Control of Job Arrivals with Processing Time Windows into Batch Processor Buffer

Transcription

1 onrol of Job Arrivals wih Processing ime Windows ino Bach Processor John Benedic. AJAN 1, Aa Iyer SIVAKUMAR 1, and Sanley B. GERSHWIN 2 1 Nanyang echnological Universiy 2 Massachuses Insiue of echnology Absrac onsider a wo-sage manufacuring sysem comosed of a bach rocessor and is usream feeder rocessor. Jobs exi he feeder rocessor and join a queue in fron of he bach rocessor, where hey wai o be rocessed. he bach rocessor has a finie caaciy Q, and he rocessing ime is indeenden of he number of jobs loaded ino he bach rocessor. In cerain manufacuring sysems (including semiconducor wafer fabricaion), a rocessing ime window exiss from he ime he job exis he feeder rocessor ill he ime i eners he bach rocessor. If he bach rocessor has no sared rocessing a job wihin he job s rocessing ime window, he job canno roceed wihou undergoing rework or validaion by rocess engineers. We generalize his scenario by assigning a reward R for each successfully rocessed job by he feeder rocessor, and a cos for each job ha exceeds is rocessing ime window wihou being rocessed by he bach rocessor. We examine a roblem where he feeder rocessor has a deerminisic rocessing ime and he bach rocessor has sochasic rocessing ime, and deermine ha he oimal conrol olicy a he feeder rocessor is insensiive o wheher he bach rocessor is under no-idling or full-bach olicy. unis; his job may need o be rerocessed, or revalidaed by rocess exers before he job can coninue o be rocessed. We refer o his scenario as he job exceeding is rocessing ime window a he bach rocessor. his is unwaned for several reasons. Firsly, a furnace s feeder rocessor frequenly also serves as he feeder rocessor for a differen rocessor M. Each job ha exceeds is rocessing ime window is equivalen o los hroughu for rocessor M, wihou increasing he furnace s hroughu. Secondly, a job ha exceeds is rocessing ime window incurs he coss of rework, or validaion by rocess engineers. Increased rework or validaion also increases he cycle ime for a leas he exired jobs, which is undesirable, due o he larger yield losses correlaed wih higher cycle imes. Figure 1 shows a subsecion of a wafer fab rocess flow involving he furnace and is feeder rocessors. he furnace can have more han one feeder rocessor, and he feeder rocessors yically feed more han one rocessor. lean Ech Index erms Wafer Fabricaion, Oimal onrol, Bach Processor, Processing Window I. INRODUION n ignored face of conrolling diffusion furnaces in A wafer fabs is he resence of rocessing ime windows for jobs ha are queuing in fron of he bach rocessor. Uon exiing he rocessor immediaely usream of he bach rocessor (which we call he feeder rocessor), a job is deemed o be conaminaed if i has waied in fron of he bach rocessor for a leas ime Manuscri received November 20, his work was suored by he Singaore-MI Alliance (SMA), Singaore. J.. ajan is a PhD suden wih he Singaore-MI Alliance under he Innovaions in Manufacuring Sysems and echnology rogram (hone: (65) ; aja0001@ nu.edu.sg) A. I. Sivakumar is an Associae Professor wih he Mechanical and Aerosace Engineering Dearmen, Nanyang echnological Universiy, Singaore. ( msiva@nu.edu.sg) S. B. Gershwin is a Senior Research Scienis wih he Mechanical Engineering Dearmen, Massachuses Insiue of echnology, MA, USA. ( sbgershwin@mi.edu). hin Film Diffusion Furnace Sri Figure 1: A subsecion of a rocess flow inside he wafer fab. he diffusion furnace is fed work-in-rogress by eiher lean or Ech. However, boh lean and Ech can also rocess jobs o feed oher rocessors. he lean saion can also rocess jobs for hin Film, while Ech can also rocess jobs for Sri. he conrol of he feeder rocessor, in accordance wih he exisence of rocessing ime windows a he bach rocessor is a roblem ha has been largely ignored by researchers. We address he roblem of conrolling job flow ino he bach rocessor queue, considering he rocessing ime window resen in jobs.

