]{%
17 \marginpar{\textbf{[FIXME]} {\footnotesize #1}}}
18 \newcommand{\FIXME}[2][]{%
19 \ifx #2\relax\relax \FIXMEmargin{#1}%
20 \else \textbf{$\triangleright$\FIXMEmargin{#1}~#2}\fi}
22 \newcommand{\abs}[1]{\lvert#1\rvert} % \abs{x} -> |x|
24 \newenvironment{algodata}{%
25 \begin{tabular}[t]{@{}l@{:~}l@{}}}{%
28 \newcommand{\VAR}[1]{\textit{#1}}
30 \newcommand{\besteffort}{\emph{best effort}}
31 \newcommand{\makhoul}{\emph{Bertsekas and Tsitsiklis}}
37 \journal{Parallel Computing}
39 \title{Best effort strategy and virtual load for\\
40 asynchronous iterative load balancing}
42 \author{Raphaël Couturier}
43 \ead{raphael.couturier@univ-fcomte.fr}
45 \author{Arnaud Giersch\corref{cor}}
46 \ead{arnaud.giersch@univ-fcomte.fr}
49 \ead{mourad.hakem@univ-fcomte.fr}
52 FEMTO-ST Institute, Univ Bourgogne Franche-Comté, Belfort, France}
54 \cortext[cor]{Corresponding author.}
57 Most of the time, asynchronous load balancing algorithms have extensively been
58 studied in a theoretical point of view. The Bertsekas and Tsitsiklis'
61 {bertsekas+tsitsiklis.1997.parallel} is undeniably
62 the most well known algorithm for which the asymptotic convergence proof is given.
64 practical point of view, when a node needs to balance a part of its load to
65 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
66 propose a new strategy called \besteffort{} which aims to balance the load
67 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
68 asynchronous iterative algorithms are less sensitive to communications delays
69 and their variations \cite{bcvc07:bc}, both load transfer and load information messages are dissociated.
70 To speedup the convergence time of the load balancing process, we propose {\it a clairvoyant virtual load} heuristic which allows
71 %asynchronous iterative algorithms, in which an asynchronous load balancing
72 %algorithm is implemented, can dissociate, most of the time, messages concerning
73 %load transfers and message concerning load information. In order to increase
74 %the converge of a load balancing algorithm, we propose a simple heuristic
75 %called \emph{virtual load}. This heuristic allows
76 a node receiving a load
77 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
78 its neighbors taking into account the virtual load it will receive in the subsequent time-steps. Based on SimGrid simulator, series of test-bed scenarios are considered and many QoS metrics are evaluated to show the usefulness of the proposed algorithm. %In order to validate our approaches, we have defined a
79 % simulator based on SimGrid which allowed us to conduct many experiments.
83 % %% keywords here, in the form: keyword \sep keyword
88 \section{Introduction}
90 Load balancing algorithms are widely used in parallel and distributed
91 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
92 computing systems to wireless sensor networks (WSN).
93 The objective of load balancing is to orchestrate the distribution of the global load so that
94 the load difference between the computational resources of the network is
95 minimized as low as possible. Unfortunately, this problem is known to be {\bf NP-hard} in its
96 general forms and heuristics are required to achieve sub-optimal solutions but in
97 polynomial time complexity.
99 In this paper, we focus on asynchronous load balancing of non negative real numbers of {\it divisible loads}
100 in homogeneous distributed systems. Loads can be divided in arbitrary {\it fine-grain} parallel parts size
101 that can be processed independently of each other. This model of divisible loads arises in
102 a wide range of real-world applications. Common examples among many, include signal processing,
103 feature extraction and edge detection in image processing, records search in a huge databases,
104 average consensus in WSN, pattern search in Big data and so on. % c'est pout toi raphael ;-)
107 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.
109 This algorithm 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.
111 %in order to reduce the execution times. They can be applied in
112 %different scientific fields from high performance computation to micro sensor
113 %networks. In a distributed context (i.e. without centralization), they are iterative by nature.
114 %In literature many kinds of load
115 %balancing algorithms have been studied. They can be classified according
116 %different criteria: centralized or decentralized, in static or dynamic
117 %environment, with homogeneous or heterogeneous load, using synchronous or
118 %asynchronous iterations, with a static topology or a dynamic one which evolves
119 %during time. In this work, we focus on asynchronous load balancing algorithms
120 %where computing nodes are considered homogeneous and with homogeneous load with
122 %In this context, Bertsekas and Tsitsiklis have proposed an
123 %algorithm which is definitively a reference for many works. In their work, they
124 %proved that under classical hypotheses of asynchronous iterative algorithms and
125 %a special constraint avoiding \emph{ping-pong} effect, an asynchronous
126 %iterative algorithm converges to the uniform load distribution. This work has
127 %been extended by many authors. For example, Cortés et al., with
128 %DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous}, propose a
129 %version working with integer load. This work was later generalized by
130 %the same authors in \cite{cedo+cortes+ripoll+al.2007.convergence}.
131 %\FIXME{Rajouter des choses ici. Lesquelles ?}
132 Although Bertsekas and Tsitsiklis describe the necessary conditions to
133 ensure the algorithm's convergence, there is no indication or any strategy to really implement
134 the load distribution. %In other word, a node can send some amount of its load to one or many of its neighbors while all the convergence conditions are followed.
