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Cycle Time Approximations for the G/G/m Queue Subject to Server - - PowerPoint PPT Presentation

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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

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SLIDE 1

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

1

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

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SLIDE 2

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

2

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
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SLIDE 3

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

3

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)

Lots arrive Queue m servers Completed lots exit

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SLIDE 4

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

4

Cycle Time Approximations

  • Popular approximation for the G/G/m queue

( ) ( ) 

                + + 

+ + 

r r m m 1 2 1 1

2 2 1 2 2

m C C CT E

m A S

Appears in the text Factory Physics Service time Variation in the system Grows as loading increases

where

– (1/m) is the mean service time – CS is the coefficient of variation of the service time (std/mean) – CA is the coefficient of variation of the interarrival time (std/mean) – m is the number of servers – System loading r = ( l / m m ) < 1

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SLIDE 5

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

5

The Approximation is Exact in Some Cases

  • For the M/G/1 queue this

expression is exact

  • (1/m) is the mean service time
  • System loading r = ( l / m ) < 1
  • CS is the coefficient of variation
  • f the service time

( ) ( )

r r m m          + + = 1 2 1 1 1

2 S

C CT E

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1

Normalized Cycle Time Loading (Utilization of Capacity)

Normalized Cycle Time Comparison: M/D/1 vs M/M/1 Increasing variation

  • r fewer tools
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SLIDE 6

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

6

Approximations Incorporating Tool Failure

  • Popular approximation which is exact for the M/G/1 queue with failures

( )

* * 2 2 , * *

1 2 1 1 r r m m          + + 

A E S

C C CT E

Appears in the text Factory Physics

where

– m* = m A –

  • effective CS

2

– CA = sA / ( 1 / m ), coefficient of variation of interarrival time – System loading r* = l / ( m A ) < 1

  • A tool may be subject to random failures

– Time to failure is exponentially distributed (mean mF) – Time to repair is generally distributed (mR, sR and CR = sR/mR) – Mean availability is A = mF / (mR + mF) – Production is resumed following repair

( )(

)

m

R R S E S

Am A C C C  + + = 1 1

2 2 2 ,

Service time is inflated by availability

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SLIDE 7

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

7

Approximations for G/G/m Queue With Tool Failures

  • Natural generalization suggested by the previous approximations

( )

( ) ( )

* 2 2 1 * 2 2 , * *

1 2 1 1 r r m m          + + 

+ + 

m C C CT E

m A E S

where

– m* = m A –

  • effective CS

2

– CA = sA / ( 1 / m ), coefficient of variation of interarrival time – System loading r* = l / ( m m A ) < 1

  • Consider G/G/m queue with exponential time to failure for each tool

– Time to repair is generally distributed (mR, sR and CR = sR/mR) – Mean availability is A = mF / (mR + mF) – Lots remain with the failed server and production resumes upon repair

( )(

)

m

R R S E S

Am A C C C  + + = 1 1

2 2 2 ,

Can be inferred from the text Factory Physics

Service time is inflated by availability

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SLIDE 8

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

8

M/M/2 Queue Subject to Tool Failure

  • Comparison of the

approximation with exact results for the M/M/2 queue

– Exponential repair – mF = 16 hours – mR = 4 hours – Process time (1 / m ) = 1 hour

  • The simpler intuitive Martin

approximation is obtained by substituting

0.2 0.4 0.6 0.8 1 Utilization of Capacity

1 2 3 4 5 6 7 8 9 10

Normalized Cycle Time (XF) M/M/2 Exact Normalized Cycle Time - E(CT)/(1/mu) Factory Physics Style Approximation Martin Style Approximation MTTF = 16 h, MTTR = 4 h, process time = 1 h (all exponential)

A Comparison of Approximate and Exact Cycle Time Performan

M/M/2 Queue with Random Failure and Repair

( ) ( )

* 2 2 1 *

1 r r 

+ + 

m

m

( ) ( )

m m * *

1 r r 

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SLIDE 9

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

9

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 sW)

