Performance Modeling of Hierarchical Memories

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1 Performne Modelng of Herrl Memores Mrwn Slemn, Lester Lpsky, Ksor Konwr Deprtment of omputer Sene nd Engneerng Unversty of onnetut Storrs, T Eml: {mrwn, lester, ksor}@engr.uonn.edu Abstrt As te modern omputng envronment expnds memory erry from PU regsters nd lol memory to network storge, te optml gol of omputer rtet beomes to desgn memory erry tt mxmzes te overll desgn of s mne wt mnml ost. Ts requres dedng on te number, speed nd sze of te errl lyers. As te gp between proessor nd memory speed s growng exponentlly, t beomes more mportnt to develop n nlytl model to pture ll tese errl levels nd optmze te memory ess tme to mke good utlzton of bot PU nd memory. In ts pper we study te performne of systems wt mult-level errl memores by modelng ter ess tme w elps te desgner optmze te ost nd ess tme. We use lner-lgebr queung teory ppro to eve our gol nd we expln wy prevous ttempts fled to provde urte models. Our model dffers from ll te prevous relted work by beng globl nd generl nd by usng some probblst equtons tt sow te nterdependene between te dfferent levels nd by usng te P-K formul to dstngus between te memory ess tme nd queung tme. Our ppro s ndependent of te pplton usng te memory wle lssl pproes were progrm dependent. Moreover, our model eves ger levels of ury wle beng expndble to multple levels. Keywords: Memory Herry, Performne Anlyss, Mrkov model, Queung, Aess Tme. INTRODUTION Beng ble to mke urte estmtes of ow long memory ess wll tke to fns n mult-level errl memory system s of prmry nterest n te performne ommunty. In su errl envronment, we ve multple levels of memory strtng from PU regsters nd extendng to es, mn RAM memory, lol dsks, nd network storge [6]. As te memory erry extends to network nd nternet storge pssng by mddle-ter rteture nd ng, te problem of optmzng te memory ess tme beomes more llengng. Dt s stored nd eld n e level untl t s used or repled. E memory level dffers from te oters by ts sze nd speed. As te memory beomes loser to te PU, t beomes fster but smller nd more expensve. Tus, wen desgnng memory erry, our m s to get te fstest desgn wle mntnng ompromse between sze, speed nd ost. Hvng found tt te prevous pproes n studyng memory ess tme were lmted wt te number of memory levels tey n represent, depended on te pplton, nd dd not provde wt g ury, we represented te errl memory by M/G/ queue [] nd we lulted te ess tme by usng lner lgebr queung teory. Ten we onsdered te queung tme of te onseutve memory requests tt n our n dtbse pplton for exmple nd we sowed te dfferene between te ess tme nd te queung tme. Ts dfferene explns te use of te nury of prevous pproes n predtng te memory ess tme. We lso study te bevor of te vrne ess tme w s gly rtl beuse t n drmtlly ffet te mss rto of te memory system nd ts performne. Te model we bult s more generl tn te lssl models beuse t n tke s nput dfferent nput prmeters lke, ess tme, t rto, ost, nd memory request dstrbuton. Te remnng of ts pper s orgnzed s follows: In seton, we present lterture survey bout prevous efforts relted to te top nd we expln our motvton. We dsuss te PK formul metod to lulte te memory request queung tme n seton 3. In seton 4, we present two ses for evlutng our metods. Ten, n seton 5, we sow te lulton results by plottng te vlues resultng from e lterntve nd we sow te dfferene between te two. Fnlly, n seton 6, we propose some tops for furter nvestgton, nd we onlude n seton 7.. BAKGROUND AND MOTIVATION Te errl memory ess tme ws studed by severl reserers but ll te prevous pproes to model nd optmze te ess tme were lmted. Te lmttons re te result eter from te dependene of te models on te pplton or from te lmttons of te nlytl model tt n not represent te deep erres. For exmple, Blsubrmonn et l. expln tt te reent memory erry orgnztons do not mt te ppltons requrements w results n degrdton n te performne []. Jn et l. develop lmted nlytl model tt ptures only two-level e [3], but we see n ter work bg dsrepny between te predted nd mesured memory performne. Most reserers foused on two-level memory s sown n rtles [8] nd [9] nd we don t see ny work tt foused enoug on deep memory erres. None of te prevous reserers tlked bout te vrne of te ess tme of te memory erry. Ts vrne s of prmry mportne beuse g vrne n te ess tme

