Journal of Applied Mathematics and Physics, 2013, 1, 21-27
http://dx.doi.org/10.4236/jamp.2013.13005 Published Online August 2013 (http://www.scirp.org/journal/jamp)
Assessment of Profit of a Two-Stage Deteriorating Linear
Consecutive 2-out-of-3 Repairable System
Ibrahim Yusuf1*, Fatima Salman Koki2
1Department of Mathematical Sciences, Bayero University, Kano, Nigeria
2Department of Physics, Bayero University, Kano, Nigeria
Received June 5, 2013; revised July 6, 2013; accepted August 15, 2013
Copyright © 2013 Ibrahim Yusuf, Fatima Salman Koki. This is an open access article distributed under the Creative Commons At-
tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-
erly cited.
ABSTRACT
Most of the researches on profit and cost evaluation of redundant system focus on the effect of failure and repair on
revenue generated. However, as these systems continue to work, their strength gradually deteriorates. Where such dete-
rioration occurs, minor an d major maintenance is employed to remedy the deterioration . Little o r no attentio n is paid on
the effect of deterioration on the impact of deterioration and their maintenance on the revenue generated. In this paper,
we study the profit generated of two-stage deteriorating linear consecutive 2-out-of-3 system. Failure, repair and dete-
rioration time are assumed exponential. The explicit expressions of availability, busy period of a repairman and profit
function are derived using Kolmogorov’s forward equations method. Various cases are analyzed graphically to investi-
gate the effect of deterioration parameters such as slow deterioration, fast deterioration, and their maintenance such as
minor and major minimal maintenance on profit generated.
Keywords: Reliability; Availability; Profit; Deterioration
1. Introduction
During operation, the strengths of systems are gradually
deteriorated, until some point of deterioration failure, or
other types of failures. Minor and major maintenance
policies are vital in the analysis of deterioration and d ete-
riorating systems as they help in improving reliability,
availability and the overall revenue generated. Both mi-
nor and major minimal maintenance are employed to
check the effect of slow and fast deterioration and return
the system to its state prior to slow and fast deterioration.
Maintenance models assume perfect repair (as good as
new), minimal repair (as bad as old) and imperfect repair
which is between perfect and minimal repair. Many re-
search results have been reported on the reliability of
2-out-of-3 redu nd ant systems. For example, [1], analyzed
reliability models for 2-out-of-3 redundant system are
subject to conditional arrival time of the server. Refer-
ence [2] presented reliability and economic analysis of
2-out-of-3 redundant system with priority to repair and [3]
studied MTSF and cost effectiveness of 2-out-of-3 cold
standby system with probability of rep air and inspection,
while [4] examined the cost benefit analysis of series
systems with cold standby components and repairable
service station. Reference [5,6] examined the cost analy-
sis of two unit cold standby system involving preventive
maintenance respectively. Reference [7] studied the cost
and probabilistic analysis of series system with mixed
standby componen ts while [8] studied cost benefit analy-
sis of series systems with warm standby components
involving general repair time where the server is not
subject to breakdowns. The failure time and repair time
are assumed to have exponential distribution. Measures
of system effectiveness such as MTSF, steady-state
availability, busy p eriod and profit fun ction are obtained.
[9] studied availability of a system with different repair
options, while [10] evaluated the reliability of network
flows with stochastic capacity and cost constraint. The
problem considered in this paper is different from the
work of [4-6]. In this paper, a linear consecutive 2-out-
of-3 repairable system with two consecutive deterioration
stages (slow and fast) is studied with minor and major
minimal maintenance at slow and fast deterioration re-
spectively. In this paper, a two-stage deteriorating linear
consecutive 2-out-of-3 system was constructed and de-
rived its corresponding mathematical models. The main
*Corresponding author.
C
opyright © 2013 SciRes. JAMP

I. YUSUF, F. S. KOKI
22
contribution of this paper is two fold. The first is to de-
velop the explicit expressions for system availability,
busy period and profit function. The second is to perform
a parametric investigation of various system parameters
on profit function and capture their effect on the profit
function.
The rest of the paper is organized as follows. Section 2
is the description and states of the system. Section 3
deals with models formulation. The results of our nu-
merical simulations are presented and discussed in Sec-
tion 4. The paper is concluded in Section 5.
2. Description and States of the System
We consider a 2-out-of-3 system with three modes: nor-
mal, deterioration and failure. The deterioration mode
consists of two consecutive stages: slow and fast. It is
assumed that the system transits from normal to slow and
later to fast deterioration with rate 1

and 2

respect-
tively. It is also assumed that the two consecutive units
never fail simultaneously. Whenever the system deterio-
rate with rate 1

