The 4 standard failure models
-to be used in maintenance optimization, with focus
on state modelling
Professor Jørn Vatn
1
Situations and maintenance tasks
1. Observable gradual failure progression


Inspect at regular intervals (or with shorter and shorter intervals)
Replace when degradation is high
2. Observable “sudden” failure progression


Inspect at regular intervals
Replace if failure progression is detected
3. Non-observable failure progression

Replace based on age
4. Shock

Perform functional test to identify hidden failures
2
Failure progression
1 - Observable gradual failure progression
Failure
Critical failure progression
Maintenance limit
Time
Tmaint Tcrit
3
Examples, observable gradual failure progression




The break disks on a train
The wear on a railway rail
The corrosion on a pipe
Cracks in an airplane structure
 The level of degradation determines the next inspection,
and whether a repair action is required
4
Failure progression
2 - Observable “sudden” failure progression
F Failure
Critical failure progression
P
PFinterval
5
Time
Examples: observable “sudden” failure progression
 Cracks in a train wheel
 Isolation resistance in a signalling cable
6
Failure progression
3 - Non-observable failure progression
Failure
Critical failure progression
Time
Tcrit
7
Failure progression
4 - Shock
F
Failure
Critical failure progression
P
Time
8
Multistate systems
 Multistate systems are described by performance
measures
 We use a state variable, Y(t), to describe the state of the
system at time t, e.g.,
 Performance (pump capacity, compressor efficiency etc)
 For binary systems Y(t) reduces to take only the values 0
and 1; Y(t) = 1 represents a functioning state, and Y(t) = 0
represents a fault state
 Y(t) is a random quantity, i.e. expressed in probabilistic
terms, involving model parameters
9
Content of the state variable Y(t)
 Y(t) was introduced as a performance variable
 However, we will let Y(t) be more general, and Y(t) will be
used to express the state of the system at time t, i.e.;
 the direct performance of the system, capacities etc., or
 a direct measure of wear, or
 an indication of wear or increased failure probability
 We use W(t) as a general quantity that simply is related to
degradation of the system:
10
Degradation quantities of interest
 WP(t): Quantities that are direct performance measures ($!!!)
 E.g., the pumping capacity of a pump
 WI(t): Quantities that are only indicators of the degradation of the
component
 E.g., the bearing temperature
 WD(t): Quantities that represent measurable degradation
 Examples are crack shape and size, corrosion level, geometrical defects
(inclusive wear)
 WS(t): Stressors that influence the degradation process
 Examples could be the cyclic loads and corrosive medium
 The stressors them selves do not measure the likelihood of failure, but is
important for the forecasting of the failure progression
 WP(t), WI(t) and WD(t) will be (probabilistic) modelled by the state
variable Y(t)
11
Y(t)
Challenges in
failure modelling
Failure limit
 How to measure Y(t)?
 For quantities that could be measured:
 Use the quantity directly, i.e., crack length
 Transformations, for example FFT (Fast Fourier Transform)
 Non measurable quantities
 Define patterns for similarity comparison
 What is the relation between the readings from the
measurements and the real physical state?
 Reliability of the measurement techniques
 To model failure (fixed failure limits rarely exist)
 To model failure, we generally specify the failure probability as a
function of the value of the state variable, i.e., p = p(y)
 A simplification would be to assume that a failure occurs the first
time the state variable reaches a fixed limit (failure limit)
12
Time
Purpose of modelling – binary systems
 We want to establish a mathematical model describing the relation
between
 the effective failure rate, E, and
E(,l)
 the maintenance, i.e.,
 the inspection interval, , and
 the intervention level, l
l=6
l=3
 E = E(,l)

