Static Timing: Back to Our Roots R. Chen , E. A. Foreman ,P. A. Habitz , J. G. Hemmett , K. Kalafala , J. S. Piaget , P. Qi , N. Venkateswaran , C. Visweswariah , J. Xiong and V. Zolotov IBM Microelectronics East Fishkill, NY ABSTRACT !#"$%"&'($)+*,#-%.*/012*354 76869 %"$* :;1.* <!*3=+* +&>?@3"/%AB*:/"* $*C D&E+*F!2*3;:G&"H+*DIKJ 323CLCL%!3 AM6 .*CN%+.O/+% &*>* $(* *21"D!*3PQ: &"+R&*@&"ST&*32UIJ .CNCL%! : +"H+.P*3V&"@"D+..P*U!.2W+%$%# !X#Y'/>[Z3CL%"*"\.C .*GA 'ZE69.]3W! :*1P?9D!H2.%/!^12*3_A #`%/!*a'(>.*C %"("D!$Y"b$-%*2`&*Y6.Y" Z %#$c.*(.,I d .32*3CL%! .]: &"X&*a2["D+..P*U!..e$2a'/`3+* 1?1+* /&"2gf+H&M+3DA$?h*b'0CL%/!..WiN "D!.*EjbJUk<ljm*C!JU.bk0!".C &* nM%-%*A o)6\3%!"pZ ."VSU! CL' +]J 1P*]qT+ q&..&srM1D:+lJq^qmr\nM,.*;3ED!.%[."DIt76B$:"DA U!3Cu' +=+P".>&Z v($"c?w"xZ G#"..*/!">%.#y.H8+* &"Y*;+* +' cW* %"#1*D!9..*(Iz9 V +("V{ "& C %".'(%1" *.:P@EZ .Cu%$"*$"0 +3Cu' +#z+ "(!%+ AU+* 3*3"&*0|*7: 1/3P.P@?w"TT+* ..*)&2/ "H(M? : +"H+.P*W62e.* !"CN21/A}* %"#1*D!RA}&D+.%/+|12*3%+C *32pZ DIq&.~"$Z%$*O!* ,&( Z ,"&Z32!"E"*/ *cH6|,!*3.* Z &"R+g%! .c$+.*DI 1. INTRODUCTION IBM Microelectronics IBM Research Essex Junction, VT Yorktown Heights, NY JUP"*3"CL'TD&2%0.2*31 +* 1'(*>0'("/%+VX#%+ * p/Z ?2P"M2.*3@:"{ %!.*V?( .P.D+*P"D&%#."%Z DI^rM%$*/A.*C %"+&.*@: !"R!' ..HU!^%+T&2.* ) |6B..CLZ*3"/) C /I}k0&"X&'..2Rb%/+*E'(B%R!{ $+T%+. CNCL%! .Ylw"T'T&XnU+* 6..* Cu%+32lw"[%/!RnI d 2X .*a%! 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STATISTICAL TIMING METHODOLOGY .%$2*F %"'G#z;*3D:@D+.%/!b.* O R*/X"&* .%/f!.P'U! :+"R!.* JUP"*3#"E JUP"*3#"E Û #/,!X%S:+"H+.P* Û Ù Ú JT"*"[ +" R* (* $*3.)"D!*3:+"H&2* J "* "[ UJ P"*3"[ R! / &"2 JU"* #"E R+ ¡N¢G£U¤¥ÜG§1ÝS©¤D¡h©«#¡w¬GT© ®ªU¥®N©Æ]ÅW¬}ª ¥®N¡NU¢G² U! T.*%"1*/7!RA7(&T.2*33+"T%.#y.HA !* ["$Z3%#T(C .,I 3.1 Variational delay modeling |j Z " / P.PSZ#U^: !"R&2*T+/ .2*3%+$^ "(P .*,+ ?h"[email protected]%$P"* #"M.*GI}T._I D69mX9: !"R+.P*1P?( #RX 3Z9b* 9Z"%$\?_:&"X&.P* Ig\ R[1 %P*.mP?(3"# %(* */DI]\*O*"* &.