A General Framework for Accurate Statistical Timing Analysis Considering Correlations

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1 A Geeral Framework for Accurate Statistical Timig Aalysis Cosiderig Correlatios 7.4 Vishal Khadelwal Departmet of ECE Uiversity of Marylad-College Park Akur Srivastava Departmet of ECE Uiversity of Marylad-College Park 1. ABSTRACT The impact of parameter variatios o timig due to process ad evirometal variatios has become sigificat i recet years. With each ew techology ode this variability is becomig more promiet. I this work, we preset a geeral Statistical Timig Aalysis (STA) framework that captures spatial correlatios betwee gate delays. Our techique does ot make ay assumptio about the distributios of the parameter variatios, gate delay ad arrival times. We propose a Taylor-series expasio based polyomial represetatio of gate delays ad arrival times which is able to effectively capture the o-liear depedecies that arise due to icreasig parameter variatios. I order to reduce the computatioal complexity itroduced due to polyomial modelig durig STA, we propose a efficiet liear-modelig drive polyomial STA scheme. O a average the degree-2 polyomial scheme had a 7.3x speedup as compared to Mote Carlo with uits of rms error w.r.t Mote Carlo. Our techique is geeric ad ca be applied to arbitrary variatios i the uderlyig parameters. Categories ad Subject Descriptors: B.8.2 [Hardware: Performace Aalysis ad Desig Aids Geeral Terms: Algorithms, performace, verificatio Keywords: Statistical timig, variability, correlatio 2. INTRODUCTION AND MOTIVATION Statistical Timig Aalysis has become a widely researched area with icreasig impact of process ad evirometal variatios o deep-submicro desigs. The growig sources of variatios alog with the delay correlatios they itroduce i the desig make it icreasigly hard to perform fast ad accurate timig aalysis. Traditioal desig-corer based static timig aalysis has become iaccurate due to pessimistic timig yield estimates. Mote-Carlo based statistical timig approaches become expesive i the presece of such large umber of sources of variability. The cetral idea i STA is to capture the variability by modelig delays as distributios ad performig timig aalysis statistically o these distributios while capturig possible correlatios that could exist betwee gate delays. A lot of recet work i statistical timig aalysis tries to cosider the process ad evirometal variabilities i performace aalysis. Some approaches propose bouds o the statistical timig iformatio [3, 2, 11 which ca be computed efficietly for quick statistical timig estimatio. Other approaches explicitly compute the timig statistically, makig approximatios at every step for curtailig the data explosio ad improvig the rutime. The authors i [6 propose a first order approximate delay model that takes ito accout both the correlated ad idepedet radom- Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. DAC 2005, Jue 13 17, 2005, Aaheim, Califoria, USA. Copyright 2005 ACM /05/ $5.00. ess from differet sources of variatio. A similar strategy is preseted i [8, where the authors preset a efficiet PERT-like traversal based statistical timig algorithm which cosiders the effects of the correlatios of itra-die parameter variatios by imposig a approximatio similar to [6. A momet based approach for capturig correlatios is preseted i [9. I this paper we preset a ovel, geeral framework for accurate STA. I our approach we model each gate delay ad arrival time distributio as a polyomial usig Taylorseries expasio o the uderlyig parameters. The degree of the polyomial depeds o the magitude of the variatios ad the desired level of accuracy. Our techique also calculates the arrival time at each gate as a polyomial i the uderlyig parameters. As compared with ruig Mote- Carlo simulatios to geerate such timig iformatio at each gate, we have sigificatly lower memory requiremet as well as lower rutime. We do ot make ay assumptios about the distributio of the gate delays or arrival times i the circuit. Ay arbitrary distributio will work i our geeral framework. I this paper we also preset a strategy for computig the MAX of arrival time sigals represeted as polyomials. Usig regressio, we approximate the result of MAX back to a polyomial with miimum impact o error. Sice all timig variables are approximated as polyomials i global parameters, the correlatios are iheretly cosidered. As the degree of the polyomial approximatio is icreased, the computatio complexity of STA ca become high. We also propose a ovel liear-regressio drive polyomial-modelig STA scheme. The computatioal complexity of this scheme is similar to that of the liear STA scheme. Hece, efficiet STA usig higher order polyomials ca be doe through our proposed approach. There are several ways i which our approach is superior to existig approaches. 