Statistical Circuit Optimization Considering Device and Interconnect Process Variations

Transcription

1 Statstcal Crcut Optmzaton Consderng Devce and Interconnect Process Varatons I-Jye Ln, Tsu-Yee Lng, and Yao-Wen Chang The Electronc Desgn Automaton Laboratory Department of Electrcal Engneerng Natonal Tawan Unversty March 17, 2007 NTUEE 1

2 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 2

3 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 3

4 Interconnect Process Varaton Interconnect delay and relablty hghly affect VLSI performance. The varablty of nterconnect parameters wll rase up to 35%. Srvastava et al., Sprnger, The worst-case corner models cannot capture the worst-case varatons n nterconnect delay. Lu et al., DAC 2000 The nterconnect optmzaton guded by statstcal analyss technques has become an nevtable trend. Vsweswarah, SLIP 2006 NTUEE 4

5 Prevous Work n Statstcal Optmzaton Statstcal gate szng wth tmng constrants usng Lagrangan Relaxaton. Cho et al., DAC Statstcal power mnmzaton by delay budgetng usng second order conc programmng. Orshansky et al., DAC Statstcal gate szng usng geometrc programmng Patl et al., ISQED No statstcal optmzaton work consder both nterconnect and devce szng. NTUEE 5

6 Comparson wth Prevous Work Szng varable Delay Model Objectve Constrant Orshansky s work (DAC 2005) Gate only Lnear model (lnear term) Power Tmng Our work Gate and wre Elmore delay model (nonlnear term) Area Power, tmng, thermal Due to the nonlnear term ntroduce by the Elmore delay model, the optmzaton usng both gate and wre szng wll be much harder to solve. NTUEE 6

7 Delay Model Our delay model and tmng constrant: Elmore delay model D = R g ( C w X w L w + C g X ) /X j + R w L w ( C w X w L w /2 + C g X )/(X w ) Tmng constrant: a = arrval tme of gate a a + j D Hgher order (quadratc) terms! Delay model and tmng constrant used n prevous work n DAC 2005: 0 a a + d + d lnear terms! d 0 = delay due to the szng for maxmum slack d = slack added to node due to the loadng j NTUEE 7

8 Statstcal Crcut Optmzaton wth SOCP Second-order conc programmng (SOCP) Mnmze f T x subject to A x + b Fx = g 2 c T x + d Convex optmzaton Theoretcal runtme O(N 1.3 ) Orshansky (DAC 2005), Davood (DAC 2006) Second-order conc constrant: T A x b c x + d + 2 Approxmaton Lnear terms! method Nonlnear (quadratc) terms are not applcable! NTUEE 8

9 Approxmaton Method Fx the gate sze n the tmng constrant. Reduce the tmng constrant from quadratc order to lnear order. Approxmate the gate szes by a two-stage flow. Iteratvely reduce the approxmaton errors. The flow s smlar to Sequental Lnear Program (SLP). no Solve the SOCP problem under current constrants (gate sze fxed) Update the gate sze of tmng constrants and form a new SOCP problem convergence or max teratons yes Fnsh NTUEE 9

10 Our Contrbutons The frst work of statstcal optmzaton on crcut nterconnect and devces Prevous work consders only crcut devces (gates). Statstcal optmzaton for consderng both nterconnect and devces s much harder. The frst work that statstcally optmzes the area wth thermal- and tmng-constraned parametrc yelds Most exstng statstcal optmzaton consders only tmng. The frst work capable of analytcally transformng the statstcal RC model nto an SOCP Prevous work uses lnear delay model NTUEE 10

11 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 11

12 Tmng Constrant D R 1 D R L C 1 17 Tmng constrant: D R L C 2 # of paths may grow exponentally to the crcut sze. To reduce problem sze, we dstrbute the tmng nformaton to each node. NTUEE 12

13 Thermal Constrant Electron Mgraton (EM) lfetme relablty of metal nterconnects s governed by the well-known Black s equaton: The desgn s relable when TTF: tme-to-fal perod A* : a constant j : average current densty Q : actvaton energy KB : Boltzmann s constant. Tm : metal temperature j 0 : specfc current densty T ref : specfc metal temperature NTUEE 13

