Stability analysis of semiconductor manufacturing process with EWMA run-to-run controllers

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1 Sabiliy analysis of semiconducor manufacuring rocess wih EWMA run-o-run conrollers Bing Ai a, David Shan-Hill Wong b, Shi-Shang Jang b a Dearmen of Comuer Science, Universiy of exas a Ausin, exas, USA b Dearmen of Chemical Engineering, Naional sing-hua Universiy, Hsin-Chu, aiwan ABSRAC: In he semiconducor manufacuring bach rocesses, each se is a comlicaed hysiochemical bach rocess; generally i is difficul o erform measuremens on-line or carry ou he measuremen for each run, and hence here will be delays in he feedback of he sysem. he effec of he delay on he sabiliy of he sysem is an imoran issue which needs o be undersood. Based on he exonenially weighed moving average (EWMA) algorihm, we roose wo kinds of conrollers, EWMA-I and II conrollers for single roduc rocess and mixed roduc rocess in semiconducor manufacuring in his aer. For he single roduc rocess, he sabiliies of sysems wih boh conrollers which undergo differen kinds of merology delays are invesigaed. Necessary and sufficien condiions for he sochasic sabiliy are esablished. Rouh-Hurwiz crierion and Lyaunov s direc mehod are used o obain he sabiliy regions for he sysem wih fixed merology delay. By using Lyaunov s direc mehod, he sabiliy region is esablished for he sysem wih fixed samling merology and wih sochasic merology delay. We also exended he heorems of single roduc rocess o mixed roduc rocess. Based on he roosed heorems, some numerical examles are rovided o illusrae he sabiliy of he delay sysem. Keywords: exonenially weighed moving average (EWMA), run-o-run conrol, merology delay, sochasic sabiliy, single roduc rocess, mixed roduc rocess. 1

2 1. Inroducion In he field of semiconducor manufacuring indusry, run-o-run conrol is now widely acceed as a means of roducion fabricaion faciliies o imrove he efficiency and wasage in roducion. I s a form of discree rocess and machine conrol in which he roduc recie wih resec o a aricular machine rocess is modified ex siu, i.e., beween machine runs, so as o minimize rocess drif, shif and variabiliy [1]. wo of he mos basic run-o-run conrol algorihms used oday in he semiconducor manufacuring indusry are exonenially weighed moving average (EWMA) algorihm and redicor-correcor conroller (or double EWMA) algorihm. Boh of hese algorihms need he insananeous informaion of he rocess ouu; however, in semiconducor manufacuring, each se is a comlicaed hysiochemical bach rocess, and measuremens are almos exclusively erformed off-line, ofen slow, inconsisen, or skied by oeraors. his caused delayed, inconsisen, and infrequen measuremen of he rocess ouu. herefore, one roblem in he alicaion of run-o-run conrol is how merology delay would affec he sabiliy of he rocess. ime delay sysem has been exensively sudied in he las few years. For he sysem wih sochasic ime delay, as oined ou in [2], many manufacuring rocess can be modeled by Markovian jum linear sysems. And he resuls of oimal conrol, robus conrol and sabiliy for such kind of sysem can be widely seen in he recen lieraures, such as [3]-[7]. However, he oimizaion roblem, he robus conrol as well as he sabiliy roblem for Markovian jum linear sysems usually change ino he roblem of solving a se of linear marix inequaliies (LMIs), and deailed algorihms o solve he LMIs can be found in [8]-[11]. he ioneer work on he sabiliy of EWMA run-o-run conroller wihou delay was carried ou by Ingolfsson and Sachs [12]. hey found ha he rediced rocess gain mus no be oo small relaed o he rue rocess gain o granee he sabiliy of he rocess. R. Good e al. examined sabiliy bounds for he discoun facors of boh single inu single ouu (SISO) and mulile inus mulile ouus (MIMO) 2

3 double EWMA run-o-run conrollers when here is lan-model mismach and delay beween roduc manufacuring and roduc merology [13]. heir work showed ha for he SISO sysem, he size of he sabiliy region decreases as merology delay and lan-model mismach increase and as he discoun facors decrease. hey also oined ou ha when merology delay exiss in he sysem, inenionally overesimaing rocess gain could lead o insabiliies, unlike he sysem wihou delay. On he oher hand, for he MIMO sysem, increasing he discoun facors does no necessarily increase sabiliy when a large lan-model mismach exiss. A few years laer, afer heir firs work on sabiliy analysis of double EWMA run-o-run conrollers, Good e al. analyzed he sabiliy of MIMO EWMA run-o-run conroller wih merology delay by using he generalized Rouh-Hurwiz sabiliy crierion [15]. hey derived he necessary and sufficien condiions for sabiliy wih merology delays u o wo runs, and develoed a sufficien condiion for he sabiliy of he MIMO sysem wih merology delay longer han wo runs. he sufficien condiion hey go is ha if all of he eigenvalues of a model-mismach marix fall inside a circle wih uni radius and cenered a {1, 0} on he comlex lane, hen he closed-loo sysem is sable. M.F. Wu e al. analyzed he influences of merology delay on boh he ransien and asymoic roeries of he roduc qualiy for he case when a linear sysem wih an iniial bias and a sochasic auoregressive moving average disurbance is under EWMA run-o-run conrol. hey oined ou ha merology delay is only imoran for rocesses ha exerience nonsaionary sochasic disurbance. Based on he sudy of numerical oimizaion resuls of he analyical closed-loo ouu resonse hey develoed guidelines o uning he discoun facor. All he aforemenioned works on EWMA or double EWMA run-o-run conrol sysem are based on he assumion ha he merology delay is fixed. However, he semiconducor manufacuring indusry characerized by hysical and chemical environmens makes measuremen in many of hese environmens difficul or ime-consuming, and his combined wih he fac ha many rocess ools are no designed for he addiion of in siu sensor, resuled in measuremen aken less frequenly han every run, or a sochasic runs. o he bes of auhors knowledge, u 3

4 ill now, in he field of EWMA run-o-run conrol, here is only one work available which discusses he issue of he sysem wih sochasic merology delay [16]. he ioneer work by B. Ai e al. in [16] assumes ha he sysem subjec o he sochasic merology delay, one kind of EWMA conroller, which we call EWMA-I conroller in his aer, is roosed o rejec he sysem disurbances. However, someimes EWMA-I conroller should no be udaed if here is no new informaion available. So in his aer, we should modify EWMA-I conroller. Also, in [16], he ransiion robabiliy marix is given, bu in he real manufacuring rocess, he ransiion robabiliy marix canno be known direcly and should be calculaed from he robabiliy disribuion of he daa we received, hence we will discuss how o calculae he ransiion robabiliy marix from he secific robabiliy disribuion. Wha we have discussed is based on he assumion ha here is only one kind of roduc in he manufacuring line which is far from realisic. Because of he high caial coss associaed wih he rocess equimens, i is a common racice in oday s semiconducor manufacuring o have many differen roducs and rocesses run on each rocessing ool, i.e., high-mix manufacuring. he firs work of he sabiliy of he high-mix manufacuring is done by Y. Zheng e al. in [20] where hey sudied a model wih wo roducs manufacured on he same ool, and roosed wo kinds of conrol mehod: ool-based and roduc-based aroaches; and, hey found ha he ool-based aroach is unsable when he lan is non-saionary and he lan-model mismach arameers are differen for differen roducs, while he roduc-based aroach is sable. However hey made he misake in deriving he ouu of he sysem for roduc-based conrol, see Lemma 2, equaions (22) and (23) in [20] for deails. Also he assumion ha only wo roducs are manufacured on he roducion line is far from realisic, herefore B. Ai e al. correced and exended Y. Zheng s work [21]-[27] o a more comlicaed case, where hey assumed ha a number of differen kinds of roducs are manufacured on he same ool wih variable manufacuring cycles, and he camaign lengh and break lengh of each cycle are also variable. hey found ha for mixed roduc drifed rocess, if he break lengh of a roduc is large, hen a he beginning runs of each cycle, he rocess 4

