Performance Implications of Tolerating Cache Faults

Transcription

1 Prformnc Implcon of Tolrng Cch Ful Andr Frd Pour Mrk D. Hll Compur Scnc Dprmn Unvry of Wconn Mdon 1210 W Dyon Sr Mdon, Wconn ABSTACT Mcroprocor r ncrngly ncorporng on or mor on-chp cch. Th cch r occupyng grr hr of chp r, nd hu my b h locu of mnufcurng dfc. Som of h dfc wll cu ful n cch g or d mmory. Th ful cn b olrd by dblng h cch block h conn hm. Th pproch l chp wh dfc b ud whou rqurng on-chp cch o hv rdundn row or column or o u rror corrcng cod. Dblng block, howvr, ypclly ncr cch m ro. Th ppr nvg how much cch m ro ncr whn block r dbld. I how how h mn m ro ncr cn b chrcrzd funcon of h m ro of rld cch, dvlop n ffcn pproch for clculng h xc drbuon of m ro ncr from ll ful prn, nd ppl h pproch o h ATUM rc [1]. ul rvl h h mn rlv m ro ncr from fw ful dcr wh ncrng cch z, nd nglgbl (< 2% pr dfc) unl complly dbld by ful. Th mxmum rlv ncr lo ccpbl (< 5% pr ful) f no nrly dbld. Indx Trm: compur rchcur, cch prformnc, rc-drvn mulon, ful olrnc, on-chp cch, mcroprocor. Th prlmnry vron of h work pprd Unvry of Wconn Compur Scnc Tchncl por #991, Jnury Th work uppord n pr by h Nonl Scnc Foundon (MIPS nd CC ), A.T.& T. Bll Lboror, Cry rch Foundon nd Dgl qupmn Corporon.

2 Inroducon Commrcl nd cdmc mcroprocor rchcur r ncrngly ncorporng cch on h procor chp lf o vod off-chp lnc [3, 6, 8, 12]. Th on-chp cch r currnly mll, bu h rnd owrd lrgr z o hd rlvly lowr off-chp mmory pd; hu, h chp dvo n ncrng poron of hr r o h mmory (g nd block) of h cch. A cch chp r bcom lrg, o wll h frcon of mnufcurng dfc h lnd n h cch. A mnufcurng dfc cu ful n cch f mpr h corrc opron of h cch. W wll udy ho ful h mk b n h cch unbl o rn h vlu wrn o, bu h do no ohrw prurb h opron of h cch (.g., do no cu n lcrcl hor crcu). A ful cu n rror f cu h ym o nr logcl ohr hn h on nndd. W cn prvn ful n n on-chp cch from cung rror by (1) dcrdng chp wh uch ful, (2) ung rdundn mmory, or (3) dblng cch block h conn ful. Th dvng of dcrdng chp, mhod (1), h work for ny dfc. I ddvng, howvr, h by rducng yld ncr chp co. dundn mmory, mhod (2), cn b mployd o olr ful n l hr wy: () dd xr mmory word (row) h r lcd nd of fuly on, (b) dd xr b pr word (column) h r lcd nd of fuly on, or (c) dd xr b pr word o or n rror corrcng cod. Th dvng of h pproch h hy work for ny mmory. Two ddvng, howvr, r () h co or opporuny co of h xr mmory, nd (b) pobl mmory cc m ncr cud by mplmnng hm. Dblng cch block h conn ful, mhod (3), pplcbl only o cch. Snc cch mrly kp copy of d from mn mmory, ll mmory n cch rdundn. Thu, nd of buldng rdundncy on op of h rdundn mmory n cch, h mhod ju cu h cch o vod ung h mmory h conn fuly b. Cch r buffr ud o hold d from rcnly-ud pr of mn mmory [14]. D uully rnfrrd from mn mmory n lgnd block (lo clld ln). Th numbr of by n block h block z. A block ord n cch wh mmory h hold conn, g (pr of mn mmory ddr) nd om b, ncludng vld b h ndc whhr block prn. Th block n cch r uully dvdd no. vry block mp o on, o h only block n h mu b rchd on rfrnc. A cch wh ocvy n h n block n ch. A cch drcmppd f n 1, fully-ocv f n qul h numbr of block n h cch, or n-wy -ocv ohrw. Cch z h numbr of block n cch mulpld by h block z. Fnlly, h numbr of h cch z dvdd by h produc of h block z nd ocvy. Cch r uully chrcrzd by hr block z, ocvy nd cch z. On pproch o mplmn h dblng of cch block, propod by Pron, l. [11], o u cond vld b. ch cch block normlly conn vld b h whn block brough n nd r whn h block nvldd. A cch h occur whn h ddr g mch h ddr rfrncd by h procor nd h vld b. W cn mplmn mhod (3) by ddng cond vld b o ch block whch or r wh mmory mchnm, 1 bu lf unchngd durng norml cch opron. Wh h nhncmn, cch h now rqur g mch nd boh vld b o b. On r. 1 Th mmory mchnm cn b nvokd whn chp d, or cn b bul n nd run whnvr h ym

