Reliability System Reliability (cont'd) · 2018. 6. 6. · AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdVl002VF+4BaSjKd1mBeTCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9NAwy6TivBWtufmFxqbhcWlldW98ob261siQXjDdZEiai43sZD4OYN2UgQ95JBfciP+Rt/+ZMxdu3XGRBEl/Kccp7kTeKg2HAPEnURad/1C9XnKqjlz0LXAMqMKuRlF9whQESMOSIwBFDEg7hIaOnCxcOUuJ6mBAnCAU6znGPEmlzyuKU4RF7Q98R7bqGjWmvPDOtZnRKSK8gpY090iSUJwir02wdz7WzYn/znmhPdbcx/X3jFRErcU3sX7pp5n91qhaJIU50DQHVlGpGVceMS667om5uf6lKkkNKnMIDigvCTCunfba1JtO1q956Ov6mMxWr9szk5nhXt6QBuz

ReliabilitySystem Reliability (cont’d)

Jean-Yves Dauxois

INSA-Toulouse & IMT, FRANCE

4-8 June 2018

SystemReliability II

J.Y. Dauxois,4-8 June 2018

CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity

Maintenanceefficiency

HPP

Definitions

Properties

StatisticalInference

MaintenancePlanification

2/52

Plan

1 Counting ProcessesRepairable systemsDefinitionPropertiesRate of Occurence of FailureFailure intensityMaintenance efficiency

2 Homogeneous Poisson Processes (HPP)Equivalent Definitions of an HPPPropertiesStatistical InferenceMaintenance Planification



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



3/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



4/52

Repairable systems

We consider a system subject to recurrent failures. We supposethat when it fails, it is repaired to a functioning state. Weassume that the repair time is negligible.

Let us denote by (Tn)n≥0 :

0 = T0 ≤ T1 ≤ T2 ≤ · · · ≤ Tn ≤ · · ·

the successive times of failures.

This series of increasing times is called a Point Process.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



5/52

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Figure: Different representations of the failure times of a repairablesystem



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



6/52

Inter-arrival times

The times between failures

X1 = T1, X2 = T2 − T1, . . . ,Xn = Tn − Tn−1, . . .

are called the inter-arrival times.

In general, the Mean Time Between Failures (MTBF)depends on the index i , i.e. MTBFi = EXi .

Of course, it is equivalent to know (or modelize) the Pointprocess (Tn)n≥0 or the series of inter-arrival times (Xn)n≥1

since we have:

Tn = X1 + X2 + . . .+ Xn, ∀n ≥ 1.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



7/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



8/52

Counting Process

One can also consider the process N = (Nt)t≥0 defined on R+

by

Nt =+∞∑n=1

1lTn≤t

and which, at every time t, gives the number of failuresobserved on [0, t].

N = (Nt)t≥0 is called the Counting Process associated to thepoint process (Tn)n≥0.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



9/52

Again, it’s equivalent to know the distribution of the pointprocess (Tn)n≥0 or the distribution of the counting processN = (Nt)t≥0 since we have:

P(T1 ≤ t1,T2 ≤ t2, . . . ,Tn ≤ tn) =

P(Nt1 ≥ 1,Nt2 ≥ 2, · · · ,Ntn ≥ n).



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



10/52

DefinitionA Counting Process is a stochastic process which satisfies:

1 Nt ≥ 0, for all t ≥ 0,

2 Nt is integer valued,

3 If s < t, then Ns ≤ Nt ,

4 for s < t, the integer Nt − Ns represents the number ofevents (here failures) observed on the interval ]s, t].

The trajectories of a counting process are increasing stepfunctions, right continuous with left limits.

We have of course: N0 = 0.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



11/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



12/52

Main Properties of a CountingProcess

• A counting process is said with independent incrementsif

∀k ∈ N∗,∀0 < t1 < · · · < tk , the r.v.

Nt1 ,Nt2 − Nt1 . . . ,Ntk − Ntk−1

are mutually independent.

This implies that the number of failures on distinctintervals are independent. It might be a strongassumption...



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



13/52

• A counting process is said with stationary increments ifthe distribution of the number of events (failures) on aninterval ]s, t] depends only on the length t − s of thisinterval.In other words, the distribution of the r.v. Nt − Ns andNt+h − Ns+h are the same, for all h.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



14/52

• A counting process is said regular if, for all t ≥ 0 and∆t > 0:

P(Nt+∆t − Nt ≥ 2) = o(∆t).

That is to say:

lim∆t→0

P(Nt+∆t − Nt ≥ 2)

∆t= 0.

In this case, no more than one failure can be observed attime t.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



15/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



16/52

Rate of Occurence of Failure

DefinitionThe mean function of the counting process N = (Nt)t≥0 isdefined by:

m(t) = E(Nt), ∀t ≥ 0.

