a semi-markovian model allowing for inhomogenities with respect to process time

A semi-Markovian model allowing for inhomogenities with respect toprocess time

G. Beckera,* , L. Camarinopoulosb, G. Zioutasc

aRISA Sicherheitsanalysen GmbH, Krumme Strabe 55, Berlin, GermanybUniversity of Piraeus, 18534 Piraeus, Greece

cAristotelian University of Thessaloniki, Faculty of Technology, General Department, Greece

Received 25 June 1999; accepted 15 March 2000

Abstract

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or smallsystems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneousMarkov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather thantransition probabilitiesQij �t�; which are used in normal mathematical text books to define semi-Markov processes, transition rateslij � � areused, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and onsojourn time in statei. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much thesame way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, whichwould result inN2 integral equations, a system ofN integral equations for frequency densities of reaching states is considered. Once thissystem is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 stateshas been solved as an illustration using the approach.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: State graph analysis; Semi-Markov process; Inhomogeneous; Transition rates

1. Introduction

A semi-Markov process can be considered as an exten-sion of an ordinary Markov process with discrete states andcontinuous time [1]. For an ordinary Markov process, theMarkovian property is given at all times. This leads to tran-sition rates between the states, which are either constant, orat most depend on the process time starting with the begin-ning of observation (subsequently referred to as calendartime). Transition rates may not depend on the duration acertain state is assumed, as this would violate the Markovianproperty. Time dependent transition rates can be used tomodel phenomena like scheduled inspection or mainte-nance, or certain phased mission problems.

The transition rates are defined as

lij �t� dt � pr{Z�t 1 dt� � juZ�t� � i} �1�

whereZ�t� is the (discrete) state at timet, andt the processtime. So, this is the probability, that the process will transfer

to statej in an infinitesimal time interval�t; t 1 dt�; giventhat it is in statei at timet. Random life and repair times aremodeled using constant transition rates. These correspond toconstant failure resp. repair rates, i.e. to exponentiallydistributed life and repair times.

For a semi-Markov process, the Markovian property isrequired only for the transition points of time. Thus, distri-bution of the duration in the states can be arbitrary [1]. Asemi-Markov process is described by transition probabilitiesdefined as

Qij �t� � pr{Sn 2 Sn21 # t > Z�Sn� � juZ�Sn21� � i} �2�whereSn is the time of next transition andSn21 the time oflast transition (or 0) with respect tot. So, in this case, thevariablet is not process time, but sojourn time in statei.

Though the distribution of the duration, when the processis in a given state, is arbitrary, this is in general not true forthe life times and repair times of the components, which arethe basis of the process considered. As soon, as there is morethan one component involved, transition to another statewill only involve a change of state of one component. So,the life times of the other components will continue, andthere is a dependency from the past beyond the time of the

Reliability Engineering and System Safety 70 (2000) 41–48

0951-8320/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0951-8320(00)00044-2

www.elsevier.com/locate/ress

* Corresponding author.E-mail addresses:[email protected] (G. Becker), lkamarin@uni-

pi.gr (L. Camarinopoulos), [email protected] (G. Zioutas).

last transition. For this reason, many semi-Markovprocesses used for reliability evaluations (those involvingmore than one component, and where the system is notrenewed, when a single component fails) will have a lotof transitions governed by exponentially distributed dura-tions and only few governed by arbitrary distributions. Asan exponential distribution will lead to a constant transitionrate, it appears sensible to attempt to describe a semi-Markov process by transition rates (which in general willdepend on sojourn time), because this will facilitate calcula-tions in case, these rates are constant.

2. Transition rates of semi-Markov processes

In the following, transition rates are derived for a generalsemi-Markov process. These transition rates will depend onthe sojourn times in the states; they have to be distinguishedfrom the transition rates of an ordinary Markov process,which are either constant or a function of process time.The transition probabilitiesQij given basically are distribu-tion functions. So, a definition similar to the failure rate of alife time distribution should be used, as the desired feature isthat constant rates are obtained in the case of exponentialdistribution. In the following, the semi-Markov transitionratesl ij are defined in terms of theQij. Subsequently, theQij are reconstructed based on thel ij to show that the trans-form is bidirectionally unique.