 Establish a cost model:
 PM cost  (inspection interval)-1
 Renewal cost increases with a restrictive intervention level
 CM cost/unavailability cost increases with increasing inspection interval
 CM cost/unavailability cost decreases with a restrictive intervention level
 Example 
13
Classes of probabilistic models used
 PF model 
 Failure progression is defined between a potential failure (P) and a failure (F)
 The Wiener process 
 During an arbritary time interval t, the “failure progression” is increased by a
normally distributed quantity with mean t and variance  2t
 A failure occurs the first time the failure progression passes the critical value 
 The Gamma process
 Similar to the Wiener process, but the increments are gamma distributed
 The shock model 
 The system is exposed to shocks, and each shock causes a damage Xi
 When the accumulated damage increases, so does also the failure probability
 The Markov state model 
 The failure progression is approximated by a discrete set of states
 The transitions between the sates are assumed to follow a Markov process
 The model is very flexible, and allows for modeling a large range of situations
14
Markov model 
The PF model
 The objective of the inspection is to detect e.g., a crack
(potential failure) before it develops to a breakage (critical
failure)
 The time from a crack is detectable (P) until the e.g., the
rail breakage is a fact (F), is denoted the PF interval
Failure progression/crack size, Y(t)
Critical failure
progression
(Breakage)
F
Detectable
failure
progression
P
t
Tinit
T
det
T crit
PF interval
15


Variation in the PF interval
 The length of the PF interval is assumed to vary from time
to time
 cracks can be initialised in different places of the component
 crack propagation depends on several different factors such as
load, structure quality, temperature etc
 The cracks that propagate very fast represent the largest
risk of not being detected by the ultrasonic inspection
 The objective of the modelling is
to obtain the probability, Q, of not
detecting the crack in due time
as a function of the inspection
interval 
35 %
30 %
25 %
20 %
15 %
10 %
5%
0%
 Q = Q()
0
30
60
90
120
150
time [months]
16

Determining Q0 (simplified)




TPF PF interval (random variable)
PF Probability distribution function of TPF
q Failure probability of one inspection
 Inspection interval
 Qt Failure probability for fixed value, TPF = t
 Q0 Failure probability of given strategy
17
The argument
 Assume PF-interval is fixed, i.e., TPF = t
 Let n = int(t/)
 Number of opportunities for inspection:
P
Best
Worst
F
t





 






n + 1 opportunities
n opportunities
 Δ = 𝑡 −𝑛⋅𝜏
 We get an extra inspection, if the first inspection after the
«P» comes before  time units, i.e.,
 Probability of n+1 opportunities: Δ/𝜏 = t/ − n
 Qt(,q,t) = (n + 1 − t/) ⋅ qn + (t/ − n) ⋅ q(n+1)

 𝑄0 𝜏, 𝑞, 𝜉PF = 0 𝑄𝑡 𝜏, 𝑞, 𝑡 𝜉PF 𝑡 𝑑𝑡
18
Cost elements - Optimization
 The most important cost elements are:




The cost per inspection, CI
The (unavailability) cost per system failure, CF
The cost of repairing a system failure, CCM
The cost of renewing the system upon a potential failure, CRC
 The total cost per unit time is then
C() = CI/ + (CF+CCM)E() + CRC()
 The objective is now to minimize C() wrt maintenance interval and
intervention level
 E()  Q0 / (MTTF-E(TPF) )
 E()  (1-Q0 )/ (MTTF-E(TPF) ) = renewal rate
19
Failure progression, Y(t)
The Wiener process
Failure
Failure limit, 
t = expected drift
t
Time
20

The shock model
Accumulated damage, Y(t)
Failure
Xi damage caused by ith shock
ith shock
Time
The shocks represent WS(t)
The magnitude of the shock also represents WS(t)
The impact Xi represents WD(t)
21

The Markov state model
State
Failure
yr
r
yr-1
r-1
.
….
:
:
2
y2
1
y1
y0
0
T0
T1
…..
T2
22
Time
Tr-1

State
yr
Model assumptions
r
:
:
y2
y1
y0
2
1
0
 The state variable, Y(t), describes the state of the system
at time t, Y(t) is a random quantity
 The state variable could take one of the values y0, y1,…, yr
 The values could either be numerical, or a qualitative
description of a state or phenomenon
 The system starts in state y0, and jumps to a higher state
(yi to yi+1) with a time independent intensity i
 There is generally a cost assossiated with being in state yi
 The system fault state is yr
 The system is inspected at intervals of length  (offline)
 The system is renewed if Y(t)  yl at an inspection
23
t
Maintenance
 r-1
yr
yl
:
:
0
y1
y0
l
Maintenance limit
y2
1
2
3
4
r
1
0

