Measures of different reliability parameters for a complex redundant system under head of-line repair

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
318
Measures of Different Reliability Parameters for a Complex
Redundant System Under Head-of-Line Repair
Dr. Rekha Choudhary
Department of Mathematics, Govt. Engineering College, Bharatpur,
Shyyorana, NH-11, Bharatpur, Rajastahan, India-321303
Tel: +91-9414543795 E-mail: rekhaparth2003@yahoo.co.in
Dr. Viresh Sharma
Department of Mathematics, N.A.S. (PG) College, Meerut
Sector-C/113, Mangal Pandey Nagar, Merrut, Uttarpradesh, India-250004
Tel: +91-9412060658 E-mail: vireshsharma1@yahoo.com
Sonendra Kumar Gupta
Department of Mathematics, Oriental College of Technology, Bhopal
22-Minal Complex Phase-III, Govind Garden, Bhopal, India-462023
Tel: +91-9893455006 E-mail: sonendrag@gmail.com
Abstract
The authors have considered a complex system consisting of two subsystems designated as ‘A’ and ‘B’
connected in series. Subsystem ‘A’ consists of N non-identical units in series, while the subsystem ‘B’ consists of
three identical components in parallel redundancy.
Keywords: Availability/Reliability Analysis, Repairable Parallel System, Laplace transform, cost profit function,
Head-of-line Repair,
1. Introduction
In this paper the authors have considered a complex system consisting of two subsystems designated as ‘A’
and ‘B’ connected in series. Subsystem ‘A’ consists of N non-identical units in series, while the subsystem ‘B’
consists of three identical components in parallel redundancy. In this model it is considered that the system goes
to complete breakdown state if any unit of subsystem ‘A’ fails or more than 1 unit of subsystem ‘B’ is in the
failed condition. Also, the system works with reduced efficiency if one unit of subsystem ‘B’ failed. The system
as a whole can also fail from normal efficiency state if there is any failure due to environmental reasons.
Supplementary variable technique and Laplace transforms have been utilized to obtain various state
probabilities and the cost incurred for the system is obtained. Failure and repair times of the units follow
exponential and general time distributions respectively. Head-of-line policy is being adopted for the repair
purpose. Some particular cases have also been taken to highlight the practical importance of the model. This
research is a step towards explaining the reliability application on a repairable system with three types of failure
under ‘head-of-line’ repair policy and Gumbel-Hougaard family copula.
So in earlier research [19, 20, 21, 22], different techniques have been applied to evaluate the reliability
of distribution system, including distributed generation such as an analytical technique using the load duration
curve, distributed processing technique, Characteristic function based approach for computing the probability
distributers of reliability indices, probabilistic method for assessing the reliability and quantity of power supply
to a customer, composite load point model, practical reliability assessment algorithm, validation method and
impact of substation on distribution reliability respectively.
2. Assumption
(i). Initially, all units are good.
(ii). A failed unit is repaired at a single service channel.
(iii). The parallel subsystem is composed of three identical units, while series subsystem is composed of
N non-identical units.
(iv). Failures are statistically independent.
(v). Environmental failure rates are constant.
(vi). After repair, units work like new.
(vii). Repairs follow general time distribution.
(viii).First come first served (Head-of-line) repair policy is being adopted.

319
3. Notations
/ /i Ef f f′ :Constant failure rates of any unit of subsystem / th
B i unit of
subsystem A/environmental failure.
1 2 3 4( ) / (y) / ( ) / ( )r x r r z r α :Repair rates with general time distribution from states 4 0toS S
1 0toS S or 3 4toS S , 2 0toS S , 5 0toS S
3
( )NP t :Probability that at time ‘t’ the system is operating in the state of normal
efficiency.
2
(y, )NP t ∆ :The probability that at time ‘t’, the system is in degraded state due to
the failure of one unit of subsystem B. The elapsed repair time lies in
the interval (y, y )+ ∆
( , )F
NP z t ∆ : The probability that at time ‘t’, the system is in failed state due to the
failure of more than one unit of subsystem B, the elapsed repair time
lies in the interval ( , )z z + ∆
3
( , )iP x t ∆ :The probability that at time ‘t’, the system is in failed state due to the
failure of ith
unit of subsystem A. The elapsed repair time lies in the
interval ( , )x x + ∆
2
(y, )iP t ∆ :The probability that at time ‘t’ the repair time lies in the interval
(y, y )+ ∆
( , )EP tα ∆ :The probability that at time ‘t’, the system is in failed state, due to the
environmental failure, the elapsed repair time lies in the interval
( , )α α + ∆
Figure 1 represents the state transition diagram of the system.
4. Formulation of the Mathematical Model
Viewing the nature of the problem, we obtain the following set of difference-differential equations:
3 2
2 3
0 0
3 ( ) (y, ) (y) y ( , ) ( )F
e N i Nf f f P t P t r d P z t r z dz
t
∞ ∞
 ∂
′+ + + = + 
∂ 
∫ ∫
Figure 1: Transition State Diagram
Good State
Degraded
State
Failed State
( ),EP tα ( ),F
NP z t
( )2
y,iP t
( )3
,iP x t
( )3r z
5S
ef 0S ( )2 yr 1S
2 'f
2S
( )3
NP t ( )2
y,NP t
3 'f( )4r α
f ( )1r x f
4S 3S
( )2 yr