2 here are oher sysems in which rocessing ime windows are resen. Processing ime windows also exis in manufacuring and ransor sysems where he roduc is erishable, for examle. When here is a considerable amoun of roducs waiing o be ransored, i may be oimal for he manufacuring faciliy o so rocessing roducs wih finie shelf life. II. LIERAURE REVIEW Bach rocessors, as covered in exising lieraure, can be broadly classified ino wo yes: bach rocessors wih comaible jobs, and bach rocessors wih incomaible jobs. For bach rocessors wih comaible jobs, he comosiion of a bach is consrained only by he caaciy of he bach rocessor; however, he rocessing ime of he bach rocessor is se o be he maximum of he minimum rocessing ime requiremens for all he jobs inside a aricular job. he burn-in oven used o simulae reeaed usage of semiconducor chis is a good examle of bach rocessors wih comaible jobs. On he oher hand, diffusion and oxidaion ovens are bach rocessors wih incomaible jobs. Each job belongs o a aricular job family, and only jobs from he same family can be bached ogeher. Processing ime of he bach is only deenden on he job family being rocessed. For his roblem, we make he simlifying assumion ha all jobs belong o he same job family. For breviy, we will simly refer he ineresed reader o a review of bach rocessor conrol lieraure by [Mahirajan and Sivakumar 2006]. If we know he arrival imes of each job ino he bach rocessor queue, hen i is ossible o incororae rocessing ime windows ino he analysis by assuming a rocessing sequence a he feeder rocessor, and if feeder rocessor rocessing imes are deerminisic, he arrival imes for each job o he queue in fron of he bach rocessor can be generaed. he due imes for each job can hen be generaed, and we can hen minimize he number of ardy jobs. o he bes of our knowledge, only [Makis 1985] has exlicily considered he rocessing ime window of jobs ha are being rocessed by a bach rocessor. He looked ino he roblem of conrolling a bach rocessor wih infinie caaciy. he sysem has an affine service cos funcion, and he holding cos funcion is nonnegaive. Jobs have o be rocessed wihin a rocessing ime window. He deermined ha he oimal olicy is a hreshold olicy, and roceeds o examine he seady-sae characerisics of he sysem under he oimal olicy. Our work differs from wha has been done rimarily in he locaion of he conrol. Previous research reas he arrival imes of jobs o be a consrain, and aem o oimize he bach rocessor conrol olicy given his se of consrains. We looked ino conrolling he feeder rocessor direcly usream of he bach rocessor. his is because even an oimal olicy a he bach rocessor can erform badly if here is an exremely large amoun of jobs queued u in fron of i, due o a sub-oimal feeder conrol olicy. III. PROBLEM SAEMEN A wo-sage manufacuring sysem is comrised of an usream feeder rocessor feeding a downsream bach rocessor via a buffer of infinie caaciy. he buffer level does no include he jobs ha are being rocessed by he bach rocessor. In wafer fabricaion, he downsream bach rocessor corresonds o a diffusion oven, while he feeder rocessor can be eiher a serial rocessor (ech saion) or a bach rocessor wih caaciy wo (clean saion). We iniially assume ha he feeder rocessor is a serial rocessor. he bach rocessor can rocess u o Q jobs a a ime, and is rocessing ime is indeenden of he number of jobs loaded ino he bach rocessor. Usream of he serial rocessor is an infinie source of jobs, and downsream of he bach rocessor is an infinie sink for jobs. Boh rocessors canno be reemed. See Figure for an illusraion of he manufacuring sysem model. he manufacuring sysem is assumed o oerae on a discree ime scale. Infinie source of jobs Serial rocessor 1 Bach rocessor (caaciy