5 ouu will far deviae from he arge value. hey roosed cycle reseing algorihm for discoun facor (CR-EWMA) algorihm, and cycle forecasing EWMA (CF-EWMA) o reduce he large deviaions as well as o achieve he minimum asymoic variance conrol; hey also roosed a discoun facor reseing faul oleran (RF) aroach and faul oleran cycle forecasing EWMA (FCF-EWMA) algorihm o handle he se faul. Alhough hey had exended heir models o more sohisicaed condiions, he models are sill no well maching he real manufacuring siuaions. Again o he bes of auhors knowledge, u ill now, in he field of semiconducor manufacuring, here is no work available which discusses he sabiliy of mixed roduc rocess subjec o sochasic merology delay. So in his aer, we shall ry o esablish he model for mixed roduc rocess wih sochasic merology delay and exend he heorems obained for single roduc rocess o mixed roduc rocess. he sabiliy of mixed roduc rocess will also be analyzed. For beer of resenaion, he remainder of he aer is organized as follows: From Secion 2 o Secion 4, we will focus on single roduc rocess, i.e., in Secion 2, wo kinds of EWMA conrollers, i.e., EWMA-I conroller and EWMA-II conroller, are roosed for single roduc rocess. In Secion 3, we will discuss how o obain he ransiion robabiliy marix from a secific robabiliy disribuion. Numerical examles are rovided in Secion 4 o obain he sabiliy regions for he sysems subjec o differen merology delays. In Secion 5, we will discuss mixed roduc rocess, and he heorems obained for single roduc rocess are exended o he mixed roduc rocess. he conclusion remarks are resened in Secion EWMA Run-o-Run Conrollers in Single Produc Process In semiconducor manufacuring, he same roducs are usually manufacured on he same ool, i.e., single roduc rocess. he mos widely used algorihm in his rocess is EWMA algorihm which needs he informaion of he ouu of he sysem for feedback. However, ouus someimes are no available imely because he measuremen is usually ime-consuming, so here will be delays in he sysem. In his 5

6 Secion, we will firs discuss EWMA conroller wihou merology delay, and hen we will roose wo kinds of EWMA conrollers for he sysem wih merology delay. he sabiliy of he conrollers will be examined in he las ar of his Secion EWMA Run-o-Run Conroller wihou Merology Delay A yical EWMA run-o-run observer assumes a saic linear model beween conrol variable Y, and maniulaed variable u, i.e., Y u a (1) where is he rocess gain beween he rocess inu and ouu, and a is he insananeous disurbance a run. Given he rediced model of he rocess Yˆ bu aˆ (2) where b is he model gain, and a ˆ is he esimaed offse a run for he rocess. When informaion of he curren run is available wihou delay, an EWMA udae of he offse is given by: where is a discoun facor beween zero and one. aˆ ˆ 1 ( Y bu ) (1 ) a (3) A conrol law is used o deermine he conrol recie for he nex run, i.e., aˆ u (4) b where is he desired arge. Wihou loss of generaliy, in his aer, we assume 0. Fig. 1 shows he srucural diagram of he above algorihm. 6

7 Disurbance a arge + Conroller u Process β Model b Resonse Y EWMA filer wihou merology delay aˆ ˆ 1 ( Y bu ) (1 ) a Fig. 1. Srucural diagram of EWMA run-o-run conroller wihou merology delay 2.2. EWMA Run-o-Run Conrollers wih Merology Delay As menioned reviously, in an acual manufacuring lan, measuremen delay is a common henomenon which consiues a sochasic rocess. When he measuremen delay haens, in ime ouu of he sysem is no longer available, so he original EWMA filer described in (3) does no hold. Oher EWMA filer should be roosed o rejec he disurbance of he sysem. Fig. 2 is he general srucural diagram for EWMA run-o-run conrol wih merology delay. Disurbance a arge + Conroller u Process β + + Resonse Y Measuremen Saion Model b + EWMA filer wih merology delay Fig. 2. General srucural diagram for EWMA run-o-run conrol wih merology delay 7

8 Suose ha here is a sochasic merology delay a run, hen a his ime, wo kinds of EWMA run-o-run conrollers, EWMA-I and II conrollers, can be used o esimae he disurbance: For EWMA-I conroller, which was firs roosed by B. Ai e al. in [16], he conrol acion is also chosen as (4), and he EWMA filer is aˆ ( Y bu ) (1 ) aˆ (5) 1 Combining (1), (2), (4) and (5), we rewrie EWMA-I conroller in he form of (6), ˆ ˆ ˆ (6) a 1 (1 ) a (1 ) a where / b reresens he lan-model mismach of he rocess gain. From Equaions (5) and (6) we know ha he EWMA-I conroller is always udaing is sae a ˆ regardless of he availabiliy of he ouu Y, i.e., in each recursion, a ˆ is udaed by he available ouu which may no be he laes ouu of he sysem. However, in some rocesses, if he newes ouu is no available, he conroller will no udae is sae, and we name his conroller as EWMA-II conroller, i.e., aˆ +1 (1 ) aˆ (1 ) ˆ a, if 1 ; aˆ, oherwise. (7) For examle, a he h roducion run, we have ouu of 3 Y for feedback, i.e., 3 : if 1 3, i.e., no new ouu is available a he roducion run, hen he sae of he sysem should no be udaed; however, if 1 2, hen a he 1 h roducion run, he laes ouu of he sysem is Y 1, and a his ime he sae of he sysem should be udaed by using Y 1. Remark 1: For he fixed merology delay, EWMA-I and II conrollers are equivalen because boh of he conrollers are in he form of 1 f f 1 h aˆ ( Y bu ) (1 ) aˆ (8) for fixed f runs merology delay. 8

9 2.3. Sabiliy of EWMA Run-o-Run Conrollers Sabiliy is a fundamenal requiremen for any sysem. An unsable conrol scheme should no be imlemened. In oher words, he conrol scheme should lead he rocess under conrol o a sable sae. In his subsecion, we shall examine he sabiliy of EWMA run-o-run conrollers subjec o fixed and sochasic merology delay Fixed Merology Delay For he sysem wih fixed f runs merology lags, combining (1), (2), (4) and (8), we have he ouu of closed-loo sysem, Y f 1 1 (1 ) BB a f 1 (9) 1 (1 ) B (1 ) B where B is he backshif oeraor. Equaion (9) yields he characerisic equaion of he closed-loo sysem f 1 G( B) 1 (1 ) B (1 ) B 0 ; hen he sysem described by (1), (2), (4) and (8) is sable if and only if he roos of GB ( ) lie ouside he uni cycle. Wih he well-known bilinear ransformaion 1W B (10) 1 W he ouside of he uni cycle is maed o he oen lef half-lane, and hen he Rouh-Hurwiz crierion will be used on (11) o find he sabiliy boundaries of sysems wih merology delays of f runs. f 1 1W 1W GW ( ) 1 (1 ) (1 ) 0 1W 1W (11) Remark 2: he Jury s es can also be imlemened direcly on GB ( ) o check wheher or no is roos lie ouside he uni cycle Sochasic Merology Delay For he sysem wih sochasic merology delay, neiher Rouh-Hurwiz crierion 9

