Some Foundations of Weibull-Based Generalized RenewalProcesses

Ricardo Jose Ferreira 1

Paulo Renato Alves Firmino2

Claudio Tadeu Cristino3

1 Introduction

Let us considerer a system that suffers from restoration along the time, by which planned or

unplanned interventions may arise. This kind of situation is common in several areas. In Engi-

neering, production machines must operate in adequate conditions to yield the target production

levels. In Economics, corporations in general are considered as systems to be maintained at high

levels of production. In Human Health Sciences, the challenge is to maintain the welfare of the

individuals. In these three situations, one can highlight as a common characteristic the eventual

presence of deterioration of the system. In engineering andhuman health sciences, equipments

and people can be seem as systems that unarguable deteriorate through the time.

Anyway, the main tool for handling systems deterioration has been based on planned or even

unplanned interventions. In other terms, the deterioration of systems has demanded planned

and unplanned interventions policies in order to maintain the adequate performance of systems;

these systems must receive some kind of intervention to extend their lives.

It is not hard to see, then, the importance of intervening against systems deterioration. It

can be done in several ways, with several existing methodologies in literature. Between them,

one can cite generalized renewal processes (GRP). The studyof GRP has been paramount for

the understanding, modelling and maintenance of dynamic systems in the most diverse areas

of knowledge. They deal with the challenge of explicitly andsimultaneously modelling two

stochastic sources: deterioration and restoration. For that, a GRP extends a family of probabil-

ity distributions by appending in their parameter set a rejuvenation parameter, sayq, dedicated

to reflect the age of the system after planned or unplanned interventions. This idea was firstly

presented by Kijima & Sumita [4] who introduced the concept of virtual age, a function that op-

erates on the real age of the system viaq. Thus, asystemcan generically represent a biological,

social, financial, economic phenomenon and so on. For the sake of illustration, epidemiology

studies are frequently dedicated to develop optimal vaccination policies against specific diseases

1Doutorando pelo Programa de Pos-Graduacao em Biometriae Estatıstica Aplicada - UFRPE, professor doIF-PB, e-mail: ricardo.jferreir@gmail.com

2Professor do Departamento de Estatıstica e Informatica -UFRPE, e-mail: paulo.firmino@deinfo.ufrpe.br3Professor do Departamento de Estatıstica e Informatica -UFRPE - e-mail: ctcristino@gmail.com

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whilst in production engineering the focus is on optimal maintenance policies for equipments.

In both cases, the termoptimal is associated with policies compromised with the minimization

of intervention costs as well as the maximization of system performance according to a given set

of metrics. Several examples can be seen in literature [5, 6,8, 11]. Thus, the decision maker is

challenged to determine the respective interventions policy, taking into account for the behavior

of the system after such interventions.

In GRP literature it has been usual to adjust a Weibull-GRP (WGRP) model to the sampled

interventions series [1, 12, 13] in order to infer the performance of the system in terms of

deterioration and restoration parameters. In their analyses, WGRP practitioners have interpreted

the rejuvenation parameterq as the one that solely reflects the restoration process (see [13, 8, 7]

for instance). It has also been common that the shape (β) and scaling (α) parameters are solely

dedicated to model the deterioration of the system, regardless of the the restoration process

[1, 12, 13]. On the other hand, it has yet been usual to assume that best intervention practices

would lead to best performances, regardless of the deterioration condition of the system. The

present paper demystifies these reasoning by studying the relationships betweenq, α andβ in

WGRP models. Therefore, interpretations for the systems performance in terms of the quality

of (i) the maintenance crew and (ii) the supporting technologies are reviewed.

Specifically, the paper presents an overview about the applications and mathematical prop-

erties of the WGRP. In this way, it highlights (i) a set of definitions, notations and relationships

in WGRP, (ii) the unexplored branches of the area, allowing to speculate about future develop-

ments, (iii) some interesting results around reliability functions showing important relationships

between the parameters aforementioned.

In Section 2, we bring a discussion about the GRP structure, practical meaning and proper-

ties of their parameters. In Section 3, we present some points to be tackled by studying WGRP

with more details, including simulations and a wide discussion about possibilities and expan-

sions of this process. Finally, in Section 4, we show some conclusions.

