=Paper=
{{Paper
|id=Vol-1256/poster1
|storemode=property
|title=Bayesian Networks for the Evaluation of Complex Systems Availability
|pdfUrl=https://ceur-ws.org/Vol-1256/poster1.pdf
|volume=Vol-1256
|dblpUrl=https://dblp.org/rec/conf/vecos/MokhtarKC14
}}
==Bayesian Networks for the Evaluation of Complex Systems Availability ==
https://ceur-ws.org/Vol-1256/poster1.pdf
113
Bayesian networks for the evaluation of
complex systems' availability
Ait Mokhtar El Hassene, Laggoune Radouane Chateauneuf Alaa
Unité de recherche LaMOS Institut Pascal
Université de Bejaia 06000, Bejaia Algérie Université blaise Pascal, Clermont- Ferrand, France
aitmokhtar_elhassene@hotmail.com
Unlike the simple systems, very few methodologies treat the evaluation of the dependability of
complex systems, especially those configured as networks, where it is difficult to take into
consideration the different links and factors that can affect the availability and reliability of such
systems. In this context, Bayesian networks is a very interesting tool. In fact, they permit the
modelling of systems configured as network and the computation of marginal probabilities of the
nodes of the system using prior and conditional probabilities. In this paper, we propose an original
approach based on the factor of conditional availability for the evaluation of the availability of the
drinkable water distribution network of Bejaia city. And this by taking into consideration the different
links and interaction between the pumping stations of this network.
Availability evaluation, Bayesian networks, Complex systems, Availability reduction factor.
1. INTRODUCTION distribution. The output variable is the overall system
reliability; the obtained results can be updated after
In the most current papers, the evaluation of the availability of new data by adding binary nodes
dependability methods (evaluation of reliability and (yes/no) that describe the system state on a given
availability) are generally reserved for the simple time.
systems (series and parallel systems) or for the
components. But, for the most part of industrial Dynamic oriented object Bayesian networks
systems, their components are configured as (DOOBN) is another type of BNs which is also used
networks, where the interactions between the in dependability analysis of complex systems [4], the
components are defined by logical or physical links study proposed in [4] allows to simulate failures of
which complicate the evaluation of the dependability different components of a complex system in order
of these kinds of systems. In this framework, to evaluate its reliability. In [5], the authors have
Bayesian networks (BNs) are very useful since they used simulation technique to estimate the
permit a qualitative and quantitative representation availability of a complex system according to four
of the relations between the variables of the model. different scenarios. In the first scenario, the
The structure of the network reflects the conditional conditional probability tables (CPT) are known, in
dependencies between the variables, while the prior the second the CPT are unknown, the third case
and conditional probabilities are used to quantify shows the contribution of adding additional data,
them [1]. and in the last case, they estimate the reliability
using data collected and added over the time.
Many papers have proposed approaches to
evaluate the availability and reliability of complex BNs can be also used to evaluate the availability of
systems by using BN modelling. In [2], the authors systems. In [6], authors have applied hybrid
have combined the evidence theory with BNs in Bayesian networks (HBN) since the different causes
order to create an effective tool for the reliability that have influence on the availability assessment
analysis of systems under random uncertainties. are continuous variables (time to repair,
The system reliability is evaluated the basic of programmed preventive maintenance times and
“Dempster Shafer” theory. In [3], the studied system delays). BNs are also used for redundant systems
is constituted of two parallel sub-systems and each with improvements of the complex systems
sub-system is composed of two components in modelling by adding a “coverage factor” [7], this
series. The data used are the time to failure knowing factor represents the probability that a simple failure
that two components follow the Weibull distribution of a redundant component causes the overall
and the two others follow the exponential system failure. It can be modelled by FT [8], but
114
according to [7], it seems to be more useful and components or events on the system reliability,
more meaningful to use BNs. unlike other methods as fault tree (FT) or Petri
networks.
The first section of this paper is reserved to the
Bayesian networks and the modelling of complex 2.2 Bayes theorem
systems using Bayesian networks. In the second
one, we present the Bayesian inference which aims The Bayesian networks are developed thanks to the
to compute the marginal probabilities of the nodes, Bayes theorem. It is a basic result in probability
in this section we introduced a new notion based of theory, and comes from the works of Thomas Bayes
the “availability reduction factor” for the creation of (1702 - 1761).
the conditional probability tables. The third and last
section is an application. It aims on the evaluation of P B A P A
average availability of the water distribution system P A B (1)
of Bejaia city by applying the methodology P( B)
developed here.
