using an empirical queueing approach to predict future flow times

Using an empirical queueing approach to predict future¯ow times

Yi-Feng Hung*, Ching-Bin Chang

Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan

Abstract

The traditional method used to describe a steady-state manufacturing system is a queueing model;whereas the common tool used to predict the future performance of a dynamic manufacturing system isa simulation model. This study proposes an empirical queueing approach to obtain the ¯ow timeperformance measures of a complex dynamic manufacturing system, such as semiconductor waferfabrication. This approach is easier to implement than a simulation model and more straightforwardthan a queueing model. Initially, the empirical queueing curve of each work station, which de®nes therelationship between the utilization rate and the expected waiting time of a small time period, isobtained from the historical database. Then, an iterative scheme is used to predict the future systembehavior. Several latest researches have reported that the prediction of future ¯ow times is important inthe operation management of a complex manufacturing system. The approach proposed in this studycan be easily implemented for such purposes, and the experimental results show that this approach canaccurately predict the future ¯ow times. 7 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Production management; Flow time prediction; Semiconductor manufacturing

1. Introduction and literature review

In a steady-state manufacturing environment with the product mix being constant, theexpected waiting times and the expected ¯ow times are time-independent. If the assumptions ofa particular queueing model [3,4] are appropriate, it could be applied to obtain the long-term

Computers & Industrial Engineering 37 (1999) 809±821

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* Corresponding author. Tel.: +886-3-574-2939; fax: +886-3-572-2685.E-mail address: [email protected] (Y.-F. Hung).

average measures of the steady-state system. However, a dynamic manufacturing system, whichinvolves time-varying product mix, time-varying expected waiting time and time-varyingexpected ¯ow times, would probably never enter steady state. In such an environment,queueing models are normally not appropriate for obtaining short-term measures of thesystem, since most queueing models assume long-run steady-state condition. An typicalexample of a dynamic manufacturing system is a multi-product semiconductor manufacturingfacility, whose product mix shifts quickly from time to time, because of the short product lifecycle and quick change in demands. A discrete-event simulation [10] is normally used to modelthis system and to predict the system performance. However, it takes engineers a lot of e�ortto build a detailed and satisfactory simulation model, which can accurately describe a complexmanufacturing system. The di�culties in using simulation in semiconductor manufacturingsystems are discussed by Johri [9]. Our study, proposes an alternative method to predict thefuture ¯ow times of a system based on the idea from queueing theory.Flow time information is essential to production planning and scheduling for long ¯ow time

systems, such as semiconductor manufacturing. The pioneering work on how to consider non-integer ¯ow times in production planning model is proposed by Leachman [11], whosemodeling technique is now popular in today's semiconductor industry [12]. Since the planningproblem deals with future production, the prediction of future ¯ow times is necessary. Aniterative computation between the production planning model and simulation was proposed byHung and Leachman [8]. They use simulation to predict the future time-dependent ¯ow timesthrough which the production planning model can compute accurate production plan. Inaddition, Lu et al. [15] and Lin [13] use the predicted future ¯ow times to compute the slack,and then, apply the least-slack-rule [15,13] to dispatch the waiting jobs in queues. This method,has proved to be very e�ective in reducing the mean and standard deviation of ¯ow times,which is one of the top priorities of semiconductor manufacturing management. Lu et al. usedsimulation models to predict ¯ow times, while Lin used exponential smoothing to updatewaiting time forecasts.A lot is a group of the same product that travel together through the factory. A work station is

a group of identical machines used to perform operations that add value to in-process material.An operation is the performance of a required processing activity on a lot by a particular workstation. A route is a sequence of operations required to produce ®nished products. Insemiconductor wafer fabrication, a route can consist of more than hundreds of operations. Flowtime is the di�erence between the time when a lot is released into shop-¯oor and the time whenthe lot is ®nished. A ¯ow time consists of the waiting time and processing time of eachconsecutive operation on the route. We assume here that transportation times are trivial, andprocessing times are known parameters, while waiting times are the only unknown parameters,which require estimation. Periods are small equally-spaced time intervals on time axis.Various common queueing models have curves describing the relationship between the

expected waiting time and the utilization rate of the system. Tables and graphs of such can befound in [5]. However, these relationship curves are drawn based on the calculation of themathematical equations of a particular queueing model with certain assumptions andsimpli®cations. These analytical models are normally useful for system design dealing with thelong-term steady-state capacity issue. However, as for the system operation (productionplanning and scheduling) mentioned previously, the analytical model because of its limitation