1. The approaches i [6, 1, 8 model the depedece of gate ad arrival time delay at each ode i the circuit as a liear combiatio of global variatios which are take to be gaussia i ature. It is quite clear that this liear approximatio ca iject large amout of error i statistical timig estimate especially whe the parametric variatios become more sigificat. I our approach we represet all delay ad timig sigals as polyomials ad therefore do ot pay the pealty i accuracy. 2. We approximate each timig sigal to be a polyomial i global variables. The approach of [6, 8 also assume each timig sigal to be a liear combiatio of global variables. Their approach although will be valid oly if these variables have a gaussia distributio. Our approach o the other had is trivially geeralizable to ay distributio of the uderlyig variables. 3. We explicitly evaluate the arrival time at the output of each gate as a polyomial expressio. This iformatio ca be used to perform optimizatio o the bechmark. As opposed to Mote-Carlo simulatios, we do ot have a memory overhead to geerate this iformatio at each gate. Our experimetal results have show that the proposed polyomial gate delay ad arrival time modelig scheme has o a average a rms error of i the output CDF as compared to from liear gate delay ad arrival time 89

2 modelig (which is doe by most existig STA schemes) whe compared with accurate Mote Carlo CDFs. This clearly brigs out the effectiveess of polyomial modelig (assumig quadratic polyomials) of gate delays ad arrival times to better capture the variability i timig due to parameter variatios. The average rutime speedups for the polyomial scheme over Mote-Carlo was 7.3x, while that from liear scheme was 7.5x. The rest of the paper is orgaized as follows: Sectio 3 describes modelig scheme used for parameter variatios, correlatio hadlig ad gate delay computatio. Sectio 4 cotais the proposed STA framework alog with the error maagemet strategy used i this work. Sectio 5 presets our ovel liear modelig based STA drive polyomial modelig STA scheme. Sectio 6 presets our experimetal results ad sectio 7 presets the coclusios draw from this work. 3. MODELING PARAMETER VARIATIONS AND SPATIAL CORRELATIONS I this sectio we will discuss the methodology that we impose for modelig the statistical correlatios betwee the gate delay variables. We assume that the gate delay is depedet o a umber of locatio-depedet parameters which are assumed to be mutually idepedet radom variables. Let P i, Q i ad R i deote three such parameters (although our approach is very geeral ad ca be trivially exteded to havig more sources of variatios also). Therefore, the delay of a gate i ca be modeled as a fuctio of these idepedet parameters as give by equatio 1: D i = F (P i,q i,r i) (1) We ote here that F ca be a o-liear fuctio of the parameters. Eve if the uderlyig variables P i,q i,r i are gaussia distributios, the distributio i delay will ot be gaussia. Most state of the art techiques for statistical timig assume the ode delays to be gaussia either directly or idirectly. I our formulatio we do ot eed to make ay such approximatio o either the delay distributio or o the distributios of P i, Q i ad R i. As it has bee idicated i several other statistical timig techiques, spatial correlatio would exist betwee delay variables of differet gates due to spatial proximity. This occurs predomiatly due to the fact that the uderlyig variables P i, Q i ad R i for two gates i close spatial proximity would show correlated behavior. 