Priority Cycle Time Behavior Modeling for Semiconductor Fabs

Shi-Chung Chang1,2

scchang@cc.ee.ntu.edu.twBo-Jiun Liao1

r93546017@ntu.edu.twYu-Ting Kao1

r97546010@ntu.edu.twArgon Chen1

achen@ntu.edu.tw1 Graduate Institute of Industrial Engineering, National Taiwan University

2 Dept. of Electrical Engineering, National Taiwan UniversityTaipei, Taiwan, R.O.C. 10617

Abstract Semiconductor wafer fabrication factories (fabs)often offer wafer manufacturing services of multiple prioritiesin terms of differentiated cycle time-based targets and fabproduction must be planned accordingly. This paper aims todevelop modeling methods and fab behavior models to capturecycle times of differentiated manufacturing services forsemiconductor supply chain management. A novel, hybriddecomposition approximation-based priority queueingnetwork model is designed to characterize how cycle times ofindividual priorities (PCTs) are affected by wafer release ratesand fab capacity utilization. The approach integrates queueingnetwork analyzer that approximates production flow behaviorsamong tool groups, sequential decomposition approximationamong priorities, fixed point iteration to exploit the re-entrantflows and an empirical data-based model tuning technique.Model tuning and validation by comparison with simulationresults demonstrate that the decomposition approximation-based models yield quick and good quality estimations of PCTsand related performance indices. The fab models thusconstructed enable efficient what-if analysis for supply chainmanagement of differentiated manufacturing services.

Keywords- priority; cycle time; modeling; semiconductor fab;hybrid decomposition approximation, queueing network

I. INTRODUCTIONA supply chain is a system of nodes that provides

manufacturing services in fact, a variety of services.Services differentiation, namely, prioritization, is common inoperations of semiconductor supply chains (SSC, Figure 1).It affects how to allocate resources and charge prices. Such anew paradigm of manufacturing services requires newmethods of operation control. Challenges are incurred byservice prioritization - how to evaluate cycle-time based fabperformance with priority with reasonable accuracy in shortcomputation time, and how to achieve model scalability andpredictability with respect to differentiation of services andvariability that are exacerbated by the increasing productvarieties and process variations in the chains.

Priority service discipline is common in foundry faboperations and is critical to service differentiation. Toprovide service with differentiable ensures that the quality ofservice (QoS), allocation of manufacturing capacity andpricing of services have to be dependent of and differentiatedby QoS requirements. Product, process and operationvariability affects the performance of individual servicenodes such as tool groups and fabs and chains/networks ofservice nodes. In order to predict the behavior of a SSC thatprovides differentiated services, research is needed in thefollowing aspects: predictable and scalable performance

metrics with respect to the chain structure, and fundamentalunderstanding of the behavior of service nodes and chainswith respect to variability.

Among the SSC service nodes, fab is the most expensive,complicated, and important. So this study is aimed at thebehavior modeling of a fab and provides a cornerstone modelfor supply chain management. The model characterizes howcycle time of individual priorities (PCT) is affected byrelease rate and fab capacity utilization. The behavior modeldescribes the relationship between output performancemetrics and inputs, which are characterized by not only meanvalues but also variability.

In this paper, we investigate the behavior modelingproblem and develop fab behavior models and modelingmethods that enhance the scalability and predictability ofSSCs with respect to varieties, and differentiability ofservices. The input/output relationship of a fab is modeledas a network of priority service nodes. Such a prioritynetwork captures factors and effects of variations throughoutthe network model. The model is scalable and allowschain/network performance metrics to be decomposed intoper node and per priority metrics. It has also predictabilitythat allows very quick evaluation of mean and variability ofboth nodal and system output performance metrics such ascycle times under various priority input, resource allocationand sources of variations. The fab behavior model thusprovides a cornerstone model to SSC management.

The remainder of this paper is organized as follows. InSection II, a fab is modeled as a priority open queueingnetwork (OQN). An innovative, hybrid decompositionapproximation-based approach for such a priority OQN isthen developed. Section III presents the validation results ofthe modeling method and a model tuning method. SectionIV concludes the paper.

II. FAB BEHAVIOR MODELING BY HYBRIDDECOMPOSITION OF PRIORITY NETWORK

Consider a fab with multiple part priorities, failure pronemachines, and re-entrant process flows. We model the fab asa failure-free, batch-free, deterministic-feedback and priorityopen queueing network (OQN). Then we develop aninnovative, hybrid decomposition approximation-basedapproach for such a priority OQN with a focus on capturingoperation priority and variations in a fab.

