CHAPTER 2 PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION
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1 24 CHAPTER 2 PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION The present chapter proposes an IPSO approach for multprocessor task schedulng problem wth two classfcatons, namely, statc ndependent tasks and dynamc tasks wth and wthout load balancng to mnmze the makespan of the schedule. 2.1 INTRODUCTION PSO s a popular swarm based optmzaton technque that mmcs the brd s flockng behavor. Though PSO shows better performance, premature convergence and local optma are the two major problems faced by the PSO approach for schedulng problem. To overcome ths dffculty, an IPSO s proposed, n whch a splt up s made n the cogntve behavor. That s the partcle s made to remember ts worst poston also. Ths modfcaton helps n explorng the search space very effectvely to dentfy the promsng soluton regon. The ntroducton of the worst partcle called bad experence component n the cogntve component yelds a sgnfcant mprovement n the results, when appled to the task schedulng problem. The ncluson of the worst partcle plays a major role n the achevement of a better soluton than wth only usng the good experence component called best partcle n the velocty equaton.
2 25 In the present chapter an IPSO approach to obtan better schedule wth mnmum convergence tme s presented. The effectveness of the proposed IPSO approach s demonstrated wth datasets and s gven n Tables 2.1 to PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION APPROACH In the proposed IPSO algorthm, the velocty updaton equaton s slghtly modfed by splttng the cogntve component of the general PSO nto two dfferent components to obtan better results for the multprocessor task schedulng problem. The frst component s called as good experence component. Ths means that the partcle has a memory about ts prevously vsted best poston. Ths s smlar to the standard PSO method. The second component s gven the name, bad experence component. The bad experence component helps the partcle to remember ts prevously vsted worst poston. To calculate the new velocty, the bad experence of the partcle s also taken nto consderaton. On ncludng the characterstcs of Pbest and Pworst n the velocty updaton process along wth the dfference between the present best partcle and current partcle respectvely, the convergence towards the soluton s found to be faster and an optmal soluton s reached n comparson wth conventonal PSO approaches. The new velocty update equaton s gven by, V w V C r P S P C r S P 1g 1 best best 1b 2 worst P C r G S (2.1) worst 2 3 best
3 26 The poston update equaton s gven by, S 1 S V (2.2) where, C 1g C 1b : self-confdence factor, whch accelerates the partcle towards ts best poston; : self-confdence factor, whch accelerates the partcle away from ts worst poston; C 2 : Swarm confdence factor, P worst : worst poston of the partcle, r 1, r 2, r 3 S S + 1 V V +1 : unformly dstrbuted random numbers n the range [0 to 1], : current poston : modfed poston, : current velocty and : modfed velocty The postons are updated usng Equaton (2.2). The ncluson of the worst experence component n the behavour of the partcle gves the addtonal exploraton capacty to the swarm. By usng the bad experence component, the partcle can bypass ts prevous worst poston and try to occupy the better poston. Fgure 2.1 shows the concept of IPSO searchng ponts.
4 27 Fgure 2.1 Concept of proposed mproved partcle swarm optmzaton search pont The Proposed IPSO Algorthm Step1 : Select the number of partcles, generatons, tunng acceleratng coeffcents C 1g, C 1b, and C 2 and random numbers r 1, r 2, r 3 to start the optmal soluton searchng. Step2 : Intalze the partcle poston and velocty. Step3 : Select partcles ndvdual best value for each generaton Step 4 : Step 5 : Step 6 : Select the partcles global best value, whch means partcle near to the target among all the partcles s obtaned by comparng all the ndvdual best values. Select the partcles ndvdual worst value. Update partcle ndvdual best (Pbest), global best (Gbest), partcle worst (Pworst) n the velocty Equaton (2.1) and obtan the new velocty.