3.1 Spatial Correlatio Modelig P1 P3 Gate i R1 R2 R3 R4 Figure 1: Grid-Based Spatial Correlatio Model We ow preset a modelig strategy to capture the spatial correlatios betwee the parameter variables P i, Q i ad R i for each gate. Therefore, this techique captures the delay correlatio betwee gates. Let us suppose that we are give a placed etlist as show i figure 1. We impose a uiform grid o the placemet to partitio the gates ito spatial regios. Let us ow cosider the parameter P ad assume that its variatio ca be represeted as a liear combiatio of four idepedet radom compoets amely P 1, P 2, P 3 ad P 4 that are zero mea ad fiite variace. These four radom variables correspod to the four corers of the chip (as illustrated i figure 1). For ay gate j, we model its correspodig parameter P j as give by equatio 2: P j = a 1P 1+a 2P 2+a 3P 3+a 4P 4+a 0 (2) where a 0 is the omial value of parameter P j. For ay gate j i the etlist, we ca compute the grid-based radial R3 R1 Gate j R2 R4 P2 P4 distace for the gate from the corers of the placemet. This is represeted by R1, R2, R3 adr4 for gate j as show i the figure. The coefficiets a 1, a 2, a 3 ad a 4 ca be computed by usig these radial distaces. Depedig o the ature of the uderlyig variability parameter P j (which ca be obtaied by aalyzig the actual variability data), we ca use a appropriate fuctio H(R) to compute these coefficiets as follows: a 1 = H(R1); a 2 = H(R2); a 3 = H(R3); a 4 = H(R4) (3) The uderlyig radom variables P 1,P2,P3,P4cahave ay arbitrary distributio depedig o the distributio of the parameter P j. Therefore, we ca see that if two gates i ad j are far apart, they will get differet cotributios from each of the four compoets P 1, P 2, P 3adP4 ad will have a weak correlatio. If they are placed close together, the their coefficiets will be similar ad strog correlatio will exist betwee them. I this way, we model spatial correlatios for each of the remaiig parameters i the system (Y ad Z i this case). Note that a similar strategy was proposed by [1 but the umber of uderlyig radom variables that capture the correlatios was sigificatly higher. I our case we eed oly four variables per parameter (P i,q ietc.) to capture the spatial closeess of two gates for capturig their correlatios. 3.2 Gate Delay Modelig We will ow illustrate a gate delay model that icorporates the spatial correlatio model described i the previous sub-sectio. We have represeted our gate delay as a fuctio of the idepedet parameters as give by equatio 1. Each of P i, Q i ad R i ca be represeted as a liear combiatio of their uderlyig radom compoets as give by equatio 2. Hece, we ca represet our gate delay as a fuctio of these variables as: D i = G i(p 1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4) (4) For simplificatio i represetatio, let us represet these variables as Y 1, Y 2, Y 3, Y 4, Y 5, Y 6, Y 7, Y 8, Y 9, Y 10, Y 11 ad Y 12 respectively. We ca use Taylor-series expasio about the mea values o this relatio ad obtai gate delay D i as a sum of a series of multiple-order compoets as give by equatio 5. The omial values for the gate delay happes whe all Y i variables are zero (essetially o variace). Therefore D i(omial) =G i(0) G i = G(0) + (Yk)G (0) + 1/2!( (Yk)) 2 G (0)... (5) k=1 k=1 The approach i [8 presets a similar strategy i which the delay for each gate is simplified accordig to Taylor series. Their approach however arbitrarily igores the higher order polyomial terms ad simply represets each gate delay as a a liear combiatio of the radom variables (the Y i terms i our case). Such a simplificatio is show below D i = Y 1+c 2Y 2+c 3Y 3+c 4Y 4+c 5Y 5+c 6Y 6+c 7Y 7 + c 8Y 8+c 9Y 9+0Y Y Y 12 + G i(0) (6) Typical gate delay models have terms which illustrate a high degree of o liear sesitivity. Such a liear approximatio ca iject a large amout of error i gate delay modelig (ad therefore the statistical timig estimate) itself. I this work we choose ot to igore the higher order terms i the expaded Taylor series. Therefore, we model the gate delays as a geeral polyomial i the global variables Y i. The order of this polyomial decides the degree of accuracy i the delay estimate. Note that this polyomial also has cross terms of the form Y iy j etc. A geeral secod degree polyomial represetig the gate delay would have the followig structure D i = Y 1+c 2Y Y 12+3Y c 24Y degree 2 cross terms + G i(0) (7) It ca be see that as we icrease the degree of the approximatig polyomial, the umber of terms icrease ad the error i approximatio reduces. Therefore it could be expected that there would be a tradeoff betwee rutime of statistical timig aalysis ad its accuracy. This tradeoff 90