%Y"V.P'T&\%$(P*3*/V%/&Z3"V%! .C Cu%+32]: +"H+.P*UI#3$,!X%c:+"H+.P*]%/+ Z "#]iL"3%D+' $o &%"PCL%!3:G&"H+*O+**3%2Z3$ET&R! %(* */0P?M 3#Rz RIOM.* !./Am*3(* *.z"D!*3Ì:G&"R!.* +)*3 %""H+.P*3)+* %/!*'(cr0 7, D6*OST&TI9 [%!U!"D+%$"2+C .*E"/%$Z3"B"p/Z "E3"/Z3%BZ%+E/ *:P.:P$T%"/+*3 pZ +{ %/+.P*1%! |69.V(%H+ &|+%"b+m,!*[.%3+.P*3^* 35%+. 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Y21* WP"D& A|"D+* P : !"R+' $B62),&2#* &..:3..@&"3"Z* T(lw%/n_%/!*3P*32%/&(?wP"1 +""1*| +"&?w",!T&)&mV:1"*3b?G$"PCN:G&.Z * 2.:$DI d .Y@3&Z"DA3*/" Z%.*?M* 6F"D+* P : !"R+' $1.*zX>%./%!;"#)Z"*31Z [,(e"R!X:$2`,+ 7:P"33+ A}&\ 76\*5.*cb"$Z.9%#X*TI 3.3 Proposed timing methodology T._IBV7691 5,&2*z 1P?9 5* D:#B.* ; 3I E{",X5Z Yb"&.CN' +#O+ "(+%+;) !* .1T&R! %""+$H&2*lw&..H+"ba.A|ù!RA(P"3D!..b*` * $y3VD%* nI )%$P* W+H#e.[Y%#*:P"@V:+"H+.*T!mR`1|*/X [%/&**.%/!M&D+.%/+m #RX,/ $Sln^P*CN CN4,a+2*W/C ,&2%5:G&"R!'..2uz +"H&2.aP"1?hZ..zXe".O"D+* Ì: &"X&'.I [ ",,20,!3.*>?mP".,"D!*3P: !"R+'.#05%"3&Y 'T&299P?U* ($*3*/\"D&* : !"R+' #9?2P"9D+%!}..* _I j^*3%@Xb/ .* T+\'($*Y3P* 3Ab*3#y9Y.\.*3" Z% *$6="D&* : +"H+' $Ec"# "&*@.* (*3*/b"D!*31*$+5& +mGP*R!m?w"P*.#"b(2*@?^V%./%!,X"#I1\3V*$yE$W )%#* Z %' /%+CL'T+$;&7+%/&TX21* )2*,"0P?MEP.' ! PZ3"%#mP?(: !"R!.*TAD9.*$(*3*/|"2V"D&*3P: !"R+'.#DAG+* 3b*6)*/X"/ Z %#5: +"H+'.[&\3:"P$*%S(2*9? 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Z+* ;69.O+*,* (*3#*.Y"D!*3C +"|P?}3HX: +"&R+.P*UD&2.%/+G%/!*3%#.R!.*)769*` + +%$%#"Z 9693#*c329.9Z "* cP* I 9&'.b& 769S31&R+%+]+*OJ9q(¾=E%# +".&* S'(H6|* EZ .Cu%$"*$"}.* [lwGJq^qmr\ng!*3b&7!..%3+.*El7 MEnI! {"TR6^0%#.Z* T*3PB B%* +"V!*31 *TI_9 *3y/T" %2Z3*T 76aX|.v("#*3%B'(u6B*c+&XCL(* &R+%+/DA7Z * 6^"CN%3+ "½2%*>*>3%/!0P?^D MEIP0'.P$@lDi +y/3C .vMo(nAS,+.Ol7i * .vmoGn!* X:"D!a v("* %lDih:/ vMoGn +"c+3D6*TI]T.* +A| aJ\q(¾2c*`I@&^t0Y q76B$" rT%/fP#/&P q_C+Pþ9+%+.*0^769* Ij^*`X:P"D&P/A DUm4 D6 "33Z%$B..*E"Z.B6.).#BG&. +*) EZ .Cu%$"*$"m43D6]$:*6X3"3½h%$*1)|6|P"M%"* "DA+* +%+.:b^&G#3ZV?