10 nor Jury s es is valid in obaining sabiliy region. In he following, we ry o find oher ways o ge he sabiliy region for such kind of sysem. where Augmen he sae variable a ˆ a run ino a vecor as 1 max X aˆ aˆ aˆ aˆ (12) max is he maximum delay ossible, and he udae equaion in (6) or (7) can be rewrien as X 1 1 (, ) X (13) where ( 1, ) is deermined by he acual ye of conroller and he maximum delay of he sysem. Combining (6) or (7) wih (12), we can wrie ( 1, ) for sysems of any merology delay wih eiher EWMA-I or II conroller. For examle, if he conroller of he sysem is EWMA-I conroller, hen he sysem marix ( 1, ) is relevan o, bu irrelevan o 1 :if he sysem is delay free, i.e., =0, and ( 1, ) (0, 0) 1 ; if he maximum delay of he 1 0 sysem is 1, if =0, hen ( 1, ) (0,0) (1,0), and if =1, (1 ) hen ( 1, ) (0,1) (1,1). 1 0 However, for he sysem wih EWMA-II conroller, he sysem marix of he sysem is relevan o boh 1 and : if he sysem is delay free, hen (0,0) 1 ; if he maximum delay of he sysem is 1, hen 1 0 (0,0), (0,1) (1,0) and (1 ) (1,1) 1 0 ; if he maximum delay of he sysem is 2, hen (0, 0) 1 0 0,

11 1 0 0 (0,1) , (1, 2) , (1, 0) , (2, 0) , 1 (1 ) 0 (1,1) , 1 (1 ) 0 (2,1) , 1 0 (1 ) (2, 2) Remark 3: hrough he augmenaion echnique, i is clear ha he sochasic merology delay sysem described by (1), (2), (4) and (6) or described by (1), (2), (4) and (7) is ransformed ino a delay-free discree-ime jum linear sysem modeled by a homogeneous Markov chains as is exressed by (13) Sabiliy Crierion of he Sysem wih Variable Delay In his aer, we will use he following definiion for sochasically sable. Definiion 1: he sysem in (13) is sochasically sable, if for every finie X 0 and iniial mode, and S 0,1,, 0 S being he se of all ossible delays, he following condiion max is saisfied. E 2 X X 0, 0 (14) 0 Le ransiion robabiliies marix beween merology delays be P [ ij ], and ij is defined by ij 1 Prob j i (15) where i, j{0,1,, }, hen, 0 1 and ij j0 ij 1. he following heorem gives sufficien and necessary condiion o guaranee he sochasic sabiliy of sysem (13). heorem 1: Sysem (13) is sochasically sable if and only if here exiss a osiive-definie marix 11

12 Q (, ) 0 for, 0,1,,, saisfying he following marix inequaliies: 1 1 max Proof: max 2 1 1, L(, ) (, ) Q(, ) (, ) Q (, ) 0 (16) Sufficiency: Consruc he sochasic Lyaunov funcion,, V X as follows: 2 1 hen E V X E V X 0 V ( X,, ) X Q(, ) X (17) , 2, 1 1, 1, V X, 2, 1,, E V X,, EV X , 1,, 2, 1, 2, 1 E V X X V X max max 0,, 1, , X Q X X Q X X, Q,, X X Q, X 2, 1 2, 1, max X 1, 1, 1, 1, 2, Q X Q 1 X 0 X L X X L X X X X min 2 1 min where min 2, 1 is he minimum eigenvalue of 2, 1 inf, min min 2 1 2, 1S ossible combinaions of 2 and 1. 2 (18) L and is he minimum value of, min 2 1 for all Given he inequaion (18), we can derive he following recursive relaion,, E, E V X V X X ,, E,, E V X V X X (19) Hence X E X EV,, V,, X

13 1, 1, 0, 0 E V X V X X (20) 1 2 i.e., 2 X V X V X 1 1 E (, ) (,, ) V( X0, 0) which imlies 2 1 lim E 0, 0 E ( 0, 0) X X V X 1 1 X0 Q( 0) X0 i.e., sysem (13) is sochasically sable. Necessiy: Define E X -, 2, 1 k k 2, k 1 k, 2, Q E X X R X X 1 (21) k wih R, is a se of osiive definie marix. Le r R i, j k2 k1 x x 0 x x 2x x, we have and given ha ki kj ki k j ki k j ij k2 k1 max max max max max 2 k k 2, k 1 Xk xk irijxk j xk -jrjj 2 xk irijxk j i0 j0 j0 i0 j0 i j X R max max max max xk -jrjj 2 rij xk i xk j c jxk j cmax X k (22) j0 i0 j0 j0 i j wih cmax max c0,, c max. Hence 2 E X -, 2, 1 max k, 2, Q E c X X X 1 k Since he sysem is sochasic sable, we have lim E, 2 X X 0 0 (23) 1 13

14 E X Q, 2, 1 X will be bounded and is asymoic value is given by:, k k 2 k 1 k 2 1 EX Q 2 1 X lim E[ X R(, ) X X,, ] (24) Similarly, we can wrie k E[ X Q, X X,, ] lim E[ X Q 1,, X ] Subracing (24) from (25), we have 0 k 1 lim E[ X R, X X,, ] k k 2 k 1 k 2 1 E E[ X Q, X X,, ] [ X Q, X ] X R(, ) X Hence we have max 2 1 X (, ) Q(, ) (, ) X X Q(, ) X 1, (25) (26) max 2 1 1, R(, ) (, ) Q(, ) (, ) Q(, ) L(, ) o sum hose u, heorem 1 holds. (27) Since L ( 2, 1) are nonlinear in he sysem marix, i is difficul o check wheher (16) is feasible or no. o his end, we have he equivalen condiion for (16), i.e., Proosiion 1: he marix inequaliy L ( 2, 1) 0 in (16) is equivalen o he following marix inequaliy Q( ˆ 2, 1) ( 1) V ( 1) Q( 1) 0 ˆ ( ˆ 1) V ( 1) ( 1) ( 1) Q Q where ( 1) ( 1,0) ( 1,1) ( 1, max ), (28) Q ˆ ( ) diag Q (,0), Q (,1),, Q (, ), max V ( ) I, I,, I, and I is an ideniy marix wih a 1 diag 1,0 1,1 1, max roer dimension. Proof: In order o rove Proosiion 1, we need o use Schur comlemen [28], which says ha for symmeric 14