Key ingredients of the hybrid decomposition approachfor modeling a priority fab OQN are as follows:1. Decompose the fab network into many independent

service nodes and their networking relationship by

2009 First International Conference on Advances in System Simulation

978-0-7695-3773-3/09 $26.00 © 2009 IEEEDOI 10.1109/SIMUL.2009.35

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adopting the decomposition approximation of queueingnetwork analyzer (QNA, [3]).

2. Model single service node behavior by sequentialdecomposition approximation (SDA) of couplinginteractions among priorities in one node ([2], Figure 2);

3. Combine the networking relationship among service nodeswith a fixed point iteration to approximate re-entrant flowline performance.

More details will be given in the following subsections.To develop fab behavior models, let us first define some

notations.Notationsji, : Priority index, i I ; j ; where is

the number of priorities;n : Node index, n=1,2, ,N, where N is the number of nodes;iS : Priority i job s own service time;

iH : Special service time of priority i (arrive into emptyqueue);

iG : General service time of priority i (arrive into non-emptyqueue);

rjS : Remaining service time when priority j job is under

service;ijq* : Probability of a priority j job is under service based on

the condition that there is no priority i in the system;ieq* : Probability of no job is served based on the condition

that there is no priority i in the system;i : Aggregated queue index, aggregate all higher priority

queues to one queue when analyzing queue i,i {1,2, ,I};

ei : External arrival rate of priority-i to node 1;2eiC : Inter-arrival time SCV of priority-i external arrivals to

node 1sni : Mean service time of priority i at node n;2sniC : Service time SCV of priority-i at node n;ani : Mean arrival rate of priority i to node n;2aniC : Inter-arrival time SCV of priority-i to node n;'ni : Adjusted utilization of serving priority-i in node n;

niW : Waiting time of priority-i at service node n;

niL : Oueue length of priority-i in the node n (including onthe service);

niv : Number of visits of priority-i job to node n;

A. Nodal Model: Sequential Priority DecompositionThe behavior modeling of priority single service node

provides a cornerstone for the fab behavior modeling. Wedevelop the nodal behavior model with priority by treatingeach service node independently as a GI/G/1 non-preemptivepriority queue, and then adopt the sequential decompositionapproximation (SDA) proposed by [2] among priorities forour behavior modeling. SDA decomposes the couplingamong priorities into approximately independent queues for

individual priorities as shown in Figure 3 by the notion ofequivalent service time.

SDA determineseach queue by taking the interactions with other queues intoconsideration. The most important feature to consider in eachindividual queue is that the service time of a job arriving intoan empty queue differs from one arriving into a non-emptyqueue. The calculation of equivalent service time is brieflysummarized as follows.

Highest priority queue ( i =1 )When a priority 1 job arrives into an empty queue, the

special service time only consists of its own service time; butif the whole system is not empty, the special service timeconsists of the remaining service time of currently processedlower priority job and its own service time.

IjqSS

qSH

jrj

e

,...,3,21*1

1*1

1 (1)

When a priority 1 job arrives into a non-empty queue, thegeneral service time is the same as the original service time,because the server does not serve any lower priority jobswhen there is highest priority job in the system.

11 HG (2)And the first two moments of the special and general

service times of priority 1 are calculated by:I

j

rjj SEqHE

2

1*11 )()( (3)

11 )(GE (4)

))(2)())((()()(2

121

21*21

1*21

I

j

rj

rjje SESESEqSEqHE

(5))()( 2

121 SEGE (6)

Any lower priority queue ( i =2,3, ,I )There are three situations in calculating special service

time when a job arrives into an empty queue. The first one isthat if the whole system is empty, it only consists of its ownservice time. The second is that if a higher priority job isunder service, it has to wait for the remaining service time ofcurrently processed job, the time required by the aggregatedhigher priority queue to become empty, and its own servicetime. And the last situation is that if a lower priority job isunder service, it has to wait for the remaining service time,the time required by the aggregated higher priority queue tobecome empty starting from the number of arrivals occurredduring the period that lower priority job is processed, and itsown service time.