2
5
24
6
7
8
Time
Par. Spec. Calculation
Markov differential equations
 Introduce Pi(t) = Pr(the system is in state i at time t)
 Consider the change in a small time interval t:
 Standard Markov considerations gives:
r
Pi(t+t) = Pi(t)(1-it) + Pi-1(t) i-1t
 i-1
(*)
i
i
i-1
 Equation (*) could now be used to obtain the state probabilities,
Pi(t), as a function of time by numerical integration
25
The easy situation: no maintenance
 If no maintenance is carried out then
 integrate equation (*)
 starting from the initial state
 Mean time to failure is given by:
 MTTF = t=0: R(t) dt = t=0: [1-Pr(t)]dt
 in fact a sum …
 To verify our calculations we should verify the analytical
result:
 MTTF = i=0:r-1MTTFi = i=0:r-11/i
26
Calculation procedure: with maintenance
 The system is inspected at intervals of length 
 The system is renewed if Y(t)  yl at an inspection (Fig.)
 The model is integrated as before, but when t equals ,
2, 3,… special considerations are necessary
 Procedure
1. Define the initial conditions: P0(0) = 1, Pi(0) = 1, i > 0
2. Set f = 0, t = 0, t = sufficient small
3. Integrate Equation (*) one step, and let t = t + t
4. Let f = f + Pr(t)
5. If t =, 2, 3,…, then let P0(t) = P0(t)+ il Pi(t), and Pi(t) = 0, il
6. Loop to Step 3 until t is sufficient large
7. System failure frequency now equals E(,l) = f/t
27
Essential source code in VBA
Do While t < MaxT ‘ Main loop
nFail = nFail + IntegrateDt(dt)
P(0) = P(0) + P(r)
Function IntegrateDt(dt As Single)
For i = r To 1 Step -1
P(i) = P(i) * (1 - lam (i) * dt) _
P(r) = 0
+ P(i - 1) * lam (i - 1) * dt
t = t + dt
Next
If t > inspection Then
P(0) = P(0) * (1# - lambda(0) * dt)
inspection = inspection + tau
IntegrateDt = P(r)
nRenewal = nRenewal + Inspect(L, q)
End Function
End If
Loop
Function Inspect(L As Integer, q As Single)
rr = 0
For i = L To r - 1
rr = rr + P(i) * (1 - q)
P(0) = P(0) + P(i) * (1 - q)
P(i) = P(i) * q
Next i
DoInsp = rr
End Function
28
Specification of model parameters
 In principle we need to specify all transition rates, i.e.
 0, 1,…, r-1
 We also need the probability of erroneous classification
 Qij = Pr(Classify into state i when the real state is j)
 In order to get numerical values (estimates) of the model
parameters, we utilise:
 Experience data
 Expert and engineering judgements
 Degradation modelling, i.e. fracture mechanics, FEM etc
 For r > 4-5 this will be a huge number of parameters
 We want to simplify the parameter specification procedure 
29
Simplified parameter specification
 We specify the parameters in the situation without
maintenance, i.e.
 What will the mean time to failure (MTTF) be if no maintenance is
carried out? (Fig. )
 Is the transition rate between states constant, or increasing?


If it is increasing then we specify the ratio:
V = r-1/0 = how much faster failure progression is just before failure
compared to initially (Fig. )
 We also need to specify
 The number of states in the model (r)
 The probability q that an inspection does not reveal that the
system is in a critical state
30
Calculation example 
MTTF without maintenance
MTTF without maintenance
yr
r
yr-1
r-1
.
….
:
:
2
y2
1
y1
y0
Failure
0
Time
31

Calculation example
 Input parameters:
Input values
MTTF
r
V=r-1/0

Intervention, l
q
Time horizon
 Result
Output result
v
120
8
0
MTTF-verify
MTTF(,l)
8
12
4
0,05
4800
A(l)
Ren. Rate
MTBR
32
1,35
0,0294
119,98
2480,14
0,00040
0,01008
99,25
MarkovStateModel.xls
The effect of maintenance
 We have established (by means of the Excel model) the
relation between maintenance ( and l) and i) the
effective failure rate, E(,l), and ii) the renewal rate (,l)
 Example results
Effective failure rate, E(,l)
0,006
0,005
0,004
Intervention: l = 6
0,003
0,002
Intervention: l = 4
0,001
0
3
6
9
12
15
Inspection interval, 
33
18
21
24
Cost elements - Optimization
 The most important cost elements are:




The cost per inspection, CI
The (unavailability) cost per system failure, CF
The cost of repairing a system failure, CCM
The cost of renewing the system at state l, CRC
 The total cost per unit time is then
C(,l) = CI/ + (CF+CCM)E(,l) + CRC(,l)
 The objective is now to minimize C(,l) wrt maintenance interval
and intervention level
34
Extension of the Markov model
 More advanced maintenance strategies could be applied
 Reducing inspection interval as we approach the maintenance
limit, l
 Conduct non perfect repair before the maintenance limit
 Models have been developed for hydro power plant
35
The gamma process
 Stationary gamma process
 Background: X is said to be gamma distributed with shape
parameter v, and scale parameter u if the PDF is given by:
 Ga(x|v,u)=uvxv-1e-ux/(v)
 Let Y(t) be the degradation level at time t
 Y(t) follows a stationary gamma process if
 Y(0) = 0
 Y(s) - Y(t) ~ Ga([s-t ]v,u), s>t
 Y(t) has independent increments
36
Mean time to failure in the gamma process
 Assume that the component fails as soon as the failure
progression exceeds the value 
 Let T denote the time to failure
 It follows that
FT(t) = Pr(T<t) = Pr(Y(t) > ) = (vt, u)/(vt)
 Where (a, x) is the incomplete gamma function
 Welte (2008) reports the following:
 E(T)  u/v + 1/(2v)
 Var(T)  u/v2 - 1/(12v2)
37
Non-stationary gamma process
 The gamma process could be extended to a nonstationary process by letting the shape parameter be a
function of time, i.e., v(t) is the shape function, and we
have:
 Y(0) = 0
 Y(s) - Y(t) ~ Ga(v(s)-v(t),u), s>t
 Y(t) has independent increments
 The CDF now reads
FT(t) = Pr(T<t) = Pr(Y(t) > ) = (v(t), u)/(v(t))
 The expected time to failure, and variance in time to
failure could be found by numerical methods
38
Comparison – Discrete model, vs gamma process
 For the discrete model we need to fix the number of states
 If the degradation is continuous, this seems not very
natural, hence a gamma process is more appealing
 Degradation rate
 In the discrete model, the degradation rate (in terms of transition
rates) depends on the state of the system, and not on the age
(time)
 In a gamma process the degradation rate could also be modelled
by a non-constant value, but degradation rate depends on the age,
and not on the state
39
Exercise
 Verify E(T)  u/v + 1/(2v) by numerical integration, i.e.,

E(T) = 0 R(t)dt
40
Non-stationary gamma process
 The gamma process could be extended to a nonstationary process by letting the shape parameter be a
function of time, i.e., v(t) is the shape function, and we
have:
 Y(0) = 0
 Y(s) - Y(t) ~ Ga(v(s)-v(t),u), s>t
 Y(t) has independent increments
 The CDF now reads
FT(t) = Pr(T<t) = Pr(Y(t) > ) = (v(t), u)/(v(t))
 The expected time to failure, and variance in time to
failure could be found by numerical methods
41
Integration of the gamma process
 Let S|t,dt = Y(t+dt) - Y(t) be the degradation during a small
time interval dt after time t
 S|t,dt ~ Ga(v(t+dt)-v(t),u)
 Further, let g(s | t, dt) denote the pdf of S|t,dt
 If the pdf of Y(t) is known, we may obtain the pdf of Y(t+dt) by a
convolution argument:
 𝑓(𝑦|𝑡 + 𝑑𝑡) =
𝑦
𝑓(𝑦
𝑠=0
− 𝑠|𝑡) 𝑔(𝑠|𝑡, 𝑑𝑡)𝑑𝑠
(*)
 Assume the system is inspected every  time unit, and
renewed whenever Y > yM
 To find the effective failure rate, we integrate (*) from t = 0
to , and whenever t = k, probability mass is moved to 0
42