135 Consequently, we propose a new strategy called \besteffort{}
136 that tries to balance the load of a node to all its less loaded neighbors while
137 ensuring that all the nodes involved in the load balancing phase have the same
138 amount of load. Moreover, %when real-world asynchronous applications are considered,
139 %using asynchronous load balancing algorithms can reduce the execution
141 most of the times, it is simpler to dissociate load information messages
142 from data migration messages. Former ones allow a node to inform its
143 neighbors about its current load. These messages are in fact very small and can be sent
144 often very quickly. For example, if a computing iteration takes a significant time
145 (ranging from seconds to minutes), it is possible to send a new load information
146 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 have
147 sense or not that nodes try to balance a part of their load at each computing
148 iteration. But the time to transfer a load message from a node to another one is
149 often much more longer that the time to transfer a load information message. So,
150 when a node is notified
151 %receives the information
152 that later it will receive a data message,
153 it can take this information into account in its load's queue list for preventive purposes.
154 %and it can consider that its new load is larger.
155 Consequently, it can send a part of its predictive
158 neighbors if required. We call this trick the \emph{clairvoyant virtual load} transfer mechanism.
161 The main contributions and novelties of our work are summarized in the following section.
163 \section{Our contributions}
164 \label{contributions}
166 \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.
168 \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.
169 This leads to a noticeable speedup of the global convergence time of the load balancing process.
171 \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.
173 %\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 ...
177 %{\bf The contributions of this paper are the following:}
179 %\item We propose a new strategy to improve the distribution of the
180 %load and a simple but efficient trick that also improves the load
182 %\item we have conducted many simulations with SimGrid in order to
183 %validate that our improvements are really efficient. Our simulations consider
184 %that in order to send a message, a latency delays the sending and according to
185 %the network performance and the message size, the time of the reception of the
186 %message also varies.
189 The reminder of the paper is organized as follows.
190 In Section~\ref{sec.related.works}, we review the relevant approaches in the literature. Section~\ref{sec.bt-algo} describes the
191 Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. %Moreover, we present a possible problem in the convergence conditions.
192 Section~\ref{sec.besteffort} presents the best effort strategy which provides
193 efficient local loads equilibrium. %This strategy will be compared with the one presented in Section~\ref{sec.other}.
194 In Section~\ref{sec.virtual-load}, the clairvoyant virtual load scheme is proposed to speedup the convergence time of the load balancing process.
195 We provide in Section~\ref{sec.simulations}, a comprehensive set of numerical results that exhibit the usefulness of our proposal when we deal with realistic models of computation and communication. Finally, we give some concluding remarks in Section~\ref{conclusions-remarks}.
198 \section{Related works}
199 \label{sec.related.works}
200 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.
202 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.
204 Cybenko~\cite{Cybenko89} propose a {\it diffusion} approach for hypercube multiprocessor networks.
205 The author targets both static and dynamic random models of work distribution.
206 The convergence proof is derived based on the {\it eigenstructure} of the
207 iteration matrices that arise in load balancing of equal amount of
208 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.
210 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
211 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.
214 Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
215 approaches combine bidding approach for task selection and consensus procedure scheme for
216 decentralized conflict resolution. The developed algorithms are proven to converge to a conflict-free assignment in
217 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.
220 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.
222 Several LB techniques based on artificial intelligence have been also 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. Simulations results show that the proposed scheme perform near optimal solutions compared to other existing techniques in terms of fairness.
227 \section{Bertsekas and Tsitsiklis' asynchronous load balancing algorithm}
230 In this section, we present a brief description of Bertsekas and Tsitsiklis' algorithm~\cite{bertsekas+tsitsiklis.1997.parallel} using its original notations.
231 A network is modeled as a connected undirected graph $G=(N,A)$, where $N$ is set
232 of processors and $A$ is a set of communication links. The processors are
233 labeled $i = 1,...,n$, and a link between processors $i$ and
234 $j$ is denoted by $(i, j)\in A$. The set of neighbors of processor $i$ is denoted by $V(i)$.
235 %In this work, we consider that
236 %Processors are considered to be homogeneous for the sake of simplicity. It is easily extendable to the case of heterogeneous platforms by scaling the processor's load by its computing power~\cite{ElsMonPre02}.
237 %In order prove the convergence of asynchronous iterative load balancing
238 %Bertsekas and Tsitsiklis proposed a model
239 %in~\cite{bertsekas+tsitsiklis.1997.parallel}. Here we recall some notations.
240 %Consider that $N={1,...,n}$ processors are connected through a network.
241 %Communication links are represented by a connected undirected graph $G=(N,A)$
242 %where $A$ is the set of links connecting different processors.
244 %consider that processors are homogeneous for sake of simplicity. It is quite
245 %easy to tackle the heterogeneous case~\cite{ElsMonPre02}.
246 Load of processor $i$
247 at time $t$ is represented by $x_i(t)\geq 0$.
248 %Let $V(i)$ be the set of neighbors of processor $i$.
249 Each processor $i$ has an estimate of the load of
250 each of its neighbors $j \in V(i)$ denoted by $x_j^i(t)$ and this estimate
251 may be outdated due to %. According to
252 asynchronism and communication delays.
253 %, this estimate may be outdated.
255 %consider that the load is described by a continuous variable.
