System Reliability in .NET

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50.3 System Reliability
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Item Frequency of Failure
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y units with a failure rate of l16 x units with a failure rate of l10
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z units with a failure rate of l27
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Operating Interval
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t1 MTBM (Scheduled Maintenance) DSM Cumulative Failures PDF 100%
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Time
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MTTF (Reliability)
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Where: DSM = Design Safety Margin MTBF = Mean-Time-To-Failure MTBM = Mean-Time-Between-Maintenance PDF = Probability Density Function
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Time 5
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Figure 50.6 Maintenance Interval Concepts
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ing a failure rate of l1. The last failure, a single instance, occurs at tn indicating a failure rate of ln. Based on this analysis, we appropriately label the mean, m, as the Mean Time To Failure (MTTF). So, what is the point of this discussion It provides insights about an entity s failure frequency distribution. So, if we are asked, how many failures can we expect at tx, we can point to Figure 50.5 at time tx and observe that based on a sample testing of 50 lightbulbs, a speci c number of failures can be expected to fail. In this regard, we refer to this as the instantaneous failure rate at time, tx. Now suppose that we are asked WHAT percentage of the lightbulbs can be expected to fail by tx. This brings us to our next topic, the probability density function (PDF) of cumulative failure. Cumulative Failure Probability Density Function (PDF). Beginning with t1 when the rst bulb fails, we plot the cumulative failures over time until all have failed by tn. Since the area under the Normal Distribution curve is normalized to 1.0, we can exploit the characteristic pro le to derive probabilities. As a result, we plot the cumulative probability density function of the frequency distribution as illustrated in Panel B of Figure 50.6. Thus, we can say that at tm, there is a probability of 0.50 that 25 lightbulbs can be expected to fail between t = 0 and that instant in time. MTBF, TTF, and MTTR Relationships. Engineers often interchange terms that have subtle but important differences. Such is the case with two reliability terms, Mean-Time-Between Failures (MTBF) and Mean-Time-To-Failure (MTTF). MTTF, as the name implies, encompasses the elapsed time duration for an item from installation until failure. Once the item fails, there is a Mean-Time-To-Repair, (MTTR) to perform a corrective action. In contrast, MTBF is the summation of MTTF and MTTR as indicated in Eq. 50.10:
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Reliability, Availability, and Maintainability
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MTBF = MTTF + MTTR The mean failure rate, m, is the reciprocal of MTTF: m Since MTTF >> MTTR, we can say that: MTBF @ MTTF Therefore: m@ 1 MTBF 1 MTTF
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(50.10)
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(50.11)
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(50.12)
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Modeling Reliability Con gurations
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Given a basic understanding in entity reliability, we now shift our attention to modeling con gurations of entity reliabilities. Three basic constructs enable us to compute the reliability of networked entities. These constructs are: 1) series, 2) parallel, and 3) series parallel. Let s explore each of these individually. Series Network Con guration Reliability. The rst reliability network construct is a series network con guration. This con guration consists of two or more entities connected in SERIES as shown in Panel A of Figure 50.7. Mathematically, we express this relationship as follows: RSeries = ( R1 )( R2 )( R3 ) K ( Rn ) where RSeries = overall reliability of the series network con guration R1 = reliability of Item 1 R2 = reliability of Item 2 ... Rn = reliability of Item n Substituting Eq. (50.6) into Eq. (50.13) using a NEGATIVE exponential distribution: RSeries = e - ( l + l
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1 2 +l3
(50.13)
Kl n )( t )
(50.14)
where: ln = unique failure rate of the nth item t = NOMINAL operating cycle duration Parallel Network Con guration Reliability. The second reliability network construct is based on a set entities connected in parallel as illustrated in Panel B of Figure 50.7. We refer to this construct as a Parallel Reliability Network and express this relationship as follows: RParallel = ( R1 ) + ( R2 ) - ( R1 )( R2 ) for one out of two redundancies (50.15)