Chapter 4 Ordinal Transformation and BRA
4.3 Summary
In this chapter, we first introduce the central idea of ordinal transformation, transforming the original space to an ordinal space by a simplified model used for ranking. We develop a BRA and take the first step to analyze single GI/G/m queue with 2 designs. BRA applies to the case of QNA as the simplified model of single GI/G/m queue. Our contributions in this chapter are as follow:
(1) Bound analysis shows that QNA is bounded by the upper and lower bounds of cycle time of single GI/G/m queue proposed by Kingman, and Brumelle and Marchal respectively.
(2) Assumed actual cycle time is uniformly distributed between its upper bound and lower bound, QNA approximation is demonstrated to provide the probability of correct ranking Pc, Pc > 0.5.
(3) Probability of correct ranking is increasing with increase of the ratio of surely-win and surely-lose region, RW and RL.
(4) With the variation of QNA, the least variation of upper bound is derived. This
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facilitates us to derive a better probability of correct ranking α, α >12+𝜌4−2𝜌3+(2𝜌−1)𝐶𝜌2(𝐶𝑠2𝜆2𝜏2+𝜆2(𝐶𝑎2+𝜆𝜏)
𝑎2+𝐶𝑠2)
∆𝐴𝐶𝑇 (𝑈𝐵2)>12.
(5) Capturing the heterogeneous SCVs is beneficial to recognize designs, especially top designs.
Finally, the derivations under the assumption of normal distributions of actual cycle times please refer to Appendix A-A.1 for more details. In the next chapter, we extend the ranking analysis to general re-entrant lines with M workstations and N designs.
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Chapter 5
Extensions of BRA to General Re-entrant Lines
In this chapter, we extend the discussion of QNA approximation of single GI/G/m queue in Section 4.3 to a general re-entrant line with M workstations.
Because congestion measures of QNA for a network are obtained by assuming that the stations are stochastically independent given the approximate flow parameters, the bound and ranking analysis of single GI/G/m queue is generalized to multiple GI/G/m queues using superposition of their upper and lower bounds.
We first consider 2-workstation re-entrant line with a pair of designs and there are four different cases of their bounds due to the superposition of bounds of independent workstations, as shown in Figure 5.2. We analyze the probability of being a concordant pair in these four cases. Based on the analysis of 2-workstation re-entrant line, it is seamless to generalize the result to M-workstation re-entrant lines.
In the last, we extend the scope of 2 designs (a pair of designs) to N designs (1
2N(N − 1) pairs in total), and validate our analysis and conclusions by an experiment of a five-workstation re-entrant line.
5.1 BRA of QNA and JNA for General Re-entrant Lines
Because JNA is a special case of QNA, here we focus on the analysis of QNA.
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We first consider a 2-workstation re-entrant line and there are two designs, D1={M1, M2}, which means 𝑚1 machines at workstation 1 and 𝑚2 machines at workstation 2, and another design D2={M1-1, M2+1}. Let their true mean cycle times be ECT1 and ECT2 which are random variables.
Let us define a notation, ACTi.m, which means the approximated mean cycle time of workstation 𝑚 given design Di. Thus, ACT1= ACT1,1+ACT1,2 and ACT2= ACT2,1+ACT2,2. Actually, D2 is equivalent to D1 moving one machine from workstation 1 to workstation 2. Due to decreasing capacity of workstation 1 and increasing capacity of workstation 2, there is growth and decline of approximated mean cycle time, a simple diagram shown in Figure 5.1.
Figure 5.1 ACT of 𝐷1 and 𝐷2 w.r.t. number of machines
Most importantly, because in this case there are interactions among stations, various machine allocation designs bring about different network configuration. The approximated parameters of QNA vary from machine allocation designs, for example, 𝐶𝑎2 and 𝐶𝑠2. Therefore, the relation between upper and lower bounds of D1
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and D2 is not always consistent with Lemma 4.5. Recall that Lemma 4.5 says if ACT1
< ACT2 then upper and lower bound of D1 would be smaller than them of D2 for a single GI/G/m queue.