257 %Since we deal with large {\it fine grain} parallelism of divisible loads,
258 %the processor's load is represented by a continuous variable for notational
261 When a processor sends a part of its load to one or some of its neighbors, the
262 transfer takes time to be completed. Let $s_{ij}(t)$ be the amount of load that
263 processor $i$ has transferred to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
264 amount of load received by $j$ from $i$ at time $t$. Then
265 the amount of load of processor $i$ at time $t+1$ is given by:
268 x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
273 %Some conditions are required to ensure the convergence. One of them can be
274 %called the \emph{ping-pong} condition which specifies that:
276 The asymptotic convergence is derived based on the {\it ping-pong} awareness condition which specifies that:
279 x_i(t)-\sum _{k\in V(i)} s_{ik}(t) \geq x_j^i(t)+s_{ij}(t)
282 for any processor $i$ and any $j \in V(i)$ such that $x_i(t)>x_j^i(t)$.
283 %This condition aims at avoiding a processor to send a part of its load and being
284 %less loaded after that.
287 This condition prohibits the possibility that two nodes keep sending load to each
288 other back and forth, without reaching equilibrium.
291 Nevertheless, we think that this condition may lead to deadlocks in some
292 cases. For example, consider a linear chain graph network of only three processors and that processor $1$
293 is linked to processor $2$ which is also linked to processor $3$, but processors $1$ and $3$ are not neighbors.
294 %(i.e. a simple chain which 3 processors).
296 \noindent Now consider that we have the following load values at time~$t$:
303 %{\bf RAPH, pourquoi il y a $x_3^2$?. Sinon il faudra reformuler la suite, c'est mal dit}
305 Owing to the algorithm's specification, processor $2$ can either send
306 load to processor $1$ or processor
307 $3$. If it sends load to processor $1$, it will not satisfy condition
308 \eqref{eq.ping-pong} because after that sending it will be less loaded than
309 $x_3^2(t)$. So we consider that the \emph{ping-pong} condition is probably too
310 strong. %Currently, we did not try to make another convergence proof without this condition or with a weaker condition.
313 Despite this, we conjecture that a weaker condition may exist since we
314 have never seen any scenario that is not leading to convergence, even with
315 load-balancing strategies that are not exactly fulfilling the authors' own conditions. %se two conditions.
317 %It may be the subject of future work to express weaker conditions, and to prove
318 %that they are sufficient to ensure the convergence of the load-balancing
323 Even though this approach is interesting, several practical
324 questions arise when dealing with realistic models of
325 computation and communication. As reported above, the
326 algorithm's description is too succinct and no details are
327 given on what is really sent and how the load balancing decision
328 are taken. To our knowledge, the only first attempt for a possible
329 implementation of this algorithm is investigated in~\cite{bahi+giersch+makhoul.2008.scalable} under the same conditions. Thus, in order to assess the performances
330 of the new \besteffort{}, we naturally chose to compare it to this anterior
331 work. More precisely, we will use the algorithm~2 from
332 \cite{bahi+giersch+makhoul.2008.scalable} and, through out the paper, we will
333 reference it under the original name {\it Bertsekas and Tsitsiklis} for the sake of convenience and readability.
336 Here is an outline of the main principle of the borrowed algorithm. When a given node $i$ has to take
337 a load balancing decision, it starts by sorting its neighbors by non-increasing
338 order of their loads. Then, it computes the difference between its own load, and
339 the load of each of its neighbors. Finally, taking the neighbors following the
340 order defined before, the amount of load to send $s_{ij}$ is computed as
341 $1/(|V(i)|+1)$ of the load difference%, with $n$ being the number of neighbors
342 . This process is iterated as long as the node is more loaded than the considered
346 \section{Best effort strategy}
347 \label{sec.besteffort}
349 In this section, we describe a new load-balancing strategy that we call
350 \besteffort{}. First, we explain the general idea behind this strategy,
351 and then we present some variants of this basic strategy.
353 \subsection{Basic strategy}
354 The description of our algorithm will be given from the point of view a processor~$i$.
355 The principle of the \besteffort{} strategy is that each processor
356 that detects it has more load than some of its neighbors, sends some load to the
357 most of its less loaded neighbors, doing its best to reach the equilibrium
358 between the involved neighbors and itself.
360 More precisely, %when a processor $i$ is in its load-balancing phase,
361 at each iteration of the load balancing process, processor~$i$
364 \item First, the neighbors are sorted in non-decreasing order of their
365 known loads $x^i_j(t)$.
367 \item Then, this sorted list is used to find its largest
368 prefix such as the load of each selected neighbor is smaller than:
370 \item the load of processor $i$, and
371 \item the mean of the loads of the selected neighbors and processor i.
373 Let $S_i(t)$ be the set of the selected neighbors, and
374 $\bar{x}(t)$ be the mean of the loads between the selected neighbors and processor $i$ which is given as follows:
376 \bar{x}(t) = \frac{1}{\abs{S_i(t)} + 1}
377 \left( x_i(t) + \sum_{j\in S_i(t)} x^i_j(t) \right)
379 so that the following properties hold: %{\bf RAPH : la suite tombe du ciel :-)}
382 S_i(t) \subset V(i) \\
383 x^i_j(t) < x_i(t) & \forall j \in S_i(t) \\
384 x^i_j(t) < \bar{x} & \forall j \in S_i(t) \\
385 x^i_j(t) \leq x^i_k(t) & \forall j \in S_i(t), \forall k \in V(i) \setminus S_i(t) \\
390 \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) =
393 %From the above equations, and notably from the definition of $\bar{x}$, it can easily be verified that:
396 In this way we obtain:
400 x_i(t) - \sum_{j\in S_i(t)} s_{ij}(t) = \bar{x} \\
401 x^i_j(t) + s_{ij}(t) = \bar{x} & \forall j \in S_i(t)
408 \subsection{Leveling the amount of load to move}
410 With the aforementioned basic strategy, each node does its best to reach the
411 equilibrium with its neighbors. However, one question should be outlined here:
412 How can we handle the case where two (or more) node initiators that may send
413 concurrently some amount of load to the the same less loaded neighbor? Indeed,
414 %since each node may take the same kind of decision at the same time,
415 there is a risk that a node will receive load from
416 several of its neighbors, and then is temporary going off the equilibrium state.