2 Basic Concepts

This section introduces some concepts underlying GRP and the notations to be used through-

out the paper for labelling random variables and constants.Following general orientation from

literature, random variables and constants will be represented respectively by upper and lower

case letters. These notations are focused on the problem of fitting GRP models to performance

datasets involving the occurrence of events of interest in agiven system. Specifically, theevents

of interestwill be considered as (un)intendedinterventionson the condition of the system and

the focus will be on modeling the response of the system to these interventions. These interven-

tions might represent a single event (e.g. preventive and corrective actions).

2

2.1 Time Concept

Without loss of generality, the wordtime will represent any unit measure over which the

interventions are observed (e.g. meters, seconds, kilograms, and so on). Besides, the time on

which the interventions occur is considered negligible,i.e. just point process [10] are taken into

account. Finally, it is also considered that systematic increasing (decreasing) times between

interventions characterize improvements (deterioration) of the system.

Definition 2.1. Let Ti be the time when theith intervention occurs (the actual time untili) and

let Xi be the time between the(i −1)th and theith interventions (X0 = 0).

From both Definition 2.1 and the Point Process Foundation, wecan see thatTi = ∑ij=1Xj is

the age of the system when the ith intervention occurs. A direct consequence is thatT0 = 0. It

must also be highlighted thatTi as well asXi can be characterized as random variables and thus

subject to statistical modeling via GRP. As follows it is presented the main concept underlying

GRP models: virtual age.

Definition 2.2. Let Vi be the virtual age of the system reflecting its restoration condition after

i interventions. Thus,Vi is a function of{X1, . . . ,Xi} and an appending parameter, sayq: Vi =

v(X1,X2, . . . ,Xi |q).

From Definition 2.2, one can see thatq is a parameter primarily responsible to represent

the quality of the performed interventions. Thus, in the light of the performance data set of the

system{x1, . . . ,xi}, one hasvi−1 = v(x1,x2, ...,xi−1 |q) and for a givenxi(> 0) the respective

virtual age isvi−1+xi . Under this reasoning, Kijima (1989) proposes two linear models forVi

widely known in GRP literature as Kijima types I and II models:

Vi =Vi−1+qXi (1)

Vi = q(Vi−1+Xi) (2)

In Equation 1, theith intervention only operates onXi . On the other hand, in Equation 2,

the ith intervention operates onXi and on the previous updated times between interventions,

composing a geometric propagation of the quality of the interventions on the restoration of the

system. Anyway, in both models three situations can be highlighted:

• q = 0: The virtual age is reset and the system is renewed by the interventions. In other

terms, the restoration is considered perfect leading the system to an “as good as new”

condition, reflecting a Renewal Process (RP) [9];

• q= 1: The virtual age equal the actual age,i.e. the restoration is minimal and lead the sys-

tem to an “as bad as before intervention” condition, characterizing a Non-Homogeneous

Poisson Process (NHPP) [9];

3

• 0< q< 1: The interventions are considered imperfect,q leads the system to an interme-

diate condition of restoration. This reveals the generalized facet of GRP.

Authors like Kijimaet. al [3] also emphasize that GRP allows to model two other patterns

of restoration of the system: “better than new” ifq< 0 and “worst than before intervention” if

q> 1. However, these situations are not usually found in literature.

Next, it is presented a brief definition about the Weibull distribution, aiming to know its

behavior according with its parameters values.

Definition 2.3. A random variable, sayX, is modeled by a Weibull distribution if its Probability

Density Function (PDF) is given as follows

fX(x;α,β) =βα

( xα

)β−1exp

{

−( x

α

)β}

, x,α,β > 0 (3)

Furthermore, the Cumulative Distribution Function (CDF) and Reliability/Survivor function

can be easily obtained from this PDF.

Besides the notations for time variables, it is important toshow the interpretations made

for the Weibull parameters. Studying the Weibull distribution, some situations can be captured

according to the values that its parameters assume. Some of these situations are shown in next

Section.