3 - 3 - cch m, h ncomng d hould b lodd no block who cond vld b. If ll block n r fuly, hn on cn hr () dcrd h chp, (b) byp h cch nd nd h rqud d drcly o dnon (CPU or mmory), or (c) v h d n pcl buffr, uch vcm cch [7]. Clrly, pplyng opon () o drc-mppd cch quvln dcrdng ll fuly chp. Dblng cch block offr wo dvng ovr ung rdundn mmory, bu lo uffr wo ddvng. Th fr dvng h, unlk wh rdundn mmory, mplmnng cond vld b do no ncr cch cc m on h. Th cond dvng h h mhod llow ll non-fuly block o b ud, whr rdundn mmory ud only whn om mmory fuly. Th ddvng of h mhod r h cn ncr boh h mn nd vrbly of h cch m ro, whr ung rdundn mmory lv h m ro unchngd. Condr dblng on block n of four-wy -ocv, 6-by cch wh 32-by block ung h LU (lrcnly-ud) rplcmn lgorhm. frnc o wll wh ocvy hr, whl rfrnc o h ohr 511 wll bhv normlly. frnc h would hv h n h fourhrcnly-ud block of n n nc cch wll now m; ll ohr rfrnc prform h m. Nvrhl, dblng cch block cn ld o br vrg cc m hn ung rdundn mmory. Th vrg cc m wh rdundn mmory, mhod (2), : mhod(2) ( cch + ) + m mmory, whr cch h cc m o cch whou rdundn mmory, h ddonl m ndd o mplmn rdundn mmory, m h m ro of h cch, nd mmory h cc m for mn mmory. For dblng cch block, mhod (3), h vrg cc m : mhod(3) cch + m (1 + δ) mmory, whr cch, m, nd mmory r bov nd δ h rlv ncr n h cch m ro du o ful. rrngng h rm, w fnd h dblng block nfror o rdundn mmory only whn: < δ m mmory. Snc our rul how h vlu for δ (from on or wo ful) r ofn wo o four prcn ( Scon 4) nd cch > m mmory for mo cch, mu b l hn wo o four prcn of cch for mhod (2) o xhb br prformnc hn mhod (3). Furhrmor, mhod (3) clrly fr for chp wh no ful (whr δ 0), nd lkly h prformnc cn b mprovd gnfcnly wh vcm cch [7]. To h b of our knowldg, h only ppr o do dld nvgon of h ffc on m ro of dblng cch block by Soh [15]. Soh nvgd h dgrdon n cch prformnc by rndomly njcng ful no h cch nd hn runnng rc-drvn mulon. For ch cch, Soh rpor h vrg m ro of vrl mulon wh dffrn ful prn. H prn rul for h numbr of fuly block rngng from 0% o 50% of h block n cch of hr z (256,, nd by), hr ocv (drc-mppd, wo-wy -ocv, nd fully ocv) nd hr block z (8, 16, nd 32 by). Th ppr xnd Soh work n wo ky wy. Fr, w how h h drbuon of m ro ncr cn b clculd from LU dnc probbl for ch, whl Soh ppr dd no condr h u. On mplcon of our quon o confrm h nuon h h mn of h m ro for cch wh nd ngl ful quvln o 1 ng n unprurbd cch nd ngl

4 - 4 - ng n ocvy dcrd by on. 2 For xmpl, l m 0 b h m ro for 6-by cch wh ocvy four, block z of 32 by nd no ful nd m 1 b h m ro for 4-by cch wh ocvy hr, block z of 32 by nd no ful. W how h mn m ro for 6by cch wh ocvy four, block z of 32 by nd on ful (511/) m 0 + (1/) m 1. Scond, w how how ll-ocvy mulon [5] cn b xndd o collc nformon for fndng h ffc of ll pobl prn of ful on cch wh mny ocv nd z (bu on block z) n on p hrough n ddr rc. Soh, on h ohr hnd, prformd mulon for ch ful prn n ch cch. For h ron, Soh m h mn m ro ncr from mll frcon of h pobl ful prn 3 nd lm h vry of cch xmnd. Our pproch, on h ohr hnd, llow u o clcul h xc mn, mxmum nd ndrd dvon of m ro ncr for vrl ful nd wdr rng of cch. W concnr on cch wh fw ful, bcu w blv h chp wh mny ful n h cch wll uully hv ful n ohr crcl rourc, nd hu wll b dcrdd nywy. ul wh h ATUM rc [1] ugg h h mn rlv m ro ncr from fw ful dcr wh ncrng cch z, nd uully mll (< 5% pr ful). Furhrmor, f no complly dbld, mn dgrdon for lrg cch nglgbl. Conqunly, lkly h h ffcv cc m of cch wh om block dbld wll b l hn h of cch ung rdundn mmory. Th mxmum rlv m ro ncr for ngl cch ful or for wo cch ful n dnc ccpbl f h ocvy of h cch wo or grr nd h block z 8 or 16 by. Lrgr block z uffr lghly lrgr m ro dgrdon whn block r dbld. Wh drcmppd cch, howvr, hr probbly (lb mll wh lrg numbr of ) h h xcung progrm hvly rfrnc h fuly block(), hrby vrly dgrdng h cch prformnc. Nvrhl, w xpc h h ovrll mpc of h wor-c bhvor wll no b mporn on mchn ud o run mny dffrn progrm. Th r of h ppr procd follow. Scon 2 dvlop quon for h mpc of cch ful on h m ro. Scon 3 dcu how d for mny ful prn n mny cch cn b ghrd wh ngl p hrough n ddr rc. Scon 4 prn h rul of h nvgon, nd Scon 5 conclud our dcuon. 2. Th Impc of Fuly Block W now urn o h mpc of dblng fuly cch block on h cch m ro. W how how h mpc cn b xprd from pr- mulon d for ny prn of ful nd hn drv quon for om mpl c. Th quon how wh d mu b ghrd n rc-drvn mulon o h m ro for ny ful prn cn b clculd. No h h followng drvon mk no umpon bou h drbuon of h rfrnc rm. Aum cch h lbld 0 hrough 1, nd rfrncd by dynmc rfrnc rm of rfrnc. Aumng ll block whn r ordrd ccordng o om ck rplcmn lgorhm [10] (uch LU), dfn D (j) o b h numbr of rfrnc o h j-h block n h -h 2 Aum h h m ro of cch wh ocvy zro on. 3 For m<<, h numbr of wy o plc m ful n O ( m )