5 28 Step 7 : Step 8 : Update new velocty value n the Equaton (2.2) and obtan the new poston of the partcle. Fnd the optmal soluton wth mnmum F by the updated new velocty and poston. n Fgure 2.2 The flowchart for the proposed model formulaton scheme s shown Start Intalze the populaton Input number of processors, number of jobs and populaton sze Compute the objectve functon Invoke IPSO If E < best E (P best) so far Yes A No B Fgure 2.2 Flowchart for the proposed IPSO
6 29 B For each generaton For each partcle Current value = new p best A Choose the mnmum F of all partcles as the g best Calculate partcle velocty usng (2.1) Calculate partcle poston usng (2.2) Search s termnated optmal soluton reached Update memory of each partcle End End Return by usng IPSO Stop Fgure 2.2 (Contnued)
7 BASIC CONCEPTS OF TASK SCHEDULING Schedulng s the allocaton of deal jobs to resources at partcular tmes whch mnmzes the total length of the schedule. The ntenton of schedulng s to dstrbute the tasks among varous processors n such a way that the total executon tme or the entre schedule length or makespan s mnmzed. The executon tme depends on the allocaton of tasks and the schedulng polcy appled to tasks. Total Fnshng Tme (TFT) and Average Watng Tme (AWT) are two computable crtera to mnmze the makespan n multprocessor archtecture. Total fnshng tme s defned as the maxmum of each processor s fnshng tme whch s the tme that the processor fnshes ts job. Watng Tme s defned as the average of tme that each job wats n a ready queue. Smultaneously mnmzng these two crtera s a Mult Objectve Optmzaton (MOO) Problem. The scheme of job schedulng s shown n Fgure 2.3. Arrvng requests / jobs Pre- -empted jobs Scheduled jobs CPU Completed Queue Scheduler Jobs Pendng Requests / jobs Fgure 2.3 Schematc of job schedulng A Task/job schedulng polcy uses the nformaton assocated wth requests to decde whch request should be servced next. All requests watng to be servced are kept n a lst of pendng requests. Whenever schedulng s to be performed, the scheduler examnes the pendng requests and selects one for servcng. Ths request s handled over to the server. A request leaves the server, when t completes or when t s pre-empted by the scheduler, n whch
8 31 case t s put back nto the lst of pendng requests. In ether stuaton, the scheduler performs schedulng to select the next request to be servced. The scheduler records the nformaton concernng each job n ts data structure and mantans t all through the lfe of the request n the system. 2.4 BASIC CONCEPTS OF MULTI OBJECTIVE OPTIMIZATION The man objectve of Mult Objectve Optmzaton (MOO) algorthms s to fnd a set of solutons whch optmally balances the trade-offs among the objectves of a Mult Objectve Problem (MOP). The task s to fnd a set of non-domnated solutons, referred to as the Pareto-optmal set. In the followng, domnaton and Pareto-optmal, set concepts wll be descrbed, as follows, (Moattar et al 2008). Let P mx Q denote the m x dmensonal search space. The search space, P s also referred to as the decson space and F the feasble space. Wth no constrants, the feasble space s the same as the search space. Let X x1, x2,... referred to as a decson vector. A sngle x m x objectve functon, mn f n X s defned as f Q Q. Let n : f X f,... mx 1 X, f 2 X fm X O Q be an objectve vector n contanng m n objectve functon evaluatons, O s referred as the objectve space Domnaton: A decson vector, X 1 domnates a decson vector, X 2 (denoted by X 1 X 2 ) f and only f, X 1 s not worse than X 2 n all objectves,.e. f X... n 1 fn X 2, n 1 mn and
