introduction to reliability theory (part 2)most statistical methods for estimation of parameters in...

Introduction to Reliability Theory(part 2)

Frank Coolen

UTOPIAE Training School II, Durham University

3 July 2018

(UTOPIAE) Introduction to Reliability Theory 1 / 21

Outline

Statistical issues

Software reliability

Decision making

Survival signature

Imprecision

Resilience


Statistical Issues

Random failure time T > 0 with CDF F (t) = P(T ≤ t)

Survival Function S(t) = P(T > t) = 1− F (t)

pdf f (t) = F ′(t) = −S′(t)

Hazard Rate h(t) = f (t)/S(t)

Interpretation, for small δt : h(t)δt ≈ P(T ≤ t + δt |T > t)

Cumulative Hazard Function (CHF) H(t) =∫ t

0 h(x)dx

H(t) =∫ t

0f (x)S(x)dx = − ln S(t)

S(t) = exp{−H(t)} = exp{−∫ t

0 h(x)dx}


Most statistical methods for estimation of parameters in assumedmodels (e.g. Weibull) use the Likelihood Function, either maximising itor combined with a prior distribution in Bayesian analysis.

Reliability data are often affected by right-censoring: an item has notfailed during a particular period of time.

Let t1, . . . , tn be observed failure times, and c1, . . . , cm right-censoredobservations. For a parametric model with parameter θ, the likelihoodfunction is

L(θ|t1, . . . , tn; c1, . . . , cm) =n∏

j=1

f (tj |θ)m∏

i=1

S(ci |θ).

This assumes that the censoring mechanism is independent of thefailure process.


Regression models for reliability data

For regression models in reliability applications Weibull models areoften used, with the survival function, depending on a vector ofcovariates x , given by

S(t ; x) = exp{−(

tαx

)ηx}.

Some simple forms are often used for the shape and scale parametersas functions of x , e.g. the loglinear model for αx , specified vialnαx = xTβ, with β a vector of parameters, and similar models for ηx .

The statistical methodology is then pretty similar to general regressionmethods (e.g. fitting by MLE or Bayesian methods) and implemented instatistical software packages.


Failure processes

A system fails at certain times, there may be some actions during thisperiod which affect failure behaviour, e.g. minimal repairs to allow thesystem to continue its function, or replacement of some components,or other improvements of the system. Normally this should not includefull replacement of the system.

Let the random quantities T1 < T2 < T3 < . . . be the failure times ofthe system, let Xi = Ti − Ti−1 (with T0 = 0).


Rate of OCcurrence Of Failure (ROCOF)

Let N(t) be the number of failures in the period (0, t ], then the ROCOFis:

v(t) =ddt

EN(t).

Increasing (decreasing) ROCOF models a system that gets worse(better) over time.

Note that the ROCOF is not the same as the hazard rate (thedefinitions are clearly different!), although intuitively they are a bitsimilar. If we consider a standard Poisson process, with iid timesbetween failures being exponentially distributed, then the ROCOF andhazard rate happen to be identical.


Non-Homogeneous Poisson Process models (NHPP)

The crucial assumption in these models, as for standard Poissonprocesses which are just a special case, is that the numbers of failuresin distinct intervals are independent if the process characteristics areknown.

A NHPP with ROCOF v(t) is easiest defined by the property that thenumber of failures in interval (t1, t2] is a Poisson distributed randomquantity, with mean

m(t1, t2) =

∫ t2

t1v(t)dt .


Suppose we have observed the system over time period [0, r ], andhave observed failures at times t1 < t2 < . . . < tn ≤ r , then thelikelihood function is

L =

{n∏

i=1

v(ti)

}exp

[−∫ r

0v(t)dt

].

This enables inference as usual.

Two basic, often used parametric ROCOFs are

v1(t) = exp(β0 + β1t)

andv2(t) = γηtη−1,


Software Reliability

Many models that have been suggested, during the last five decades,for software reliability, are NHPPs which model the software testingprocess as a fault counting process.

Many models are variations to a basic model which assumes:

(1) Software contains an unknown number of bugs, N.

(2) At each failure, one bug is detected and corrected.

(3) The ROCOF is proportional to the number of bugs present.


The basic model is a NHPP with

v(t) = (N − i + 1)λ, for t ∈ [Ti−1,Ti),

for some constant λ.

N and λ are both considered unknown, and estimated from data,where N tends to be of most interest, or in particular the number ofremaining bugs.


Software reliability is a very important topic area, in particularstatistical support for software testing is challenging.

Book to appear (Wiley, August 2018): Analytic Methods in Systemsand Software Testing, edited by Keneth, Ruggeri and Faltin.

Includes chapter by Wooff, Goldstein and Coolen on the use ofBayesian Graphical Models for software testing, including manyaspects of implementation.

