The two types of maintenance performed on repairable systems are condition maintenance and preventative maintenance. Condition maintenance involves actions performed after a failure has occurred in the system and maintenance has to be performed in order to restore the system to functioning condition. Preventative maintenance, on the other hand, involves any action associated with maintenance performed on a system with the aim to prevent or delay a system failure [40]. The rate of occurrence of failure (ROCOF) or repair rate is defined as the rate at which failures occur during the lifetime of the system. Failure rates, on the other hand, are only applicable to the first failure in a non-repairable system; it is therefore incorrect to use this term in the context of repairable systems [171]. Time in a repairable system is measured as the number of time units during which the system has been in operation, from initial turn-on to the end of system life, also known as the global time. The system experiences failures as it ages and is repaired to operational condition after each failure [40, 171].
Repair models are either classified as renewal models or as minimal repair models [40]. In a renewal model, it is assumed that after each failure the system is repaired to an “as good as new” condition, whereas in a minimal repair model it is assumed that the system is only repaired to the state it was in just before the occurrence of the last failure. The latter type of model is more commonly used in very constrained systems [40]. The three main repair models used to represent repair rates in repairable systems are reviewed in this section. These models are the Homogeneous poisson process (HPP), the Non-homogeneous poisson process (NHPP) following an exponential law and the NHPP following a power law.
3.4.1 The Homogeneous poisson process
The HPP is recognised as one of the simplest models for representing and predicting failures in a repairable system [40, 167]. This model is widely used despite its simplicity and is justified by the shape of the Bathtub curve in Figure 3.1. Complex systems generally operate in the “Useful Life” portion of the Bathtub curve for most of the system’s lifetime during which a constant repair rate prevails. The HPP is one of the only models for repairable systems that applies to this portion of the Bathtub curve by exhibiting a constant repair rate [171, 173]. Therefore, the ROCOF in the HPP model is expressed by
h(t) = λ, (3.32)
which does not depend on the lifetime of the system. The repair rate may be estimated by calculating the mean time between failures
MTBF = Total system operating time Total number of observed failures =
1
λ (3.33)
for the system. The PDF of the basic HPP is the same as the PDF for the exponential model for non-repairable systems as presented in (3.11). The CDF in the basic HPP model is
F (t) = 1 − e−λt.
Therefore, the reliability of a system that is represented by an HPP model may be calculated as
From (3.32), which is similar to the exponential model for non-repairable systems described in §3.3.1, the expected number of failures to have occurred by a certain time t may be expressed as
H(t) = Z t
0
h(x) dx = λt,
where t is the continuous global time of the system in question. In order to calculate confidence band for the expected failure time Tn, confidence intervals must be calculated. This is once
again achieved in the exact same manner as for the exponential model, as explained in §3.3.1 — see (3.14) and (3.15).
3.4.2 The Non-homogeneous poisson process following an exponential law
The NHPP model following an exponential law, also known as the log-linear model or the Cox- Lewis model, has been employed successfully in a variety of applications [48, 167, 171, 173]. This model follows a ROCOF of the form
h(t) = eα0+α1t, (3.35)
where α1 is positive for a repairable system and t represents continuous global time [136]. The number of failures expected to have occurred by a certain time t may be expressed according to the model by integrating (3.35), that is
H(t) = Z t 0 h(x) dx = e ˆ α0 (eαˆ1t− 1) ˆ α1 , (3.36)
where ˆα0 and ˆα1 are the estimated values for α0 and α1. From (3.36), it is possible to calculate the timing of the n-th failure by solving for t in the equation
H(t) = e ˆ α0 (eαˆ1t− 1) ˆ α1 = n, (3.37)
where n − 1 failures have already been observed in the system. By using (3.10), it is possible to calculate the reliability of the system between times t1 and t2 as
R(t1 < x < t2) = exp −exp ( ˆα 0+ ˆα1t 2) − exp ( ˆα0+ ˆα1t1) ˆ α1 .