|!'GPZ3 yUI d ..03"Dy2.R[%#"3| "D:3# ..*@"#..?}.*)EZ XCN%#"*$"^..*GA|2|.2MB(&..&.%0"/&C */?T !R&}%#""R!.*TI B¨ ©BAU®h¥¦(§CSÄ_ûm¥¤¡wÅe¥/g«#© ®Bü#ÇP¥T© ¤D¡N¬ü#² DFEHGJIK?LNM OQP RTS U RTS V RWRHS U Y)OQLWLNGJZ K?[NM O?I]\^M P_[NK?IFEHG RI^`I^M [ba RbI`cIFM [ba VQd?dfehg RbI^\GJijG0I^\GJIk[NZ afLWKQI^\FO5l1m nO opGHP nO RXRHS V VQd5dfe.g oBGHP 5. RESULTS 6|[%"$* :0X.*@$2^6|#"03&%".'().*)X. U!("DB^EZ..CL%"&*3"^1 / .P[Z.*@30GJ9q|qmr+"23,A &*]SD+.&.%/+ 1 5Z* )' /%+CL' !$+D&.2%/& .C .*GI^U[U&:^'(#*@321*E2*@|?h"D+16B"b?)(+* ,.-0/1* 2354RA &*`.* Z &X"H+MD&%11* 5/HIb[ 2$S12: &"E%$.{a!/7658%#".H!*3Y#@'( Y.*]"D+ .%/!gZ#"9.*C "?X!%/I 9 0&%.P*"$+$*S%/+[Z32$P?MH6|e&5* ~%+32,3&*DA Ø0;&*<Ø0PIaØ.PI.#9:9 %+.a%*/D&* .*OùPA.þP1 H+%C &'.)P'}½2%AU6.`ØS1T&þùPA þþ'}½2%#X .*<&*<+"31P?EPI. 9;9 I`93cPZ3"%#?0: !"R!H2*8&%.:+z6B"aq("Z %rT.H+C '32..HFlw%[' .*T+.P*O?SÂ} +* OPXCL.#%"*v($%DnA|"$C ]:PD+Y?/&%!]?>k\V?+12..#DA&*]."D+%+3.* W'G#u6B* Â\T³^ö&*q_T³m"*XDIa9 Y+$,!.W("1V1D+ RXP"DAP($"D!Z ">&*5:D+b: +"H+.P*3IJU"*"CN' +&c1. &m %#".'(1*7%*,m6^"93D# Igk+"ZM& +H&%""H+C .*c%#* +"\6|#"b'(*%+3,!"$YP*Y'GY $+.*\Z &.* bH6| $ /.P.#DI H*O!3.* A *J!".>"#Z.S6^$"' 7!.* *a0EZ%+c,+.#" 2*TAgØPI D1]?V z3X&%/!'3aP'}½2%+"zR+"Pa!%";";1" '32%+/DA/12*5 9%/+b SX&"P"%+.YU!9?w6^"P'g½w%I ¨^©BA N® ¥ÃG§1¯¬GÅeû © ¤7¡Nü#¬G¬(Ë9¢G¯°b°@±ÿ©Uªq ¨'rS¨@² D^EJGHIFK5LXM O tY s^M i DFupvwutxyQYz{z{|PNZ KQE0}~ i^PN D^uvwu y?Ytz{z{| ~ PWGHEkS ~ PNGJES KQk\M M IF\^M vky?\M RTS U U f 5 x?d QV.U UWVhUN RTS U V UW5 x?d UWQ? ?d5Q RTS V U f .U5U x? Q? V?5QV RTS V V ?5 x? ?d UWQV? ?V5dQ RXRTS U U f UWdhU xV?V UW Q? U?U_Q RXRTS U V V?5 x0UWd?d ? UWQ? ?d5Q RXRTS V U f ?V5 x? V?d Q? V?5Q RXRTS V V ?hU x? UNd.U UWQ? ?V5dQ 5.1 Perfectly correlated variations t\#"c6^c76"D&.%/+.a3YR!%+e.v("#*3%#'(R6^$*OEZ..C %"*"[2.* &*zD+.&.%/+..* ;*]3&.*FØS.*z%#* +" #I.&*W I.PI) R*eX )%*T&"2AgV&U!XH+|:&"X&.P*W@2Z3( 6%+.CLCL%+ : +"H+' .u/AB*$">6^") >: +"H+' .R] %* #";c'([("?h%$%#""H+!%"b%+ UIET.(IMþPln 769bVR!%+,%1T+"*W62XWEZ .2CN%"* #"b.