15 marix M11 M12 M, he following hree condiions are equivalen: M 12 M M 0; 2. M 0, M M M M 0; M 0, M M M M Based on he Schur comlemen, in he following, we will esablish he relaionshi beween L ( 2, 1) 0 and Proosiion 1: L(, ) max , ( 1, ) ( 1, ) ( 1, ) ( 2, 1) Q Q (,0) Q(,0) (,0) (, ) Q(, ) (, ) Q(, ) 1 1, max 1, max 1 max 1 max 2 1 (,0) (, ) 1 1 max (,0) (, ) 1 1 max 1,0Q( 1,0) ( 1,0) Q( 2, 1) 1, ( max 1, max ) ( 1, max ) Q 1,0Q( 1,0) 1,0 I ( 1,0) Q( 2, 1) Q(, ) I ( 1, max ) 1, max 1 max 1, max 1,0 I Q( 1,0) ( 1,0) ( 1, max ) 1,0 I ( 1,0) 2 1 Q(, ) (, ) I Q ( 1, max ) 1, I max ( 1 ) 1, max 1 max V ( 1 ) Qˆ ( 1 ) ( ) V ( ) Qˆ ( ) V ( ) ( ) Q(, ) we ake M11 Q ( 2, 1), M12 ( ˆ 1) V ( 1) Q( 1) and M ˆ( ) 22 Q 1, hen he hird condiion of Schur comlemen is saisfied, and herefore 15 V ( 1 ) Q(, ) ( ) V ( ) Q ˆ ( ) M = 0 M M M11 M ˆ ( ˆ 1) ( 1) ( 1) ( 1) Q V Q ( 1 ), i.e., Proosiion 1 holds. Remark 4: he necessary and sufficiency condiions of sochasic sable for he sysem wih EWMA run-o-run conrollers (i.e., EWMA-I conroller in his aer) subjec o sochasic merology delay were firs roosed by B. Ai e al. in [16]; however hey made misakes in esablishing he relaionshi beween ransiion robabiliy marix and he sysem marix (lease see heorem in [16] for deail), and herefore he roosiion (also see [16]) hey obained for solving he marix inequaliy is wrong. In fac, (28) is a se of linear marix inequaliies, we can use Malab Robus Conrol oolbox o solve hem.

16 3. ransiion Probabiliy Marix for Single Produc Process Since he merology delay is a sochasic variable insead of being fixed in an acual manufacuring lan, how o esablish he relaionshi beween he merology lags will be sudied in his Secion. Firsly, we will show he rocess of obaining acual merology delay sequence from he original merology delay sequence, and hen we will derive he ransiion robabiliy marix from a secific robabiliy disribuion. How o calculae he average delay of he sysem will be discussed in he las ar of his Secion Original and Observed Merology Delay he rocess of samling, measuremen and reoring merology resuls is a sochasic rocess so ha he original or acual merology delay is a sequence of sochasic variable generaed by a robabiliy funcion. Le OMD, he original merology delay of he sysem a he run, be a random number generaed by a secific robabiliy disribuion. Hence he informaion of his run will be available a run OMD. Le LRA be he index of he laes run a which merology resuls are available a he run, we have: k= 1,,-1 LRA arg min OMD k +k (29) Furhermore, if he laes run a which merology resuls are available is more recen han ha of he las run, hen conroller will be noified of his new delay and merology, oherwise he delay will be increased by 1: τ = LRA (30) able 1 illusraed an examle of he rocess of calculaing resamled delay numbers from original delay numbers. he informaion of he firs roducion run is available immediaely, hence he delay is zero. A he 2 nd roducion run, he newes daa available is sill he daa of he 1 s run, hence he delay is 1. A he 3 rd run, he informaion of he 2 nd run was received; hence he delay remained o be 1. A he 6 h 16

17 run, wo revious runs (he 4 h and 5 h ) are available; only he laes (he 5 h ) will be used, hence he delay is 1. Since he 6 h and he 8 h runs were no samled, and he 7 h run was reored only afer long delay, he daa of he 5 h roducion run remained he laes informaion available for 7 h and he 8 h runs. Hence he corresonding delays for 7 h and he 8 h runs were 2 and 3 accordingly. able 1: he relaion beween original merology delay and acual merology delay by he conroller Run Number Original Merology Delay OMD Run when merology available OMD Laes Run Available LRA Acual Delay in EWMA Conroller run is no samled From he revious analysis we conclude ha he acual merology delay can increase a mos 1 a each run, i.e., 1 1 and Prob( 1 +2) 0. Since 1 is only affeced by, is a Markov Chain. If he consrain condiions for merology delay are considered, hen he srucure of he ransiion robabiliy marix will be 17

18 P [ ij ] ,0 2,1 2,2 2, ,0 1,1 1,2 1,3 1, , ,0 1,1 1,2 1,3 1, 1, 1 1, 2 0 (31) Each row reresens he ransiion robabiliies from a fixed sae o all he saes, he diagonal elemens are he robabiliies of merology sequences wih equal delays, he elemens below he diagonal indicae shorer delays, and he elemens above he diagonal are he robabiliies of encounering longer delays. Fig. 3 illusraes hree saes ransiion diagram. From he figure, we can see ha i can jum from 1 and 2 o any saes, while i canno jum from 0 o 2. Fig. 3. hree saes ransiion diagram 3.2. Calculaion of ransiion Probabiliy Marix from a Secific Probabiliy Disribuion Le η Prob OMD = j j be he robabiliies of a merology being reored afer j runs; and be he robabiliies of a run no samled a all. he following heorem calculaes he robabiliies of observing τ +1 =j and τ =i, i.e., ij given η and j : 18

19 heorem 2: he ransiion robabiliy marix ij can be calculaed from he robabiliy disribuion of he merology delay, η and j by: Proof: 0, i1 j; ij ( 1 j i) (1 ) k, j i 1; (32) k i 1 (1 ) j, 0 j i. If i, he laes daa available a run is i, i.e., he daa of run i is samled wih merology delay of i runs. Furhermore, he daa of runs i 1,, are eiher no samled, or are reored wih merology delays OMD longer han or equal o i,,1 runs resecively, i.e.: Prob i 1 i 1 k 1 k k i k i1 1 k 1 k k2 k1 (A) If i and 1 j i, he OMDs of runs beween i 1 and j has o be longer han or equal o i,, j 1 because even if he OMD of i 1 is i and arrived a 1 as did he merology of j 1, i will be disregarded as j 1 is a newer run. Furhermore, he daa of runs j 2,, 1 are eiher no samled, or are reored wih merology delays OMD longer han or equal o j,,1 runs resecively. herefore: Prob 1 j i k kj1 1 i 1 k 1 k k i k i1 1 k 1 j 1 k k j1 k j k k k2 k1 (33) (34) 19

20 Prob 1 j i ij =Prob 1 j i Prob i k k i k i k i1 i 1 k 1 k k i k i1 1 1 k k k j1 j k j 1 1 k k k j1 k j k kj k k k kj1 k2 k1 1 1 k k k j2 k 1 1 j 1 1 (B) If i and 1 j i 1, he OMDs of runs beween i 1,, 1 and has o be longer han or equal o i 1, 2,1. herefore: (C) j i 1 Prob i1 i 1 1 i 1 k 1 k k i1 k i 1 k 1 k k2 k1 Prob 1 i1 i i, i1=prob 1 i 1 i Prob i 1 1 k k i k i1 k i i 1 k 1 k k i k i1 1 1 k k k2 k1 k k k i 1 k Analysis in subsecion 3.1 showed ha he observed delay can increase by 1 bu no 1 1 (35) (36) (37)

21 more, i.e. ij 0 if j i 1. o sum hose u, we have heorem 2. Remark 5: he summaion of he ransiion robabiliies from a fixed sae o all he saes is 1: Proof: ij 1 (38) j0 i ij ij ij j0 j0 ji1 i (1 ) (1 ) k k k 0 k i1 (1 ) 1 k k 0 (39) Remark 6: he calculaion of ransiion robabiliy marix from a given robabiliy disribuion was firs derived by Z.X. Yu e al. for he roblem of daa-acke ransfer in nework conrol sysem (NCS) [17]-[18]; Y. Zheng e al. exended his work o a more general case which assumes he daa-acke will be los someimes; however, he heorems obained by Yu and Zheng e al. are wrong (see heorem 1 in [17]-[19] for deails) Asymomaic Probabiliy and Average Delay For any homogeneous Markov chain, from he relaionshi beween absolue robabiliy disribuion and iniial robabiliy disribuion, we have he following equaion: wih he consrain: P P (40) j i ij i0 Pj 1 (41) j0 where P j is he average robabiliy a sae j. 21

