187
1 INTRODUCTION
The transport is one of essential characters of human
activities, determining the stability of other systems,
that are crucial for social and economic organisms.
Ensuring of appropriate quality and continuity of
supply chains is of a key importance to maintain the
appropriate level of functioning of all activities carried
out within the modern economy [41, 22, 21].
One of the base facts confirming unique significance
of transport for surrounding systems, is its location
within the critical infrastructure, both on international
and national levels [10]. A reference can be made to the
European Council Directive 2008/114/EC on the
identification and designation of European critical
infrastructures and the assessment of the need to
improve their protection [11], or Polish Act of 26 April
2007 on Crisis Management [33].
The intensive development of modern technologies,
that can be observed last years, results in increase of
dependency of modern societies, on functioning of
systems like transport. One of the most meaningful
evidences regarding the dependency, the reference can
be made to, is recent Covid-19 pandemic, during
which, in its initial phase, the disruption of supply
chains caused discontinuance in a number of sectors of
the world economy [13, 27, 28]. Reconstruction and
reorganization of the supply chains, that was made
afterwards, has demonstrated the key importance of
proper functioning of transport systems, to quality of
functioning of social and economic organisms,
including their restoration to the level ensuring their
sustainable performance [14, 29, 6].
The intensive development of various technologies,
the transport systems base on, results also in increase
of complexity of their structure and functionality.
Reliability and Availability Analysis of Transport
System Composed of Dependent Subsystems
P. Dziula
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The paper presents reliability and availability analysis of the transport system, taking into account
its structure, that is composed of dependent subsystems. The issues introduced are basing on the assumption that
one subsystem impacts on functioning of other subsystems, meaning - disruptions occurring within the
subsystem can reduce functionality and change level of safety and inoperability of others. By means of multistate
approach to analysis, it has been assumed that the deterioration of one subsystem affects the reliability of other
subsystems and the entire system. Following this assumption, the transport system reliability function and its
basic reliability characteristics were determined. In addition, the system availability function was set out,
assuming that its renewal is carried out when its reliability falls below a certain threshold. Furthermore, the
reliability and availability analysis of the transport system were conducted, taking into account additional stress
on its particular subsystem at certain time points. The summary contains conclusions resulting from the analysis
and comparison for various additional stress levels.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 1
March 2025
DOI: 10.12716/1001.19.01.22
188
This as well causes extension of influence of
dependencies linking particular subsystems, on
functioning, reliability and availability of the entire
system [35, 17, 8].
The dependencies can be of various nature [40], and
emerge on different levels. Their character can be either
holistic, appearing within the whole system, or more
particular [16], existing in certain sectors or system’s
components level [38, 37]. For the purpose of the
transport system reliability, availability and resilience
analysis, it is studied as system of systems [37, 12, 9].
Modeling of dependencies linking subsystems and
components building the entire system, is a key issue
in its management. Recognizing and distinguishing of
the subsystems associations is a key condition to
achieve proper functioning and administration of the
entire system [4, 34].
There is a number of approaches in the literature,
aiming to distinguish particular subsystems of the
transport system, and their interconnections and
interdependencies. One of the most general
perspectives is selecting: individuals and/or products
being transported, transport means in which they are
moved, and networks through which the transport
means are moving [23, 45, 51]. Extended approaches
specify: transport infrastructure, transport means,
human capital (service providers and recipients), and
rules applicable for the whole system functioning [19,
15, 32]. For this article purposes, more detailed studies
have been adopted, selecting following subsystems of
the transport system [44, 43, 48]: infrastructure
(structures through which transport operations are
conducted), transport means (carriers transferring
goods through the transport network), complementary
appliances (equipment operating in a complementary
way to infrastructure), power supply (generators of
power necessary for operations of transport),
provisioning resources (solutions for supplying of
provisions essential for functioning of subsystems of
the transport system), control (monitoring and
communication solutions ensuring operations of
transport systems, like safety, traffic control,
information provision etc.).
