CC BY
10
PROFIT ANALYSIS OF REPAIRABLE COLD STANDBY SYSTEM UNDER REFRESHMENTS
1Ajay Kumar and 2Ashish Sharma
:SOET, Raffles University, Neemrana, Rajasthan 2Department of Pharmacy, Sushant University, Gurugram [email protected], [email protected]
Abstract
In the generation of science and technology, every company wants to increase the reliability of their products. So, they used the concept of cold standby redundancy, timely repair of the failed unit and providing limited refreshments to the available technician when required. This paper aims to explore the system of two identical units where the primary unit is operative and the secondary unit is in cold standby mode. When the primary unit fails due to any fault then secondary unit starts working immediately. Here, times of failure of unit and technician refreshment request follow the general distribution whereas times of repair of unit and refreshment follow the exponential distributions. Such types of systems are used in industries and education systems to prevent losses. The system's performance 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 function using the semi Markov process and regenerative point technique. Tables are used to explore the performance of the system.
Keywords: Cold standby, refreshment, 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 Subramanian [17] explored the idea of preventive maintenance in two distinct units system under repair. Bao and Mays [3] analyzed the hydraulic reliability of water distribution systems under demand, pipe roughness and pressure head. Gnedenko and Igor [8] explored reliability and probability studies for engineering purposes. Diaz et al. [6] threw light on the warranty cost management system. Jack and Murthy [10] discovered the role of limited warranty and extended warranty for the product. Wang and Zhang [19] examined the repairable system of two non identical components under repair facility using geometric distributions. Mahmoud and Moshref [14] described the cold standby system under hardware failure and preventive maintenance using the semi Markov process. Deswal and Malik [5] evaluate the non identical units system under different working conditions by using the semi Markov process. Kumar et al. [11] examined the stochastic behavior of two unit system where one unit in cold standby mode and subject to maximum repair time using the regenerative point technique. Kumar and Goel [12] analyzed the preventive maintenance in two unit cold standby system under general distributions. Malik and Rathee [15] threw light on the two parallel units system under preventive maintenance and maximum operation time.
Temraz [18] analyzed the performance of two parallel components system under load sharing and degradation facility. Levitin et al. [13] explored the results of optimal preventive replacement of failed units in a cold standby system by using the poisson process. Barak et al. [2] threw light on the availability and profit values of milk plant under repair facility. Agarwal et al. [1] described the reliability and availability of water reservoir system under repair facility. Chaudhary and Sharma [4] explored the parallel non identical units system that gives priority to repair over preventive maintenance. Garg and Garg [7] analyzed the reliability and profit values of briquette machine under neglected faults like sound and overheating. Jia et al. [9] explored the two unit system under demand and energy storage techniques. Sengar and Mangey [16] examined the performance of complicated systems under inspection using copula methodology.
II. System Assumptions
There are following system assumptions:
Initially, the system has two units such that one is an operative (primary) unit and the other is a cold standby (secondary) unit.
When the operative unit fails then the cold 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 and refreshment times are exponentially distributed whereas times for failure of unit and server refreshment request are general.
III. System Notations
There are following system notations:
R Collection of regenerative states Sj (i = 0,1,2,3)
O / Cs Operative unit / cold standby unit of the system
a / b The probability that the cold standby unit is working/ not working
X / ^ Failure rate of the unit/ rate by which the server needs refreshment
g(t) / G(t) PDF/ CDF of the repair time of the unit
f (t)/ F(t) PDF/ CDF of refreshments time that restores freshness to the server
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 Sj 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 Sj e R
Wr (t) Probability that up to time (t) the server is busy at the state Sr without transit
to another state Sj e R or before return to the same state through one or more non regenerative states
© /® Laplace convolution / Laplace Stieltjes Convolution
* /** /' Symbol for 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 upstate with two units such that one is operative (O) and other is cold standby (Cs).
51 This regenerative upstate has two units such that one is failed under repair (Fur) and the other is in operative mode (O).
