Simulated Annealing Algorithm Combined with Chaos for Task Allocation in Real-Time Distributed ...

14 Pages·2015·2.56 MB·English

Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 151394, 13 pages
http://dx.doi.org/10.1155/2014/151394
Research Article
Simulated Annealing Algorithm Combined with Chaos for
Task Allocation in Real-Time Distributed Systems
WenboWu,JiahongLiang,XinyuYao,andBaohongLiu
CollegeofInformationSystemandManagement,NationalUniversityofDefenseTechnology,Changsha,Hunan410073,China
CorrespondenceshouldbeaddressedtoWenboWu;[e-mail]
Received17May2014;Revised24July2014;Accepted3August2014;Published14August2014
AcademicEditor:Jun-JuhYan
Copyright©2014WenboWuetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
This paper addresses the problem of task allocation in real-time distributed systems with the goal of maximizing the system
reliability,whichhasbeenshowntobeNP-hard.Wetakeaccountofthedeadlineconstrainttoformulatethisproblemandthen
proposeanalgorithmcalledchaoticadaptivesimulatedannealing(XASA)tosolvetheproblem.Firstly,XASAbeginswithchaotic
optimizationwhichtakesachaoticwalkinthesolutionspaceandgeneratesseverallocalminima;secondlyXASAimprovesSA
algorithm via several adaptive schemes and continues to search the optimal based on the results of chaotic optimization. The
effectivenessofXASAisevaluatedbycomparingwithtraditionalSAalgorithmandimprovedSAalgorithm.Theresultsshow
thatXASAcanachieveasatisfactoryperformanceofspeedupwithoutlossofsolutionquality.
1.Introduction alargeandcomplexsystem,thenodesandcommunication
linksfailuresareinevitable.Hence,thereliabilityisacrucial
Inmanyapplicationdomains,(e.g.,astronomy,geneticengi- requirementforDS,especiallyforthereal-timeDS(RTDS).
neering, and military systems), increased complexity and Distributed system reliability (DSR) has been defined
scale has led to the need for more powerful computa- by Kumar et al. [8] as the probability for the successful
tion resources; distributed systems (DS) have emerged as completion of distributed programs which requires that all
a powerful platform for addressing this issue, alternating the allocated processors and involved communication links
to traditional high performance computing systems. DS are operational during the execution lifetime. Redundancy
consists ofa set ofcooperatingnodes (eitherhomogeneous and diversity is the traditional technique to attain better
orheterogeneous)communicatingoverthecommunication reliability[6,7,9–14].Theyprocesshardwareand/orsoftware
links.AnapplicationrunninginaDScouldbedividedinto redundancy, hence impose extra cost. Moreover, in many
a number of tasks and executed concurrently on different situations,thesystemconfigurationisfixedandwehaveno
nodesinthesystem,referredtoasthetaskallocationproblem freedomtointroducesystemredundancy.Taskallocationis
(TAP). To improve the performance of DS, several studies thealternativewaytoimproveDSreliability,andthismethod
have been devoted to the TAP with the main concern on doesnotrequireadditionalresources,neitherhardwarenor
theperformancemeasuressuchasminimizingtheexecution software.
and communication cost [1–3], minimizing the application The TAP with the goal of maximizing the DSR is a
turnaroundtime[4, 5], andachieving better faulttolerance typical combinatorial optimization problem; unfortunately,
[6,7]. this has been shown to be NP-hard in strong sense, and
Ontheotherhand,thereal-timepropertyisrequiredin the computational complexity of optimal algorithms (e.g.,
manyDS(e.g.,militarysystems).Insuchsystem,theappli- branch and bound technique) is exponential in nature. We
cation should complete its work before deadline, not only cannotobtaintheoptimalresultsinreasonabletimeforlarge
promisingthelogicalcorrectness.WhilethecomplexityofDS scale problems. Hence, several heuristic and metaheuristic
couldincreasethepotentialofsystemfailure,becauseinsuch algorithmshavebeenimplemented,suchasgeneticalgorithm
2 MathematicalProblemsinEngineering
(GA) [7, 14, 15], simulated annealing algorithm (SA) [16], of the TAP with the goal of maximizing reliability, and the
particle swarm optimization (PSO) [17], honeybee mating solution approach is presented in Section4 with some
optimization (HMO) [18], cat swarm optimization (CSO) details of implementation. Section5 discusses the perfor-
[19], and iterated greedy algorithm (IG) [20]. These algo- mance evaluation of proposed algorithm according to sev-
rithms may obtain suboptimal results, but they can sharply eral experiments and analyses. And Section6 concludes
reducethecalculationtime. thiswork.
Acommonthreadamongthesealgorithmsisthattheyall
startfromarandomlychoseninitialsolutionorasetofsolu- 2.RelatedWorks
tionsinthesolutionspaceandthenrepeattheexploration-
decision procedure until convergence and obtaining good Theideaofsimulatedannealingalgorithmwasproposedby
enough solutions (maybe suboptimal results) [21]. Here, Metropolisetal.[22]in1953,andwasappliedtooptimization
explorationmeansobtainingnewsolutionsbasedonthecur- problemsbyKirkpatricketal.[25]in1983.Toourknowledge,
rentsolution,inSAalgorithm;forinstance,newsolutionsare
thefirstapplicationofSAalgorithmtotheTAPwasmadeby
chosenfromtheneighborsofthecurrentsolution.Decision
vanLaarhovenetal.[26]in1992,whichappliedSAalgorithm
is made after exploration: a new solution is either accepted
to a job shop scheduling problem. From then on, several
or rejected according to some rules, if the new solution is
works [27–29] have been done that compare SA algorithm
accepted, then it becomes new current solution, and move
to other optimization algorithms for problems related to
on, otherwise drop it, start new exploration. According to
the TAP. Attiya and Hamam applied SA algorithm to solve
a series of the exploration-decision procedures, the quality
theTAPandcompareditwithbranch-and-boundtechnique
of the obtained solutions becomes better and better until
in 2006 in terms of maximizing the reliability of DS [16];
meetingtheterminationconditionwhichisoftendefinedas
extendingthiswork,Faragardietal.proposedimprovedSA
someconvergentsituations.Hence,theconvergencespeedof
algorithmstosolvethisproblem[30,31],usingthehybridof
algorithms is affected by the choice of initial solutions and
SA and tabu search with a nonmonotoniccooling schedule
rulesthatareappliedintheexploration-decisionprocedures.