9 32 X 1 n s strctly better than X 2 n at least one objectve,.e m : f X f X n n 1 n 2 Smlarly, an objectve vector, f 1 domnates another objectve vector f 2, f f 1 s not worse than f 2 n all objectve values and f 1 s better than f 2 n at least one of the objectve values. Objectve vector domnance s denoted by f1 f 2. Pareto-optmal: A Decson vector exst a decson vector, X X * X * * n X f n. An objectve vector f * X : f X pareto-optmal. F s pareto optmal f there does not F that domnates t. That s n s pareto optmal, f X s Pareto-optmal set: The set of all pareto optmal decson vectors form the pareto-optmal set * * * P that s P X F X * F : X X One of the approaches to deal wth Mult objectve problems s to defne an aggregate objectve functon as a weghted sum of objectves and s redefned as, Mnmze Subject to, m n n 1 n f ( x) (2.3) n g X 0, t 1,... t m g h X 0, t m 1,... m t g g m h X X mn, X max m x 0, n 1,... n m n
10 33 where, g t and h t are the nequalty and equalty constrants and [X mn, X max ] represents the boundary constrants 2.5 SIMULATION PROCEDURE The detals of the smulaton carred out for the multprocessor task schedulng problem wth the datasets are as follows, Benchmark datasets are taken from ErcTalard s ste for dynamc task schedulng. Two datasets are taken for smulaton. Dataset 1 nvolves 50 tasks and 20 processors. Dataset 2 nvolves 100 tasks wth 20 processors. The data for the statc schedulng are randomly generated such as, 2 processors wth 20 tasks, 3 processors wth 20 tasks, 3 processors wth 40 tasks, 4 processors wth 30 tasks, 4 processors wth 50 tasks, 5 processors wth 45 tasks and 5 processors wth 60 tasks. To demonstrate the effectveness of the proposed approach, the approach s run wth 30 ndependent trals wth dfferent values of random seeds and control parameters. The optmal result s obtaned for followng parameter settngs. Proposed IPSO: Max. Iteraton : 500 Swarm sze : Twce the number of tasks (Salman mn max : 0.5 C 1 b, C 1 g and C2 : 2 et al 2002)
11 34 The proposed approach IPSO s developed usng MATLAB R2009 and executed n a PC wth Intel core 3 processor wth 3 GB RAM and 2.13 GHz speed. 2.6 STATIC TASK SCHEDULING Let T= {1, 2 n} be a gven set of ndependent tasks, and P= {1, 2.m} represents the set of dentcal processors. All tasks are mutually ndependent and each task can be scheduled on any processor. All the tasks should be scheduled n such a way that one processor executes one task at a tme. Statc task schedulng requres pror nformaton about the number of tasks, number of processors, executon tme of tasks, and schedulng done before the executon. The objectve of schedulng s to smultaneously mnmze the total fnshng tme and the average watng tme of the schedule. Mnmzaton of these two objectves smultaneously makes the schedulng problem a mult objectve optmzaton problem. The consdered schedulng problem s formulated by, fn n P j exe 1 (2.4) Mn TFT max fn ( P ) j m (2.5) schedule j 1 WT Arrval fn (2.6) 1 AWT schedule m n j 1 1 n WT (2.7) where n s the number of tasks, and m s the number of processors, the varable refers tasks from 1 to n and varable j refers processors from 1 to m.
12 35 fn P j refers the fnshng tme of processor j exe represents executon tme of task TFT schedule denotes the Total Fnshng Tme.e., maxmum of each processor s fnshng tme WT refers the watng tme of task and AWT s the average watng tme of n tasks The consdered Mult objectve problem s unfed by usng weghted sum of objectves converted nto sngle objectve optmzaton problem (Moattar et al 2008) and s represented by, Mn f TFT schedule AWTschedule (2.8) where s the weght coeffcent, whch can be adjusted to compromse between two mnmzaton objectves. The frst term n Equaton (2.8) s mnmzng the Total fnshng tme, that s the makespan of the entre schedule and the second term s related to mnmzaton of the watng tme. The global mnmzaton problem s changed nto global maxmzaton problem. Through the transformng Equaton (2.8) the proper ftness functon for the multobjectve problem s represented by, F V 0 TFT schedule AWT schedule f f V V (2.9) where V should select an approprate postve number to ensure that the ftness of all good ndvduals are postve n the feasble soluton space.