5 - 5 - nd D (j) o b h numbr of rfrnc o h j-h block n ny. Thn: D (j) 1 Σ D (j), j > 0. 0 L M (n) b h numbr of m n n n-wy -ocv cch wh ccd by rfrnc. Snc m occur whn rfrnc no o on of h fr n block of : M (n) M (0). Σ n D (j), n > 0 nd j 1 Furhr, dfn F o b n -lmn ful vcor (f 0, f 1,..., f 1 ), whr f h numbr of fuly block n. Th ddonl m h ful cu n n n-wy -ocv cch r: (n, F) 1 f 1 Σ Σ D (n j), n > 0. 0 j 0 Th numbr of m for h fuly cch, M (n, F), m ro, m (n, F), nd h rlv m ro ncr wh rpc o mlr ful-fr cch, δ(n, F), r: M (n, F) M (n) + (n, F), n > 0, m (n, F) δ(n, F) M (n, F), n > 0, nd m (n, F) m (n) m (n) (n, F), n > 0. M (n) Thu, h prformnc of cch wh nd ocvy n wh ny prn of ful cn b drmnd from h vlu of D (j) for 0 o 1 nd j 1 o n. Scon 3 wll how how o prform rc-drvn mulon o ghr D (j) pror o lcng ful vcor. Th llow u o pply mny ful vcor o h m mulon rul. Nx w pply h bov quon o h mporn pcl c of on nd wo ful. Tbl 1 rp frqunly-ud noon Sngl Ful Sngl ful vcor F 1 r pcl c of ful vcor F whr h on ful nd ll ohr hv no ful: f 1, 0 1, nd f k 0, k. Th ffc of ful n of n n-wy -ocv cch o cu cch m on rfrnc o block n n, whch would no hv md whou h ful. Th ohr n 1 block n h wh h ful r unffcd, r h rmnng 1 n h cch. Th mpl h h ddonl m nducd by h ful vcor F 1 r: (n, F 1 ) D (n).

6 - 6 - Trm Dfnon h numbr of ; mplcly-ud n dfnon blow. n h ocvy. h numbr of rfrnc (rc lngh). D (j) h numbr of rfrnc o h j-h block n h -h. D (j) h numbr of rfrnc o h j-h block n ny. M (n) h numbr of m n n n-wy -ocv cch. m (n) h m ro of n n-wy -ocv cch. F n -lmn ful vcor (f 0, f 1,..., f 1 ), whr f h numbr of fuly block n. (n, F) h numbr of ddonl m n n n-wy -ocv cch wh ful ccordng o ful vcor F ovr mlr ful-fr cch. m (n, F) h m ro of n n-wy -ocv cch wh ful ccordng o ful vcor F. δ(n, F) h rlv m ro ncr of n n-wy -ocv cch wh ful ccordng o ful vcor F ovr mlr ful-fr cch; for brvy, rfrrd o h rlv ncr. Tbl 1 Frqunly-ud noon. Snc h numbr of n cch,, cn b lrg, worhwhl o dll h drbuon of (n, F 1 ) cro ll. Aum h h ful qully lkly o b prn n ny. Thn h mn (), mxmum (MAX), nd ndrd dvon (STD) of (n, F 1 ) r:

7 ?> 98 F VU ` b $# %, MAX STD (n, F 1 ) (n, F 1 ) "! (n, F 1 ) 1 1 D (n) Σ 0 MAX 1 &'( Σ 0 D (n) D (n) )) ) ) ) ) D(n),, nd * / 1 Σ 0 D (n) (1) Svrl ubuon my b md o d n undrndng h mn ddonl m, mn m ro, nd mn rlv m ro ncr: ;: (n, F 1 ) A@ m(n, F 1 ) HG δ(n, F 1 ) < < < < < D(n) m (n) + II 1 M(n 1) M (n), B B B B B B B B B B B B B B B m (n 1) m (n) J J J J J D(n) KK 1 M(n) NO LL L L L L L L L L L L L L L m(n 1) m (n) MM 1 P m (n) C C C C C 1 m (n) + DD 1 m (n 1), nd (2) Q Q Q Q Q Q Q Q m(n 1) 1 m (n) S T. (3) quon (2) confrm h nuon h h mn of h m ro wh ngl ful quvln o 1 ng n unprurbd cch nd ngl ng n ocvy dcrd by on. quon (3) ugg h h mn rlv m ro ncr wll b mll whn: ) lrg, or ) [m (n 1) m (n)] mll Doubl Ful Th c of wo ful my b ubdvdd no hr c, whr h wo ful occur () n h dffrn (wh ful vcor dnod by F 2,dff ), (b) n h m (F 2,m ), or (c) nywhr (F 2 ) Doubl Ful n Dffrn S Ful vcor F 2,dff dno h c of wo ful n dffrn nd j: f f j 1, 0, j 1, j, nd f k 0, k, k j. Lk h ngl-ful c, h numbr of m n n-wy -ocv cch ncr wh ch rfrnc o h n-h block n nd j: (n, F 2,dff ) D (n) + D j (n). Aum h h fr ful qully lkly o lnd n ny nd h cond ful qully lkly o lnd n ny ohr. Thn, 4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 Dfn Σ n x j j 0, j XW (n, F 2,dff ) Σ n j 0 x j c d x. YZY YZYZYZYZY 1 ( 1) 1 Σ 0 1 [\ Σ j 0, j ^] D (n) + D j (n)

8 Ÿ r zy Œ on ~} j ˆ lk f f f f f D(n) + gzg gzgzgzgzg 1 ( 1) hh h h h h h 2 D(n) 2 2 qp (n, F 1 ) 1 Σ ( 1) Dj (n) j 0 M (n 1) M (n) m m m m m m m m m m m m m m m m m m m. Smlrly, u m(n, F 2,dff ) { δ(n, F 2,dff ) vwv v v v 2 2 m (n) + xx m (n 1), nd 2 δ(n, F 1 ). (4) In gnrl, h mn m ro nd mn rlv m ro ncr for ful vcor wh g ful, no wo of whch mp o h m (0 g ), r: ƒ m(n, F g,dff ) Ž δ(n, F g,dff ) m(n 1) m (n) m (n) + g δ(n, F 1 ). g Š Š Š Š Š g m(n) + g m(n 1) nd (5) Doubl Ful n h Sm S Ful vcor F 2,m dno h c of wo ful n h m : f 2, 0 1, nd f k 0, k. Th ddonl m nducd by h ful vcor F 2,m n n n-wy -ocv cch qul o h numbr of rfrnc o h n-h nd (n 1)-h block n : (n, F 2,m ) D (n) + D (n 1), n 2. If ch qully lkly o hv h ful, hn: (n, F 2,m ) 1 1 š Σ 0 œ D (n) + D (n 1) m(n, F 2,m ) δ(n, F 2,m ) D(n) + D (n 1) m (n) + ªª 1 m (n 2) m (n) «««««««««««««««m(n 2) m (n) 1 m (n) ž ž ž ž ž ž ž ž ž ž ž ž ž ž ž M(n 2) M (n). 1 m (n) + 1 m (n 2), nd m(n 2) 1 m (n) ± ² ³.

9 ÚÙ ÆÅ ÒÑ Doubl Ful Anywhr Ful vcor F 2 dno h c whr wo ful my or my no b n h m : f + f j 2, 0, j 1, j, nd f k 0, k, k j. Aumng n 2, h numbr of ddonl m nducd by h ful vcor F 2 : (n, F 2 ) µ D (n) + D j (n), D (n) + D (n 1), j j. If h fr ful qully lkly o lnd n ny nd h cond ful ndpndn of h fr nd qully lkly o lnd n ny, hn h probbly h boh ful lnd n h m 1/. Thrfor, º¹ (n, F 2 ) ½¼»»»» 1 ¾ (n, F 2,dff ) ÈÇ (n, F 2,dff ) ÍÎ 1 + ÉÉËÊÌ Á 1 + ÀÀ Äà δ(n, F 2,m ) ÐÏ (n, F 2,m ) ÓÔÖÕ (n, F 2,dff ). (6) Smlr quon cn b drvd for m nd δ. For lrg, h mn bhvor wh wo ful nywhr (ful vcor F 2 ) wll b ndnguhbl from h mn wh wo ful n dffrn (F 2,dff ), bcu h fnl rm h ØØ 1 fcor Mulpl Ful Anywhr Mulpl ful vcor r h gnrl c of ful vcor F whr h numbr of ful, m, r n h rng 2 m n; hu, ll h block n h my b fuly. Th c do no llow for conc dcrpon nd rqur probblc pproch, nc xhuv mulon h m complxy of O( m ). 5 To do h clculon, ncry o drmn h probbl of crn numbr of ful occurrng n prculr gvn h ol numbr of ful. Thn, h xpcd m ro cn b dcrbd by: ÜÛ m(n, F) P[ Y 0 ] m(n) + P[ Y 1 ] m(n 1) + P[ Y 2 ] m(n P[ Y 0 ] + P[ Y 1 ] + P [ Y 2 ] +.. 2) Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý + P[ Y g ] m(n g), (7) + P[ Y g ] whr P[ Y ] h probbly h prculr h ful whn ll m ful r unformly rndomly drbud mong wh ocvy n nd mo g ful r llowd n prculr (0 g n). To clcul h probbl, w fr rvly drmnd ll pobl drbuon of f ful ovr ( pronng problm), nd hn ud mulnoml o clcul how m wy prculr drbuon occur. Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ Þ 5 Th pproxmon hold only for h rng 0 < m < 2 n ß ß ß.

