Identity of King and Flajolet & al. Formulae for LRU Miss Rate Exact Computation

Transcription

1 detty of g ad laolet & al orlae for LRU M Rate Eact otato hrta BERTHET STMcroelectroc Greoble race Abtract Th hort aer gve a detaled roof of detty betwee two clac forla for the cotato of the eact M Rate of LRU cache A eteo to the detty of two forla of the eected te of a artal collecto the coo collector roble alo reeted eyword: LRU Leat-Recetly Ued ache M Rate Eact cotato oo ollector Proble Addre all correodece to: BERTHET hrta; E-al: hrtaberthet@tco trodcto There et two forlae for the eact cotato of LRU M Rate A early a 97 W g derved a forla to cote the falt robablty rate of LRU dead agg yte g7 Ad 99 laolet et al gave a tegral forla of that robablty laolet9 ther aer laolet et al eto that g gave aother for of that robablty lyg that they ed a dfferet ethod for cotg the ae thg However to or owledge a foral evalece betwee thee two forlae ha ever bee etablhed yet th aer we how a drect evalece betwee the two by algebrac ea both cae hyothee o the LRU yte are the followg: yte aed to obey a deedet-referece Model RM e cache accee are deedet of at htory Addree of cache le or age are characterzed by a olarty dtrbto e a geeral robablty law ot ecearly for ally for reader falar wth HW deg the cache flly aocatve other word dealg wth lted aocatvty ch a HW cache rere the eed of reortg to other odel th hort aer we how the evalece of the two afore-etoed LRU M Rate forla Proof baed o a le rewrtg echa that ae e of a cobatoral detty o the ertato of a bet of robablte rove Lea a fal ecto we gve a eteo to the evalece of two other ad lar forla for the eact cotato of the eected te of a artal collecto the coo collector roble der a geeral robablty law BERTHET Page 7/5/6

2 BERTHET Page 7/5/6 g orla for LRU Notato We ae a robablty law wth geeral o-for dtrbto Probablty law ofte called olarty of the addree Orgal orla g forla for the M Rate of a LRU cache of caacty g7 a follow tle g M rate a ato over all the t-le of ze e all the ertato of all the bet of ze of the et Th orla alo gve ag aer ag77 ag78 Rewrtg of g orla Let be a bet of the et wth we trodce the otato: of ertato Notcg that deedet of the ertato of the bet g tal forla rewrtte: g laolet tegral orla Orgal forla A tegral forla of the LRU M rate of a cache of ze gve by laolet et al loolet9 Theore 5 g 9: t t t dt e e e

3 BERTHET Page 3 7/5/6 where f deote the coeffcet of the olyoal f t ca be re-wrtte g ccevely varable chage e -t dtrbto of the cotato of coeffcet ad dtrbto of the tegral: d d d d We ow defe the followg lea whch rove Aed t allow to relate the two forlae g the atty defed above Lea Let be a bet of ze of the et t hold that: d Rewrtg of laolet et al forla Ug Lea laolet et al forla : Ug the fact that for obvoly a cache of caacty ht oly for cceve accee to the ae addre otato ca be eteded to ll et wth: Alo ce M Rate ll for a caacty t hold that o ca alo be ereed a follow: At frt ght g forla ay ee ore tractable tha that of laolet et al ce the ato erfored o a gle vale ad ot o a bet or However th very argable ce both ereo rere the eerato of all bet whch tractable

4 BERTHET Page 4 7/5/6 Evalece of g ad laolet forla Evalece of thee two forla redce to rovg that: Ufor Dtrbto Let frt reare orelve all th ae ee wth a le for dtrbto / for all Ug varable chage / e - dd t follow that:!!! d d d Relato!!! b a b a d b a clacally obtaed g terated tegrato by art g orla collae to the well-ow LRU vale for a for dtrbto: Ad laolet et al o the other had: Before rovg the evalece the geeral cae we frt trodce the followg lea whch rove Aed Lea Let be a bet of the et of ze ad be a ercal fcto t hold that: Evalece of g ad laolet forla geeral cae We gve the roof of evalece for a geeral dtrbto: detty obvo for ce:

5 BERTHET Page 5 7/5/6 Alo ote that for -: or the geeral cae we roceed by dcto o the ze : ro the hyothe g ad otg dcto te : O the other had: g ro the defto of the followg relato hold: Th te drectly fro coderg the ertato of whoe lat eleet correod to the de oeetly g Lea aled to the fcto fally lead to the dered relt: g ED