417 This is particularly true with strongly connected applications.
421 In order to reduce this effect, we add the ability to level the amount of load to send.
422 The idea, here, is to make as few steps as possible toward the equilibrium, such that a
423 potentially unsuitable decision pointed above has a lower impact on the local equilibrium.
424 Roughly speaking, once $s_{ij}$ is estimated as previously explained, it is simply weighted by
425 a given prescribed threshold parameter which we call
426 %. This parameter is called
428 Section~\ref{sec.results}. The amount of data to send is then $s_{ij}(t) =
429 (\bar{x} - x^i_j(t))/k$.
430 %\FIXME[check that it's still named $k$ in Sec.~\ref{sec.results}]{}
434 %\section{Other strategies}
437 %Another load balancing strategy, working under the same conditions, was
438 %previously developed by Bahi, Giersch, and Makhoul in
439 %\cite{bahi+giersch+makhoul.2008.scalable}. In order to assess the performances
440 %of the new \besteffort{}, we naturally chose to compare it to this anterior
441 %work. More precisely, we will use the algorithm~2 from
442 %\cite{bahi+giersch+makhoul.2008.scalable} and, in the following, we will
443 %reference it under the name of naïve implementation of Bertsekas' load balancing algorithm. {\bf : RAPH j'ai renommé MAKHOUL en naive, il faut valider !!!! LE SOUCI, il faudrait refaire les figures}
445 %Here is an outline of the \makhoul{} algorithm. When a given node needs to take
446 %a load balancing decision, it starts by sorting its neighbors by increasing
447 %order of their load. Then, it computes the difference between its own load, and
448 %the load of each of its neighbors. Finally, taking the neighbors following the
449 %order defined before, the amount of load to send $s_{ij}$ is computed as
450 %$1/(n+1)$ of the load difference, with $n$ being the number of neighbors. This
451 %process continues as long as the node is more loaded than the considered
455 \section{Virtual load}
456 \label{sec.virtual-load}
458 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.
460 %use this concept, load balancing messages must be sent using two different kinds
461 %of messages: load information messages and load balancing messages.
463 precisely, a node wanting to send some amount of its load to one (or more) of its neighbors
464 can first send a load information message about the load it will send, and
465 later it can send the load message containing data to be transferred.
466 Load information messages are in fact short
468 and will be received soon.
470 In contrast, load transfer messages are often larger ones and thus
471 require more time to be transferred.
473 The concept of \emph{virtual load} allows a node receiving a load
474 information message to integrate (virtually) the future load it will receive later in its load's list
475 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
476 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
477 into account the information about the predictive loads not
481 %In fact, a node that receives a load information message knows that
482 %later it will receive the corresponding load balancing message containing the
483 %corresponding data. So, if this node detects it is too loaded compared to some
484 %of its neighbors and if it has enough load (real load), then it can send more
485 %load to some of its neighbors without waiting the reception of the load
488 %Doing this, we can expect a faster convergence since nodes have a faster
489 %information of the load they will receive, so they can take it into account.
491 %\FIXME{Est ce qu'on donne l'algo avec virtual load?}
493 %With integer load, this algorithm has been adapted by rounding the load value. In fact, we consider that the total amount of load is big enough and that it can be split with integer numbers.
498 %\FIXME{describe integer mode}
500 \section{Implementation with SimGrid and simulations}
501 \label{sec.simulations}
503 In order to test and validate our approache, we wrote a simulator
505 framework~\cite{simgrid.web,casanova+giersch+legrand+al.2014.simgrid}. This
506 simulator, which consists of about 2,700 lines of C++, allows to run
507 the different load-balancing strategies under various parameters, such
508 as the initial distribution of load, the interconnection topology, the
509 characteristics of the running platform, etc. Then several metrics
510 were considered to assess and compare the behavior of the different
511 %are issued that permit to compare the
514 The simulation model is detailed in the next section (\ref{sec.model}), and the
515 experimental contexts are described in section~\ref{sec.exp-context}. Then the
516 results of the simulations are presented in section~\ref{sec.results}.
518 \subsection{Simulation model}
521 In the simulation model the processors exchange messages which are of
522 two types. First, there are \emph{control messages} which carry only the information exchanged between processors, such as the
523 current load, or the virtual load transfers if this option is
524 considered. These messages are rather small, and their size is
525 constant. Then, there are \emph{data messages} that carry the real
526 load transferred between processors. The size of a data message
527 is a function of the amount of load that it carries, and it can be
528 pretty large. In order to receive the messages, each processor has
529 two receiving channels, one for each type of messages. Finally, when
530 a message is sent or received, this is done by using non-blocking
531 primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
532 and \texttt{MSG\_task\_irecv()}.}.
534 During the simulation, each processor concurrently runs three threads:
535 a \emph{receiving thread}, a \emph{computing thread}, and a
536 \emph{load-balancing thread}, which we will briefly describe hereafter.