10 Mhod In h l con, w howd how h D (j), h numbr of rfrnc o h j-h block n h -h of, cn b ud o clcul h m ro for ngl cch wh ny ful prn. In h con, w how how o fnd h D (j) for mny cch wh ngl p hrough n ddr rc. W lo dcrb h ddr rc h w u. All-ocvy mulon [5] n lgorhm h clcul h m ro for cch of mny z nd ocv wh ngl p hrough n ddr rc, provdd h ll cch hv h m block z, u h l-rcnly-ud rplcmn lgorhm nd do no prfchng. In n ffcn mnnr, h lgorhm xmn rc rfrnc o drmn h h rfrnc o h j-h morcnly-ud block of h -h of for cch wh mny vlu of. Th nformon uffcn o clcul h D (j). All-ocvy mulon, howvr, collp h nformon nd only clcul h D(j), nc do no nd o rn numbr o drmn h m ro of ful-fr cch. W xndd ll-ocvy mulon o rcord D (j), nd of D(j), mply by xpndng counr by fcor of o rcord h rfrncd. Our xnon h nglgbl ffc on mulon run-m. I cu only mod ncr n mulon org, bcu h org ndd for h xpndd counr ll mllr hn h org ndd for h ddr g of h cch bng muld. Onc w know ch D (j), w cn clcul m ro of cch wh ful whou ddonl rc-drvn mulon. Th ful r drbud n vry pobl wy nd h rlvn c xrcd by h quon prnd n Scon 2. Snc drc-mppd -by cch wh block z of 8 by conn 2 12, xhuvly clculng ll m ro for hr ful n h cch nvolv (2 12 ) clculon. Thrfor, w u probblc modl for mor hn wo ful. Nvrhl, w b our rul on mny mor c hn Soh, nc w cn clcul nw m ro by ummng ppropr D (j) rhr hn by prformng compl rc-drvn mulon. W vldd our rul by comprng hm wh rul of h xhuv mulon. W u h ATUM rc, bcu hy wr h only vlbl rc h ncludd oprng ym rfrnc nd mulprogrmmng ffc [1]. Tbl 2 how h numbr of nrucon fch, d rd, nd d wr for ch of h rc ud, wll brf dcrpon of hr orgn. Du o h lrg numbr of rc, w gv rul only for combnd rc, dnod by ll. W conrucd h combnd rc by lrnng h ndvdul rc nd cch fluh. For rul from wo rprnv ndvdul rc, Pour nd Hll [13]. W only mul cch mllr hn 6 by, bcu h ndvdul rc r no long nough (ypclly, 400,000 rfrnc) o proprly xrc lrgr cch. 4. ul Th con prn mulon rul for on, wo nd mny ful. Mo of h nly u h rlv m ro ncr cud by nroducng ful (δ(n, F)). For brvy, w wll uully rfr o h mrc h rlv ncr. Unl ohrw ndcd, h block z wll b 16 by Sngl Ful Fgur 1 how h mn nd mxmum m ro of h ll mulon wh on ful (F 1 ) for vrou ocv (numbr of block pr ) nd cch z. Th m ro onbly follow h bhvor of ful-fr cch m ro [2, 4, 5] : for ll ocv, h m ro dcr wh

11 à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á D D Inrucon Nm d Wr Fch Dcrpon â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â â dc DCSIM bhvorl mulon of om cch hrdwr dc Sm prvou for FOTAN complr complng rco.for forf Two FOTAN complon: 4x1x5.for nd lnpck.for forf Sm prvou forf Sm prvou forf Sm prvou fxzz ULTIX fl ym xrcr, 20 k vx Inrconnc vrfy mcr Mcro mblr mblng lnpck2.mr mmxx ULTIX mmory xrcr, 10 k mul Mulprocng 2 job: ALLC (Mcrocod ddr llocor, b rng nnr loop) nd SPIC (Spc mulng oupu buffr) mul Sm prvou mul Sm prvou mul Mulprocng 8 job: Unknown job mul Mulprocng 8 job: Unknown job mul Mulprocng 8 job: Unknown job vc ULTIX C complr u UTP (Ur nvronmn T Progrm, VMS dgnoc), 2 k u UTP, 10 k u UTP, 20 k ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã Tbl 2 Trc ud n h mulon. å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å å ncrng cch z nd for gvn cch z, h m ro dcr wh ncrng ocvy. Th com no urpr, nc quon (2) prdc h mn m ro o b quvln o cch wh 1 unprurbd nd on wh n ffcv ocvy of n 1. Thu n h wor c (n 1) h mn m ro wll dgrd by 1 ææ.