13 Results and Dscusson The proposed IPSO approach s appled to the statc ndependent tasks. The obtaned results are depcted n Table.2.1. The results reveal that the proposed approach IPSO smultaneously reduces the total fnshng tme and the average watng tme. Table 2.1 Total fnshng tme and average watng tme usng IPSO No of Processors No of jobs Total Fnshng Tme (TFT) Proposed IPSO Average Watng Tme (AWT) The proposed approach IPSO produces total fnshng tme as 69.04s and average watng tme as 41.03s for 3 processors and 40 tasks. For the dataset 4 processors wth 30 tasks, the acheved result for total fnshng tme s 70.97s and average watng tme s 29.74s. For the dataset 5 processors wth 60 tasks, the result acheved for total fnshng tme s 72.31s and average watng tme s 36.56s Performance Comparson In order to valdate the performance of the proposed algorthm IPSO, comparsons have been made wth the exstng algorthms such as LPT, SPT GA and standard PSO for the same datasets and are reported n Table 2.2.
14 37 Table 2.2 Comparson of total fnshng tme and average watng tme of jobs usng LPT, SPT, GA, PSO and IPSO No. of Processors No. of jobs LPT SPT GA PSO Proposed IPSO AWT TFT AWT TFT AWT TFT AWT TFT AWT TFT For the dataset 4 processors wth 30 tasks, the Longest Processng Tme algorthm (LPT) produces the total fnshng Tme and average watng tme as 62.7s and 48.2s. Shortest Processng Tme (SPT) algorthm produces 75.36s as the total fnshng tme and 32.64s as average watng tme for the same dataset of 4 processors and 30 tasks. Genetc Algorthm (GA) produces 32.91s and 74.26s as average watng tme and total fnshng tme respectvely for the same dataset. Standard PSO produces total fnshng tme as 72.18s and the average watng tme as 30.65s. The proposed approach IPSO produces the average watng tme as 29.74s and the total fnshng tme 70.97s. LPT reduces the total fnshng tme whereas the average watng tme s drastcally hgh and n SPT the total fnshng tme s drastcally hgh and reduces the average watng tme. Genetc Algorthm reduces the total fnshng tme and average watng tme smultaneously when compared wth LPT and SPT algorthms. Standard PSO also smultaneously reduces the total fnshng tme and the average watng tme than GA. Further, the ntroducton of the proposed approach IPSO reduces both the total fnshng tme and average watng tme smultaneously. These results prove that the proposed approach IPSO produces comparatvely better results.
15 38 The varatons n total fnshng tme and watng tme usng conventonal methodologes LPT, SPT and evolutonary approaches, GA, PSO and IPSO are shown n Fgures 2.4 to Fgure 2.4 Total fnshng tme and average watng tme for 2 processors wth 20 jobs usng LPT, SPT, GA, PSO and IPSO Fgure 2.5 Total fnshng tme and average watng tme for 3 processors wth 20 jobs usng LPT, SPT, GA, PSO and IPSO
16 39 Fgure 2.6 Total fnshng tme and average watng tme for 3 processors wth 40 jobs usng LPT, SPT, GA, PSO and IPSO Fgure 2.7 Total fnshng tme and average watng tme for 4 processors wth 30 usng LPT, SPT, GA, PSO and IPSO
17 40 Fgure 2.8 Total fnshng tme and average watng tme for 4 processors wth 50 jobs usng LPT, SPT, GA, PSO and IPSO Fgure 2.9 Total fnshng tme and average watng tme for 5 processors wth 45 jobs Usng LPT, SPT, GA,PSO and IPSO
18 41 Fgure 2.10 Total fnshng tme and average watng tme for 5 processors wth 60 jobs usng LPT, SPT, GA, PSO and IPSO Thus, t s observed from the results that the proposed algorthm IPSO s better than the conventonal methodologes LPT, SPT and the other optmzaton technques, such as GA, standard PSO. 2.7 DYNAMIC TASK SCHEDULING The present research examnes Dynamc Task Schedulng wth the followng scenaro. The processors n the system are dstrbuted computng envronment and the tasks are scheduled as when they arrve n the queue. Load balancng of dynamc tasks s partcularly useful n a system, where processor utlzaton s the major ssue, nstead of mnmzng executon tme of the applcatons. The objectve s to mnmze the total executon cost encountered by the assgnment of all tasks allocated to the processors. A partcle s evaluated by calculatng ts ftness functon. Ftness functon ndcates the goodness of the schedule. The cost functon n case of optmalty determnaton computes