22 he average delay E() can be obained by aking he execaion of saes in he sae-se. i.e., If i 0 for i saes 0,1, wih wih E( ) j Pj (42) j0, he above equaions can be runcaed o a se of finie delay j Pi ij (43) j0 P ij ij 0 i,0 j ; j i,0 j. (44) Proof: If i 0 for i, all rows wih i can be runcaed since Prob i 0 and hence ij is undefined. Furhermore, if we runcae all columns of robabiliy ransiion marix from he column j 1 onwards, he reduced ransiion robabiliy marix mus be normalized accordingly ij ij = 1 j 1 ij (45) However, ij 0 for all i, j 1, herefore, we have ij = ij for ; 0 j. When i, we have 0 i, 1= 1 (46) j j 1 hus herefore (44) holds., j 1, j j j 1 1 he asymoic average delay is given by: (47) E( ) j P (48) j0 j 22

23 he value of of. can be chosen so ha changes in E( ) is negligible wih increase 4. Numerical Examles for Single Produc Process In his Secion, we will invesigae he sabiliy of sysems wih EWMA-I and II conrollers subjec o differen kinds of merology delays. In subsecion 4.1, he sabiliy regions for sysems wih fixed merology delay will be obained; we will consider he sabiliy regions for he sysems wih EWMA-I and II conrollers wih measuremen aken in a aricular samling inerval in subsecion 4.2; he sabiliy regions for sysems wih EWMA-I and II conrollers subjec o sochasic merology delay will be discussed in subsecion Sabiliy Analysis for Sysems wih Fixed Merology Delay heorem 1 which is obained in subsecion is based on Lyaunov s direc mehod. In his subsecion, we will comare he sabiliy regions obained by Lyaunov s direc mehod and Rouh-Hurwiz crierion (or Jury s es) for he sysems wih fixed merology delays. Fig. 4 - Fig. 13 are he simulaion resuls for sysems wihou merology delay and wih fixed one o fixed nine runs merology delay. From he figures, i is clear ha for he same merology delay sysem, he sabiliy region will be he same desie of which kind of conroller, EWMA-I conroller or EWMA-II conroller, is adoed. his resul coincides wih he heoreic resul obained in subsecion 2.3. Also Lyaunov s direc mehod and Rouh-Hurwiz crierion arrive a he same sabiliy regions for he same sysems. 23

24 Fig. 4. Sabiliy regions for sysems wihou merology delay Fig. 5. Sabiliy regions for sysems wih fixed one run merology delay 24

25 Fig. 6. Sabiliy regions for sysems wih fixed wo runs merology delay Fig. 7. Sabiliy regions for sysems wih fixed hree runs merology delay 25

26 Fig. 8. Sabiliy regions for sysems wih fixed four runs merology delay Fig. 9. Sabiliy regions for sysems wih fixed five runs merology delay 26

27 Fig. 10. Sabiliy regions for sysems wih fixed six runs merology delay Fig. 11. Sabiliy regions for sysems wih fixed seven runs merology delay 27

28 Fig. 12. Sabiliy regions for sysems wih fixed eigh runs merology delay Fig. 13. Sabiliy regions for sysems wih fixed nine runs merology delay Fig. 14 is he comarison figure which shows he sabiliy regions of sysems wih differen fixed merology delays. From he figure, i is noiced ha alhough he sysems have differen fixed merology delays, hey have similariies in he sabiliy region: when he esimaed rocess gain is greaer han half of he rue rocess gain ( 2 ), he sysem is guaraneed closed-loo sable for any discoun facor beween 0 o 1; however, if 2, is decreasing in o kee he closed-loo sysem sable. In addiion, he size of sabiliy region will shrink wih he increase of 28

29 merology delay. Fig. 14. Comarison of sabiliy regions for sysems wih differen fixed runs delays 4.2. Sabiliy Analysis for Sysems wih Fixed Samling Merology In semiconducor manufacuring, in siu measuremens are usually aken a he aricular samling inerval; hus, here is a need o analyze he sabiliy of he sysem wih such kind of merology delay. aking one run as he samling inerval for examle, if a he h roducion run, we have aken he merology, hen a he 1 h roducion run, we will use he informaion we go a he h roducion run, i.e., he sysem will be of one run delay, and herefore On he oher hand, if a he h roducion run, no merology is aken, hen a he 1 h roducion run, a new merology will be aken, i.e., he sysem is delay free, and Based on he above analysis, we know ha he ransiion robabiliy marix is P he same analysis can be done for he sysem wih oher samling merology inervals. 29

30 And he ransiion robabiliy marix is P and P for he samling inerval wo and hree resecively. Fig. 15 shows he simulaion resuls for sysems wih differen fixed samling merology inervals. From he figure, we noice ha when he model mismach 2, he sysem wih eiher conroller is guaraneed closed-loo sable for any discoun facor beween 0 o 1. For he sysem wih EWMA-I conroller, he larger he samling merology inerval is, he smaller he size of sabiliy region will be; while for he sysem wih EWMA-II conroller, he sabiliy region is irrelevan o he lengh of samling inervals and his resul does no coincide wih our inuiion. In fac, for he fixed samling merology delay, EWMA-II conroller (7) is (1 ) x, if 0; equivalen o EWMA-III conroller, x +1. If he merology x, oherwise. is aken in he form of fixed samling, hen he sabiliy condiion for sysem wih EWMA-III conroller is he same as ha of he sysem wihou merology delay. here are wo reasons for his resul: firs, he daa used o udae he conroller are he measured daa which are free of delay, and his haens wih robabiliy 1; second, when delay aears, EWMA-III conroller will no udae. For he sysem of fixed samling merology, if he samling inerval is d, hen d from (40) and (42), i is easy o ge ha he average merology delay E() and 2 1 P0 P1 P d. From Fig. 15, we know ha i is wrong o use ceil(e(τ)) 1 as d 1 he delay for he sysem wih EWMA-I conroller o ge he sabiliy region, while if we do so for he sysem wih EWMA-II conroller, we will obain conservaive sabiliy regions. 1 ceil(a) rounds he elemens of A o he neares inegers greaer han or equal o A. 30

31 Fig. 15. Sabiliy regions for delay sysems wih differen samling inervals 4.3. Sabiliy Analysis for Sysems wih Sochasic Merology Delay In his subsecion, we will discuss he sabiliy of he sysem wih EWMA-I and II conrollers. We firs assume ha he ransiion robabiliy marix is unknown, bu can be calculaed from Poisson disribuion (for oher robabiliy disribuions he calculaions are also sraighforward); and hen we will find ou he sabiliy regions for he sysem wih his kind of merology delay. he conclusion for sabiliy of sysems wih EWMA-I and II conrollers will be given in he las ar of his subsecion ransiion Probabiliy Marix Calculaed from Poisson disribuion In he real manufacuring rocess, we someimes hardly do he measuremen for each run, or here is no need o obain he daa of every run. We call hese cases as he samling merology. In his subsecion, we firs give an examle wih he samling merology o verify heorem 2. hen we will discuss how o make an aroximaion of he average delay of he sysem, i.e., how o choose a roer 31, o accelerae he comuaion. And he sabiliy analysis will be done on he sysem wih ransiion robabiliy marix calculaed from Poisson disribuion.