2.1. Definition of transition rates

In analogy to the definition of the failure rate of a compo-nent with the given life time distribution, the followingdefinition of semi-Markov transition ratesl ij is suggestedusing the same notation as above

lij �t� dt � pr{Sn 2 Sn21 [ �t; t 1 dt�> Z�Sn� � juZ�Sn21�� i > Sn 2 Sn21 . t} (3)

This means the probability that a transition to statej occursimmediately after the process has been in statei for a dura-tion t, given that no transition leaving statei has occurredbefore. From the definition of transition probabilitiesQij �t�;

it can be derived trivially that

lij �t� �_Qij �t�

1 2 Qi�t� �4�

where

Qi�t� � pr{Sn 2 Sn21 # tuZ�Sn21� � i} �X�k�

Qik�t� �5�

is the distribution function of sojourn time in statei. Simi-larly, it is obtained that

Qij �t� �Zt

0lij �z�exp 2

Zz

0li�x� dx

� �dz �6�

where

li�t� �X�k�

lik�t� �7�

Thus, the information provided by semi-Markov transitionrates is equivalent with the information provided by semi-Markov transition probabilities.

2.2. Transition rates and failure rates

For ordinary Markov processes, as applied in reliabilitytheory, it is a well known result, that transition rates willcorrespond to constant failure or repair rates of the compo-nents, which can fail or become repaired in a given state. Itis an interesting question, whether the same would hold forthe semi-Markov transition rates introduced so far.

Consider the following case: if the system enters a statei,a number of independent random timesTk with distributionfunctionsFk�t�; and probability density functionfk�t�; andfailure ratelk�t� will begin, as shown in Fig. 1, which maycorrespond to life-times or repair-times of some compo-nents. The process will transfer to statej, if the realizationof Tj is the smallest of all these variables, and the sojourntime in state i will just be this smallest realization.This situation is typical for most applications of semi-Markov processes, especially, when reliability problemsare considered.

The derivative of the semi-Markov transition probabilityQij(t) can be obtained by

_Qij �t� �Y�k±j��1 2 Fk�t��fj�t� �8�

i.e. statej is assumed, ifTj elapses and none of the otherrandom durations has elapsed before. For the failure rate, itholds

lj�t� �fj�t�

1 2 Fj�t� �9�

hence

_Qij �t� �Y�k��1 2 Fk�t��lj�t� �10�

The distribution function of the sojourn timeQi(t) can be

G. Becker et al. / Reliability Engineering and System Safety 70 (2000) 41–4842

i

1

2

k

λi1

λi2

λik

Fig. 1. A general subgraph withk competing transitions.

obtained by the argument, that sojourn time will be less thant, if any of the randomT is less thant. This leads to

Qi�t� �Y�k��1 2 Fk�t�� 11�

Substituting the last two equations into Eq. (9) yields

lij �t� � lj�t� �12�It can be concluded that with the definition (3) of semi-Markov transition rates, these latter will correspond to theunderlying failure rates in the same way, as this is true forthe transition rate of ordinary Markov processes, whichshould be a great help, when constructing semi-Markovmodels for reliability problems. Of course, this only holds,if all durations start at the time, when statei is entered(otherwise, the process would not be a semi-Markov processany more).

These relations have been proved by Becker et al. [2]. It isinteresting that Mode [3] reaches similar results for acompletely different area of application (Biostatistics)using a competing risk approach.

3. Finding state probabilities given transition rates

In the following sections, first the homogeneous case isconsidered, which subsequently is extended to the inhomo-geneous case. Finally, a formulation as an initial valueproblem is treated, which has the potential to reduce thenumber of integral equations to be solved simultaneouslyfrom N2 to N.

3.1. The homogeneous case

State probabilities of a semi-Markov process are given by[1]

pij �t� � pr{Z�t� � juZ�0� � i} �13�which is the probability, that the process is in statej at timetgiven that it has been started in statei. These are defined bythe system of integral equations

pij �t� � dij �1 2 Qi�t��1X�k�

Zt

0

_Qik�t�pkj�t 2 t� dt �14�

whered ij is the Kronnecker delta. Substituting Eqs. (4)–(7),one obtains

pij �t� � dij exp 2Zt

0li�x� dx

� �

1X�k�

Zt

0lik�t�exp 2

Zt

0li�x� dx

� �pkj�t 2 t� dt

�15�This form can be interpreted in the following way: the firstterm in the sum describes the event, that statei have notbeen left untilt; it will only contribute topii �t�: The second

part describes the event that statei is not left until a timet ,then a transition to some statek occurs, and then the processmoves from statek to statej in t 2 t time units.