538 For the sake of simplicity, a few details were voluntary omitted from
539 these descriptions. For an exhaustive presentation, we refer to the
540 actual source code that was used for the experiments%
541 \footnote{As mentioned before, our simulator relies on the SimGrid
542 framework~\cite{casanova+giersch+legrand+al.2014.simgrid}. For the
543 experiments, we used a pre-release of SimGrid 3.7 (Git commit
544 67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
545 \url{https://github.com/simgrid/simgrid})}, and which is
547 \url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.
549 \subsubsection{Receiving thread}
551 The receiving thread is in charge of waiting for incoming messages, either on the
552 control channel, or on the data channel. Its behavior is sketched by
553 Algorithm~\ref{algo.recv}. When a message is received, it is pushed in a buffer
554 of received messages, to be later consumed by one of the other threads. There
555 are two such buffers, one for the control messages, and one for the data
557 The buffers are implemented with first-in, first-out queues (FIFO).
560 \caption{Receiving thread}
564 \VAR{ctrl\_chan}, \VAR{data\_chan}
565 & communication channels (control and data) \\
566 \VAR{ctrl\_fifo}, \VAR{data\_fifo}
567 & buffers of received messages (control and data) \\
570 wait for a message to be available on either \VAR{ctrl\_chan},
571 or \VAR{data\_chan}\;
572 \If{a message is available on \VAR{ctrl\_chan}}{%
573 get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
575 \If{a message is available on \VAR{data\_chan}}{%
576 get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
581 \subsubsection{Computing thread}
583 The computing thread is in charge of the real load management. As outlined in
584 Algorithm~\ref{algo.comp}, it iteratively runs the following operations:
586 \item if some load was received from the neighbors, get it;
587 \item if there is some load to send to the neighbors, send it;
588 \item run some computations, whose duration is a function of the processor's current
591 Practically, after the computation, the computing thread waits for a
592 small amount of time if the iterations are looping too fast (for
593 example, when the current load is near zero).
596 \caption{Computing thread}
600 \VAR{data\_fifo} & buffer of received data messages \\
601 \VAR{real\_load} & current load \\
604 \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
605 wait until a message is pushed into \VAR{data\_fifo}\;
607 \While{\VAR{data\_fifo} is not empty}{%
608 pop a message from \VAR{data\_fifo}\;
609 get the load embedded in the message, and add it to \VAR{real\_load}\;
611 \ForEach{neighbor $n$}{%
612 \If{there is some amount of load $a$ to send to $n$}{%
613 send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
616 \If{$\VAR{real\_load} > 0.0$}{
617 simulate some computation, whose duration is function of \VAR{real\_load}\;
618 ensure that the main loop does not iterate too fast\;
623 \subsubsection{Load-balancing thread}
625 The load-balancing thread is in charge of running the load-balancing algorithm,
626 and exchange the control messages. As shown in Algorithm~\ref{algo.lb}, it
627 iteratively runs the following operations:
629 \item get the control messages that were received from the neighbors;
630 \item run the load-balancing algorithm;
631 \item send control messages to the neighbors, to inform them about the
632 processor's current load, and possibly the future virtual load transfers;
633 \item wait a minimum (configurable) amount of time, to avoid to
638 \caption{Load-balancing}
641 \While{\VAR{ctrl\_fifo} is not empty}{%
642 pop a message from \VAR{ctrl\_fifo}\;
643 identify the sender of the message,
644 and update the current knowledge of its load\;
646 run the load-balancing algorithm to make the decision about load transfers\;
647 \ForEach{neighbor $n$}{%
648 send a control messages to $n$\;
650 ensure that the main loop does not iterate too fast\;
654 %\paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
655 % par ex, donner l'idée générale de l'implémentation. l'idée générale est déja
656 % décrite en section~\ref{sec.virtual-load}}
658 \subsection{Experimental contexts}
659 \label{sec.exp-context}
661 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.
663 \subsubsection{Load balancing strategies}
665 Several load balancing strategies were compared. Experiments with
666 the \besteffort{}, and with the \makhoul{} strategies have been performed. First the \emph{best
667 effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$. Then,
668 each strategy was run in its two variants: with, and without the management of
669 \emph{virtual load}. Finally, each configuration with \emph{real},
670 and with \emph{integer} load values is considered.
672 To summarize the different load balancing strategies, we have:
674 \item[\textbf{strategies:}] \makhoul{}, or \besteffort{} with $k\in
676 \item[\textbf{variants:}] with, or without virtual load
677 %\item[\textbf{domain:}] real load, or integer load
680 %This gives us as many as $4\times 2\times 2 = 16$ different strategies.
682 \subsubsection{End of the simulation}
684 The simulations were run until reaching the global equilibrium threshold.
685 %the load was nearly balanced among the participating nodes.
686 More precisely the simulation stops when each node holds
687 an amount of load at least less than 1\% of the load average.
688 %, during an arbitrary
689 %number of computing iterations (2000 in our case).
691 %Note that this convergence detection was implemented in a centralized manner.
692 %This is easy to do within the simulator, but it is obviously not realistic. In a
693 %real application we would have chosen a decentralized convergence detection
694 %algorithm, like the one described in \cite{ccl09:ij}.
696 \subsubsection{Platform}
698 %In order to show the behavior of the different strategies
700 %settings, we simulated the executions on two sorts of platforms. These two
701 %sorts of platforms differ by their network topology. On the one hand,
702 %we have homogeneous platforms, modeled as a cluster. On the other hand, we have
703 %heterogeneous platforms, modeled as the interconnection of a number of clusters.