320
3
1 4
0 0
( , ) ( ) ( , ) ( )i EP x t r x dx P t r dα α α
∞ ∞
+ +∫ ∫ (1)
2
22 (y) (y, ) 0
y
Nf f r P t
t
 ∂ ∂
′+ + + + = 
∂ ∂ 
(2)
3 ( ) ( , ) 0F
Nr z P z t
t z
 ∂ ∂
+ + = 
∂ ∂ 
(3)
3
1( ) ( , ) 0ir z P x t
t x
 ∂ ∂
+ + = 
∂ ∂ 
(4)
2 2
2 (y) (y, ) (y, )
y
i Nr P t f P t
t
 ∂ ∂
+ + = 
∂ ∂ 
(5)
4 ( ) ( , ) 0Er P t
t
α α
α
 ∂ ∂
+ + = 
∂ ∂ 
(6)
4.1 Boundary Conditions
2 3
(0, ) 3 ( )N NP t f P t′= (7)
2
(0, ) 2 ( )F
N NP t f P t′= (8)
3 3 2
20
(0, ) ( ) (y, ) (y) yi N iP t f P t P t r d
∞
= + ∫ (9)
2
(0, ) 0iP t = (10)
3
(0, ) ( )i e NP t f P t= (11)
4.2 Initial Conditions
3
(0) 1NP = , Otherwise zero (12)
5. Solution of the Model
Taking Laplace transforms of equations (1) through (11) and using initial conditions one may obtain:
[ ]
3 2
2 30 0
3 ( ) 1 (y, ) (y) y ( , ) ( )
F
N N Nes f f f P s P s r d P z s r z dz
∞ ∞
′+ + + = + +∫ ∫
3
1 40 0
( , ) ( ) ( , ) ( )N EP x s r x dx P s r dα α α
∞ ∞
+ +∫ ∫ (13)
2
22 (y) (y, ) 0
y
Ns f r P s
 ∂
′+ + + = 
∂ 
(14)
3 ( ) ( , ) 0
F
Ns r z P z s
z
∂ 
+ + = ∂ 
(15)
3
1( ) ( , ) 0is r z P x s
x
∂ 
+ + = ∂ 
(16)
2 2
2 (y) (y, ) (y, )
y
i Ns r P s f P s
 ∂
+ + = 
∂ 
(17)
4 ( ) ( , ) 0Es r P sα α
α
∂ 
+ + = ∂ 
(18)
2 3
(0, ) 3 ( )N NP s f P s′= (19)
2
(0, ) 2 ( )
F
N NP s f P s′= (20)
3 3 2
20
(0, ) ( ) (y, ) (y) yi N iP s f P s P s r d
∞
= + ∫ (21)
2
(0, ) 0iP s = (22)
3
(0, ) ( )E NeP s f P s= (23)
After solving the above equations, we get finally