3 could be geerated by cotrollig the degree of the polyomial used i represetig the timig variables. Moreover, all delay variables i the circuit would share the same global variables Y i. This would eable effective capturig of the correlatios betwee them. 4. STATISTICAL TIMING ANALYSIS FRAMEWORK We will ow describe our geeral STA framework. We use a block-based STA approach that traverses the circuit topologically from the primary iputs to the primary outputs. There are two basic operatios that are performed at each gate durig this traversal. We first perform a SUM operatio o the arrival time at a fai ad the correspodig gate delay. This SUM operatio is repeated for each fai of the gate. We the perform the MAX operatio o the result of the already computed SUM operatios. This gives us the arrival time at the output of the gate. As described i sectio 3, each gate delay is represeted as a polyomial i the idepedet/global parameters. Followig a similar strategy we would like to approximate each arrival time sigal as a polyomial too. The approach i [6 proposes a similar strategy for represetig all arrival time sigals as liear combiatios of global variables. At the ed of the topological traversal of the circuit, the STA data has bee geerated. Let us ow try to uderstad the two basic operatios that are performed repeatedly i STA. Figure 2 shows a typical gate i the circuit that has K fais ad a polyomial gate delay represetatio D. The arrival time at fai i of the gate is deoted by A i, which is also a polyomial represetatio similar to D. A 1 A 2 A K 1 2 K D Figure 2: SUM ad MAX Computatio D = poly(y 1,Y2,..., Y 12) (8) A 1 = poly(y 1,Y2,..., Y 12) (9) A 2 = poly(y 1,Y2,..., Y 12) (10) (11) A K = poly(y 1,Y2,..., Y 12) (12) 4.1 SUM Operatio It is very simple to compute the result of the SUM operatio. Sice arrival time ad gate delay are both polyomials i the same idepedet parameters, the result of the SUM operatio is also a polyomial. The coefficiet of each term i the resultig polyomial is the sum of the coefficiets of the correspodig terms i A i ad D. For each fai i, we deote the result of the SUM operatio by A io: A 1o = A 1 + D (13) A 2o = A 2 + D (14) (15) A Ko = A K + D (16) We ote that this is a accurate computatio ad o approximatio has bee made at this step. 4.2 MAX Operatio We ow compute the MAX operatio i our proposed framework. We perform a MAX of K polyomials to get thearrivaltimesigala o at the output of the gate. We would like to represet A o as a polyomial too. Sice all timig variables are represeted as a polyomial i global variables, the correlatios are effectively captured. A o = MAX(A 1o,A 2o,..., A Ko) (17) = poly(y 1,Y2,..., Y 12) (18) It is kow that the MAX operatio itroduces the complexity i STA. It is very hard to efficietly geerate a accurate result of the MAX operatio. We propose a regressio based strategy to compute the resultig polyomial A o by O A O performig least square fittig. Assumig we kow the degree of the polyomial that we wat A o to be approximated i, least square fittig will try to fid the best polyomial of that degree that has the smallest error with the actual data of the MAX operatio. Let us suppose that we are tryig to approximate A o with a degree two polyomial as idicated i equatio 19. We eed to evaluate all coefficiets such that the resultig polyomial has smallest error whe compared with the actual MAX data. A o = Y 1+c 2Y Y 12+3Y c 24Y degree 2 crossterms + c 91 (19) Now we will formalize the regressio strategy that is used to compute these coefficiets. Let us assume that we are give a set of samplig vectors for the parameters (Y1,...,Y12) (these samples will ot be a very large set). We ca evaluate the exact value of the MAX result at these samplig vectors. This could be doe by evaluatig all the polyomials A io ad calculatig their MAX. Let the ith value be represeted by z i. We ca defie a residual R for least square fittig as [ R 2 = z i (Y 1 i Y 12 i + 3Y 1 2 i c 24Y 12 2 i + 66 degree 2 cross terms + c 91) 2 (20) This residue essetially is the root mea square error betwee the actual data of MAX z i ad the oe predicted by the polyomial. I order of miimize the residual, we evaluate the partial derivative wrt. each coefficiet i the polyomial ad equate the result to zero. This ca be represeted as : = 2 c 2 = 2 [ z i (Y 1 i +...) Y 1 = 0 (21) [ z i (Y 1 i +...) Y 2 = 0 (22)... =... (23) [ = 2 z i (Y 1 i +...) Y 1 2 = 0 (24) 3... =... (25) [ = 2 z i (Y 1 i +...) 1 = 0 (26) c 91 We ca re-orgaize these to get equatios : Y 1 iy 1 i +c 2 Y 1 iy 2 i +c 2 Y 2 iy 1 i c 91 Y 1 i = Y 2 iy 2 i Y 1 2 i Y 1 i +... z iy 1 i (27) Y 1 2 i Y 2 i c 91 Y 2 i = z iy 2 i (28) Y 1 iy 1 2 i +c 2 Y 2 iy 1 2 i Y 1 2 i Y 1 2 i c 91 Y 1 2 i = z iy 1 2 i (29) Y 1 i + c 2 Y 2 i Y 1 2 i c 91 = z i (30) 91