706 %The clusters are modeled by a fixed number of computing nodes interconnected
707 %through a backbone link. Each computing node has a computing power of
708 %1~GFlop/s, and is connected to the backbone by a network link whose bandwidth is
709 %of 125~MB/s, with a latency of 50~$\mu$s. The backbone has a network bandwidth
710 %of 2.25~GB/s, with a latency of 500~$\mu$s.
712 In order to make our experiments, an heterogeneous grid platform descriptions were created by taking a subset of the
713 Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
714 Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
715 \texttt{g5k.xml} distributed with SimGrid. Note that the heterogeneity of the
716 platform here only comes from the network topology. Indeed,
717 processors are considered to be homogeneous for the sake of simplicity.
718 It is easily extendable to the case of heterogeneous platforms
719 by scaling the processor's load by its computing power~\cite{ElsMonPre02}.
721 %algorithms currently do not handle heterogeneous computing resources,
723 processor speeds were normalized, and we arbitrarily chose to fix them to
724 1~GFlop/s. Each type of platform with four different numbers of computing
725 nodes: 16, 64, 256, and 1024 nodes is built in a similar way.
727 \subsubsection{Configurations}
729 The distributed processes of the application were then logically organized along
730 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)
731 tests were performed with the total load initially on only one node%(at one end for the line topology)
732 , ii) and other tests for which the load was initially randomly distributed across all the
733 participating nodes of the platform. The total amount of load was fixed to a number of load
734 units equal to 1000 times the number of node. The average load is then of 1000
737 For all the previous configurations, the
738 computation and communication costs of a load unit are defined. We chose them, such as to
739 have two different computation to communication ratios (CCR), and hence characterize
740 two different types of applications:
742 \item mainly communicating, with a CCR of $1/10$;
743 \item mainly computing, with a CCR of $10/1$.
744 %\item balanced, with a computation/communication cost ratio of $1/1$.
747 % To summarize the various configurations, we have:
748 % \begin{description}
749 % \item[\textbf{platforms:}] homogeneous (cluster), or heterogeneous (subset of
751 % \item[\textbf{platform sizes:}] platforms with 16, 64, 256, or 1024 nodes
752 % \item[\textbf{process topologies:}] line, torus, or hypercube
753 % \item[\textbf{initial load distribution:}] initially on a only node, or
754 % initially randomly distributed over all nodes
755 % \item[\textbf{computation/communication cost ratio:}] $10/1$, $1/1$, or $1/10$
758 % This gives us as many as $2\times 4\times 3\times 2\times 3 = 144$ different
761 % Combined with the various load balancing strategies, $16\times 144 =
762 % 2,304$ distinct settings have been evaluated. In fact, as it will be shown later, only configurations with a maximum number of 1,024 nodes are considered in order to limit the time of experiments.
765 \subsubsection{Metrics}
768 In order to evaluate and compare the different load balancing strategies, several metrics were considered. Our goal, when choosing these metrics, is to have
769 something tending to a constant value, i.e. to have a measure which is not
770 changing anymore once the convergence state is reached. Moreover, we want to
771 have some normalized value, in order to be able to compare them across different
772 settings. With these constraints in mind, we define the following metrics:
775 \item[\it{average idle time:}] that is the total time spent, when the nodes
776 do not hold any share of load, and thus have nothing to compute.
778 %time is divided by the number of participating nodes, such as to have a number
779 %that can be compared between simulations of different sizes.
780 %This metric is expected to give an idea of the ability of the strategy to
781 %diffuse the load quickly.
782 A smaller value is better.
784 \item[\it{average convergence time:}] that is the average of the times when
785 all nodes reached the final balanced load distribution. Times are measured as a number
786 of (simulated) seconds from the beginning of the simulation.
788 \item[\it{maximum convergence time:}] that is the time when the last node
789 reached the final stable equilibrium.
790 %These two dates give an idea of the time needed by the strategy to reach the
792 A smaller value is better.
794 % \item[\textbf{data transfer amount:}] that is the sum of the amount of all data
795 % transfers during the simulation. This sum is then normalized by dividing it
796 % by the total amount of data present in the system.
798 % This metric is expected to give an idea of the efficiency of the strategy in
799 % terms of data movements, i.e. its ability to reach the equilibrium with fewer
800 % transfers. Again, a smaller value is better.
805 \subsection{Experimental results}
808 In this section, the results for the different simulations are presented,
809 and our observations are explained.
811 % \subsubsection{Cluster versus grid platforms}
813 % As mentioned earlier, different algorithms have been simulated on two kinds of
814 % physical platforms: clusters and grids. A first observation,
815 % is that the graphs we draw from the data have a similar aspect for the two kinds
816 % of platforms. The only noticeable difference is that the algorithms need a bit
817 % more time to achieve the convergence on the grid platforms, than on clusters.
818 % Nevertheless their relative performances remain generally similar.
820 % This suggests that the relative performances of the different strategies are not
821 % influenced by the characteristics of the physical platform. The differences in
822 % the convergence times can be explained by the fact that on the grid platforms,
823 % distant sites are interconnected by links of smaller bandwidth.
