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1. C A S E S T U D Y
An optimized technique for reliability analysis of safety‐
critical systems: A case study of nuclear power plant
Pramod Kumar1
| Lalit Kumar Singh2
| Chiranjeev Kumar1
1
Computer Science and Engineering, IIT
(ISM), India
2
Computer Science and Engineering, IIT
(BHU), India
Correspondence
Pramod Kumar, Computer Science and
Engineering, IIT (ISM), India.
Email: pramod.16dr000212@cse.ism.ac.in
Abstract
Stochastic models are extensively used in quantifying the reliability of safety
critical systems. These models use the state‐space model for reliability quanti-
fication. Markov chain is comprehensively used in describing a sequence of
possible events of any system in which the probability of each event depends
only on the state attained in the previous event. Markov chains are convenient
to model the software system of the SCS with the help of Petri Nets, a directed
bipartite graph widely used for the verification and validation of real‐time sys-
tems. However, the stochastic model suffers from the state‐space explosion
problem. In this paper, we proposed a technique for reliability analysis of safety
critical systems, excavating into the coherent optimization of Markov chain.
The approach has been validated on 17 safety critical systems of nuclear power
plants.
KEYWORDS
Markov chain, reliability, SCS, SDS2
1 | INTRODUCTION
Safety‐critical systems, on which we daily bet our lives,
have become increasingly more complex, networked,
and distributed. In combination with the growing profes-
sionalism of adversarial teams, this demands not only for
safe systems but systems that remain safe while under
attacks. The reliability analysis of a safety‐critical system
is a challenging task due to having negligible failure rate.
However, failure of such systems can lead to catastrophic
effects, including the loss of economy, death or injury to
humans, or harm to environment. Continuous research
and effective techniques are being proposed to address
the reliability of safety critical systems through diversified
modeling techniques, like Software Reliability Growth
Models, Reliability Block Diagram, etc. Software Reliabil-
ity Growth Model is a black box approach based on unre-
alistic assumptions while Reliability Block Diagram
computes the reliability only when reliabilities of the
components are known. The shortcoming of these
approaches is that they ignore the internal structure of
the application, and hence, the reliability and perfor-
mance behavior of the various parts of the application
are not individually and explicitly captured. The reliabil-
ity knowledge guides the developers to a more suitable
development strategy.
The reliability prediction in the development process
of an SCS has led to some approaches based on Markov
chain (MC). However, the number of states is much com-
plex for safety critical systems and hence their interac-
tions. Due to increase in the number of states, the
computational complexity of the reliability computing
algorithm is very high and sometimes even lead to com-
putation failure. The problem is well known with the
term, “State‐space Explosion Problem”. Despite many
worthy efforts from the researchers, as shown in the
related works of Section 2, this problem is still open and
challenging.
This paper contributes a novel approach to address
the problem of state explosion for reliability analysis of
Received: 23 March 2018 Accepted: 9 May 2018
DOI: 10.1002/qre.2340
Qual Reliab Engng Int. 2018;1–9. © 2018 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/qre 1

2. safety critical systems. The proposed technique is vali-
dated and demonstrated with a case study of nuclear
power plant system.
The organization of the paper is as follows. In the fol-
lowing section, existing approaches that can be improved
for reliability prediction using state space optimization is
briefly recalled. Section 3 discusses the case study: Shut-
down system (SDS)‐2, on which the state‐space explosion
optimization technique is shown. Section 4 describes our
generic framework for optimizing the MC for predicting
the reliability. In Section 5, the validation of our approach
is done using non optimized MC and also using operational
profile data of 720 days. Section 6 concludes this paper.
2 | RELATED WORKS
Reliability of safety critical systems has gained much
attention due to its critical function. Numerous reliability
assessment techniques using MC have been proposed
so far.
Lalit Kumar Singh et al1
proposed a technique to
quantify the reliability of SCS. The technique was demon-
strated and validated on NPP safety critical systems. This
paper gives a mathematical approach for calculating the
transition probability between the states of the MC. How-
ever, this paper remains silent on optimization of MC,
when the number of states is very large.
