Proceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 17-19, 2005 (pp55-60)

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1 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) A eural etwork Realzaton of Schedulng n Grd Computng Envronment MOHAMMAD KALATARI, KAZEM AKBARI Computer Engneerng Department Amrkabr Unversty of Technology (Tehran Polytechnc) TEHRA-IRA Abstract: - The Computatonal Grds provde a promsng platform for effcent executon of computatonal and data ntensve applcatons. Schedulng n such envronments s challengng because target resources are heterogeneous and ther load and avalablty vares dynamcally. In ths paper, we propose a mathematcal neural network based schedulng soluton for grd computng envronment. Usng mathematcal method guarantees rapd convergenc that s essental for such envronments wth prolferaton of resources. Key-Words: - Schedulng, Grd Computng, eural etworks. Introducton Grd schedulng s ntrnscally more complcated than local schedulng of resources, because t must manpulate large-scale resources across management boundares. In such a dynamc dstrbuted computng envronment, resource avalablty vares dramatcally, so schedulng becomes qute challengng. There have been extensve research actvtes on schedulng problems n dstrbuted systems that must be extended for the purpose of grd computng envronment [,,3]. Most problems n schedulng area are P- Complete. Ths fact mples that an optmal soluton for a large schedulng problem s qute tmeconsumng. Therefore, some researchers translated the job-schedulng problem nto a format of lnear programmng or K-out-of- rule and mapped t nto an approprate neural network structure to obtan a reasonable soluton [4,5,6]. eural networks for combnatoral optmzaton problems were frst ntroduced by Hopfeld and Tank n 985 [7]. They used the predefned energy functon E whch follows the quadratc form: E = = j = W VV j j + = V I () where W s the strength of a synaptc lnk between j the th and the jth neuron where the condton of W j = W j must be always satsfed. ote that I s constant bas of the th neuron. Hopfeld gves the moton equaton of the th neuron: du U E = () dt τ V where the output follows the contnuous, nondecreasng, and dfferentable functon called sgmod functon: V = f ( U ) = ( tanh( λ 0 U ) + ) (3) where λ 0 s constant and s called gan whch determnes the slope of the sgmod functon. Snce Wlson and Pawley strongly crtczed the neural network methods (specfcally Hopfeld model) for optmzaton problems [8], and n addton, after the publcaton of dscouragng report of Paell [9] regardng the drawbacks of Hopfeld nets (e.g., convergence to local mnma, lmted capacty of network, and dsablty for solvng hardlearnng problems), t has been wdely beleved that the neural network methods are not sutable for optmzaton problems. But Takefuj and others n ther contnuous and unfalng efforts have demonstrated the capablty of the artfcal neural networks (.e., Hopfeld-lke method) for solvng optmzaton problems, over the best known algorthms and methods. They found that the use of decay term ( U / τ ) n Eq.() ncreases the computatonal energy functon E under some condtons nstead of decreasng t [0]. In hs method, Takefuj explots the topology of Hopfeld net n conjuncton wth both mathematcal method of McCulloch-Ptts (wth or wthout

2 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) hysteress) and Maxmum (wnner-take-all) functon to tackle and solve the problems [0]. Hs works cover a wde varety of professonal felds ncludng game theory, computer scence, graph theory, molecular bology, VLSI computer aded desgn, communcaton, and computer networks. In [] Yueh-Mn Huang et al. represented a Hopfeld neural network based soluton to schedulng multprocessor job wth resource and tmng constrans. In ther model, they assume that all resources are homogeneous and avalable for schedulng at tme 0, and durng schedulng, no resources are added to or deleted from system, whch s ratonal for such a system. However n Grd computng envronments, resources are heterogeneous and can be added or deleted dynamcally. Ths work ams to overcome these new constrants by usng mathematcal neural model rather than Hopfeld neural method. The rest of ths paper s organzed as follows. Secton contans an overvew of the mathematcal neural network model. In secton 3 our grd computng envronment s descrbed. In secton 4 the schedulng problem s descrbed n