4 We ca combie these equatios to give a more compact matrix represetatio as: Y 1iY 1i... Y 12 i Y 1 i... Y 1i Y 1iY 12 i... Y 12 i Y 1 2 i... Y 12 i Y 1i... Y 12 i... ziy 1i =... ziy 12 i... c 91 zi Essetially, we have represeted the polyomial regressio as the system YC = Z where we eed to solve for the C matrix. There are several well kow techiques for solvig such a system of matrix, ay of which could be used. This approach essetially selects the coefficiets i such a way that the polyomial approximatio of A o has miimum error with the real data set z i. This polyomial re-approximatio is performed every time a MAX operatio is computed. The regressio strategy used i MAX operatio has two sources of complexity. The first oe is the size of the samplig values. Icreasig the umber of samples at each MAX operatio icreases the computatioal cost of this operatio but improves the accuracy of the polyomial fit. Also, we ote that as we icrease the degree of polyomial approximatio, the dimesios of matrix Y also icreases. For first order liear regressio, this matrix is a matrix while for degree two approximatio, this is a matrix. Thus, we ca clearly see a trade-off betwee accuracy ad computatio rutime through the order of polyomial approximatio used. We also poit out here that the geerality of our STA approach to hadle all kids of parameter variatio distributios, gate delay distributios ad arrival time distributios is made possible by ot makig ay distributio based approximatio i the MAX operatio. Polyomial regressio ca be applied to ay arbitrary distributio of the parameter variables Y i ad the accuracy cotrolled through the degree of the polyomial ad the umber of samplig vectors used. After the topological traversal of the circuit, the arrival time at the primary output is represeted as a polyomial i global variables. It ca be see that we have preseted a geeric statistical timig methodology that is ot costraied by ay assumptios o uderlyig distributio. 5. REDUCING COMPLEXITY IN POLYNOMIAL REGRESSION We ote that the computatioal complexity i polyomial STA comes primarily from the MAX operatio as described i sectio 4.2. The size of the polyomial regressio matrix formed i this step is grows expoetially with the degree of the polyomial approximatio used. Hece, this step becomes the ru-time determiig step of the STA scheme. Ideally, we would like to maitai the accuracy obtaied from usig a higher degree polyomial (chose to be degree 2 i this paper) while keepig a rutime that is comparable to a STA scheme with liear delay/arrival time models. The advatage of usig regressio is the geerality i the scheme to hadle timig distributios of ay ature (ot gaussia oly) ad the mathematical accuracy iheret i regressio. I order to achieve the desired level of accuracy as well as rutime behavior, we propose a scheme that uses liear-modelig based STA to drive the polyomial STA. 5.1 Liear Regressio Drive Polyomial STA Polyomial modelig based STA is more accurate i geeratig the PDF/CDF of arrival time distributio because of two primary reasos: firstly, because polyomial gate delay modelig is better able to capture the ature of distributio due to the uderlyig parameter variatio ad secodly, because polyomial arrival time modelig is able to represet the PDF/CDF more accurately tha liear modelig. However, the mea ad variace of the arrival time distributios are captured with reasoable accuracy i the liear modelig based STA. We will ow propose a polyomial modelig based STA techique that is drive by liear modelig based STA (which has lower rutime). X Y Ax Ay Delay = D G Aout OUT Figure 3: STA techique at Gate G We traverse the circuit topologically ad at each gate, we ru liear STA ad the use liear STA results to drive polyomial STA. Liear STA correspods to performig liear regressio assumig a liear model for arrival time ad gate delay. Thus, we geerate ad store both liearly ad polyomially modeled timig values at each gate. Let us suppose we are evaluatig the arrival time at output of gate G with two fais (X ad Y ) as show i figure 3. For each iput X ad Y, we are give both liear ad polyomial modelig values for the sigal arrival times A x ad A y respectively. Let us deote the liear arrival times as A l x ad A l y respectively ad the polyomial arrival times as A p x ad A p y respectively. The liear