Lalit Kumar Singh and Hitesh Rajput,2
focuses on
reliability and safety analysis of a Shutdown system
(SDS)‐2 of a SCS using Petri Nets. The proposed tech-
nique utilizes the modeling power of Petri net,
converting it into MC for quantification. Authors have
efficiently devised a technique for state space reduc-
tion using linear programming. However, the compu-
tation of reliability metrics is based on system
throughput, which is computed through Petri net
model. Therefore, model to model (MC to MC) verifi-
cation and validation for state space reduction tech-
nique is not possible.
Lalit Kumar Singh et al3
proposed a technique for
early prediction of software reliability. The technique is
shown on a NPP system and validated across 38 opera-
tional datasets with average accuracy of 99.67%. How-
ever, the focus was given to compute the transition
probabilities among the states with high accuracy during
the design phase itself, rather than state space
optimization.
Vinay et al4
proposed a technique for safety analysis
of SCS by deriving Petri net model from UML techniques,
thereafter converting it into MC. The technique has been
shown on Emergency Core Cooling System of NPP. The
converted MC can be used for quantification of reliability.
However, nothing has been thought on minimizing states
of MC.
Peng Li et al5
modeled an algorithm, capable to
automatically construct state space models of large sys-
tems. The proposed state space modeling algorithm
showed better performance than the Modified Nodal
Analysis formulation for less complex systems. However,
it has been observed that for complex systems, the auto‐
construction of model have issues. Also, the work
remains silent on the minimization of the states. Fur-
ther, the models validation requires a strong mathemat-
ical background.
Vinay et al6
proposed a technique for transformation
of deterministic models into state space models for safety
analysis of safety critical systems with a case study of
NPP. The technique provided a strong validation, how-
ever no optimization technique for the constructed model
was discussed and hence will be very computationally
complex for large‐scale systems.
Lalit Singh et al7
proposed a technique to estimate
model parameters for quantification of software reliabil-
ity, in which MC and Petri nets have been used as a reli-
ability modeling technique. The technique works fine for
software system. However, it can be extended for system
reliability as well, where number of states would be very
large and hence would be difficult to solve.
Raj Kamal et al8
proposed a security analysis technique
for safety critical and control systems with a case study of
NPP. A very small module is picked up for case study in
which Petri net modeling technique is used, which can
be extended further for quantification of the reliability.
The model can be transformed into MC, which may con-
tain several states depending on the size of the system.
3 | A CASE STUDY: SDS‐2 AND ITS
MARKOV MODEL
The safety systems of NPP are operated and maintained
following strict rules and high reliability requirements,
established by Atomic Energy Regulatory Board of the
respective country. NPP have multiple safety systems that
ensure 3 basic functions:
1. Controlling the Reactors
2. Cooling the fuel
3. Containing radiation
These systems are maintained and inspected regularly
and upgraded when necessary to ensure plants meet or
exceed safety standards. When the reactor is operating,
the power level is controlled by adjustor rods and by vary-
ing the water level in vertical cylinder. Sensitive detectors
2 KUMAR ET AL.

3. constantly monitor the different aspects like temperature,
pressure, and reactor power level. When needed the
nuclear power plant reactors can safely and automatically
shut down within seconds. Nuclear reactors have 2 inde-
pendent, fast acting, and equally effective shutdown sys-
tems, SDS. The first shutdown system, SDS‐1, consists of
rods that drop automatically and stop the nuclear reac-
tion if something irregular is detected. The second shut-
down system, SDS‐2, injects a liquid or poison called
gadolinium nitrate inside the reactor to immediately stop
the nuclear reaction. Both systems work without power
or operators intervention. However, they can also be
manually activated. These systems are regularly and
safely tested. The NPP remains in shut down state until
any manual action by operators takes place. SDS‐2 has
been taken as case study for the demonstration of our
methodology.
3.1 | SDS‐2
Figure 1 shows a SDS2‐Liquid Poison Injection System.