32 We assume ha he merology delay of he real manufacuring rocess follows Poisson disribuion wih mean arameer 1, and he robabiliy of he curren run no being measured is which is chosen from 0 o 0.9 in 0.1 incremens. Based on he calculaion rocess discussed in able 1, Poisson random numbers are generaed, as he original measuremen delays of he sysem, o ge he resamled delay numbers. he simulaions are done 50 imes for each o obain he ransiion robabiliy marix P and he average delay of he sysem E(). During he simulaions, P is runcaed ino a 3 3 marix in each calculaion 2. Fig. 16 is he simulaion resuls for P and E(). Each elemen of ransiion robabiliy marix calculaed by heorem 2 is denoed by Calculaed ij, and Observed ij is comued from he simulaions. From he figure, we know he differences beween Calculaed ij and Observed ij are very small, less han 3%. Also he relaionshis beween E( ) Calculaed, calculaed from (48), and E( ) Observed, comued by simulaion, are linear wih sloe 1. hese simulaion resuls verify heorem 2. Fig. 16. Simulaion es of P and E(τ) Fig. 17 shows he relaionshi beween runcaion of he ransiion robabiliy marix and he average delay of he sysem. he comuaion in his simulaion is 2 he runcaion will no affec he resul, because boh P Calculaed and P Observed are normalized a he same ime, hence E() Calculaed and E() Observed are normalized a he same ime. 32

33 based on (32) and (48). From he figure, we know he minimum, denoed as min, is increasing wih he increase of, o make he average delay of he sysem converge o is limiaion. For examle, if 0, i.e., every daa of he sysem will be measured for feedback, min should be 4; and min 31 for 0.7. Fig. 17. Relaionshi beween rancaion of he ransiion robabiliy marix and average delay of he sysem Fig. 18 shows he relaionshi beween sysem. I is clear ha he relaionshi beween and he average delay of he and E() is nonlinear, and he nonlineariy becomes srong esecially when aroaches o 1. 33

34 Fig. 18. Relaionshi beween and E(τ) Remark 7: If is greaer han 5, hen i s ime consuming (o solve LMIs) o ge sabiliy regions for he delay sysem (more han 13 hours for 8; more han 3 days for 9, and he LMIs are unsolvable if 10 by MALAB), so we shall make some aroximaion on he average delay of he sysem o choose a reasonable average delay,. able 2 gives he reasonable for each. able 2: he aroximaion of E( ) Calculaed o E() Aroximaed ( min, E( ) ) E() Aroximaed (, E( ) ) Calculaed 0 (4, ) (4, ) 0.1 (6, ) 0.94 (4, ) 0.2 (6, 1.093) 1.1 (4, 1.079) 0.3 (8, 1.288) 1.3 (5, 1.27) 0.4 (13, 1.544) 1.5 (5, 1.477) 0.5 (15, 1.895) 1.9 (8, 1.859) 0.6 (22, 2.414) 2.4 (11, 2.371) 0.7 (31, 3.267) 3.3 (20, 3.251) 34

35 0.8 (55, 4.955) 5 (41, 4.95) 0.9 (123, 9.977) 10 (76, 9.951) Fig. 19 is he comarison figure of sabiliy regions which are obained by using min and as he maximum delay of he sysem. From he figure we can see ha he sabiliy regions go by using min and as he delays of he sysem are he same. So a roer can be used insead of min o accelerae he numerical comuaion. Fig. 19. Comarison of sabiliy regions obained by using min and as he maximum delay of he sysem Sabiliy Regions for Single Produc Proce Fig. 20 shows he sabiliy regions of sochasic merology delay sysems wih EWMA-I and II conrollers. Since he measuremen delay of he sysem follows Poisson disribuion wih arameers 1, , , , , , , , , and k 0 for k 8. he is from 0 o 0.5 in 0.1 incremen. he roer is chosen from 35

36 able 2, he ransiion robabiliy marix and he average delay of he sysem are calculaed by (32) and (48) for each. For insance, if 0, hen we should runcae he robabiliy marix P ino a 5 5 (i.e., 4 ) marix P o obain he convergen average delay, E( ) , of he sysem; if 0.3, P should be a marix of 6 6 (i.e., 5 ), E( ) P and In Fig. 20(a), i.e., he sysem wih EWMA-I conroller, i is clear ha wih he increase of, he sabiliy regions decrease. For he sochasic case, he increasing of E() will deeriorae he sabiliy region. However, i is someimes wrong o use ceil( E( )) as he delay of he sysem o ge he sabiliy region, which can be seen from he sabiliy regions obained when 0.1 or 0.5. For he sysem wih EWMA-II conroller, he simulaion resuls in Fig. 20(b), lay ricks wih our inuiion: Firs, he increasing of (or E() ) has minor effecs on he sabiliy of he sysem; second, he sabiliy regions lie beween hose wihou delay and hose wih fixed one run delay desie of wha he (or E() ) is, in oher words, (or E() ) conveys lile informaion abou sabiliy of he sysem wih EWMA-II conroller. Using ceil( E( )) as he delay of he sysem o ge he sabiliy regions will lead o conservaive sabiliy regions for he sysem, 36

37 esecially when E() has a large value. Comaring Fig. 20(a) and (b), we can also noe ha he sysem wih EWMA-I or II conroller is sable for any ransiion robabiliy marices ( or E() ) if 2, when akes value beween 0 o 1. Also, for he same sochasic merology delay, he sysem wih EWMA-II conroller has a bigger sabiliy region comared wih he sysem wih EWMA-I conroller. Fig. 20. Sabiliy regions for sysems wih P calculaed from Poisson disribuion From able 2, i is clear ha alhough he aroximaions on E() are made, we sill canno ge he sabiliy regions for sysems when 0.6 because he LMIs are unsolvable by Malab for large. However, we can affirm ha he sabiliy regions for sysems wih EWMA-II conroller lie beween he sabiliy regions of delay free sysem and hose of he sysem wih fixed one run delay. For he sysem wih EWMA-I conroller, he sabiliy region may lies around he sabiliy region obained by using ceil( E( )) as he fixed delay. As an examle o illusrae our affirmaion, Fig. 21 gives he resonses of sae variables of boh conrollers under he same sochasic merology delay which follows Poisson disribuion wih =1, 0.9 and he corresonding 37

38 E( ) Aroximaed 10. Fig. 21(a) and Fig. 21(b) are he resonses of sae variables of sysem wih EWMA-I conroller under differen, : in Fig. 21(a),, is chosen as 2.6,0.34 under which he sysem wih fixed nine runs delay is unsable, and from he same figure, we can see ha he sae variable is divergen; while in Fig. 21(b),, is 2.6,0.14 which guaranees he sabiliy of he sysem wih fixed en runs delay, also from he same figure, we noice ha he sae variable converges o 0. And he resuls obained from hose wo figures convince us ha we can use he sabiliy region of fixed en runs delay sysem o aroximae he sabiliy region of he sysem wih EWMA-I conroller under his kind of sochasic merology delay. Fig. 21(c) and Fig. 21(d) are he resonses of sae variables of sysem wih EWMA-II conroller wih differen, : he, in Fig. 21(c) is 2.6,0.78 which will cause he delay free sysem o be unsable, and herefore if his air of, is used for he sysem wih EWMA-II conroller, he sysem is unsable by all means (he sae variable divergen, see Fig. 21(c)); in Fig. 21(d),, is chosen such ha he sysem wih fixed one run merology delay is sable, and from he same figure we noe ha he sae variable of he sysem wih EWMA-II conroller is convergen. Also he combinaion of hose wo figures ells us ha if he delay follows Poisson disribuion, no maer wha is, he sabiliy region of he sysem wih EWMA-II conroller always falls beween he sabiliy region of delay free sysem and he sabiliy region of he sysem wih fixed one run delay. 38