Where P A is the prior probability, P B is the
2. BAYESIAN NETWORKS, DEFINITIONS AND observations (or evidence) and the posterior
PROPERTIES probability is given by P A B .
2.1 Definition 1 (Bayesian networks) 2.3 Complex systems modelling as Bayesian
network
A Bayesian network B , is defined by
In the literature, several papers discuss the methods
A directed acyclic graph X , E where of BN construction. When modeling of complex
X is a set of nodes (or vertices) and E is a systems in order to optimize the maintenance or to
evaluate their reliability and availability, two
set of directed links (or edges);
information sources are generally considered: the
A probability space (Ω, ) ; expert judgment and the statistical data on the
system [7]. It is also important to specify that the
A set of random variables X X 1 X n modeling of complex systems by BNs is a difficult
and very time consuming task. The steps of BN
associated with the graph’s nodes (Ω, )
construction are:
X X X Pa X
n
such as 1 n i i (i) Specify what we want to model: identify
i 1 the limits of the study by defining what to
where Pa X i is the set of the parent’s include and what is not.
(ii) Definition of variables: Select the
nodes of the node X i in . important variables of the system to take into
consideration in the BN. At this step, we
In other words, A Bayesian network is a graph where have also to specify the range of continuous
the nodes represent random variables (continuous variables and the states of discrete
or discrete) and the edges represent the influences variables.
between the variables of the graph. We associate (iii) Qualitative step: This step aims at
the random variable X to its modalities ( connecting the different nodes to each other,
X x1 ; X x2 ; X xn if X can takes n values). by directed edges, in order to express the
About the edges, they represent the causalities dependencies and independencies between
which can be deterministic or probabilistic. For an the nodes.
(iv) Quantitative step: It consists in creating the
edge, linking the fact A and the fact B , there is a
probability tables: the prior probability tables
relation which is the conditional probability noted
for the root nodes and the conditional
P B A , it represents a probabilistic relation of a probability tables for the other nodes. To do
node known its nodes parent. For the nodes without it, we can use the statistical data of the
parents, named “root” nodes, a prior probability will system or the estimations of the experts.
be assigned to them. Generally Bayesian networks Note that their value must be normalized;
(BNs) are mostly used as an efficient framework for they must be between 0 and 1 and their sum
decision-making with uncertain knowledge [7]. They must be equal to 1.
describe the system as a directed acyclic graph (v) Verification: It is generally made by
(DAG), and not as a tree, and represent a powerful performing sensitivity analysis and behavior
mathematical formalism to model the complex tests by simulating know scenarios.
stochastic processes. They allow for exact
calculation of the influences of dependent
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Note: to facilitate the determination of the different We can note that the modelling objective of the BN
linking and dependencies between the nodes of the and PN is the same but the way to deal with the
BN, we can use the FMECA analysis (Failure issue is very different.
Modes, Effects and Criticality Analysis) or Fault tree
analysis. In this paper, we have opted for the use of Bayesian
networks since they permit:
2.4 Bayesian network and dependability analysis
assessment of complex systems The use of imprecise of historical data.
The use of expert judgement to complete the
In literature, except the BNs, there are three lack of data.
traditional methods used in dependability of complex The use of multi-states variables which are
systems; the fault tree (FT), Markov chains (MC) and
useful to model the event with several
Petri networks (PN) [9]
effects.
2.4.1. Fault tree
3. BAYESIAN INFERENCE
This method gives important results of modelling
since it permits the integration of different kinds of Bayesian networks are essentially used to compute
knowledge (organisational, decisional, technical and the marginal and posterior probabilities of events
human aspect) and allows considering the connected between each other by relations of cause
dependencies between events. It also gives an and effect. And this, by using prior probability tables
exact computation thanks to its Boolean for root nodes and conditional probability tables for
representation of the elementary events. the other BN nodes, this use is called “inference”.