6 Alcato to P eectato of a artal collecto laolet et al tegral orla for P eectato or a geeral o-for dtrbto laolet et al have gve a forla for the eected watg te to collect te ot of artal collecto of the coo-collector roble: t t Eected te laolet9 3a 6: E e e dt where the ze of the collecto ad f deote the coeffcet of the olyoal f t obvo that E ad E Let otce that forla ha a le recrrece relato: t t E E e e dt whch ca be rewrtte wth varable chage e -t cotato of coeffcet ad dtrbto of the tegral: E E d the eel we deote th ereo E Ug Lea ad otato t follow that: E ad E Let otce that for a for dtrbto: E gvg the well-ow E H H where H the -th haroc ber ther aer laolet et al alo gve a yetrc fcto ereo or a varat of t after de chage E P where P the of the robablte of the bet Alteratve forla fro errate et al A alteratve forla for the eectato of a artal collecto oted E evalet of laolet E - ad baed o codtoal robablte gve Prooto age 7 of aer errate t very lar to g orla for the LRU rate calclato ce t rere the eerato of all the ertato of all the bet Eectato for a artal BERTHET Page 6 7/5/6

7 BERTHET Page 7 7/5/6 collecto of ze ot of defed a: Ε E where E g ther otato for the ertato: E Aga there a very le recrrece relato E E-E ag E Evalece of laolet et al E ad errate&al E or a for robablty!!!! E hece E Ε or whatever evalece readly obtaed by cotato of the of laolet et al yetrc fcto ereo: E P P P E More geerally detty of laolet et al ad errate et al relato for ay ca be tated ly by otcg that E hece Ε E ED oclo Ug algebrac ea we have how the detty of two forla for the eact cotato of LRU rate ad two other forla for the eectato of a artal collecto P Both dette ae e of a cobatoral relato o the ertato of a bet roved Aed Acowledget The athor tha Rell May for h very helfl coet

8 Referece W g Aaly of dead agg algorth Proc of the P ogre 97 North- Hollad Pblhg oay Roald ag Aytotc rato over deedet referece oral of oter ad Syte Scece Vole 4 e Arl 977 Page 5 htt://wwwcecedrectco/cece/artcle//s ag R & Prce T G 978 Effcet calclato of eected rato the deedet referece odel SAM oral o otg laolet Phle; Gardy Daèle; Thoer Loÿ 99 "Brthday arado coo collector cachg algorth ad elf-orgazg earch" Dcrete Aled Matheatc 39 3: 7 9 do:6/ errate M & rgo N O the eected ber of dfferet record a rado ale arv rert arv:9459 BERTHET Page 8 7/5/6

9 BERTHET Page 9 7/5/6 Aed : Proof of Lea Let be a bet of ze of the et We wat to rove that: of ertato d or ae of lcty of otato we ote the bet detty ealy rove for ad by eadg ad evalatg the tegral or : d ad for : d We rove the geeral cae by dcto o ad we ote d R The: R d d R We trodce the varable chage z hece d dz ad d dz z R At th ot we ae the dcto hyothe hold for the two tegral both of the beg deed by or the frt tegral otce that: of ertato of ertato dz z We e the teredate varable ; ; to ae the roof ore readable hece: of ertato d R Ad:

10 BERTHET Page 7/5/6 of ertato of ertato of ertato R Now ce a bet of ze there are ertato for each ertato of a bet of ze we ca eerate all the oble oto of eleet : of ertato of ertato Lea hold f ad oly f we ca rove that R ad of ertato are the ae ereo other word for each ertato of the followg hold: Whch lfe to: Hece Lea hold f ad oly f:

11 BERTHET Page 7/5/6 3 Wthot lo of geeralty we ly ote the ertato ad coeetly the teredate varable Notg ad wth the coveto that a rodct eal to f t a ety rodct e the de rage ety revo ealty evalet to: t hold f ad oly f the olyoal coeffcet oted a al of both de are eal LHS RHS for ay Th obvoly tre for ce: or we obta the detty: or the geeral cae oe obta o the left had de: LHS ad o the rght had de: RHS The -th coeffcet of the frt ll for de ad that of the ecod ll for - hece otg ad : RHS hagg the de of the ecod ato to - gve: RHS The two ato dffer oly by ther er bod th:

12 BERTHET Page 7/5/6 LHS RHS ce by coveto Th colete the roof of detty ED Aed : Proof of Lea Let be a bet of ze of the et ad a ercal fcto We wat to rove that: or : the geeral cae ha ter: The deotg the et : ED