* c+R&%!/@* 3!yV&*6|P"CN%3+E"3½h%$+D&.2%/& .* )&R+%+30* 3 ¼&y/DIb[.PS*;TITþc+* @ :+'(Z )A PcU!7 (.*/c! %I )"&Z [*3.%/!`/a,&%! IaØ|.v("#* %$# STAT Slacks (ns) STAT Slacks (ns) 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 -3 -3 -2.5 -2 -1.5 -1 -0.5 0 gCPPR Slacks (ns) -3 -2.5 -2 -1.5 -1 -0.5 0 gCPPR Slacks (ns) 0 -0.5 -1 -1.5 -2 -2.5 -3 STAT Slacks (ns) STAT Slacks (ns) ln\7%* +". #I. lw'Un9D%*U!". I ¡N¢G£U¤¥G§ ¯¬GÅeû © ¤¡wü#¬GAm¥/«!É@¥/¥Å;£T®w«#¡ DÇP¬G¤D ¥¤K©Uª~ü#«©F «¡Nü#«$¡hÇ©®b«#¡hÅW¡N ¢KÉ¡h«#Ê© ¡N ¡N«¥]ÇP¬G¤D¤7¥/®h©«#¡h¬(ªU¡Nü«#© Ç¥R©p É¡w«ÊT¬G£ «0¡N ª ¥/ûM¥3Uª ¥/g«¤D© ªT¬GÅ`T¥ü#ükf0AÉ¡N«$ÊW¡N ª ¥/ûM¥/ ª ¥/_« ¤7©Uª ¬(ÅW ¥/ü#ü#² 0 -0.5 -1 -1.5 -2 -2.5 -3 -3 -2.5 -2 -1.5 -1 -0.5 0 gCPPR Slacks (ns) -3 -2.5 -2 -1.5 -1 -0.5 0 gCPPR Slacks (ns) lPn7%* +". #I. lw'Un9D%* +". I ¡N¢G£U¤¥G§ ¯¬GÅeû © ¤¡wü#¬GAm¥/«!É@¥/¥Å;£T®w«#¡ DÇP¬G¤D ¥¤K©Uª~ü#«©F «¡Nü#«$¡hÇ©®|«¡NÅW¡N ¢`É1¡N«ÊO©,ǬG¤7¤¥®N© «¡N¬GªU¡hü#«#© Ç¥Y¬GË9Ãww 9 R©p É¡w«ÊT¬G£ «0¡N ª ¥/ûM¥3Uª ¥/g«¤D© ªT¬GÅ`T¥ü#ükf0AÉ¡N«$ÊW¡N ª ¥/ûM¥/ ª ¥/_« ¤7©Uª ¬(ÅW ¥/ü#ü#² &"5#*a.*a 5"D&*W C!+P3A\& XA!*3F+".&53Z>,$*3 32v#"*%$V+* E.v("$*31+ "G!%!30D!*) +*3E&U!R+ %""H+.P*UI\U$6\.3AT.(ITþlw'TnM76S31%T&"2*;6 *,.*C 3#G* *m"D!*31*&mmZ"* P*UIU9(."Z %.*V. *3P.%/!'3S2*5.9%/+&/I 5.2 Monte Carlo result @ :P$"?w 0D+.&.%/+("&Z32HA( */1J9+".Y&*T&2&.B6b&B%*C 3Z%$P*,^,&2."m3&*,ØSb&*.* b"$+Z3X|6|#"%PU!" *c .* \X#} !} D6^$E3^R&"PU .v("*%$|'(R6^$*EEZ..C %"*"[..*O+* a+#!.%3&B..*_IY935"R!.:5"""1?9 +*>P?} &R+%+V."' Z *56S+(Ic!*35 6B"Cu%+& "C ½2%YR!%+)6b!69..*ePI.jFP? H P*>J!"."$+Z3XI 5.3 Uncorrelated variations T._I0lnb769> ,%P1 !".P*P*$2*ØSc'(R6^#*EZ..C %"*"\H+%+3\P*Y +y/.!*3D!..%/!}+R&%!/\P*Y&y3.DA &+Z3* WE%$""H+.P*,.D!*3%#EP?S&P 9WIU.P6/AGT._ITlw' n D6c;2ZU!X*69.<.* (*3$*35"D+* 1*YZ3"*$<* I d 3*b !XH+P%""H+.P*S} +* bD+.&.%/+./A$H+"#"g(. " Z %$2*cP' D+* TI 5.4 A comment on memory consumption u*V%#*U!".^ H #I.@+*1 #I.PA +R!%$""#R+.P*6b+mT+* .V'[%"C &* V"F+* ]*"/ Z %.*)* 6xih:3."Z +.o`"D+* Ñ: !"R!'3 ?2P"9/!