Simulatedannealingalgorithmisoneoftheearliestand in 2012 and adding systematic search of neighborhood and
most wildly used optimization approaches; it introduces memorytoSAalgorithmin2013.
randomfactorinsearchingprocessandmodelstheannealing COAoftencombineswithotheroptimizationalgorithm
of solids as Metropolis process [22]. SA algorithm accepts toovercomeitsdrawbacksandtakeadvantageofitsbeneficial
“worse” solution with some probabilities related to the cur- property such as ergodicity, for example, chaotic simulated
renttemperature,soitcanescapefromthelocaloptimaand annealing (CSA) [32], chaotic particle swarm optimization
find the global optimal solution. The convergence speed of (CPSO) [23], and chaotic improved imperialist competitive
SAalgorithmisdependingonitsinitialsolutionandcooling algorithm(CICA)[33]etal.CSAwasproposedbyChenand
schedule. Aihara[32]tosolvecombinatorialoptimizationproblemsin
Chaos is a bounded unstable dynamic behavior that
1995, which used Hopfield neural networks. Then, several
exhibits sensitive dependence on initial conditions and
other papers have expanded this work [34–37]. Mingjun
includes infinite unstable periodic motions [23]. Although
and Huanwen have proposed another version of CSA to
it appears to be stochastic, it occurs in a deterministic
solvetheoptimizationproblemofcontinuousfunctions[38].
nonlinear system under deterministic conditions. In recent
Mostoftheseworksfocusontheapplicationofcontinuous
years, chaotic optimization algorithm (COA) has aroused
functionsoptimization,whilethemethodproposedbyChen
intense interests due to its ergodicity, easy implementation,
andAiharaisactuallybasedonartificialneuralnetwork,not
and ability to escape local optima [24]. However, COA is
SAalgorithm.FerensandCook[39]adaptedCSAdeveloped
lack of heuristic, mostly needs a large number of iterations
by Mingjun and Huanwen into the TAP in 2013, where
to reach the global optimum, which means its convergence
chaos was infused into a solution by setting the number
speedisslow.
of perturbations made by the value of a chaotic variable.
In this paper, we propose a combinational algorithm
calledXASA(ChaoticAdaptiveSimulatedAnnealing),where However,thismethoddoesnotmakefulluseofthebeneficial
X alludes to the Greek spelling of chaos (𝜒𝛼𝑜𝜍) and which property of COA and does not improve the convergence
is proposed to solve the TAP in RTDS with the goal of speedeither.
maximizingtheDSR.Wetakeintoaccountseveralkindsof The present paper differs from the above mentioned
constraintsincludingdeadline.XASAstartsfromCOAand researchesbecauseitcancombinetheadvantagesofbothtwo
obtainsseveraloptimaviaitsergodicity,thenSAalgorithm algorithms.Firstly,withtheergodicityproperty,COAcanget
willoperatebasedontheseoptimainrelativesmallerranges theskeletonofsolutionspacebychaoticwalkinginit,thereby,
to find the best solution. This method can overcome the preventingtheresultfromfallingintolocaloptima.Secondly,
slowconvergenceofSAalgorithmandCOAwithoutlossof based on the results of COA, we can easily determine the
solutionquality. coolingscheduleofSAalgorithmwhichisveryimportantto
The rest of this paper is organized as follows. Section2 theperformanceofSAalgorithmbuthardtodealwith.Lastly,
presentstherelatedworkintheapplicationofSAalgorithm severaladaptiveschemesareusedinSAalgorithm;allthese
andChaoticOptimizationtotheTAP,andthecontributions schemesincludingCOApreliminaryresultscanincreasethe
ofthispaperarestated.Section3describestheformulation convergencespeedsignificantly.
MathematicalProblemsinEngineering 3
Processor S1 {10,16,11} {14,5,30}
Memory M1 [1;2] [2;2]
Node1 t1 3 t2
Processor S2 {12,17,25}
Memory M2 4 [5;3] 2 5{11,8,12}
[4;1]
Node2 t3
t
Processor S3 2 {4,[621;4,1]3} 4
Memory M3 1
t
Node3 5
(a) Adistributedsystem (b) Ataskgraph
Figure1:Resultsoftheevaluation.