14 Average Temperature of the Chp The average temperature of the chp, T avg, can be estmated by: Power P tot : total power consumpton of the chp T ar : ambent temperature R n : thermal resstance of the substrate and the package A : chp area Banerjee et al., ISPD NTUEE 14

15 Power Constrant Need to constran chp s temperature under a reasonable bound durng the optmzaton: For smplcty, consder the dynamc power consumpton only. P B: the power bound of the gate c : the downstream capactance of the gate I α : swtchng actvty of component I NTUEE 15

16 Determnstc Formulaton l : gate unt area or wre length x : gate or wre sze (szng varable) Thermal constrant Tmng constrant Power constrant f: workng frequency; α : swtchng actvty of component I; c : load capactance of component I; ω: path n the path set Ω. NTUEE 16

17 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 17

18 Varaton Models Introduce two process parameters as the varaton sources: Inter-layer delectrc (ILD) thckness (H), and metal thckness (T). R and C can be approxmated by the frst-order Taylor expresson: R nom /C nom : nomnal value of R/C T/ H : random devaton of metal thckness/ild thckness a1, b1, b2 are senstvtes calculated by the dfferental dfferentaton of: C gnd /ε: normalzed capactance S: space between parallel lnes Srvastava et al., Sprnger NTUEE 18

19 RC Varablty Assume T and H are Gaussan, the varablty magntude of R and C can easly be calculated by: σ σ 2 R 2 C = a σ = b σ 2 T 2 T + b 2 2 σ 2 H Apply the nterconnect delay varaton metrc to calculate the varablty of the product of R and C. Well captured by a normal dstrbuton wth 1.2% average error of the mean delay and 3.8% average error of the standard devaton. Blaauw et al., DAC NTUEE 19

20 NTUEE 20 Statstcal Formulaton δ/ζ/η: Thermal/Tmng/Power yeld constrant Determnstc formulaton Statstcal formulaton B B j j B m s n s U x L P c D a a D a a D T T to subject x l Mnmze = ' 1 α B B j j B m s n s U x L P c Prob ( D Prob (a a D Prob (a a Prob (D ) T Prob (T to subject x l Mnmze = η α ζ ζ ζ δ ' 1 ) ) )

21 Transformaton nto SOCP Theorem: Gven ndependent Gaussan random vectors a and bound vectors b, the parametrc yeld (η) problem s as follows: the problem can be reformulated as an SOCP: Φ -1 : the cumulatve densty nverse functon Boyd and Vandenberghe, Cambrdge, NTUEE 21

22 Transformaton Flow varance: mean: zero mean unt varance Gaussan varable cumulatve densty functon NTUEE 22

23 Thermal & Power Constrants n SOCP Form Thermal constrant: Power (Thermal dstrbuton) constrant: NTUEE 23

24 Tmng Constrant n SOCP Form X j : sze of the drvng gate X : sze of the loadng gate X w : wdth of the nterconnect L w : length of the nterconnect (constant) D = R g ( C w X w L w + C g X ) /X j + R w L w ( C w X w L w /2 + C g X )/ X w Only Xw s the szng varable D = R j C w X w L w + R j C + R w C w (L w ) 2 /2 + R m L w C / X w Tmng constrant: NTUEE 24

25 Statstcal Problem Formulaton usng SOCP Thermal constrant Tmng constrant Power constrant NTUEE 25

26 Program Flow Begn Formulate the problem wth the RC varaton Assgn the values of the current gate szes to the gate sze varables n the tmng constrant Formulate the problem nto SOCP wth gate sze as fxed value n the tmng constrant Solve t wth the nteror pont method no Convergence or Max teratons yes End Iteratvely reduce the approxmaton errors. NTUEE 26

