Bathtub curve

Figure 2.5-2 depicts the force of mortality as a bathtub curve for the life-death history of a component without repair. The reasons for the near universal use of the constant X exponential distribution (which only applies to the mid-life region) are mathematical convenience, inherent truth (equation 2.5-19), the use of repair to keep components out of the wearout region, startup testing to eliminate infant mortality, and detailed data to support a time-dependent X. 

Explain why a plant accident is more likely to happen during startup of a new plant or a retro-fit process. Refer to Chapter 20 and careful review the presentation or tlie bathtub curve tliat is represented by the Weibull distribution. 

Monitoring the trends of a machine-train or process system will provide the ability to prevent most catastrophic failures. The trend is similar to the bathtub curve used to schedule preventive maintenance. The difference between the preventive and predictive bathtub curve is that the latter is based on the actual condition of the equipment, not a statistical average. 

Throughout this book reviews have been made on products that literally are used in many different markets. This action fits the usual statement that this is the World of Plastics Important with all the cost analysis is that profits have to be included. Influencing factors that involve profits are summarized in Figs. 9-10 to 9-13. The life-history curve, Fig. 9-11, shows the basic format of a typical product cycle for an infinite number of products. It is also called a bathtub curve. 

Early failures and wear out failures are reflected in the curve known as the bathtub curve (see Figure 11.14). 

FIGURE 11.14 Failure bathtub curve statistic. The early failure drop is generally canceled with design, type, and routine tests. 

The Weibull distribution provides a inatlieiiiatical model of all tluee stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tluee stages of tlie bathtub curve is... 

To illustrate probability calculations involving Uie exponenUal and Weibull distributions introduced in conjunction with Uie bathtub curve of failure rate, consider first Uie case of a transistor having a constant rate of failure of... 

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. 

The LINUX cluster had been managed under continually increasing load for a period of months. Systems administrators had responded by incremental addition of PCs, but the performance had shown continued decline. The system was loaded near capacity, which is often far below theoretical, 70% of maximum being a common benchmark. Another feature of the system, which is especially relevant to clusters, was the impact of component failure. For computers, this is likely the disk drive. Drives have a mean time between failure of one to two years and follow a bathtub curve of burn-in failure, followed by reliable performance and then high failure near the end of the lifecycle. Cluster design must take into account the need for graceful degradation. 

Fig. 5.9.6 Schematic bathtub curve of failure rates over time with increasing reliability, the curves are lower and less steep at the end of a product s life...

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 ... 

Develop bathtub curve based on the failure rate data and the test data. 

Pattern A. Referred to as a bathtub curve with three identifiable regions, namely (i) the initial period of high probability of failure (ii) region of constant and low probability of failure (iii) a wear-out region of high probability of failure. 

The lifetime of a population of units at the component, board, box, or system level can be divided into three distinct periods. This is most often defined by the so-called reliability bathtub curve (Fig. 6.16). The bathtub curve describes the cradle-to-grave failure rates or frequency of failures as a function of time. The curve is divided into three distinct areas early failure rate (also known as infant mortality), the useful life period, and the wearout failure period. The infant-mortality portion of the curve, also known as the early life period, is the initial steep slope from the start to... 

The graphical results of various Ss on the Weibull analysis can be seen in Fig. 6.17. If all curves are taken together, the result is the bathtub curve. 

Figure 6.17 Weibull plots and the bathtub curve (a) infant-mortality period, (b) infant mortality and useful life period and (c) useful life and end-of-life period.

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