3.ProblemStatement We assume that all components of the node except
processorareperfect,whichmeansthereliabilityofthenode
WeconsideraheterogeneousDSthatrunsareal-timeappli- equals the reliability of its processor. The reliability of the
cation.EachnodeofDSmayhavedifferentprocessingspeed, processorattime𝑡is𝑒−𝜆𝑘𝑡 [16];underataskassignment𝑋,
memorysize,andfailurerate.Moreover,thecommunication the total execution time of tasks that are assigned to 𝑝𝑘 is
lsitnaktesomfanyoadlessoahnadvecodmiffmeruennitcbaatinodnwliindkthsaisneditfhaeilruoreperraateti.oThnael ∑𝑀𝑖=1𝑥𝑖𝑘𝑒𝑖𝑘,sothereliabilityofthenode𝑝𝑘is
orfailed,andthefailureeventsarestatisticallyindependent. 𝑅 (𝑋)=𝑒−𝜆𝑘∑𝑀𝑖=1𝑥𝑖𝑘𝑒𝑖𝑘. (1)
Wealsoassumethatthefailureratesofnodesandcommuni- 𝑘
cationlinksareconstant. Similarly, the reliability of the communication link 𝑙𝑘𝑏
Thereare𝑀taskstobeexecutedonaDSwith𝑁nodes, at time 𝑡 is 𝑒−𝜑𝑘𝑏𝑡, under a task assignment 𝑋; the total
and 𝑀 > 𝑁 for most cases. Tasks executed on the node communicationtimevia𝑙𝑘𝑏is∑𝑀𝑖=1∑𝑖=𝑗̸ 𝑥𝑖𝑘𝑥𝑗𝑏(𝑐𝑖𝑗/𝜔𝑘𝑏),sothe
require resources, including processing load resources and reliabilityofthecommunicationlink𝑙𝑘𝑏is
memoryspace.Additionally,twotasksexecutedondifferent
nodes require communication bandwidth to communicate 𝑅 (𝑋)=𝑒−𝜑𝑘𝑏∑𝑀𝑖=1∑𝑖≠𝑗𝑥𝑖𝑘𝑥𝑗𝑏(𝑐𝑖𝑗/𝜔𝑘𝑏). (2)
𝑘𝑏
with each other.Figure1 illustratesa simple case consisting
of5tasksand3nodes.AnapplicationrunningonDScanbe Hence,weobtainthereliabilityofthesystem:
representedbyataskinteractiongraph𝐺(𝑉,𝐸)asshownin
𝑁 𝑁−1 𝑁
tFhigeuirnet1e(rba)c,twiohnesreb𝑉etwreepernesetwntosatassektso.fEtaasckhsatnasdk𝐸𝑖rep∈res𝑉entiss 𝑅(𝑋)=∏𝑅𝑘(𝑋)∏ ∏ 𝑅𝑘𝑏(𝑋)=𝑒−𝑌(𝑋), (3)
𝑘=1 𝑘=1𝑏=𝑘+1
associated with two properties: {𝑞𝑖1,𝑞𝑖2,...,𝑞𝑖𝑁} represents
the execution time of the task at different node and [𝑠𝑖;𝑚𝑖] where
representstheprocessingloadandmemoryrequirementsof 𝑁 𝑀
thetask.Andthelabelofeachedge(𝑖,𝑗) ∈ 𝐸representsthe 𝑌(𝑋)= ∑∑𝜆𝑘𝑥𝑖𝑘𝑒𝑖𝑘
communicationrequirementsamongtasks. 𝑘=1𝑖=1
(4)
The purpose of this work is to find a task assignment
𝑁 𝑁 𝑀 𝑐
thatall𝑀tasksareassignedto𝑁nodes(notethatonetask +∑ ∑ ∑∑𝜑 𝑥 𝑥 ( 𝑖𝑗 ).
𝑘𝑏 𝑖𝑘 𝑗𝑏 𝜔
should be assigned to one and only one node, while one 𝑘=1𝑏=𝑘+1𝑖=1𝑖≠𝑗 𝑘𝑏
node can execute multitasks or none), so that the overall
system reliability is maximized, the deadline and other
3.3.Constraints. Inordertoachieveasatisfactoryallocation,
requirementsoftasksaresatisfied,andthecapacitiesofthe
there are several basic constraints of the TAP in RTDS
systemresourcesarenotviolated.
that should be met. Traditionally, the aim of allocation
constraints is devoted to not violate the availability of the
3.1. Notations. The notationsthat are used to formulate the systemresources,likememorycapacity.Whilethedeadline
problemarelistedinAbbreviationSection. requirementshouldbemetforreal-timepropertyaswell.
3.2.SystemReliability. Reliabilityofadistributedsystemmay (i)MemoryConstraint.Thetotalamountofmemoryrequire-
bedefinedastheprobabilitythatthesystemcanruntheentire ments of tasks assigned to a node should not exceed the
applicationsuccessfully[7,14,40].Duetotheindependence capacityofthenode.Thatis,
ofthefailuresofthenodeandpath,thesystemreliabilityisthe
𝑀
productofthecomponentsreliabilities.Whichistheproduct ∑𝑚𝑖𝑥𝑖𝑘 ≤𝑀𝑘, ∀𝑘∈[1,𝑁]. (5)
ofthereliabilitiesofallnodesandcommunicationlinks.
𝑖=1
4 MathematicalProblemsinEngineering
(ii) Computation Resource Constraints. The total amount of Step3. Settheinitialsolutionastheoptimal𝑓∗ ←𝑓𝑠,𝑋∗ ←
computation resource requirements of tasks assigned to a 𝑋𝑠.
nodeshouldnotexceedthecapacityofthenode.Thatis,
𝑀 Step4. Initializethetemperature(𝑇=𝑇0).
∑𝑠𝑖𝑥𝑖𝑘 ≤𝑆𝑘, ∀𝑘∈[1,𝑁]. (6) Step5. Selectaneighbor(𝑋𝑛)of𝑋𝑠.
𝑖=1
Step6. Calculatethecost(𝑓𝑛)of𝑋𝑛.
(iii)CommunicationResourceConstraints.Thetotalamount Step7. If𝑓𝑛 ≤ 𝑓𝑠,then𝑋𝑠 ← 𝑋𝑛and𝑓𝑠 ← 𝑓𝑛,otherwisego
of communication resource requirements of tasks via a
toStep9.
communication link should not exceed the capacity of the
link.Thatis, Step8. If𝑓𝑛 <𝑓∗,then𝑋∗ ←𝑋𝑛and𝑓∗ ←𝑓𝑛,gotoStep10.