Y.-F. Hung, C.-B. Chang / Computers & Industrial Engineering 37 (1999) 809±821810

of steady-state assumption may not be appropriate for a dynamic production environment.The complexity of the semiconductor manufacturing are discussed by Johri [9]. Also, Connoret al. [2] classi®ed the complex manufacturing methods and proposed a detailed analyticalmodel for the semiconductor manufacturing under steady-state and single-productassumptions. Connor's model is complex and requires many input parameters for computation.Thus, building the model for a dynamic and multiple-product manufacturing environment willbe even more di�cult, if not impossible. Normally, under such circumstances, a simulationmodel is used to predict the future factory status. This study proposes using an empiricalutilization rate vs. waiting time curve for each work station. Previously, this curve is providedby a queueing model, which is derived under certain assumptions and simpli®cations of thesystem. The assumptions and simpli®cations of queueing models have been the major obstaclesin their application. This empirical queueing approach bypasses this di�culty of building ananalytical model for a complicated actual system, and obtains the relationship curve betweenutilization rate and waiting time directly from the historical database.

2. Constructing the empirical curve

There are two steps in constructing the empirical curve of each work station. Section 2.1outlines the ®rst step, which collects the empirical data points; while Section 2.2 discusses thelinear programming model used for constructing the empirical approximation curve, whichdescribes the relationship between the utilization rate and expected waiting time of a workstation.

2.1. Collecting the empirical data points

The ®rst step involves collecting the empirical data points, which are the historical datapoints of utilization rate vs. waiting time. These data points can be computed from themanufacturing database, which is called WIP (Work-In-Process) Tracking system popularlyimplemented in semiconductor industry [17,18]. A WIP Tracking system records the arrivaltime at the work station, the starting time and the ending time of the processing, as well as thedeparture time from the work station for each operation of every lot produced in the past.A common queueing model provides the relationship between server utilization and expected

waiting time. Most of the queueing models deal with a system under steady state. However, ina complex manufacturing system, owing to changing product mix, long processing route andunreliable machines, it may not be proper to assume steady state in the long run. However, itmay be possible to assume steady state within a small time period, say within a half day. Evenso, a queueing model is still not ideal because of three reasons. First, owing to the samereasons mentioned above, it is common that the bottleneck may shift from one work station toanother. Further, the required machine utilization rate within a particular period may be morethan 1.0; i.e., the workload requirement is more than the capacity available within a small timeperiod. Most queueing models cannot deal with a utilization rate greater than 1.0 s, for mostqueueing models, such as M/M/1 or M/M/s, as the utilization rate approaches 1.0, the waitingtime will approach in®nity. Third, Hou [6] pointed out that, based on his experiment, the

Y.-F. Hung, C.-B. Chang / Computers & Industrial Engineering 37 (1999) 809±821 811

queueing model cannot predict waiting time accurately in a dynamic manufacturingenvironment.A waiting time is de®ned as the time between the time point of arrival and the time point of

operation initiated. A utilization rate is de®ned as the workload required in a period divided bythe number of machines and the length of a period. The empirical curve of a particular workstation plays the same role as the relationship curve between utilization rate and expectedwaiting time provided by a queueing model. However, a queueing model is usually used in along-term steady state; whereas, the empirical curve here is intended for use in short timeperiods. We also implicitly assume that the waiting time at the work station is independent onthe operations to be processed and the product type of the lots. To describe how to obtainsuch empirical curve the following notations are used:

t time period index; t � 1, 2, . . . ,T;k work station index; k � 1, 2, . . . ,K;b the length (hours) of a small time period;nk the number of identical machines in work station k;akt the cumulative total workload (machine � hours) arrival at work station k in period t;wkt the cumulative waiting time of lots arriving at work station k in period t;ukt the cumulative total number of lots arriving at work station k in period t;qkt the amount of the queue (machine � hours) of work station k at the end of period t;pk the up time ratio of work station k;ckt the capacity (machine � hours) of work station k in period t;xkt the utilization rate of work station k in period t;ykt the average waiting time of lots arriving at work station k in period t.

The following procedures are needed to obtain the historical data:

1. Initialize variables.set akt � 0, wkt � 0, and ukt � 0, 8k 8t:

2. From a WIP Tracking database system, we can collect the arrival time at work stations ofall lots and their processing times. Based on these data, we can compute the cumulativearrival workload akts, and their corresponding cumulative total waiting times wkts.