Q) Infinie sink for jobs Figure 2: Simle 2-sage manufacuring model. Wihou a cos associaed wih jobs exceeding heir rocessing ime windows, he oimal olicy a he feeder rocessor (in his case, he feeder rocessor is iniially assumed o be a serial rocessor) would be o rocess a job a each insance he feeder rocessor is available. here is a rocessing ime window beween he bach rocessor and is feeder rocessor. If a job has no sared rocessing a he bach rocessor ime unis afer i has exied he feeder rocessor, a cos is incurred. his cos quanifies los hroughu a he feeder rocessor, yield loss due o longer cycle ime and oher ancillary effecs of rejecing a job. For each job ha he feeder rocessor rocesses (including jobs ha end u exceeding he rocessing ime window), a reward R is generaed. his reward is he incremenal benefi of rocessing a job for he bach rocessor, as comared o feeding a differen

3 rocessor. We wish o deermine when i is aroriae o rocess a he feeder rocessor, given he number of jobs in fron of he bach rocessor, and he saus of he bach rocessor. I is assumed ha >R, oherwise, i would be always rofiable o rocess a job, regardless of he robabiliy of he job exceeding is rocessing ime window. IV. PROBLEM MEHODOLOGY We look a he robabiliy ha he rosecive job a he serial rocessor will have o queue for a leas ime unis in fron of he bach rocessor before i can sar rocessing a he bach rocessor. We obain exressions for he execed reward of rocessing a job, and so rocessing a he feeder rocessor when i is no longer rofiable o rocess a job, given he reward R and he ossibiliy of incurring a cos. A. Assumed Bach Processor onrol Policies We assume ha he bach rocessor is oeraed according o wo olicies: a.) no-idling olicy whenever he bach rocessor is idle and here are jobs waiing in fron of he bach rocessor, he bach rocessor will immediaely sar a bach, regardless of he number of jobs ha can be rocessed. his is equivalen o a minimum bach size (MBS) olicy wih X=1. b.) full-bach olicy a bach is sared only if he bach rocessor is idle and a full bach can be rocessed. his is equivalen o an MBS olicy wih X=Q. Analysis of he roblem when he bach rocessor is under full-bach olicy enails esimaing when addiional jobs afer he rosecive job is rocessed will arrive. We make he assumion ha he feeder rocessor will coninue o rocess jobs unil i is old o so. V. MODEL ONE ASSUMPIONS AND DEVELOPMEN he following assumions are made in model one: Boh machines erfecly reliable Serial rocessor rocessing ime is deerminisic. I akes one ime uni o rocess one job a he serial rocessor. Bach rocessor rocessing ime is geomerically disribued. Probabiliy (bach being finished in curren ime insance) = P. A any ime insance, le Z be he curren buffer level in fron of he bach rocessor, and S be he curren sae of he bach rocessor, wih one meaning he bach rocessor is busy and zero meaning he bach rocessor is idle. hen, he sae of he sysem X can be described as X = (Z, S ). he serial rocessor akes one ime uni o rocess a job. We can omi racking he sae of he serial rocessor, as i is guaraneed o be available a he sar of each ime insance. A. When he bach rocessor is under no-idling olicy We divide he model develomen ino hree differen cases, based on he sae of he sysem. ase 1: X = (0, 0) he rosecive job a he serial rocessor will have zero queue ime. Once i eners he bach rocessor queue, i is immediaely rocessed by he bach rocessor. In his case, i is rofiable o rocess he job. ase 2: X = (0 o Q-1, 1) We rocess he rosecive job only if R>*Probabiliy (curren bach will ake a leas +1 unis o rocess). We rocess he rosecive job only if: R > ( 1 ) (1) ase 3: X = (>Q, 1) Le x be he larges ineger less han or equal o x. (onversely, le x be he smalles