27 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 27

28 Expermental Setup Crcut Name c17 c432 c499 c880 c1355 c1908 c2670 c3540 c5315 c6288 Crcut Sze #Gate #Wre #Total Implemented n C++ & appled the MOSEK optmzaton tool to solve t. Tested on the commonly used ISCAS85 benchmark crcuts n ths area. Used Desgn Compler & Astro wth UMC 0.18¹m technology lbrary to synthesze and place the crcuts. NTUEE 28

29 Expermental Results Acheve 51%, 39%, and 26% area reductons for 70%, 84.1%, and 99.9% yeld constrants, respectvely. Avg. / Max. # of the runnng teratons: 5.6 / 10 Tmng constrant error bound: 2% Crcut name area (µm 2 ) Determnstc Runtme / te. (s) Total runtme (s) area (µm 2 ) 70% yeld Area mprov. Runtme / te. (s) Total runtme (s) c % c % c % c % c % c % c % c % c % c % Avg % NTUEE 29

30 Expermental Results of 84.1% and 99.9% yeld The lower the yeld constrants, the better the area optmzaton. All constrants (tmng, power, thermal) are met. Crcut name area (µm 2 ) 84.1% yeld Area mprov. Runtme / te. (s) Total runtme (s) area (µm 2 ) 99.9% yeld Area mprov. Runtme / te. (s) Total runtme (s) c % % c % % c % % c % % c % % c % % c % % c % % c % % c % % Avg % 25.60% NTUEE 30

31 Delay, Power and Temperature Performance Though the delay and the maxmum metal temperature are ncreased, they all meet the gven bounds. Crcut Name Fully utlzed the constrant bound to get the best optmzaton results. Bound Delay (ns) Before After Before Power (mw) After Max T ncrease () Before After c c c c c c c c c c Comparson NTUEE 31

32 Introducton Determnstc Algorthm Statstcal Algorthm Expermental Results Conclusons Outlne NTUEE 32

33 Conclusons Presented the frst statstcal work for area mnmzaton under thermal and tmng constrants by gate and wre szng. Obtaned much better results than those of the determnstc method. Formulated statstcal RC model by SOCPs whch can be solved effcently and effectvely. Used more accurate delay model (Elmore delay model) Solved the problem by a two-stage approxmaton flow Nonlnear terms are not applcable to SOCP NTUEE 33

34 Thank You NTUEE 34

35 Backup Sldes NTUEE 35

36 Temperature Dstrbuton Applyng the Fnte Dfference Method (FDM), we can dvde the whole chp nto m mesh nodes and calculate each node s temperature by P : th mesh node s power dsspaton T : th mesh node s temperature g: power densty of the heat sources (W/m3) Chapman, Heat Transfer, New York: Macmllan, 1984 Vol., 4 th Ed.. NTUEE 36

37 Temperature Dependent Delay An nseparable aspect of electrcal power dstrbuton and sgnal transmsson through the nterconnects Resstance s dependent of Temperature ( 1+ T ( )) r( x) = ρ0 β x ρ 0 : the resstance per unt length at reference temperature β: the temperature coeffcent of resstance (1/ C) NTUEE 37

38 Interconnect Temperature Calculaton The nterconnect temperature s gven by x : wre wdth θ nt : the thermal mpedance of the nterconnect lne to the chp σ: duty cycle V cross : cross voltage of wre t ox : the total thckness of the underlyng delectrc t m : the thckness of the wre K ox : the thermal conductvty l : wre length R m : the temperature dependent unt resstance ψ : the heat spreadng parameter Not lnear functons Least Square Estmator NTUEE 38

39 Least Square Estmator (LSE) Least squares solves the problem by fndng the lne for whch the sum of the square devatons (or resduals) n the d drecton (the nosy varable drecton) are mnmzed. Apply Cramer Rule to fnd the A 1 and A 0, whch mnmzes the square devatons y = A 1 x +A 0 Cramer Rule: NTUEE 39

40 Approxmaton for Thermal Constrant Let N = 5 and pck fve szes of x, we can approxmate the thermal constrant by Least Square Estmator (LSE). Banerjee et al., DAC 1999 NTUEE 40