12 úù ïî ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç 0.30 Aoc1 Aoc2 Aoc4 Aoc Aoc1 Aoc2 Aoc4 Aoc M 0.15 o 0.10 M o Cch Sz Cch Sz Fgur 1 Mn m ro of h ll mulon wh on ful for vrou ocv Fgur 1b Mxmum m ro of h ll mulon wh on ful for vrou ocv èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéè Mor rpd ngh no h ffc of nroducng ful cn b gnd by udyng h rlv (m ro) ncr. Fgur 2 nd 2b how h mn nd mxmum rlv ncr for h ll mulon. Fgur 2 rvl h h mn rlv ncr g mllr h ocvy or numbr of g lrgr. For drc-mppd cch (ocvy on), on ful dbl whol ; hu, on vrg, D(1) êëê ê ê ê ddonl rfrnc wll m. A h ocvy ncr, h ffc of h ful o rduc h ocvy of prculr by on. Locly of rfrnc rduc h mpc of h wh lrgr ocv. A h xrm of fully-ocv cch, h cch z mrly rducd by on block, rulng n nglgbl mpc on h m ro. A h cch z ncr wh gvn ocvy, h mn rlv ncr g mllr for ll ocv muld. I do o, bcu ncrng h cch z (whl holdng h block z nd ocvy conn) ncr h numbr of n h cch. From quon (3) w cn h h mn rlv ncr proporonl o ìì 1. Furhrmor, h frcon í í í í í í í í m(n 1) lo nd o dcr m(n) wh ncrng cch z [5]. Fgur 2b llur h mxmum rlv ncr for on ful: MAX ñð δ(n, F 1 ) òò ò ò ò ò 1 M (n) MAX óô öõ D (n) For ocv of wo nd lrgr, cch z g lrgr, h mxmum rlv ncr gnrlly g mllr bcu MAX ø D (n) dcr fr hn h ol numbr of m (M (n)) dcln. ch.

13 ÿ û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û 0.10 Aoc1 Aoc2 Aoc4 Aoc Aoc1 Aoc2 Aoc4 Aoc l 0.06 v 0.15 l v 0.10 I n 0.04 c r 0.02 I n c r Cch Sz 0.00 Cch Sz Fgur 2 Mn rlv m ro ncr of h ll mulon wh on ful for vrou ocv Fgur 2b Mxmum rlv m ro ncr of h ll mulon wh on ful for vrou ocv üéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéüéü doublng of cch z doubl h numbr of, hrby ypclly rducng h numbr of c whr n block mp h m. vnully, MAX ýþ D (n) rch zro, whn h cch z lrg nough h no n block. Wh drc-mppd cch, howvr, mxmum rlv ncr g lrgr for lrgr cch. Th bcu, cch z ncr, MAX D (1) pproch h numbr rfrnc n h ddr rc o h mo-rcnly rfrncd block (nd of pprochng zro dd for -ocv cch), whl h ol numbr of m ll dcln. vn hough h mxmum rlv ncr g lrgr wh cch z, h mxmum m ro ll dcr ( Fgur 1b). For brvy, w wll no prn numrcl rul for ndrd dvon. Howvr, for cch wh ocv grr hn on h gnrl rnd for h ndrd dvon of h rlv ncr dy, mll dcr cch g lrgr. Wh drc-mppd cch, h rnd owrd lgh ncr cch g lrgr. Th ndrd dvon n rlv ncr lo g mllr wh lrgr ocvy. Fgur 3 dpc h ffc of chngng h block z on h rlv ncr. All h grph n Fgur 3 nclud block z of 8 by, 16 by, nd 32 by. Fgur 3 dply h mn rlv ncr for ocv of on nd wo, whl Fgur 3b do o for ocv of four nd gh. Qulvly, h rul do no dgr from h obrvon md bov; h ln for block z of 16 r n Fgur 2. Th mo nobl pc of h grph com no urpr: h lrgr h block z, h grr h rlv ncr. Dblng 32-by block for ll prccl purpo quvln o dblng wo djcn 16-by block. Thu on h vrg m ronbl o xpc n pproxm doublng

14 n h ddonl m cud by dblng B-by block wh rpc o B /2-by block; om dcrpncy cud by h fc h h m ro for qul zd B-by block nd B /2-by block cch wll dffr. Fgur 3c nd 3d how mxmum rlv ncr. No h h cl of h wo fgur dffr by fcor of n. Agn, h ln for block z 16 r n Fgur 2. No qulv dffrnc r found hr, hr. A wh h mn, h mxmum rlv ncr pproxmly proporonl o h block z Doubl Ful Fgur 4 how rlv ncr for cch wh wo ful. Fgur 4 nd 4b how h mn, whl Fgur 4c nd 4d how h mxm. Fgur 4 nd 4c rqur h h wo ful b plcd n dffrn (ful vcor F 2,dff ), whr Fgur 4b nd 4d pu hm n h m (F 2,m ), nd hu om drc-mppd cch whch hv only on block pr. If h wo ful r drbud ndpndnly, hn Fgur 4 nd 4c r mo rlvn, nc lmo ll pr of ndpndnly drbud ful lnd n dffrn ( Scon 2.2.3). For h ron, d for wo ful plcd nywhr (F 2 ) r ndnguhbl from h F 2,dff c, nd w do no dply prly. Th mn rlv ncr follow h m qulv rnd for h ngl ful c. A prdcd by quon (5), h vlu nd o b wc ho of h ngl ful c. Th mxmum rlv ncr wh wo ful n dffrn r mllr hn corrpondng ngl ful numbr, nc h um of h wo lrg D (n) lmo lwy l hn wc h lrg D (n).