ineger greaer han or equal o x.) he number of baches he bach rocessor needs o rocess before he rosecive job can be loaded Z is + 1. he rosecive job should be rocessed Z only if: R > * Pr obabiliy( + 1 baches will ake a leas +1 unis o rocess). he robabiliy ha Z +1 baches will be rocessed in exacly x ime unis has a Pascal disribuion. We sum u he robabiliies of Z +1 baches being rocessed in exacly x ime unis, Z for + 1<x< o obain he robabiliy of no incurring a cos, and his is subraced from uniy o obain he robabiliy of incurring a cos. x Le x! y =, hen ( y!)( x y)! Z Z + 1 i 1 R i 1 ( )( P). Z Z + 1 (2 ) Z > is a necessary condiion for he rosecive job o have a osiive robabiliy of avoiding a cos. Since he

4 sysem runs in discree ime, i will ake a leas + 1 ime unis o rocess all he baches in fron of he incoming job (including he bach currenly being rocessed by he bach rocessor). he serial rocessor akes one ime uni o rocess he bach. he rosecive Z job will be joining bach + 2. B. When he Bach Processor is under Full-Bach Policy When he bach rocessor is under full-bach olicy, i is only rofiable o rocess a full bach if he ime i akes o form a full bach will no cause a job o exceed is rocessing ime window. hus, a necessary condiion for he full-bach olicy o be rofiable is >Q-1. If <Q-1, hen i is no rofiable o form a full bach, regardless of how many jobs are queued u in fron of he bach rocessor. >Q-1 will be assumed hroughou he hree cases. We consider hree differen cases: ase 1: X = (0 o Q-1, 0) he rosecive job will have queue ime equal o he ime i akes o form a full bach. Given our addiional assumion ha >Q-1, hen we will always choose o rocess under case 1. ase 2: X = (0 o Q-1, 1) he rosecive job can incur rocessing ime in wo ways. Firsly, he curren bach may no be finished before he job arrives a he buffer. Secondly, he job may be forced o wai for addiional jobs o arrive in order o form a full bach. Prosecive job s waiing ime = MAX (rocessing ime of he curren bach-rosecive job s rocessing ime a he serial rocessor, ime rosecive job has o wai o form a full bach) Prosecive job should be rocessed only if: >Q-Z -1 (his is covered by revious assumion on he rofiabiliy of he full-bach olicy) AND R>*Probabiliy (curren bach will ake a leas +1 ime unis o rocess). Prosecive job should be rocessed only if: R > ( 1 ) (3) ase 3: X = (>Q, 1) Waiing ime for rosecive job = MAX (rocessing ime of all full baches ahead of rosecive job-rosecive job s rocessing ime a he serial rocessor, ime rosecive job has o wai o form a full bach). Le B be he number of addiional jobs he rosecive job will have o wai for o form a full bach. Le Z Z ( Z \ Q) = Z * Q Q. If (Z \Q)=0, hen he rosecive job sars a new bach, and B =Q-1. Oherwise, B =Q-(Z \Q)-1. Prosecive job should be rocessed only if: Z >B AND R > * robabiliy( + 1 baches will ake a leas +1 unis o rocess) R (4) Z + 1 ( i 1 Z )( P) Z + 1 VI. MODEL ONE ANALYSIS Z i 1 For model one, he oimal olicy will be idenical regardless of wheher he bach rocessor olicy is noidling or full-bach, assuming ha >Q-1. he oimal olicy is: If he bach rocessor is idle, always rocess he rosecive job Else If here is less han a full bach in fron of he bach rocessor and he bach rocessor is busy, rocess he rosecive job if R > ( 1 ). If his condiion is no me, hen no job should wai in fron of he bach rocessor when he bach rocessor is no idle. Else here is a leas one full bach in fron of he bach rocessor. Process he rosecive job if: Z Z R + 1 i 1 i 1 ( )( P). Z Z + 1 he oimal olicy a he feeder rocessor is a hreshold olicy. his is because he robabiliy of a rosecive job incurring a queuing ime of a leas ime unis is a nondecreasing funcion wih resec o he number of jobs in fron of