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In prior chapters, the analysis was focused on determining the reliability of the system i. The prior formulations provided us with the probability of success of the entire system, up to a point in time, without looking at the question: "What happens if a component fails during that time and is then fixed? These actions change the overall makeup of the system. These actions must now be taken into consideration when assessing the behavior of the system because the age of the system components is no longer uniform nor is the time of operation of the system continuous.

In attempting to understand the system behavior, additional information and models are now needed for each system component. Our primary input in the prior chapters was a model that described how the component failed its failure probability distribution. When dealing with components that are repaired, one also needs to know how long it takes for the component to be restored. That is, at the very least, one needs a model that describes how the component is restored a repair probability distribution.

In this chapter, we will introduce the additional information, models and metrics required to fully analyze a repairable system. To properly deal with repairable systems, we need to first understand how components in these systems are restored i. In general, maintenance is defined as any action that restores failed units to an operational condition or retains non-failed units in an operational state.

For repairable systems, maintenance plays a vital role in the life of a system. It affects the system's overall reliability, availability, downtime, cost of operation, etc. Generally, maintenance actions can be divided into three types: corrective maintenance, preventive maintenance and inspections. Corrective maintenance consists of the action s taken to restore a failed system to operational status. This usually involves replacing or repairing the component that is responsible for the failure of the overall system.

Corrective maintenance is performed at unpredictable intervals because a component's failure time is not known a priori. The objective of corrective maintenance is to restore the system to satisfactory operation within the shortest possible time. Corrective maintenance is typically carried out in three steps:.

Preventive maintenance, unlike corrective maintenance, is the practice of replacing components or subsystems before they fail in order to promote continuous system operation. The schedule for preventive maintenance is based on observation of past system behavior, component wear-out mechanisms and knowledge of which components are vital to continued system operation.

## Research on Spare Part Requirement for Repairable System Based on the Quasi-Renewal Model

Cost is always a factor in the scheduling of preventive maintenance. In many circumstances, it is financially more sensible to replace parts or components that have not failed at predetermined intervals rather than to wait for a system failure that may result in a costly disruption in operations. Preventive maintenance scheduling strategies are discussed in more detail later in this chapter.

Inspections are used in order to uncover hidden failures also called dormant failures. In general, no maintenance action is performed on the component during an inspection unless the component is found failed, in which case a corrective maintenance action is initiated. However, there might be cases where a partial restoration of the inspected item would be performed during an inspection.

For example, when checking the motor oil in a car between scheduled oil changes, one might occasionally add some oil in order to keep it at a constant level.

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The subject of inspections is discussed in more detail in Repairable Systems Analysis Through Simulation. Maintenance actions preventive or corrective are not instantaneous. There is a time associated with each action i. This time is usually referred to as downtime and it is defined as the length of time an item is not operational. There are a number of different factors that can affect the length of downtime, such as the physical characteristics of the system, spare part availability, repair crew availability, human factors, environmental factors, etc.

Downtime can be divided into two categories based on these factors:. These downtime definitions are subjective and not necessarily mutually exclusive nor all-inclusive. As an example, consider the time required to diagnose the problem. One may need to diagnose the problem before ordering parts and then wait for the parts to arrive. That is, the time-to-repair is a random variable, much like the time-to-failure.

The statement that it takes on average five hours to repair implies an underlying probabilistic distribution.

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Distributions that describe the time-to-repair are called repair distributions or downtime distributions in order to distinguish them from the failure distributions. However, the methods employed to quantify these distributions are not any different mathematically than the methods employed to quantify failure distributions. The difference is in how they are employed i. As an example, when using a life distribution with failure data i. In the case of downtime distributions, the data set consists of times-to-repair, thus what we termed as unreliability now becomes the probability of the event occurring i.

Using these definitions, the probability of repairing the component by a given time, , is also called the component's maintainability. Maintainability is defined as the probability of performing a successful repair action within a given time. In other words, maintainability measures the ease and speed with which a system can be restored to operational status after a failure occurs.

In maintainability, the random variable is time-to-repair, in the same manner as time-to-failure is the random variable in reliability. As an example, consider the maintainability equation for a system in which the repair times are distributed exponentially. Its maintainability is given by:. Note the similarity between this equation and the equation for the reliability of a system with exponentially distributed failure times.

However, since the maintainability represents the probability of an event occurring repairing the system while the reliability represents the probability of an event not occurring failure , the maintainability expression is the equivalent of the unreliability expression,.

