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PROFIT ANALYSIS OF REPAIRABLE WARM STANDBY SYSTEM UNDER IMPERFECT SWITCH
1Nishant Yadav, 2*Shiv Kant, 3Shashi Kant, 4Arunita Chaukiyal, 5Bindu Jamwal
�
1Department of Mathematics, Yaduvanshi Degree College, Narnaul, Haryana
2*Department of Computer Science and Engineering (AI & DS), GNIOT, Greater Noida
3Department of Computer Science and Engineering, Sharda University, Greater Noida
4Department of Computer Science, Acharya Narendra Dev College, University of Delhi
5School of Engineering and Technology, Raffles University, Neemrana
[email protected], [email protected], [email protected], [email protected], [email protected]
*Corresponding Author
Abstract
In this paper, the performance of two non identical units repairable system are analyzed by using regenerative point graphical technique. Generally, the system has one operative unit and one warm standby unit. Fuzzy concept is used to find the reliability measures under imperfect switch. Regenerative point graphical technique and semi markov process are used to evaluate the reliability measures. Primary, secondary and tertiary circuits are used to describe the base state. The system is repaired by the available technician when any unit is failed or switch is under imperfect mode. The priority in repair is given to switch before working units. In this paper, the failure time and repair time follow general distributions. The tables are used to explore the reliability measures such that mean time to system failure, availability and profit values.
Keywords: Base state, warm standby, imperfect switch, availability and profit values.
I.Introduction
To meet the increasing demand for their products, manufacturers need to constantly improve their products which can be achieved by improving their production procedures. The MTSF, availability and profitability of two non identical units system with priority in switch repair are discussed by using the regenerating point graphical technique under specific circumstances. In the real time system, it is impossible to ignore the concept of spare unit switching failure when it is utilized to replace a failed operative unit.
Balagurusamy [1] described the terms related to the system meantime, failure, repair, redundancy, maintainability and availability. Gopalan and Bhanu [3] examined the behaviour of two unit repairable system under online preventive maintenance using regenerative point technique. Gupta and Singh [4] threw light on the profit and availability values of redundant system with imperfect switch. Lim [7]. Kumar and Sirohi [6] examined the reliability, availability and profit values of two unit system under cold standby approach with delays in repair of partially failed unit. Kumar and Goel [5] evaluated the behaviour of two unit cold standby system under general distribution. Taneja et al. [10] discussed on the comparative study of profit values two reliability models under varying demands. Bhardwaj et al. [2] analyzed the reliability measures of two unit cold standby system under standby failure and arbitrary distribution for
repair and replacement times. Sadeghi and Roghanian [8] discussed on the availability and profit values of the two dissimilar unit repairable system under two cases of imperfect switch. Singh et al. [9] evaluated the behaviour of the complex repairable system with two subsystems connected in series with switch facility.
II.System Assumptions
To describe the system, there are following assumptions
�The system has two distinct units where unit A is in operative mode and unit B is
in warm standby mode.
�There is an imperfect switching.
�The sequence of repair is switch, operative main unit and warm standby unit.
�A technician is available to repair the failed unit.
�Failure time and repair time follow the general distribution.
�The repaired unit functions just like a brand-new one.
III.System Notations
To explain the juice plant, there are following notations
jirS..>.
r
iffs..>..
A directed simple failure free path from state . to state �i�.
cyclem.
A circuit (may be formed through regenerative or non regenerative / failed state) whose terminals are at the regenerative state �m�.
cyclem.
A circuit (may be formed through the unfailed regenerative or non regenerative state) whose terminals are at the regenerative �m� state.
kkU,
Probability factor of the state �k� reachable from the terminal state �k� of �k� cycle.
kkU,
The probability factor of state �k� reachable from the terminal state �k� of . cyclek
i.
Mean sojourn time spent in the state �i� before visiting any other states.
i..
Total unconditional time spent before transiting to any other regenerative state while the system entered regenerative state �i� at t=0.
i.
Expected waiting time spent while doing a job given that the system entered to the regenerative state �i� at t=0.
aA/
The first unit is in the operative state/failed state.
bBB//
The second unit is in the operative state/ standby state/ failed state.
The third unit is in the operative state/reduced state/failed state.
21,..
Fixed failure rate of the unit A/B respectively.
qp/
Probability of switch properly working/not working.
S/s
The switch is in perfect mode/Imperfect mode.
21,..
Fixed repair rate of the unit A/B after respectively.
.
Fixed repair rate of the switch.