2 orresponds to ger mss rto nd unexpeted dely - w s undesrble. In prevous work [4], we ve sown tt te ess tme for errl memory wt n nfnte dept s power tled []. In ts pper, we expnd our prevous work nd work on optmzng te memory desgn by tryng to buld memory wt mnml response tme nd mntnng mnml ost. We lso sow nlytlly te use of dsrepny between te nlytl nd mesured vlues for prevous reserers. Our Model s bsed on Mrkov n nlyss w s ndependent of te dstrbuton of memory request tt depends on te pplton. We onsder errl memory tt onssts of L levels nd lowest memory level m s sown n fgure. We model ts erry n stte dgrm s sown n fgure. E pysl memory level,, n fgure orresponds to two sttes n fgure : Te upper stte orresponds to te lookup tme wle te lower stte orresponds to te memory ess tme. Te frst stte orresponds to te memory request from te proessor wle te lst stte orresponds to te lowest memory n te erry. PU Aess Tme t rto n every level nd te ess tme T t every stte. Ts errl memory system s M/G/ queue. In order to buld te lner lgebr model for ts system [], we defne te followng terms: X s te rndom vrble representng te system tme tt orresponds to te totl memory ess tme troug ll memory stges. P s te sub-stost mtrx tt orresponds to te trnston from one stte to noter one. p s te entrne vetor tt orresponds to te stte of te system wen t te frst memory request. p s row vetor of sze L +, were L s te number of ntermedte levels. ε s te unt olumn vetor of sze L +. M s te trnston rte mtrx; t orresponds to te rtes of levng e stte. M s dgonl mtrx of te sme sze s P. I s te dentty mtrx of te sme dmenson s P nd M. B M(I P) Level T V B - s te nverse of B. Level Level l m T T l T m P l l Fgure. Herrl Memory Model: PU wt L ntermedte levels of memory nd mn memory m. - T - (-) T (l-) T l- T T T l Fgure. Stte dgrm of errl memory system wt L ntermedte levels: Intermedte memory levels re represented by two sttes. Te fgure sows te l l l - l M T T 0 0 T m ε We ve sown n [4] tt te memory ess tme s gven by te frst moment of V nd t s ndependent of te dstrbuton of bot te memory ess request (w s dependent on te ompler) nd te serve tme of te nodes (w depend on te rdwre speftons of te memory levels). Te men memory ess tme s gven by: E( X ) x p Vε ()

3 However te vrne of te ess tme s dependent on te dstrbuton of te serve tme of te memory nodes. It s dependent on te frst nd seond moments of V. For exponentl dstrbutons, t s gven by: σ p V ε - (pvε ) () ex For non-exponentl dstrbutons, te vrne s gven by: σ σ + p V T Γ ε X ex Were Γ dg( v-, v-,, vl-) Were v E X of stte n Fg.. ( ) x x s te oeffent of vrton In n pplton tt s multple onseutve memory requests lke dtbse pplton for exmple, te memory requests wll be queued nd must wt to get serve from busy memory, so neter te prevous models nor te bove model wll be suffent to predt te ext tme. Tus we use te P-K formul n te next prgrp to fnd te ext queung tme. 3. THE P-K FORMULA Te Pollzek-Kntne formul (lled P-K formul) [7] gves te expeted verge number of ustomers n queue nd n proess n M/G/ queues. Te P-K formul ws ombned wt Lttle s teorem [] to sow tt te men tme spent by ustomer n n M/G/ queue s gven by: Were, x xρ T + ρ ρ s oeffent of vrton, ρ s te utlzton ftor, nd λ s te rrvl rte. ρ λ x, σ ex, In ts pper we use equton (3) to predt te queung tme for our errl memory system sown n fgure w beomes s sown n fgure 3. Te queung ours wen we ve system wt multple onseutve memory requests tt n not be essed by te sequentl memory t te sme tme, so te memory requests re