4.1. Introduction

To take into account the uncertainty with the temporal aspect, system behavior is modeled by a random variable, which takes its values in finite states corresponding to the system states. The state space method is well known in dependability literature [COR 75, VIL 88, ANS 94, COC 97, AVE 99, GER 00] and also in industrial standards [IEC 06]. The models obtained estimate the failure probabilities of systems throughout their lifetime.

This method gives a graphical representation [VIL 88, p. 303; COC 97, p. 282] whose complexity depends on the hypotheses made that correspond to the real stochastic process. However, the complexity increases tremendously when the number of components increases. Indeed, the state space describing the system is built from the Cartesian product of the component states.

To reduce this problem, state aggregation techniques are proposed [COC 97, p. 282]. The Markov chains (MCs) obtained are usually compact. For instance, in [WEB 06, WEB 08, POU 09, WEB 12a] it was applied to wind turbine modeling and gives models with fewer states than the initial model. A new modeling approach has to satisfy such a compactness target.

Dynamic fault trees (DFTs) [DUG 92, MES 02] are one solution allowing us to describe a MC of great dimension. The DFT modeling is based on a graphical description language of the component combination (for instance, a passive redundancy structure, i.e. “spare gate”). The MC is generated by compiling the DFT model.

It is now necessary to model increasingly complex relations to estimate the dynamic behavior of systems. The models aim to estimate the state probability distributions of the system, taking into account the component age, the maintenance and control actions, and the context evolution. A holistic modeling of systems and their interaction with the context leads us to take into account a huge number of modeling states. This number of states makes the MC modeling very hard to realize because of the combinatory explosion of states. The model, even if it can be made by automaton combination, is very hard to read, understand, interpret and maintain [DE 92]. DFTs also have their limitations because they are based on a graphical language with the binary state hypothesis, and thus cannot conveniently model multi-state systems.

In this situation, BN are of interest because they extend the usual modeling capacity of graphical modeling to handle complex multi-state systems. Dynamic Bayesian networks (DBN) are known to be able to formalize stochastic processes in a compact form [MUR 02]. Thus, they solve the main limits of usual modeling methods. The first research works on the application of DBN to the reliability and system availability are given in [WEL 00]. In 2002, [WEB 02, WEB 03] applied DBN to the availability analysis of a system and showed the ability of a DBN to model a multi-state MC in a compact way. With this compactness property, a better model can be computed by reducing the modeling difficulty, to address more complex systems.

The goal of this section is to show the application of DBN in the dependability area and to explain the extension they allow according to the MC modeling and also their limits.

4.2. Component modeled by a DBN

A DBN can take into account the temporal dimension of the system states’ evolution along their lifetime by factorizing and discretizing the state space of each independent random variable at each time instant. A stochastic process is represented at time k by a variable with a finite number of states . The state of variables with the same value of k constitutes the time slice k [HUN 99, BOU 99].

A DBN can model the evolution of discrete random variables by defining the conditional dependence of a time slice k + 1, given the states of the random variables at the previous time slice k. The definition of the dependence linking the variables at different time slices can model various complex stochastic processes. This time-based stochastic process is modeled by a CPT. Figure 4.1 shows a particular case where a variable, , is defined conditionally to itself at the previous time slide . This is the Markovian case.

From an observed situation at any time k or from the initial conditions with k = 0, the inference mechanism in the DBN allows us to compute the state probability distribution of all variables for each time slice. To compute this, it is necessary to memorize the state probability distribution of all the variables in all time slices. The solution consists of developing the time slices for the entire desired time horizon, i.e. to duplicate all the variables for each time period. However, the BN size increases proportionally to the computing horizon [KJA 95]. This solution is not convenient for system dependability analysis because the process should be studied for a large time horizon. It conducts to a combinatory explosion of variables that cannot be handled by current inference mechanisms.

4.2.1. DBN model of a MC

In the case of Markovian processes, the Markov property is used to simplify the inference mechanism. For instance, in a Markovian process, the CPT is time invariant. The inference can be realized iteratively without explicitly defining a variable for each time slice. The DBN model is then compact and only two successive time slices are modeled, as shown in Figure 4.2. A DBN with two time slices noted 2-TBN [BOY 98] allows us to define all the necessary parameters to model the MC. The first slice contains the variables at the current time step k, while the second allows us to compute the distribution of variables at the time step (k + 1) by inference. A variable is defined conditionally to its states in the current time step . The CPT, , is constant whatever the value of k (Table 4.1). This CPT is defined from the transition probability matrix between the states of the MC. With this model, the future states at (k + 1) are conditionally independent of the past given the present states at time (k). The CPT clearly shows a MC [KJA 95].