IijqSSNRS

qSNRS

qS

Hijij

rj

ii

r

iei

i

i

iii

,1)( *

*

*

(7)

39

The general service time of priority i job consists of itsown service time and the time required by the aggregatedhigher priority queue to become empty starting from thenumber of arrivals occurred during the period that theprevious job with the same priority is processed

)( iiii SNRSG . (8)And the first two moments of the special and general servicetimes of lower priority i are calculated by:

I

ij

rjj

ij

riii

SESNREq

NRESEqHE

i

iii

1

*

*

)())((

)()(

(9)

))(()( iii SNREGEi

(10)

)(22

)(2)()(

)()(

1

22*

2*2

iii

iii

i NRESENRE

SESENRESEq

SEqHE

r

rii

ri

iiei

I

ij jijrj

rjiji

rji

jSNRESNRESE

SESNRESESEq

ii

i

1

222*

))((2))(()(2

)(2)()()(

(11)))((2))(()()( 222iiiii SNRESNRESEGE

ii(12)

Due to decomposition of our priority system, theseparated queues can be analyzed independently but it stillhas to consider the influences from another priorities. So wecalculate the probability of priority i job arriving into a non-empty queue by the adjusted-utilization of serving priority i( '

i ). Define iS~ as the equivalent service time of priority i,

which is calculated by'' )1(~iiiii GHS (13)

By applying the SDA, the waiting time parameters can becalculated for each priority class independently, from thequeue length moments of the corresponding priority queue.The waiting time has two components. One is the queueingdelay: the job has to wait in the buffer till there are no jobs inthe same queue arrived earlier; and the other is servervacation: if it is the only job in its own priority queue, it hasto wait till the server can process it due to the possiblepresence of other jobs belonging to other priorities. Thewaiting time is thus

iii VWqW (14)and the server vacation can be approximated by

))(1()( ''iiiiiii SHSGV (15)

Given the equivalent service time parameters and partrelease parameters, the means and variances of the cycletimes and the departure process of individual priorities canthen be obtained.

B. Decomposition Approximation-based QueueingNetwork Modeling of FabIn combining SDA with QNA [3], a priority open

queueing network (OQN) [4] is first developed for a fab withmultiple part types, multiple priorities, failure prone

machines, and re-entrant process flows. This re-entrant OQNis analyzed by using a class of approximate decompositionmethods. The decomposition methods decompose an OQNinto individual network nodes and use two types ofparameters to characterize the stochastic arrival, service anddeparture processes of each node: one describing the rate andthe other describing the variability. The aggregate externalarrival and the flow rates and variability within the networkare as below.

Traffic rate equationsWhen the network is in a steady state, there is a flow rate

balance relationship among network nodes, i.e.

NnqN

mmnianineiani 1

1

(16)

where n = 1 if n = 1 and n = 0 otherwise.

Traffic variability equationsThe departure flow out of a node is split into a few sub-

flows of different destination nodes according to the routingmatrix.

NnbCaCN

mmniaminiani ,...,2,1

1

22 (17)

Equation (17) describes the approximate relationshipamong the inter-arrival time SCVs of all nodes. In theequation, the term an captures the SCV of the external arrivalprocesses, bmn approximates the SCV due to internaltransitions.

There is an OQN for each priority. OQNs of individualpriorities are coupled through competition of service noderesources. The priority coupling is handled by application ofthe SDA procedure to sequentially solving the equivalentservice times from the highest priority in a node. We applyQNA to deal with interactions among nodes: splitting,merging, and deterministic feedback with priority. Figure 3depicts a two node-example. The departure parameters ofone node are equal to the arrival parameters of next node intandem queues. We can directly use QNA to separate nodesin the network, and apply SDA to sequentially analyze theperformances of each priority in each node. But in a fab,there are re-entrant flows and the arrival parameters of onenode are affected by the departure parameters of many othernodes. Figure 4 depicts a simple re-entrant example of twonodes and two priorities. We deal with the re-entrant flows ina priority fab network by combining fixed point iteration [5]over SDA and QNA.

Based on this priority network model with deterministicrouting, we can estimate system cycle time. Once arrival andservice parameters ),,,,( 22

snisnianiani CC at each node areavailable, we can utilize them as input data to calculate manynodal performance measures, such as the mean and varianceof waiting time, the first second moments of queue length,and the server utilization.

40

III. MODEL VALIDATION AND TUNING

A. Model ValidationTo validate the hybrid priority network model, we

consider a small example of two machine groups and fourstages as depicted in Figure 4. Figure 5 shows the cycletimes obtained by the hybrid SDA+QNA algorithm over theexample. Although there exist difference in standarddeviation for priority 2, the difference does not increase asthe number of nodes increases.