𝑀
∑∑𝑐𝑖𝑗𝑥𝑖𝑘𝑥𝑗𝑏 ≤𝐶𝑘𝑏, 1≤𝑘<𝑏≤𝑁. (7) Step9. Ifrandom(0,1)<𝑒−(𝑓𝑛−𝑓𝑠)/𝑇,then𝑋𝑠 ←𝑋𝑛and𝑓𝑠 ←
𝑖=1𝑗≠𝑖 𝑓𝑛.
Step 10. Repeat Step5 to Step9 for a given number of
(iv)DeadlineConstraints.Alltasksshouldcompleteexecution iterations.
before their deadline. Since there is no priority of tasks, all
tasksthatareassignedtoanodecanexecuteinanyorder.We Step 11. Reduce the temperature via some cooling function
shouldtakeaccountoftheworstcase,thatis,consideringtask 𝑇=𝑓(𝑇).
alwaysexecutedatlast.Hence,thedeadlineconstraintsis
𝑁 𝑀 Step12. Iftheterminationconditionissatisfied(e.g. 𝑇≤𝑇𝑓),
∑𝑥 ∑𝑒 𝑥 ≤𝑑, ∀𝑖∈[1,𝑀]. thengotoStep13,otherwisegotoStep5.
𝑖𝑘 𝑗𝑘 𝑗𝑘 𝑖 (8)
𝑘=1 𝑗=1
Step13. Outputthesolution.
3.4. Problem Formulation. According to the above discus- Note that the neighborhood defines the procedure to
sion, we can tell that maximizing the reliability of RTDS is move from a solution point to another solution point [16].
equivalenttominimizingtheobjectfunction𝑌(𝑋),withall Inthispaper,aneighborisobtainedbyrandomlychoosinga
constraints mentioned before. Hence, we can formulate the task𝑖among𝑀tasksandreplacingitscurrentassignednode
TAPbythefollowingcombinatorialoptimizationproblem: withanotherrandomlyselectedone.
min 𝑌(𝑋)
(9)
subject to (5)∼(8). 4.2. Cooling Schedule. It has been shown that the SA algo-
rithmconvergestotheglobaloptimalwithprobability1[41],
4.TaskAllocationSolution which needs a sufficiently slow cooling schedule (i.e., suffi-
cienthotinitialtemperature,sufficientlowfinaltemperature,
ThissectiondescribesbasicSAalgorithmbrieflyatfirst,with and sufficient slow cooling speed). However, the required
thediscussionofthecoolingschedulewhichhasasignificant slow cooling schedule may lead to an unacceptably long
effectonitsconvergencespeedthenpresentstheXASAand solutionconvergencetimewhichcanbeexponential.
explainsthedetailsofhowitcanbeappliedintotheTAPin The cooling schedule is a set of parameters, which
termsofthestatementproposedinthispaper. controls the procedure of SA algorithm so that it can be
asymptoticconvergetoasuboptimalinreasonabletime.The
4.1.BasicSimulatedAnnealingAlgorithm. TheSAalgorithm coolingscheduleismadeupoftheseparameters.
startsfromarandomlychoseninitialsolutionandgenerates
(i) The Initial Value of the Control Parameter (i.e., Tem-
a series of Markov chains according to the descent of
the control parameter (i.e., temperature). In these Markov perature) 𝑇0. The initial temperature represents one of the
most important parameters in SA algorithm. If the initial
chains,anewsolutionischosenbymakingasmallrandom
temperature is very high, it will take very long time to be
perturbationofthesolution,and,ifthenewsolutionisbetter,
convergent.Ontheotherhand,poorsolutionsareobtained
thenitiskept,butifitisworseitiskeptwithsomeprobability
iftheinitialtemperatureislow.Abasicprincipleofchoosing
relatedtothecurrenttemperatureandthedifferencebetween
initial temperature is that the acceptance probability of
the new solution and the previous solution. According to a
worse solutions is close to 1. Which means the exchanging
seriesofiterationofsolutions,anoptimalonewasfound.The
of neighboring solutions should be almost freely at first.
SAalgorithmappliedintheTAPislistedasfollows.
Hence,wecandeterminetheinitialtemperature𝑇0viainitial
Step1. Chooseaninitialtaskarrangement(𝑋𝑠)atrandom. acceptanceprobabilityofworsesolution𝑃0.Inthispaper,𝑃0
issettobe0.9.𝑃0 =𝑒−Δ/𝑇0,whereΔisthedifferencebetween
Step2. Calculatethecost(𝑓𝑠)of𝑋𝑠. theneighboringsolutions,so𝑇0 =−Δ/ln𝑃0.Wecanusethe
MathematicalProblemsinEngineering 5
Δ = 𝑓max−𝑓minastheestimationofΔ,where𝑓maxand𝑓min Start
is the maximum and minimum of energy function among
randomlychosen𝐾solutions.Sotheinitialtemperature𝑇0 GenerateKchaotic vectors at random,
isdeterminedbythefollowingformula: each vector containsMvariables
𝑇0 = 𝑓minln−𝑃𝑓0max. (10) Calculate correspondingKsolutionsAand their costs
Search the solution space via COA
(ii)TheCoolingFunction𝐹(𝑇).Thecoolingfunctiondefines
thecoolingmethodoftemperature𝑇.Acommonlyusedtype The new solutionsis better No
than the worst one ofA
ofcoolingfunctionisexponentialdescentfunction:𝐹(𝑇) =
𝛼𝑇,where𝛼isconstantand𝛼<1.Sothatthecoolingspeed Yes
dependsontheparameter𝛼,andwecallitthecoolingfactor. Replace the worst solution ofAbys
Alargevalueof𝛼representsslowcooling,whichyieldsgood
solution,butexpensiveintime.𝛼issettobeslightlylessthan No
1inthemostreportedliteratures,anditischosentobe0.95 Termination of COA
inthispaper. Yes
Determine the cooling schedule of SA
(iii) The Final Value of Control Parameter 𝑇𝑓, Termination according toKoptimized solutions
ConditioninAnotherWord.Thecriterionforterminationcan
Choose the best one ofKoptimized
beeitherfinaltemperatureorsteadystateofthesystem.The
solutions to be the initial solution of SA
former one can control the total calculation time, but not
thesolutionquality:agivenfinaltemperaturemayintroduce
Search the optimal solution by SA
calculation redundancy for small scale problem but obtain
poorsolutionforlargescaleproblem.Ontheotherhand,the
End
latteronecantakeintoaccountbothtimeandquality.Inthis
paper,theSAalgorithmwillterminateifthesolutionremains Figure2:TheflowchartofXASA.
