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A bivariate process model for maintenance and inspection planning M.J. Newby *, C.T. Barker Centre for Risk Management, Reliability and Maintenance, City University, London EC1V 0HB, UK

Abstract The paper describes decision making about monitoring and maintenance of systems described by a general stochastic process. The system is monitored and preventive and corrective maintenance actions are carried out in response to the observed system state. The decision process is simplified by using an associated process as well as the underlying state as decision variables. The bivariate approach allows a wide class of models to be considered including long term memory within a simple probability structure. Both average cost and life-cycle cost models are used as the basis for decision making. The models generalize age replacement and other simple maintenance strategies. The approach can deal with failures that prevent the system functioning further, failures defined by regulation, or by economic considerations. The unified framework developed allows the inclusion of covariates and imperfect inspection or repair. q 2006 Elsevier Ltd. All rights reserved. Keywords: Bivariate process; Bessel process; Renewal-reward; Fredholm equation

1. Introduction We consider the problem of inspecting, maintaining and replacing a system. The system can be a single component or a more complex system. The system state is described by a stochastic process Xt. The arguments are developed with few restrictions on the process. The arguments thus allow systems with monotone or non-monotone trajectories to be analyzed. Non-monotone trajectories usually result when system usage is followed. For example, gas turbines have a maximum power output rating and users are advised not to exceed a fixed percentage of the maximum in normal operation. Use above the advised output indicates extra maintenance at the next service action. Built in test equipment records this type of information and report it to a diagnostic computer. We address the problem with two tools, firstly, the standard method of seeking the regeneration points of the stochastic process, secondly, by considering an associated stochastic real valued process. We thus begin with the underlying process Xt and work with a bivariate process (Xt,Yt) where Yt is a performance metric. The process Yt may be given or may be constructed by applying a functional to the basic process, YtZA(Xt). The advantages of the approach are that decisions can be based on failures defined * Corresponding author. Tel.: C44 20 7040 8347; fax: C44 20 7040 8597. E-mail addresses: [email protected] (M.J. Newby), [email protected] ac.uk (C.T. Barker).

0308-0161/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2006.02.009

by the entry of Xt into a critical region or on the pair (Xt,Yt). The associated process Yt can incorporate the process history or some other important aspect of the process. The system is inspected to determine its state (Xt,Yt) and a repair or replacement is chosen on the basis of the system state at the inspection. The aim here is to show, how optimal policies for inspection and maintenance can be derived for the system described by a stochastic process Xt. Initially, we make no strong assumptions about the process, the properties of the process are made explicit in particular examples. 1.1. Motivational background Many examples are available in risk analysis for engineering projects. Van Noortwijk [18] determined optimal maintenance policies for sea defenses; fatigue crack growth in pressure vessels and in aircraft structures has been described by Sobczyk [15] and Newby [8,9]; optimal inspection and maintenance policies for degradation processes are studied by Newby and Dagg [10,11] and that study forms the main background to the developments described here. Similar approaches are used in epidemiology: Jewell and Kalbfleisch [6] use marker processes in the study of CD4 counts in HIV infected patients; Betensky [3] used Wiener process models in designing sequential tests for differences in treatment effect of drugs. All of the examples mentioned in this paragraph construct a decision making process through using a stochastic process and using an associated process, usually a transform of the underlying process, as a decision variable.

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1.2. The model The approach in this paper differs through the use of the bivariate process (Xt,Yt) motivated by an extension of the methodology in an earlier paper by one of the authors [9] and the consideration of marker processes [6,19]. Natural examples of an associated process are: (a) (b) (c) (d) (e) (f)

the maximum process, Yt Z sup0%s%t Xs ; in a multivariate process YtZkX Ð tk; an accumulation process Y Z t 0%s%t Xs ds; Ð a usage measure Yt Z 0%s%t jXs jds; errors in measurement YtZu(Xt,3) where 3 is a noise term; covariate processes where a distribution F(XtjYt) describes the dependence of Xt on covariate Yt.