Even though we know ACT1 > ACT2, we cannot definitely conclude that the upper bound of ACT1 must be greater than upper bound of ACT2. Instead, there are four possible cases of the bounds of D1 and D2 as shown in Figure 5.2. We analyze the probability of D1 and D2 being a concordant pair in each case.
Case 1: UB1 < UB2 and LB1 < LB2 Case 2: UB1 < UB2 and LB1 > LB2
Case 3: UB1 > UB2 and LB1 < LB2 Case 4: UB1 > UB2 and LB1 > LB2 Figure 5.2 Four possible cases of bounds given ACT1 < ACT2
Note that we derive BRA in single GI/G/m queue under the assumption of uniform distribution, but here the bounds obtained from the superposition of several
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stations are not be uniformly distributed anymore. For the ease of BRA, we assume true mean cycle time still follows uniform distribution even though the superposition and discuss the influence on rank correlation later.
Analysis of Case 1
Case 1 is the same as the case of single GI/G/m queue. Thus, analysis of single GI/G/m queue is directly applicable to Case 1. The probability of being a concordant
pair is P{ECT1 < ECT2 | ACT1 < ACT2} as Equation (4.3) in Chapter 4.
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=12𝑈𝐵(𝑈𝐵22−2𝐿𝐵1𝑈𝐵2+𝐿𝐵12
2−𝐿𝐵2)(𝑈𝐵1−𝐿𝐵1)
=12(𝑈𝐵 (𝑈𝐵2−𝐿𝐵1)2
2−𝐿𝐵2)(𝑈𝐵1−𝐿𝐵1) (5.4)
Maximum of Equation (5.4) is equal to 12, which means in Case 4 the probability of being a concordant pair is not greater than that of being a discordant pair.
In this section, we respectively discuss the four possible cases of relations of upper and lower bounds with 2 designs. In fact, because there are still only four possible cases of their bounds for a pair of designs, the above analyses are seamlessly applicable to more-than-two workstations re-entrant lines. Thus, the BRA is applicable to a general M-workstations re-entrant line.
Even though there are four possible cases of the bounds of any pairs of designs, we believe that the sum of approximated mean cycle time of workstations is essentially a representative performance index to describe their bounds. At least in single GI/G/m queue, Lemma 4.5 shows higher approximated mean cycle time (ACT) absolutely leads to higher upper bound and lower bound.
We hypothesize that most of pairs belong to Case 1 and we validate the above hypothesis by an experiment of 5-workstation re-entrant line with 415 designs in Chapter 6 and show the statistical result of four possible cases here.
Case 1 2 3 4
Ratio 89.91% 4.13% 3.89% 2.07%
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As we can see, most of pairs still hold for Lemma 4.5. For Case 2 and Case 3, there are some specific conditions to make the probability of being a concordant pair greater than 0.5 and the statistical results are as follow.
Case 2 3
Ratio of Pc > 0.5 99.72% 2.06%
Ratio of Pc < 0.5 0.28% 97.94%
The result of statistics shows that most of design pairs belong to Case 1 as our above analysis, and in fact the ratio of pairs whose Pc greater than 0.5 is more than 94%.
Discussion about assumption of uniform distribution after superposition
It is known that sum of uniform distributions is not a uniform distribution anymore. Here we use a sum of two uniform distributions, X = 𝑈1+ 𝑈2, 𝑈1, 𝑈2 uniformly distributed between [0,1]. This is a special case (n=2) of Irwin-Hall
distribution, and X follows a triangular distribution:
𝑓𝑋(𝑥) = { 𝑥, 0 ≤ x ≤ 1 2 − 𝑥, 1 ≤ x ≤ 2
If we assume X is still uniformly distributed in its support, [0,2], as same as our BRA, then its probability density function is
𝑓𝑋(𝑥) =1
2, 0 ≤ x ≤ 2.