High‐pressure helium contained in the tank pressurizes
the poison for rapid injection into the moderator. The
helium tank and helium header which services the poison
tanks have 4 Fast Acting Valves (FAVs) between them.
For ensuring FAV openings with high reliability and on
demand, it is air‐to‐close and spring to‐open. The poison
tanks are mounted on the outer wall of the reactor vault.
All the poison tanks are connected to nozzle in order to
inject the poison into the moderator. All poison tanks
are connected by stainless steel pipes to a horizontal in
core injection tube nozzle that spans the calandria and is
immerse in the moderator. As soon as the injection is ini-
tiated, the helium pressure transfers the poison to the
calandria and the ball, which is at the top of the poison
tank, falls to the tank bottom. In the bottom position,
the ball sits at the poison tank outlet and prevents the
release of high‐pressure helium to the calandria.
3.2 | Markov model
Figure 2 shows the equivalent MC for the SDS‐2 system.
The process of designing the equivalent MC for the
SDS‐2 is given in Singh and Rajput.2
The circles in the
MC represent the states, and the arrows represent the fir-
ing rate of transitions.
4 | THE PROPOSED METHOD FOR
OPTIMIZED TECHNIQUE WITH ITS
APPLICATION
We extend our work to optimize the existing
approaches of MC for reliability estimation. We demon-
strate the method for SDS‐2 of a NPP. Stochastic pro-
cess is chosen because of the abstractions like
internal architecture of the operating system, hardware,
dynamic operational profile, etc. on which the system
reliability depends. Our framework contains 4 phases
as described below.
FIGURE 1 Shutdown system 2‐liquid
poison injection system [Colour figure can
be viewed at wileyonlinelibrary.com]
KUMAR ET AL. 3

4. 4.1 | Phase 1: Optimized MC model
creation
The optimized MC model is created using “Merge” algo-
rithm. Merge indicates that some states of the MC have
to be grouped together based on the values of the transi-
tion probability.
Merge (Mi,j ∣ Mi, Mj) over G = (V, E) produces a new
graph G′
= (V′
,E′
). In this new graph, a new node Mi,j
is introduced to V′
= V − {Mi, Mj} ∪ {Mi,j}. The algorithm
optimizes the MC based on 3 cases, which are described
below:
Case 1. Let Mi and Mj be a pair of unary
states with ti and tj as their transition rate,
respectively. Then, merge Mi and Mj to
form Mi,j as the new state. The new
transition rate of the merged state will be
equivalent to the individual sum (ti + tj)
of the transition rates of the 2 unary states.
Case 1 of Figure 3 shows the pictorial
representation.
Case 2. Let Mi and Mj be a pair of 2 states
having same transition rate as ti, from differ-
ent parents as Ma and Mb, respectively. Then,
merge Mi and Mj to form Mi,j as the new
state, keeping the transition rate as
unchanged. Case 2 of Figure 3 shows the pic-
torial representation.
FIGURE 3 Different case for merge algorithm FIGURE 4 Optimized MC of Figure 2
FIGURE 2 Equivalent Markov chain for SDS2‐LPIS
4 KUMAR ET AL.

5. Case 3. Let Mi and Mj be a pair of 2 absorb-
ing states with Ma and Mb be their parents,
respectively. Mi has the transition rate as ti
from Ma and tj from Mb. Also, ti is a transition
rate to Mj from Mb. Then, merge Mi and Mj to
form state Mi,j and replace the transition rate
by (ti + tj). Case 3 of Figure 3 shows the pic-
torial representation.
The optimized MC drawn using the above method is
shown in Figure 4.
4.2 | Phase 2: Transition rate matrix
computation
The transition rate matrix T is calculated by solving (1). The
term tij represents the transition firing rate from state i to j.
The firing rates of transitions are calculated using TimeNET
tool,9
given by Table 1. The transition rate for a given state
should add to zero, yielding the diagonal elements to be
tii ¼ −∑
j≠i
tij (1)
The transition rate matrix T is given in Equation 2.