After the first inference, the distribution is injected as the a priori distribution for . The next inference allows us to compute the distribution for the next time step. An exact inference computes the probability distribution of the random variable for the time step k + 1, from the distribution at time step k. The probability distribution for the next time steps k + 2, k + 3, … are computed by successive inferences [WEL 00]. For a time horizon of size h, h inferences are necessary. This computing method is equivalent to the Chapman–Kolmogorov equation.

4.2.2. DBN model of non-homogeneous MC

The extension to non-homogeneous MC is possible by introducing time-indexed CPT. By working with the Bayesia company (http://www.bayesia.com/) this possibility has been introduced in the BayesiaLab software. The parameters defined in the CPT can be indexed to an exogenous variable k that represents time.

Here we illustrate the concept. Let us consider valves with three states: a normal functioning state and two failure states, i.e. a remained closed state {1} and a remained open state {2}. In the case of varying parameters, the principle is illustrated by combining two Weibull laws for the valve x₁. The failure rates are time varying and defined according to Weibull laws with the following parameters:

• – for the transition to the remained closed state {1}, the failure rate is defined as follows:

[4.1]

• – for the remained open state, the failure rate is defined as follows:

[4.2]

The DBN model of the valve x₁ is shown in Figure 4.3. The probability distribution on the valve states is computed over 1,000 hours with 1,000 iterations, i.e. with a time step of 1 hour, as shown in Figure 4.4.

4.2.3. Stochastic process with exogenous constraint

As shown in [WEB 04], a hidden Markov model (HMM) [RAB 89] can represent the degradation of components. The modeling of component degradation by HMM has also been used in [MOG 12, ROB 13, LE 14].

Time is usually considered as a conditional factor in component reliability, as shown in the previous section. It can be insufficient [SIN 95]. The conditions of use and the environmental context (like humidity, temperature, etc.) can alter the component reliability. All factors that alter component reliability are called co-variables or exogenous variables [BAG 01]. As described in [COX 55], the component reliability can be modeled precisely by taking into account the effects of exogenous variables.

In [WEB 04], several models of MC are defined according to the operational context of the component. A Markov switching model (MSM) is introduced to model the switching from one MC to another subsequent to the state variation of the exogenous variables. These models are also considered as conditional MC where the transitions are conditional to exogenous variables.

The MSM models are non-stationary because of the fast modifications of parameter values [BEN 99, p. 147]. A MSM represents the conditional distribution given the input state sequence , where represents the state of the exogenous variable. The simulation of the MSM is based on discontinuous changes of parameters associated with each modification of the exogenous variable state. It is very hard to obtain an analytical solution as with MC, and it is quite simple to use a simulation.

The modeling solution by a DBN is really simple [WEB 04]. One or several exogenous variables modeling the constraints or the operational conditions are added as new discrete variables in the time slice k. The CPT that defines the transition between two consecutive time steps, , is defined conditional on , as shown in Figure 4.5 for one variable.

Moreover, if the variable is not observed and only a sequence of the exogenous variable is observed, the stochastic process model of the component and its environment is formalized as a HMM conditioned by an input sequence u_i(k). This model is called an input–output hidden Markov model (IOHMM). The exogenous variable is an input that models the constraints or the variations of the component environment. The impact of the hidden process on the failure mode is defined by the output . The variables and induce the behavior of the hidden process that describes the component degradation but remains unobservable [BEN 06]. The formal model of the IOHMM is well suited to model complex stochastic processes such as [BEN 99, p. 145].

As proposed in [BEN 06], the DBN model of an IOHMM can easily be applied to the components’ reliability (Figure 4.6). In this model, has three states:

• – ‘0’, the component is not used;
• – ‘1’, the component is used in normal conditions;
• – ‘2’, the component is used outside its nominal conditions.

A simulation of the component state evolution and of the function realized by the component is given for a sequence of defined from k = 0 to k = 600. The reliability of the component is defined as (Figure 4.7).

4.3. Model of a dynamic multi-state system

A DBN is particularly interesting when dealing with several components in a system. The DBN presented in section 4.2 allows us to represent several multi-state stochastic processes in a system model. A multi-state model, as presented in Chapter 3, can easily combine the models of dynamic multi-state components as presented in section 4.2 to give a whole model of the dynamic multi-state system. The computation in a DBN with several stochastic processes is solved by different inference mechanisms well suited to this problem and to the conditions of use of the models.