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Figure 5.3 Actual p.d.f. and the p.d.f under our assumption
From Figure 5.3, we observe that the p.d.f. under our assumption is higher than actual p.d.f. while 2 ≥ x ≥ 1.5 and 0.5 ≥ x ≥ 0. Using the notation of D-3 in Section 4.2.2, it implies that we overestimate the probability of surely-win region and probability of surely-lose region, RW and RL, so the derived probability of correct ranking is over-optimistic in case of multiple stations. However, the range of true mean cycle time is often large so the value of density is essentially low and the extent of over-estimate has no major influence. The BRA is still applicable for re-entrant
Sum of 2 uniform distributions between [0,1]
Assumed uniform distribution between [0,2]
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designs in total, the 2-design analysis is actually also applicable because rank correlation is based on the pair-wise comparisons. For design Di and design Dj ,their
approximated mean cycle times by QNA are ACTi and ACTj , where i= 1,2,…, N and j=i+1,…, N . (ECTi, ACTi) and (ECTj, ACTj) still has the probability of being a
concordant pair, 𝑃𝑐(𝑖, 𝑗), and the probability of being a discordant pair, 1 − 𝑃𝑐(𝑖, 𝑗).
Thus, if we rank designs completely in accordance with QNA approximations, then the expected number of concordant pairs is ∑𝑁𝑖=1∑𝑁𝑗=𝑖+1𝑃𝑐(𝑖, 𝑗), and the expected number of discordant pairs is ∑𝑁𝑖=1∑𝑁𝑗=𝑖+1(1 − 𝑃𝑐(𝑖, 𝑗)). Since there are 12N(N − 1)
pairs in total, the expected rank correlation is
E[RC] =𝐸[# of concordant pairs]−𝐸[# of discordant pairs]
Total number of pairs
=∑𝑁𝑖=1∑𝑁𝑗=𝑖+1𝑃𝑐(𝑖,𝑗)1−∑𝑁𝑖=1∑𝑁𝑗=𝑖+1(1−𝑃𝑐(𝑖,𝑗))
2N(N−1)
=∑𝑁𝑖=1∑𝑁𝑗=𝑖+11 [2𝑃𝑐(𝑖,𝑗)−1]
2N(N−1) Because of ∑𝑁𝑖=1∑𝑁𝑗=𝑖+11=12N(N − 1),
E[RC] = 2 ×∑𝑁𝑖=11∑𝑁𝑗=𝑖+1𝑃𝑐(𝑖,𝑗) 2N(N−1) − 1
We define the average probability of being a concordant pair of all pairs is 𝑃̅ , 𝑐 𝑃̅ =𝑐 ∑𝑁𝑖=11∑𝑁𝑗=𝑖+1𝑃𝑐(𝑖,𝑗)
2N(N−1) . Then, E[RC] = 2𝑃̅ − 1. 𝑐
5.3 Discussion of Variability
QNA utilizes heterogeneous SCVs to more delicately characterize network flows
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but Jackson network approximation (JNA) only utilizes unity SCVs because of its exponential assumption. Here we further discuss about the effects of variability while approximating network performance and also investigate what advantages for ranking can acquire if taking the second order statistics into consideration. JNA ignores the differences in variance terms and uses only their mean values for evaluation and comparison, which intuitively reduces the recognition between designs, especially for those designs whose mean values are similar.
(1) Performances of D
1and D
2are similar
We discuss in the case of a single GI/G/m queue and also assume there are two allocation designs, 𝐷1 and 𝐷2. The only difference between 𝐷1 and 𝐷2 is the variance of inter-arrival time. However, because the exponential assumptions of arrival and service processes ignores the unique difference between 𝐷1 and 𝐷2, their JNA performances are the same as shown in Figure 5.4 (a).
For QNA approximation, different network configuration resulted from 𝐷1 and 𝐷2 leads to different characterization of traffic flows which approximated by traffic
variability equations as Equation (3.14). QNA utilizes the variance terms to capture more information about arrival and service processes and the minor difference of variabilities is enough to identify which design is better.