However, when the system is affected by multiple The model represented by a BN is not a statistical
failures with several consequences (which is closed model; in fact, we can integrate new
generally the case of the industrial complex information. By changing the likelihood of certain
systems) the model needs a representation with nodes, the posterior probability of the system will be
multi-state variables. In this case FT can’t be used. changed (data updating) [12]. This property
We can also add that the FT method permit the (updating) is very interesting of the diagnostic
analysis of one event. Contrariwise, BN allows the application, where its appreciation will change
use of multi-states variables and the analysis of according to one or many observations [13].
several events in the same model.
Many papers have proposed methods that permit There are two kind of Bayesian inference, exact and
the transformation of FT to BN [9]. approximate inference method. For the first kind, we
can find two classes: message passing method
2.4.2. Markov chains introduced by Pearl [14], which is used for networks
This method is adequate for the reliability and
configured as tree or poly-tree and the methods
availability analysis of systems; it allows exact using grouping nodes like the junction tree method
analysis of the failure probability even when the
of Jensen [15]. The main problem of the direct
system components are dependent between them.
inference methods is the computing time, since the
It also permits the representation of multi-state BNs are generally used for complex systems with e
variables, however, to reproduce the different
great number of variables, so the BN size of this kind
interdependencies and links between the system
of the complex systems is very large. And the
variables we need to use a very large number of execution time of the exact inference algorithms is
variables and the modelling becomes very difficult
very important according to the complexity of the
and leads to a combinatory explosion of the number
graph (the number of variables and their modalities)
of states [10], this gap is the main defect of this [16]. To deal with this problem, the approximate
method. According to [11], thanks to the use of
inference method is very interesting then the exact
conditional probability tables, BN permit to avoid this inference methods regarding the computation time.
combinatory explosion.
We have to note also that for some kind of BNs (BN
2.4.3. Petri Networks
It is a traditional method of the dependability that contain continuous and discrete nodes: hybrid
modelling; it is also used in the domain of dynamic BNs), we can just use the approximate inference
reliability and maintenance optimization policy. It is methods. These methods are generally based on
based on the simulation procedures like Monte Carlo stochastic methods type MCMC (Monte Carlo
analysis and other variants of this method which Markov Chain) [17].
leads to the following constraints [9]:
In this paper we have opted for the use of the
Inefficient consideration of low frequency Junction tree method (also called clustering or
events (accidents). clique-tree propagation algorithm) introduced by
They do not allow easily integrating Jensen in 1990 [15]. This method can be applied for
evidence. all the DAG structures.
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3.1 Junction tree algorithm 1. For each clique and separator 𝑋 fix ∅𝑋 (𝑋)
to 1;
This algorithm can be divided into two phases, the 2. For each variable 𝑉 of the BN; assign to 𝑉
junction tree construction phase and the message one clique that contain its family (𝑉 and its
propagation phase. parents), then multiply ∅𝑋 by 𝑃(𝑉|𝑃𝑎(𝑉));
This step must verify the following equation:
3.1.1. Construction of the junction tree
∏𝑁
𝑖=1 ∅𝑋𝑖
Moralisation: the first step of the transformation of = 𝑃(𝑈) (2)
the graph is the moralisation. It consists on ∏𝑁−1
𝑗=1 ∅𝑆𝑖
connecting two by two the parents of each node by
non-directed edges. After having moralised the ii. Global propagation
graph, we finished the transformation by deleting the
direction of each edge. In this step, we perform an ordered series of local
manipulations, called “message passes”. The
Triangulating the moral graph: a non-directed message passes rearrange the junction tree
graph is triangulated if every cycle of length four or potentials and they become locally consistent; thus,
greater contains an arc that connects two the result of the global propagation is a “consistent”
nonadjacent nodes in the cycle. It is made according junction tree. This step can be divided into two
to the following steps: phases: the collect and distribution phase. In the first
i. Associate to each node of the BN 𝑿𝒊 a phase, the messages are sent from the leaf cliques
“weight” equal to the product of the to the chosen clique. In the second phase, the
modalities of 𝑿𝒊 and its neighbours; messages are sent from the chosen clique to the leaf
ii. Select the node 𝑿𝒊 whose the weight is cliques.