When the underlying process Xt is a Wiener process (a) above is well known, (b) is a Bessel process, and (c) is the Kolmogorov diffusion [7]. 2. The structure of the model

271

0%x 0 %x. In most cases y 0 Z0 because the decision variable will be reset; in the case when Yt is the maximum y 0 Zx 0 . Clearly x 0 Z0 corresponds to perfect repair and x 0 Zx implies no repair. The planned inspection period ends normally with the planned inspection or is terminated by the failure of the system. The time to system failure, starting from [Xt,Yt]Z(u,v) is the hitting time Tu,v of the critical sets, one of: † T u;v Z infftjðXt ;Yt Þ 2A * !B * g † TUu;v Z infftjXt 2A * g † TRu;v Z infftjXt 2B * g where A* and B* are critical sets. We shall write the hitting distribution starting from (u,v) as Gu,v(t) and assume it possesses a density gu,v(t). The probabilities required are the state probabilities u;v pi;j Z P ðXt ;Yt Þ 2Ai !Bj jX0 Z u;Y0 Z v ðð Z E 1fðXt ;Yt Þ2Ai!Bj jX0Zu;Y0Zvg ftu;v ðx;yÞdxdy Ai Bj

The bivariate process (Xt,Yt) is defined on a product space U !R, the state transitions are described by a transition density ftu;v ðx;yÞ where

and the failure probability u;v pu;v F Z G ðtÞ

ftu;v ðx;yÞdxdy ZP½Xt 2ðx;xCdxÞ;Yt 2ðy;yCdyÞjX0 Z u;Y0 Zv 4. The inspection cycle Failures are generally defined by the basic process Xt entering a critical set, that is the system works if Xt2G and fails at time T, the time of first entry into Gc ;T Zinf fsR0jXs 2Gc g. By using the bivariate model more possibilities are available. The decision process is thus driven by the excursions of the bivariate process ðXt ;Yt Þ2U!R. Decisions can then be made by partitioning the state space. The process space U is partitioned into sets Ai and R into sets Bi an inspection reveals the system state as (Xt,Yt)2Ai!Bj. The action is then chosen on the basis of the set Ai!Bj. For example, partition U into three exclusive subsets B0 (perfect), B2 (working), B3 (failed) and R into exclusive subsets A0 (perfect), A2 (working), and A3 (failed). 3. Inspection policies The decision maker inspects the system according to a policy P. The policy is a list of inspection epochs PZ {t1,t2,.tn} and we assume for the moment that inspection is perfect. Repairs are assumed instantaneous. The decision maker’s actions are determined by the system state (Xt,Yt); we develop two approaches: perfect repair and partial repair as described in Stadje and Zuckerman [16,17]. In perfect repair the revealed state on inspection (Xt,Yt) falls in an interval Ai!Bj and the system is returned to the perfect working state with cost Ci. In partial repair the system found with state (Xt,Yt)Z(x,y) is restored to the state (Xt,Yt)Z(x 0 ,y 0 ) where (x 0 ,y 0 )Zd(x,y). The function d(x,y) describes the repair and since the system cannot be made better than new we see that

The inspection and repair actions are assumed to occur at the beginning of each interval. This choice allows linking of the chain of decisions required in the dynamic programming solutions later. 4.1. Perfect repair The policies are determined by the intervals between inspections. We consider first the perfect repair case. Consider a single cycle where the time to the next inspection is t. The decision maker’s actions are: (1) do nothing if the system is in a ‘good’ state (Xt,Yt)2A0! B 0; (2) the system state is Ai!Bj, repair to the perfect working state with cost Ci,j and probability px;y ij ; (3) replace on failure with cost CF and probability px;y F . The cost of planned and unplanned repairs is X x;y ct ðx;yÞ Z Ci;j px;y i;j C CF pF i;j

Considering the whole cycle, if the total cost starting in state (x,y) is Vtx;y then E Vtx;y Z E VtXt ;Yt 1fðXt ;Yt Þ2A0!B0 Þg C ct ðx;yÞ where the first term in the expectation arises because the system state is left unchanged when the system is found in the

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‘good’ state A0!B0. Writing vtðx;yÞ Z E½Vtx;y it is clear that ð ð vt ðx;yÞ Z ct ðx;yÞ C vt ðu;wÞftx;y ðu;wÞdudw (1) A0 B0