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(a) JNA performance (b) QNA performance
Figure 5.4 D1 and D2 are similar, (a) JNA (b) QNA Define the following notations for JNA performance and so as QNA:
∆𝑈𝐵1 : Derivation of upper bound of D1, 𝑈𝐵1𝑄𝑁𝐴− 𝑈𝐵1𝐽𝑁𝐴
∆𝑈𝐵2 : Derivation of upper bound of D2, 𝑈𝐵2𝑄𝑁𝐴− 𝑈𝐵2𝐽𝑁𝐴
Here we ignore the difference of lower bound because that has minor effect in comparison with upper bound. Based on Equation (4.3), if use the JNA performance
of D1 and D2 for ranking, the probability of being a concordant pair is
1
2+12(𝑈𝐵2
𝐽𝑁𝐴−𝑈𝐵1𝐽𝑁𝐴)
(𝑈𝐵2𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴) = 12. If use the QNA performance for ranking, upper bounds of D1 and D2 change ∆𝑈𝐵1 and ∆𝑈𝐵2 respectively, with the variations of bounds, the probability of being a concordant pair is 1
2+12 (∆𝑈𝐵2−∆𝑈𝐵1)
(𝑈𝐵2𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴+∆𝑈𝐵2), which is greater than 12. Because of the additional variability information of each design, variabilities press the differences between those designs with similar performances, andthis assists to differentiate which one design is better and improve the probability of correct ranking. Clearly, the more we capture of these key factors, the greater is our chance of recognizing some differences between designs. Furthermore, it is significant that JNA
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ignores the differences caused by variability and it is a special case of QNA when the true network is exponential distributions (CV=1). Therefore, it is no surprise that QNA has better rank correlation than JNA.
(2) Performances of D
1and D
2have distinct differences
Taking variability into consideration is good for pressing the differences between similar designs but it may sometimes deteriorate ranking. For example, there are two designs whose JNA performances have distinct differences as shown in Figure 5.4(a) and the probability of being a concordant pair is 12+12(𝑈𝐵2
𝐽𝑁𝐴−𝑈𝐵1𝐽𝑁𝐴)
(𝑈𝐵2𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴). If use the QNA performance for ranking, upper bounds of both D1 and D2 change ∆𝑈𝐵1 and ∆𝑈𝐵2 respectively as shown in Figure 5.4(b). With the variations of bounds, the probability of being a concordant pair is 12+12(𝑈𝐵2
𝐽𝑁𝐴+∆𝑈𝐵2−𝑈𝐵1𝐽𝑁𝐴−∆𝑈𝐵1)
(𝑈𝐵2𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴+∆𝑈𝐵2) . Thus, if ∆𝑈𝐵2 is smaller than 𝑈𝐵2
𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴
𝑈𝐵1𝐽𝑁𝐴−𝐿𝐵2𝐽𝑁𝐴∆𝑈𝐵1, then the variation of bounds caused by the increased variability would deteriorate the probability of correct ranking and also cloud the judgment on ranking at the same time. Fortunately, in most of cases ∆𝑈𝐵2 is much greater than ∆𝑈𝐵1 because the upper bound is exponentially increasing with utilization.
We shortly summarize the above discussion here. First, variability benefit press the differences between similar designs and also improve the probability of correct ranking. Second, at the same time, variability may deteriorate the probability of
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correct ranking because of blurring the recognition between designs with distinct differences. But fortunately, the second one is not common.
(a) JNA performance (b) QNA performance Figure 5.5 D1 and D2 have distinct differences, (a) JNA (b) QNA
In this section, we compare JNA and QNA in their ranking performance and also investigate how variability influences the goodness of ranking. In viewpoint of OT, the resulting benefit of variability is significant because it facilitates us to recognize the relative orders of top designs whose performances are similar more precisely.
