1.1 Background
1.1.5 Introduction to RAMS
Real life systems – e.g. chemical plants, waste management systems – are very complex, containing hundreds or even millions of parts (Ireson et al., 1996). System reliability can generally be calculated as the product of the reliabilities of system components. However, this task can be more complicated sometimes. The number of system components has to be considered both in series and parallel systems. As the number of components increases, the probability of failures increases as well, as shown in Table 1.1 (Kuo and Zuo, 2003).
Generally, the higher is reliability, the more expensive are the processes and the less expensive are maintenance actions. The main problem is to find appropriate reliability parameters for specific scenarios while considering overall system cost and the environmental impact.
Table 1.1 Affect of complexity to the reliability of series equipment.
Number of
Availability is the probability of the successful operation of a system in a determined period of time. It can be calculated by the ratio between life time and total time between failures of the equipment (de Castro and Cavalca, 2006).
life time life time MTBF
total time lifetime + repair time MTBF + MTTR
A= = = (1.1)
where
MTBF is the Mean Time Between Failures, the inverse of the failure rate, MTTR is the Mean Time To Repair, the inverse of the repair rate.
There are three frequently-used terms defined by Ireson et al. (1996) and elsewhere:
Inherent availability:
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Further specifications exist as well, for example the ones given by Hosford (1960):
• Pointwise availability is the probability that the system will be able to operate within the tolerances at a given instant of time.
• Interval availability is the expected fraction of a given interval of time that the system will be able to operate within the tolerances.
o A special type of interval availability was defined by Barlow and Proschan (1996), called limiting interval availability. It is the expected fraction of time in the long run that the system operates satisfactorily.
Ai is the availability of the components of subsystem i yi is the number of redundant components in subsystem i Comparing downtimes is an intuitive way to express availability.
Table 1.2 Comparison of availability values and corresponting downtimes.
Availability “Nines” According to Table 1.2, 90% availability means 36.5 days downtime a year, while 99.9999%
availability corresponds to the downtime of 31 s/y.
Component availability can be increased by improving reliability and maintainability. If reliability is increased, the system can work for longer periods of time. If the maintenance program is improved, the system can be repaired quickly (de Castro and Cavalca, 2006).
Several methods exist for availability calculations, including weakest link, blocking, probabilistic, simple stochastic, and stochastic. The choice depends on the system, the data available, the degree of accuracy required and the time devoted to the analysis.
1.1.5.2 Reliability
Reliability is the probability that a system will perform satisfactorily for at least a given period of time t when used under stated conditions (Kuo and Zuo, 2003).
Interval reliability is the probability that at a specified time, the system is operating and will continue to operate for an interval of duration x (Barlow and Hunter, 1961). The interval reliability R x T
(
,)
for an interval of duration x starting at time T was mathematically given by Barlow and Proschan (1996) as(
,) ( )
1,R x T =P X t = T t T x≤ ≤ + (1.6)
Reliability is often expressed as Eti et al. (2007) defined, i.e.
Introduction This function is used for computing reliabily values and referred as the reliability function. For a given value of t, R t
( )
is the probability that T t≥ . The probability of failure occurance before time t can be defined as( )
1( )
Pr{ }
F t = −R t = T t≤ where F t
( )
≥0, 0F( )
=0 (1.9)This function is called the cumulative distribution function (CDF) and used for failure probability calculations.
The probability density function (PDF) describes the shape of the failure and defined as:
( )
dF t( )
dR t( )
System reliability function is an analytical expression that describes the reliability of the system as a function of time based on the reliability functions of its components.
In order to understand reliability calculations, a deeper analysis of probability theory is beneficial. Many features in reliability engineering are related to probability. The time for event occurrence, also known as the failure time, can be defined as a non-negative random variable.
Conditional probabilities of event occurrences should be considered for reliability analyses.
The performed analyses are, in many cases, based on assumptions (e.g., Kijima, 1989). Future probability of failure or additional cost can be expected only. Beyond those analyses that are based on failure history data, reliability estimations, predictions and simulation widely use probabilistic values. This is the reason for using applied probability described by Asmussen (1987) and others. Real life data sets deal with stochastic variables (Osaki, 2002).
Assume the following scenario. A subsystem has two components and the subsystem fails if either component fails (or both fail). Then, A is the event of Component 1 failure and B is the event of Component 2 failure. The system probability of failure (unreliability) is
( ) ( ) ( ) ( )
Pf =Q P A B= ∪ =P A +P B P A B− ∩ (1.11)
If there is independence between the components, the following equation holds:
( ) ( ) ( ) ( ) ( )
Pf =P A B∪ =P A +P B P A P B− ⋅ (1.12)
i.e. the probability of failure can be calculated by subtracting the product of the probabilities A and B from the summary of probabilities. Mathematically,
( ) ( ) ( )
no failure_ system
P =P A B∩ =P A P B⋅ =R (1.13)
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Using these formulas, the probability of system failure Pf can simply be written as
f 1
P = −R (1.14)
1.1.5.3 Maintenance and maintainability
Maintenance covers those activities that are undertaken to keep the system operational or restore it to operational condition after a failure occurrence (Ireson et al., 1996). There are several classifications of maintenance. Here are the most important ones:
• Breakdown maintenance: an item of the system would be repaired each time it breaks down (Mechefske and Wang, 2003)
• Condition-based maintenance (CBM): the critical components are monitored for deterioration and the maintenance is carried out just before the failure occurs (Mechefske and Wang, 2003)
• Preventive (scheduled) maintenance: the plant is stopped at intervals, often annually, and partly stripped and inspected for faults (Mechefske and Wang, 2003)
• Reliability-centered maintenance (RCM): a procedure to identify preventive maintenance (PM) requirements of complex systems (Cheng et al., 2008)
Maintainability is the measure of the ability of an item to be retained in or restored to specified condition when maintenance is performed by personnel having specified skill levels, using prescribed procedures and resources, at each prescribed level of maintenance and repair (Ireson et al., 1996).De Castro and Cavalca (2006) defined it asthe ability to renew a system or component in a determined period of time, enabling it to continue performing its design functions.
Dependability ratio can be defined by
MTBF d µ MTTR
= λ = (1.15)
The lower the value the greater the need for maintenance in the system (de Castro and Cavalca, 2006).