4.3 | Phase 3: Transition probability
computation
The transition probability pij of MC is computed using
the transition rate matrix T. The ratio of transition
rate tij (of going from state i to j) to that of the
sum of all transition rates except it transits to itself
gives the transition probability (pij) from 1 state to
other. However, if it transits to itself in a loop then
it will not be ergodic and for this case pij will become
zero, ie,
pij ¼
tij
∑
k≠i
tik
ifj ≠ i
0 otherwise
8
>
<
>
:
(3)
Using this, the transition probability matrix P is writ-
ten as:
P ¼ I−D−1
T T; where
DT ¼ diag T
f g is the diagonal matrix of T
The transition probability is calculated, and the tran-
sition matrix is given in Equation 4.
4.4 | Phase 4: Reliability estimation
The specified time for successfully poison injection is
1 second. In view of the significance of mission time,
the reliability analysis is done. The state M15,17 undergoes
transition tpd to reach state M0,1. M8,10 and M15,17 are
absorbing states, as in both these states either FAV is in
open state or Logic Circuit is in off state. Firing of ttsfav
T ¼
M0;1 M2 M3 M4 M5 M6;7 M8;10 M9 M11 M12 M13 M14 M15;17 M16
M0;1 −399 399 0 0 0 0 0 0 0 0 0 0 0 0
M2 0 −199:02 199 0:02 0 0 0 0 0 0 0 0 0 0
M3 0 0 −199 0 197 2 0 0 0 0 0 0 0 0
M4 0 0 0 −2 0 2 0 0 0 0 0 0 0 0
M5 0 0 0 0 −4 0 0 2 2 0 0 0 0 0
M6;7 0 0 0 0 0 −396:02 199:02 197 0 0 0 0 0 0
M8;10 0 0 0 0 0 0 0 0 0 0 0 0 0 0
M9 0 0 0 0 0 0 0 −2:02 0 0 2 0:02 0 0
M11 0 0 0 0 0 0 0 0 −4 2 2 0 0 0
M12 0 0 0 0 0 0 0 0 0 −2 0 0 2 0
M13 0 0 0 0 0 0 0 0 0 0 −2:02 0 2 0:02
M14 0 0 0 0 0 0 0 0 0 0 0 −2 0 2
M15;17 0 0 0 0 0 0 0 0 0 0 0 0 0 0
M16 0 0 0 0 0 0 0 0 0 0 0 0 2 −2
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(2)
TABLE 1 Firing rate of transitions
λtpd λtLCh λtLCoff λtvvo
200 197 2 2
λtfavc λtsfav λtvvc λtfavo
2 0.02 199 199
KUMAR ET AL. 5

6. leads the SDS2 to unreliable state. So, states M4, M8,10,
M14, and M16 are the failure states in the optimized MC,
other states are behavioral states. Let pi(t) be the probabil-
ity that a state is in state i at time t. As the execution time
approaches to infinity (t → ∞), the probability converges
and leads to stationary distribution.10
p
*
¼ p M0;1
; p M2
ð Þ; p M3
ð Þ; p M4
ð Þ; p M5
ð Þ;
p M6;7
; p M8;10
; p M9
ð Þ; p M11
ð Þ; p M12
ð Þ;
p M13
ð Þ; p M14
ð Þ; p M15;17
; p M16
ð Þ;
(5)
Also,
∑
i∈M
p i
ð Þ ¼ 1 (6)
p
*
¼ p
*
P (7)
Equations 5, 6, and 7 are simple linear equations which
can be solved to estimate the reliability of SDS2. Hence, the
estimated reliability of SDS2 can be written as,
Rest
SDS2 ¼ 1− ∑
i¼4; 8;;10
ð Þ;14;16
p Mi
ð Þ: (8)
From Equation 7, we get Equation 9.