Remark. The mean function is not necessarily discontinuous.

DefinitionThe rate of the counting process N = (Nt)t≥0 is defined by:

w(t) = m′(t) = lim∆t→0

m(t + ∆t)−m(t)

∆t= 0.

In Reliability, this process is called the RoCoF (Rate ofOccurence of Failure). It represents the mean number of failureper unit at time t.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



17/52

From the definition of the RoCoF one can write:

m(t) =

∫ t

0w(s)ds.

In case of a regular process we have, for “small” t:

w(t) ≈ P(Nt+∆t − Nt = 1)

∆t.

This is why in some books the RoCoF is defined as

w(t) = lim∆t→0

P(Nt+∆t − Nt = 1)

∆t.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



18/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



19/52

Failure intensity

Let us denote Ft the history of the counting process up totime t.

• Roughly speaking, this history is given by the trajectory{s 7→ Ns , for all s ≤ t} of the process, i.e. the record ofall the failure times before time t.

• Mathematically, this history is defined by the σ-fieldFt = σ(Ns ,∀s ≤ t). Since the trajectory of a countingprocess is a step function, we also have:Ft = σ(Nt ,T1, . . . ,TNt ).

• The family (Ft)t∈R+ is called a Filtration with mainproperty the inclusion Fs ⊂ Ft for s ≤ t.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



20/52

DefinitionThe conditional rate of the counting process N = (Nt)t≥0 isdefined by:

λt = lim∆t→0

P(Nt+∆t − Nt = 1|Ft−)

∆t.

In Reliability, this process is called the Failure intensity.

One can write:

P(Nt+∆t − Nt = 1|Ft−) = λt∆t + o(∆t).

Remind that, in case of a regular process,

P(Nt+∆t − Nt = 1) = w(t)∆t + o(∆t),

Remark. Note that the rate function t 7→ w(t) isdeterministic, whereas the conditional rate t 7→ λt is generallystochastic.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



21/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



22/52

Maintenance efficiencyRemind that we observe a system subject to recurrent failures.After each failure, a repair of the system is operated. Thequestion is to know in which state is the system after repair.

Mathematically, one wonders to know• the distribution of the inter-arrival time Xi , after the

failures and repairs at times T1, . . . ,Tti−1 ,• or equivalently the failure intensity function λt for

t ≥ Tti−1 .







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Figure: Occurence of the failure times of a repairable system



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



23/52

Thus the question under consideration is to judge themaintenance efficiency. Different models of repair efficiencyare considered by engineers, including:

• As Good As New (AGAN): after a repair, the system is asnew. This is Perfect Repair ⇒ Homogeneous PoissonProcesses, Renewal processes.

• As Bad As Old (ABAO): after repair, the system is in thesame state as it was before failure. This is MinimalRepair ⇒ Non Homogeneous Poisson Processes

• Some times, the system is between these extreme cases.This is Imperfect Repair

• Brown et Proschan (BP)• Virtual Ages• . . .

We will consider successively these different models.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



24/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



25/52

Plan





CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



26/52

Definitions of an HomogeneousPoisson Processes

We give three equivalent definitions of an HomogeneousPoisson Processes.

Definition (1)

A Point Process (Tn)n≥0, or equivalently its Counting ProcessN = (Nt)t≥0, is called an Homogenous Poisson Process(HPP) with intensity λ if the inter-arrival times (Xn)n≥1 arei.i.d. with exponential distribution E(λ).

The link with the Poisson distribution is given by the followingProposition.

Proposition

Let N = (Nt)t≥0 be an HPP with intensity λ. For all t > 0,the r.v. Nt has a Poisson distribution with parameter λt, i.e.

P(Nt = n) =(λt)n

n!e−λt .



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



27/52

Let us recall a result about the distribution of the orderstatistics

X(1) < . . . < X(n)

of a sample (X1, . . . ,Xn).

LemmaLet X1, . . . ,Xn be n i.i.d. continuous r.v. with probabilitydensity function (p.d.f.) f . Let X(1) < · · · < X(n) be the nassociated order statistics. The multivariate p.d.f. of therandom vector (X(1), . . . ,X(n)) is

f(X(1),...,X(n))(u1, . . . , un) = n!n∏

i=1

f (ui )1lu1<u2<···<un .



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



28/52

Proposition

Let (Tn)n≥0 be an HPP with Counting Process (Nt)t≥0. Wehave the following results.

i) For all n ≥ 1, the p.d.f. of the random vector (T1, . . . ,Tn)is

λne−λtn1lt1<t2<···<tn .

ii) For all n ≥ 1, the conditional distribution of (T1, . . . ,Tn)given {Nt = n} is the same as the distribution of the orderstatistics of n i.i.d. r.v. with uniform distribution on [0, t],i.e.

f Nt=n(T1,...,Tn)(t1, . . . , tn) =

n!

tn1lt1<t2<···<tn≤t .