The objective of the paper is to signify, that
considering dependencies linking subsystems of the
transport system, can be of great meaning for analysis
of its reliability and availability, and consequently
safety of its functioning. Moreover, article
considerations include analysis of deteriorations of the
system performance, resulting from additional stress
that can appear in one of its subsystems. The additional
stress on one or more subsystems, coming from
internal or external sources, increases their
interdependencies, consequently can result with so
called “domino-effect”, and significantly reduce safety
and operational capabilities of the entire system.
2 RELIABILITY OF THE TRANSPORT SYSTEM
COMPOSED OF DEPENDENT SUBSYSTEMS
Relations interconnecting subsystems of the major
system, can be identified and described according to
different approaches. One of the propositions is to
identify nodes and links connecting particular
subsystems, then, by taking into account their
functional characteristics, the importance and rankings
of particular links and nodes can be determined. This
can allow to find the significance of particular
subsystems, and distinguish the most and the least
significant ones [26, 24, 25]. Another approach suggests
to represent the whole system as a multilayer structure,
built of layers representing particular interconnections
linking the subsystems. The layers represent horizontal
links within each subsystem showing flow connectivity
in that part of system. Layers are interconnected by
vertical links, denoting the interdependencies between
the subsystems. This leads to specify the relations
connecting particular subsystems by a combination of
horizontal and vertical links [50, 7, 42]. There are also
studies introducing interdependency matrices to
analyse relations between interconnected subsystems.
By generating various types of the matrices,
bidirectional interlinks, joining the subsystems, can be
specified, and then the impact of failure of one
subsystem, on another one can be determined [39, 49,
31].
For this article purposes, methodology assuming
the major system as the network of interconnected
nodes representing subsystems, has been adopted.
Interdependencies between particular subsystems are
related to probability of inoperability that one
contributes to other one. The approach lets to evaluate
engineering resilience and interdependency for
subsystems of a networked infrastructure [36, 30, 20].
The subsystems of the transport system,
distinguished earlier, have been denoted as below:
− Infrastructure – SUB1;
− Transport means – SUB2;
− Complementary appliances – SUB3;
− Power supply – SUB4;
− Provisioning resources – SUB5;
− Control – SUB6;
The above subsystems, and interdependencies
existing in the system, are presented in Figure 1.
Figure 1. Interdependencies within particular subsystems of
the transport system.
It has been assumed, that for proper functioning of
the entire system, all the subsystems should be in the
appropriate working state. Therefore, reliability and
availability analysis of the transport system, is basing
on the assumption, that subsystems SUBi, i = 1, 2,..., 6,
the system is built of, are connected in series.
189
The transport system and its subsystems are
analyzed as multistate structures, and their following
four reliability states have been specified:
− state 3 of full ability – meaning system is fully
operational, and working without any
perturbations;
− state 2 of partial reliability – standing for situation
when some disruptions in the system appear, but its
functionality is still maintained at the appropriate
level;
− state 1 of limited reliability – expressing the
situation when the disruptions in the system result
in falling of its exploitation parameters below
allowed limits
− state 0 of complete unreliability – occurring in case
of failure stopping system’s operation.
It has been also assumed, that reliability functions
of subsystems SUBi, i = 1, 2,..., 6, are exponential:
( ) ( ) ( ) ( )
, 1, ,1 , ,2 , ,3 , 0, 1,2, ,6,
i i i i
R t R t R t R t t i
= =
(1)
and their coordinates Ri(t,u), u = 1,2,3, are defined as the
probability of subsystem staying in subset {u, u + 1,...,3}
of reliability states at the moment t, under the
assumption that it was at full reliability state (state 3)
at the moment t = 0, and are given by:
( ) ( )
, exp , 1,2,3, 1,2, ,6,
ii
R t u u t u i
= − = =
, (2)
where λi(u), u = 1,2,3, denote the intensities of
departure of subsystems SUBi, i = 1, 2,…, 6, from the
subset {u, u + 1,...,3}.