unchanged(neitherupgradingnordowngrading)foragiven
numberofiterations.Thenumberofiterationsthatsolution
remains unchanged is chosen to be 𝑀 × 𝑁. Furthermore,
thevalidationofthefinalsolutionshouldbesatisfiedaswell,
Step 2. Generate the solution vector A via Z, then generate
whichwillbediscussedlater.
thetaskassignment𝑋viaAandcalculatethecostfunction
(iv)TheLengthofMarkovChain𝐿𝑘.Itisthenumberofinner 𝑓.
looprepetitions.Thisparameterischosentobe𝑀×(𝑁−1), Step3. Settheinitialsolutionastheoptimal𝑓∗ =𝑓,Z∗ =Z.
whichisthesizeofsolutionneighborhood,becauseeachtask
canbeassignedtoanother𝑁−1nodes. Step4. CalculatenewchaoticvectorZviaformula[9].
The cooling schedule has a significant effect on the
resultsofthealgorithm,especiallyontheconvergencespeed. Step5. Generate𝑋asStep2,calculatethecostfunction𝑓.
Besides, the initial solution of SA algorithm can affect the
convergencespeedaswell. Step6. If𝑓<𝑓∗,then𝑓∗ =𝑓,Z∗ =Z.
4.3. Chaotic Optimization Algorithm. The chaotic variables Step7. RepeatSteps4,5,and6,until𝑓∗remainsunchanged
areproducedbythefollowingwell-knownone-dimensional foragivennumberofiterations.
logisticmap:
NotethattheiterationnumberinStep7ischosentobe
𝑧𝑘+1 =𝜇𝑧𝑘(1−𝑧𝑘), 𝑘=0,1,..., (11) assameasSAalgorithm,whichisdiscussedbefore,whilethe
validationofthesolutionisnotrequiredinCOA,sinceitis
where 𝜇 = 4, 𝑧𝑘 ∈ [0,1]. The logistic map has special notheuristicanditwilltakeverylongtimetogetconvergence
characters such as the ergodicity, stochastic property and iftherearefewvalidsolutionsinlargescalecases.
sensitivity dependence on initial conditions. The chaotic
optimization algorithm applied in this paper is listed as
follows. 4.4. Simulated Annealing Algorithm Combined with Chaos.
Thebasicideaofourproposedalgorithmissimulatedanneal-
Step 1. Initialize the chaotic vector Z at random, note that ingcombinedwithchaoticsearchandaddingsomeadaptive
thevalueofchaoticvariablescannotbe0,0.25,0.5,0.75,and schemes to the cooling schedule so that we can improve
1.0,whicharethefixedpointsoflogisticmap,andallchaotic the convergence speed without loss of solution quality. The
variablesaredifferentfromeachother. flowchartofXASAisshowninFigure2.
6 MathematicalProblemsinEngineering
Thereare4schemestospeeduptheconvergenceofSA All constraints are formulated as penalty functions as
algorithm. follows:
First,weapplyCOAtofind𝐾optimizedsolutions,whichis 𝑁 𝑀
a preliminary search in the solution space and the solution 𝐸𝑀 = ∑max(0,∑𝑚𝑖𝑥𝑖𝑘−𝑀𝑘),
𝑘=1 𝑖=1
distribution can be found according to the ergodicity of
chaossystem.Hence,wecansearchoptimalsolutionviaSA 𝑁 𝑀
algorithm in a relative smaller range with optimized initial 𝐸𝑆 = ∑max(0,∑𝑠𝑖𝑥𝑖𝑘−𝑆𝑘),
solution. 𝑘=1 𝑖=1
(13)
𝑁 𝑁 𝑀
Soencothned,rtehsuelitnsiotifalCtOemApbeercaatuursee𝑇w0eccaannarlespolabceestmheall𝑓emrabxaasnedd 𝐸𝐶 =𝑘∑=1𝑏=∑𝑘+1max(0,∑𝑖=1∑𝑖≠𝑗𝑐𝑖𝑗𝑥𝑖𝑘𝑥𝑗𝑏−𝐶𝑘𝑏),
𝑓minwiththatof𝐾optimizedsolutions.
𝑀 𝑁 𝑀
Third, the length of Markov chain 𝐿𝑘 is constant in SA 𝐸𝐷 =∑max(0,∑𝑥𝑖𝑘∑𝑒𝑗𝑘𝑥𝑗𝑘−𝑑𝑖).