buffered n queue; ts n be te se of sred memory on prllel mne, smple dtbse ess pplton, or ppelned PU. In te lst two ses te queung uses bottlenek nd ffets te performne of te system beuse t nreses te PI n ppelned proessor [9] x (3) nd nreses te query exeuton tme n dtbse pplton [0]. Te vrne of te queung tme of te model sown n fgure 3 s te sme s tt of te model sown n te prevous prgrp w s sown n equton (). It s obvous from bot equtons () nd (3) tt te men memory ess tmes depend on severl prmeters nludng t rto nd ess tme T t e memory level. λ T T T l Fgure 3. Queung dgrm of errl memory system wt L ntermedte levels: Now te rrvng memory requests rrve wt rte λ nd re queued before te PU. 4. SAMPLE ASES FOR TEST AND EVALUATION In order to sow te dfferene between te men memory ess tme n equton () nd te men queung tme for memory nput/output requests n equton (3), we pply equtons () nd (3) to severl ses ten ompre te results for e se. Our m s to sow tt te men ess tme s not te sme for bot metods nd s dfferent mnm. We suppose tt we ve errl memory system we re buldng nd we ssume tt te system s ost. Przybylsk [] used Agrwl s e mss model [] to sow tt te e mss rto s nversely proportonl to ts sze; so te t rto of e memory level, w s te probblst omplement of te mss rto, s proportonl too to te nverse of ts sze nd tus ts ost. But extendng ts observton to mult-level memory s lttle omplted nd requres more lultons; so we defne te followng prmeters for te system n Fg.: s te probblty of fndng dt n te ntermedte memory level. S s te sze of e ntermedte memory level. s te ost per unt of sze of e ntermedte memory level. β, were β s onstnt. S - T - (-) T (l-) T l- Te totl ost of te L-levels errl system beomes: L S (4) l l l - l m

4 4.. TWO-LEVEL ASH MEMORY SYSTEM We frst onsder te -level s memory system sown n fgure 4. - T T 3 - T 5 - l- - 3 We defne te followng terms: 3 m S s te sze of memory level, M s te ost per unt of sze of memory level, M S s te sze of memory level, M s te ost per unt of sze of memory level, M Y s te rndom vrble representng te probblty of fndng dt n memory level M M Fgure 4. Stte dgrm of two-level s memory: In ts fgure we dstngus between te memory t rto nd te probblty of fndng dt n e level. We nme te frst level M nd te seond level M. Pr( Y M / Y M) Pr( Y M ) T - T 3 3 Pr( Y M M ) Pr( Y M ) ( ) ( ) Pr( Y M / Y M) Were, nd re respetvely te t rtos of memory level nd memory level. Te ost of ts system s gven by: S S + S (5) T T THREE-LEVEL ASH MEMORY SYSTEM Ten we onsder te 3-level s memory system sown n Fg.5. - m l T T 4 T 6 M M M 3 Fgure 5. Stte dgrm of tree-level s memory: In ts fgure we dstngus between te memory t rto nd te probblty of fndng dt n e level. We nme te frst level M, te seond level M, nd te trd one M 3. Agn we defne te followng terms: S s te sze of memory level, M s te t rto t memory level, M, Y M Y M Pr( / ) s te ost per unt of sze of memory level, M s te ost per unt of sze of memory level, M b s te ost per unt of sze of memory level 3, M 3 Y s te rndom vrble representng te probblty of fndng dt n memory level We lso defne te followng probbltes: H H b Pr( Y M ) Pr( Y M ) Pr( Y M ) Pr( Y M ) 3 3 H Pr( Y M ) 3 3 We n proof by smple lulton tt te t rtos re gven by: H H H b HbH( H ) H H H b H( Hb) 3 H H b Te ost of te memory system s gven by: S S + S + S (6) b 3 It s ler from equtons () nd (3) nd from te bove dervtons n ts seton tt te men tme s funton of te t rto nd sze of e ntermedte level, so to optmze