ad polyomial models for gate delays are give as D l ad D p respectively. The liear arrival time A l out at the output of gate G is give by: A l out = MAX(A l x + D l,a l y + D l ) (31) Durig liear STA we perform regressio based MAX operatio based o liear gate delay ad arrival time models as give by equatio 31. I sectio 4, we have discussed the details of the proposed regressio based STA scheme. This eables the time cosumig regressio i the MAX step (sectio 4.2) to be much faster tha the polyomial case (degree 2 or higher). The liear regressio output gives us the arrival time (A l out) at the output of gate G as a liear combiatio of parameters: A l out = c 0 + Y 1+c 2Y Y 12 (32) where Y1, Y2,..., Y12 are the idepedet parameter variables as discussed i sectio 3. We kow the distributio of these radom variables ad hece ca calculate the mea ad the variace of the arrival time A l out as: Mea(A l out) =c 0 + Mea(Y 1) Mea(Y 12) (33) Var(A l out) =c 2 1 Var(Y 1)+c 2 2 Var(Y 2)+...+c 2 12 Var(Y 12) (34) We will ow assume that the mea ad variace of the output arrival time after liear regressio is accurate. We will ru polyomial STA by matchig the mea ad variace (first two momets) of the polyomial arrival time with the liear regressio output. Let us ow uderstad the scheme i more detail. The polyomial arrival time at the output of gate G (say A p out) isgiveby: A p out = MAX(A p x + D p,a p y + D p ) (35) where A p x ad A p y are the sigal arrival times at the iputpis X ad Y respectively ad D p is the polyomial gate delay. Now let us suppose that we kow the probability p such that arrival time (A p x + D p ) arrival time (A p y + D p ). We ca calculate the probability p = Prob(A p x + D p A p y + D p ) durig the liear STA ru at gate G. We ca ru polyomial STA o gate G by utilizig this probability p to geerate a output polyomial A p out, which will the be scaled to match its first two momets to the values evaluated from liear regressio based STA as give by equatios 33 ad 34. Let the output arrival time polyomial A p out be geerated as follows: 92

5 A p out = p (A p x + D p )+(1 p) (A p y + D p ) (36) where A p x ad A p y are the polyomial arrival times of the sigal at the fai pis X ad Y (which have already bee calculated previously). After this step, we eed to match thevariaceofa p out to the variace of A l out from liear regressio. For simplicity of uderstadig, we will keep this discussio limited to A p out beig a polyomial of degree 2 as give by: A p out = c 0+Y 1+c 2Y Y 12+3Y c 24Y degree 2 cross terms (37) But the aalysis that follows ca trivially be exteded to higher order polyomial modelig as well. Sice we kow the distributio of each uderlyig parameter variatio (Y 1 to Y 12), we kow their mea ad variace values. We ca evaluate the mea ad variace of A p out as follows: Mea(A p out) =c 0+ Mea(Y 1) Mea(Y 12) + 3 Mea(Y 1 2 )+... other terms (38) Var(A p out) =c 2 1 Var(Y 1) c 2 12 Var(Y 12) + c 2 13 Var(Y 1 2 ) c 2 24 Var(Y 12 2 ) + c 2 25 Var(Y 1 Y 2) +... other cross terms +2c 2 Cov(Y 1,Y2) + 2c 3 Cov(Y 1,Y3) +... all other covariace terms (39) We will first match the variace of A p out (from equatio 39) with that of A l out (from equatio 34) by scalig A p out with a factor α such that: α 2 = Var(A l out)/v ar(a p out) (40) A p out = α A p out (41) The mea of the ew scaled polyomial will be: Mea(A p out) =α Mea(A p out) (42) Hece, to match the mea of the polyomial arrival time expressio with that obtaied from liear regressio (equatio 33), we ca add a costat factor β to the costat term c 0 of A p out such that: β = Mea(A l out) Mea(A p out) (43) c 0 = c 0 + β (44) Hece the fial polyomial arrival time at the output of gate G ca be give by A poly out : A poly out = α A p out + β (45) This completes our liear regressio drive polyomial STA techique. We have avoided the complexity of solvig a large polyomial regressio problem at each gate (durig the MAX operatio) by solvig a smaller liear regressio problem ad the performig momet matchig (first two momets) as explaied i this sectio. The rutime complexity of this scheme will be of the order of the rutime for liear regressio. 