Constant failure rate

A considerable assumption in the exponential distribution is the assumption of a constant failure rate. Real devices demonstrate a failure rate curve more like that shown in Eigure 9. Eor a new device, the failure rate is initially high owing to manufacturing defects, material defects, etc. This period is called infant mortaUty. EoUowing this is a period of relatively constant failure rate. This is the period during which the exponential distribution is most apphcable. EinaHy, as the device ages, the failure rate eventually increases. 

Confidence Estimation for the Constant Failure Rate Model... 

Thi.s method assumes that X, the total constant failure rate for each unit, can be expanded into independent and dependent failure contributions (equation... 

A type of time dependence that is available in most codes evaluates the exponential distribution at specified times. This is the constant failure rate - constant repair rate approximation ( Section 2.5.2). This may not be realistic as indicated by Figure 2.5-2 in which the failure rate is not constant. Furthermore, Lapides (1976) shows that repair rates are not constant but in many casc. appear to be lognormally distributed. 

Weibull distribution This distribution has been useful in a variety of reliability applications. The Weibull distribution is described by three parameters, and it can assume many shapes depending upon the values of the parameters. It can be used to model decreasing, increasing, and constant failure rates. 

Equations 11-1 through 11-5 are valid only for a constant failure rate fi. Many components exhibit a typical bathtub failure rate, shown in Figure 11-2. The failure rate is highest when the component is new (infant mortality) and when it is old (old age). Between these two periods (denoted by the lines in Figure 11-2), the failure rate is reasonably constant and Equations 11-1 through 11-5 are valid. 

Constant conduction heat pipes, 13 227 Constant failure rate, 13 167 Constant-field scaling, of FETs, 22 253, 254 Constant-modulus alloys, 17 101 Constant of proportionality, 14 237 Constant pressure heat capacity, 24 656 Constant rate drying, 9 103-105 Constant rate period, 9 97 23 66-67 Constant retard ratio (CRR) mode, 24 103 Constant slope condition, 24 136-137 Constant stress test, 13 472 19 583 Constant-voltage scaling, of FETs, 22 253 Constant volume heat capacity, 24 656 Constant volume sampling system (CVS), 10 33... 

As described in Problem HZA.7, the failure rate of equipment frequently exhibits three stages a break-in stage with a declining failure rate, a useful life stage characterized by a fairly constant failure rate, and a wearout period characterized by an increasing failure rate. Many industrial parts and components follow this path. A failure rate curve exhibiting these three phases is called a bathtub curve. 

Midlife or premature failures roughly constant failure rate due to random factors... 

In contrast, the midlife stage has a constant failure rate, because failures mainly occur randomly due to external circumstances. This period is often re-... 

With shape parameter / < 1 a decreasing failure rate can be simulated, insuch as on the left side of the bathtub curve (Fig. 5.9.6). For / =1 the Weibull distribution reduces to a pure exponential with a constant failure rate that is given by the reciprocal of the characteristic time T ... 

Note that the simplest and most obvious initial assumption of a constant failure rate has actually led to an exponential curve. That is because the exponential curve is simply a succession of evenly spaced data points (very close to each other), that are in simple geometric progression, that is, the ratio of any point to its preceding point is a constant. Most natural processes behave similarly, and so e is encountered very frequently. 

Useful life period. After the weaker units die off in the infant-mortality period, the failure rate becomes nearly constant and the assemblies have entered the normal or useful life period. This period is characterized as a relatively constant failure rate and is also referred to as the system life of a product. Mean time between failures is calculated in this section of the curve. Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a system during operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system. Failure rates calculated from MIL-HDBK-217 and Telcordia-332 apply only to this period. 

Constant failure rate or useful life period. 

The most widely used type of failure rate is the constant—or exponential—distribution. The use of the word exponential for constant failure rate seems contradictory. However, the constant failure rate refers to h(t), not fit). In other words, as time progresses the number of items that were operable at time f(0) will decline. However, those items that do survive to time f have the same rate of failure as they had at time f(0). 