algorithm,whileitisadaptiveinXASA.Thealgorithmwill 𝑖=1 𝑘=1 𝑗=1
jumpoutoftheinnerloopiftherejectionsofnewsolution
exceed a given threshold 𝜃. The threshold is given by the As all constraints are of the same importance, we use a
formula:𝜃=min(𝐿𝑘×𝛿1,𝜃×𝛿2)ateachtemperature.Because common coefficient 𝛾 for all penalty function. Hence, the
𝑃0 is set to be 0.9, we can set the initial value of 𝜃 to be energyfunctionis
i⌈n𝐿𝑘th×i0s.0p5a⌉p.e𝛿r1,ritepisresseetnttostbhee0m.6a;x𝛿im2 uremprtehsreensthsotlhde,ainndcr𝛿e1as<in1g; 𝑓(𝑋)=𝑌(𝑋)+𝛾(𝐸𝑀+𝐸𝑆+𝐸𝐶+𝐸𝐷). (14)
speedof𝜃,thisissimilartothecoolingfactor𝛼,soitissetto Thecriterionofchoosingpenaltyfunctioncoefficient𝛾is
be1.05. thatitshouldscalethevaluesofthepenaltyfunctionstothe
comparablevaluesasthatofobjectfunction𝑌(𝑋)suchthat
Fourth,thecoolingfactor𝛼isadaptiveinXASAaswell:the
theproceduresofthealgorithmwillbetowardthedirection
moresolutionsareaccepted(bothbetterandworsesolutions)
of penalty avoiding; hence, the valid solution can be found
at the current temperature, the smaller the cooling factor
withhighprobability.Besides,thevalidationofasolutioncan
is, and vice versa. The rationale is that high temperature at berepresentedas𝑓(𝑋)=𝑌(𝑋).
thebeginningofthealgorithmgeneratesnumeroussolution
acceptances; thus, a rapid reduction of temperature can be
5.PerformanceEvaluation
made,whilefewersolutionswillbeacceptedasthetemper-
atureiscoolingdown;thus,aslowcoolingspeedshouldbe
Toevaluatetheperformanceoftheproposedalgorithm,both
appliedsinceweneedcarefullyasolutionsearch.Hence,the
SAandXASAarecodedinMatlabandtestedfornumerous
cooling factor of XASA is 𝛼󸀠 = 𝛼 × 𝑒−𝜅/(𝜅+4×𝑛), where 𝜅 is randomlygeneratedtasksetsthatareallocatedontoaRTDS.
theacceptancenumberoflastinnerloop,and𝑛istheactual TherearetwovariationsofSAalgorithmimplementedinthis
lengthofMarkovchains.Notethat𝜅∈[0,𝑛],so𝑒−𝜅/(𝜅+4×𝑛) ∈ paper,thetraditionalone(SA1)andtheimprovedone(SA2).
[0.8187,1]and𝛼󸀠 ∈[0.7778,0.95]. SA2 applies the last two adaptive schemes of XASA, that
To implement the algorithm, some details should be is, adaptive lengthof Markov chains and coolingfactor. All
presentedasfollows. othercomponentsofSA2areassameasSA1,includinginitial
solution,initialtemperature,andterminationcondition.The
(1) Solution Representation. In this paper, solutions are pre- usedcomputationsystemisMatlab7.11.0,withIntelCorei7-
sentedwithavectorA(M,1);eachelementrepresentsatask, 2600@3.40GHzand16GbmainmemoryunderaWindows
itsvalueisbetween1and𝑁,denotingwhichnodethistask 7environment.
isassignedto.InordertoapplyCOA,weuseachaoticvector
ZrelatedtoA(M,1),whereA = [Z×(𝑁−1)+1].Atask 5.1. Experiment Parameters Settings. All DS parameters are
assignment𝑋oftheTAPisgeneratedby𝐴asfollows: followed by the former researches [16, 17, 40]. The failure
rates of processors and communication links are given in
theranges[0.00005–0.00010]and[0.00015–0.00030],respec-
1, 𝑘=𝐴(𝑖), 𝑖=1,2,...,𝑀,
𝑋(𝑖,𝑘)={0, 𝑘≠𝐴(𝑖), 𝑘=1,2,...,𝑁. (12) tively.Thetimeofprocessingataskatdifferentprocessorsis
givenintherange[15–25].Thememoryrequirementofeach
taskandnodememorycapacityisgivenintherange[1–10]
and[100–200],respectively.Thetaskprocessingloadversus
node processing capacity is given in the ranges [1–50] and
(2) Energy Function. We integrate the object function 𝑌(𝑋) [100–300]. The value of data to be communicated between
andallconstraintsintoacostfunctiontofittheSAalgorithm tasks is given in the range [5–10]. The bandwidth and load
framework. And the cost function is used as the energy capacityofcommunicationlinksaregivenintheranges[1–4]
function. and[100–200].Therangeoftaskdeadlinevalueis[10–200].
MathematicalProblemsinEngineering 7
Table1:Simulationresultsforthecaseof12nodes.
Cases XASA SA1 SA2 Δ𝑡 Δ𝑡 Δ𝑡 Δ𝑅 Δ𝑅
𝑁 𝑀 𝑅 𝑡 𝑅 𝑡 𝑅 𝑡 1 2 3 1 2
avg avg avg avg avg avg
16 0.9349 3.336 0.9361 39.266 0.9353 10.409 91.50% 67.95% 73.49% 0.13% 0.04%
18 0.9135 3.914 0.9152 50.188 0.9151 13.237 92.20% 70.43% 73.62% 0.18% 0.17%
12 20 0.8948 4.014 0.8992 57.456 0.8968 14.454 93.01% 72.23% 74.84% 0.49% 0.22%
22 0.8733 4.918 0.8781 55.815 0.8777 14.474 91.19% 66.02% 74.07% 0.55% 0.50%
25 0.8299 5.325 0.8373 84.469 0.8367 22.463 93.70% 76.30% 73.41% 0.89% 0.81%
40 0.6000 23.187 0.6048 377.104 0.6028 96.961 93.85% 76.09% 74.29% 0.79% 0.47%
45 0.5320 36.523 0.5334 484.005 0.5329 123.667 92.45% 70.47% 74.45% 0.25% 0.16%
25 50 0.4537 36.300 0.4553 606.927 0.4565 161.550 94.02% 77.53% 73.38% 0.35% 0.61%
55 0.3732 57.156 0.3753 720.068 0.3726 195.631 92.06% 70.78% 72.83% 0.58% −0.14%
60 0.2866 157.777 0.2886 795.438 0.2860 217.155 80.16% 27.34% 72.70% 0.71% −0.19%
Average 91.42% 67.51% 73.71% 0.49% 0.27%
Table2:Simulationresultsforthecaseof12nodes.
