This section gives a description of some of the commonly used terms in the thesis. The following information is based on Andrews and Moss (2002) which also contains further reading on these topics.
1.3.1 Hazard Rate
The hazard rate, or conditional failure rate, is a measure of the rate at which a component or system fails. By plotting the hazard rate against time, the curve of the graph usually follows that of the bath-tub curve. Figure 1.1 shows a generalised view of the bath-tub curve.
Figure 1.1 shows three distinct phases of a component or system’s life-cycle. The first, burn-in, shows a decreasing hazard rate as component manufacturing defects are most likely to present themselves early in the component’s life-cycle. The second phase, useful life, shows a constant failure rate as a result of random failures. The final phase, wearout, shows an increasing failure rate as the component/system deteriorates with age.
1.3.2 Reliability and Unreliability
The reliability, R(t), of a component or system is the probability that an item (component, equipment, or system) will operate without failure for a stated period of time under specified conditions. This is a measure that the item under consideration is successful over a given period of time. Equation 1.3.1 represents the reliability of a system with a constant hazard rate, λ. Constant hazard rate is also referred to as the failure rate.
R(t) = e−λt (1.3.1) The unreliability, F (t), of a component or system is the probability that a com- ponent/system fails to work continuously over a specified time period, under specified conditions. The relationship between reliability and unreliability is given as follows:
F (t) = 1 − R(t) (1.3.2)
1.3.3 Availability and Unavailability
The availability, A(t), of a component or system is defined as the probability that the component or system is working at a particular instant. Alternatively, it is the fraction of the total time the component or system is able to undertake its required function. The availability of a system is an important measure of the performance of the system. This value is calculated when a system failure can be tolerated and repair can be initiated. Equation 1.3.3 represents the availability of a system.
A = M T T F
M T T F + M T T R (1.3.3)
Where the Mean Time to Failure (MTTF) is defined as the reciprocal of the failure rate
1 λ
and the Mean Time to Repair (MTTR) is defined as the average time taken from the failure of a system to its start-up, τ . The MTTR is also defined as the reciprocal of the repair rate ν1.
The unavailability, Q(t), of a component or system is the counterpart to availability and is the probability that a component or system does not perform its required function for time t. Unavailability has the following relationship:
Q(t) = 1 − A(t) (1.3.4)
1.3.4 Maintenance Policies
There are three types of maintenance policies for systems or components: no repair, scheduled maintenance and unscheduled maintenance. Each of these is given in detail below.
1.3.4.1 No Repair
For this type of policy there is no maintenance once a system fails. If a system is said to be working at a given time t, then the system must have been working continuously up
to time t. Therefore the reliability and availability are equal. Equation 1.3.1 would be applicable for analysing a system with this policy.
1.3.4.2 Scheduled Maintenance
Faults in systems do not always become apparent the moment they have failed. This can occur when a system is dormant and can only be detected when there is a demand on the system, or is discovered during a scheduled maintenance. This can be quantified by finding the probability of the system being in a failed state at any time by equation 1.3.5.
QAV = λ θ 2+ τ (1.3.5)
Where λ is the unrevealed failure rate of the system and θ is the test interval.
Equation 1.3.5 can be approximated to equation 1.3.6 when the mean repair time, τ , is much shorter than the test interval, θ.
QAV =
λθ
2 (1.3.6)
For the scheduled maintenance policy an inspection is carried out after a fixed time interval. When a failure is discovered during this inspection, repair is initiated. As this maintenance is based on the time between inspections, θ, the unavailability is a function of this time. Therefore, equation 1.3.5 is used for the average unavailability of the system. Another form of the average unavailability can be found from integrating between the interval times as shown below. This is more accurate than the simplified equation given in equation 1.3.6. Equation 1.3.7 shows the integral between t = 0 and t = θ, the first inspection period. This equation represents the unavailability of a system (or component). Between these inspection intervals the system (or component) is non-repairable.
QAV = 1 θ Z θ 0 1 − e−λtdt (1.3.7)
By integrating, equation 1.3.7 this gives equation 1.3.8.
QAV = 1 − 1 λθ 1 − e−λθ (1.3.8) 1.3.4.3 Unscheduled Maintenance
This policy initiates any repairs when a failure occurs. For this type of maintenance policy the analysis is only dependent on the failure and repair rate of the system (or component), as the fault is known as soon as it occurs; therefore there is no detection time. By the use of Laplace transforms it can be shown that the unavailability of a system (or component) is given by equation 1.3.9.
Q(t) = λ λ + ν
1 − e−(λ+ν)t (1.3.9)
For components that have settled down into their steady state, taking t → ∞ in equation 1.3.9 gives the steady state equation, equation 1.3.10.
Q = λ
λ + ν (1.3.10)
This can be simplified further, given that the MTTF will be significantly larger than the MTTR, therefore reducing equation 1.3.10 to equation 1.3.11.
Q = λτ (1.3.11)
1.3.5 Cut Sets and Minimal Cut Sets
A failure mode, or system failure mode, is the failure of a system that can occur through the failure of a single component or a combination of components in that system. These failure modes can be defined by cut sets. Cut sets are a list of basic components or combinations of basic components that, should they fail, would cause a system failure event. Minimal cut sets are an extension of this concept that expresses the minimal set of components that is sufficient to cause each failure event.
1.3.6 Implicants and Prime Implicants
Implicants and Prime Implicants are similar to cut sets and minimal cut sets, in that they show what components, or combination of components, cause a system failure event. The difference is that implicants are combinations of working and failed components that cause a system failure event. Prime implicants are the minimal, but sufficient combinations of working and failed components required to cause a failure event.