Eq. (15) obviously becomes simpler, if transition rates areconstant. Even a mixture of Eqs. (14) and (15) can be used,if desired, in cases, where the transition probabilities have asimple form for some states, whereas in other states, thereare constant rates. A significant advantage of the form basedon transition rates is that it can be readily extended to rates,which depend both on process time and sojourn time.

3.2. Transition rates depending both on process time andsojourn time

A state graph model has been developed, which imple-ments transition rates, which depend on sojourn time.Another one is known, which depends on process time. Asboth types of transitions occur in real problems, they mayalso occur simultaneously.

Thus, it is interesting to develop a model, where a transi-tion rate may depend on sojourn time as well as on processtime. In this case, a distinction is necessary between at leasttwo ways, how the process time influences transition rates.One possibility is that the transition rate immediatelydepends on process time. An example would be scheduledtests or maintenance, which occur at fixed times, indepen-dent from the duration, the process has been in a given state.

Another possibility is that the transition rate depends onthe process time, when the present state has been reached.An example is a tolerable down time of a physical system,where this tolerable down time depends on process time. Inthis case, the tolerable down time will begin, when a certainfailed state is assumed [4,5].

In order to be flexible for both possibilities, two variableswill be used to represent the two types, the transition ratecan be affected, the variablet for an immediate dependencyon process time, and the variablettr, if the transition ratedepends on the process time of the last transition. For thedependency from the sojourn time, the variablex shall beused. With this, the following definition of a transition rateis obtained:

lij �x; t; ttr� dx� pr{Sn 2 Sn21 [ �x; x 1 dx�> Z�Sn� � juZ�t�

� i > Sn 2 Sn21 . x > Sn21 � ttr > process time� t}

�16�Note, that botht and ttr are process time, butttr is processtime of the last transitionSn21 such thatt � ttr 1 x.

In order to obtain state probabilities for this process,consider that this process is not homogenous with respectto process time any more. So,pij will not only depend on thestate, where the process starts, but also on the time, when itstarts. Hence

pij �t; s� � pr{Z�t� � juZ�s� � i > S0 � s} �17�With this definition, it is easy to find a descriptive model for

G. Becker et al. / Reliability Engineering and System Safety 70 (2000) 41–48 43

pij using Eq. (15). Care must be taken that the three vari-ables, which the transition rates depend on, assume thecorrect ranges during integration. One obtains

pij �t; s� � dij exp

2Zt

sli�x 2 s; x; s� dx

!

1X�k�

Zt

slik�t 2 s; t; s�exp

2Zt

sli�x 2 s; x; s� dx

!pkj�t; t� dt

�18�So, a semi-Markov model can be developed, which is inho-mogeneous with respect to process time. Clearly, theMarkovian property still is given at the points of transition.The solution of this system of integral equations, however,is more complex, than in the case of a homogenous semi-Markov process, because the integrals involved are no moreof convolution type, and also, because thepij to be deter-mined depend on two parameters.

Frequently, the task is given to determine state probabil-ities given an initial state, such that for a given starting states, all pis have to be determined, i.e.N items, if the processhasN states. It appears sensible, to assess, whether a formu-lation of this initial value problem can be given.

3.3. Formulation as an initial value problem

The initial value problem could be stated in the form:determinepj�t� givenpj�0�: For an ordinary Markov processwith discrete states and continuous time, this leads to asystem ofN ordinary differential equations, i.e. a systemof equations, which involve only known resp. given quan-tities, like the transition rates, andpj. Unfortunately, a simi-lar system of orderN cannot be given in the case of morecomplex processes; note that Eqs. (14) and (15) are of orderN2. This appears to be so, because knowledge ofpj is notsufficient to determine subsequent behavior of the process.There must be room in the model for the memory involved[6,7].

