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PROFIT ANALYSIS OF REPAIRABLE WARM STANDBY SYSTEM
1 Shiv Kant, 2Shashi Kant, 3*Mohit Yadav, 4Arunita Chaukiyal, 5Bindu Jamwal
department of Computer Science and Engineering (AI & DS), GNIOT, Greater Noida ^Department of Computer Science and Engineering, Sharda University, Greater Noida 3*Department of Mathematics, University Institute of Sciences, Chandigarh University, Mohali, Chandigarh "Department of Computer Science, Acharya Narendra Dev College, University of Delhi 5School of Engineering and Technology, Raffles University, Neemrana [email protected], [email protected], [email protected], arunita@yahoo. com, binduj amwal1810@gmail. com *Corresponding Author
Abstract
In the generation of science and technology, every company wants to increase the reliability of their products. So, they used the concept of warm standby redundancy, timely repair of the failed unit. This paper aims to explore the system of two non identical units where the primary unit is operative and the secondary unit is in warm standby mode. When the primary unit fails due to any fault then secondary unit starts working immediately. Here, times of failure of unit and times of repair of unit follow general distributions. Such types of systems are used in companies to prevent losses. The system's behaviour is calculated by using concepts of mean time to system failure, availability, busy period of the server, expected number of visits made by the server and profit values using the semi Markov process and regenerative point technique. Tables are used to explore the performance of the system.
Keywords: Warm standby, non identical units, regenerative point, semi Markov process
I. Introduction
Reliability and maintainability are the essential parameters of items and products that satisfy customers' requirements. In today's era, several approaches for performance improvement of maintainable systems have been adopted by scientists and engineers during designing them. A large amount of research work has been done on repairable systems such that Jack and Murthy [4] discovered the role of limited warranty and extended warranty for the product. Wang and Zhang [8] examined the repairable system of two non identical components under repair facility using geometric distributions. Deswal and Malik [3] evaluate the non identical units system under different working conditions by using the semi Markov process. Malik and Rathee [7] threw light on the two parallel units system under preventive maintenance and maximum operation time. Levitin et al. [6] explored the results of optimal preventive replacement of failed units in a cold standby system by using the poisson process.
Agarwal et al. [1] described the reliability and availability of water reservoir system under repair facility. Chaudhary and Sharma [2] explored the parallel non identical units system that gives priority to repair over preventive maintenance. Jia et al. [5] explored the two unit system under demand and energy storage techniques.
II. System Assumptions
There are following system assumptions:
• Initially, the system has two non identical units such that one is operative and the other is warm standby.
• When the operative unit fails then the warm standby unit starts working.
• An expert repairman is always available to repair the failed unit.
• The failed unit behaves like a new one after repair.
• Repair times and failure time follow general distribution.
III. System Notations
There are following system notations:
R Collection of regenerative states st (i = 0,1,3)
O / Ds Operative unit / warm standby unit of the system
X / w Failure rate of the unit/ rate by which the server needs refreshment
qr s (t)/ Qr s (t) PDF/ CDF of first passage time from rth to sth regenerative state or sth failed state without halting in any other Si e R in (0,1]
Mr(t) Represents the probability of the system that it initially works Sr e R at a time
(t) without moving through another state Si e R Wr(t) Probability that up to time (t) the server is busy at the state Sr without transit
to another state Si e R or before return to the same state through one or more non regenerative states © /® Laplace convolution / Laplace Stieltjes Convolution
* /* */' Laplace Transform/ Laplace Stieltjes Transform/ Function's derivative
O / • / □ Upstate/ regenerative state/ failed state
IV. State Descriptions The individual state description is given by the table 1:
Table 1: State Descriptions
States_Descriptions
50 It is a regenerative up state with two units - Ao and Bs .
51 This regenerative up state has two units such that (A Fur) and (B o).
52 It is a down state where one unit is failed under repair continuously from previous state (A fur ) and the other is failed under repair (BFur).
S 3 It is a regenerative up state with two units such that one is operative (Ao) and the
other is failed under repair (BFur). S4 It is a down state where one unit fails under repair continuously from previous state ( Afur ) and the other unit is failed under repair (BFur).