Solving Equation 9, we get the following linear
equations:
M0,1 = 1M15,17 (i)
M2 = 1M0,1 (ii)
M3 = 0.999M2 (iii)
M4 = 0.0001M2 (iv)
M5 = 0.989M3 (v)
M6,7 = 0.101M3 + M4 (vi)
M8,10 = 0.503M6,7 (vii)
M9 = 0.5M5 + 0.5M6,7 (viii)
M11 = 0.5M5 (ix)
M12 = 0.5M11 (x)
M13 = 0.5M11 + 0.99M9 (xi)
M14 = 0.009M9 (xii)
M15,17 = M12 + 0.99M13 + M16 (xiii)
M16 = M14 + 0.009M13 (xiv)
Also, using Equation 6,
M0;1 þ M2 þ M3 þ M4 þ M5 þ M6;7 þ M8;10 þ M9 þ M11
þ M12 þ M13 þ M14 þ M15;17 þ M16 ¼ 1
(xv)
P ¼
M0;1 M2 M3 M4 M5 M6;7 M8;10 M9 M11 M12 M13 M14 M15;17 M16
M0;1 0 1 0 0 0 0 0 0 0 0 0 0 0 0
M2 0 0 0:999 0:0001 0 0 0 0 0 0 0 0 0 0
M3 0 0 0 0 0:989 0:101 0 0 0 0 0 0 0 0
M4 0 0 0 0 0 1 0 0 0 0 0 0 0 0
M5 0 0 0 0 0 0 0 0:5 0:5 0 0 0 0 0
M6;7 0 0 0 0 0 0 0:503 0:5 0 0 0 0 0 0
M8;10 0 0 0 0 0 0 0 0 0 0 0 0 0 0
M9 0 0 0 0 0 0 0 0 0 0 0:991 0:009 0 0
M11 0 0 0 0 0 0 0 0 0 0:5 0:5 0 0 0
M12 0 0 0 0 0 0 0 0 0 0 0 0 1 0
M13 0 0 0 0 0 0 0 0 0 0 0 0 0:990 0:009
M14 0 0 0 0 0 0 0 0 0 0 0 0 0 1
M15;17 1 0 0 0 0 0 0 0 0 0 0 0 0 0
M16 0 0 0 0 0 0 0 0 0 0 0 0 1 0
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
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6
6
6
6
6
6
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6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
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7
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7
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7
7
7
7
7
5
(4)
6 KUMAR ET AL.

7. Solving the above 15 equations (i to xv), we get:
M0;1 ¼ M2 ¼ M15;17 ¼ 0:1592157; M3 ¼ 0:1590977; M4
¼ 0:00001599997; M5 ¼ 0:1574909; M6;7
¼ 0:01600158; M8;10 ¼ 0:00841592; M9
¼ 0:08672578; M11 ¼ 0:078725; M12
¼ 0:0393625; M13 ¼ 0:1252296; M14
¼ 0:0008594438; M16 ¼ 0:002099342:
P, being a sparse matrix can be improved to take less
space and time complexity. So, using Equation 5, we get
p
*
¼ 0:159215; 0:1592157; 0:1590977; 0:00001599997;
½
0:1574909; 0:01600158; 0:00841592; 0:08672578;
0:078725; 0:0393625; 0:1252296; 0:0008594438;
0:1592157; 0:002099342Š:
Hence, the estimated reliability of SDS2 (Rest
SDS2), using
Equation 8 is given by:
Rest
SDS2 ¼ 1−p M4
ð Þ−p M8;10
−p M14
ð Þ−p M16
ð Þ
¼ 1−0:00001599997−0:00841592−0:0008594438
−0:002099342 ¼ 1−0:01101638 ¼ 0:9889836
(10)
Rewriting the reliability,
Rest
SDS2 ¼ 0:9889836
5 | RESULTS AND VALIDATION
We validate our approach using 2 results. In first, we use the
reliability value of SDS2, calculated using non‐optimized
MC. In second, we use the reliability calculated using oper-
ational profile data of 720 days.
5.1 | Validation using non‐optimized MC
In,2
the authors have calculated the reliability of SDS2
using non‐optimized MC (RNOMC
SDS2 ). So, we opted this data
for our validation.