, minor minimal maintenance is invoke
with rate 1

to regain th e system to its early stage prior
to slow deterioration stage or the deterioration will be
faster with rate 2

where major minimal maintenance
will be done with rate 2

. Unit I fail with rate 1

and
is under minimal repair with rate 1

and unit III is
switch on. It is assumed that the switch from standby to
operation is perfect. Similarly, unit II fails with rate 2

and is minimally repaired with rate 2

. The system
failed when unit I and II have failed. The system is at-
tended by one repair man.
States of the System
0: Units I and II are in operation, unit III is in st
Standby, the system is operational.
S
State 1: The system is under slow deterioration and
is receiving minor minimal maintenance.
S
State 2: The system is under fast deterioration and is
receiving major minimal maintenance.
S
State 3: Unit I failed and is under repair, units II and
III are in operation, the system is in slow deterioration
stage and Operational.
S
State 4: Unit I failed and is under repair, units II and
III are in operation, the system is in fast deterioration
stage and Operational.
S
State 5: Units II failed and is under repair, the sys-
tem failed.
S
3. Models Formulation
Let


Pt be the probability row vector at time , then
the initial conditions for this problem are as follows:
t


  

012345
0
0,0, 0,0, 0,0
1,0,0,0,0,0
P
PPPPPP





we obtain the following system of differential equations:
 
0110111
PtPt Pt Pt  
 3
4
0
4
 
 
11121
1022 13
Pt Pt
PtPt Pt


 

 
2122211
PtPt PtPt  
 
 
 
322131
112 425
2PtPt Pt
Pt Pt Pt
 


 
 
 
42124
12 2325
Pt Pt
Pt PtPt


 

 
525232
2PtPt PtPt 
 (1)
The differential equations in (1) above is transformed
into matrix as
PTP


(2)
where





11 11
11122 1
212 1
11 2212
12212
22
00
00
00
02
00
000
T
 

 
 

2
2
2
0
0
2




  
 

 
 

 

 
 

 

 
 
3.1. System Availability Analysis
For the availability case of Figure 1 using the initial con-
dition in section 3 for this system,















123456
0 0,0,0,0,0,0
1,0,0,0,0,0,0
P PPPPPP






I. YUSUF, F. S. KOKI 23
The system of differential equations in (1) for the sys- tem above can be expressed in matrix form as:





00
11 11
11
111221
2 2 212 1
3 11 22122
3
4 122122
45
22
5
00
00
00
02
00
000 2
PP
PP
P P
P
PP
PP
P
 

 
 




  

 


  

 

  


 

 
 
 


 2
0
0
Let be the time to failure of the system. The
steady-state availability is given by
V
 


01234V
APP PPP

(3)
In steady state, the derivatives of state probabilities
become zero, thus (2) becomes


0TP

 (4)
which in matrix form is





0
11 11
1 11122 1
2
212 1
3 11 22122
4 122122
5 222
0 00
0
00
0 00
0 02
0
00
0 000 2
P
P
P
P
P
P
 

 
 


  
 

 
 
 
 

 
 

 

  
  
0
0
0
0
using the normalizing condition
 
012345
1PPPPPP (5) we substitute (5) in the last row of (4) following [2,3,5].
The resulting matrix is





0 11 11
1
11122 1
2 212 1
3 11 22122
4 122122
5
0
00
0 00
0
00
0 02
0 00
1 111 111
P
P
P
P
P
P
 
 
 
 

 
 

 
 
 
 

 
 

 
   
  
Expression for thus is:
V
A
22V
A
ND




2 2
212112 12112 212 122 121112 12122122
22 222
1212212121211122 1212112212122
1
2
22112212112
444322222
24 2424 222
322
N    
  
 
 
  



2
12121112 112 112 12221212
22 22
1221212212112121212111212212112
22 2
121 121 11221 121211212
224242
2223222 2
2222222

  
   


 
  



2212221 122121
2
1222121221212122 112121112 112 122221
2222
212122122121121 221212121 21
2
22 242
22222

 
 


 


I. YUSUF, F. S. KOKI
24
22 222222223
212221122 212212121212 2122122122122
22 322
212212 12122 12122121212122 122122112
2
2112221
22244
22622 2
22
D































 


 


 222
12211221 21221 2121 2 1212 121221
222 2 2222
21 2121 2121 2121122122211212212121
222323
1212122112121 21 22
22 4
22222 22
2



 

 




 

 
 
  
 
 3322222
2122122121221 22
22 222222
1 211 21212212121211 22121 2121 21212
22222 2
1212122112121212 1212 1212
222
2 2634
4424444
2




 
  


 

2
1221 11212
1211212112 1121211221122121222112212
12122 12121 12112 121211212212121
2
26 33
336336




      
       



3.2. Busy Period Analysis
Using the same initial condition in section 3 above as for
the reliability case















123456
0 0,0,0,0,0,0
1,0,0,0,0,0,0
PPPPPPP 


and (4) and (5) the busy period is obtain ed as follows:

2
0
0
In the steady state, the derivatives of the state prob-
abilities become zero and this will enable us to compute
steady state busy period:
The system of differential equations in (1) for the sys-
tem above can be expressed in matrix form as:





00
11 11
11
111221
2 2
212 1
3 11 22122
3
4 122122
45
22
5
00
00
00
02
00
000 2
PP
PP
P P
P
PP
PP
P
 
 
 
 




 
 

 


  

 

  


 

 
 
 



Let be the time to failure of the system. The
steady-state busy period is given by
V
 