minimal and which caused the least number
of edge to add (to form a clique 𝑪𝒊 of cycle 1. ∅∗𝑆𝑖 = ∑𝐶𝑖∖𝑆𝑖 ∅𝐶𝑖 , 𝑖 = 1, … , 𝑛
lower or equal to 3); ∅∗
2. ∅∗𝐶 = ∅𝐶 ∏𝑛𝑖=1 𝑆𝑖
iii. Remove the selected node and its adjacent ∅𝑆𝑖
edges and update the weights of the rest of 3. ∅𝑆𝑖 = ∅∗𝑆𝑖 , 𝑖 = 1, … , 𝑛
the nodes. 4. ∅𝐶 = ∅∗𝐶
Repeat this operation until there are no nodes. The
𝑪𝒊 are the cliques of the junction tree. The junction tree is consistent if the following
equation is verified:
Construction of an optimal tree:
i. For each pair of clique 𝑿 and 𝒀, create a ∑𝑖 ∅𝐶𝑖
𝑃(𝑈) = (3)
separator 𝑺𝑿𝒀 equal to 𝑿 ∩ 𝒀 (we will have ∏𝑗 ∅𝑆𝑗
𝒏 − 𝟏 separators, where 𝒏 is the number of
cliques).
ii. Select the separator 𝑺𝑿𝒀 with the greatest
weight and inset it between the cliques 𝑿
and 𝒀. Repeat the operation until all the iii. Marginalisation
separators will be inserted.
iii. The resulted graph is called “junction tree“ Since the junction tree is become consistent, we can
now compute the marginal probability 𝑃(𝑉) of each
Note: when two or many separators have the same node of the BN as the following:
weight, we choose the separator with the smallest
cost. 1. We define a clique or separator that contain
“The cost” of 𝑺𝑿𝒀 is the weight of 𝑿 plus the weight the node 𝑉;
of 𝒀. 2. We compute 𝑃(𝑉) by marginalising ∅𝑋 as
the following equation:
3.1.2. Inference on the junction tree 𝑃(𝑉) = ∑ ∅𝑋 (4)
𝑋∖{𝑉}
In this phase, potentials are attributed to the
components of the junction tree, then a series of
4. METHODOLOGY AND APPLICATION
calculation is performed in order to compute the
marginal probabilities of the BN nodes. The different
steps of this phase are developed below. In this part, we present a methodology that aims to
evaluate the availability of complex systems using
i. Initialisation Bayesian networks. This methodology is applied on
a real system (the water distribution system of Bejaia
In this step, we assign potentials for the junction tree city).
by using the probability tables of the BN.
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4.1 Construction of the Bayesian network state that permit to perform a required function under
a given conditions (the availability).
To model the studied system as a BN, three kinds of
data has been used; the arrangement of pumping 𝑨=
𝒂𝒗𝒂𝒊𝒍𝒂𝒃𝒍𝒆 𝒕𝒊𝒎𝒆
(5)
𝒂𝒗𝒂𝒊𝒍𝒂𝒃𝒍𝒆 𝒕𝒊𝒎𝒆+𝒖𝒏𝒂𝒗𝒂𝒊𝒍𝒂𝒃𝒍𝒆 𝒕𝒊𝒎𝒆
stations scheme of the system and the expert advice
for the creation of the BN structure. We have also For a BN, we have to establish a probability tables
used the statistical data and the expert judgment for for each node. The prior probability tables for the
the creation of the probability tables of the BN. On root nodes and the conditional probability tables for
the fig1 and fig2 are represented the water the other nodes. For the prior probability tables, we
distribution system and its corresponding BN. can use directly the relation (5). So, the probability
table of a given root node ‘’x’’ is as follow:
Table 1: Prior probability table of a root node ’’x”
x 0 1
1−𝐴 𝐴
Where 0 and 1 represent the component state: 1 for
the operation state and 0 for the failure state.
For the other BN nodes, the conditional probability
concept will be applied. So, the availability of a given
node has to be evaluated by knowing the state of its
Figure 1: Water distribution system scheme. parent nodes. For that, a new concept is introduced
herein: the factor of availability reduction due to the
failures and the factor of availability reduction due to
1 the PM actions.
Availability reduction factors
2
The complex systems are subjected to various kinds
of failure. By considering the causes of these
failures, it is usually found that most of them are
caused by the failure of another component or sub-
4 3 system. So, it becomes very important to take in
consideration this observation to compute the
availability. For this reason, we have introduced a
new concept; the availability reduction factor for the
6
5 creation of the conditional probability tables. This
factor can be defined as the proportion of availability
of a given node (component or sub-system) affected
by the failure of its parent nodes and not by its own
7 failure or the failure of one of its components. This
factor is computed from the historical data of the
maintenance actions and the PM plan.