3.3.1. Exploiting the approach of DevooghtFrom Devooght and Smidts [6] stems a formulation of the

initial value problem, where the state probabilities maydepend on an arbitrary number of arguments apart fromt.They use (in a slightly different notation)

pj�x; t� �Z

upj0�u�d�x 2 gj�t;u��exp

2Zt

0lj�gj�s; u�� ds

!du

1X�i�

Zt

0

Zud�x 2 gj�t 2 t;u��lij �u�

exp

2Zt 2 t

0lj�gj�s; u�� ds

!pi�u; t� dt du (19)

wherex is a vector of arbitrary variables, which the transi-tion rates depend upon (in the original work: physical vari-ables). It is assumed that the vectorx has the valuex�0� � gj�t; u�; if the process starts att � 0 in statej withan initial valuex�0� � u: (Note that the sequence of integra-tion in the second term of the sum has been modified a bit.This was necessary, because in the case treated here,u willdepend ont ; so the transition ratel ij also has to beintegrated with respect tot ; also, the sum over the statesi does not exclude statej, as in the context of transitionrates depending on sojourn times, an immediate transitionfrom a state to itself is realistic, as it will reset sojourntime). Transition ratesl ij are given in dependence onx:The inhomogeneous semi-Markov process treated in thepreceding sections has transition rates depending on threevariabless, t, andttr. So, if one wants to exploit Eq. (19),one has to definex� �xs; xt; xtr�; and to find suitablefunctions gj, such that these variables bear the intendedmeaning (sojourn time forxs, process time forxt, andtime of last transition forxtr). This can be obtained bychoosing

gj�t; �us; ut;utr�� t

t 1 ut

utr

0BB@1CCA �20�

Eq. (20) has the important implication, that the work ofDevooght and Smidts also applies for systems, wherethe failure behavior of the components is described bya semi-Markov process [5], if—in addition to thephysical variables—sojourn times are introduced andthe corresponding component ofgj is set as givenabove. Subsequently, however, it is assumed, that—apart from the sojourn times—no other variables areconsidered.

Eq. (20) means, sojourn time will always bet independentfrom a possible start value, process time will increasestarting from the start valueut, whereas the time of thelast transition will not be changed, when sojourn timechanges. Of course, a sojourn time and a time of lasttransition is required for each state, and correspondingcomponents ofgj will be zero for the sojourn times ofother states and have been omitted in Eq. (20). Now,taking into account the renewal property of a semi-Markov process, it can be concluded, thatpj will notdepend on the other sojourn times. Eachpj will dependon its specific sojourn time and time of the last transi-tion, and on process time, which is common to all. Alsonote,xt is equivalent to the process timet and if processtime and sojourn time are known, the time of last tran-sition is given as the difference of the two, such that infact, only dependency from sojourn time in addition toprocess time is required.

Considering this and substituting Eq. (20) into Eq. (19)


allows solving the integral overu yielding

pj�xs; t� � pj0�xs�exp

2Zt

s�0lj�s; s; 0� ds

!

1X�i�

Zt

t�0exp

2Zt 2 t

s�0lj�s; s1 t; t� ds

!

�Zt

s�0pi�s; t�lij �s; t; t 2 s� dsdt (21)

with pj0�xs� � pj0d�xs�: Integrating this equation overxs willyield the state probabilities desired.

3.3.2. An approach involving transition frequency densitiesConsidering frequency densities of reaching states, it is

attempted to obtain a representation of semi-Markovprocesses, which involves onlyN (N: number of states)functions of one variable, which should simplify numericalintegration methods.

Let Hrj �t� be the expected value of the number of timesthat statej of the given process is reached in the interval(0,t). Then, ifHrj �t� is differentiable, dHrj �t� � hrj �t� dt willdefine a corresponding density function.

As the process has the property that at most one transitioncan occur in some interval�t; t 1 dt�; the frequency densitymay be interpreted as the probability, that a transition occursinto statej in �t; t 1 dt�; i.e.:

hrj �t� dt � pr{statej reached in�t; t 1 dt�} �22�

Then, it holds

hrj �t� �X�i�

pi�0�exp

2Zt

0li�s; s;0� ds

!lij �t; t;0�

1X�i�

Zt

0hri �t�exp

2Zt 2 t

0li�s; t 1 s; t� ds

!

� lij �t 2 t; t; t� dt (23)

This means, statej is reached in a given time interval, if:

• the process has been in statei initially, which has notbeen left untilt, and then, a transition fromi to j occurs;or

• the process has reached statei at time t , has remainedthere fort 2 t time units, and subsequently transfers tostatej.