825 % Therefore, in the following, we only discuss the results for the grid
828 \subsubsection{Main results}
832 \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-line}%
833 \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-line}
834 \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-torus}%
835 \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-torus}
836 \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-hcube}%
837 \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-hcube}
838 \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
839 time, the second part represents the average convergence time, and then the third part represents the maximum convergence time.}
845 \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-line}%
846 \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-line}
847 \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-torus}%
848 \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-torus}
849 \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-hcube}%
850 \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-hcube}
851 \caption{Real mode, random initial distribution, CCR = $10/1$ (left), or $1/10$ (right).}
855 The main results for our simulations on grid platforms are presented in Figures~\ref{fig.results1} and~\ref{fig.resultsN}.
857 The results in Figure~\ref{fig.results1} are when the load to balance is
858 initially on only one node, while the results in Figure~\ref{fig.resultsN} are
859 when the load to balance is initially randomly distributed over all nodes.
860 On both figures, the CCR is $10/1$ on the left
861 column, and $1/10$ on the right column. %With a computation/communication cost
862 %ratio of $1/1$ the results are just between these two extrema, and definitely
863 %don not give additional information, so we chose not to show them here.
864 On each of Figures~\ref{fig.results1} and~\ref{fig.resultsN}, the results
865 are given for the process topology being, from top to bottom, a line, a torus or
868 Finally, the vertical bars show the measured times for the evaluated metrics
869 %each of the algorithms
870 . These measured times are, starting at $t=0$ and from bottom to top, the average idle
871 time, the average convergence time, and the maximum convergence time (see
872 Section~\ref{sec.metrics}). The measurements are repeated for the different
873 platform sizes. Some bars are missing, especially for large platforms. This is
874 because the algorithm did not reach the convergence state in the
878 %\FIXME{annoncer le plan de la suite}
880 \subsubsection{The \besteffort{} and \makhoul{} strategies without virtual load}
882 The {\it simple} ({\it plain}) version of each strategy is defined as the load balancing
883 algorithm without virtual load's transfers. For each strategy, we compare the simple
884 version (without virtual load) and the improved one (with virtual load).
885 Each algorithm is evaluated in terms of achieved idle time and convergence time.
887 Before looking at the different variations, we will first show that the simple
888 \besteffort{} strategy is valuable, and may be as good as the \makhoul{}
889 strategy. On Figures~\ref{fig.results1} and~\ref{fig.resultsN},
890 these strategies are respectively labeled ``b'' and ``a''.
892 We can see that the relative performance of these strategies is mainly
893 influenced by the application topology structure. It is for the line topology that the
894 difference is the more important. In this case, the \besteffort{} strategy is
895 readily faster than the \makhoul{} strategy. This can be explained by the
896 fact that the \besteffort{} strategy tries to distribute the load fairly between
897 all the nodes and are in a good agreement with the line topology since it is easy
898 to load balance the load efficiently.
900 In contrast, for the hypercube topology, the \besteffort{}' performances are lower than
901 the \makhoul{} strategy. In this case, the \makhoul{} strategy which
902 tries to give more load to few neighbors reaches the equilibrium faster.
904 For the torus topology, for which the number of links is between the line and
905 the hypercube, the \makhoul{} strategy is slightly better but the difference is
906 more nuanced when the initial load is only on one node. The only case where the
907 \makhoul{} strategy is really faster than the \besteffort{} strategy is with the
908 random initial distribution when the communication are slow.
910 Generally speaking, the number of interconnection is very important. Indeed, the more
911 the interconnection links are, the faster the \makhoul{} strategy is because
912 it distributes quickly significant amount of load, even if this is unfair, between
913 all the neighbors. However, the \besteffort{} strategy distributes the
914 load fairly when needed and is is better for sparse connected applications.
920 \subsubsection{With virtual load}
922 The impact of virtual load scheme is most of the time really significant compared to
923 the simple version of the algorithm with the same configuration. %Sometimes it has no effect but, based on our observations, it has never a negative effect on the load balancing we tested.
924 For instance, as can be seen from Figure~\ref{fig.results1}, when the load is initially on one node, it can be
925 noticed that the average idle times are generally longer with the virtual load
926 than the simple version. This can be explained by the fact that, with virtual load,
927 processors will exchange all the load they need to exchange as soon as the
928 virtual load has been balanced between all the processors. As a consequence, they
929 cannot compute at the beginning. This is especially noticeable when the
930 communication are slow (on the left part of Figure ~\ref{fig.results1}).
933 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
934 that for small CCR values, high communication costs plays quite a significant role. Moreover, the impact of
935 communication becomes less important as the CCR values increases, since larger CCR values result in smaller communication times. We also tested the impact of CCR values on the performance of each algorithm in terms of idle times. From Figures~\ref{fig.results1} and ~\ref{fig.resultsN} we can find that our virtual load scheme achieves
936 a 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 the most of times compared to fine grain applications.
939 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.
940 These results indicate that local load balancing decisions has a significant impact on the global
941 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.
944 Unlike {\it Betsekas and Tsistlikis} algorithm, our approach is not really sensitive when
945 we deal 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 load difference between the computational resources is minimized as low as possible.
948 Comparing the results of the extended version (with virtual load) to the results of the simple one, we observe 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 optimize 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}.
951 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 reveals 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.
954 Finally, it is worthwhile noting from Figures~\ref{fig.results1} and ~\ref{fig.resultsN}, that the algorithm's convergence time increases together with network's size. 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}),
955 as expected. In addition, it is interesting to note that when the number of nodes increases, there is not 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.
957 %For the hypercube, in any case, the effect of the virtual load is visible. It is more visible when communications have a more important role (i.e. with the mainly communicating case).