While (there are lot arrival data left)remove and use the next lot arrival datat � d the arrival time of the lot

b e /� compute the time period number �/akt�akt� the processing time of the lotwkt�wkt� the waiting time of the lotukt � ukt � 1

end while loop

3. We assume the capacity to be period-independent and estimate the average capacity of workstation k by:

ckt � nk � b� pk, 8t 8k:


4. We can estimate the queue amount at the end of each time period. The queue at the end ofa period (the start of next period) is equal to the queue at the start of the period plus thearrival workload during the period minus the capacity during the period. However, it isimpossible to have a negative queue.

For k � 1 to KFor t � 1 to T

qkt � maxÿ0, qk, tÿ1 � akt ÿ ckt

�:

5. For a particular work station k, the total workload of a particular period is the sum ofinitial queue and the arrival workload within the period; thus, the utilization rate can becomputed as follows:

xkt � �qk, tÿ1 � akt�nk � b

, 8t 8k:

6. For all the lots arriving at work station k in period t, we can compute their average waitingtime:

ykt � wkt

ukt, 8t 8k:

2.2. Fitting the empirical approximate curves

Here, we have a data point �xkt, ykt� for each past period t of each work station k. For workstation k, we can use the following linear programming model LPk and data points �xkt, ykt�,t � 1, . . . ,T to ®t an empirical piecewise linear curve, as shown in Fig. 1.

1. Index:t: data point number (previously, period number);i: break point number on the piecewise linear approximation curve.

2. Parameters:xt = the utilization rate of data point t; it equals to xkt for LPk;yt = the average waiting time of data point t; it equals to ykt for LPk;

max utilization rate � maxt

xt;

max waiting time � maxt

yt;

ri = the utilization rate of the ith break point, i � 1, . . . ,H,where r0 � 0, r0 < r1 < r2 < � � � < rH and rHÿ1 < max utilization rateRrH:

3. Decision variables:Ei = the expected waiting time of the ith break point, whose utilization rate is ri:


4. Notations:yt = the y-value on the piecewise linear curve, when x-value is xt:

5. Objective function:

MinimizeXTt�1jyt ÿ ytj

We seek to minimize the sum of the absolute deviation between the waiting time projectedby the piecewise linear curve �yt� and the actual waiting time �yt� of each data point.

6. Constraints:(1) The y-value projection of xt on piecewise curve:

yt � Ei � Ei�1 ÿ Ei

ri�1 ÿ ri� �xt ÿ ri�, in which i satisfies riRxt < ri�1, 8t:

(2) The slopes are non-decreasing as the utilization rate increases:

Ei�1 ÿ Ei

ri�1 ÿ rirEi ÿ Eiÿ1

ri ÿ riÿ1, i � 1, 2, � � � ,Hÿ 1:

(3) The maximum waiting time constraint:

EHRmax waiting time:

(4) The slopes are non-negative:

E1rE0:

(5) The non-negativity constraints:

Eir0, 8i � 0, 1, 2, � � � ,H:

Fig. 1. The linear programming model used to ®t the piecewise linear curve.


After solving the LPk, we have the break points �ri, Ei), i � 1, 2, � � � ,H: These pointsempirically depict the relationship between the utilization rate and waiting time for workstation k. We believe that this curve is dependent only on the manufacturing method of theequipment, the number of machines in work station, and the machine failure pattern (thedistribution of the time-between-failure and the distribution of the time-to-repair). Therefore,this curve need to be reconstructed only when the above dependent factors have changed. Wedo not need to resolve the LP every time when we do the iterative computations.In the above formulation, �ri ÿ riÿ1� is the x-distance between two consecutive break points.

The smaller the distance, the more break points, the more accuracy of the empirical curve.However, the longer it takes to compute in both obtaining and applying the curve. In ourexperiment, we use 0.04 as the x-distance of two consecutive break points. We believe 25points is su�cient to approximate the curve when the utilization rate is between 0.0 and 1.0. Ittook less than 2 min on a SUN UNIX computer work station to solve one LP problem, whichcovers the operation data of a work station for three years.