This is a system of integral equations, which involvesNfunctions of one variable. If this system is solved simulta-

neously, the state probabilitiespj�t� can be found as

pj�t� � pj�0�exp 2Zt

0lj�s; s; 0� ds

� �

1Zt

0hrj �t�exp 2

Zt 2 t

0lj�s; t 1 s; t� ds

� �dt �24�

This means, the process is in statej at some timet, if:

• the process has been in state i initially, which has notbeen left untilt; or

• the process has reached statej at time, and has remainedthere for at least the subsequentt-units of time.

This is a system of non-coupled integrals, i.e. they can besolved independently, and all terms on the right hand sideare known. This overhead must be seen in relation to thefact, that the basic system involves onlyN equations in onetime variable.

3.4. Summary of efforts

To obtain state probabilities of a inhomogeneous SMPwith given transition rates and given initial conditions, thefollowing effort is involved:

Method E1 E2 E3

Classical method of first step N2

Specialization of Devooght’s equation N NVia frequency densities N N

where with given number of statesN, E1 is the number ofcoupled integral equations with two variables,E2 thenumber of coupled integral equations with one variableandE3 the number of straight forward integrations.

4. Application

4.1. Continues tolerable down time and system failure rates

In the present application, system failure is caused onlywhen, in some states, the sojourn time exceeds tolerabledown time. The semi-Markov models which have beendeveloped in this work are convenient to face the problemwith a stochastic tolerable down time which is a continuousrandom variable,Tj, with the given distributionFj.

Suppose, the system enters statei where, except compo-nent life or repair times, a tolerable down timeTj begins.The process will transfer to statej (system failure), if therealization ofTj is the smallest of all other time variables,and the sojourn time in statei will just be this smallestrealization. Similarly, based on the argument of Section2.2, system failure rates will correspond to semi-Markovtransition rates.


Inferentially, semi-Markov transition rates correspond tolife failure rates, repair rates and system failure rates. This isvery helpful for implementing the frequency densitymethod, as described in Section 3.3.2, for the followingreliability example.

4.2. Description of an example

The example has been used to describe the behavior of adistrict heating net with two sources over the period of half ayear. It has been taken from Becker et al. [4], where anordinary non-homogenous Markov model has been used.A district heat network has a tolerable down time, becauseif repair occurs fast, the users will not even notice a servicedisruption, but if repair lasts a long time, houses will cooldown, and the users will feel uncomfortable (at least).

The tolerable down time is a random variable and willdepend on the environmental temperature; this has beenimplemented as a time dependent tolerable downtime byusing average environment temperatures for differentphases of heating period.

In the state graph a heating period consisting of fivemission phases given in Table 1 with 14 states as definedin Table 2 has been modeled. Clearly, this is a phased

mission problem because in the beginning of the heatingperiod, first one (states 1, 2, 3) then the other source (states4, 5, 6) will be out of the service for scheduled maintenance.Afterwards both sources are available (unless failed), butonly one is required (states 7, 8, 9, 10). Then the environ-mental temperature decreases to a point that both sourcesare required (states 11, 12, 13, 14). Finally climate becomesmilder, and the sources act as redundancies again (states 7,8, 9, 10). Failure and repair rates are assumed to be constant;the according transitions are represented by solid lines, asshown in Fig. 2. Transitions between the mission phases areassumed to occur at fixed points of time. These are indicatedby thin lines. States 2, 5, 9, 12 represent system failure,where the tolerable down time has not yet been exceeded.The tolerable down transitions are indicated by wavy lines.Note that for internal reasons it has been assumed, that inmission phase 4, where both sources are required (states 11,12, 13, 14), failure of the second source will lead to systemfailure immediately.

4.3. Data for the example and results

Evaluation of the example has been performed using thefollowing data:

• failure rate of a unit: 1024/h;• repair rate of a unit: 0.05/h;• duration of mission phases according Table 1.