4.2. Partial repair The state space is subdivided more simply, there are now only two states, ‘working’ and ‘failed’ and the decision maker carries out a repair d(x,y) on finding the system in state (x,y), d(x,y)/(x 0 ,y 0 ),. The cost of this repair is c(d(x,y)). Using a similar argument to above, it is clear that vt ðx;yÞ Z cðdðx;yÞÞ C fvt ð0;0Þ C CF gpFdðx;yÞ ð ð C vt ðu;uÞftdðx;yÞ ðu;uÞdudu

If the inspection intervals are allowed to change with the evolution of the system state, a non-periodic policy PZ {t1,t2,.tn} can be defined. The construction of the interval cost functions with the inspection and action at the beginning, makes the step from one interval to the next straightforward. With this construction the function vt(x,y) is the value function for a dynamic programming problem as described in Bather [2]. The optimality equation for the perfect repair version is 9 8 > > ð ð = < vt ðx;yÞ Z inf ct ðx;yÞ C vt ðu;wÞftx;y ðu;wÞdudw tO0 > > ; : A0 B0

(2)

A0 B0

where vt(0,0) arises from the replacement which resets all processes to zero. 5. Optimal policies

In the partial repair case the programming problem becomes 8 > < vt ðx;yÞ Z inf cðdðx;yÞÞ C fvt ð0;0Þ C CF gpFdðx;yÞ tO0 > : 9 > ð ð = dðx;yÞ C vt ðu;wÞft ðu;wÞdudw > ; A0 B0

The interval costs and the expected length of an interval can be used to construct an optimal average Q cost solution for a fixed maintenance interval with policy ZfktjkZ 1;2;.;ng. 5.1. Average cost criterion Q If we take a fixed policy with ZfktjkZ 1;2;.;ng the sequence of entries into the critical set defines an embedded renewal process and the average cost per unit time can be obtained using the renewal-reward theorem. For this, we need the expected length of an interval. For perfect restoration the expected interval length satisfies ðt ð ð x;y [t ðx; yÞ Z 1KG ðsÞ ds C [t ðu; wÞftx;y ðu; wÞdudw (3)

If costs are discounted with rate r the value function is modified and the perfect repair dynamic programming problem becomes 8 > < dðx;yÞ vt ðx;yÞ Z inf eKrt cðdðx;yÞÞ C fvt ð0;0Þ C CF gpr;F tO0 > : 9 > ð ð = dðx;yÞ C vt ðu;wÞft ðu;wÞdudw > ; A0 B0

where pu;w r;F Z

ð

eKrs gu;w ðsÞds

B0

A0 B0

0

5.3. Obtaining solutions

and for partial restoration ðt [t ðx;yÞ Z 1KGdðx;yÞ ðsÞ ds

The optimization problems above contain integral equations of the Fredholm type (cf. Eqs. (1)–(4)) so that discretization of the state space and application of quadrature rules produce equivalent matrix equations with the general form

0

ð ð C

[t ðu;wÞftdðx;yÞ ðu;wÞdudw

(4)

A0 B0

where 1KGd(x,y)(s) is the interval survival function. Thus, using Eqs. (1) and (2) (or Eqs. (3) and (4) for the case of partial repair), we can calculate the average cost per unit time v ðx;yÞ Cðx;tÞ Z t [t ðx;yÞ The optimum policy for a system starting in state X0Zx can then be determined as t- Z argminfCðx;tÞg t

5.2. Lifecycle cost criterion

v Z c C Mv which are readily solved numerically [12]. The dynamic programming problems translate in the same way and allow a policy improvement algorithm [2] to be applied to develop the optimal policy. Convergence proofs for the algorithms are given by Dagg in his thesis [5]. 6. Examples The models depend on obtaining the joint transition density u;v fX;Y ðx;yÞ of the process starting from X0Zu and Y0Zv.