RNOMC
SDS2 ¼ 0:9986
Rest
SDS2 ¼ 0:9889
Comparing the estimated reliability (Rest
SDS2Þ and calcu-
lated reliability using non‐optimized MC (RNOMC
SDS2 ) of the
SDS2, we get:
RDiff 1
Comm ¼ RNOMC
SDS2 −Rest
SDS2
RDiff 1
Comm ¼ 0:9986−0:9889 ¼ 0:0097
(11)
M0;1; M2; M3; M4; M5; M6;7; M8;10; M9; M11; M12; M13; M14; M15;17; M16
¼ M0;1; M2; M3; M4; M5; M6;7; M8;10; M9; M11; M12; M13; M14; M15;17; M16
M0;1 M2 M3 M4 M5 M6;7 M8;10 M9 M11 M12 M13 M14 M15;17 M16
M0;1 0 1 0 0 0 0 0 0 0 0 0 0 0 0
M2 0 0 0:999 0:001 0 0 0 0 0 0 0 0 0 0
M3 0 0 0 0 0 0:101 0 0 0 0 0 0 0 0
M4 0 0 0 0 989 1 0 0 0 0 0 0 0 0
M5 0 0 0 0 0 0 0 0:5 0:5 0 0 0 0 0
M6;7 0 0 0 0 0 0 0:503 0:5 0 0 0 0 0 0
M8;10 0 0 0 0 0 0 0 0 0 0 0 0 0 0
M9 0 0 0 0 0 0 0 0 0 0 0:991 0:009 0 0
M11 0 0 0 0 0 0 0 0 0 0:5 0:5 0 0 0
M12 0 0 0 0 0 0 0 0 0 0 0 0 1 0
M13 0 0 0 0 0 0 0 0 0 0 0 0 0:990 0:009
M14 0 0 0 0 0 0 0 0 0 0 0 0 0 1
M15;17 1 0 0 0 0 0 0 0 0 0 0 0 0 0
M16 0 0 0 0 0 0 0 0 0 0 0 0 1 0
2
6
6
6
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5
(9)
KUMAR ET AL. 7

8. Therefore, error percentage can be computed as:
error% ¼
RDiff1
Comm
RNOMC
SDS2
× 100 ¼
0:0097
0:9986
× 100 ¼ 0:971%
∴ Accuracy ¼ 100−error% ¼ 100−0:971
⇒Accuracy ¼ 99:03%
(12)
Equation 12 shows that our method predicts the reli-
ability of SDS2 with accuracy of 99.03%, which demon-
strates the validity of our proposed optimized software
reliability technique.
5.2 | Validation using profile data
In Singh and Rajput,2
the authors have also used opera-
tional profile data of 720 days for calculating the reliabil-
ity of SDS2 (ROP−Data
SDS2 ).
ROP−Data
SDS2 ¼ 0:99603
Rest
SDS2 ¼ 0:98898
Comparing the estimated reliability (Rest
SDS2Þ and calcu-
lated reliability (ROP−Data
SDS2 ) of the SDS2 using operational
profile data, we get:
RDiff 2
Comm ¼ ROP−Data
SDS2 −Rest
SDS2
RDiff 2
Comm ¼ 0:99603−0:98898 ¼ 0:00705:
(13)
Therefore, error percentage can be computed as:
error% ¼
RDiff 2
Comm
ROP−Data
SDS2
× 100 ¼
0:00705
0:99603
× 100 ¼ 0:708%
∴ Accuracy ¼ 100−error% ¼ 100−0:708
⇒ Accuracy ¼ 99:292%:
(14)
Equation 14 shows that our method predicts the reli-
ability of SDS2 with accuracy of 99.29%, which demon-
strates the validity of our proposed optimized software
reliability technique.
6 | CONCLUSION
In this paper, we explored the reliability analysis of SCS
using optimized MC. SDS2 of NPP was chosen as the case
study for analysis. From the literature survey, we found
that there exists a gap of MC optimization. MC has been
used by many researchers for the reliability analysis. But
to the best of our search, we could not find a single paper
which has used optimized MC for reliability analysis.
