Memory Size Estimation of Supercomputing Nodes of Computational Grid using Queuing Theory

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1 Inernonl Journl of Compuer Applcons ( ) Volume 8 No., Ocober 00 Memory Sze Esmon of Supercompung Nodes of Compuonl Grd usng Queung Theory Rhul Kumr Insue of Envronmen & Mngemen Lucknow, Ind. Dr. I. A. Khn Inegrl Unversy Lucknow, Ind. Dr. S. P. Trph Insue of Engneerng & Technology Lucknow, Ind. Dr. V. D. Gup Inegrl Unversy Lucknow, Ind. ABSTRACT Grd compung prncples focus on lrge-scle resource shrng n dsrbued sysems n flexble, secure nd coordned fshon. The mos wdespred conemplon s performnce, becuse compuonl grd servers mus offer cos-effecve nd hgh-vlbly servces n he elonged perod, hus hey hve o be scled o mee he expeced lod. Performnce mesuremens cn be he bse for performnce modelng nd predcon. Wh he help of performnce models, he performnce mercs (lke buffer esmon, wng me) cn be deermned he developmen process. Ths pper descrbes he possble queue models hose cn be ppled n he esmon of queue lengh o esme he fnl vlue of he memory sze. Boh smulon nd expermenl sudes usng synheszed worklods nd nlyss of rel-world Gewy Servers demonsre he effecveness of he proposed sysem. Generl Terms Grd Compung, Compuonl Grd, Dsrbued Compung, Cloud Compung. Keywords Grd Compung, Queung Model.. INTRODUCTION The compuer ulzes s growng bly for offces, ndusres, busness orgnzons, corporons; scenfc urbnzon, nsuons, publc dmnsrons nd ohers re usng compuers for beer mngemen. Every orgnzon hs s own compung sysem for exchnge of nformon. Inerne hs been ld down whch cn collec or send d globlly. There re problems where huge d re requred o be rnsferred from one compuer o oher compuer. If we look for he locons of nformon d, hey re physclly nd geogrphclly dsrbued. Thus source of d s dfferen plce. Mny mes d processng s requred o be done n shor me. For some problems even Super Compuers cnno mee he processng speed for problem. A em hs been consued n USA n 00 o fnd mehodology for lrge d compuons n shores me. In Foser -4 crred ou he sudy nd suggesed o hve compuonl grd for USA Compuonl grd shll be ble o provde () Trnsfer of d n Ggbyes per second. (b) Processng speed n Terbyes per second. (c) Sorng cpcy of grd n Pebyes. To mee he bove menoned speed nd hndlng of very lrge d prllel processng of Super Compuer s one of vble soluon. Foser suggesed h super Compuer be clled node. Mny super Compuers n he form of node cn be pr of grd. If good number of supercompuers become nodes we requre hvng n operng sysem progrm for compung d where super compuer wll process n prllel. Hence mxmum prllel compuon cn be possble provded grd orgnzon s esblshed.. GRID ORGANIZATION In Grd Orgnzon he enre super Compuers re o be conneced usng Opcl Fber Sysem. The bes grd s when Mesh Topology s used. We know h n nework prllel d movemen s convered o serl d movemen beween wo compung uns. Ths mkes he workng of nework slow. Inerne lms he processng o clen compuers only. Clen compuer cn receve he d nd even pckge progrms from server or from fellow clens. Nework hence provdes he fcly of nformon exchnge bu hs poor speed of processng. When supercompuers form nodes hen s processng speed s very hgh. The modulr srucure of progrm enbles o process sepre modules sepre nodes. Afer he compuon of nodl progrms fnl compuon s crred ou. Ths mehodology enbles compuers o hndle huge d wh shores me. The d beween wo nodes should be