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273

The examples give some instances of the way in which they can be derived.

with error (here a simple additive error is assumed)

6.1. Maximum process

where 3t represents the error. To be specific assume here that the inspection error is normal, with mean zero and variance n2. The underlying process is the Gamma process with increments distributed as [1]

An illustration of the bivariate process is obtained by taking a basic process Xt and constructing the bivariate process (Xt,Yt) with the maximum process Yt Z supfXt j0% s% tg. A nonmonotonic process with continuous sample paths is defined by the Wiener process XtZsBtCmt with drift m and variance parameter s and where Bt is a standard Brownian motion. The state space is UZR and is divided into intervals at points K N!s1!.,sj,.,sn so that YtRsn or XtRsn indicate failure. The required densities and distributions can be deduced from results in Rogers and Williams [14]; the joint density of the process and its maximum, with X0Zu, is 2ð2yKxKuÞ ðxKuKmtÞ2 u ft ðx;yÞ Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp K 6 3 2s2 t 2ps t 2ðyKuÞðyKxÞ !exp K s2 t The marginal distribution of Yt is yKuKmt 2mðyKuÞ Ky C uKmt u pﬃﬃﬃ pﬃﬃﬃ Ft ðyÞ Z F Kexp F s t s t s2 which gives an inverse Gaussian distribution (Chikkara and Folks [4]) as the hitting time distribution. 6.2. The integrated process When the underlying process Xt is a Wiener process the integrated process Yt makes up the two-dimensional Kolmogorov diffusion process {X t,YÐt}. For a Brownian motion Bt the transition density of Bt ; Bs ds starting from (u,v) is following McKean [7] pﬃﬃﬃ 3 ðvKyKtuÞ2 ðxKvÞðyKvKtuÞ ftu;v ðx;y;tÞ Z 2 exp K6 C6 pt t2 t3 ðuKyÞ2 K2 t The linearity of the integral shows that Yt is also Gaussian and its moments are easily obtained from first principles. The moments of the integrated process when the basic process Xt has drift m and volatility s are 1 E½Yt Z mt2 2

1 V½Yt Z s2 t3 3

Yt Z Xt C 3t

Xtj KXti wGaðaðtj Kti Þ;bÞ and error distributed as 3t wNð0;n2 Þ

ct

The decision process needs the conditional distribution of the true level of degradation given the observation at that time. Then it is elementary that PðYt % yjXt Z xÞ Z PðXt C 3t % yjXt Z xÞ

yKx Z Pð3t % yKxÞ Z F n

So that YtjXtwN(Xt,n)2. The distribution of the true level of degradation at a particular instant, given the observed degradation level is also needed. The distribution is derived assuming that 0%X%c where c is the critical level defining failure. Thus pﬃﬃﬃﬃﬃﬃ f ðxjyÞ Z exp K12n2 ðyKxÞ2 n 2p½FðynÞKFðyKcnÞ because YtZXtC3t and 3twN(0,n2). The density of Xtu is given by

bat ðxKuÞatK1 eKbðxKuÞ f Xt Z xjX0 Z u Z GðatÞ

Imperfect inspection provides another example of associated processes. In this case, Yt is simply the observed level of degradation subject to error. The simplest model of imperfect inspection is when the system degradation Xt is observable, but

(6)

The results given in Eqs. (5) and (6) combine to give the joint distribution of the future observed and true values of the degradation

fYt ;Xt jX0 y;xju ZfYt jX0 ;Xt yju;x fXt jX0 xju ZfYt jXt yjx fXt jX0 xju Plugging in the densities yields the joint density of the observed and true level of degradation, conditional on the true initial level of degradation, 1 1 ftu ðy;xÞZ pﬃﬃﬃﬃﬃﬃ exp K 2 ðyKxÞ2 2n n 2p !

bat ðxKuÞatK1 expfKbðxKuÞg GðatÞ

The distribution of the hitting time of the critical set from an initial level of degradation u is pxF ZPðTcu !hÞZ

6.3. Models for imperfect inspection

(5)

Gðah;bðcKuÞÞ GðahÞ

6.4. Bessel processes The simplest approach is illustrated by the norm functional YtZkXtk which generates a Bessel process from a multivariate

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Brownian process. The basic process is based on an d-dimensional Brownian motion, Bt Z ½B1 ðtÞ;B2 ðtÞ;.;Bd ðtÞ, and the process YtZkBtk2 or more explicitly " #1=2 d X Yt Z B2i ðtÞ iZ1