The constant failure rate curve shown in Figiue 16.9 is a straight line corresponding to the exponential failure rate aheady discussed. It represents random events that occur independently of time. For example, operator error that can take place at any time. 

Given this data, a conservative way to estimate the constant failure rate is to calculate the average time to failure. It can be shown that a constant failure rate is related to MTTF (mean time to failure) according to the equation below (Ref 6, pg. 73). 

Given those numbers, the average value for time to failure (MTTF) equals 6479 hours. The constant failure rate would be 0.000162 failures per hour. This is calculated by adding the actual time to failure for each of the nineteen failures, plus the number 8761 for the seven successful units, and dividing by nineteen, the number of failures. This is shown in Table 3-3. 

A module has an MTTF of 75 years for all failure modes. Assuming a constant failure rate, what is the total failure rate for aU failure modes ... 

Constant Failure Rate— When a constant failure rate is assumed (which is valid during the useful life of a device), then the relationship between reliability, unreliability, and MTTF are straightforward. If the failure rate is constant, then ... 

Figure 4-5 shows the reliability and unreliability functions for a constant failure rate of 0.001 failures per hour. Note that the plot for reliability looks the same as Figure 4-2, which shows the probability of successful operation given a probability of success for one hour of 0.999. 

Problem A device has a constant failure rate of 5000 FITS during its useful life. What is the MTTF ... 

These results can converted into one of the most well known equations in reliability engineering. Remember that for a single component with a constant failure rate ... 

Since state 0 is the success state, reliability is equal to So(t) and is given by Equation D-11. Unreliability is equal to Sj(t) and is given by Equation D-12. This result is identical to the result obtained when a component has an exponential probability of failure. Thus, the Markov model solution verifies the clear relationship between the constant failure rate and the exponential probability of failure over an interval of time. 

It is also assumed that the single board PEC is a series system (the failure of any component is considered a failure of the unit) with constant failure rates. The failure rates may therefore be added to obtain failure rates for the PEC. 

Constant failure rates and repair rates are assumed. 

This period with its constant failure rate represents random failures of such nature that future failures do not depend on past operation. These random failures are caused mainly by random fluctuations of operating and environmental conditions which cause loads exceeding design strength. In addition failures may occur due to unprofessional maintenance and hidden design and manufacturing flaws which have remained despite the measures mentioned above. 

As already mentioned, the use of components in plants usually begins after the period of early failures and is usually terminated before aging effects become manifest. Hence, there is good reason to assume that components in operation are inscribed in period II and therefore characterized by a constant failure rate () (t) = X = const.). This then leads to exponentially distributed lifetimes which are treated below. They are generally used in safety and risk studies (cf. [27-29]). 

Hence, the average lifetime is equal to the inverse of its failure rate in case the failure rate is constant. The important property of the exponential distribution is that the probability of the failure of a component in the time interval [t, t + At] does not depend on its preceding operating time but only on the value of its failure rate X and the duration of the time interval At. This is shown by inserting Eq. (9.24) in Eq. (9.25). Furthermore, it can be proved that the exponential distribution is the only one with a constant failure rate [30]. 

It must be emphasized that a component whose lifetime is exponentially distributed cannot be improved by maintenance. For an improvement would imply a reduction of its failure rate. In the present model it is ensured that the unavailability is equal to zero after every functional test. This is achieved by determining in the first place whether it is still capable of functioning or has failed. In the latter case the component is either repaired or replaced. If it is still capable of functioning it is as good as new because components with a constant failure rate do not age by definition. If it has to be repaired, as good as new is a hypothesis usually corroborated in plants with a good safety culture. 

In this article, the basic concepts of E-L-M model are formulated mainly for on demand working systems. That does not mean that continually operated systems should be put out of the analysis. Diversity application in the design of continually operated systems appears to be additional, often evenmoie complex topic, going beyond the scope of this paper. It can be discussed, for example, that the traditional assinnp-tions of constant failure rate Poisson model may not be fulfilled within the context of these diversity effects... 

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