5.2 Gaussia Approximatio The liear regressio based techique discussed earlier i this sectio is applicable to ay give gate delay distributio. If the gate delay distributios are kow to be gaussia, the we could avoid the geeric regressio based liear STA, ad use faster techiques (uder the gaussia approximatio) to drive the polyomial STA. I [6, the authors have proposed a first order approximate delay model based STA uder the assumptio that all uderlyig parameters have a gaussia distributio. Their scheme approximates the arrival time distributio after the MAX operatio to be gaussia as well. For brevity, we do ot go ito the details of their scheme. As explaied i the previous subsectio, to drive polyomial STA usig liear STA, we eed to evaluate three quatities for each MAX operatio: the probability p that oe arrival time is larger tha the other, the mea of the output arrival time distributio ad the variace of the output arrival time distributio. I our geeric scheme we use liear regressio to get these quatities. Uder the gaussia assumptio, the authors i [6 use the results from [5, 10 to perform the MAX operatio. They represet each arrival time as a liear combiatio of gaussia radom variables (represetig the uderlyig parameter variatios). The distributio of the timig sigal after the MAX operatio is reapproximated back as a liear combiatio of the uderlyig gaussia variables (to maitai the gaussia form). Aalytical expressio proposed i [5, 10 are used to evaluate the mea ad variace of the result of performig the MAX operatio o two joitly gaussia arrival time distributios. They use the probability p of oe arrival time beig larger tha the other to geerate a expressio for the resultig liear arrival time. Usig the variace value obtaied from the aalytical expressio, they match the variace of the resultig output arrival time to get the fial output arrival time expressio as a liear combiatio of the uderlyig parameter variatios. We ca utilize the scheme preseted i [6 to drive our polyomial STA uder the assumptio that uderlyig parameters are gaussia ad by imposig a first order approximatio o gate delay. This is a faster techique tha performig our more geeric regressio based STA to drive polyomial STA ad ca be used whe the uderlyig parameters are give to be gaussia i ature. 6. EXPERIMENTAL RESULTS The proposed STA framework was implemeted i sis [7 assumig three uderlyig parameters. For the gate delay model i equatio 46, we assumed the parameters supply voltage (V dd ), threshold voltage (V th )adthevelocitysaturatio idex for short chael effects (α) as the uderlyig sources of variability. We used a academic placemet tool (CAPO [4) to get a valid placemet for each bechmark. This placemet iformatio was used to geerate the V dd,v th,α variatios at each gate as idicated i equatio 2. This automatically captures correlatios due to spatial proximity. We imposed 10%, 20% ad 7% variability o correspodig mea values of 1.8V, 0.5V ad 1.3 respectively. Furthermore, each of these parameters were assumed to have a uiform distributio to see the effects of relaxig the gaussia distributio assumptio to cosider a more geeral approach. C LV dd D i (46) (V dd V th ) α I sectio 4, we have preseted the geeral regressio based STA scheme. As the degree of the polyomial approximatio is icreased, the computatio complexity of regressio makes this scheme impractical. I sectio 5, we have proposed a ovel liear regressio drive polyomial STA scheme. The computatioal complexity of this scheme is similar to that of liear regressio. Hece, efficiet STA usig higher order polyomials ca be doe through this scheme. We assumed the gate delays were a secod order polyomial of the parameters. This secod order polyomial for each gate delay was geerated usig best fit regressio with Mote Carlo data for gate delay. The Mote Carlo data for the gate delay was calculated usig the delay model idicated i equatio 46 with differet parameter istaces. We geerated accurate timig CDFs for each bechmark usig equatio 46 for gate delays through Mote Carlo simulatios. All rutimes ad error comparisos are are made with Mote Carlo. We experimeted with the followig cases: 1. Usig liear gate delay ad arrival time models, we performed regressio based STA (as described i sectio 4). This approach is similar to the oe proposed by state of the art STA techiques like [6, 8 where all delay ad arrival time variables are assumed to be liear approximatios of uderlyig global parameters. 93