( ) ( ) ,

* 1 * 2 1 1 ) (

1 ) 1 ( 2 2 2 ,

r r m m          + + 

 +

m C C CT E

m A E S e e

( )   ( )

1 2 2 2 2 2 ,

1 / 1 ) 1 )( 1 ( / 1

 W

      W +        W +  + + W + + = A A m A A C C

e R R S E S

m m m m s s  

where r* = l(W + 1/m)/(mA) < 1

Production speed is reduced Loading is increased

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SLIDE 10

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

10

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

( ) ( ) ,

* 1 * 2 1 1 ) (

1 ) 1 ( 2 2 2 ,

r r m m          + + + + + 

 +

m C C P H T CT E

m A E S 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

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SLIDE 11

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

11

Loyalty to a Failed Tool

  • Recall that lots were assumed to remain with a failed tool
  • nce they begin production at that tool
  • If the tools exhibit 80% availability the approximation yields
  • Even in very low loading conditions (r* = 0)
  • Inappropriate model for some toolsets as lots may defect

from a failed server in favor of an available one!

( )

( ) ( )

* 2 2 1 * 2 2 ,

1 2 25 . 1 25 . 1 r r m m          + + 

+ + 

m C C CT E

m A E S

( )

         m 1 25 . 1 CT E

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SLIDE 12

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

12

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 ) 1 ( 1 ) ( lim +  + 

+

m m A CT E

R m

m

r

Residual down time when all tools fail Probability that an arriving lot sees all tools in failure

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SLIDE 13

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

13

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

( ) ( ) ( )

* 1 * 2 1 1 1 1 ) ( ) (

1 ) 1 ( 2 2 2 ,

r r m m          +         W + +         W + +       +  + + + 

 +

m C C m m A P H T CT E

m A E S R m

Cycle time offsets All tools fail Process time Queueing

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SLIDE 14

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

14

Alternate 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

( ) ( ) ( )

m m A E S R m

C C m m A P H T CT E * 1 * 2 1 1 1 1 ) ( ) (

2 2 ,

r r m m          +         W + +         W + +       +  + + + 

Intuitive & simpler queueing approximation Cycle time offsets All tools fail Process time

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SLIDE 15

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

15

Applying the Cycle Time Approximation: First Example

  • For a tool set operating in

IBM’s 200mm fabricator

  • For a biweekly period:

– Measure statistics (CA

2, A,

r*, W, T, …) – Measure actual cycle time performance – Compare!

  • Dominant factors

– Cycle time offsets – Idle with WIP – Low loading

Measured and Predicted Mean Cycle Time

1 2 3 4 5 6 7 20 40 60 80 100 Loading (Utilization of Capacity) Normalized Cycle Time (X Factor) Actual X Factor Standard Curve (M/D/1) Cycle Time Prediction Approximation

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SLIDE 16

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

16

Applying the Cycle Time Approximation: Second Example

Measured and Predicted Mean Cycle Time

1 2 3 4 5 6 7 0% 20% 40% 60% 80% 100%

Loading (Utilization of Capacity) Normalized Cycle Time (X Factor) Actual X Factor Standard Curve (M/D/1) Alternate Prediction

Approximation

  • For a tool set operating in

IBM’s 200mm fabricator

  • For a biweekly period:

– Measure statistics (CA

2, A,

r*, W, T, …) – Measure actual cycle time performance – Compare!

  • Dominant factors

– Cycle time offsets – Idle with WIP – Multiplicity of tools – Low variability

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SLIDE 17

IEEE/SEMI Advanced Semiconductor Manufacturing Conference

May 22-24, 2006 ASMC 2006 – Boston, Massachusetts

17

Concluding Remarks

  • Queueing models for manufacturing system performance evaluation
  • Standard approximations for the mean cycle time in a G/G/m queue
  • Extensions to the G/G/m queue

– Idle with WIP – Cycle time offsets – Defection of lots from failed servers

  • Application to toolsets in IBM’s 200mm semiconductor fabricator
  • Future directions: Apply to all toolsets, rollup to fab performance

curve, more rigorous lot defection analysis