A confidence band may be calculated around the expected failure time in terms of the variance of h(t). This is accomplished by using the so-called Fisher information matrix1 for the two estimates ˆα0 and ˆα1, expressed by
Var( ˆα0) Cov( ˆα0, ˆα1) Cov( ˆα0, ˆα1) Var( ˆα1) =" − ∂2` h ∂( ˆα0)2 − ∂2` h ∂ ˆα0∂ ˆα1 − ∂2`h ∂ ˆα0∂ ˆα1 − ∂2`h ∂( ˆα1)2 #−1 ,
where `h is the log-likelihood function. Using the Fisher information matrix, the variance of
h(t) can be calculated as
Var(ˆh(t)) = eα0+α1t2Var( ˆα0) +teα0+α1t2Var( ˆα1) +2 eα0+α1t teα0+α1t Cov( ˆα0, ˆα1). (3.38) 1
The Fisher information matrix is used in statistics as a measure of the amount of information a random variable contains about a certain parameter of a distribution which may be used to model the random variable.
From this expression it is now possible to calculate an upper confidence limit tu,n and a lower
confidence limit t`,n around the expected next failure estimation calculated in (3.37). This is
achieved by solving for tu,n and t`,n, respectively, in
Z tu,n tr h h(x) − Zβ p Var(h(x))idx = 1 (3.39) and Z t`,n tr h h(x) + Zβ p Var(h(x))idx = 1, (3.40)
where Var(h(x)) is calculated using (3.38) and Zβ is the corresponding Z value for the required
significance level β, since in general, the maximum likelihood estimates of the parameters are asymptotically normal for a large enough data set [178]. It is important to note that the upper limits of the integrals are only defined for values larger than zero. In cases where the distribution does not fit the data well enough, the integral will not converge to 1, in which case no upper limit can be quantified.
3.4.3 The Non-homogeneous poisson process following a power law
The NHPP model following a power law, also known as the Duane model or the United States Army Materials System Analysis Activity (AMSAA) model [171], is another model that has proven successful for a variety of applications. This model follows a ROCOF of the form
h(t) = λδtδ−1, (3.41)
where δ is positive for a repairable system and t represents continuous global time [182]. As with the NHPP following an exponential law, the number of failures expected to have occurred over a certain time period may be calculated by integrating (3.41) over the time period in question, that is
H(t) = Z t
0
h(t)dt = ˆλtˆδ, (3.42)
where t here denotes the end of the time period and where ˆλ and ˆδ are estimates for the values of λ and δ. From (3.42), it is possible to calculate the timing of the n-th failure by solving for t in
H(t) = ˆλtδˆ= n, (3.43)
where n − 1 is the total number of failures already observed in the system. Using (3.10), it is possible to calculate the reliability of the system between times t1 and t2 as
R(t1 < x < t2) = e−λ(t
δ 2−tδ1).
A confidence band may be calculated around the expected failure time of the NHPP following a power law in terms of the variance of h(t) as for the NHPP following an exponential law model. This may be accomplished by again using the Fisher information matrix for the two estimates ˆ λ and ˆδ, expressed by Var(ˆλ) Cov(ˆλ, ˆδ) Cov(ˆλ, ˆδ) Var(ˆδ) = " −∂2`h ∂ ˆλ2 − ∂2` h ∂ ˆλ∂ ˆδ −∂2`h ∂ ˆλ∂ ˆδ − ∂2` h ∂ ˆδ2 #−1 = " n−1 ˆ λ2 t ˆ δ n−1ln tn−1 tδn−1ˆ ln tn−1 n−1δˆ2 + ˆλt ˆ δ n−1ln2tn−1 #−1 ,
where `h again denotes the log-likelihood function [178]. Using the Fisher information matrix
for the NHPP following a power law, the variance of h(t) is given by
Var(ˆh(t)) = δtδ−12Var(ˆλ) +λtδ−1+ λ(δ − 1)tδ−12Var(ˆδ)
+2δtδ−1 λtδ−1+ λ(δ − 1)tδ−1Cov(ˆλ, ˆδ). (3.44) From this expression it is again possible to calculate an upper confidence limit tu,n and a lower
confidence limit t`,n around the expected next failure estimation calculated in (3.43). This is
achieved by solving for tu,n and t`,n in (3.39) and (3.40) using the NHPP power law model’s
variance in (3.44). As with the NHPP following an exponential law, it is important to note that the integral upper limits are only defined for positive values. In cases where the distribution does not fit the data well enough, the integral will again not converge to 1, in which case no upper limit can be quantified.