IEEE/SEMI Advanced Semiconductor Manufacturing Conference Cycle Time Approximations for the G/G/m Queue Subject to Server Failures and Cycle Time Offsets with Applications James R. Morrison Central Michigan University Department of Engineering and Technology Donald P. Martin Productivity Engineering IBM, Systems and Technology Group ASMC 2006 – Boston, Massachusetts May 22-24, 2006 1
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Presentation Overview • Standard approximations for the mean cycle time in a G/G/m queue • Extensions to the G/G/m queue: Idle with WIP • Extensions to the G/G/m queue: Cycle time offsets • Extensions to the G/G/m queue: Defection of lots from failed servers • Application to toolsets in IBM’s 200mm semiconductor fabricator • Concluding remarks ASMC 2006 – Boston, Massachusetts May 22-24, 2006 2
IEEE/SEMI Advanced Semiconductor Manufacturing Conference The G/G/m Queue • G/G/m queue contains – m equivalent tools – Random service times (General distribution with mean 1/ m) – Random time between lot arrivals (General distribution, mean 1/ l) – System loading r = l / ( m m ) (Utilization of capacity) m servers Queue Completed Lots lots arrive exit ASMC 2006 – Boston, Massachusetts May 22-24, 2006 3
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Cycle Time Approximations • Popular approximation for the G/G/m queue + + + r 2 2 1 2 m 2 ( ) 1 1 C C Appears in the text + S A E CT ( ) Factory Physics m m r 2 m 1 Grows as Service Variation in loading time the system increases where (1/ m ) is the mean service time – – C S is the coefficient of variation of the service time (std/mean) – C A is the coefficient of variation of the interarrival time (std/mean) – m is the number of servers System loading r = ( l / m m ) < 1 – ASMC 2006 – Boston, Massachusetts May 22-24, 2006 4
IEEE/SEMI Advanced Semiconductor Manufacturing Conference The Approximation is Exact in Some Cases Normalized Cycle Time Comparison: • For the M/G/1 queue this M/D/1 vs M/M/1 expression is exact 8 7 + r 2 ( ) 1 1 1 C Normalized Cycle Time = + 6 S E CT ( ) m m r Increasing variation 2 1 5 or fewer tools 4 3 (1/ m ) is the mean service time • 2 System loading r = ( l / m ) < 1 • 1 • C S is the coefficient of variation 0 of the service time 0 0.2 0.4 0.6 0.8 1 Loading (Utilization of Capacity) ASMC 2006 – Boston, Massachusetts May 22-24, 2006 5
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Approximations Incorporating Tool Failure • A tool may be subject to random failures – Time to failure is exponentially distributed (mean m F ) Time to repair is generally distributed (m R , s R and C R = s R /m R ) – – Mean availability is A = m F / (m R + m F ) – Production is resumed following repair • Popular approximation which is exact for the M/G/1 queue with failures + r 2 2 * C C ( ) 1 1 + Appears in the text S , E A E CT m m r Factory Physics * * * 2 1 where m * = m A – Service time ( ) ( ) = + + m – 2 2 2 2 - effective C S C C 1 C 1 A Am is inflated S , E S R R C A = s A / ( 1 / m ), coefficient of variation of interarrival time by availability – System loading r * = l / ( m A ) < 1 – ASMC 2006 – Boston, Massachusetts May 22-24, 2006 6
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Approximations for G/G/m Queue With Tool Failures • Consider G/G/m queue with exponential time to failure for each tool Time to repair is generally distributed (m R , s R and C R = s R /m R ) – – Mean availability is A = m F / (m R + m F ) – Lots remain with the failed server and production resumes upon repair • Natural generalization suggested by the previous approximations ( ) + + + r 1 2 m 2 2 2 * C C Can be inferred ( ) 1 1 + S , E A ( ) E CT from the text m m r * * * 2 m 1 Factory Physics where m * = m A – ( ) ( ) = + + m – Service time 2 2 2 2 - effective C S C C 1 C 1 A Am S , E S R R is inflated C A = s A / ( 1 / m ), coefficient of variation of interarrival time – by availability System loading r * = l / ( m m A ) < 1 – ASMC 2006 – Boston, Massachusetts May 22-24, 2006 7