rnsferred usng Opcl fbers. Opcl fbers presenly vlble 44 prllel phs hrough whch Opcl sgnls cn be rnsmed. Ths requres un n every brnch of rnsmsson o conver nlycl sgnl o opcl sgnl. Opcl sgnl re consruced usng lser bem. Hence f he compuer word lengh, whch s o 8 bs presenly, s used hen enre word cn be rnsmed prllel by sngle pulse commnd. Ths mkes rnsmsson speed very fs, whch s very cler h nodl sysem uses prllel pors of Super Compuers for nework whle nerne uses serl pors. 3. MEMORY SIZE AT GRID COMPUTING NODES The mos mporn resource, whch he bsc fcor for deermnng he performnce of Grd, s he sorge spce mngemen. Snce he memory of supercompuer s lmed 4

2 Inernonl Journl of Compuer Applcons ( ) Volume 8 No., Ocober 00 nd here s huge moun of d, whch s o be shred mong he dfferen supercompuers n Vrul Orgnzon, he formon of wng lnes or queues s mmnen. The ro beween he rrvl re of d prculr node n Grd usully denoed by bs/sec nd he servce re or deprure re of d from he node denoed by bs/sec, gves he Grd s performnce mercs known s Quly of Servce (QoS). 5 The wo hngs requred for he workng of node re ) I should be ergodc 6.e. sble b) Sze of memory for he spce complexy Le he d presen memory s 'n' megbyes. Then he probbly h here re 'n' d me s P n (). Le me s ncresed by smll vlue. Snce he me nervl s smll, so here cn be only one rnson me whch s eher rrvl or deprure. Se rnson cn be explned s follows. Le he re of rrvl of d be bs/sec fer secs wll rrve. So he probbly of rrvl of one d = The probbly h no d wll rrve = - Smlrly, he probbly of deprure of one d = The probbly h no d wll depr = - Therefore ny gven pon n me here re hree possble ses: ) No d rrves he node nd no d deprs from he node b) One d rrves he node. c) One d deprs from he node The probbly h here re 'n' d me s P n (). The me s ncresed by smll vlue. Probbly (no d rrves, no d deprs) = P n () (- ) (- ) Probbly (one d rrves) = P n- () ( ) Probbly (one d deprs) = P n+() ( ) Hence he combned probbly of ll he hree cses s: P n ( + ) = P n () (- ) (- ) + P n-() ( ) + P n+() ( ) P n(+ )= P n ()-P n()( )-P n ()( )+P n() )+P n-()( )+ P n+() ( ) Neglecng erm we ge [P n (+ ) - P n ()]/ = - P n () - P n () + P n-() + P n+() Now s very smll.e. 0 so, kng lm Lm 0 Pn ( For sbly d [P n ()]/d=0 ) - Pn () =- P n()- P n()+ P n-()+ P n+().e. - Pn ()-P n () + P n-() + P n+() =0... () Consder se rnson me nd n=0. n hs cse wo condons rse ) No d rrves n he memory buffer b) One d deprs from he memory buffer. Probbly (no d rrves) = P 0 () (- ) Probbly (one d deprs) = P () ( ) Therefore he combned probbly of success P 0 (+ ) = P 0 () (- ) + P () ( ) or, P 0 (+ ) = P 0 () - P 0 () + P () ( ) or, P 0 (+ ) - P 0 () = - P 0 () + P () ( ) or,[p 0(+ )- P 0( )]/ = - P 0()+ P () For sbly, P ( ) - P () Lm 0 0 =0 0 - P 0 ()+ P ()= 0 or, P () = ( P 0 ().... () From equons () & () P 0 () = ( / ) 0 P 0 () P () = ( / ) P 0 ().. P n () = ( / ) n P 0 () Bu he sum of ll probbles should be one.e. n 0 P n () = ( ) 0 P 0()+( ) P 0() +( ) P 0 () +..+ ( / ) n P 0 () = [+( / )+( ) + +( n ] P 0 () =.. () Now le = x The sum of nfne seres + x + x + x 3 + +x n s gven by S = /-x Subsung x= / n bove equon we ge S=/ (- ) Subsung he vlue of S n equon () we ge Now f [/ (- / ) P 0 () = <, he sysem s sble for ergodc condon P 0 () = [- ] The probbly densy funcon (pdf) of d n memory for he ergodc se s gven by P n()=( ) n [- ]. (v) The bove equon, whch s known s he probbly densy funcon of he d n memory cn be used for esmon of verge vlue or men vlue of d syng n he memory. Men d Q (L) = n.p n() 5