3 Interpretations of WGRP

As described in Section 2, GRP use some specific notations, usually connected with time

variables and a virtual age function to measure the ageing ofthe system. Besides, the Weibull

distribution proves to be quite faithful to assess time random variables. Thus, the use of the

Weibull distribution linked with GRP seems to be a suitable modeling for treating systems

with deterioration characteristics. Firstly, it is presented a brief introduction to the Weibull

parameters relationship. Then, the link of this distribution with the Weibull distribution shows

the importance of considering the connection between the parameters. From PDF, one can

obtain the Reliability/Survivor function as follows.

Definition 3.1. A random variable, sayX, has the following reliability/survivor function derived

from the PDF in Equation 2.3:

RX(x;α,β) = exp

{

−( x

α

)β}

, x,α,β > 0 (4)

From definition 3.1, it is easy to analyze the influence of the parameters.

4

3.1 Study of scale and shape parameters

The first step is to identify the influence of the scale and shape parameters on the relia-

bility/survivor function. As illustrated in Figure 1, thisfunction decreases quickly or slowly,

depending on the value assumed byα. Basically, its scale is changed – this is the reason thatαis considered thescaleparameter.

Figure 1:Reliability/Survivor function varying according with values ofα

In Figure 2, the curve changes in its form by the variation of the values ofβ. Thus, it is

common to callβ as theshapeparameter.

Figure 2:Reliability/Survivor function varying according with values ofβ

Based on the observation of the reliability/survivor function, one important issue can be

discussed. Using a fixed value for the scale parameter (α) and varying the shape parameter

(β), it is possible to notice an important feature about the probabilities of occurrence of an

intervention. This is what can be seen in Figure 2. More specifically, the probability of a

system survive until its half-life characteristic is 36,78% approximately. Thus, in that instant,

all systems have the same probability of survival regardless their shape parameter.

5

Furthermore, regardless this feature, one can notice that whenβ < 1, the reliability function

is lower compared with a curve withβ > 1. Thus, one can argue that for small periods systems

with a low value forβ tend to suffer more interventions than systems with high values forβ. On

the other hand, this scenario is inverted as the time evolves.

A practical implication of this issue rests the virtual age of a system, which is discussed

next.

3.2 The WGRP: joint influence of the Weibull parameters and the virtual

age parameter

As said before, it is not correct to consider the isolated effects of Weibull parameters, spe-

cially in treating systems that suffer from some level of restoration. Furthermore, restoration

itself needs special attention - which is treated when including the GRP rejuvenation parameter,

q.

First of all, a discussion about the impact ofq in a system with a lower or higher values of

β. As said before, this contributes to the performance of the system according with half-life

characteristics – lower values tend to stabilize with more time, whereas higher values bring the

system to undesirable situations more quickly after the half-time characteristic.

However, since these systems suffer from restoration, it isnot correct to consider a perfect

recover of them after an intervention. For the sake of illustration, as time evolves, equipments

and human body loses resistance even with some precautions (interventions). Thus, it is com-

mon to think of imperfect intervention, which is more realistic in real systems.

To study these situations, a conditional reliability/survivor function is defined in GRP.

Definition 3.2. A random variable, sayX, has a conditional Weibull reliability/survivor function

iff its form is as follows:

RX(x+v|α,β,v) = exp

{

( vα

)β−

(

x+vα

)β}

, x,α,β,v> 0 (5)

wherev is the predefined virtual age of the system.

Intuitively, this discussion leads to the point that it is necessary to link the three parameters

mentioned here - in some way, scale, shape and virtual age areconnected. It only remains to

explore how it happens.

A similar graphic to Figure 2 is presented, now involving theGRP parameter. In Figure 3,

the value of the virtual age is considered as 100 measure units, whereasα = 500, as before.

One can see that the previous relationship is not the same here - the presence of the virtual age

parameter disturbs the simple-direct relationship between the shape and scale parameters in the

reliability/survivor curve.

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Figure 3:Reliability/Survivor function behaviour with the influence of GRP parameter

The meaning of curves previously scratched is wide and implies in important interpretations

in each analyzed case. Some analytical results is presentednext to support the figures shown

and to create a generalization.