Even though taking variability into consideration makes the rankings of those designs with distinct differences a little blurred at the same time, this is however the secondary concern.
5.4 Summary
In this chapter, we first extend the BRA of a single GI/G/m queue in Section 4.3 to a general re-entrant line with M stations. Because parametric decomposition regards each station as an independent node given the approximate flow parameters,
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therefore we generalize BRA by superposition of individual stations.
Unfortunately, unlike a single GI/G/m queue, there are interactions among stations in a general production line which causes Lemma 4.5 does not always hold in case of M stations. There are four possible cases of relations of upper and lower bounds and BRA derives the probability of being a concordant pair of these four cases.
Our experimental statistics shows that most of pairs (over 89%) still holds Lemma 4.5 and over 94% pairs have probability of being a concordant pair greater than 0.5. Then, the BRA result is extended from 2 designs to N designs using rank correlation. The effects of variability are also discussed by means of comparing JNA and QNA. From our discussion, variability benefits ranking those top designs because variability presses their differences even though it also blurs some rankings at the same time.
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Chapter 6
Machine Capacity Allocation Experiments
Machine capacity allocation is to determine how best to assign machines in each workstation of a queueing network in order to minimize the expected total mean cycle time. We study systems with several types of products and general inter-arrival/service time distributions. For each product type, process flow is re-entrant through systems, which means product routing makes multiple visits to individual workstations and may also compete for the finite capacity of a workstation.
For such networks there are no analytical formulas for mean cycle time performance.
OO-based method is used to address this kind of problem. Here we conduct an experiment to analyze how the selection of simplified models affects the ranking for OO and compare two simplified models, QNA and JNA, where the main difference is the characterization of variability, one being heterogeneous SCVs and the other being unity SCVs.
Section 6.1 summarizes the flowchart of our experiment. Section 6.2 discusses how to select promising designs after fast evaluations of simplified model for further steps. Section 6.3 describes the simulation model and experiment factors considered in this set of experiments. Section 6.4 shows the experiment results and several
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comparisons on ranking between QNA and JNA. Section 6.5 discusses the efficiency.
6.1 Overview
There are two comparable simplified models, QNA and JNA. In the procedures of parametric decomposition method, it needs the following inputs: product routings, product releases, and processing steps of each product. With above information, QNA
derives all the means and SCVs of both inter-arrival and service times of a re-entrant line. Parameter (𝜆𝑎𝑚, 𝐶𝑎𝑚2 , 𝜏𝑚, 𝐶𝑠𝑚2 ) of each node can be utilized to estimate the
node-level measures and the system-level measures. Due to exponential assumption of JNA, each node is characterized using two parameters (𝜆𝑎𝑚, 1, 𝜏𝑚, 1) which
assumes unity SCVs. Based on these two simplified models, we can quickly take a glance over the whole solution space and rank all designs in terms of approximated performances respectively. According to these rankings, we apply ordinal transformation on the original solution space. Then in the ordinal space, we screen top-ranking designs for further evaluation by DES simulation model. Finally, we simulate all screened designs by detailed DES simulation model and find the optimal one as our best design. This flowchart of experiment is shown in Figure 6.1.
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Figure 6.1 Flowchart of Experiment
6.2 Selection of Top Designs in Ordinal Space
After OT, we select top designs in the ordinal space for further evaluations by DES simulation model, and the resulting set of selected top designs is denoted as S*.
This section emphasizes on how to determine the number of selected designs, |S*|.
First, we show a boundary, k, to guarantee the quality of selected designs in simplified model in Lemma 6.1.
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Lemma 6.1: Given n designs in total and the rank correlation τ between simplified model and detailed model, there exists a boundary, k= ⌊√
(1−𝜏)𝑛(𝑛−1)4
⌋+1, to make sure that in the top-k designs of simplified model there is at least one design also in the
top-k designs of detailed model.