%+Y"Y%RIm932\.*%"/&&0H@"c"pZ ."#*DIm9 .*/X"/Z3%*aP?9: !"R+' $1,""$+$*1.*3#G#*3*/E"D!* * &5(*/H&S?w"*$"5(P.*/X5*K3,%#.%!O.".'Z.*K"O!. .*%$"/+&5"/I R*K,%/!;?5ØSA9*"/ Z %* z$`"D&*C 3P: !"R!'329' D6\Z Y b#"5"p/Z32"$*/9?wP"9b.*3 c?h" þ a;I G#&bBI3t0D6|#:"DAUT+"@{ $"2*3V%+C *32pZ 56b+132$*/X#a69 #"$'e,!.B&* .:/..#+"&> "&Z3* vl+* ;%$P['3* ;.*/531* (* #*."D+* ",n0!S ..*;"%$# I#2*3%#>%/!*3P* %/&?h"+"c"#z*FVT&"+ ?w",+DA("Cu: !.Z `$*.2:.S/+;c1"DX:/.*I d . * 2.:R5{ X#".*V3"& cP?\Df=>^A/P">%*Z1X* " Z %#zOP& A}69.F)R!%+W%+T&*P>?1V1"1.&!Y+ ..*,X#DI d . Z@{ $"+.* (AU356|"&CN%/!).`!*W+U!% %.#y.H)6BZ.c'()lb|nU2*5 b"&5#I c"#Z.`+'(7: * "D!X Y(&.2Ñ"# Z %.P*%/&C T&'..2uP?m&D+.2%+M..*c6 *;T+H+M%""H+.P*;b"+ D+.%/!../Im93@P")D:" /!Y.\%*/"H+' .E'>T&"&b{.C ".*GI d \3PZ.;+.&b*|\.PmP?(9?&&%| +m9D!X.C %/& 3/ P.P5%$:"0 @*2"@3"/%\+U!%5lw..P@ @EZ..C %"*"@+ "(!%+TnXAP' Z BB.*%$"*/D+RIU9.B!^2@/>%/!*3C . +c?w"PZ3 * `/.%/!&*/ $+.DAM./&*8&*z{y/CNZ I Ø|R+* P.%EZ%+K+E%$".%3&u.!S!*3e.e"D+ *52+|%/+* +;'(5 23?hZ3Z "* `../!.*TIq#1Ñ" Z %$2*+* ..*E".+[%3+*)'(\'T+H+* %)'/glXPn_%! P.*c%#*": !.:T&C R!S%""#R+.P*K?hZ * %* Vlw'Tn>!..3%+* ].Y?@ `/,+X2% : !"R+*z,& !R&"D!* s:&"X&'..*z>T&D)*3R6B"}lw%n r97* VP*32>E "/%\:G&"X&'.\T+1!"@*769*Y)'G@* C (* */,&*] " ½h%.* ` c"&;."*3.:/ Z +6B"CL%/&+ %"*"I 6. CONCLUSIONS AND FUTURE WORK .B +("|"$*3$>R6^@#X / P.P.B?w"b+%$%#Z "D!$1 !*3.* %+.CNCu%+. A_69. *3Cu%+.`T+R!0&*W6X * CL%!32;* (*3#*. "D+* ö: &"X&* I>93){"@[email protected]%P"*3#"E.*3#X/3C 3A/69.c#y$*.P*30?T >J9qmq|rF+"&.3V !*3.@T&R! %""+$H&2*TA#.*3(* *T"D+* * #!* E&D+.%/+/"/&*/ ? SZ'3&0?m3"&%&\:G&"X&'.DIE%P*32SD+.&%+ .* 2/A 6932%+Y(+$+$0,+*c'G#*3{DA3%!32$?\!* V '(.*5.* /+"b"Z *`H2Y!*3`?hZ3.* %"*/D&m("D!X* AG"+$.*.C %*/?. #"1TXV "&."1W3z+ :*/?$"D+* jbJUk~%C -%$2*DIZ Z "16^"YS.*%$Z ET&*..*>?m** CL +"D!'32 PZ3"%#9?T: +"H+.P*e!*3Y"Zc"/!1*9?\+gPZ "%#$\?\+* /&"+.gf!R&5 E* %.Z * ,* P.>.T&%1*]H2.* (A r<"Pa?2C ?w#%$DA}&*c1P* I 7. 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