detal and mapped onto a neural network, followed by smulaton results n secton 5. Fnally, we wll summarze the outcomes and future work n secton 6. Mathematcal eural etwork Model The mathematcal model of the artfcal neural network conssts of two components; neurons and synaptc lnks. The output sgnal transmtted from a neuron propagates to other neurons through the synaptc lnks. The state of the nput sgnal of a neuron s determned by the lnear sum of weghted nput sgnals from the other neurons where the respectve weght s strength of the synaptc lnks. Every artfcal neuron has the nput U and the output V. The output of the th neuron s gven by V = f ( U ) where f s called the neuron s nput/output functon. The nterconnectons between the th neuron and other neurons are determned by the moton equaton. The change of the nput state of the th neuron s gven by the partal dervatons of the computatonal energy functon E wth respect to the output of the th neuron where E follows an n- varable functon: E ( V V,..., ) equaton of the th neuron s gven by:, V n. The moton du E( V, V,..., Vn ) de = = (4) dt V dv In general, the goal of neural computaton s to optmze the fabrcated computatonal energy functon. The energy functon not only determnes how many neurons should be used n the system but also t specfes the strength of synaptc lnks between neurons. Indeed, energy functon s constructed from nformaton n the gven problem, consderng the requred constrants and/or cost functon. Practcally, t s usually easer to calculate the moton equaton (partal dfferental of the energy functon) than the energy functon tself. The superorty of the moton equaton over energy functon can be artculated as follows: ts smplcty (step by step computatons and ease of formulaton), bnary behavor, ease of applcaton, and the flexblty n whch all constrants can be ncorporated. It also resolves the defcences of the Hopfeld net whch was dscussed earler. The energy functon, however, can be defned: du E = de = dv (5) dt In order to numercally solve the partal dfferental equaton or the dfferental equaton to determne the value of moton equaton, the frst order Euler method s wdely used where t s the smplest among the exstng numercal methods. Therefore, based upon the frst order Euler method the value of U ( t +) s determned as below: U ( t + ) = U ( t) + U ( t). t (6) where U (t) s gven by Eq. (4) and termnaton condton s gven by: U ( t) = 0 (7) The condton of the Eq. (7) mples that the constrants are all satsfed. 3 System Model The Grd computng envronment n ths work (shown n Fg.) conssts of n stes, S, S,... S ; n where a ste S ( n) conssts of a number of heterogeneous computatonal resources. Wthn each local ste, there s a forecastng system such as PACE (Performance Analyss and Characterzaton Envronment) toolkt [,] to predct the Job s executon tme on the canddate resources pror to run tme. Selecton of canddate resources s

3 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) accomplshed va a selector component n each ste. Canddate resources are those whch are avalable at the tme or wthn a predetermned tme. To avod the race condton we can reserve the canddate resources. The schedulng mechansm n our work s done n batch-mode, n whch arrvng jobs are collected at prescheduled tmes and are scheduled as a meta-task when a schedulng event s trggered. The followng two steps are accomplshed for schedulng; frst scheduler sends the characterstcs of jobs to each ste; and then n each ste, selector and predctor components return whatever requred for schedulng to the scheduler (.e. a lst of canddate resources and the executon tme of each job on each resource). Scheduler uses a neural network based schedulng mechansm (see secton 4) along wth the nformaton for the schedulng purposes. For the sake of smplcty, we gnore the communcaton overhead n ths work. Therefore, schedulng parameters comprse the lsted jobs, the requred machnes, and tme varables as depcted n Fg., where the x axs denotes the job varable wthn a range from to (the total number of jobs to be scheduled) and the y axs represents the machne varable wthn a range from to M (the total number of machnes) and the z axs s for the tme varable and k represents a specfc tme whch should be less than or equal to T, the amount of total schedulng tme. Thus, a state varable V s defned to ndcate whether or not the jk job s executed on machne j at a certan tme k. In