Figure 2: The Bayesian network corresponding to
the figure 1. For a node “x” knowing that “y” is one of its parent
nodes, the availability reduction factor is given by:
One node in the BN can include several components
(water storage tank, pumps, pipes …). The node “1” 𝑻𝑰𝑫𝒙|𝒚
represents the source station, the node “2” is the 𝑷𝒙|𝒚 = 𝟏 − (6)
𝑻−𝑻𝑰𝑫𝒙
central pumping station, the nodes “3, 4, 5, 6”
represent the secondary pumping stations which are
linked to the node “7” which represent the client. Whit: 𝑇 is the inspection period, 𝑇𝐼𝐷𝑥 is the
unavailable time of the node ‘’x” caused by its failure
4.2 Probabilities assessment and 𝑇𝐼𝐷𝑥∖𝑦 is the unavailable time of the node ‘’x”
The Bayesian theory is essentially based on the caused by the failures of its parent node ‘’y’’.
mathematical concept of probability. In this paper,
we talk about the capability of a system to be in the
118
For the studied system, the availability factors of the
BN nodes are computed from the historical data of
the different pumping stations of the system. The
availability reduction factors, caused by failures, of
the BN nodes are in the table 2. node 5 node 6
node node
Table 2: Availability reduction factors caused by failures 0 1 0 1
3 4
0 0,083 0,917 0 0,1247 0,8753
Node Factor Value
2 𝑃2|1 0,9324 1 0,0444 0,9556 1 0,04 0,96
3 𝑃3|2 0,954
4 𝑃4|2 0,9532
5 𝑃5|3 0,9596 node 7
6 𝑃6|4 0,9118 node node node node
0 1
𝑃7|3 0,9608 3 4 5 6
𝑃7|4 0,9149 0 0 0 0 0,2587 0,7413
7
𝑃7|5 0,9593 0 0 0 1 0.1989 0,8011
𝑃7|6 0,9254 0 0 1 0 0,2272 0,7728
0 0 1 1 0,1649 0,8351
So, the conditional probability tables of a given node 0 1 0 0 0,1897 0,8103
are as the following: 0 1 0 1 0,1244 0,8756
0 1 1 0 0,1553 0,8447
Table 3: Conditional probability table of a node “x”
knowing the state of its parent node “y” 0 1 1 1 0,0872 0,9128
1 0 0 0 0,2284 0,7716
node x
1 0 0 1 0,1662 0,8338
y 0 1
1 0 1 0 0,1957 0,8043
0 𝑃𝑥|𝑦 (1 − 𝐴) 𝑃𝑥|𝑦 𝐴 1 0 1 1 0,1308 0,8692
1 1−𝐴 𝐴 1 1 0 0 0,1567 0,8433
1 1 0 1 0,0887 0,9113
1 1 1 0 0,1209 0,8791
1 1 1 1 0,05 0,95
4.3 Conditional probability tables of the studied
system 4.4 Application results
The conditional probability tables of the nodes of the For the availability evaluation of the studied system, we
studied system are as the following: have opted for the Bayesian inference by using the
junction tree algorithm of Jensen. We have programmed
Table 4: Conditional probability tables of the studied this algorithm on the mathematical computing software
system MATLAB by using BNT tools. From this algorithm, we
have computed the marginal probability of the node “7”
node 1 node 2 which represents the client node. This probability
node represents the availability of the client node. It represent
0 1
1
0 1 also the availability of the studied system, it is computed
by taking into account the different interactions and links
0,0116 0.9884 0 0,1216 0,8784 between the nodes (pumping stations) of the water
1 0,0579 0,9421 distribution system of Bejaia city. This availability is equal
to 0.9352.
5. CONCLUSION
In this paper we have proposed an original methodology,
node 3 node 4 based on the availability factor caused by the
node node maintenance actions, for the evaluation of the availability
2
0 1
2
0 1 of the complex systems by using the Bayesian networks.
Thanks to the Bayesian networks, we are able to model
0 0,118 0,882 0 0,1246 0,8754 the real complex systems by including the different links
1 0,0754 0,9246 1 0,0817 0,9183 and causalities that can exist between the system
components. The Bayesian inference permit the
119
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