Concept of Reliability Prepared By Dr. M. S. Memon Department of Industrial Engineering and Management Mehran University of Engineering and Technology Jamshoro, Sindh, Pakistan
RELIABILITY Reliability is the probability of a product performing its intended function for a stated period of time under certain specified conditions. Four aspects of reliability are apparent from this definition. First, reliability is a probability-related concept; the numerical value of this probability is between 0 and 1. Second, the functional performance of the product has to meet certain stipulations. Third, reliability implies successful operation over a certain period of time. Fourth, operating or environmental conditions under which product use takes place are specified.
LIFE-CYCLE CURVE AND PROBABILITY DISTRIBUTIONS IN MODELING RELIABILITY Most products go through three distinct phases from product inception to wear-out. Figure shows a typical life-cycle curve for which the failure rate λ is plotted as a function of time. This curve is often referred to as the bathtub curve; it consists of the debugging phase, the chance-failure phase, and the wear-out phase. FIGURE: Typical life-cycle curve
RELIABILITY The debugging phase, also known as the infant-mortality phase, exhibits a drop in the failure rate as initial problems identified during prototype testing are ironed out. The chance failure phase, between times t 1 and t 2, is then encountered; failures occur randomly and independently. This phase, in which the failure rate is constant, typically represents the useful life of the product. Following this is the wear-out phase, in which an increase in the failure rate is observed. Here, at the end of their useful life, parts age and wear out.
Probability Distributions to Model Failure Rate Exponential Distribution The life-cycle curve shown in above figure shows the variation of the failure rate as a function of time. For the chancefailure phase, which represents the useful life of the product, the failure rate is constant. As a result, the exponential distribution can be used to describe the time to failure of the product for this phase. The probability density function of exponential distribution is given by where λ denotes the failure rate
Probability Distributions to Model Failure Rate The mean time to failure (MTTF) for the exponential distribution is given as Thus, if the failure rate is constant, the mean time to failure is the reciprocal of the failure rate. For repairable equipment, this is also equal to the mean time between failures (MTBF). There will be a difference between MTBF and MTTF only if there is a significant repair or replacement time upon failure of the product.
Probability Distributions to Model Failure Rate The reliability at time t, R(t), is the probability of the product lasting up to at least time t. It is given by where F(t) represents the cumulative distribution function at time t. Adjacent Figure shows the reliability function, R(t), for the exponential failure distribution. At time 0, the reliability is 1, as it should be. Reliability decreases exponentially with time. FIGURE: Reliability function for the exponential time-to-failure distribution
Probability Distributions to Model Failure Rate In general, the failure-rate function r(t) is given by the ratio of the time-to-failure probability density function to the reliability function. We have For the exponential failure distribution implying a constant failure rate, as mentioned earlier. FIGURE: Reliability function for the exponential time-to-failure distribution
Example 1 An amplifier has an exponential time-to-failure distribution with a failure rate of 8% per 1000 hours. What is the reliability of the amplifier at 5000hours? Find the mean time to failure.
Example 2 What is the highest failure rate for a product if it is to have a probability of survival (i.e., successful operation) of 95% at 4000 hours? Assume that the time to failure follows an exponential distribution.
Availability The availability of a system at time t is the probability that the system will be up and running at time t. To improve availability, maintenance procedures are incorporated, which may include periodic or preventive maintenance or condition-based maintenance. An availability index is defined as
Availability Downtime may consist of active repair time, administrative time (processing of necessary paperwork), and logistic time (waiting time due to lack of parts). It is observed that maintainability is an important factor in influencing availability. Through design it is possible to increase the reliability and hence operational probability of a system. Further, downtime can be reduced through adequate maintenance plans. For a steady-state system, denoting the mean time to repair (MTTR) to include all the various components of downtime, we have In the situation when the time-to-failure distribution is exponential (with a failure rate λ) and the time-to-repair distribution is also exponential (with a repair rate μ), the availability is given by μ /(λ + μ).
SYSTEM RELIABILITY Most products are made up of a number of components. The reliability of each component and the configuration of the system consisting of these components determines the system reliability (i.e., the reliability of the product). Although product design, manufacture, and maintenance influence reliability, improving reliability is largely the domain of design. One common approach for increasing the reliability of the system is through redundance in design, which is usually achieved by placing components in parallel: As long as one component operates, the system operates. Here we demonstrate how to compute system reliability for systems that have components in series, in parallel, or both.