5 te men ess tme, we wll ve to optmze () nd (3) versus tese prmeters. For te two systems we sow ere, we suppose tt we ve onstnt ost nd we try to optmze te ntermedte memory ost nd sze to get te fstest possble desgn s sown n te next seton. 5. ALULATIONS AND RESULTS Now tt we ve te nlytl model to lulte te memory ess tme nd queung tme, we wrote Mtlb ode to mplement our equtons nd to verfy tt wt we mentoned s urte. We ve rred out n exustve set of progrm runs over severl prmeters. Sne te results re onsstent wt e oter we present only few ere. but fnsng too erly s lso undesrble beuse we wste our memory resoures. We remrk ere tt, wle te memory tme s onvex bevor, te vrne dereses s we nrese te memory sze nd ts s norml beuse t depends on te seond moment of te memory ess tme [4] w nreses fster s te memory sze nreses. To empsze more on te dfferene between E(X) nd E(T), we lulte te dfferene between te vlue of te mnmum of E(T) nd te vlue of X t te sme vlue of S. We ll ts dfferene DffT. We plot DffT versus te nput rte λ n fgure 8. We selet te vlues of λ to keep te system utlzton ρ ftor between zero nd one []. We frst onsder te -Level memory system n fgure 4 nd we ssume tt t s n rbtrry fxed ost. To study te men response tme versus S, te sze of memory level, we ssume tt S s n n rbtrry ntervl (4.4<S<7.8). So S, te sze of memory level, wll be gven by dret dervton from Equton (5): S S We plot bot te men memory tme, E(x), nd te queung tme, E(T), versus te sze of te level memory n fgure 6. We remrk tt tey ve dfferent mnm - w onfrms our ssumpton bout te dfferene between tem. Fgure 7. Vrne of te memory ess tme versus te sze S of te Level memory, M, for -Level errl memory system. Te vrne dereses s te memory sze nreses. Fgure 6. Men memory ess tme E(X) nd men queung tme E(T) versus te sze S of te Level memory, M, for -Level errl memory system. E(x) s ts mnmum for S 6.4, wle E(T) s ts mnmum for S 6.8 Ten we plot te vrne of te memory ess tme obtned from equton () for te two-level memory system n fgure 7. Te bevor of te vrne s s mportnt s tt of te men memory tme nd queung tme beuse t uses devton from te men tme. Devton from te e sde of te men tme s undesrble: fnsng too lte s obvously undesrble beuse t n drmtlly ffet te performne (lke nresng te PI n ppelned proessor for exmple), Fgure 8. Dfferene between Mn(T) nd te vlue of X for te sme vlue of S for dfferent vlues of nput rte λ, for - Level errl memory system. We remrk n fgure 8 tt te dfferene s more sgnfnt s te trff nput rte nd system utlzton nreses. Ts vlue goes up to.5 % of te mnmum vlue of te men tme for utlzton ftor lose to.

6 To sow tt our results re ndependent of te number of level n errl memory, we repet wt we dd for te -level s memory to te 3-level s memory sown n fgure 5. Now te system s more omplted beuse we ve more vrbles. Here too, we ssume tt te system s n rbtrry ost nd we ssume tt S nd S 3, te szes of te seond nd trd memory levels, re n rbtrry ntervls (<S <6 nd 6<S 3 <8). So, S, te sze of memory level, wll be gven by dret dervton from Equton (6): bs S S 3 We plot bot te men memory tme, E(x), nd te queung tme, E(T), versus te szes of te memory levels nd 3 n fgure 9. We remrk ere too tt bot surfes re smlr nd tey ve dfferent mnm. sze of te upper level. If we ompre fgure to fgure 7, we remrk te vrne s ger n fgure w mens tt ddng one more level to te erry nreses te vrne nd ts observton s very mportnt beuse te omputer rtet must tke t nto onsderton wen desgnng systems senstve to te vrne. Fgure 0. Dfferene between Mn(T) nd te vlue of X for te sme vlue of S for dfferent vlues of nput rte λ, for 3- Level errl memory system. Fgure 9. Men memory ess tme E(X) nd men queung tme E(T) versus te szes of memory levels nd 3 for 3-Level errl memory system. Bot surfes look onve. E(x) s ts mnmum of.59 for S 4.5 nd S 3 6, wle E(T) s ts mnmum of 3.68 for S 4.75 nd S 3 6. Te plot of te dfferene between te vlue of te mnmum of E (T) nd te vlue of X versus λ n fgure 0 sows ere more sgnfnt dfferene equl to % of te mnml vlue of te memory ess tme. Ts dfferene explns te dfferene between te predted nd mesured performne tt Jn et l. get n ter pper bout performne predton on sred memory progrms (3) n effet te utors tere lulte te men ess tme wle te vlues tey mesure re tose of te queung tme, ts s wy tey get 0% dfferene! Fnlly to ompre te performne of te two memory systems we plot te vrne of te memory ess tme for te tree-level memory system n fgure. We remrk n fgure tt te vrne s more senstve to te upper level memory nd t dereses fster s we nrese Sb, te Fgure. Vrne of te memory ess tme versus te szes of te ntermedte Levels for 3-Level errl memory system. Te vrne dereses s te memory szes nreses. 