3 Inernonl Journl of Compuer Applcons ( ) Volume 8 No., Ocober 00 where Q(L)= n.( ) n [- ].. (v) n = D n memory = D rrvl re = D deprure re Le probbly of success p= Probbly of flure q =-p = Tble : Clculon of Averge Queue D(n bs) Probbly Pn() n.p n() 0 p 0.q 0 p.q p.q p.q.p.q 3 p 3.q 3. p 3.q n p n.q n. p n.q Men d Q (L) = n.p n() Q(L)=0 + p.q +.p.q +3. p 3.q+.+ n. p n.q Q(L)=p.q[+.p+3.p +..+np n- ]....(v) The sum of nfne seres +.p+3.p +4.p 3 + +n.p n- s gven by Le S=+.p+3.p +4.p 3 + +n.p n- S = n.p n-... (v) Mulplyng by p on boh sdes of equon (v) we ge S.p=p+.p +3.p Subrcng equon (v) from (v) we ge S-Sp= +p+p +. or, S(-p) = /(-p) or, S = /(-p) Subsung he vlue of S n equon (v) we ge Q (L) = p. q. (/ (-p) ) Subsung p= = p / (-p), we ge (v) Q (L) =. (x) The verge vlue of he queue lengh cn be used o esme he Sndrd Devon of he queue lengh, hereby he sze of he memory. If he Sndrd Devon s dded o he verge vlue of queue, hen he sze of memory wll be Sze of Memory = Q (L) + SD Sndrd Devon s gven by where = - Sndrd Devon = vrnce = men or verge The probbly densy funcon (pdf) of d n memory for he ergodc se s gven by Le, Probbly of success p= Probbly of flure q =-p=- n.p n ()= n p n q Vrnce = p (+p)/ (-p) P n()=( ) n [- ] = /n [( n P n () - Q (L)) ] =[p.(+p)/(-p) -p/(-p)] = [p / (-p) ] p / (-p) = Hence, Sndrd Devon S.D. = - = p / (-p) [p/ (-p)] = p/ (-p) Subsung p= Sndrd Devon S.D. = If he Sndrd Devon s dded o verge queue lengh hen he sze of memory Sze of Memory = Q(L) + S.D. = = + 4. G/M/ SIMULATION MODEL In he G/M/ Queue 7-6 smulon model, he ner-rrvl mes nd he servce mes of he processes hve generl dsrbuon. For hs model, n exc nlyss s n generl mpossble. pproxmons re gven for he performnce mesures lke he expeced me n he queue, he expeced me n he sysem, he expeced number of clens n he queue, he expeced number of clens n he sysem, ec. In he model, he rrvl process for whch he ner rrvl mes (A, A...) of clens re denclly dsrbued rndom vrbles 6

4 Inernonl Journl of Compuer Applcons ( ) Volume 8 No., Ocober 00 wh n rbrry dsrbuon funcon. Servce mes of clens (B, B...) re denclly dsrbued rndom vrbles wh rbrry dsrbuon funcon. There s sngle server nd he cpcy of he queue s nfne. Jus s n he M/M/, M/G/ nd G/M/ queue, he sbly condon for he G/G/ queue s h he moun of work offered per me un o he server should be less hn he moun of work he server cn hndle per me un. The rrvl re for he ske of compuon convenences re ken s =5000, 7500, 0000, 500, 5000 nd for =500, 750, 000, 50, 500. The compued resuls re reveled n he subsequen ble: Tble : Esmon of Queue Lengh Re of Arrvl ( ) Re of Deprure ( ) Q (L) for G/M/ We hve clculed Queue Lengh for 5 res of rrvls. These 5 pons re on he curve of Queue Lenghs. Queue Lengh very hgh re wll be lso on he curve of Queue Lengh. Le he equon of he curve s represened s: y 0 (x) Now, usng Non-lner regresson echnques, we hve o clcule he vlue of 0,, nd, for he mnml error. These leds o develop formon equons s gven below: n y.. (x) 0 3 y... (x) y.. (x) 0 There re hree unknown vrbles.e., 0,, nd nd hree equons for gven vlues of. These vlues re compued nd gven s: Tble 3: Regresson Tble for hgher vlues of Queue Lengh x y x x 3 x 4 xy x y *0 7.5*0 6.5*0 4.55*0 7.7* *0 7 4.