/ /
[1]Balagurusamy, E. (1984). Reliability Engineering. Tata McGraw-Hill Education.
[2]Bhardwaj, R. K., Kaur, K., and Malik, S. C. (2017). Reliability indices of a redundant systemwith standby failure and arbitrary distribution for repair and replacement times. InternationalJournal of System Assurance Engineering and Management, 8, 423-431.
[3]Gopalan, M. N., and Bhanu, K. S. (1995). Cost analysis of a two unit repairable system subjectto on-line preventive maintenance and/or repair. Microelectronics Reliability, 35, 251-258.
[4]Gupta, V. K., and Singh, J. (2007). Behavior and profit analysis of a redundant system withimperfect switch. Journal of Mathematics and System Science, 3, 16�35.
[5]Kumar, J. and Goel, M. (2016). Availability and profit analysis of a two-unit cold standbysystem for general distribution. Cogent Mathematics, 3(1): 1262937.
[6]Kumar, P., & Sirohi, A. (2015). Profit analysis of a two-unit cold standby system with thedelayed repair of the partially failed unit and better utilization of units. International Journal ofComputer Applications, 117(1), 41-46.
[7]Lim, T. (2000). Analysis of System Reliability with Dependent Repair Modes. IEEE Transactionon Reliability, 49(2), 153-162.
[8]Sadeghi, M., and Roghanian, E. (2017). Reliability analysis of a warm standby repairablesystem with two cases of imperfect switching mechanism. Scientia Iranica, 24(2), 808-822.
[9]Singh, V.V., Poonia, P.K., and Adbullahi, A.H. (2020). Performance analysis of a complexrepairable system with two subsystems in series configuration with an imperfect switch.Journal of Mathematical and Computational Science, 10 (2), 359�383.
[10] Taneja, G., Malhotra, R., and Chitkara, A.K. (2016). Comparative profit analysis of tworeliability models with varying demand. Arya Bhatta Journal of Mathematics and Informatics, 8(2),305-314.
Upstate/ reduced state/ failed state.
IV.Circuits Descriptions
Primary, secondary and tertiary circuits are used to find the base state such that
Table 1: Circuit Descriptions
i
(C1)
(C2)
(C3)
0
(0, 2, 0)
(0, 1, 2, 0)
(0, 2, 4, 3, 0)
Nil
Nil
(4, 3, 4)
(3, 4, 3)
Nil
Nil
Nil
Nil
1
1, 2, 0, 1
(0,2,0)
(0,2,4,3,0)
(2,0,2)
(2,4,3,0,2)
Nil
(4,3,4)
(3,4,3)
Nil
(4,3,4)
(3,4,3)
2
(2,0,2)
(2,0,1,2)
(2,4,3,0,2)
Nil
Nil
(3,4,3)
(4,3,4)
Nil
Nil
Nil
Nil
3
(3,0,2,4,3)
(0,2,0)
(0,1,2,0)
(2,0,2)
Nil
Nil
Nil
4
(4, 3, 0, 2, 4)
(0,2,0)
(2,0,2)
Nil
Nil
Figure 1 State Transition Diagram
where, , , , ,SBAS=0
aBSS=1
sBaS=2
abSS=3
AbSS=4
V.Transition Probabilities
There are following transition probabilities
,, ,pp=1,0
qp=2,0
)/(2110,1...+=p
)/(2123,1...+=p
,, (1) 14,31,2==pp
)/(1220,4...+=p
)/(1213,4...+=p
It has been conclusively established that
, , , , 12,01,0=+pp
13,10,1=+pp
11,2=p
14,3=p
13,40,4=+pp
VI.Mean Sojourn Time
For the particular state, it becomes
, ,, 10/1..=
)/(1211...+=
../12=
13/1..=
(2) )/(1)(124...+=t
VII.Evaluation of Parameters
Using the circuit table, �0� is used as the base state to calculate the reliability using the regenerative point graphical technique. The probability factors of all the reachable states from the base state �0� are given below
, , 1)0,1,0(0,0==U
1,01,0pU=
2,02,0pU=
, 3,11,22,03,11,03,0pppppU+=
4,33,11,22,04,33,11,04,0pppppppU+=
I.Mean Time to System Failure
The regenerative un-failed states (i=0, 1) to which the system can transit (with initial state 0) before entering to any failed state (using base state .=0) then MTSF becomes
()().................................>.................=................................>.=...................SrkkVksffSrprikkVkiisffSrprSrT22102)00(11011101.)0(0.