19 42 the best possble soluton. Ths cost functon (for ex. best cost, worst cost) wll be the soluton for the objectve functon consdered. As a result, t does not nfer to the regular cost value, whch has unt Rupees or dollars. Snce the termnology used here refers to optmal best cost, worst cost and average cost, t does not possess an unt of ts own. Dynamc task schedulng can be done under two cases namely,. Consderng only the arrval tme of the tasks (wthout load balancng). Consderng both the tme of arrval of each task and effcent processor utlzaton. (wth load balancng) Ftness functon to mnmse the total executon tme, that means schedule length vares for the above sad two cases, hence s dealt ndvdually n the followng sub-sectons Dynamc Task Schedulng wthout Load Balancng Dynamc Task Schedulng s adopted to schedule the tasks, n the way n whch they are arrvng. Dynamc load balancng does not requre the pror knowledge of the tasks as lke statc. That s, t need not be aware of the run-tme behavour of the applcatons before executon. In Dynamc Schedulng, tasks are redstrbuted among processors durng the executon tme. Ths redstrbuton s done by transferrng tasks from heavly loaded processors to the lghtly loaded processors (called load balancng), to mprove the processor utlzaton and performance. To mnmse the makespan of the schedule, the equaton s formulated and represented as,
20 43 TFT P j arrval n 1 fn (2.10) fn exe arrval exe f arrval otherwse fn 1 (2.11) Fnshng Tme. Ftness functon s formulated to mnmse the average of Total F X m TFT Pj mn j 1 m (2.12) where n s the number of tasks m s the number of processors TFT P j = Total Fnshng Tme of Processor j arrval = arrval tme of task fn =fnshng tme of task and exe =executon tme of task Results and Dscusson The results obtaned for the proposed algorthm IPSO have been tabulated and s shown n Table 2.3. The Table 2.3 gven below shows the detals of the obtaned best, worst and average cost. GA produces the best cost as 3018 for 50 tasks for the
21 44 dataset 1 and t s 5928 for dataset 2. PSO obtans 2972 as best cost for dataset 1 and t s 5552 for dataset 2. The proposed IPSO approach acheves best cost for dataset 1, as 2374 and 4527 for dataset 2. From the results, the best cost, worst and average costs are better n the proposed IPSO when compared to the other approaches. The convergence tme for the proposed methodology for dataset 1 s s and s for dataset 2. Table 2.3 Comparsons of best cost, worst cost, average cost and convergence tme usng GA, PSO, and proposed IPSO for dynamc task Schedulng wthout load balancng Methods GA PSO Proposed IPSO No of tasks Best cost Worst cost Average cost Convergence tme n seconds than the other algorthms. The results from Table 2.3 nfer that the IPSO performs better The best cost comparson for dataset 1 and 2 usng GA, standard PSO and IPSO are shown n Fgures 2.11 and 2.12.
22 45 Fgure 2.11 Best cost for 50 tasks and 20 processors usng GA, PSO and IPSO Fgure 2.12 Best cost for100 tasks and 20 processors usng GA, PSO and IPSO Performance Comparson The performance of the proposed IPSO approach s compared wth the prevously proposed (Vsalaksh and Svanandam 2009) varant PSO methods namely, PSO wth Fxed (PSO-FI) and PSO wth Varable Inerta (PSO-VI) for multprocessor dynamc task schedulng wth same datasets.