RT&A, No 4(80) Volume 19, December, 2024
Figure 1: State Transition Diagram
Where, S0 = {A0, Bs), Sx = (AFur, B0 ), S2 = (AFUR, BFur )
S3 = (A0, BFur ), S4 = (AFur, BFUR )
V. Transition Probabilities
The transition probabilities are calculated
A2 wi
p0,1 = ^ 7~ , p0,3 = ^ 7~ , p1,0 =-r
Ai + Aj Ai + Aj Wi + Aj
w2 A1
P3,0 =—-, P3,4 =—-, Pj,3 = P4,1 = 1
A1 + w2 A1 + w2
It has been conclusively established that
P0,1 + P0,3 = 1, P1,0 + P1,2 = 1, P3,0 + P3,4 = 1
Pl,2 =
An
Wi + A2
(1)
(2)
VI. Mean Sojourn Time
Let / represents the mean sojourn time. Mathematically, time consumed by a system in a particular state is, / =Z mi, j = J P(T > t) dt. Then
J
0
1 1 1
ß0 = m0,1 + m0,3 = „ . „ , ßl = m1,0 + m1,2 =-r- , ß2 = m2,3 =■
ß3 = m3,0 + m3,4 =
A1 + A2
1 1
--, A4 = m4,1 =-
A1 + W2 W2
W1 + A2
W
1 1
ß 1 = m1,0 + m1,3.2 =-r- , A3 = m3,0 + m3,1.4 =
W1 + A2
A1 + W2
Mo + PoM + Po3M3 MTSF = —- ' — ,3(6)
Shiv Kant, Shashi Kant, Mohit Yadav , Arunita Chaukiyal, Bindu Jamwal
PROFIT ANALYSIS OF REPAIRABLE WARM RT&A, No 4(80) STANDBY SYSTEM_V°lume 19 Decemto; 2024
VII. Reliability Measures Evaluations
I. Mean Time to System Failure (MTSF)
Let the cumulative distribution function of the first elapsed time be <Pi (t) from the regenerative state St to the failed state of the system. Treating the failed states as an absorbing state then the repetitive interface for p (t) being
Po (t) = Qo ,i(t) ® Pi (t) + Qo ,3 (t) ® P3 (t) Pi(t) = Qi ,o(t) ®Po(t) + Qi,2(t)
P3(t) = 03,o(t) ®Po(t) + 03,4(t) (4)
**
Taking LST of the relation (4) and solving for po (s) then
**
MTSF = limi-^ (5)
s ^o s
System reliability can be obtained by using the inverse LT of equation (5). We have Mo + Po,iMi + Po,3M3 1 - po,i pi,o - po,3 p3,o
II. Availability of the system
From the transition diagram, the system is available at the regenerative up states So, Si and S2 . Let Aj (t) is the probability that the system is in upstate at time (t) specified that the system arrives at the regenerative state Sj at t = 0. Then the repetitive interface for Aj (t) is Ao (t) = Mo (t) + qoi (t) ® Ai (t) + qo,3 (t) © A3 (t) Ai (t) = Mi (t) + qi,o (t) © Ao (t) + qi3.2 (t) © A3 (t)
A3 (t) = M3 (t) + q3,o (t) © Ao (t) + q3,i.4 (t) © Ai (t) (7)
where, Mo(t) = e~(Ai+A2)t, Mi(t) = , M3(t) = e+"2)? (8)
Using LT of the above relation (7), there exist
. Na Mo[i - Pi,3 p3,i] + Mi^,Po,i + Po,3 p3,i] + M3[Po,3 + Po,iPi,3^ ^ .'. Ao = lim—A =-,-,-,-,-,-,-,-,- (9)
s^0 D'i Mo[i - Pi,3P3,i] + M'i [_Po,i + Po,3P3,i] + M'3 [-Po,3 + PoiP\,3]
III. Busy Period of the Server
Let Bj (t) is the probability that the repairman is busy due to the repair of the failed unit at time 't' specified that the system arrives at the regenerative state Sj at t = 0. Then the repetitive interface for Bt (t) is
Bo (t) = qo,i (t) © Bi (t) + qo,3 (t) © B3 (t)
Bi (t) = Wi (t) + qi,o (t) © Bo (t) + qi,3.2 (t) © B3 (t)
B3 (t) = W3 (t) + q3,o (t) © Bo (t) + q3i .4 (t) © Bi (t) (10)
where, Wi(t) = Wie "(w'+22) t +.....