Subsystems SUBi, i = 1, 2,..., 6, constituting the
transport system, are interdependent. Defects, failures
or disruptions in one of the subsystems, do have
influence on functioning of others, consequently, on
the reliability of entire system. Therefore, reliability
analysis of the transport system as a structure of series
and dependent subsystems, is carried out. It has been
assumed the relationships between the subsystems can
be unidirectional or bidirectional. Deterioration of the
reliability state of one of subsystems may cause
fluctuations of functioning of others, and result in
deterioration of their reliability characteristics. To
identify layout of interconnections of the subsystems,
it is recommended to determine their mutual impact in
case of their failure, and specify their behavior before,
during and after the appearance of the disruption in
one of them [5, 18, 46].
Therefore, by adopting the local load sharing
dependency model for a series structure [1 - 3], the
expanse of dependencies between subsystems is
expressed by the influence coefficients q(υ, SUBj, SUBi),
i,j = 1,2, … ,6, i ≠ j. The influences between the
subsystems do not have to be symmetrical. Hence, it
has been assumed that q(υ, SUBj, SUBi) reflects the
effect of changes in reliability state {u, u + 1,...,3}, u =
1,2,3, in SUBj, j = 1,2,…,6, on lifetimes in the subset {υ,
υ + 1,...,3}, υ = 1,2, of SUBi, i = 1,2,…,6, i ≠ j. Assuming
that subsystems SUBi, i = 1, 2,…, 6, have exponential
reliability functions, defined by formulas (1)-(2),
assuming as described above, the transport system
reliability function is defined as follows [2, 3, 47]:
( ) ( ) ( ) ( )
, 1, ,1 , ,2 , ,3 , 0,
dep dep dep dep
t t t t t
=
R R R R
(3)
where
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( )
6 6 6
66
1 1 1
11
6 6 6
1 1 1
21
1
,1 exp 2 exp
1 1, ,
21
1
exp 2 1 , 0,
1 1, ,
jj
i
dep i
i j i
ji
ii
ii
i
ii
i i i
ji
t t t
q SUB SUB
tt
q SUB SUB
R
= = =
==
= = =
−
= − + − −
−
−
− − +
−
(4)
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( )
6 6 6
66
1 1 1
11
6 6 6
1 1 1
32
2
,2 exp 3 exp
1 2, ,
32
2
exp 3 2 , 0,
1 2, ,
jj
i
dep i
i j i
ji
ii
ii
i
ii
i i i
ji
t t t
q SUB SUB
tt
q SUB SUB
R
= = =
==
= = =
−
= − + − −
−
−
− − +
−
(5)
( ) ( )
6
1
,3 exp 3 , 0,
dep i
i
t t tR
=
= −
(6)
For the article purposes, to demonstrate outcomes
of the reliability and availability analysis of the entire
transport system, reliability parameters shown in Table
1 have been adopted, and subsystems impact
coefficients given in Table 2, have been applied.
Table 1. Intensities λi(1), λi(2) and λi(3) of the SUBi, i =
1,2,…,6, subsystem departure from the safety states subsets
{1,2,3}, {2,3}, and {3}, respectively [year
-1
].
SUBi
λi(1)
λi(2)
λi(3)
SUB1
0.06667
0.1
0.28571
SUB2
0.1
0.13333
0.5
SUB3
0.13333
0.2
0.4
SUB4
0.06667
0.1
0.33333
SUB5
0.13333
0.4
1
SUB6
0.2
0.5
2
Table 2. Coefficients q(υ, SUBj, SUBi), i,j = 1,2,…,6, υ = 1,2, of
the SUBj subsystem impact on lifetimes and their mean
values in the subsets {1,2,3} and {2,3} of the SUBi subsystem.