3. Iterative procedure to predict future ¯ow time

Machine loading (utilization rate) will a�ect waiting time, which in turn determines the timeof lot arrival at the following work station; that is, the machine loading of the following workstation is changed. Then, the waiting time of the following work station is also changed. Thedetermination of waiting times and utilization rates are dependent on each other; thus, thesechain e�ects form a vicious circle. In order to resolve this problem, an iterative approach isproposed here. Initially, we assume the expected waiting time in each period on each workstation to be zero. By using the processing times, we can then determine the arrival time at thework station of each operation for each lot, from which we can compute the arrival workload(machine � hours) of each time period at each work station. Then, we can compute theestimated queue at the end points of periods, and the utilization rate in each period of eachwork station. Applying the piecewise curve obtained previously, we then compute a projectedwaiting time in each period of each work station from the utilization rate. At the end of aniteration, an exponential smoothing formula is used to update expected waiting times. Thereason of using exponential smoothing is explained in Section 4. Once we have the newexpected waiting time in each period of each work station, we can start another iteration. Thatis to recalculate the arrival time at the work station of each operation for each lot; thus,revising the arrival workload in each period of each work station. Then, the revised queue, therevised utilization rate, and revised expected waiting time can be computed. Through a seriesof iterations, we could converge to an accurate ¯ow time prediction for each lot. The followingis the outline of the iterative procedure:

Notationss variable for the iteration numberS maximum number of iterationsl variable for lot identi®cation


e time point variabler variable for operation identi®cationU a set of lots

Beginwkt � 0 8k, 8tfor s � 1 to S

akt � 0 8k, 8txkt � 0 8k, 8tput all the planned lots and WIP lots into set Uwhile (set U is not empty)

l = remove a lot from set Uif (l is a WIP lot)

e � current time

else

e � the planned release time

endifwhile (there is operation left on l )

r = remove next operation of lot lk = the work station performing rt � b ebc /� operation r start waiting on work station k in period t �/e � e� wkt /� delay for waiting time �/e � e + the processing time of r /� delay for processing time �/akt � akt + the processing time of r /� cumulate the workload �/

end while loop /� operation �/end while loop /� lot �//� Note: here we have the revised arrival workload �akt� array �/for k � 1 to K

for t � 1 to Tqkt�maxf0; qk; tÿ1�aktÿcktg /� revise queue �qkt� �/xkt��qk; tÿ1�akt��nk � b� /� revise utilization rate �xkt� �/wtemp = the projected waiting time from utilization rate xkt using the empiricalcurve of work station kwkt � wkt � a�wtemp ÿ wkt�; /�use exponential smoothing to revise wkt �/

end for t

end for k

end for s

End


4. Experiment

A series of experiments have been done to validate our approach. We started with thedetermination of the parameters used; then, compared the ¯ow times predicted by ourapproach with those from a simulation model.

4.1. Determining the parameters

This section focus on how to determine the values for a, b and S? We used the same data setas [7,8], in which there are 30 work stations and 10 product types, each of which follows aprede®ned distinct route. The average number of operations on a route is more than 130operations. A particular machine type may be visited by the same lot several times. We variedthe exponential smoothing factor �a� from 0.1, 0.2, 0.3, . . . through 1.0, and the length of smallperiod b from 0.25, 0.5, 1.0, through 2.0 days, and we iterated 15 times.Let us assume that N is the total number of lots, and gsl the computed ¯ow time of lot l in

iteration s.The closeness of two consecutive iterations is often used as a convergence criterion in

numerical methods [16]. To measure this closeness, we de®ne

Ds �(XN

l�1

�gsl ÿ gsÿ1, l

gsÿ1, l

�=N

)� 100%:

Fig. 2 shows the Ds values of various a values when b � 0:5 days. Letting a � 1:0 (noexponential smoothing) is equivalent to ®xed point iteration [16]. According to our experiment,when the a is larger than 0.7 (including 1.0), the Ds curves are too high (thus, not shown in the®gure), and the iteration scheme could not converge. We conjectured that the function whichgoverns the relationship between the machine loading of all machines and ¯ow times of all lotsis not continuous, and is very complicated. The traditional ®xed point iteration would over-react to the changes between two iterations; thus, it could not converge. It can be seen that atiteration 9, the Ds value does not decrease greatly with increasing number of iterations; i.e., theiteration scheme converges after about nine iterations. Thus, in the following experiment, weset the number of iteration S � 9:To measure the accuracy of the piecewise curve iteration approach, a simulation model that

involved random machine failures was used. Thirty simulation runs with di�erent initial seedswere executed to collect the mean ¯ow time of each individual lot. Let fl be the averagesimulated ¯ow time of lot l. These simulated ¯ow times were compared with those computedby the iteration approach. Two metrics were used to determine the accuracy of the iterativescheme with s iterations. They were the average ¯ow time di�erence of iteration s, de®ned as

Ms �(XN

l�1

�gsl ÿ fl

fl

�=N

)� 100%,

and the average absolute ¯ow time di�erence of iteration s, de®ned as


As �(XN

l�1

� jgsl ÿ fljfl

�=N

)� 100%:

Let p be the product type index, Fp the average ¯ow time of product type p from simulation, spthe standard deviation of ¯ow time of product type p from simulation, and rp the number oflots that are product type p; thus, N �Pp rp: A weighted average percentage of standarddeviation of ¯ow time is computed to estimate the con®dence level for our ¯ow timeestimation:

SD � 100%�Xp

�spFp� rp

N

�:

Figs. 3 and 4 show the M9 and A9 values, respectively, of various a and b values. It can beseen that the performances are all satisfactory when a is between 0.2 and 0.7 and b is less than2 days. However, in the following experiments, we set a � 0:5 and b � 0:5 days, whose M9

value is almost equal to zero.