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Furthermore, the single model parameter is now referred to as the repair rate, which is analogous to the failure rate, , used in reliability for an exponential distribution. This now becomes the mean time to repair instead of the mean time to failure. The same concept can be expanded to other distributions. In the case of the Weibull distribution, maintainability, , is given by:.

While the mean time to repair is given by:. It should be clear by now that any distribution can be used, as well as related concepts and methods used in life data analysis. The only difference being that instead of times-to-failure we are using times-to-repair. What one chooses to include in the time-to-repair varies, but can include:.

In the interest of being fair and accurate, one should disclose document what was and was not included in determining the repair distribution. This metric is availability. Availability is a performance criterion for repairable systems that accounts for both the reliability and maintainability properties of a component or system. It is defined as the probability that the system is operating properly when it is requested for use. That is, availability is the probability that a system is not failed or undergoing a repair action when it needs to be used.

For example, if a lamp has a Note that this metric alone tells us nothing about how many times the lamp has been replaced. For all we know, the lamp may be replaced every day or it could have never been replaced at all. Other metrics are still important and needed, such as the lamp's reliability. The next table illustrates the relationship between reliability, maintainability and availability. For a repairable system, the time of operation is not continuous. In other words, its life cycle can be described by a sequence of up and down states. The system operates until it fails, then it is repaired and returned to its original operating state.

It will fail again after some random time of operation, get repaired again, and this process of failure and repair will repeat. This is called a renewal process and is defined as a sequence of independent and non-negative random variables. Each time a unit fails and is restored to working order, a renewal is said to have occurred. This type of renewal process is known as an alternating renewal process because the state of the component alternates between a functioning state and a repair state, as illustrated in the following graphic.

A system's renewal process is determined by the renewal processes of its components. For example, consider a series system of three statistically independent components. Each component has a failure distribution and a repair distribution.

Since the components are in series, when one component fails, the entire system fails. The system is then down for as long as the failed component is under repair. The following figure illustrates this. One of the main assumptions in renewal theory is that the failed components are replaced with new ones or are repaired so they are as good as new, hence the name renewal.

One can make the argument that this is the case for every repair, if you define the system in enough detail. In other words, if the repair of a single circuit board in the system involves the replacement of a single transistor in the offending circuit board, then if the analysis or RBD is performed down to the transistor level, the transistor itself gets renewed.

In cases where the analysis is done at a higher level, or if the offending component is replaced with a used component, additional steps are required. We will discuss this in later chapters using a restoration factor in the analysis. For more details on renewal theory, interested readers can refer to Elsayed [7] and Leemis [17]. The definition of availability is somewhat flexible and is largely based on what types of downtimes one chooses to consider in the analysis.

As a result, there are a number of different classifications of availability, such as:. Instantaneous or Point Availability,. Instantaneous or point availability is the probability that a system or component will be operational up and running at any random time, t.

This is very similar to the reliability function in that it gives a probability that a system will function at the given time, t. Unlike reliability, the instantaneous availability measure incorporates maintainability information. At any given time, t, the system will be operational if the following conditions are met Elsayed [7] :. The item functioned properly from to with probability or it functioned properly since the last repair at time u, , with probability:.

With being the renewal density function of the system. Average Uptime Availability or Mean Availability ,. The mean availability is the proportion of time during a mission or time period that the system is available for use. It represents the mean value of the instantaneous availability function over the period 0, T] and is given by:. Steady State Availability,. The steady state availability of the system is the limit of the instantaneous availability function as time approaches infinity or:. In other words, one can think of the steady state availability as a stabilizing point where the system's availability is a constant value.

However, one has to be very careful in using the steady state availability as the sole metric for some systems, especially systems that do not need regular maintenance. A large scale system with repeated repairs, such as a car, will reach a point where it is almost certain that something will break and need repair once a month. However, this state may not be reached until, say, , miles. Obviously, if I am an operator of rental vehicles and I only keep the vehicles until they reach 50, miles, then this value would not be of any use to me. Similarly, if I am an auto maker and only warrant the vehicles to miles, is knowing the steady state value useful?

Inherent Availability,. Inherent availability is the steady state availability when considering only the corrective downtime of the system. This gets slightly more complicated for a system. To do this, one needs to look at the mean time between failures, or , and compute this as follows:.

## Generalized renewal process for repairable systems based on f..|INIS

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First, the failure mechanism of one component may depend on other components when considering component failure dependence. Second, imperfect repair actions can have accumulated effects on the repaired components and these accumulated effects are difficult to measure. In this paper, we propose a parametric statistical model to capture the failure dependence information with general component repair actions.