15 Aoc1, BS8 Aoc1, BS16 Aoc1, BS32 Aoc2, BS8 Aoc2, BS16 Aoc2, BS Aoc4, BS8 Aoc4, BS16 Aoc4, BS32 Aoc8, BS8 Aoc8, BS16 Aoc8, BS l v l v 0.02 I n c r 0.05 I n c r Cch Sz Cch Sz Fgur 3 Mn rlv m ro ncr wh on ful for ocv of 1 nd 2 nd vrou block z Fgur 3b Mn rlv m ro ncr wh on ful for ocv of 4 nd 8 nd vrou block z 0.4 Aoc1, BS8 Aoc1, BS16 Aoc1, BS32 Aoc2, BS8 Aoc2, BS16 Aoc2, BS Aoc4, BS8 Aoc4, BS16 Aoc4, BS32 Aoc8, BS8 Aoc8, BS16 Aoc8, BS32 l v l v 0.02 I n c r 0.1 I n c r Cch Sz Cch Sz Fgur 3c Mxmum rlv m ro ncr wh on ful for ocv of 1 nd 2 nd vrou block z Fgur 3d Mxmum rlv m ro ncr wh on ful for ocv of 4 nd 8 nd vrou block z

16 Aoc1 Aoc2 Aoc4 Aoc Aoc2 Aoc4 Aoc l v 0.10 l 0.15 v I n c r 0.05 I n 0.10 c r Cch Sz Cch Sz Fgur 4 Mn rlv m ro ncr for wo ful n dffrn Fgur 4b Mn rlv m ro ncr for wo ful n h m 0.4 Aoc1 Aoc2 Aoc4 Aoc8 0.4 Aoc2 Aoc4 Aoc8 l v l v I n c r 0.1 I n c r Cch Sz Cch Sz Fgur 4c Mxmum rlv m ro ncr for wo ful n dffrn Fgur 4d Mxmum rlv m ro ncr for wo ful n h m

17 Mn,Aoc1 Mx,Aoc1 Mn,Aoc2 Mx,Aoc2 Mn,Aoc4 Mx,Aoc Mn,Aoc2 Mx,Aoc2 Mn,Aoc4 Mx,Aoc4 M 0.6 M 0.6 o 0.4 o Numbr of Ful Numbr of Ful Fgur 5 Mn nd mxmum m ro wh numrou ful, bu wh mo on ful pr Fgur 5b Mn nd mxmum m ro wh numrou ful, bu wh mo wo ful pr 1.0 Mn,Aoc4 Mx,Aoc4 Mn,Aoc8 Mx,Aoc8 1.0 Mn,Aoc8 Mx,Aoc M 0.6 M 0.6 o 0.4 o Numbr of Ful Numbr of Ful Fgur 5c Mn nd mxmum m ro wh numrou ful, bu wh mo four ful pr Fgur 5d Mn nd mxmum m ro wh numrou ful, bu wh mo x nd gh ful pr ; ocvy 8

18 Two ful n h m (Fgur 4b nd 4d) r lkly o r only f ngl flur cu boh ful (.g., conrol gnl o boh block fl). Th prncpl ffc of movng h wo ful from dffrn o h m on wo-wy -ocv cch. Snc h cch now hv n nr dbld, hy nd o bhv lk drc-mppd cch wh on ful (whch lo hv n nr dbld). A h cch g lrgr, mn rlv ncr no longr pproch zro (Fgur 4b) nd mxmum ncr g lrgr Mulpl Ful Fgur 5 xmn h mpc of mny ful on h mxmum nd mn m ro for cch wh 16-by block nd 64. Th mxmum numbr of ful llowd pr on, wo, four nd gh n Fgur 5, 5b, 5c nd 5d, rpcvly. ch fgur vr ocvy from h mxmum numbr of ful llowd pr o gh, nd w clcul ch d pon from ll pobl rlvn drbuon of ful. D for mllr numbr of ful mor mporn hn h for lrg numbr, nc chp wh lrg numbr of fuly b r mor lkly o hv drucv flur h forc hm o b dcrdd. Snc h grph dpc cch wh h numbr of conn, cch of ocvy 2 n r wc lrg ho wh ocvy n. Th lm h uly of compron bwn dffrn ocv. Fgur 5 dply d for mo on ful pr. A prdcd by quon (5), mn m ro ncr lnrly wh h numbr of ful lop of [(m(n 1) m (n))]/. Th lop for h drcmppd cch h lrg, nc n nr dbld wh ch ful. Mxmum m ro r qul o mn m ro h ndpon, whr non or ll of h hv fuly block. Bwn h ndpon, mxmum vlu r modly wor hn mn vlu. Mn m ro wh mor ful pr bgn wh h m lop bov (nc wh h fr ful hr mxmum of on ful n ), bu hn ncr ccordng o polynoml wh dgr boundd by h mxmum numbr of ful pr. Mxmum m ro r much wor hn mn ro whnvr h mxmum numbr of ful pr qul h ocvy, llowng n nr o b dbld. 5. Concluon W hv mpd o provd ngh no h ffc on cch m ro of olrng non-crcl ful n n on-chp mcroprocor cch. Snc cch non-crcl rourc prmrly ud o ncr h prformnc of, no nur h corrc opron of, procor bcom ronbl lrnv o u mcroprocor chp h conn procng ful n h d block or g b. Dong o ncr ffcv chp yld, nd hrfor rduc chp co. Whl chp wh dbld cch block wll uffr lrgr m ro, hy my ll produc fr mmory ym hn chp h olr cch ful by uffrng h cc m ovrhd nroducd by rror corrcng cod or rdundn row or column. W fr howd how h m ro of cch wh ny ful prn cn b clculd from h numbr of rfrnc o h j-h block n h -h of. W hn dcrbd how o xnd llocvy mulon [5] o clcul h mrc for mny cch wh ngl p hrough n ddr rc. Fnlly, w ppld h chnqu o h ATUM rc [1]. ul ugg h h mn rlv m ro ncr from fw ful nglgbl f no r complly dbld nd mll n ny c (< 5% pr ful). Conqunly, lkly h h ffcv