4.2. Experimental validation

This study used the production planning model proposed by Leachman [11] to obtain themonthly production plan within a planning horizon of two years. As long as there is demand,this model will fully utilize the resource capacity, which is the case in our experiment. Since the

Fig. 2. Ds values of various a values.


demand data are time-varying, the planned production mix also varies from month to month.We called this production plan ``changing product mix''. If we sum up the planned productionquantity of all months within the two-year planning horizon of a particular product type, andthen, evenly release it within the planning horizon, we called this case the ``®xed product mix''.We can clearly see that ``®xed product mix'' production plan will drive the manufacturingsystem into steady state after a warm-up period; whereas, under the ``changing product mix''production plan and the ¯ow time being as long as months, the system will not enter a steady

Fig. 3. M9 values of various a and b values.

Fig. 4. A9 values of various a and b values.


state, and thus, being a dynamic system. In order to experiment the various level of machineutilization rate (due to the various level of demand rates), we scaled the production quantity ofeach month by various percentages: 80%, 85%, 90%, 95%, and 100%. Owing to theunavailability of the actual factory waiting time data, simulation was used to generate therequired data for constructing the relationship curve in our experiment.The ¯ow times obtained by the simulation of various production schedules were compared

with those predicted by the iterative approach. Tables 1 and 2 show the case of ®xed productmix and changing product mix, respectively. From these tables, we can see that the di�erencesbetween the ¯ow times obtained by simulation and the ¯ow times predicted by the iterativeapproach was very small with mean absolute di�erence less than 5%. They were all within onestandard deviation of the simulated ¯ow time. If assuming normal distribution and using 95%con®dence interval, whose boundary are plus and minus two standard deviation away from themean value, the ¯ow times estimated by the iterative approach were well within the con®denceinterval. The risk of estimation error for our iterative approach is very small. The result showsthat our approach performs well in both steady state environment (®xed product mix) anddynamic environment (changing product mix); and its performance is as good as that of thequeueing models with a steady state assumption [1,2], which use complex mathematical modelsto derive measures.We used a SUN UNIX work station to run our experiments. The execution time of our

proposed iterative approach is less than 10 s/iteration. Thus, it took about 1.5 min to convergein nine iterations. The simulation programs were written in C++ [14]. It took about 10 min/simulation run and about 5 h to run 30 simulation runs to obtain the average ¯ow times. Thetime of a simulation run is about 60 times that of an iteration run in our approach.

Table 1A comparison of the simulation ¯ow time and the iterative ¯ow time in the case of ®xed product mix (%)

Release scale 80% 85% 90% 95% 100%

M9 2.87% 1.98% 1.21% 0.72% 0.22%A9 3.87% 3.23% 2.79% 2.83% 3.06%D9 0.10% 0.08% 0.10% 0.21% 0.25%

SD 4.36% 4.98% 4.87% 5.08% 5.04%

Table 2A comparison of the simulation ¯ow time and the iterative ¯ow time in the case of changing product mix (%)

Release scale 80% 85% 90% 95% 100%

M9 2.42% 2.02% 1.55% 1.86% 0.19%A9 3.66% 3.59% 4.62% 4.13% 5.10%D9 0.16% 0.25% 0.33% 0.68% 0.89%

SD 6.49% 6.65% 7.14% 8.58% 15.65%


5. Conclusion and future direction

Our approach bypasses the di�culty of the mathematical modeling for various complexmanufacturing methods. All analytical models aim to obtain the relationship between themachine utilization rate and the waiting time, which in this study are empirically obtained bydirect construction from the historical data of machine utilization rate and waiting time. Basedon the simulation experiment, our method works well in predicting the ¯ow times, which areimportant to the production planning and schedule for complex manufacturing system likesemiconductor manufacturing. The industrial application of the iterative approach in realisticproduction plan and schedule problems requires further study.

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using an empirical queueing approach to predict future flow times

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