23 46 Table 2.4 Performance comparson of varous PSO based approaches Methods PSO-FI (Vsalaksh and Svanandam 2009) PSO-VI (Vsalaksh and Svanandam 2009) Proposed IPSO Number of tasks Best cost Worst cost Average cost Convergence tme n seconds The proposed IPSO approach gves the best cost for dataset 1 as 2374 and for dataset 2, t s PSO-FI produces the best cost for dataset 1 as 2814 and for dataset 2, t s PSO-VI produces 2592 as the best cost for dataset 1 and t s 4893 for dataset 2. The convergence tme for the proposed approach s s for dataset 1 and for dataset 2. The above mentoned comparson reveals that the proposed method IPSO yelds better results wth faster (0.2 to 0.3s ) convergence compared wth PSO-FI and PSO-VI. The result nfers that the ncluson of the bad experence partcles plays a major role for the mprovement of the achevement Dynamc Task Schedulng wth Load Balancng The drawback n dynamc schedulng wthout load balancng s that some of the processors may be lghtly loaded and others are heavly loaded. To overcome the drawback and also to mprove the processor utlzaton and performance, load balancng s consdered. To acheve load balancng n dynamc task schedulng, tasks have to be allocated n such a way that they
24 47 are arrvng. Our man goal s to mnmse the makespan of the entre schedule and to mprove the processor utlzaton. To acheve better load balancng, tasks have to be assgned to the processors n such a way to reduce the makespan and to ncrease the utlzaton. Hence, equatons are formulated and represented as follows, (Zomaya et al 1999, Zomaya and The 2001, (Vsalaksh and Svanandam 2009). 1 TAQ P j arrval t1 Avg. Utlzaton (2.13) maxspan Utlzaton P j CT P max span j (2.14) m Utlzaton ( Pj j 1 Avg. Utlzaton m (2.15) where TAQ Pj s used to evaluate the Qualty of task assgnment n a processor. arrval t 1 s the arrval tme of task 1 max span s the total fnshng tme of the schedule CT P j s the Completon Tme of processor j Avg. Utlzaton s the sum of all processors utlzaton dvded by the total number of processors and m s the number the processors Effectve utlzaton of processors support the concept of load balancng. If all the processors are used to ther maxmum, the loads, whch
25 48 are the measures of dleness of processors are effectvely reduced. Hence, the ftness functon calculates the average of the total executon tme of the set of tasks allocated to the processors as, F X m TAQ Pj max j 1 m (2.16) Results and Dscusson The results obtaned usng the proposed method IPSO have been compared wth GA and standard PSO and shown n Table 2.5. Table 2.5 Comparsons of best cost, worst cost, average cost and convergence tme usng GA, PSO, and proposed IPSO for dynamc task schedulng wth load balancng Method GA PSO Proposed IPSO Number of tasks Best Cost Worst Cost Average Cost ConvergenceTme (n seconds) The acheved results best cost, worst and average cost values usng the proposed approach IPSO are compared wth GA and standard PSO. The best cost s the best ftness value acheved. The proposed approach IPSO produces the best cost as for dataset 1and t s for dataset 2. The average cost produced by IPSO s for dataset 1 and for dataset 2.