The tolerable downtimeT is a random variable with alognormal distribution:

fikt�t� � 1��2pztp exp 2

12

ln t 2 m�t�z�t�

� ��25�

where

m�t� � E{ln �T�t��} and z�t� � ��Var{ln�T�t��}p �26�

The mean value of tolerable down timeE{ T�t�} has been setto values between 20 and 2 h, decreasing continuously forthe first interval of 14 weeks. These correspond to environ-mental temperatures between110 and2158C and havebeen found by a thermodynamic simulation of the net.Also, 10% of the mean value is considered as a standarddeviation for the variableT. A service disruption of the nethas been considered tolerable if the temperatures in thebuildings attached to this net will not fall below 188C. Aninteresting measure for this system is unavailability,because during the time the system is unavailable, inhabi-tants will use auxiliary heating devices, e.g. electric heaterswith a constant consumption of electric power. Hence, thecost will be roughly proportional to the time the system isunavailable and tolerable down time is exceeded. Thesystem is unavailable, if the process is in states 3, 6, 10,13 or 14. The resulting unavailability is given in Fig. 3, andto a different scale, in Fig. 4.


Table 1Mission phase for district heating net example

Phase States Duration(weeks)

Description

1 1, 2, 3 2 Start of heating period;unit 2 under maintenance

2 4, 5, 6 2 As above, but unit1 under maintenance

3 7, 8, 9, 10 8 Both units in service;one required

4 11, 12, 13, 14 2 Both units in service;both required

5 7, 8, 9, 10 8 As phase 3

Table 2List of states for district heating net example

State Phases Description

1 1 Unit 1 operating2 1 Unit 1 failed, tolerable down time not exceeded3 1 as above, but tolerable down time exceeded4 2 Unit 2 operating5 2 Unit 2 failed, tolerable down time not exceeded6 2 as above, but tolerable down time exceeded7 3, 5 Both units operating8 3, 5 One unit operating, the other failed9 3, 5 Both units failed, tolerable down time not exceeded10 3, 5 as above, but tolerable down time exceeded11 4 Both units operating12 4 One unit operating, the other failed,

tolerable down time not exceeded13 4 as above, but tolerable down time exceeded14 4 as above, but both units failed

5. Conclusions

A description of the semi-Markov processes using transi-tion rates has been developed, where the transition ratesdepend on sojourn times in the states. It has the basic advan-tage that there is a correspondence between the transitionrates and the failure rates or the repair rates of the under-

lying components, much like it is well known for ordinaryMarkov processes with constant rates. This description canbe extended to allow for transition rates, which depend notonly on sojourn times, but also on process times, leading to asemi-Markov model, which is inhomogeneous with respectto process time.

This model has the potential to allow for scheduled tests


Fig. 2. A state graph for a tolerable downtimes problem involving phased mission.

Fig. 3. Unavailability plot for the example.

and maintenance, which has only been possible for ordinaryMarkov processes before. It is also useful for some phasedmission problems, as it was demonstrated by an example.

As it is based on frequency densities, it can be formulatedas an initial value problem ofN integral equation, if theprocess hasN states.

References

[1] Howard RA. Dynamic probabilistic systems, vols. 1/2. New York:Wiley, 1971.

[2] Becker G, Camarinopoulos L, Micheler M, Zioutas G. The role oftransition rates in semi-Markov processes. In: Janssen J, Limnios N,editors. Proceedings of the Second International Symposium on semi-

Markov Models: Theory and Applications, Universite´ de Compie`gne,9–11th December 1998.

[3] Mode CJ. An overview of methods for applying semi-Markovprocesses in biostatistics. In: Janssen J, Limnios N. editors. Proceed-ings of the Second International Symposium semi-Markov Models:Theory and Applications, Universite´ de Compie`gne, 9–11th December1998.

[4] Becker G, Camarinopoulos L, Zioutas G. An inhomogeneous stategraph model and application for a phased mission and tolerable down-time problem. Rel Engng Syst Saf 1995;49:51–7.

[5] Vaurio JK. Reliability characteristics of components and systems withtolerable repair times. Rel Engng Syst Saf 1997;56:43–52.

[6] Devooght J, Smidts C. Probabilistic reactor dynamics—I: the theory ofcontinuous event trees. Nucl Sci Engng 1992;111:229–40.

[7] Labeau PE. Probabilistic dynamics: estimation of generalized unrelia-bility through efficient Monte Carlo simulation. Ann Nucl Energy1996;23(17):1355–96.


Fig. 4. Unavailability plot using a different scale.

a semi-markovian model allowing for inhomogenities with respect to process time

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