3.3 Some analytical results

By observing Figure 2, it is important to explore how that relationship is given. Some math

can be done to obtain the following equality, consideringβ1 < β2.

R(x;α,β1) = R(x;α,β2) ⇔( x

α

)β1=

( xα

)β2. (6)

Sinceβ1 < β2 andx> 0, this equality holds only ifx= α, which coincides with the results

shown in Figure 2. Furthermore, whenx < α, the left side of the equality turns into a larger

value, whereasx> α represents smaller values of the left side. In other terms, this means that

systems will be more susceptible to interruptions before time α if they are modelled byβ1.

Similarly, systems will be more susceptible to interruptions after timeα if they are modelled by

β2.

This kind of interpretation is only valid for non-perishable systems. When deterioration is

introduced, the rejuvenation parameterq arises as a good estimator of the effect of interventions

made on systems. Observing Figure 3, one can see that the relationship between different

reliability curves results in a different interception point where those curves shift position in

deterioration. By doing a similar analysis as made in (6):

R(x;α,β1,v) = R(x;α,β2,v) ⇔( v

α

)β1−

(

x+vα

)β1

=( v

α

)β2−

(

x+vα

)β2

. (7)

7

By doing some analysis, the Equation 7 holds if those roots are equal to zero or

0.5

(√

vα

α−√

−3vα+4α2

)√

vα−2v.

Sincev,α > 0, the condition inside the second root has to be satisfied:v <4α3 . To study the

cases where one curve is the greatest, similar analyzes mustbe done with these roots.

This analysis shows that the relationship between two different reliability curves considering

deterioration present some complexity due to the presence of the GRP parameter.

00

00

200200

400400

600600

800800

10001000

5050100100

vv

10001000150150 800800600600

��

400400200200200200

Figure 4:Analyzing the root of the equality of reliability curves with deterioration

Figure 4 exhibits the relationship in (7). This is an important feature, since one can induce

that, oncev and α gets large, times between intervention gets lower even more. Thus, old

systems tends to involve earlier interventions, in such a way that in the future the time between

interventions will vanish.

These kind of results open opportunities to study other features of the WGRP. It is inter-

esting, for example, to study the expectation ofX, which is theme of actual researches of the

authors.

4 Discussions

This paper discusses the importance of adequately dealing with systems that suffer from

restoration. In this way, stochastic process literature highlights GRP as a good approach to deal

with this kind of problem once it involves a parameter able tocapture the effects of the planned

and unplanned interventions in the system ageing.

Some definitions and notations are presented in a general way, since a system can be defined

8

broadly. The study of deterioration in Engineering, Agrarian, Health Sciences, and Social Sci-

ences is emphasized, and so forth. The most important feature is about the best way to define

systems interventions and measure performance.

The time until unplanned interventions is the target, sinceageing is acting directly on that.

Thus, some probabilistic measures can be studied to assess some important features as the re-

liability/survivor behaviour of these systems. The paper also highlights that the most used dis-

tribution in literature is the Weibull-based GRP (WGRP) distribution and, in this way, attention

on the interpretation of the WGRP parameters is dedicated. In summary, WGRP parameters

can be interpreted as the half-life and intensity parameters (α andβ, respectively).

Thus, one can interpret two parameters set. The Weibull reliability/survivor function char-

acterizes an important feature where two different systems- consideringβ - have different

behaviour regarding the value ofα. For a time less than thisα, smallerβ’s represent a less

reliable system, whereas for times greater thanα, increasingβ’s represent this situation.

However, the introduction of the GRP parameter changes abruptly the interpretations, since

the ageing acts direct in the actual time. These influences can be seen in the set of figures

in Section 3.2 and 3.3. Those results brings some important discussions. Furthermore, the

obtained results provide some mathematical appeal to a better understanding of the shifting of

behaviour of two different systems that suffer with deterioration.

Besides, analysing the roots of this shift helps to plan optimal maintenance policies trying to

maximize the life of the system and minimizing costs. One important measure to deal with this

is the analysis of the expected time between interventions.The authors are developing some

calculus to present some moments of this distribution, aiming to provide better interventions

planning.

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