*0 3.6* * * *0 8.00*0.00*0 6.0*0 8.0* *0 8.95*0.44*0 6.59*0 8.99* * *0 5.06*0 6.9* *0 Le, x= & y=q (L) x= , y= , x =5.63*0 8, x 3 =6.88*0, x 4 =8.88*0 6, xy=5.73*0 8, x y=7.0*0 Usng he Abbreved Doolle Process, we ge Tble 4: Abbreved Doolle Process for Regresson Anlyss 3 y * * *0 5.73* * *0 A * *0 4 B *0 8.0*0 4 A * *0 7 B * *0 A3-6.38* *0 B3.00* *0 3 Wh he help of bove ble we cn fnd ou he vlues of 0,,.e. =B3y= =By-B3 = =By-B-B3=689.5 Bsed on he obned vlue, Q(L)= where, s he re of rrvl nd Q(L) s he Queue Lengh. The resuls for he Queue Lengh nd hence esmed memory sze s s gven n he followng ble Tble 5: Compuon of Queue Lengh w.r.. G/M/ model Re of Arrvl Q (L) for G/M/ Memory sze [G/M/] *0 4 [ *0 4 ] *0 6 [ *0 6 ] *0 8 [ *0 8 ] 5. CONCLUSION We proposed h for supercompung nodes, he buffer esmon s conemporry need n crfng he blueprn of fuure compuonl grd. The sudy delnees h for sble funconng of supercompuer, ech node of he compuonl grd should work n ergodc envronmen. The oher ssue h dmges he mplemenon of compuonl grd s freezng of d n some node becuse of ndeque sze of memory. In crcumsnces of mul-chnnel rrvls nd sngle chnnel deprure, kn o Queue Model, G/M/ hs been suded n fce by smulng queues lower re of rrvls. The consequence so ned s exrpoled o compue he sze of memory very owerng re. The oucome demonsres h we requre few 7

5 Inernonl Journl of Compuer Applcons ( ) Volume 8 No., Ocober 00 hundred err byes for rrvl re of 0byes. Exensve smulon sudy llusres poneerng mehod h cn provde smooh performnce conrol nd beer rck n compuonl grd sysems. 6. REFERENCES [] Foser I. nd Kesselmn C., "The GRID: Blueprn for new Compung Infrsrucure," Morgn Kuffmn Publshers, 5 Edon 999, nd Edo03. [] Foser I., Kesselmn C., Tuecke S., "The Anomy of he Grd: Enblng Sclble Vrul Orgnzons," In. Journl of Supercompuer Applcons, 5(3), 00l. [3] Genzsch W., Enerprse Resource Mngemen: Applcons n Reserch nd Indusry, In: In Foser nd Crl Kesselmn, The Grd: Blueprn for new compung nfrsrucure, nd Edon, Morgn Kufmnn Publsher 003. [4] Krl Czjkowsk, In Foser & Crl Kesselmn, Resource Co-Allocon n Compuonl Grds,Proceedngs of he 8h IEEE Inernonl Symposum on Hgh Performnce Dsrbued Compung,999. [5] Foser I., Kesselmn C., Nck J.M., Tuecke S., Grd Servces for Dsrbued Sysem Inegron, Compuer, June 00. [6] Per, F.J. Mzumdr, R.R.: An ergodc resul for queue lengh processes of se-dependen queueng neworks n he hevy-rffc dffuson lm, Communcon,Conrol And Compung, pp ,008. [7] Brford,P., Crovell,M.: Generng represenve web worklods for nework nd server performnce evluon,n Mesuremen nd Modelng of Compuer Sysems,pp.5-60,998. [8] Mensce,D., Almed,V.: Cpcy Plnnng for Web Servces Mercs,Models,nd Mehods,Prence Hll PTR,00. [9] Lzowsk, E.D.: Qunve sysem performnce: compuer sysem nlyss usng queung nework models,prence-hll, Inc,984. [0] Anderson & Drrell: A Cse for Buffer Servers, p.8, IEEE Sevenh Workshop on Ho Topcs n Operng Sysems, 999. [] Robers & Jm W.: Trffc Theory nd he Inerne, IEEE Trnscons on Communcons, 00. [] Zr & Mzen: Undersndng nd Reducng Web Delys, IEEE Journl for Elecroncs nd Compuer Scence, pp.30-37, Vol.34, No., 00. [3] Seven H. & Srkn R.: A Mhemcl Frmework for Desgnng Low-Loss, Low-Dely Inerne, IEEE Trnscons, 00. [4] Chen X. & Mohopr P.: Performnce Evluon of Servce Dfferenng Inerne Servers, pp , Vol. 5, No., 00. [5] Yng Le: Globl Sbly of Inerne Congeson Conrollers wh Heerogeneous Delys, IEEE Trnscons on Communcons, 003. [6] Klenrock L.: Queueng Sysems, Vol., Applcons. John Wley Publcons, NY,

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