26 49 Thus the result nfers that the proposed IPSO produces the best results when compared wth GA and standard PSO. The convergence tme s slghtly ncreased (0.02 to 0.05s) usng IPSO than wth the standard PSO. The best cost comparson for dataset 1 and for dataset 2 are shown n Fgures 2.13 and 2.14 usng GA, PSO and IPSO Fgure 2.13 Best cost for 50 tasks and 20 processors usng GA, PSO and IPSO Fgure 2.14 Best cost 100 tasks and 20 processors usng GA, PSO and IPSO
27 Performance Comparson The performance of the proposed IPSO approach s compared wth that of the prevously proposed (Vsalaksh and Svanandam 2009) varant PSO methods namely PSO wth Fxed (PSO-FI) and Varable Inerta (PSO- VI) for the same datasets and for the same dynamc task schedulng problem. Table 2.6 Performance comparsons of varous PSO based approaches Method PSO-FI (Vsalaksh and Svanandam 2009) PSO-VI (Vsalaksh and Svanandam 2009) Proposed IPSO Number of Tasks Best cost Worst cost Average cost Convergence tme n seconds The proposed approach IPSO produces the best cost for dataset 1 as and for dataset 2, t s PSO-FI produces the best cost for dataset 1 as and for dataset 2, t s PSO-VI produces the best cost for dataset 1 as and for dataset 2, t s The convergence tme for the proposed method for dataset 1, s s and for dataset 2 s s. The convergence tme for dataset 1and dataset 2 usng PSO-FI are s and s respectvely. The convergence tme for dataset 1and dataset 2 usng PSO-VI are s and s respectvely. Ths comparson reveals that the proposed approach IPSO acheves better results faster (0.003 to 0.02 tmes) than PSO-FI and PSO-VI. Thus, the result nfers that the ncluson of the bad experence component plays a major role n the
28 51 mprovement and achevement of a near optmal soluton when appled to the task assgnment problem wth dynamc tasks. 2.8 CONCLUSION The chapter two brought out a detaled study of the proposed IPSO. The concept of the proposed IPSO s presented to solve the multprocessor task schedulng wth two cases, namely, Statc Schedulng wth ndependent tasks and Dynamc Task Schedulng wth and wthout load balancng, to reduce the makespan of the schedule. The proposed method IPSO s valdated, for Statc Schedulng by comparng wth the tradtonal algorthms such as Longest Processng Tme algorthm (LPT), Shortest Processng Tme (SPT), Genetc Algorthm (GA) and standard PSO. For the dataset, 4 processors wth 30 tasks, LPT produces total fnshng Tme and average watng tme as 62.7s and 48.2s respectvely. SPT produces 32.64s as average watng tme and 75.36s as total fnshng tme. GA produces 32.91s and 74.26s as average watng tme and total fnshng tme respectvely. Standard PSO produces the total fnshng tme as 72.18s and average watng tme as30.65s, and the proposed approach IPSO produces total fnshng tme as 70.97s and average watng tme as 29.74s. The proposed method IPSO s valdated for Dynamc Task Schedulng by comparng wth the prevously proposed varant PSO methods namely PSO-FI and PSO-VI for the same datasets. The proposed IPSO approach for dynamc task schedulng wthout load balancng gves the best cost for dataset 1, as 2374 and for dataset 2, t s PSO-FI produces best cost for dataset 1, as 2814 and for dataset 2, t s PSO-VI produces 2592 as best cost for dataset 1 and 4893 for dataset 2. The convergence tme for the proposed approach s s for dataset 1 and s for dataset 2 and the proposed approach IPSO s 0.2s faster than the PSO-VI.
29 52 For Dynamc Task Schedulng wth load balancng, the proposed approach IPSO produces the best cost as for dataset 1 and for dataset 2, t s PSO-FI produces the best cost for dataset 1 as and for dataset 2, t s PSO-VI produces the best cost for dataset 1 as and for dataset 2, t s The convergence tme for the proposed method IPSO for dataset 1 s s and for dataset 2 s s. The convergence tme for dataset 1 and dataset 2 usng PSO-FI are s and s respectvely. The convergence tme for dataset 1and dataset 2 usng PSO-VI are s and s respectvely. The proposed approach IPSO s 0.02 tmes faster than PSO-VI. Thus, the smulaton results show that the proposed IPSO works well and produces better results for statc and dynamc task schedulng problem n a multprocessor system. Further, to mprove the performance of IPSO approach, a hybrd approach IPSO wth SA s proposed n the next chapter.
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