case of V jk =, t denotes that the job s run on machne j at the tme k; otherwse, V jk = 0. It should be noted that, each V corresponds to a neuron of the jk neural network. Scheduler Selector & Predctor Component..... Ste Ste Fg. System Model Ste n 4 Problem Formulaton Wth eural etwork Method Our schedulng approach consders jobs to be run on some machnes (resources) and the followng assumptons are made regardng the problem doman. Frst, the executon tme of each job on each machne s predetermned (see secton 3). Second, a job can not be assgned to dfferent machnes, mplyng that no job mgraton s allowed between machnes. Thrd, resources are added to or deleted from envronment dynamcally and the tme of addton or deleton s estmated by predctor component. The constrants mposed to our model are a deadlne( d ) for each job and a processng tme along wth avalable resources on systems. Fg. Three-dmensonal modelng of problem Mappng constrants nto the moton equaton s as follows: du dt jk * = = A V V - st nhbtory factor jk M T * jk j= k= j j jk B V jk V - nd nhbtory factor T C * L * V jk P j -3 rd nhbtory or st k= exctatory factor. D * K * V jk -4 th nhbtory factor = E * H k d -5 th nhbtory factor ( ( ))

4 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) ( ) F * H P j d -6 th nhbtory factor (8) Moreover, the effects of overall nhbtory and exctatory factors n the moton equaton can be summarzed as : The frst nhbtory factor states that a machne j can only execute one job at a certan tme k. The second nhbtory factor states that no mgraton s allowed, on the other word, job can only be executed on machne j or machne j at any tme. The thrd nhbtory factor or the frst exctatory factor states that the tme spent by job should be equal to P, where j P s the j estmated executon tme of job on machne j. The fourth nhbtory factor states that only one job can be executed on a specfed machne and at a certan tme. The ffth and sxth nhbtory factors state that no volaton of deadlne s allowed. The functons L, K and H s defned as follow: α > 0 L ( α) = 0 α = 0, α α 0 K ( α ) =, 0 α < 0 - α < 0 α > 0 H ( α ) = 0 α 0 we have used hysteress McCulloch-Ptts neuron model where the nput/output functon of the th hysteress neuron s gven by: f U > UTP V = f ( U ) = 0 f U < LTP (9) unchenged otherwse where UTP and LTP are upper trp pont and lower trp pont respectvely, as well as a modfed form of Maxmum euron Model. Maxmum neuron model s composed of M cluster where each cluster conssts of n neurons. In ths model wnner-take-all functon s embedded. Ths mples that one and only one neuron out of n neurons n every cluster wth a maxmum value s encouraged to be fred. The correspondng nput/output functon of the th neuron n the mth cluster s gven by: V m f Um = max m = Um Umj 0 otherwse { U,..., U } mn and for > j (0) In the maxmum neuron model t s always guaranteed to generate satsfactory solutons [0]. The advantages of the maxmum neural model can be summarzed [0]: Every local mnmum s one of the acceptable solutons, whle other exstng neural models cannot guarantee that. Tunng of coeffcent parameters for the actvaton functon s not requred. The termnaton condton of the equlbrum state s clearly defned by a smple mathematcal formula. It should be noted that there are some mportant ponts n our method, whch can be summarzed n the followng: Although t s possble to fnd the exact form of energy functon E, but we do not need t n the process because of the moton equaton du / dt = de / dv. When we use du / dt to fnd a new state of the system, the energy functon wll be employed mplctly. The termnaton condton and the maxmum neuron model together cause the convergence of system n the global mnmum. By usng the moton equaton method, n fact we are employng a sparsely connected network whenever we need that, as opposed to the Hopfeld s fully connected networks. Therefore, for the bg problems there s no need to employ a huge number of synaptc connectons whch would cause a spurous convergence. The correspondng algorthm s gven n Fg.3. 5 Smulaton Results The neural network based schedulng algorthm descrbed n the prevous sectons has been mplemented n C and executed on computer wth Pentum IV processor and 5 mega byte RAM. For the evaluaton of the system s performance, dfferent examples of all szes: small, medum and large were run for 0 tmes. Some of the experments and the overall results are lsted below.