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



29/52

Definition (2)

Let N = (Nt)t≥0 be a Counting Process. It is an HPP withintensity λ if

i) N0 = 0;

ii) N has independent increments;

iii) for all 0 ≤ s < t, the r.v. Nt − Ns has a Poissondistribution with parameter λ(t − s).



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



30/52

Definition (3)

Let N = (Nt)t≥0 be a Counting Process. It is an HPP withintensity λ if

i) N0 = 0,

ii) N has stationary and independent increments,

iii) N is regular,

iv) P(Nt+∆t − Nt = 1) = λ∆t + o(∆t).



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



31/52

Plan





CountingProcesses

Repairablesystems

Definition

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RoCoF

Failure intensity


HPP

Definitions

Properties



32/52

Properties

Proposition

Let N = (Nt)t≥0 be an HPP with intensity λ.

• The RoCoF is w(t) = λ, thus is constant.

• The failure intensity is equal to the RoCoF, i.e. λt = λ.Thus it is deterministic and constant.

• The mean number of failures in time interval ]s, t] isw(t)− w(s) = λ(t − s).

• The MTBF doesn’t depend on the index i of theinter-arrival times and is equal to 1/λ = EXi .

• The distribution of the nth failure Tn is Gamma(n, λ), i.e.with p.d.f.:

fTn(t) =λn

(n − 1)!tn−1e−λt , for all t ≥ 0.

• N is a Markov Process.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



33/52

Sum and Decomposition of HPP

Proposition

• Suppose that N1 and N2 are two independent HPP withparameter λ1 and λ2. Then N1 + N2 is an HPP withparameter λ1 + λ2.

• Reversely, if N is an HPP where the occurence are oftype 1 with constant (over time) probability p and of type2 with probability 1− p, then the counting process N1

(resp. N2) of the number of events of type1 (resp. type 2)is an HPP with parameter pλ (resp. (1− p)λ). The twoHPP N1 and N2 are furthermore independent.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



34/52

The first point could correspond to the observation of twotypes of failures on a system, where a failure of type 1 occursat a rate λ1, independently to the other type which appears ata rate λ2.

The second point could modelize the situation where, in thefailure occurrences of a system, we want to distinguish differentcauses or different severities, with constant over timeprobabilities of being of each type.

Note that the previous results can be generalized to a numbern of sum or decomposition of HPP.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

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35/52

Compound HPP

Proposition

Suppose that N is an HPP with parameter λ and that at eachoccurence time Ti of the counting process, we observe arandom variable Ci , for i ∈ N. If (Ci ) is a series of i.i.d.random variables with mean c and variance σ2, then, for allt ≥ 0, the r.v.:

St =Nt∑i=1

Ci

has mean µλt and variance = λ(µ2 + σ2)t.

The random variables (Ci ) could represent the repair costsafter failure.



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



36/52

Plan





CountingProcesses

Repairablesystems

Definition

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RoCoF

Failure intensity


HPP

Definitions

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37/52

ObservationsThere are two different types of observation for a Countingprocess.• Either we observe the process on a fixed interval [0, t].












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Figure: Observation of a counting process on a fixed interval [0, t]

• Or observe it up to the occurence of the nth failure.













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Figure: Observation of the n = 6 first occurrences of a countingprocess



CountingProcesses

Repairablesystems

Definition

Properties

RoCoF

Failure intensity


HPP

Definitions

Properties



38/52

Maximum Likelihood Estimation

Proposition

Depending on the type of observation we have, the likelihoodof a trajectory of a HPP is

• in case of Type 1 censoring, i.e. when the process isobserved on an interval [0, t]:

λne−λt1lt1<t2<···<tn<t ,

• in case of Type 2 censoring, i.e. when we observe the nfirst failures on the system:

λne−λtn1lt1<t2<···<tn .



CountingProcesses

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HPP

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39/52

Let us denote by TTT, the Total Time on Test:

TTT =

{t with Type I obsservations

Tn with Type II observations.

In both cases, the Maximum Likelihood Estimator of λ canbe written as

λ̂MLE =NTTT

TTT.

These two estimators are nothing but what we have obtainedthrough the maximum likelihood estimation of lifetime datawith exponential distribution with censored (resp. complete)data.



CountingProcesses

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HPP

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40/52

Asymptotic Properties of the MLE

Let us consider separately the two cases given by the type ofobservation we have.

Under Type I observation, the time t is fixed and Nt

random. We can prove the following result.