321
3 1
( )
( )
NP s
A s
= (24)
2
2 3
( ) ( 2 )
( )
N r
f
P s D s f f
A s
′
′= + + (25)
2 3
2
6
( ) ( 2 ) ( )
( )
F
N r r
f
P s D s f f D s
A s
′
′= + + (26)
2 2
2 3
( ) ( ) ( 2 )
(2 ) ( )
i r r
f f
P s D s D s f f
f f A s
′
 ′= − + + ′ +
(27)
{ }2 2
3 3
( ) 1 ( ) ( 2 ) ( )
( ) 2
i r r
f f
P s S s S s f f D s
A s f f
η
′ 
′= + + + ′ + 
(28)
4
( ) ( )
( )
e
E r
f
P s D s
A s
= (29)
Where, 2 32
( ) 3 3 ( 2 ) 6 ( 2 ) ( )r re rA s s f f f f S s f f f D s f f S s′ ′ ′ ′ ′= + + + − + + − + +
{ }2 2 4
3
1 ( ) ( 2 ) ( ) ( )
2
r r re
f
f S s S s f f S s f S s
f f
η
′ 
′− + − + + − ′ + 
(30)
It is interesting to note that sum of relation (24) through (29) =
1
s
6. Ergodic behaviour of the system
Using Abel’s Lemma
0
lim ( ) lim ( ) (say)
s t
s F s F t F
→ → ∞
= = , provided the limit on the R.H.S. exists, the time
independent probabilities are obtained as follows by making use above lemma in the relations (24) through (29)
3 1
(0)
NP
A
=
′
(31)
2
2 3
(2 )
(0)
N r
f
P D f f
A
′
′= +
′
(32)
2 3
6
(2 )
(0)
F
N r r
f
P D f f M
A
′
′= +
′
(33)
3 2
2 3
(2 )
(2 ) (0)
i r r
ff
P M D f f
f f A
′
 ′= − + ′ ′+
(34)
3
(0)
i
f
P M
A
η=
′
(35)
4
(0)
e
E r
f
P M
A
=
′
(36)
Where,
0
(0) ( )
s
d
A A s
ds =
 ′ =  
 
and kM = Mean time to repair kth
unit
7. Evaluation of up and down state probabilities
We have,
1 3
( ) 1
3 2
up
e
f
P s
s f f f s f f
′ 
= + ′ ′+ + + + + 
(37)
On inverting

322
{ } { }
3 3
( ) 1 exp (3 ) exp (2 )up e
e e
f f
P t f f f t f f t
f f f f
′ ′ 
′ ′= − − + + + − + ′ ′+ + 
(38)
( ) 1 ( )down upP t P t= − (39)
8. Cost Analysis
We have,
1 20
( ) ( )
t
upG t C P t dt C t= −∫ (40)
Where,
( )G t = Expected cost for total time,
1C = Revenue cost per unit up time and 2C = Service cost per unit time
{ } { }
1 1 2
1 exp (3 ) 1 exp (2 )3 3
( ) 1
3 2
e
e e e
f f f t f f tf f
G t C C C t
f f f f f f f f f
′ ′ − − + + − − + ′ ′   
= − + −      
′ ′ ′ ′+ + + + +       
(41)
9. Numerical Computation
Substituting 1 20.001, 0.002, 0.003, 2, 1ef f f C C′= = = = = and all repair rates are zero.
Availability
( ) 0.2exp ( 0.010 ) 1.2exp ( 0.005 )upP t t t= − − + −
Cost function analysis
1 exp ( 0.010 ) 1 exp ( 0.005 )
( ) 0.4 2.4
0.010 0.005
t t
G t t
− − − −   
= − + −   
   
10. Interpretation
10.1 Table 1 outlines the variation of availability of the model with time and their corresponding curve
S.No. t Pup(t)
1 0 1
2 1 0.996005
3 2 0.9920201
4 3 0.9880452
5 4 0.9840805
6 5 0.980126
7 6 0.9761817
8 7 0.9722477
9 8 0.9683241
10 9 0.9644107
11 10 0.9605078
12 11 0.9566154
13 12 0.9527334
14 13 0.9488619
15 14 0.9450009
16 15 0.9411506
Table 1: Availability as function of time

323
10.2 Table 2 exhibits expected cost function with respect to time and their corresponding curve
S.No. t G(t)
1 0 0
2 1 0.9960033
3 2 1.9840267
4 3 2.9640903
5 4 3.9362144
6 5 4.9004192
7 6 5.8567252
8 7 6.805153
9 8 7.7457231
10 9 8.6784561
11 10 9.603373
12 11 10.520494
13 12 11.429841
14 13 12.331435
15 14 13.225296
16 15 14.111446
Figure 2: Availability as function of time
Availability V/S Time
Table 2: Cost Profit as function of time

324
11. Conclusion
Table 1 and Figure 2 provide information how availability of the complex engineering repairable
system changes with respect to the time when failure rate increases availability of the system decreases.
Table 2 and Figure 3 when revenue cost per unit time C1 and C2 are fixed, then one can conclude by
observing this graph that as cost increases, when time increases.
Hence the present study clearly proves the importance of head-of line repair policy in comparison of [17-18]
which seem to be possible in many engineering systems when it is analyzed with the help of the copula. The
further research area is widely open, where one may think of the application of other members of copula family,
MTTF and sensitivity analysis.
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