Template:Using power law model rsa

Using the Power Law Model to Analyze Complex Repairable Systems
The Power Law model is often used to analyze the reliability for complex repairable systems in the field. A system of interest may be the total system, such as a helicopter, or it may be subsystems, such as the helicopter transmission or rotator blades. When these systems are new and first put into operation, the start time is $$0$$. As these systems are operated, they accumulate age, e.g. miles on automobiles, number of pages on copiers, hours on helicopters. When these systems fail, they are repaired and put back into service.

Some system types may be overhauled and some may not, depending on the maintenance policy. For example, an automobile may not be overhauled but helicopter transmissions may be overhauled after a period of time. In practice, an overhaul may not convert the system reliability back to where it was when the system was new. However, an overhaul will generally make the system more reliable. Appropriate data for the Power Law model is over cycles. If a system is not overhauled, then there is only one cycle and the zero time is when the system is first put into operation. If a system is overhauled, then the same serial number system may generate many cycles. Each cycle will start a new zero time, the beginning of the cycle. The age of the system is from the beginning of the cycle. For systems that are not overhauled, there is only one cycle and the reliability characteristics of a system as the system ages during its life is of interest. For systems that are overhauled, you are interested in the reliability characteristics of the system as it ages during its cycle.

For the Power Law model, a data set for a system will consist of a starting time $$S$$, an ending time  $$T$$  and the accumulated ages of the system during the cycle when it had failures. Assume the data exists from the beginning of a cycle (i.e. the starting time is 0), although non-zero starting times are possible with the Power Law model. For example, suppose data has been collected for a system with 2000 hours of operation during a cycle. The starting time is $$S=0$$  and the ending time is  $$T=2000$$. Over this period, failures occurred at system ages of 50.6, 840.7, 1060.5, 1186.5, 1613.6 and 1843.4 hours. These are the accumulated operating times within the cycle and there were no failures between 1843.4 and 2000 hours. It may be of interest to determine how the systems perform as part of a fleet. For a fleet, it must be verified that the systems have the same configuration, same maintenance policy and same operational environment. In this case, a random sample must be gathered from the fleet. Each item in the sample will have a cycle starting time $$S=0$$, an ending time  $$T$$  for the data period and the accumulated operating times during this period when the system failed.

There are many ways to generate a random sample of $$K$$  systems. One way is to generate $$K$$  random serial numbers from the fleet. Then go to the records corresponding to the randomly selected systems. If the systems are not overhauled, then record when each system was first put into service. For example, the system may have been put into service when the odometer mileage equaled zero. Each system may have a different amount of total usage, so the ending times, $$T$$, may be different. If the systems are overhauled, then the records for the last completed cycle will be needed. The starting and ending times and the accumulated times to failure for the $$K$$  systems constitute the random sample from the fleet. There is a useful and efficient method for generating a random sample for systems that are overhauled. If the overhauled systems have been in service for a considerable period of time, then each serial number system in the fleet would go through many overhaul cycles. In this case, the systems coming in for overhaul actually represent a random sample from the fleet. As $$K$$  systems come in for overhaul, the data for the current completed cycles would be a random sample of size  $$K$$.

In addition, the warranty period may be of particular interest. In this case, randomly choose $$K$$  serial numbers for systems that have been in customer use for a period longer than the warranty period. Then check the warranty records. For each of the $$K$$  systems that had warranty work, the ages corresponding to this service are the failure times. If a system did not have warranty work, then the number of failures recorded for that system is zero. The starting times are all equal to zero and the ending time for each of the $$K$$  systems is equal to the warranty operating usage time, e.g. hours, copies, miles. In addition to the intensity function $$u(t)$$  given by Eqn. (intensity) and the mean value function given by Eqn. (expected failures), other relationships based on the Power Law are often of practical interest. For example, the probability that the system will survive to age $$t+d$$  without failure is given by:


 * $$R(t)={{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}}$$

This is the mission reliability for a system of age $$t$$  and mission length  $$d$$.