he bach rocessor. Figure 3 illusraes a yical grah of he robabiliy of he rosecive job incurring a cos, wih resec o he number of jobs in fron of he bach rocessor. he robabiliy of a job incurring a cos is a se funcion, wih resec o he number of baches in fron of i. If we know ha i is rofiable o rocess a job when here are n*q jobs in he buffer (his means ha he rosecive job will end u saring a new bach), hen we immediaely know ha i is also rofiable o rocess a job when he number of jobs in he buffer are from n*q+1 o (n+1)*q-1. his simlifies he search for he hreshold

5 value as we only need o es for hreshold values ha are ineger muliles of Q. robabiliy Probabiliy of job incurring cos number of jobs in fron of rosecive job Figure 3: Probabiliy of he rosecive job incurring a cos when he bach rocessor is under no-idling olicy, =5, Q=5 and P = 1/3, assuming ha he bach rocessor is already busy. When he bach rocessor is idle, he robabiliy of incurring a cos is zero. he arrows oin o he insances when he rosecive job will form a new bach. When he job forms a 6 h bach queuing in fron of he bach rocessor, he robabiliy of incurring a cos becomes 1. Since he bach rocessor is busy, he sysem will have o rocess 6 baches in =5 ime insances for he rosecive job no o incur a cos, and his is no ossible, due o he roblem assumions. bach rocessor is under eiher a full-bach or a no-idling olicy. Furhermore, we have shown ha he oimal conrol olicy is a hreshold olicy, wih he hreshold being an ineger mulile of he bach rocessor caaciy. Given a rofiabiliy assumion on he full bach olicy, he oimal olicy is idenical for he wo bach rocessor conrol olicies. Unforunaely, hese roeries rarely remain rue when he model assumions are changed. For examle, when he serial rocessor rocessing ime remains deerminisic bu becomes larger han one, boh roeries no longer hold rue. his is looked ino in furher deail in fuure work. REFERENES [1] Mahirajan M. and Sivakumar A.I., "A lieraure review, classificaion and simle mea-analysis on scheduling of bach rocessors in semiconducor," Inernaional Journal of Advanced Manufacuring echnology 29 (2006) [2] Makis, V., "Oimal conrol of a bach service queueing sysem wih bounded waiing ime," Kyberneika 21 (1985) VII. EXENSIONS O OHER MODELS wo qualiies of he oimal olicy for Model 1 are highly desirable: (1) he oimal hreshold is an ineger number of he bach rocessor caaciy, and (2) he oimal olicy is insensiive o he bach rocessor olicy. Unforunaely, boh of hese roeries no longer hold when he assumions are slighly changed. For examle, when he rocessing ime of he serial rocessor is deerminisic bu larger han one, boh roeries are no longer rue. his is furher discussed in oher work. VIII. ONLUSION An imoran subsysem inside he wafer fab is he oxidaion/diffusion oven, a bach rocessor, and is feeder rocessor. An imoran face of scheduling sysems wih bach rocessors ha exiss in wafer fabs is he exisence of rocessing ime windows. Jobs exiing he feeder rocessor yically have o be rocessed by he bach rocessor wihin a cerain eriod of ime afer i has exied he feeder rocessor. Jobs ha exceed heir rocessing ime windows may have o undergo rework or validaion, due o conaminaion concerns. We generalize his scenario by associaing a reward R wih each job rocessed by he serial rocessor, and a cos wih each job ha exceeds is rocessing ime window. We use a wo-sage model, wih a serial rocessor feeding a bach rocessor via an infinie buffer, and assume his wo-sage sysem has an infinie source and an infinie sink. Problem 1 assumed a deerminisic serial rocessor rocessing ime, wih a geomerically disribued bach rocessor rocessing ime. We derived mehods o obain he oimal conrol olicy a he feeder rocessor, when he