5 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) The nput data for the frst, the second, and the thrd experments and ther correspondng results have been shown n Fg.4, 5 and 6 respectvely. In the frst example, some machnes are not avalable at tme 0. We compute the maxmum executon tme of each job on each machne and use these values as nputs to the algorthm. Job 0 Job Job Job 3 deadlne Job 4 Job 5 Job 6 deadlne Job Job3 Job4 Algorthm: begn ntalze U jk s randomly. whle (a set of conflcts s not empty) do begn for := to do for j:= to M do for k:= to T do begn Ujk = Ujk + Ujk V jk = f ( Ujk ) end for := to do begn. fnd the maxmum U jk. determne a segment (j) n whch the maxmum U jk exsts. If there s te remove t randomly. 3. The output of the other segment s neurons s set to zero. end end end. Job0 Job Tme Fg.4 The frst experment However f the deadlnes of all jobs are set to 5 (The mnmum makespan), our algorthm can t be converged. In Fg.6 no constrants on avalablty mposed but the problem sze has been ncreased and the deadlnes of all jobs have been set to 8. m 0 m m Job Job Job Job Job Job6 Job5 m 0 m m tme of avalablty Job 0 Job Job Job 3 Job 4 deadlne Fg.3 The Schedulng Algorthm 3 Job Job3 Job4 m 0 m m Job Job 5 Job 0 0 Job 3 4 Job 4 7 Job Job maxmum fnshng tme of m each job 0 m m on each machne Job Job 8 4 Job Job Job Job Job Job0 Job Tme Fg.5 The second experment m 0 m m tme of avalablty 3 0

6 Proceedngs of the 5th WSEAS Int. Conf. on SIMULATIO, MODELIG AD OPTIMIZATIO, Corfu, Greece, August 7-9, 005 (pp55-60) m 0 m m m 3 m 4 m 5 Job Job Job Job Job Job Job Job Job Job Job Job0 Job3 Job4 Job5 Job9 Job8 Job6 Job Job Tme Job0 Fg.6 The thrd experment Table shows the average number of teratons and executon tme of algorthm for dfferent examples of dfferent szes. The number of teraton and the executon tme of program decrease or ncrease smoothly when the problem sze ncreases, as shown n Table. Job7 o. # of Job # of machnes Avg # of Iteraton Avg Conv. Tme(sec.) < < < Table : The overall results 6 Concluson and Future Work We have presented a neuro-based schedulng soluton for grd computng envronment. As mentoned before, f all machnes are avalable at tme 0, our algorthm always gves the schedulng soluton wth respect to the gven deadlnes. Anyway, rapd convergence s an mportant characterstc of the proposed soluton. In the future, a neuro-based schedulng algorthm whch ncludes communcaton overhead wll be presented. References: [] Y. M. Huang and R. M. Chen, Schedulng multprocessor job wth resources and tmng constrants usng neural networks. IEEE Transacton on system, man, and cybernetcs, Vol. 9, o. 4, August 999. [] T. D. Braun, Howard Jay Segel, oah Beck, A comparson of eleven statc heurstcs for mappng a class of ndependent tasks onto heterogeneous dstrbuted computng systems, Journal of Parallel and Dstrbuted Computng, 00. [3]. Fujmoto and K. Haghara, A comparson among grd schedulng algorthms for ndependent coarse-graned tasks. IEEE Int. Symp. on Applcatons and the Internet Workshops (SAITW 04), 004. [4] Y. P. S. Foo and Y. Takefuj, Integer lnear programmng neural networks for job-shop schedulng, IEEE Int. Conf. eural etworks, 99, pp [5] C. Y. Chang and M. D. Jeng, Expermental study of a neural model for schedulng job shop, IEEE Int. Conf. System, Man, Cybernetcs, 995, Vol., pp [6] J. M. Gallone, F. Charpllet, and F. Alexandre, Anytme schedulng wth neural networks, Proc. RIA/IEEE Symp [7] J. J. Hopfeld and D. W. Tank, eural computaton of decson n optmzaton, Journal of Bologcal Cybernetcs, Vol. 5, 985, pp [8] G. V. Wlson and G. S. Pawley, On stablty of the travelng salesman problem algorthm of Hopfeld and Tank, Bologcal Cybernetcs, Vol 58, 988, pp [9] R. A. Paell, Smulaton tests of the optmzaton method of Hopfeld and Tank usng neural networks, ASA Techncal Memorandom 0047, 988. [0] Yoshyasu Takefuj, eural etwork Parallel Computng, Kluwer Academc Publshers, 99. [] G. R. udd, D. J. Kerbyson, E. Papaefastathou, J. S. C. Perry and D. V. Wlcox, PACE: A toolset for the performance predcton of parallel and dstrbuted systems, In Internatonal journal of Hgh Performance Computng, 999. [] L. He, S. A. Jarvs, D. P. Spooner, X. Chen and G. R. udd, Dynamc schedulng of parallel jobs wth Qos demands n multclusters and Grds, In Proc. of 5 th IEEE/ACM Int. Workshop on Grd Computng, 004.