39 Fig. 21. Resonses of sae variables when =0.9 I is worh menioning ha for oher kinds of merology delay, he sabiliy region can be go by he heorems roosed in Secion 2. Remark 8: he simulaion resuls in Fig. 20(b) and Fig. 21(d) ell us ha if he ransiion robabiliy marix of he sysem is calculaed from he Poisson disribuion, hen he sabiliy regions of he sysem wih EWMA-II conroller lie beween he regions of sysems wihou delay and hose wih fixed one run merology delay. However, we canno affirm ha he sabiliy region of sysem wih EWMA-II conroller, for any ransiion robabiliy marix, lies beween he regions of delay free sysem and hose of fixed one run delay sysem. he following examles are cases in oin o illusrae his affirmaion. Fig. 22 is he simulaion resuls of he sabiliy regions for sysems wih EWMA-I and II conrollers wih differen ransiion robabiliy marices which are randomly chosen as P , P , P , and P ; he corresonding average delays are E( ) , E( ) , E( ) , and E( ) resecively. From Fig. 22(a), i 39

40 is obvious ha wih he increase of average delay, E(), he sabiliy regions of he sysem wih EWMA-I conroller are decreasing. While his resul holds for sysems wih EWMA-II conroller only rovided ha he sysems are subjec o sochasic merology delay. Using ceil( E( )) as he delay of he sysem will lead o conservaive sabiliy regions for boh conrollers, bu i is much more conservaive o do his for EWMA-II conroller. For he same ransiion robabiliy marix (he same average delay), he sabiliy regions for he sysems wih EWMA-I and II conrollers are comared in Fig. 23. From he figure, we can find ha he sysem wih EWMA-II conroller has a bigger sabiliy region comared wih he sysem wih EWMA-I conroller for he same ransiion robabiliy marix. Also he sysem wih eiher conroller is guaraneed sable for 2 and 0 1. Fig. 22. Sabiliy regions for sysems wih differen ransiion robabiliy marix 40

41 Fig. 23. Comarison of sabiliy regions for sysems wih EWMA-I and II conrollers wih differen ransiion robabiliy marix Conclusions for Sysems wih Sochasic Merology Delay Comaring he resuls obained in subsecion 4.3.2, we can infer he following conclusions: 1. For he sysem wih sochasic merology delay, we have he delayed daa of he sysem, and hese delayed daa deeriorae he sabiliy of he sysem. 2. For sochasic case, he sabiliy region of he sysem wih EWMA-I conroller is decreasing wih he increase of average delay; however one canno always exec o use average delay as a benchmark o deermine he sabiliy region of he sysem wih EWMA-II conroller. 3. Using ceil( E( )) as he delay of he sysem will lead o conservaive sabiliy regions for sysems wih EWMA-II conroller, bu i is someimes wrong o do his for sysems wih EWMA-I conroller. 4. he sysem wih eiher conroller is sable for any beween 0 o 1 if 2. 41

42 5. EWMA Run-o-Run Conrollers in Mixed Produc Process Wha we have discussed in he revious Secions are for he single roduc rocess. However, i is a common racice in oday s semiconducor manufacuring o have many differen roducs and rocesses run on each rocessing ool, i.e., high-mix manufacuring. he engineers are ineresed in he sabiliies of each kind of roduc on he secific ool, i.e., mixed roduc rocess. In his secion, we will exend he heorems obained for single roduc rocess o analyze mixed roduc rocess. We firs esablish he sysem model for mixed roduc rocess, and hen we will discuss he ransiion robabiliy marix for he same kind of roduc. he numerical examles for his kind of rocess are rovided in he las ar of his secion Sysem Model Suose ha a number of roducs are randomly manufacured on a secific ool. aking roduc N as an examle, he run number of roduc N is denoed as n and he corresonding run number of he oal rocess is exressed as n. Fig. 24 gives an examle of mixed roduc rocess. Fig. 24. An examle of mixed roduc rocess Assume ha he inu-ouu relaionshi for roduc N on he secific ool is linear wih sloe N, hen he rocess model is: Y u a (49) n N n n Or 42

43 Y u a (50) N, n N N, n N, n where Y YN, n is he ouu of roduc N a he n h n roducion run, and a n a is he insananeous disurbance a run n. N, n he rediced model for roduc N is Yˆ b u aˆ (51) n N n n Or Yˆ b u aˆ (52) N, n N N, n N, n where b N and a ˆ (or a ˆN, n ) are model gain and offse arameers esimaed for n roduc N. We also suose ha here is a sochasic merology delay = N, n S a run n, wo kinds of EWMA run-o-run conrollers, EWMA-I and II conrollers, can be used o esimae he disurbance: For EWMA-I conroller, he conrol acion is also chosen as aˆ aˆ u or u (53) n N n N N, n N, n bn bn n where N is he arge for roduc N. And he EWMA filer wih discoun facor N for roduc N is aˆ ( Y b u ) (1 ) aˆ n1 N n N n n N n n n or aˆ ( Y b u ) (1 ) aˆ N, n+1 N N, n- N, n N N, n N N, n-n, n Combining (51)-(54), we rewrie EWMA-I conroller in he form of (55), aˆ (1 ) aˆ (1 ) aˆ n1 N n N N n n or aˆ (1 ) aˆ (1 ) aˆ N, n+1 N N, n N N N, n- Nn, (54) (55) where N is he lan-model mismach for roduc N. For he same reason as in he single roduc rocess, in he mixed roduc rocess, EWMA-II conroller should also be considered, i.e., 43

44 aˆ aˆ Nn, +1 n1 (1 ˆ ˆ N) a N(1 ), ; n N an if n n1 n aˆ, oherwise. n or (1 N ) aˆ ˆ N, n N (1 N ) an, n, if,, -1, ; Nn N n N n a ˆ Nn,, oherwise. (56) Augmening he sae variables a run n, as wha we have done in subsecion 2.3, he delay sysem in (55) or (56) can be changed ino a delay free sysem X X (, ) X n1 N n1 n n or (, ) X N, n+1 N N, n-1 N, n N, n (57) where (, ) and N ( N, n-1, N, n) are deermined by which kind of EWMA N n1 n conroller he sysem has adoed. If we do he same analysis as ha of subsecion 2.3.2, (, ) and N ( N, n-1, N, n) are easy o wrie ou. he sabiliy roblem N n1 n of (57) can also be solved by heorem 1 in subsecion Remark 9: If only he same roducs, say roduc N, are manufacured on he same ool, hen n n, n 1 n 1 1, and he model of roduc N in mixed roduc rocess becomes he same as ha of single roduc rocess. I is worh oining ou ha similar analysis can be done for oher roducs ransiion Probabiliy Marix In order o ge he sabiliy regions for mixed roduc rocess, we have o know he ransiion robabiliy marix of he merology delay for each kind of roduc. We also ake roduc N for examle. he robabiliy of roducing his roduc in he oal rocess is q N. For he oal rocess, we use i as he robabiliy of he daa a ime being delayed by i runs and he robabiliy of curren run no being measured is. he disance beween curren run, n, and he las run, n 1, for roduc N, is n n n 1. hen for any roducion run, he robabiliy of he disance beween adjacen roduc N being r is 44