Numerical experiments are also conducted on a simplebut full-scale fab model [6] to examine the efficiency,accuracy and application potential of the hybriddecomposition approximation-based queueing networkmodeling. Discrete event simulations are also developed forvalidation of the fab behavior model. The fab example hastwo parts with two priorities classes, and the numbers ofprocessing steps of products P1 and P2 are 32 and 60,respectively. In a special fab model (SFM), all the servicetimes of a node have exponential distributions. Details ofSFM data are listed in Appendix A. Node and system levelcycle times of the SFM are given in Figure 6, Figure 7, andTable 1 respectively. The differences of mean cycle times oftwo priorities and the cycle time standard deviation of P2 aremostly within 3%. Although the difference of cycle timestandard deviation of P1 is high (up to 45%), the absoluteerror is only 1.627.

In a general fab model (GFM), the service times ofindividual nodes have general distributions, such as uniform,erlang and exponential (see Appendix A). Node and systemlevel cycle times of the GFM are given in Figure 8, Figure 9,and Table 2 respectively. The differences of mean cycletimes of two priorities and the cycle time standard deviationof P2 are mostly within 10%. The cycle time standarddeviation of P1 has a high relative error of 55%. But again,the absolute error, 1.214, is still very small as compared tothe mean. Comparisons of numerical results with simulationin these two fab models show that our network modelingmethodology provides good approximations in most nodalperformances

B. Model TuningTo further fit the decomposition-based priority queueing

network model to the actual priority system and reduce thedifference caused by approximation, we design a modeltuning step for the behavior model. In the hybridSDA+QNA model, the mean cycle time of one priority jobconsists of the mean waiting time and the service time,where the waiting time is divided into queueing and theapproximate server vacation time because of higher priorityjobs as given in Eq. (14). Consider the cycle timeperformance of priority 1. Figure 8 shows that the hybridSDA+QNA provides a good approximation. But for priority2, Figure 9 shows that there are positive errors in mean cycletimes. We reason that the potential source of error to priority2 is caused by the estimation of vacation time. So we use twoparameters and for each machine group to compensate

the mean and variance of the vacation time in Eqs. (9) and(11) respectively.

Take node 5 and node 12 for example. By adjusting theparameters of vacation time, the PCT approximation errorsare reduced and lead to a close fit of the cycle time ofpriority 2 as compared to simulation results listed in Table 3.

C. Computation EfficiencyApplication of hybrid decomposition approximation to

each model (listed in Table 4) only requires less than 4seconds of CPU time on a 2.8 GHz personal computer. Thisleads to fast calculation with respect to changes of systeminput options. Consequently, both the accuracy andcomputing efficiency of hybrid decompositionapproximation-based network modeling support its potentialfor applications of real fab with service differentiation.

IV. CONCLUSIONSResponding to rapidly changing complex SSC

requirements, SSC planners/managers need an effectiveperformance evaluation/prediction tool to do what-if analysisbetween SSC inputs and outputs. Given a set of priority mix,capacity, mean and variability of wafer release, mean servicetime and variability of tools, the hybrid decompositionapproximation-based network model allows very quickevaluation of mean and variability of nodal and fabperformance metrics with good accuracy.

Effective evaluation of various input options in terms ofcapacity allocation, priority mix, wafer release policy, tooladjustment, etc., may serve as a behavior model for SSCperformance optimization. The mean and variability statisticsat individual tool groups may also be utilized by six-sigmamanagement for on time and quick delivery, where the meanvalues can be used to derive control targets while variabilityvalues for calculation of control limits.

ACKNOWLEDGEMENTThis work was supported in part by the Semiconductor

Research Corporation and International SemiconductorManufacturing Initiative under FORCe-II project 1214 andby the National Science Council, Taiwan, R.O.C., undergrant NSC95-2221-E-002-346-MY3.