19 cc m of cch wh om block mrkd fuly wll b l hn h for h lrn mhod of olrng cch ful. Th mxmum rlv m ro ncr for ngl cch ful, or for wo cch ful n dnc, ccpbl f h ocvy of h cch wo or grr nd h block z 8 or 16 by. Lrgr block z uffr grr pnl wh prmnn block nvldon. Wh drc-mppd cch, howvr, hr probbly (lb mll wh lrg numbr of ) h h xcung progrm hvly rfrnc h fuly block(), vrly dgrdng h cch prformnc. W xpc h h ovrll mpc of h wor-c bhvor wll no b gnfcn for mchn ud o run mny dffrn progrm. 6. Acknowldgmn W hnk h Condor projc h Unvry of Wconn [9] for provdng u wh h compuonl rourc rqurd for h lrg numbr of lnghy mulon prformd nd G. Soh for rdng nd mprovng drf of h ppr. frnc 1. Agrwl, A., S,., nd M. Horowz, ATUM: A Nw Tchnqu for Cpurng Addr Trc Ung Mcrocod, Proc. h 13h Annul Sympoum on Compur Archcur, pp , Tokyo, Jpn, Jun Agrwl, A., Horowz, M., nd J. Hnny, An Anlycl Cch Modl, ACM Trn. on Compur Sym, vol. 7, no. 2, pp , My Brnbum, A. D., Colbry, B. W., D.. Dzl,. D. Frmn, H.. McLlln, K. J. O Connor, nd M. Shoj, CISP: A Pplnd 32-b Mcroprocor wh 13-kb of Cch Mmory, I Journl of Sold-S Crcu, vol. SC-22, no. 5, pp , Ocobr Goodmn, J.., Ung Cch Mmory o duc Procor-Mmory Trffc, Proc. Tnh Inrnonl Sympoum on Compur Archcur, pp , Sockholm, Swdn, Jun Hll, M. D. nd Smh, A. J., vlung Aocvy n CPU Cch, I Trn. on Compur, vol. C-38, no. 12, pp , Dcmbr Horowz, M., Chow, P., D. Srk,. T. Smon, A. Slz, S. Przybylk, J. Hnny, G. Gulk, A. Agrwl, nd J. M. Ackn, MIPS-X: A 20-MIPS Pk, 32-b Mcroprocor wh On-Chp Cch, I Journl of Sold-S Crcu, vol. SC-22, no. 5, pp , Ocobr Joupp, Normn P., Improvng Drc-Mppd Cch Prformnc by h Addon of Smll Fully-Aocv Cch nd Prfch Buffr, Proc. 17h Annul Sympoum on Compur Archcur, Compur Archcur N w, vol. 18, no. 2, pp , ACM, Jun Kdo, H., Myk, J., I. Okbyh, T. Md, T. Okmoo, M. Nkjm, nd K. Kgw, A 32-b CMOS Mcroprocor wh On-Chp Cch nd TLB, I Journl of Sold-S Crcu, vol. SC-22, no. 5, pp , Ocobr Lzkow, M., Lvny, M., nd M. W. Muk, Condor A Hunr of Idl Workon, Procdng of h 8h Inrnonl Confrnc o Drbud Compung Sym, Sn Jo, Clforn, Jun Mon,. L., Gc, J., D.. Schulz, nd I. L. Trgr, vluon Tchnqu for Sorg Hrrch, IBM Sym Journl, vol. 9, no. 2, pp , Pron, D. A., Grron, P., M. Hll, D. Loup, C. Nybrg, T. Sppl, nd K. Vn Dyk, Archcur for VLSI Inrucon Cch for ISC, Th Tnh Annul Sympoum on Compur

20 Archcur, vol. 11, no. 3, pp , Sockholm, Swdn, Jun 13-17, Phllp, D., Th Z80000 Mcroprocor, I Mcro, pp , Dcmbr Pour, Frd nd Hll, Mrk D., Prformnc Implcon of Tolrng Cch Ful, Compur Scnc Tchncl por #991, Unv. of Wconn, Jnury Smh, A. J., Cch Mmor, Compung Survy, vol. 14, no. 3, pp , Spmbr Soh, G., Cch Mmory Orgnzon o nhnc h Yld of Hgh-Prformnc VLSI Procor, I Trncon on Compur, vol. 38, no. 4, pp , Aprl 1989.