(3) )]1,0,1(1[11,000,00.+=..UUT
II.Availability of the system
The system is available for use at regenerative states j=0, 1, 4 with .=0 then the availability of system is defined as
..............=..............................>..............................................>.=..4022102'.)0(11101..)0(0ikkVkiiSrprSrjkkVkjjfjSrprSrA..
(4)44,033,022,011,000,044,011,000,00........UUUUUUUUA++++++=
III.Busy Period of the Technician
The Technician is busy due to repair of the failed unit at regenerative states j= 1, 2, 3, 4 with . = 0 then the fraction of time for which the server remains busy is defined as
..............=..............................>..............................................>.=..4022102'.)0(11101.)0(0ikkVkiiSrprSrjkkVkjjSrprSrB..
(5) 44,033,022,011,000,044,033,022,011,00.........UUUUUUUUUB+++++++=
IV.Estimated number of visits made by the Technician
The technician visits at regenerative states j= 1, 2 with .=0 then the number of visits by the repairman is defined as
..............=...............................>...............................................>.=..4022102'.)0(11101)0(0ikkVkiiSrprSrjkkVkjSrprSrV.
(6) 44,033,022,011,000,022,011,00.......UUUUUUUV+++++=
V.Profit Analysis
The profit of the system is analyzed by
(7) 020100VEBEAEP..=
where, 15000 (Revenue per unit uptime of the system) =0E
500 (Cost per unit time for which technician is busy due to repair) =1E
100 (Cost per visit of the technician) =2E
VIII.Discussion
Regenerative point graphical technique plays an important role to determine the reliability metrics of the repairable system. Here, .i are the failure rate and .j are the repair rate. Let all .i=. (i=1, 2) and .j=. (j=1, 2) then tables 2, 3 and 4 describe the nature of mean time to system failure, availability and profit values of the two unit system under imperfect switch having an increasing trend corresponding to increment in repair rate (.).
In these tables, the values of parameters change such that .=0.3, 0.4, 0.5 and .=0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95 respectively. When .=0.3 changes into 0.4, 0.5 then MTSF, availability and profit values have decreasing trends.
Table 2: MTSF vs. Repair Rate (. )
.
.=0.3
.=0.4
.=0.5
0.5
3.628692
3.333333
3.090278
0.55
3.675035
3.368794
3.132184
0.6
3.720609
3.403509
3.173516
0.65
3.765432
3.4375
3.214286
0.7
3.809524
3.47079
3.254505
0.75
3.852901
3.503401
3.294183
0.8
3.895582
3.535354
3.333333
0.85
3.937583
3.566667
3.371965
0.9
3.97892
3.59736
3.410088
0.95
4.019608
3.627451
3.447712
Table 3: Availability vs. Repair Rate (. )
.
.=0.3
.=0.4
.=0.5
0.5
0.623324
0.604782
0.542904
0.55
0.628307
0.609813
0.547959
0.6
0.633159
0.614717
0.552904
0.65
0.637887
0.619499
0.557741
0.7
0.642494
0.624164
0.562476
0.75
0.646985
0.628716
0.56711
0.8
0.651365
0.633159
0.571646
0.85
0.655637
0.637497
0.576089
0.9
0.659806
0.641734
0.58044
0.95
0.663876
0.645873
0.584703
Table 4: Profit vs. Repair Rate (. )
.
.=0.3
.=0.4
.=0.5
0.5
2438.338
2386.076
2019.52
0.55
2467.262
2415.876
2049.467
0.6
2495.431
2444.927
2078.759
0.65
2522.874
2473.257
2107.417
0.7
2549.618
2500.892
2135.461
0.75
2575.691
2527.857
2162.912
0.8
2601.117
2554.178
2189.787
0.85
2625.919
2579.875
2216.104
0.9
2650.121
2604.972
2241.881
0.95
2673.744
2629.49
2267.135
IX.Conclusion
The performance of the two non identical unit system is discussed using the regenerative point graphical technique. The above tables explore that when the repair rate increases then the MTSF, availability and profit values also increase but when the failure rate increases then the MTSF, availability and profit values decrease.
It is clear that RPGT is helpful for industries to analyze the behaviour of the products and components of a system.
X.Future Scope
It is observed that the role of the regenerative point graphical technique for the two non identical unit system will be beneficial and also used by the management, manufacturers and the persons engaged in reliability engineering and working on analyzing the nature and performance analysis of the system like soft drink, paper industry.
References