Systems with Components in Series Figure shows a system with three components (A, B, and C) in series. For the system to operate, each component must operate. It is assumed that the components operate independent of each other (i.e., the failure of one component has no influence on the failure of any other component). In general, if there are n components in series, where the reliability of the i th component is denoted by R i, the system reliability is A B C FIGURE: System with components in series.
Example 3 A module of a satellite monitoring system has 500 components in series. The reliability of each component is 0.999. Find the reliability of the module. If the number of components in series is reduced to 200, what is the reliability of the module?
Systems with Components in Series Use of the Exponential Model If the system components can be assumed to have a time to failure given by the exponential distribution and each component has a constant failure rate, we can compute the system reliability, failure rate, and mean time to failure. As noted earlier, when the components are in the chance-failure phase, the assumption of a constant failure rate should be justified.
Systems with Components in Series Suppose that the system has n components in series, each with exponentially distributed time-to-failure with failure rates λ 1, λ 2, λ n. The system reliability is found as the product of the component reliabilities: implies that the time to failure of the system is exponentially distributed with an equivalent failure rate of σ n i=1 λ i. Thus, if each component that fails is replaced immediately by another that has the same failure rate, the mean time to failure for the system is given by
Systems with Components in Series When all components in series have an identical failure rate, say λ, the MTTF for the system is given by
Example 4 The automatic focus unit of a television camera has 10 components in series. Each component has an exponential time-to-failure distribution with a constant failure rate of 0.05 per 4000 hours. What is the reliability of each component after 2000 hours of operation? Find the reliability of the automatic focus unit for 2000 hours of operation. What is its mean time-to-failure?
Example 5 Refer to Example 4 concerning the automatic focus unit of a television camera, which has 10 similar components in series. It is desired for the focus unit to have a reliability of 0.95 after 2000 hours of operation. What would be the mean time to failure of the individual components?
Systems with Components in Parallel System reliability can be improved by placing components in parallel. The components are redundant; the system operates as long as at least one of the components operates. The only time the system fails is when all the parallel components fail. Figure demonstrates an example of a system with three components (A, B, and C) in parallel. All components are assumed to operate simultaneously. A B C
Systems with Components in Parallel Suppose that we have n components in parallel, with the reliability of the i th component denoted by R i, i=1, 2,..., n. Assuming that the components operate randomly and independently of each other, the probability of failure of each component is given by F i = 1- R i. Now, the system fails only if all the components fail. Thus, the probability of system failure is
Systems with Components in Parallel Thus, the probability of system failure is
Systems with Components in Parallel Use of the Exponential Model If the time to failure of each component can be modeled by the exponential distribution, each with a constant failure rate λ i, i= 1,..., n, the system reliability, assuming independence of component operation, is given by In the special case where all components have the same failure rate λ, the system reliability is given by
Systems with Components in Parallel the mean time to failure for the system with n identical components in parallel, assuming that each failed component is immediately replaced by an identical component, is given by
Example 6 Find the reliability of the system shown with three components (A, B, and C) in parallel. The reliabilities of A, B, and C are 0.95, 0.92, and 0.90, respectively. Note that the system reliability is much higher than that of the individual components. Designers can increase system reliability by placing more components in parallel, but the cost of the additional components necessitates a trade-off between the two objectives.
Example 7 For the system shown in Figure, determine the system reliability for 2000 hours of operation, and find the mean time to failure. Assume that all three components have an identical time-to-failure distribution that is exponential, with a constant failure rate of 0.0005/ hour. What is the mean time to failure of each component? If it is desired for the system to have a mean time to failure of 4000 hours, what should the mean time to failure be for each component? A B C
Example 7 By placing three identical components in parallel, the system MTTF has been increased by about 83.3%.
Example 7
Systems with Components in Series and in Parallel Complex systems often consist of components that are both in series and in parallel. Reliability calculations are based on the concepts discussed previously, assuming that the components operate independently.
Example 8 Find the reliability of the eightcomponent system shown in Figure; some components are in series and some are in parallel. The reliabilities of the components are as follows: R A1 = 0.92, R A2 = 0.90, R A3 = 0.88, R A4 = 0.96, R B1 = 0.95, R B2 = 0.90, R B3 = 0.92, and R C1 = 0.93. A1 A3 A2 A4 B1 B2 B3 C1
Example 8
Example 9 Find the system failure rate and the mean time to failure for the eight component system shown in previous Figure of example 8. The failure rates (number of units per hour) for the components are as follows: λ A1 = 0.0006, λ A2 = 0.0045, λ A3 = 0.0035, λ A4 = 0.0016, λ B1 = 0.0060, λ B2 = 0.0060, λ B3 = 0.0060, and λ C1 = 0.0050.
Example 9