Proof:
In other words, given rank correlation, find the maximum of 𝑘̂ that may result in all of the top-𝑘̂ designs of simplified model being not in the top-𝑘̂ designs of
detailed model. Then, the boundary 𝑘 is ⌊𝑘̂⌋+1.
The maximum of 𝑘̂ occurs when all of the top-𝑘̂ designs of simplified model being not in the top-𝑘̂ designs of detailed model is the only source of discordant pairs, and causes that the number of discordant pairs= 𝑘̂2. From the definition of rank
correlation τ in Section 4.1, τ =Number of concordant pairs−Number of discordant pairs Total number of pairs
and total number of pairs = number of concordant pairs + number of discordant pairs,
1
2𝑛(𝑛 − 1). Thus, the number of discordant pairs is 1−τ2 ×12𝑛(𝑛 − 1).
The maximum of 𝑘̂ is the positive root of 𝑘̂2 = 1−τ2 ×12𝑛(𝑛 − 1), so 𝑘̂ = √(1−𝜏)𝑛(𝑛−1)
4 and 𝑘 = ⌊𝑘̂⌋+1= ⌊√(1−𝜏)𝑛(𝑛−1) 4 ⌋+1.
Q.E.D.
From Lemma 6.1, we can make sure that in the top-k designs of simplified model there is at least one design also in the top- 𝑘 designs of detailed model, where
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𝑘 = ⌊√(1−𝜏)𝑛(𝑛−1)
4 ⌋ +1. Therefore, if the cost of computation of 𝑘 designs is endurable, then we could set the number of selected designs to 𝑘, |S*|= 𝑘, and guarantee that in those 𝑘 designs there exists at least one design placed in the top- 𝑘 of detailed model.
In practice, the number of selected designs depends on how much computation budget we have. If total computation budget is limited and it is not enough to contain the guaranteed k designs, then an intuitive selection strategy is to fully utilize total computation budget. If assume that C(R) is the computation cost of executing R replications of DES simulation and total computation budget is limited to B, then |S*|
is set to 𝐵
𝐶(𝑅).
6.3 Re-entrant Network Models and Experiment Factors
Detailed re-entrant queueing network model is introduced and experiment factors are also discussed in this section.
6.3.1 Simulation model: 5-station and 2-product model
This simulation model, as shown in Figure 6.2, has multiple product types, re-entry, failure-prone, and general service and inter-arrival time distributions. There are two types of products, P1 and P2. Each product has a deterministic routing through the system, as shown in Figure 6.3. There are the mean release rates of 1.0
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jobs/sec for P1 and 1.25 jobs/sec for P2, and their inter-arrival time distributions are shown in Table 6.1.
Figure 6.2 A five-workstation and two-product re-entrant Line
Product 1 (P1) Routing
Enter → 1 → 2 → 3 → 5 → 4 → 3 → 1 → Exit Product 2 (P2) Routing
Enter → 2 → 1 → 4 → 5 → 3 → 4 → 5 → Exit Figure 6.3 Routing of each product
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Table 6.1 Release of each product Product Release
Product Inter-arrival time distribution Mean (sec) SCV
1 Log-normal 1 0.8
2 Log-normal 0.8 0.8
Processing times of one product type at the same workstation are different due to various processing steps. The model involves failure-prone processing workstations, each having one or more identical but independent machines. Both times between failures and times to repair follow exponential distributions, more details in Table 6.2.
Processing times of products at each processing step are shown in Table 6.3 and all are generally distributed.
Table 6.2 Workstation failure setting Workstation
Workstation MTTF1 (sec) MTTR2 (sec) Distributions
1 40 10 Both exponential
2 60 10 Both exponential
3 40 10 Both exponential
4 60 10 Both exponential
5 40 10 Both exponential
MTTF1 : Mean Time To Failure, MTTR2 : Mean Time To Repair
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Table 6.3 Processing steps of each product Product 1, P1
Step At Workstation Service time distribution Mean (sec) SCV
Step At Workstation Service time distribution Mean (sec) SCV