For simplicity, consider the process Zt Z Yt2 Z sBt s2 Z

d X

B2i ðtÞ

iZ1

The excursions of the two processes produce equivalent decision rules, and the second is simpler to handle. The process Zt is a squared Bessel process [13]. If dR2, the process never reaches 0 for tO0. The squared Bessel process is the square of the Euclidean norm of a d-dimensional Brownian motion. Because the distributions of the Bi(t) are normal, the probability density function is a chi-squared distribution if z0Z0 and dO0: the density is the chi-square with d degrees of freedom ftn ð0;zÞ

Kz zn Z e 2t 1fzO0g; tO 0 nC1 ð2tÞ Gðn C 1Þ

6.4.1. Modelling the degradation A simplification used in this paper is to regard the components as belonging to groups with a common volatility. The object is to reduce the complex system to a smaller collection of subsystems each of which can be dealt with analytically. For this, suppose that the components of the considered system can be classified in the following categories: K1,.,Kp, and ni indicates the number of components in catergory Ki i2{1,.,p}. One category of components is now examined. The squared Bessel process provides a summary description of the degradation of this category of components. The category is K1 containing n1 components, namely Cj1 ; jZ 1.n1 where the superscript indicates the category. The individual processes Wi1 ðtÞ Z m1i t C s1 Bi ðtÞ;

Ytn1 ;m Z sWt1 s2 Z 1

if z0s0 and dO0: the density is a non-central chi-square written in terms of modified Bessel functions of the first kind n pﬃﬃﬃﬃﬃﬃ z0 z 1 z 2 Kz02tCz ftn ðz0 ;zÞ Z e In 1fzO0g ;tO 0 2t z0 t where N X

z nC2n 2

n!Gðn C n C 1Þ nZ0

; n;z 2C

The model is more realistic if the Brownian motion is replaced by a Wiener process with drift. Consider now a d-dimensional Wiener process: Wt Z ðW1 ðtÞ;W2 ðtÞ;.;Wd ðtÞÞ with Wi(t)ZmitCsiBt(t) and Wi(0)Zxi. Future values are distributed as Nðxi C mi t;s2i tÞ. Now define the squared Bessel process with drift Ytd;m as the square of the Euclidean norm of the Wt process d X iZ1

where mZ

ni X

jWi1 j2

iZ1

with m1 Z

n1 X

ðm1i Þ2

iZ1

d n Z K1 2

Ytd;m Z

c i Z 1;.;n

The squared norm is the n1-dimensional squared Bessel 1 process Ytn1 ;m

where

In ðzÞ Z

Wi ð0Þ Z 0;

d X iZ1

jWi ðtÞj2

Failure occurs when the total degradation of the components in category K1 exceeds a critical value. The assumption of a common variance allows the immediate conclusion 1 n1 1 pﬃ 2 X Ytn1 ;m mi t 2 wcn1 ;l1 where l1 Z 2 s1 s1 t iZ1 Thus, for tO0 and s1s0, the probability that the summary process is above a critical value r12 O 0 at time t is " # 1 h i Ytn1 ;m r12 n1 ;m1 n1 ;m1 2 n1 ;m1 P Yt O r1 jY0 Z0 ZP Z0 O 2 jY0 s2 t st C ðN

Z

ftd ð0;yÞdy

r12 =ðs21 tÞ C ðN

Z

eK

yCl 1

2

1 pﬃﬃﬃﬃﬃ l1 y n1K 2 n1

r12 =ðs21 tÞ

2ðl1 yÞ 4

I n1K1 2

pﬃﬃﬃﬃﬃﬃﬃ l1 y

The overall system can be considered as failed if the level of degradation of at least one class exceeds a critical value or if the sum all these levels of degradation exceeds some fixed value K. This probability is " # n X n[ ;m[ Yt O K P lZ1

m2i

and the reduction to a small number of classes allows direct computation.

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6.4.2. Modelling the environment Environmental variables are assumed to act by causing the failure of a category of components. We consider a single category/component C1 and append the environmental processes. The component follows a process WðtÞ Z mt C sBðtÞ in its normal state. Covariates (temperature, pressure, humidity, etc.) or the history of the component affect the degradation process. Let z1,z2,.,znK1 be the nK1 covariates, and I1,I2,.,InK1 corresponding critical sets. If zi2Ii, the degradation of C1 switches to a new Wiener process Wi ðtÞ Z mi t C sBi ðtÞ To obtain a full description which allows overlapping time intervals during which zi2Ii, begin with an n-dimensional Wiener process: ( ) nK1 Y Wt Z 1fzi;Ii g WðtÞ ; iZ1

!