j | i
SUB1
SUB2
SUB3
SUB4
SUB5
SUB6
SUB1
0
0.7
0.5
0.8
0.6
0.25
SUB2
0.85
0
0.25
0
0
0
SUB3
0.55
0.8
0
0
0.75
0.5
SUB4
0.65
0.25
0.85
0
0.7
0.9
SUB5
0.7
0.85
0.65
0.35
0
0.2
SUB6
0.75
0.7
0.6
0.65
0.45
0
Basing on the above given data, basic reliability and
availability characteristics of the transport system
composed of dependent subsystems have been
determined. Table 3 indicates the mean lifetimes in
reliability state subsets {1,2,3}, {2,3}, {3}, and standard
deviations of the lifetimes, obtained by use of formulae
(4)-(6).
Table 3. Mean values and standard deviations of the
transport system lifetimes in safety states subsets {1,2,3},
{2,3}, and {3} [year].
Reliability states
subset
Mean value
µdep
Standard deviation
σdep
{1,2,3}
0.880
0.747
{2,3}
0.395
0.349
{3}
0.221
0.221
Reliability function Rdep(t,2) of the transport system
sojourning in reliability states {2,3}, obtained for the
above data, have been shown in Figure 2.
190
Figure 2. The coordinate of reliability function Rdep(t,2),
taking into account dependencies between the subsystems,
related to coordinate of the function Rindep(t,2), assuming the
subsystems are independent.
Figure 2 is also indicating Rindep(t,2) function,
determined for the same data, but without considering
dependencies linking the subsystems.
Figure 3. Reliability function Rdep(t,1), vs. function Rindep(t,1).
Figure 3 illustrates the reliability function Rdep(t,1) of
the system, remaining in reliability states subset {1,2,3},
determined for the same data. Similarly, the reliability
function Rindep(t,1), has been demonstrated, related to
the same reliability states subset, calculated without
taking into account the dependencies between the
subsystems.
3 AVAILABILITY OF THE RENEWABLE
TRANSPORT SYSTEM
To analyze availability of the renewable system, the
key point is to determine the threshold of probability
Pren, of its staying in certain set of reliability states,
which, if reached, results in the renewal of the system.
For this paper purposes, the renewals of the transport
system are considered within the system sojourning in
reliability states subset {2,3}. The Pren probability
determines value of the coordinate Rdep(t,2), of the
reliability function. Therefore, the coordinate of the
system availability function AFdep(t,2), is specified as
the probability of renewable transport system staying
in subset of states {2,3}. That coordinate of availability
function, assuming regular renewals of the system
after exceeding the Pren threshold, is determined as
follows:
( )
( ) ( ) ( )
( )
( )
( )
( ) ( ) ( )
,2 ,2 , 0 2
,2
2 ,2 2 1 2 , 1,...,
ren
ren ren ren
dep dep ren
dep
dep
t if t t if t
t
t if t N
=
− + =
P
P P P
R R P
AF
R
(7)
where N is the number of the system renewals and
τPren(2) is the moment of the first renewal after
exceeding the Pren threshold. The coordinate Rdep(t,2) of
the system reliability function is given by (5).
The formula (7) allows to determine the availability
function AFdep(t,2) coordinate, of the system built of
dependent subsystems renewal, after exceeding certain
threshold. Figure 4 illustrates the availability function
coordinates, determined for the Pren thresholds: 60%
(Figure 4a), 40% (Figure 4b), and 20% (Figure 4c). The
coordinates have been obtained for the same initial
data (given in Tables 1 and 2), that were used to
demonstrate Rdep(t,2), and Rindep(t,2) functions
coordinates presented in Figure 3, and under the
assumption, that renewals of the system are processed
within its sojourning in reliability states subset {2,3}.