Cases XASA SA1 SA2
𝑁 𝑀 𝑅 𝑡 𝑉 𝑉 𝑅 𝑡 𝑉 𝑅 𝑡 𝑉
std std 1 2 std std std std
16 0.0016 0.3035 91.10% 100.00% 0.0019 0.8582 99.99% 0.0013 0.6208 100.00%
18 0.0008 0.6041 85.57% 100.00% 0.0020 0.9552 98.76% 0.0018 0.8322 99.21%
12 20 0.0062 0.4114 72.39% 100.00% 0.0029 1.3870 98.12% 0.0036 0.3914 97.63%
22 0.0016 0.6186 54.30% 100.00% 0.0015 1.3489 93.23% 0.0017 0.5886 94.07%
25 0.0049 0.7613 29.27% 99.82% 0.0038 2.9824 85.99% 0.0031 1.2888 86.88%
40 0.0042 1.5797 21.67% 99.87% 0.0026 8.7429 89.62% 0.0027 4.6770 91.39%
45 0.0014 5.4829 9.85% 99.67% 0.0024 12.3919 76.51% 0.0027 3.4313 81.95%
25 50 0.0039 5.6073 2.06% 99.30% 0.0030 11.4135 71.62% 0.0025 11.0605 79.76%
55 0.0028 8.1346 0.16% 96.58% 0.0019 8.9093 58.05% 0.0023 7.7128 67.33%
60 0.0022 9.3194 0.03% 89.76% 0.0016 15.5811 55.63% 0.0022 8.4473 67.03%
Average 0.0029 3.2823 36.64% 98.50% 0.0024 6.4570 82.75% 0.0024 3.9051 86.53%
The network topology is star, 𝑁 is set to be 12 and 25, The comparative results from Table1 show that XASA
and 𝑀 is set to be [16,18,20,22,25] and [40,45,50,55,60] can sharply reduce the convergence time against the other
fortwocases,respectively.Thecoefficientofpenaltyfunction twoSAalgorithms,whilethesolutionquality(i.e.,reliability)
𝛾 is set to be 1. The number of randomly chosen solutions is slightly worse (less than 0.01 in terms of value and 1%
at the beginning of the algorithmis set to be 𝐾 = 10. And in percentage). Note that the value of Δ𝑡3 is steady in the
theinitialsolutionofthetwoSAalgorithmsisoneofthese rangeof72% ∼ 75%withsmallvariation,whichmeansthe
𝐾 solutions which is chosen at random. Because SA is a thirdandfourthadaptiveschemestakeafixedeffectonthe
stochasticalgorithm,eachindependentrunofthealgorithm SAalgorithm.Furthermore,Δ𝑡1 issteadyinallcasesexcept
the last one (𝑁 = 25, 𝑀 = 60) as well, which has an
onasameapplicationmayyielddifferentresult,we,thus,run
average value of 92.66% with standard deviation 0.1036. In
all of the three algorithms on an application 10 times and
our preliminary experiments, the value of function 𝑌(𝑋) is
obtaintheaveragevalues.
alwaysbelow2duringthewholealgorithm,whilethevalue
ofthepenaltyfunctioncanbehundreds.Besides,according
5.2. Experiment Results. Table1 summarizes the reliability
to the experiment parameters set in this paper, there is no
and calculation time of all TAPs by deploying the XASA
significantdifferencebetweennodes,nordothetasks.Hence,
and other two SA algorithms. The title 𝑅 with suffix avg
therearelotsoflocalminimainthesolutionspacewithout
representsReliabilitywiththeaveragevalueof10independent
consideringthevalidationofsolutions.Constraintsareeasy
runs, and 𝑇 represents Time where the unit is second. Δ𝑡1 tobesatisfiedwhentheproblemscaleissmall,soCOAcan
is the acceleration ratio of XASA versus SA1, where Δ𝑡1 = obtainseveralvalidsolutions,andtheinitialtemperaturefor
(𝑡SA1 − 𝑡XASA)/𝑡SA1 × 100%. Δ𝑡2 and Δ𝑡3 is the acceleration theSAalgorithminthenextstepofXASAcanberelatively
ratioofXASAversusSA2andSA2versusSA1,respectively. small; therefore, it is fast to get convergence. On the other
Δ𝑅 represents the average deviation in percentage between hand, COA can hardly obtain valid solutions in the case of
XASAandSAalgorithmsintermsofreliability,whereΔ𝑅𝑖 = large scale (e.g., 𝑁 = 25, 𝑀 = 60 in this paper), if there
(𝑅 −𝑅 )/𝑅 ×100%,𝑖=1,2. are insufficient valid solutions (less than 𝐾), a large initial
SAi XASA SAi
8 MathematicalProblemsinEngineering
0.22
0.18
n n
o o
cti cti
n 0.2 n
u u
y f y f 0.16
g g
er er
n n
e e 0.18 e e
h h
of t of t 0.14
es es
Valu 0.16 Valu
0.12
0.14
0 200 400 600 800 0 1000 2000 3000
Iterations Iterations
COA inSA
0.22 0.22
nction 0.2 nction 0.2
nergy fu 0.18 nergy fu 0.18
Values of the e 00..1146 Values of the e 00..1146
0.12 0.12
0 1 2 3 4 5 0 2000 4000 6000 8000 10000
Iterations ×104 Iterations
SA1 SA2
Figure3:Energyfunctionateachiterationfor𝑁=12,𝑀=20.
temperaturewillbesetbecauseofthegiantvalueofpenalty 5.3. Time Series Analysis. Figure3 shows the values of the
function,thiswillaffecttheconvergencespeedsignificantly. energyfunction,whicharecalculatedateachiterationofthe
Additionally,constraintsarenotsoeasytobesatisfiedwhen algorithmsforthecase𝑁=12,𝑀=20.Notethatthevalues
problemscaleislarge,sotherearemuchlesslocalminimain of the invalid solutions are set to be 0.22, because the real
thesolutionspace;thiswillslowdowntheconvergencespeed values of the invalid solutions are too large, and they may
aswell.Allthesefactorsweakenthespeedupeffectbasedon concealthedetailsofthevalidones.