960 %Dans ce cas légère amélioration de la cvg. max. Temps moyen de cvg. amélioré,
961 %mais plus de temps passé en idle, surtout quand les comms coutent cher.
963 %\subsubsection{The \besteffort{} strategy with an initial random load
964 % distribution, and larger platforms}
967 %Mêmes conclusions pour line et hcube.
968 %Sur tore, BE se fait exploser quand les comms coutent cher.
970 %\FIXME{virer les 1024 ?}
972 %\subsubsection{With the virtual load extension with an initial random load
975 %Soit c'est équivalent, soit on gagne -> surtout quand les comms coutent cher et
976 %qu'il y a beaucoup de voisins.
978 \subsubsection{The $k$ parameter}
981 As explained previously when the communication are slow the \besteffort{}
982 strategy is efficient. This is due to the fact that it tries to balance the load
983 fairly and consequently a significant amount of the load is transferred between
984 processors. In this case, it is possible to reduce the convergence time by
985 using the leveler parameter (parameter $k$). The advantage of using this
986 solution is particularly efficient when the initial load is randomly distributed
987 on the nodes with torus and hypercube topologies and slow communication. When
988 virtual load scheme is used, the effect of this parameter is also perceptible
989 with the same conditions. %However, sometimes this parameter may have a negative effect on the convergence time.
994 \subsubsection{With non negative integer load values}
995 In addition to the first tests devoted to the case of non negative real load values, we also carried out further experiments with integer load values to assess the performance of our proposal.
996 %We also performed some experiments with integer load instead of load with real
1000 the obtained results have globally the same behavior, that is why we decided not to show similar figures. The most
1001 interesting result, from our point of view, is that the virtual mode allows
1002 processors in a line topology to converge to the uniform load balancing state. Without
1003 the virtual load, most of the time, processors converge to what we call the
1004 ``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
1005 obtain 10 9 8 7 6 6 7 8 9 10 instead of 8 8 8 8 8 8 8 8 8 8).
1008 To summarize and broadly speaking, the simulation results led us to show that, with 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 corroborate 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.
1011 %Cas normal, ligne -> converge pas (effet d'escalier).
1012 %Avec vload, ça converge.
1014 %Dans les autres cas, résultats similaires au cas réel: redire que vload est
1017 %\FIXME{ajouter une courbe avec l'équilibrage en entier}
1019 %\FIXME{virer la metrique volume de comms}
1021 %\FIXME{ajouter une courbe ou on voit l'évolution de la charge en fonction du temps : avec et sans vload}
1024 % \item cluster ou grid, entier ou réel, ne font pas de grosses différences
1025 % \item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
1026 % \item makhoul? se fait battre sur les grosses plateformes
1027 % \item taille de plateforme?
1028 % \item ratio comp/comm?
1029 % \item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
1030 % \item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
1031 % \item répartition initiale de la charge ?
1032 % \item integer mode sur topo. line n'a jamais fini en plain? vérifier si ce n'est
1033 % pas à cause de l'effet d'escalier que bk est capable de gommer.
1036 % On veut montrer quoi ? :
1038 % 1) best plus rapide que les autres (simple, makhoul)
1039 % 2) avantage virtual load
1041 % Est ce qu'on peut trouver des contre exemple?
1042 % Topologies variées
1045 % Simulation avec temps définies assez long et on mesure la qualité avec : volume de calcul effectué, volume de données échangées
1046 % Mais aussi simulation avec temps court qui montre que seul best converge
1048 % Expés avec ratio calcul/comm rapide et lent
1050 % Quelques expés avec charge initiale aléatoire plutot que sur le premier proc
1052 % Cadre processeurs homogènes
1054 % Topologies statiques
1056 % On ne tient pas compte de la vitesse des liens donc on la considère homogène
1058 % Prendre un réseau hétérogène et rendre processeur homogène
1060 % Taille : 10 100 très gros
1062 \section{Conclusion}
1063 \label{conclusions-remarks}
1065 In this paper, we have presented a new asynchronous load balancing algorithm for non negative real numbers
1066 of divisible loads in distributed systems. The proposed algorithm which is called {\it best effort strategy}
1067 seeks greedily for loads imbalance detection and tries to achieve efficient local load equilibrium
1068 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.
1069 This leads to a noticeable speedup of the global convergence time of the load balancing process.
1070 Based on SimGrid simulator, we have demonstrated that, when we deal 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.
1072 \section*{Acknowledgments}
1074 This paper is partially funded by the Labex ACTION program (contract
1075 ANR-11-LABX-01-01). We also thank the supercomputer facilities of the Mésocentre de calcul de Franche-Comté.
1077 \bibliographystyle{elsarticle-num}
1078 \bibliography{biblio}
1079 %\FIXME{find and add more references}
1083 %%% Local Variables:
1087 %%% ispell-local-dictionary: "american"
1090 % LocalWords: Raphaël Couturier Arnaud Giersch Franche ij Bertsekas Tsitsiklis
1091 % LocalWords: SimGrid DASUD Comté asynchronism ji ik isend irecv Cortés et al
1092 % LocalWords: chan ctrl fifo Makhoul GFlop xml pre FEMTO Makhoul's fca bdee
1093 % LocalWords: cdde Contassot Vivier underlaid du de Maréchal Juin cedex calcul
1094 % LocalWords: biblio Institut UMR Université UFC Centre Scientifique CNRS des
1095 % LocalWords: École Nationale Supérieure Mécanique Microtechniques ENSMM UTBM
1096 % LocalWords: Technologie Bahi