Figure 1: Semiconductor supply chain

41

P1 moving

P2 moving

P1 moving

P2 moving

Node m

1

2

Node n

1

2

QNA

SDA

Figure 2: Decomposition approximation-based queueingnetwork modeling

Figure 3: Decomposition among priorities

Figure 4: Two stations with reentrant line

Figure 5: Mean and standard deviation of cycle time formulti-nodes (2~5) with reentrant line

Priority 1 Mean Cycle Time

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12Node Index

Hours SDA+QNASimulation

Figure 6: Mean Cycle Times of P1 in SFM Nodes

Priority 2 Mean Cycle Time

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12

Node Index

Hours SDA+QNASimulation

Figure 7: Mean cycle times of P2 in SFM nodes

Priority 1 Mean Cycle Time

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

1 2 3 4 5 6 7 8 9 10 11 12Node Index

Hours SDA+QNASimulation

Figure 8: Mean cycle times of P1 in GMF nodes

Priority 2 Mean Cycle Time

012345678910

1 2 3 4 5 6 7 8 9 10 11 12

Node Index

Hours SDA+QNASimulation

Figure 9: Mean cycle times of P2 in GMF nodes

Table 1: System level performance comparisons (SFM)

Mean Cycle Time Cycle TimeStandard Deviation

SDA+QNA 18.613 5.190Simulation 18.572 0.014 3.5622 0.008

Absolute Error 0.041 1.627Relative Error 0.22 % 45.68 %SDA+QNA 172.78 54.674Simulation 174.695 1.159 53.088 0.899

Absolute Error -1.915 1.586Relative Error -1.10 % 2.99 %

Mean Cycle Time

0

100

200

300

400

500

600

2 3 4 5Number of Nodes

Unit TimeSDA+QNA 1SDA+QNA 2Simulation 1Simulation 2

Cycle Time Standard Deviation

0

40

80

120

160

200

240

280

2 3 4 5Number of Nodes

Unit TimeSDA+QNA 1SDA+QNA 2Simulation 1Simulation 2

42

Table 2: System level performance comparisons of general

Mean Cycle Time Cycle TimeStandard Deviation

SDA+QNA 15.137 3.503Simulation 15.451 0.009 2.289 0.005

Absolute Error -0.314 1.214Relative Error -2.03 % 53.02 %SDA+QNA 122.22 36.392Simulation 112.015 0.668 38.133 0.424

Absolute Error 10.205 -1.740Relative Error 9.11% -4.56%

Table 3: PCT adjustment table

Machine

P2Modified

Cycle timeApprox.

Simul. error

5 3.2 -3.2 8.4268 8.4102 0.20%12 -1.6 -1.6 5.2404 5.2204 0.38%

Table 4: Comparisons of CPU times in fab modelsAverage CPU time (seconds)

Two PrioritiesFab Models Hybrid SDA+QNA Simulation (one run)

Special case 3.606 2466General case 3.622 2484

Appendix A: Fab Model Data

Enter 1 2 101 2 3 81 2 101 2 11 5 2 3 81 2 3 8 101 2 3 81 2 6 2 9 12 Exit

Figure A1: Process flow of P1 in 2-PR fab model

Enter 1 2 3 8 101 2 61 2 3 8 91 2 3 8 101 6 11 1 2 51 2 3 8 91 2 3 8 101 6 11 1 2 51 2 3 8 9 10 111 2 4 8 31 2 7 8 101 2 12 Exit

Figure A2: Process flow of P2 in 2-PR fab model

Table A1: Service processes of nodes for special fab model

Node # ofMachines

Service TimeDistribution

MPT(hr/lot)

Utilization %*

1 1 Exponential 0.125 88.382 1 Exponential 0.125 85.233 1 Exponential 0.25 91.544 1 Exponential 1.8 68.185 1 Exponential 0.9 90.916 1 Exponential 0.6 83.337 1 Exponential 1.8 68.188 1 Exponential 0.2 80.819 1 Exponential 0.6 83.33

10 1 Exponential 0.3333 88.3811 1 Exponential 0.6 83.3312 1 Exponential 1.25 78.91

Table A2: Service processes of nodes for general fab model

Node # ofMachines

Service TimeDistribution

MPT(hr/lot)

Utilization %*

1 1 Erlang Order 4 0.125 88.382 1 Exponential 0.125 85.233 1 Uniform 0.25 91.544 1 Erlang Order 3 1.8 68.185 1 Erlang Order 2 0.9 90.916 1 Erlang Order 4 0.6 83.337 1 Exponential 1.8 68.188 1 Erlang Order 3 0.2 80.819 1 Uniform 0.6 83.33

10 1 Erlang Order 2 0.3333 88.3811 1 Exponential 0.6 83.3312 1 Uniform 1.25 78.91

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[3] W. Whitt, The Queueing Network Analyzer, The Bell SystemTechnical Journal, Vol. 62, pp2779-2815, 1983.

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