IEEE/SEMI Advanced Semiconductor Manufacturing Conference M/M/2 Queue Subject to Tool Failure • Comparison of the A Comparison of Approximate and Exact Cycle Time Performan M/M/2 Queue with Random Failure and Repair approximation with exact results for the M/M/2 queue 10 9 – Exponential repair Normalized Cycle Time (XF) 8 – m F = 16 hours 7 – 6 m R = 4 hours 5 Process time (1 / m ) = 1 hour – 4 3 2 • The simpler intuitive Martin 1 approximation is obtained by 0 substituting 0 0.2 0.4 0.6 0.8 1 Utilization of Capacity ( ) ( ) + + m r r 1 2 m 2 * * M/M/2 Exact Normalized Cycle Time - E(CT)/(1/mu) Martin Style Approximation ( ) ( ) Factory Physics Style Approximation r m r * * m 1 1 MTTF = 16 h, MTTR = 4 h, process time = 1 h (all exponential) ASMC 2006 – Boston, Massachusetts May 22-24, 2006 8
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Idle Tools in the Presence of WIP • A tool may be idle even in the presence of WIP – Loading time – Operator unavailable • Model the idle with WIP as a random addition to the process time (mean W and standard deviation s W ) ( ) + + r 2 2 2 ( 1 ) 1 m C C 1 1 * + , S E A E ( CT ) ) , ( Loading is m m r 2 1 * m increased e e Production speed is reduced where r* = l(W + 1/m) /(m A ) < 1 1 s + s W 2 2 m 1 = W + + m + 2 2 S R C ( 1 C )( 1 A ) A ( ) ( ) m + W m , m + W S E 2 R e 1 / 1 / A A ASMC 2006 – Boston, Massachusetts May 22-24, 2006 9
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Cycle Time Offsets • Common manufacturing events include: – Transport of lots from one toolset to another – Hold of lots pending resolution of a process concern – Post production delay • Often independent of the queue at a particular toolset ( ) + + r 2 2 2 ( m 1 ) 1 C C 1 1 * + + + + S , E A ( ) ) , E CT T H P ( m m r 2 m 1 * e e where – T is the mean transport time for lots arriving to the toolset – H is the mean time that lots are on hold before release to the queue – P is the mean post production delay before transport to the next toolset ASMC 2006 – Boston, Massachusetts May 22-24, 2006 10
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Loyalty to a Failed Tool • Recall that lots were assumed to remain with a failed tool once they begin production at that tool • If the tools exhibit 80% availability the approximation yields ( ) + + + 1 2 2 r m 2 2 * C C ( ) 1 . 25 1 . 25 + S , E A ( ) E CT m m r * 2 1 m Even in very low loading conditions ( r * = 0) • ( ) 1 E CT 1 . 25 m • Inappropriate model for some toolsets as lots may defect from a failed server in favor of an available one! ASMC 2006 – Boston, Massachusetts May 22-24, 2006 11
IEEE/SEMI Advanced Semiconductor Manufacturing Conference Defection of Lots From a Failed Tool • Suppose lots are allowed to defect to another tool in the event that their production is interrupted by tool failure In very low loading conditions ( r * = 0) • – Service time may continue uninterrupted if another tool is up – Only if all tools have failed will the service be delayed – Roughly expect (with deterministic repair times) 1 m + m R lim E ( CT ) ( 1 A ) m + + r m 1 0 Residual down time Probability that an when all tools fail arriving lot sees all tools in failure ASMC 2006 – Boston, Massachusetts May 22-24, 2006 12
IEEE/SEMI Advanced Semiconductor Manufacturing Conference General Cycle Time Approximation • For the G/G/m queue, incorporating – Failure prone tools with deterministic repair times – Idle with WIP – Cycle time offsets – Defection of lots from failed servers + + ( ) ( ) E CT T H P Cycle time offsets ( ) m + m R 1 A All tools fail + m 1 1 + + W Process time m ( ) + + r 2 2 2 ( m 1 ) 1 C C 1 * + + W , S E A ( ) Queueing m r 2 1 * m ASMC 2006 – Boston, Massachusetts May 22-24, 2006 13
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