Repairable Systems

Background

Most complex systems, such as automobiles, communication systems, aircraft, aircraft engine controllers, printers, medical diagnostics systems, helicopters, etc., are repaired and not replaced when they fail. When these systems are fielded or subjected to a customer use environment, it is often of considerable interest to determine the reliability and other performance characteristics under these conditions. Areas of interest may include assessing the expected number of failures during the warranty period, maintaining a minimum mission reliability, addressing the rate of wearout, determining when to replace or overhaul a system and minimizing life cycle costs. In general, a distribution, such as the Weibull distribution, cannot be used to address these issues. In order to address the reliability characteristics of complex repairable systems, a process is often used instead of a distribution. The most popular process model is the Power Law model. This model is popular for several reasons. One is that it has a very practical foundation in terms of minimal repair. This is the situation when the repair of a failed system is just enough to get the system operational again. Second, if the time to first failure follows the Weibull distribution, then each succeeding failure is governed by the Power Law model in the case of minimal repair. From this point of view, the Power Law model is an extension of the Weibull distribution.

This section of the on-line reference includes the following subsections:

Distribution Example for Repairable Systems

Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each failure, the socket is put back into an "as good as new" condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a renewal process. In the illustration below, the component life is Xj and tj is the system time to the jth failure.u

Figure 10.1: Component life and system time

Each component life Xj in the socket is governed by the same distribution F(x).

A distribution, such as the Weibull, governs a single lifetime. There is only one event associated with a distribution. The distribution F(x) is the probability that the life of the component in the socket is less than x. In the illustration above, X1 is the life of the first component in the socket. F(x) is the probability that the first component in the socket fails in time x. When the first component fails, it is replaced in the socket with a like component. The probability that the life of the second component is less than x is given by the same distribution function, F(x). For the Weibull distribution:

A distribution is also characterized by its density function, such that:

The density function for the Weibull distribution is:

In addition, an important reliability property of a distribution function is the failure rate given by:

The interpretation of the failure rate is that for a small interval of time , is approximately the probability that a component in the socket will fail between time x and time , given that the component has not failed by time x. For the Weibull distribution, the failure rate is given by:

It is important to note the condition that the component has not failed by time x. Again, a distribution deals with one lifetime of a component and does not allow for more than one failure. The socket has many failures and each failure time is individually governed by the same distribution. In other words, the failure times are independent of each other. If the failure rate is increasing, then this is indicative of component wearout. If the failure rate is decreasing, then this is indicative of infant mortality. If the failure rate is constant, then the component failures follow an exponential distribution. For the Weibull distribution, the failure rate is increasing for , decreasing for and constant for . Each time a component in the socket is replaced, the failure rate of the new component converts back to the value at time 0. This means that the socket is as good as new after each failure and the subsequent replacement by a new component. This process is continued for the operation of the socket.

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Process Example for Repairable Systems

Now suppose that a system consists of many components with each component in a socket. A failure in any socket constitutes a failure of the system. Each component socket is a renewal process governed by its respective distribution function. When the system fails due to a failure in a socket, the component is replaced and the socket is again as good as new. The system has been repaired. Because there are many other components still operating with various ages, the system is not typically put back into a like new condition after the replacement of a single component. For example, a car is not as good as new after the replacement of a failed water pump. Therefore, distribution theory does not apply to the failures of a complex system, such as a car. In general, the intervals between failures for a complex repairable system do not follow the same distribution. Distributions apply to the components that are replaced in the sockets but not at the system level. At the system level, a distribution applies to the very first failure. There is one failure associated with a distribution. For example, the very first system failure may follow a Weibull distribution.

For many systems in a real world environment, a repair is only enough to get the system operational again. If the water pump fails on the car, the repair consists only of installing a new water pump. If a seal leaks, the seal is replaced but no additional maintenance is done, etc. This is the concept of minimal repair. For a system with many failure modes, the repair of a single failure mode does not greatly improve the system reliability from what it was just before the failure. Under minimal repair for a complex system with many failure modes, the system reliability after a repair is the same as it was just before the failure. In this case, the sequence of failure at the system level follows a non-homogeneous Poisson process (NHPP).

The system age when the system is first put into service is time 0. Under the NHPP, the first failure is governed by a distribution F(x) with failure rate r(x). Each succeeding failure is governed by the intensity function u(t) of the process. Let t be the age of the system and is very small. The probability that a system of age t fails between t and is given by the intensity function . Notice that this probability is not conditioned on not having any system failures up to time t, as is the case for a failure rate. The failure intensity u(t) for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, u(t) = r(t), where r(t) is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

Under minimal repair, the system intensity function is:

(1)

This is the Power Law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the Power Law model governs each succeeding system failure. If the system has a constant failure intensity u(t) = , then the intervals between system failures follow an exponential distribution with failure rate . If the system operates for time T, then the random number of failures N(T) over 0 to T is given by the Power Law mean value function.

(2)

Therefore, the probability N(T) = n is given by the Poisson probability.

This is referred to as a homogeneous Poisson process because there is no change in the intensity function. This is a special case of the Power Law model for . The Power Law model is a generalization of the homogeneous Poisson process and allows for change in the intensity function as the repairable system ages. For the Power Law model, the failure intensity is increasing for (wearout), decreasing for (infant morality) and constant for (useful life).