Figure 4. The coordinates of availability function AFdep(t,2), of
the transport system built of dependent subsystems,
obtained for: a) the Pren threshold 60% (τ0.6(2)), b) 40% (τ0.4(2)),
and c) 20% (τ0.2(2)), in relation to Rdep(t,2) reliability function,
determined for not renewable system.
191
The coordinates shown in Figure 4 allow to
determine, that the system renewals, for threshold Pren
= 60% take place every 0.24 years, for Pren = 40% every
0.38 years, and for Pren = 20% every 0.62 years. The data
indicated in Figure 4 let to conduct various types of
analysis. Basing on the system Pren threshold
determination, costs of system renewal related to the
Pren threshold, additional costs associated with
performance of not fully efficient system or its
malfunction, and costs of technical inspections,
optimal cost strategies of system maintenance and
repairs can be planned. This however is not covered by
the paper considerations, but will be subject of future
research works.
4 AVAILABILITY OF THE TRANSPORT SYSTEM
REFLECTING ADDITIONAL STRESS ON ITS
SUBSYSTEMS
The reliability of the transport system may be reduced
by unforeseen external elements, resulting with
additional negative impact on one or more of its
subsystems. To analyze the reliability and availability
of the system, affected by the external negative impact
on its subsystems, the impact is interpreted as
appearance of additional stress within them. Reduced
system’s reliability results in abridging of its sojourn
time in certain subset of reliability states. Moreover,
additional stress concerning one of the subsystems,
affects functionality and reliability of other dependent
subsystems.
To figure out the analysis of reliability and
availability it is assumed, that the additional stress
appears in one of the subsystems at certain moment TL.
The TL moment is the initial point of increase of the
entire system’s intensity of departure from intended
reliability states subset.
The coordinate of the transport system reliability
function, influenced by the additional stress, in case the
system is not renewed, is then specified as follows:
( )
( )
( )
_
_
,2 0
, ,2
,2
i
i
dep L
SUB L
L
dep
SUB L
L
dep
t if t T
tT
t if t T
=
R
R
R
(8)
where
( )
_
,2
i
SUB L
dep
tR
is the transport system, composed
of dependent subsystems SUBi, i = 1, 2,..., 6, reliability
function coordinate, reflecting its increased departure
intensity from the subset of states {2,3}, while TL is the
moment of appearance of the additional stress within
the system.
( )
_
,2
i
SUB L
dep
tR
is determined
correspondingly to the coordinate
( )
,2
dep
tR
specified
by (5), where the intensity
i(2) of sojourn of the i-th
subsystem SUBi, i = 1, 2,..., 6, in the subset of states {2,3},
is amended with the new value
i
L
(2), raised due to the
additional stress L.
To determine the availability function of the
transport system in the case the system is renewable,
similar assumption, that the additional stress in one of
the subsystems at certain moment TL, results in
increase of intensity of the entire system departure
from intended subset of reliability states, has been
adopted. However, when considering the renewals, it
must be emphasized, that after the system renewal, its
reliability parameters, i.e. intensity of departure from
intended subsets of reliability states are the same as at
the initial moment.
To conduct the analysis, it has been assumed that at
the initial moment t = 0, the transport system stays in
the reliability state 3. It has been also assumed, the
renewal of the system takes place when the probability
of its sojourning in the subset of reliability states {2,3},
assuming that subsystems are dependent, decreases
below certain threshold Pren.
Basing on the above assumptions, the coordinate of
the transport system availability function, influenced
by the additional stress at the moment TL
(1)
, is drawn
out as follows:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( ) ( )
_
(1)
1
1
_
1
11
_
, ,2
,2 0 min ; 2
,2 2
2 1 2
2 ,2
2 , 1,...,
2 1 2
2 ,2
i
ren
i
ren
ren ren
ren
ren
ren ren
i
ren
SUB L
L
dep
dep
L
SUB L
L
dep
L
dep
LL
SUB L
dep
tT
t if t T
t if T t
if t and t T or
t
t T and T N
if t an
t
=
+
−
=
+
−
P
P
PP
P
P
PP
P
AF
R
R
R
R
( )
( )
1
1
, 1,...,
ren
L
L
d t T and
TN
=
P
(9)
where N is the number of the transport system
renewals, and τPren(2) is the moment of the first
renewal, after the probability of the system sojourn in
subset of states {2,3} goes down below Pren.