W3(t) = W2 e "(li+w2) f +..........(11)
Shiv Kant, Shashi Kant, Mohit Yadav , Arunita Chaukiyal, Bindu Jamwal
PROFIT ANALYSIS OF REPAIRABLE WARM RT&A, No 4(80) STANDBY SYSTEM_V°lume 19 December, 2024
Using LT on relations (10) then we get
Nb /WP0,1 + P0,3P3,1] + /'3 [P0,3 + P0,1P1,3] n9v B0 = lim—B =- (12)
D1 /0[1 - pu ^3,1] + /'1 [p0,1 + P0,3P3,1] + /'3 [P0,3 + P01P13 ]
IV. Estimated number of visits made by the server
Let Ni (t) is the estimated number of visits made by the repairman for repair in (0, t] specified that the system arrives at the regenerative state Si at t = 0. Then the repetitive interface for Ni (t) is N0 (t) = 00,1 (t) ® [1 + N1 (t)] + 00,3 (t) ® [1 + N3 (t)] N1 (t) = 01,0 (t) ® N0 (t) + 01,3.2 (t) ® N3 (t)
N3 (t) = 030 (t) ® N0 (t) + 031.4 (t) ® N1 (t) (13)
Using LST of the above relations (13) then we get
N0 = lim ^ = _1+P4P3!_^ (14)
M0 [1- Pi,3P3,1 ] + [P0,1 + P0,3P3,1 ] + M3[ Po,3 + P0,1P1,3]
V. Profit Analysis
The profit function of the system is defined by
P = T0A - T1B0 - T2N0 (15)
where, T0 = 5000 (Revenue per unit up-time)
T1 = 600 (Charge per unit for server busy period) T2 = 100 (Charge per visit made by the server)
VIII. Discussion
Let A1 = Aj = A and w1 = wj = w then the reliability measures like MTSF, availability of the system, busy period of the server, expected number of visits made by the server and profit
_Table 2: MTSF vs. Repair Rate_
w A=0.55 A=0.6, A=0.65
I
0.55 4.76082 4.413408 4.192308
0.6 4.929006 4.585448 4.290541
0.65 5.063985 4.728682 4.36747
0.7 5.174709 4.849785 4.429348
0.75 5.267176 4.953519 4.480198
0.8 5.345557 5.043371 4.522727
0.85 5.412844 5.121951 4.558824
0.9 5.471236 5.191257 4.589844
0.95 5.522388 5.252838 4.616788
0.1 5.567568 5.307918 4.640411
Table 3: Availability vs. Repair Rate
w X=0.55 X=0.65, X=0.6
I
0.55 0.686403 0.670466 0.657986
0.6 0.69355 0.678566 0.662905
0.65 0.699041 0.684997 0.666652
0.7 0.703393 0.690227 0.669601
0.75 0.706926 0.694563 0.671982
0.8 0.709852 0.698216 0.673945
0.85 0.712315 0.701336 0.675592
0.9 0.714416 0.704032 0.676992
0.95 0.716231 0.706384 0.678198
0.1 0.717813 0.708454 0.679247
Table 4: Profit vs. Repair Rate
w X=0.55 X=0.65, X=0.6
I
0.55 2722.902 2572.193 2533.318
0.6 2788.908 2650.579 2583.419
0.65 2838.719 2711.387 2620.861
0.7 2877.66 2759.963 2649.913
0.75 2908.947 2799.676 2673.117
0.8 2934.638 2832.757 2692.08
0.85 2956.115 2860.746 2707.868
0.9 2974.335 2884.737 2721.218
0.95 2989.99 2905.531 2732.655
0.1 3003.585 2923.73 2742.563
values are calculated. It can be seen from the tables 2, 3 and 4 that the tendency of MTSF, availability of system and profit values increase smoothly with respect to increments in repair rate(6) whereas these values declines corresponding to increment in failure rate.
IX. Conclusion
It is calculated that the MTSF, availability and profit values of the two non identical unit system increase with respect to increments in repair rate but these reliability values decline when failure rate of unit is enhanced. The idea of repair facility is used by corporate sectors, industries, cybercafés, education, university systems, etc.
References
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[3] Deswal, S. and Malik, S. C. (2015). Reliability measures of a system of two non-identical units with priority subject to weather conditions. Journal of Reliability and Statistical Studies, 181190.
[4] Jack, N., and Murthy, D. P. (2007). A flexible extended warranty and related optimal strategies. Journal of the Operational Research Society, 58(12), 1612-1620.
[5] Jia, H., Peng, R., Yang, L., Wu, T., Liu, D., and Li, Y. (2022). Reliability evaluation of demand based warm standby system with capacity storage.
[6] Levitin, G., Finkelstein, M., and Dai, Y. (2020). Optimal preventive replacement policy for homogeneous cold standby systems with reusable elements. Reliability Engineering and System Safety, 204, 107135.
[7] Malik, S. C. and Rathee, R. (2016). Reliability modelling of a parallel system with maximum operation and repair times. International Journal of Operational Research, 25(1), 131-142.
[8] Wang, G. J. and Zhang, Y. L. (2007). An optimal replacement policy for repairable cold standby system with priority in use. International Journal of Systems Science, 38(12), 1021-1027.