Challenge: considering reliability of systems with hardware andsoftware, including interaction.


Decision Making

Traditionally, decisions related to reliability (e.g. planning ofreplacements or inspections, warranties, etc) were mainly based onRenewal (Reward) Theorem:

It is assumed that the same process goes on for a very long time(‘infinitely’), consisting of consecutive cycli which are stochastic copiesof each other.

Let Xi be the random length of cyclus i , and Ri a random rewardassociated with cyclus i . Assume that the Xi ’s are iid, and that the Ri ’sare iid, but allow dependence of Ri on Xi .

For a process starting at time 0, let R(t) be the random cumulativereward upto time t . Then

R(t)t→ E(Ri)

E(Xi)if t →∞.

So ‘long-run average reward per unit of time’ is used.(UTOPIAE) Introduction to Reliability Theory 13 / 21

Assumption underlying use of renewal reward theorem oftenunrealistic; optimisation over one cycle, or a few cycli, or a fixed periodof time more realistic. Analysis may be harder, and decisions typically‘less risky’.

In recent years, emphasis has shifted to service contracts of fixedlength. This is a main game-changer, with possible advantages forreliability theory (e.g. wrt data ownership) and many new challengesfor decision making, e.g. planning and location of spare parts.


Survival Signature

System with K ≥ 2 types of components

mk components of type k ∈ {1,2, . . . ,K}, with∑K

k=1 mk = m

State vector x = (x1, x2, . . . , xK ), with xk = (xk1 , x

k2 , . . . , x

kmk

) thesub-vector representing the states of the components of type k withxk

i = 1 if the i th component of type k functions and xki = 0 if not.

Structure function φ(x) = 1 if system functions with state x andφ(x) = 0 if not.


The Survival Signature Φ(l1, l2, . . . , lK ), with lk = 0,1, . . . ,mk , is theprobability that a system functions given that precisely lk of itscomponents of type k function, for each k ∈ {1,2, . . . ,K}.

There are(mk

lk

)state vectors xk with precisely lk of their mk

components xki = 1, so with

∑mki=1 xk

i = lk .

Let Sl1,...,lK denote the set of all state vectors for the whole system forwhich

∑mki=1 xk

i = lk , k = 1,2, . . . ,K . Assuming exchangeability of thefailure times of the mk components of type k

Φ(l1, . . . , lK ) =

[K∏

k=1

(mk

lk

)−1]×

∑x∈Sl1,...,lK

φ(x)


Probability system functions at time t

Let Ckt ∈ {0,1, . . . ,mk} denote the number of components of type k in

the system that function at time t > 0. If the probability distribution forthe failure time of components of type k is known and has CDF Fk (t),and we assume failure times of components of different types to beindependent, then the probability that the system functions at timet > 0 is

P(TS > t) =

m1∑l1=0

· · ·mK∑

lK=0

Φ(l1, . . . , lK )P(K⋂

k=1

{Ckt = lk}) =

m1∑l1=0

· · ·mK∑

lK=0

[Φ(l1, . . . , lK )

K∏k=1

((mk

lk

)[Fk (t)]mk−lk [1− Fk (t)]lk

)]


Presented at 1st UTOPIAE Training School (Glasgow, November2017).

Main challenges include upscaling to very large real-world systems(with ‘zooming in’), applications to systems with multiple functions andto networks with different routes through them.

For some questions of practical relevance other summaries of the fullstructure function may be required.


Imprecision

Use of imprecise probability theory (also shown at 1st UTOPIAETraining School) is attractive in reliability, particularly when consideringnew or updated systems or components, or when there is doubt aboutsome model assumptions.

Inference and decision problems with imprecise probability tend torequire solution of constrained optimisation problems, with theinference as target function and the set of probabilities as constraint.

There are more opportunities to use idea of imprecision to providerobustness, enabling simple models to be used (e.g. in AcceleratedLife Testing).

Overall goal: Make the jobs of practitioners easier or better, preferablyboth!

Many opportunities and challenges!


Resilience

In recent years there has been increasing attention to ‘Resilience’: ifthings go wrong, how can negative effect be minimized?

Hugely important topic! For example for systems with multiplefunctions or multiple phases, some critical functions or phases may stillbe possible after a failure, possibly by re-configuring the system.

Of course, this also involves the usual aspects of maintenance, repair,etc.

Very many challenges, perhaps generalizing structure function as(imprecise) probability will prove useful.


The topic of ‘Reliability Theory’ is extremely wide, including aspectsfrom very many traditional areas of mathematics and engineering.

For substantial progress it is important to develop theory and methodsclosely linked to real-world problems, as the gap between text-bookproblems and reality is often very large.

UTOPIAE ESRs are well placed to make good contributions!

Feel free to contact me for more information or references.


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