“Merge” algorithm has been used for merging the
states based on transition probability. Section 4 shows
the whole process of optimizing along with its
application. Results and its validation are done in
Section 5. The result has been validated with 2 different
reliability data. First is with the non‐optimized MC and
second is with the operational profile data. Our approach,
when compared with the non‐optimized MC approach,
gives an accuracy of 99.03%, and when compared with
the operational profile data, our approach gives an accu-
racy of 99.29% which is quite rewarding. The technique
has been applied successfully on safety critical systems
of NPP.
REFERENCES
1. Singh LK, Vinod G, Tripathi AK. Design verification of instru-
mentation and control systems of nuclear power plant. IEEE
Trans on Nucl Sci. 2014;61(2):921‐930.
2. Singh LK, Rajput H. Dependability analysis of safety critical
real‐time systems by using petri nets. IEEE Trans Contr Syst
Tech. 2017;99:1‐12.
3. Singh LK, Vinod G, Tripathi AK. Early prediction of software
reliability: a case study with a nuclear power plant system. IEEE
Comput. Jan. 2016;49(1):52‐58.
4. Kumar V, Singh LK, Tripathi AK, Singh P. Safety analysis of
safety‐critical systems using state‐space models. IEEE Comput
Soc. 2016;38‐47.
5. Li P, Yu H, Wang C, et al. State‐space model generation of
distribution networks for model order reduction application.
Proc 2013 IEEE Power Energ Soc General Meeting (PES 13).
2013.
6. Kumar V, Singh L, Tripathi AK. Transformation of determinis-
tic models into state space models for safety analysis of safety
critical systems: a case study of NPP. In: Annals of Nuclear
Energy, Elsevier. July 2017;105: 133‐143.
7. Singh L, Vinod G, Tripathi AK. An approach for parameter
estimation in Markov model of software reliability for early
prediction: A case study. In: IET Software June 2015;9(3):
65‐75.
8. Kamal R, Singh L, Pandey B. Security analysis of safety critical
and control systems: a case study of nuclear power plant
system. In: Nuclear Technology, American Nuclear Society,
Feb 2017;197(3): 296‐307.
9. Lijie C, Tao T, Xianqiong Z, Schnieder E. Verification of the
safety communication protocol in train control system using col-
ored petri net. Rel Eng Syst Safety. Apr. 2012;100:8‐18.
10. [Onine].Available: http://www.dis.uniroma1.it/~leon/didattica/
webir/pagerank.pdf
Pramod Kumar is pursuing his PhD in reliability
prediction of safety‐critical systems from the Depart-
ment of Computer Science and Engineering at the
8 KUMAR ET AL.

9. Indian Institute of Technology (Indian School of
Mines) Dhanbad, Jharkhand, India. His research
interests are reliability, safety, and mathematical
modeling. Kumar received his MTech in the year
2016 in Information Technology from Birla Institute
of Technology, Mesra, Ranchi, Jharkhand, India. Con-
tact him at pramod.16dr000212@cse.ism.ac.in.
Lalit Kumar Singh is a scientist, level E, at the
Nuclear Power Corporation of India. His research
interests are software reliability, dependability, mathe-
matical modeling, and fault tolerance. Singh received
his PhD from the Indian Institute of Technology
(Banaras Hindu University). Contact him at lalit.rs.
cse@iitbhu.ac.in.
Chiranjeev Kumar is a professor in the Department
of Computer Science and Engineering, Indian Insti-
tute of Technology (Indian School of Mines) Dhanbad,
Jharkhand, India. His research interests include Wire-
less Networks, Software Engineering and IoT. Kumar
received his PhD from University of Allahabad, India
in 2006. Contact him at kumar.c.cse@ismdhanbad.
ac.in.
How to cite this article: Kumar P, Singh LK,
Kumar C. An optimized technique for reliability
analysis of safety‐critical systems: A case study of
nuclear power plant. Qual Reliab Engng Int.
2018;1–9. https://doi.org/10.1002/qre.2340
KUMAR ET AL. 9