The exact inference algorithms are based on a junction tree [JEN 96]. This mechanism is applied to unroll up models. If all time slices are defined in the model, then the usual inference algorithm can be used to compute the exact results (Figure 4.8) but with high computation costs. The models can be of high complexity with much dependence between the components (Figure 4.9) and thus are not practical for such a modeling step. Moreover, they are not adapted for a large time horizon.

In the case of a 2TBN, the condition that warrants an exact computation with a junction tree algorithm is the independence between the root stochastic processes. This condition is verified when all the components are independent as in Figure 4.10. The marginal distribution is easily computed in a 2TBN by using the exact inference algorithm. The state probability distribution is computed for each multi-state component, as shown in Figure 4.11.

For component x₁, the process is non-homogeneous:

• – failure 1 (valve remained closed) has an occurrence probability given by equation [4.1];
• – failure 2 (valve remained open) has an occurrence probability given by equation [4.2].

Component x₂ is conditionally defined to three functioning modes by the variable :

• – , the component is not used: λ₂₁ = 0 and λ₂₂ = 0;
• – , the component is used in nominal conditions: λ₂₁ = 2.10⁻³ and λ₂₂ = 3.10⁻³;
• – , the component is used outside its nominal conditions: λ₂₁ = 4.10⁻³ and λ₂₂ = 6.10⁻³.

Component x₃ is modeled by a homogeneous MC with the following transition probabilities λ₃₁ = 3.10⁻³ and λ₃₂ = 4.10⁻³.

Then, the independent components are modeled in the DBN by a non-homogeneous MC, an IOHMM, and a homogeneous MC. The DBN models the multi-state system composed of the independent components in a compact form. The models of the processes are shown in Figure 4.10. The time slide k + 1 computes the state propagation from the components to the system y(k), to compute its reliability (Figure 4.12).

4.4. Discussion on dependent processes

The components of systems are not always independant. To decrease the model complexity in the case of dependent processes, it is possible to mix the dependent components in only one stochastic process that is combined with other independent processes by a multi-state BN, as shown previously. According to this method, the DBN models only independent processes. The whole structure of the global system is then simplified, but the number of states of some variables increases.

Nevertheless if some dependence exists between stochastic processes, as in the roll up of DBN shown in Figure 4.9, it is necessary to use a specific inference algorithm that computes the joint distribution at each time step with significant computing and memory costs. The approximate inference algorithm proposed by [BOY 98, KOL 99] or particular filtering [KOL 00] can estimate the marginal distribution with a bounded error, which is often sufficient for dependability purposes.

Unfortunately, another phenomenon introduces difficulties in computing the marginal distribution even in the case of the independent structure shown in Figure 4.10. In the analysis of the functioning scenarios, it is of interest to integrate observations or evidence like events in the DBN. If evidence about a component is introduced in the DBN for a state variable or an exogenous variable , the inference is correct until the processes are independent. However, if evidence is introduced on a variable, for instance , and this evidence introduces a dependence between the variables , then a computing problem appears. This dependence requires the use of specific algorithms. So, it is necessary to be cautious when using DBN and evidence to compute the distributions correctly by considering the right hypothesis.

4.5. Conclusion

In this chapter, DBN are introduced. The chapter shows that DBN can decrease the effects of combinatory explosion when assessing the reliability by the factorization of multi-state systems [WEB 03]. DBN are a representation formalism for MC [WEB 03], HMM and IOHMM. They are well suited for modeling and assessing component reliabilities [WEB 04, BEN 06]. DBN are also well suited to model time-varying non-homogeneous MC [WEB 04]. DBN are also able to model IOHMM by integrating exogenous variables for component degradation modeling [BEN 06].

In addition, DBN are able to model the wearing out or aging of a component. It can also integrate the impact of maintenance actions or the evolution of the component operational conditions. In [WEB 06, MUL 04], these kinds of models were applied to the reliability analysis and prognostic of a system. The DBN model is based on a multi-state probabilistic model of the system and a dynamic reliability model of the components.

DBN are also applied to aiding diagnosis [WEB 08]. In this work, a DBN integrates the reliability models into the probabilistic decision model to diagnose faulty components and to solve possible ambiguity in their isolation. In [VER 08, VER 09], BN or DBN are used as the classifier of faults for the diagnostic.

The works done in [BEN 08] address the modeling of temporally sequenced events. The application concerns the monitoring of the railway line where structural characteristics are taken into account for the diagnosis of degradations (i.e. cracks, fragments, random wears etc.). The problem is solved by an IOHMM that shows the difference between the normal situation and degradations. It has been applied to classify singular points of rail defects [BOU 04, BEN 04, SAM 07, OUK 08].