1fz12I2 g W1 ðtÞ; 1fz22I2 g W2 ðtÞ; .; 1fznK12InK1 g WnK1 ðtÞ Dealing with this n-dimensional Wiener process allows us to treat any possible combination for the presence of covariates. The indicator functions 1fzi2Ii g ; i Z 1.nK1;

and

nK1 Y

1fzi;Ii g

iZ1

act as switches as the environment evolves. The decisions are then based on the squared Bessel process associated with the underlying Wiener process. " #2 nK1 nK1 Y X Yt Z 1fzi;Ii g WðtÞ C ½1fzi2Ii g Wi ðtÞ2 iZ1

iZ1

which simplifies to ( ) nK1 nK1 Y X Yt Z 1fzi;Ii g WðtÞ2 C 1fzi2Ii g ½Wi ðtÞ2 iZ1

iZ1

7. Conclusion The models proposed have developed a unified structure for decision making where the degradation is described by a

275

process Xt has an associated process Yt that is given or is constructed. We have outlined the development of the associated processes here. Most of the models require a bivariate transition density and some simple examples have been given. The formulation of the models depends on identifying the instants of perfect repair or replacement where the probability laws are reset to time zero. The construction of the intervals also allows the sequential version of the models to be formulated as a dynamic programming problem. References [1] Abdel-Hameed MA. A gamma wear process. IEEE Trans Reliab 1975;R24:152–4. [2] Bather J. Decision theory: an introduction to dynamic programming and sequential decisions. London: Wiley; 2000. [3] Betensky RA. A boundary crossing probability for the bessel process. Adv Appl Probab 1998;30:807–30. [4] Chhikara RS, Folks JL. The inverse gaussian distribution. Theory, methodology and applications. New York: Marcel Dekker; 1989. [5] Dagg R. Optimal inspection and maintenance for stochastically deteriorating systems (PhD thesis, City University, London; 2000). [6] Jewell NP, Kalbfleisch JD. Marker processes in survival analysis. Lifetime Data Anal 1996;2:15–29. [7] McKean Jr HP. A winding problem for a resonator driven by a white noise. J Math Kyoto Univ 1963;2:227–35. [8] Newby MJ. Estimating of Paris-Erdogan law parameters and the influence of environmental factors on crack growth. Int J Fatigue 1991;13:187–98. [9] Newby MJ. Analysing crack growth. Proc ImechE Part G Aero Eng 1998; 212:157–66. [10] Newby MJ, Dagg R. Optimal inspection and perfect repair. IMA J Manage Math 2004;15(17):192–219. [11] Pitman JW, Yor M. Bessel process and infinitely divisible laws, stochastic integrals. Lecture notes in mathematics. Berlin: Springer; 1981. [12] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes in C. 2nd ed. Cambridge: Cambridge University Press; 1992. [13] Revuz D, Yor M. Continuous martingales and brownian motion. 3rd ed. Berlin: Springer; 1999. [14] Rogers LCG, Williams D. Diffusions, markov processes and martingales, volume 1: foundations. 2nd ed. Chichester: Wiley; 1994. [15] Sobczyk K. Stochastic models for fatigue damage of materials. Adv Appl Probab 1987;19:652–73. [16] Stadje W, Zuckerman D. Optimal maintenance strategies for repairable systems with general degree of repair. J Appl Probab 1991;28:384–96. [17] Stadje W, Zuckerman D. Optimal repair policies with general degree of repair in two maintenance models. Oper Res Lett 1992;11:77–80. [18] Van Noortwijk JM. Optimal maintenance decisions for hydraulic structures under isotropic deterioration, PhD thesis, Technical Univrsity Delft, Netherlands; 1996. [19] Whitmore GA, Crowder MJ, Lawless JF. Failure inference from a marker process based on a bivariate wiener model. Lifetime Data Anal 1998;4: 229–51.