45 r 1 1 P r q q (58) N N Define i as he robabiliy ha he ime delay of he curren run for roduc, which can be esimaed by he following heorem: N is i, P i i n heorem 3: For roduc N, he robabiliy ha he daa of curren run being delayed by i runs is for i 1, and 0 0 bk 1, i 1, q (59) i k N ki 1 ki i 1 N k 1 N N i.where,, 1 robabiliy of roducing i 1 roduc N in k 1 runs. Proof: (A) If he curren run, run n, is measured, hen b k i q C q q is he binomial 0 n 0 0 (60) (B) If he measuremen delay of run n is 1, hen n1 2 n1 d n1 k 1 k 1 k 1 n P P P d P k k n1 k Pn 1 l k 1 lk l1 1 q q k 1 lk k N N 1 q k N k 1 b( k 1,0, q ) k (C) If he measuremen delay of run n is 2, hen we have N (61) 45

46 2 2 n 2 n n 2 P n 3 n n 3 P d d k 2 k 2 n P d n1 n1 n2 P k k k n1 n1 n2 k P n1 l n 1 n2 l k 2 lk 1 k 2 lk k Ck 1 1 qn qn 1 qn qn lk 1 k 2 lk k Ck 1 1 q q 1 q q k 2 lk k 2 k 2 1 k 2 k Ck 1 1 qn qn b k 1,1, q k N N N N N (D) If he measuremen delay of run n is 3, hen we have 3 3 P n1 n2 n1 n2 n3 P n1 n2 n1 n2 n3 k 3 n P d d d n1 n2 n1 n2 n3 P k k k n1 n2 n1 n2 n3 k P n1 n2 l n 1 n2 n3 l k 3 lk k 3 k 3 k 3 2 k 3 2 lk k Ck 1 1 qn qn 1 qn qn lk 2 k 3 2 k Ck 1 1 qn qn b k 1,2, q k N (62) (63) (E) If he measuremen delay of run n is i, hen 46

47 i n i P i i i n1 n2 ni1 n1 n2 ni1 ni P i 1 i 1 i1 n1 n2 ni1 n1 n2 ni1 ni P d d d n1 n2 ni1 n1 n2 ni1 ni P k ki k 2 ki k n1 n2 ni1 n1 n2 ni1 ni i1 k i i-1 lk k Ck 1 1 qn qn 1 qn qn lk i1 ki i1 k Ck 1 1 qn qn b k 1, i 1, q k N k k P n1 n2 ni 1 l n 1 n2 ni 1 ni l k i lk k i (64) And hese consiue he roof. Remark 10: If q 1, he ool only roduce roduc N, i.e., he single roduc N rocess; hen from (59), we know i i. Remark 11: he robabiliies of he curren run being delayed or wihou delay can be normalized: b( k 1, i 1, q ) i 0 i 0 k N i0 i1 i1 k i b( k 1,0, q ) b( k 1,1, q ) b( k 1, d, q ) k N k N k N k 1 k 2 k d 1 b(0,0, q ) b( k 1,1, q ) b( k 1, d, q ) 0 1 d 1 N k N k N k 2 k d 1 1 Remark 12: For roduc N, since we have derived he robabiliies of he merology delay for differen runs, i.e., i, by he same analysis as we do in Secion 3, he ransiion robabiliy marix of merology delay can be easily obained as 47

48 0, i1 j; ij ( 1 j i) (1 ) k, j i 1; k i 1 (1 ) j, 0 j i... (65) Remark 13: he average delay for roduc N, E( ) N can be calculaed by combining (41), (42) and (65). Remark 14: heorem 3 can be alied o oher roducs Numerical Examles In his subsecion, we will firs ake Poisson disribuion as he measuremen delay of roducs as an examle o es he validiy of he heorems we have goen in Secion 5 for mixed roduc rocess. How o choose a roer o ge he convergen average delay will also be discussed. hen, we will esablish he sabiliy resuls for mixed roduc rocess wih he measuremen delay following Poisson disribuion. I is worh menioning ha for oher robabiliy disribuions, by using he heorems in his aer, he sabiliy regions will be obained by redoing he comuaions ransiion Probabiliy Marix Calculaed from Poisson disribuion Suose ha wo kinds of roducs, roduc M and N, are randomly manufacured on he same ool wih he robabiliy qm 0.3 and qn 0.7. We assume ha he merology delay of he real manufacuring rocess follows Poisson disribuion wih mean arameer 1, and he robabiliy of he curren run no be measured is which is chosen from 0 o 0.9 in 0.1 incremens. Based on he calculaion rocess discussed in able Poisson random numbers are generaed, as he original measuremen delays for boh roducs, o ge he resamled delay numbers. he simulaions are done 20 imes for each o obain he ransiion robabiliy marix, P M for roduc M and 48 P N for roduc N, as well

49 as he average delay E( ) M for roduc M and E( ) N for roduc N. During he simulaions, P M and P N are runcaed ino 3 3 marices in each calculaion. Fig. 25 and Fig. 26 are he simulaed resuls for each elemen of P M and P N, as well as E( ) M and E( ) N. From he figure, i is clear ha for he same roduc, boh he ransiion robabiliy marix and he average delay of he sysem which are calculaed from he simulaions are close o hose obained from heorem 2 and heorem 3, and his verifies heorem 2 and heorem 3. Fig. 25. Simulaion es of P M and E(τ) M Fig. 26. Simulaion es of P N and E(τ) N 49

50 Fig. 27 and Fig. 28 show he relaionshis beween runcaion of he ransiion robabiliy marix and he average delay of he sysem when qm 0.3 and qn 0.7 resecively. he comuaions in hese simulaions are based on (58), (59) couled wih (40)-(42). From he figure, we know ha min is increasing wih he increase of, o make he average delay of he sysem converge o is limiaion. Also he same as he single roduc rocess, in he mixed roduc rocess, he relaionshi beween and he average delay of he sysem, E(), is nonlinear, and he nonlineariy becomes srong esecially when aroaches o 1. Fig. 27. Relaionshi beween rancaion of he ransiion robabiliy marix and average delay of he sysem when qm

51 Fig. 28. Relaionshi beween rancaion of he ransiion robabiliy marix and average delay of he sysem when qn 0.7 In order o accelerae he comuaion, some aroximaions on he average delay of he sysem are made. able 3 and able 4 give he reasonable for each when q 0.3 and M qn 0.7 resecively. Comaring hese wo ables, we know ha for he same, he more he roduc are manufacured, he larger he average delay of he roduc is. For examle, when 0.3, he average delay for roduc M is E( ) which is smaller han E( ) N Comaring able 2, able 3 and able 4, we can find ha for he same kind of merology delay, each roduc in mixed roduc rocess enjoys less average delay han he single roduc rocess. M able 3: he aroximaion of E() Calculaed o E() Aroximaed, qm 0.3 ( min, E( ) ) E() Aroximaed (, E( ) Calculaed ) 0 (3, ) 0.70 (3, ) 0.1 (6, 0.811) 0.81 (3, ) 0.2 (8, ) 0.95 (4, 0.95) 0.3 (8, 1.137) 1.13 (5, 1.125) 0.4 (11, 1.38) 1.4 (6, 1.356) 0.5 (15, 1.718) 1.7 (7, 1.666) 0.6 (23, 2.223) 2.2 (10, 2.165) 51