The stress affecting certain subsystem of the
transport system can appear several times. It can be
assumed, that particular subsystem, within certain
time period, is affected by M successive stresses, at TL
(k)
moments, k = 1, ... , M, causing increased strain in the
entire system. The coordinate of the availability
function of the system, taking into account its renewals,
is then determined as below:
( )
( )
( )
( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( )
_
_
(1)
1
_
(1)
1
_
( 1)
,2
, ,2 0 1
, ,2 0 1 2 2 , 0,1,..., , 1
2 1 2 2 ,
, ,2
0,1,..., ,
i
i
i
ren ren
i
ren ren ren
SUB L
dep
SUB L
L
dep
SUB L
LL
dep
kk
SUB L
k
LL
L
dep
t
t T for t if M
t T if t and T for N M
if t and T T
tT
for N k
+
+
=
=
+ =
+
==
PP
P P P
AF
AF
AF
AF
( )
( )
( )
( )
_
( 1)
1,..., 2
22
, ,2
1,..., , 1, 1
i
ren ren
k
SUB L
k
L
L
dep
M
if t and T
tT
for N k M M
+
−
= = −
PP
AF
(10)
where AFdep
SUBi_L
(t,TL
(1)
,2) is given by (9) and
( )
( )
( )
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( )
_
()
_
, ,2
2 1 2
2 ,2
2 , 1,...,
2 1 2
2 ,2
2 , 1,...,
i
ren ren
ren
ren
ren ren
i
ren
ren
SUB L
k
L
dep
k
L
dep
kk
LL
k
L
SUB L
dep
k
L
tT
if t and t T or
t
t T and T N
if t and t T and
t
TN
=
+
−
=
+
−
=
PP
P
P
PP
P
P
AF
R
R
(11)
for k = 2, ... , M.
5 APPLICATION
The formulae and indications introduced in the article
allow to conduct various analyses concerning
reliability and availability of the transport system,
taking into account it is built of interdependent
subsystems. The analyses allow to consider influence
192
of additional stress appearing in one of subsystems, on
the entire system. Basing on the formulae (8) and (10),
and with use of data given in Chapter 2, the reliability
function
( )
_
, ,2
i
SUB L
L
dep
tTR
, and availability function
( )
_
, ,2
i
SUB L
L
dep
tTAF
), of the system, reflecting additional
stress in subsystem i, taking place at certain time TL,
can be determined. Figure 5 demonstrates reliability
(taking into account system is not renewed), and
availability (for renewable system) function
coordinates, calculated for the Pren = 60% threshold,
assuming additional stress L as triple increase of
intensities of i subsystem departure from the safety
states subsets {2,3} appearing in subsystem SUB2
(Figure 5a), and in subsystem SUB5 (Figure 5b), at time
TL = 0.5 years.
Figure 5. The coordinates of reliability function
Rdep
SUBi_L
(t,0.5,2), and availability function AFdep
SUBi_L
(t,0.5,2),
of the transport system, determined for Pren threshold 60%,
and assuming: a) the stress appears in SUB2 subsystem, and
b) the stress appears in SUB5 subsystem.
Moreover, it is possible to analyse impact of
different levels of stress appearing in one of the
subsystems, on the entire system reliability and
availability. The different levels of the stress are
represented by appropriate increase of intensities of
particular subsystem departure λi(2), from the subset
of reliability states {2,3}, given in Table 2.