6. FUTURE WORK Ts pper s prt of lrger work exmnng performne of errl memory systems wt bot nlytl nd smulton tenques. Tere re mny tops we re eter lredy nvestgtng or ope to nvestgte soon. We ntend to vldte our performne model by omprng te predted ess tmes gnst exeuton tmes mesured on rel mnes by usng benmrks. We re lso plnnng to study te bevor of vrne of te memory ess tme more n dept for bot te exponentl nd non exponentl ses. We re workng rgt now on proofng te onvexty of te men tmes obtned n equtons () nd (3) nd we re plnnng to

7 pply severl optmzton tenques on equton (3) to optmze te desgn of mult-level errl memores versus to ome out wt te fstest possble ost-effetve system. 7. ONLUSION We ve developed n nlytl model to evlute te men nd vrne of te ess tme for memory requests n errl mult-level memory envronment. We ve sown nd explned te dfferene between te men memory ess tme nd te memory requests queung tme. Ts dfferene explns te dsrepny between te nlytl vlues nd te prtl vlues obtned by prevous reserers. We ve lso sown te bevor of te vrne of te ess tme s we nrese te levels nd dd more levels to te errl memory system: Inresng te sze of memory redues te vrne; owever ddng more levels nreses t. Our observton elps te desgner dede weter to use bgger levels of memores or use more levels n s desgn. Our nlytl model sown n equton (3), s unversl model nd n represent deep memory erres tt extend beyond te onept of lol mne storge to network storge. Ts model uses Mrkov n nlyss nd n tke dfferent types of memory request dstrbutons. Our model s lso del for optmzton beuse t n tke dfferent nputs lke te t rto, sze, ost nd speed prmeters of te ntermedte memory level. Ts flexblty of tkng dfferent prmeters mkes t esy to expnd nd pture ny level of erry. [7] Dnel P Heymn nd Mttew J Sobel, Stost Models n Opertons Reser: Stost Proesses nd Opertng rtersts, ourer Dover Publtons, 004 [8] A. Smt, e Memores, omputng Surveys, 4(3): p , 98. [9] A. Smt, Dsk e-mss rto nlyss nd desgn onsdertons. AM Trnston on omputer Systems, 3(3), p 6-03, 985. [0] I. MIntyre nd B. Press, Te Effet of e on te Performne of Mult-Treded Ppelned RIS Proessor, te Engneerng Insttute of nd, 99. [] S. Mnegold, P. Bonz, nd M. Kersten, Gener Dtbse ost Models for Herrl Memory Systems, Proeedngs of te 8t VLDB onferene, Hong Kong, n, 00. [] S. Przybylsk, e nd Memory Desgn: A Performne-Dreted Appro, Morgn Kufmnn Publsers, 990. [] A. Agrwl, Anlyss of e Performne for Opertng Systems nd Multprogrmmng, P.D. tess, Stnford unversty, My 987. [3] R. Jn nd G. Agrwl, Performne Predton for Rndom Wrte Redutons: A se Study n Modelng Sred Memory Progrms, Proeedngs of te 00 AM SIGMETRIS nterntonl onferene on Mesurement nd modelng of omputer systems, Mrn Del Rey, lforn p 7-8 REFERENES [] Lester Lpsky, Queueng Teory - A Lner Algebr Appro, Mxwell Mmlln Interntonl publsng group, 99. [] Rjeev Blsubrmonn, Dvd Albonesz, Alper Buyuktosunoglu, nd Sndy Dwrkds, Dynm Memory Herry Performne Optmzton, 7 t nterntonl symposum on omputer rteture, June 000. [3] Ruomng Jn, Ggn Agrwl, Performne Predton for Rndom Wrte Redutons: A se Study n Modelng Sred Memory Progrms, Proeedngs of te 00 AM SIGMETRIS nterntonl onferene on Mesurement nd modelng of omputer systems, Mrn Del Rey, lforn, pges: 7 8, 00. [4] Ksor M. Konwr Lester Lpsky Mrwn Slemn, Moments of Memory Aess Tme for Systems Wt Herrl Memores, st Interntonl onferene on omputers nd Ter Appltons (ATA-006), Settle WA, Mr 006. [5] rstn Hrste, Dnel Lenosk, nd Jon Keen. Mesurng Memory Herry Performne of e- oerent Multproessors Usng Mro Benmrks. Proeedngs of S 97, 997. [6] Jon Wllm Togo, Te Holy Grl of Network Storge Mngement, Prente Hll, 004