TheoremLet (Nt)t≥0 be an HPP with parameter λ. We have:

i) p.s. limt→+∞

Nt

t= λ

ii)√

t

(Nt

t− λ)L→ N(0, λ), when t → +∞.

Thus the MLE λ̂MLE under Type I is asymptotically unbiasedand normal.



CountingProcesses

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HPP

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41/52

We can deduce asymptotic confidence intervals for λ from thisconvergence in distribution.

Proposition

An asymptotic confidence interval for λ is under Type IIobservation of an HPP is[

Nt

t−√

Nt

tz1−α

2,

Nt

t+

√Nt

tz1−α

2

],

where zα denotes the α-quantiles of the N(0, 1) distribution.



CountingProcesses

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42/52

Under Type II observation, the number of observation n isfixed but the Tn is random. We can prove the following result.

TheoremLet N be an HPP with parameter λ. We have:

i) p.s. limn→+∞

n

Tn= λ

ii)√

n

(n

Tn− λ)L→ N(0, λ2), when n→ +∞.

Thus the MLE λ̂MLE under Type I is also asymptoticallyunbiased and normal.



CountingProcesses

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43/52

Proposition

The following results holds under Type II observation of anHPP.

• An asymptotic confidence interval for λ is[λ̂MLE

1 + 1√n

z1−α2

,λ̂MLE

1− 1√n

z1−α2

].

• Fix λ0 a limit value under which the system is assumed tobe reliable. A test of level α of H0 : λ ≥ λ0 againstH1 : λ < λ0 is given by the rule

Reject H0 if λ̂MLE < λ0 +λ0√

nzα.



CountingProcesses

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44/52

Non Asymptotic Properties of theMLE

Here we study the properties of

λ̂MLE =Nt

t

for fixed n.

We will see that, again, the results differ according to the typeof observation we have: Type I or Type II censoring.



CountingProcesses

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HPP

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45/52

MLE Properties under Type 1observations

Remind that, under Type 1 observation, we observe theCounting Process on a fixed interval [0, t]. The number offailures Nt in this interval is random.

We have seen that in case of an HPP, the counting process Nt

is such that:Nt ∼ P(λt), for all t > 0.

We deduce from this that:

• λ̂MLE is unbiased,

• its variance is equal to λ/t,

• it is an efficient estimator of λ,

• it is the best (minimal variance) unbiased estimatorof λ.



CountingProcesses

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HPP

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46/52

MLE Properties under Type 2observations

Under Type 2 observation, we observe the Counting Process upto a number of occurence (failures). This number is fixed,equal to n, but the time of observation Tn is random.

Remind that the expression of the MLE of λ in this case is

λ̂MLE =n

Tn

We can prove that

E(λ̂MLE ) =n − 1

nλ,

which proves that the estimator is biased but asymptoticallyunbiased.



CountingProcesses

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47/52

One can prove that

λ̃ =n − 1

Tn

is:

• unbiased,

• not efficient, but asymptotically efficient.

• it is the best (minimal variance) unbiased estimatorof λ



CountingProcesses

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48/52

Exact Confidence interval for λ

Here also, we have to consider separately the two types ofobservation.

In Type I case, it is hard to find exact confidence interval orexact statistical test for the parameter λ. This is due to thefact that the distribution of Nt is discrete.

However, there exists rather simple approximative confidenceintervals. See references.



CountingProcesses

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49/52

Things are easier in case of Type II observations, thanks tothe distribution of Tn which is Gamma(n, λ).

Proposition

• An exact confidence interval for λ in case of Type Iobservation is: [

χ2α2

(2n)

nTn,χ2

1−α2

(2n)

nTn

].

• Fix λ0 a limit value under which the system is assumed tobe reliable. A test of level exactly equal to α ofH0 : λ ≥ λ0 against H1 : λ < λ0 is given by the rule

Reject H0 if 2λ0Tn > χ21−α(2n).



CountingProcesses

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Definition

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RoCoF

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HPP

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50/52

Plan





CountingProcesses

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HPP

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51/52

Maintenance Planification

Remind that we have supposed that

• after each failure a corrective maintenance is performed,

• in case of an HPP, after each maintenance, the system isconsidered as new.

This is what we call AGAN corrective maintenance.

Let us consider the question of the interest to perform apreventive maintenance on the system before that the nextfailure occurs and to replace the system by a new one.



CountingProcesses

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HPP

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52/52

We know that the exponential distribution satisfies thememoryless property, that is:

P(X 6 x + x0 / X > x0) = P(X 6 x), ∀(x , x0) ∈ R+ × R+

⇔ L(τx0) = L(X ), ∀x0 ∈ R+.

This induces that there is theoretically no reason to performpreventive maintenances between corrective maintenance incase of an HPP model.

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