Figure 6. The coordinates of reliability function
Rdep
SUB6_L
(t,0.25,2), and availability function
AFdep
SUB6_L
(t,0.25,2), of the transport system, determined for
Pren threshold 60%, obtained for the stress: a) 150% increase,
and b) 300% increase of intensities of the subsystem
departure from the safety states subset {2,3}.
Figure 6 shows reliability (taking into account
system is not renewed), and availability (for renewable
system) function coordinates, calculated for the Pren =
60% threshold, assuming additional stress appears in
subsystem SUB6, at time TL = 0.25 years, obtained under
the assumption the stress L is 150% (Figure 6a) increase
of intensities of the subsystem departure from the
safety states subset {2,3}, and 300% (Figure 6b) increase.
Furthermore, the considerations presented, allow to
take into account successive stresses emerging in one
of the subsystems at certain time moments TL
(k)
. The
impact of two successive stresses, on the reliability and
availability of the system, determined for the Pren = 60%
threshold, assuming the first additional stress appears
in subsystem SUB3, at time TL = 0.30 years, and the
second one appears in the same subsystem, at time TL
= 0.60 years, has been presented in Figure 7. The
reliability (taking into account system is not renewed),
and availability (for renewable system) function
coordinates have been determined under the
assumption the stress L is 300% increase of intensities
of the subsystem departure from the safety states
subset {2,3}.
193
Figure 7. The coordinates of reliability function
Rdep
SUB3_L
(t,TL,2), and availability function AFdep
SUB3_L
(t,TL,2), of
the transport system, determined for Pren threshold 60%,
obtained for stress L 300% increase of intensities of the
subsystem departure from the safety states subset {2,3},
appearing in subsystem SUB3 twice, at TL = 0.30 years, and at
TL = 0.60 years.
6 CONCLUSIONS
The paper presents analysis of reliability of the
transport system consisting of dependent subsystems.
The availability analysis of the system has been also
conducted, assuming renewal of the system takes place
at the moment τPren(2), reflecting the probability of its
sojourn in the subset of reliability states {2,3} goes
down below the certain threshold Pren. Furthermore,
reliability and availability analyses have taken into
account degradation of their parameters, caused by the
appearance of additional stress, in one of subsystems
forming the entire system, at the certain moment TL.
The stress can be result both of internal or external
disruptions. The stress can be as well of different level,
and can also be repeatable, taking place at successive
time moments TL
(k)
. The results introduced in Chapter
5, prove that worsening and degradation of reliability
state of one of subsystems, can significantly affect the
safety parameters of the entire transport system.
Negative effects are getting more significant meanings,
if the subsystems forming the complete system are
interdependent.
Results shown in Chapter 5 do also indicate
remarkable value of considering interactions and
interdependencies linking the subsystems, and taking
into account internal and external disturbances
degrading their reliability parameters, for reliability
and availability analyses of the entire system. The
outcomes of the paper allow to analyze and deduce,
which of the subsystems are of the key importance for
the whole system functioning. The diagrams shown in
Chapter 5 indicate clearly, that the same additional
stress level, taking place in different subsystems,
results with different impact on the entire system
reliability parameters.
Moreover, the paper content and results prove,
considering interdependencies connecting the
subsystems, and their impact on safety parameters of
the complete system, are of the key importance in case
of activities aiming to ensure respective levels of its
safety and efficiency. By analyzing of system reliability
and availability, considering various exploitation data,
optimal strategies of system renewals, maintenance
and repairs can be built.
The impact of additional stresses, on functioning
and reliability of the transport system, depends as well
on the time slots the stress takes place. By investigating
the influence of the moment the stress takes place, in
relation to interdependencies linking the subsystems
and the system renewals, on the safety parameters of
the entire system, the proper resistance of the system
can be projected, and appropriate safety management
activities can be conducted.
ACKNOWLEDGEMENTS
The paper presents results developed in the scope of the
research project “Modeling, safety analysis and optimization
of critical infrastructure systems’ operation”,
WN/2025/PZ/13, granted by GMU in 2025.
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