12345V
BP PPPP

(6)
In steady state, the derivatives of state probabilities
become zero, thus (2) becomes


0TP

 (7)
which in matrix form is





0 11 11
1 11122 1
2 212 1
3
11 22122
4 122122
5 222
0
00
0 00
0
00
0 02
0 00
0 000 2
P
P
P
P
P
P
 
 
 
 


  
 

 
 
 
 

 
 

 

  
  
0
0
0
0
using the normalizing condition
 
012345
1PPPPPP (8) we substitute (8) in the last row of (7) following [2,3,5].
The resulting matrix is





0 11 11
1 11122 1
2
212 1
3 11 22122
4 122122
5
0
00
0 00
0
00
0
02
0 00
1 111 111
P
P
P
P
P
P
 
 
 
 

 




 






  
  

I. YUSUF, F. S. KOKI 25
In the steady state, the derivatives of the state prob-
abilities become zero and this will enable us to compute
steady state busy:
 
0
1BP 
The steady state busy period is therefore:

B

3
2
N
BD 


22 2
312121 11221 1212112212122
2 2
221122 1211212121112 112112122212 12
22 2
122121221211212
42424222
322 224242
22232
N   
   
 
 
 



2
1212111212212112 121
22 2
121112 2112 12112122 212221122121122
2
2121221212 122 112121112112
22 2
22222222
22 24


  
   

 
 
 

1 22221212
2222
1 22122121121 221212121 211 211 21221 21
22
112122 122212 112212 1211212 122 112221
12
22
22 22
22

    
     


 
 


222222
1121 212 112121212 12 1221112121
2    
3.3. Profit Analysis
The system/units are subjected to minor and major mini-
mal maintenance and corrective maintenance at failure as
can be observed in states 1, 2, 3, 4, and 5. From Figure 1
the repairman is busy performing corrective maintenance
action to the units/system at failure in states 1, 2, 3, 4 and
5. According to [1-3], the expected profit per unit time
incurred to the system in the steady-state is giv e n by:
Profit = total revenue generated – accumulated cost
incurred due maintenance/repairing the failed units.
 
01
PFCAC B
(9)
where : is the profit incurred to the system.
2
: is the revenue per unit up time of the system.
PF
0
C
C1: is the cost per unit time which the system is under
repair.
4. Results and Discussions
In this section, we numerically obtained the results for
mean time to system failure, system availability, busy
S
0
S
1
S
2
S
3
S
4
S
5
1

1

2

2

1

1

2

1

1

2

2

2

1

1

2

2

Figure 1. Transition diagram of the system.
period and profit function for all the developed models.
For the model analysis, the following set of parameters
values are fixed throughout the simulations for consis-
tency:
10.1


,20.2


,10.4


,20.1

,10.1


,
20.1


,10.3


,20.4


, ,
050,000C110,000C

The impact of 1

on profit can be observed in Figure
2. From this figure it is evident that the profit decreases
as 1

increases while in Figure 3, the increases with
increase in 1

. Similar results can be observed in Fig-
ures 4 and 5 of profit with respect to 2

and 2

. From
these figures, the profit decreases as 2

increases and
increases with increase in 2

. Results of profit with
respect to 1

is given in Figure 6. It is evident from
Figure 6 that as 1

increases, the profit decreases while
from Figure 7, the profit increases with increase in 1

.
00.1 0.20.3 0.40.5 0.6 0.70.8 0.9 1
35
36
37
38
39
40
41

1
Profit
Figure 2. Effect of 1

on Profit.

I. YUSUF, F. S. KOKI
26
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
35
36
37
38
39
40
41
42

1
Profit
Figure 3. Effect of 1

on Profit.
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
35. 5
36
36. 5
37
37. 5
38
38. 5
39
39. 5
40
40. 5

2
Profit
Figure 4. Effect of 2

on Profit.
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
34
35
36
37
38
39
40
41

2
Profit
Figure 5. Effect of 2

on Profit.
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
20
25
30
35
40
45
50

1
Profit
Figure 6. Effect of 1

on Profit.
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
15
20
25
30
35
40
45

1
Profit
Figure 7. Effect of 1

on profit.
5. Conclusion
In this paper, we constructed a two-stage linear consecu-
tive 2-out-of-3 system to study the impact of deteriora-
tion and maintenance on the generated profit. Explicit
expressions of steady-state availability, busy period and
profit function were derived. We performed numerical
investigation to see the effect of slow deterioration, fast
deterioration, minor minimal maintenance, major mini-
mal maintenance, failure and repair rates on the gener-
ated profit. It is evident from the results obtained that
repair rate, minor minimal maintenance rate and major
minimal maintenance rates increase the profit generated
while slow deterioration, fast deterioration and failure
rate decrease the profit. It is evident from the results ob-
tained that deterioration makes a tremendous effect on
the generated re venue ( pr ofit ) .

I. YUSUF, F. S. KOKI
27
6. Acknowledgements
The authors are grateful to the anonymous reviewers for
their constructive comments which have helped to im-
prove the manuscript.
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