As we can see, COA cannot guarantee the validation of
COA,sotheperformanceinthelastcaseisnotsogood.
the solutions, it is completely stochastic without heuristic.
Table2showssomeoverallstaticscharactersofallthree
However, it can generate a good start for the next step of
algorithms. Where the 𝑅 and 𝑡 represent the standard
std std XASA, which is shown in the results of inSA, where all
deviation of reliability and calculation time, 𝑉1 represents solutions are valid and the convergence speed is quite fast.
the valid solutions in percentage of COA in XASA and 𝑉2 TheinSAbeginstoreachtothegoodenoughsolutionsbefore
representsthatofinSA(inSArepresentstheSAalgorithmin
2000iterations.Theothertwo SAalgorithmsstartwiththe
XASA),theothertwo𝑉columnshavethesamemeaning.As
worse condition, and spend much more iterations to reach
wecansee,XASAisthebestintermsofmeanvalueoftime to the good enough solutions. They both need thousands
standarddeviation,andthereisnocasethatSA1excelsXASA of iterations to explore the solution space which generates
inthiscriterion.𝑉2getsthebestresultcomparedtoothertwo lots of invalid solutions as COA but the cost is much more
algorithmsaswell.Notethat𝑉1showspoorresultsinthelarge expensive.Becauseoftheadaptiveschemes,SA2canquickly
cases, this is an evidence for our analysis of the large scale pass through invalid solutions as can be seen in Figure3.
issuepresentedbefore. Hence,theseschemesareeffective.
MathematicalProblemsinEngineering 9
250 250
200 200
ons 150 ons 150
ulati ulati
m m
Accu 100 Accu 100
50 50
0 0
0 10 20 30 40 50 0 20 40 60
Cooling steps Cooling steps
n 𝜅 n 𝜅
𝜃 𝜃
(a) InneriterationcharactersofinSA (b) InneriterationcharactersofSA2
0.95
0.95
0.9
ctor ctor 0.9
a a
g f 0.85 g f
n n
oli oli 0.85
o o
C C
0.8
0.8
0.75 0.75
0 10 20 30 40 0 20 40 60
Cooling steps Cooling steps
(c) CoolingfactorofinSA (d) CoolingfactorofSA2
1 1
0.8 0.8
ure 0.6 ure 0.6
at at
er er
p p
m 0.4 m 0.4
Te Te
0.2 0.2
0 0
0 10 20 30 40 50 0 20 40 60
Cooling steps Cooling steps
ASA ASA
SA SA
(e) TemperatureofinSA (f) TemperatureofSA2
Figure4:Coolingschedulecharactersfor𝑁=12,𝑀=20.
10 MathematicalProblemsinEngineering
1.7
Values of the energy function 111...567 Values of the energy function 1111....3456
1.4
1.2
0 2000 4000 6000 8000 10000 0 2 4 6
Iterations Iterations ×104
COA inSA
1.7 1.7
n n
o o
ncti 1.6 ncti 1.6
u u
y f y f
g g
er 1.5 er 1.5
n n
e e
e e
h h
of t 1.4 of t 1.4
es es
u u
Val 1.3 Val 1.3
1.2 1.2
0 1 2 3 4 0 5 10
Iterations ×105 Iterations ×104
SA1 SA2
Figure5:Energyfunctionateachiterationfor𝑁=25,𝑀=60.
Figure4showsthedetailsoftheadaptivecoolingsched- Figure6showstheresultsofthecase𝑁=25,𝑀=60as
ule schemes, which are calculated at each cooling step of Figure4.Thecoolingfactorandtemperaturecurvearenotso
the algorithms for the case 𝑁 = 12, 𝑀 = 20. The left differentbetweeninSAandSA2,andthecharacter𝑛cannot
three figures are the results of inSA, and right ones are the bring as much benefit as before. These cause the inefficient
resultsofSA2.NotethatthetemperatureinFigures4(c)and situationofthelargestcase.
4(f) is set to be unitary. At the beginning of the cooling
steps,theacceptanceratesofthenewsolutionsarebothhigh
6.Conclusions
in two algorithms; hence, the cooling factor is small, and
the temperature reduces rapidly, while the actual length of
Markov chains 𝑛 is much smaller in inSA than SA2, and Inthispaper,weconsideraheterogeneousDSthatrunsareal-
timeapplication,toachievemaximizationofsystemreliabil-
the cooling factor increases faster as well. It is caused by
itywithtaskallocationtechnique.Byformulatingthereliabil-
thedifferencesoftheinitialtemperatureandinitialsolution,
ityandconstraints,wemodelthisproblemasacombinatorial
sinceinSAandSA2areactuallythesame.Hence,COAcan
trulyimprovetheconvergencespeed. problem.Tosolvethisproblemwithfastconvergencespeed,
Figure5showsthedetailsofthecase𝑁 = 25,𝑀 = 60 we improve the well-known simulated annealing algorithm
wheretheinvalidsolutionvaluesaresettobe1.7.Theresult basedontheanalysisofthecoolingscheduleofSAalgorithm
of COA is quite bad, only three valid solutions are found; which has a significant effect on its convergence speed.
therefore,inSAcannotgetagoodstartandithastoexplore Then,weproposethealgorithmXASA,whichcombinesSA
widesolutionspaceatthebeginningwhichgenerateslotsof algorithm with chaotic optimization algorithm with several
invalidsolutions. adaptive schemes. The experimental results show that the