6 Bechmark Mote Carlo Liear-Modelig STA Liear-Drive Polyomial Modelig Rutime Rutime Speedup rms Error Rutime Speedup rms Error C X X C X X C X X C X X C X X C X X C X X C X X Average 7.5X X We performed polyomial STA usig our proposed liear regressio drive polyomial STA scheme (as described i sectio 5). All gate delays as well as arrival times i the STA were represeted as degree two polyomials. Table 1 presets the experimetal results. All rutime ad error comparisos are made wrt. Mote Carlo results. Colums 2, 3 ad 6 preset the rutime values for Mote Carlo, liear regressio based STA ad liear regressio drive - polyomial STA respectively. The correspodig speedups wrt. Mote Carlo are give i colums 4 ad 7 respectively. Oaaverage,weget7.5x ad 7.3x speedup compared with Mote Carlo rutime from the two schemes respectively. O a average, there is ad uits of rms error i the output CDFs from the two schemes respectively as compared with the accurate CDFs from Mote Carlo. These results poit out the superiority of polyomial STA as compared to liear STA. Polyomial gate delay ad arrival time models are better able to capture the distributio as compared to liear models. We also ote that the rutime from the liear regressio drive polyomial STA are comparable to that of pure liear regressio based STA. Our proposed scheme is a fast techique to perform higher order polyomial approximatios durig STA. With icreasig variability i uderlyig parameters, such a scheme would be very useful. From the rutime speedups reported i table 1, we ca see that as the bechmark size is icreasig (listed i order of icreasig umber of total gates) the speedup as compared with Mote-Carlo also icreases. We ote here that we perform regressio ad Mote-Carlo simulatios at the same umber of samples to make a fair compariso. Additioally, as poited out earlier i the paper, we geerate a polyomial expressio for arrival time at each gate which ca be used for performig optimizatio. I order to geerate this iformatio usig Mote-Carlo simulatios, we would require a very high memory overhead to save the results at each gate. These are the advatages of usig our regressio based scheme over Mote-Carlo based simulatios. Eve though the rms error umbers are small i magitude, they ca make a sigificat impact o the CDF. For example, the average rms error i liear regressio scheme is uits, so if we are lookig at the 50 percetile poit o the accurate CDF, the predicted CDF potetially be showig a value of either or 0.658, which is a very sigificat differece from the actual value of 0.5. The impact of this iaccuracy o decisios made o the desig usig these CDFs could be very drastic. Figure 4 depicts the CDF at the output of bechmark C880. We ca see that the liear regressio drive polyomial STA gives us a more accurate CDF as compared to the liear regressio based STA scheme. This clearly brigs out the superiority of polyomial STA over liear STA. 7. CONCLUSION AND FUTURE WORK I this work we have proposed a geeral framework for accurate STA. Our scheme is idepedet of the distributios of the variatios, gate delay ad arrival times. We cosider the impact of itra-die parameter variatios o gate delays ad also cosider the spatial correlatios that ca exist betwee them. We have proposed a polyomial gate delay modelig scheme where the order of the polyomial ca be decided by the desired accuracy as well as the magitude of the uderlyig variatios. We have preseted a regressio based MAX computatio techique that ca be used to represet each arrival time as a polyomial i the uderlyig Table 1: Rutime ad rms Error Compariso 94 Cumulative Probability Mote Carlo Timig CDF for Bechmark C880 Polyomial Liear Primary Output Arrival Time i ps Figure 4: CDF Result for C880 parameters. However, sice the computatioal complexity of this regressio icreases sigificatly with the degree of the polyomial, we have proposed a ovel liear regressio drive polyomial STA scheme. Our results show the advatage of usig polyomial modelig over liear modelig as doe i the existig literature. Future work would be to develop fast techiques for polyomial MAX operatio i STA. As the impact of the parameter variatios icreases, o-liearity creeps i ad we eed to develop higher order approximatio schemes which are accurate but at the same time efficiet i computatio. 8. REFERENCES [1 A. Agarwal, D. Blaauw ad V. Zolotov. Statistical Timig Aalysis for Itra-Die Process Variatios with Spatial Correlatios. I Procs of ICCAD, [2 A. Agarwal et al. Computatio ad Refiemet of Statistical Bouds o Circuit Delay. I Procs of DAC, [3 A. Agarwal, V. Zolotov ad D. Blaauw. Statistical Timig Aalysis Usig Bouds ad Selective Eumeratio. 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