52 It is a regenerative upstate under refreshment facility (sut) where one unit is failed & waiting for repair (Fwr) and the other is in operative mode (O).
53 It is a regenerative down state and the system has two units such that one is failed under repair (Fur) and the other is failed and waiting for repair (Fwr).
54 It is a down state where one unit fails under repair (FUR) continuously from the prior state and the other unit is failed & waiting for repair (Fwr).
55 It is a down state that has two units under refreshment facility (sut) such that one is failed and waiting for repair (FWR) continuously from the previous state and the other is failed and waiting for repair (Fwr).
56 At this down state, the system has two units such that one is failed under repair (FUR) continuously from the previous state and the other unit is failed and waiting for repair (FWR) continually from the prior state.
57 This down state has two units under continuous refreshment facility (SUT) such that one is failed and waiting for repair (Fwr) and the other is failed and waiting for repair (FWR) continuously from the previous state.
58 This down state has two units under refreshment facility (sut) such that one is failed and waiting for repair (Fwr) and the other is failed and waiting for repair (FWR) continuously from the previous state.
Figure 1: State Transition Diagram
STANDBY SYSTEM UNDER REFRESHMENTS Volume 18, December 2023
V. Transition Probabilities
The transition probabilities are calculated using f (t) = в e , g (t) = фе $t and get
b Ф И
P01 = a , P03 = b, P10 =-,-т, p\2 =-,-т, P14 =-
i + ^ + A' $ + ^ + fi + ^ + A,
0 X d u 6 u
P21 , P27 = ^~7 , P31 =——, P38 =——, P41 =——, p45 =T~
0 + A 0 + A 9 + ^ 9 + ^ 9 + ^ 9 + ^
ft № 1 /1 N
P61 =-, P65 =-, P56 = P76 = P86 = 1 (1)
9 + ^ 9 + ^
It has been conclusively established that
P01 + P03 = 1, P10 + P12 + P14 = 1, P21 + P27 = 1, P31 + P38 = 1 p41 + p45 = 1 , P56 = P76 = P86 =1, P31 + P318{65f = 1
P10 + P12 + m.4 + pnA{56)n =1, P21 + p 21J(65) n = 1 (2)
VI. Mean Sojourn Time
In the cold standby redundant system, ^ represents the mean sojourn time. Mathematically, time consumed by a system in a particular state is, Hi = £ mi j =
J P(T > t) dt. Then j 0
1 1 1
M0 = mo1 + mo3 = —, M1 = m10 + m12 + m14 =~,-r, 2 = m21 + m27 =-
1 0 + 0 + A
1 1 1
M3 = m31 + m38 =-,-, V4 = m41 + m45 =-, M6 = m61 + m65 =~,-
9 + ^ 9 + n 9 + n
1 0i + A(0 + u)
M5 =V7 = M8 =~, M 1 = m10 + m12 + m11.4 + mn 4(56)n = "77771-77
0 11.4(56) 00(0+^+A)
0</> + A(0 + 0 + ^) (0 + M)
ii2 = m21 + m,, _ „ =--— , M 3 = m31 + m n = --— (3)
^2 21 211(65)" GtftG + X) 3 31 31.8(65)n dtp '
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 cp{ (t) being
n>(t)=Qo,3(t) + Qo,1(t) ®n(t)
q\(t)=Q\,4(t)+Q1,2(t) ®n(t) + Q1,o (t) ® vo (t)
<P2(t) = Q2,7(t) + Q2,1(t) ®q\(t) (4)
**
Taking LST of the relation (4) and solving for (5) then
**
MTSF = lim 1 ~V0 (S) (5)
System reliability can be obtained by using the inverse LT of equation (5). We have
MTSF = [1 - P12 P21]^0 + P01M1 + P01P12M2 (6)
1" P01P10 " P12P21
2 2 9(/> + X9 + $X + ^X + X + aX9 + aX + a^X MTSF =----(7)
9<f>+X9 + $X + + X -a9<f>-aX<j> II. Availability of the system
From the transition diagram, the system is available at the regenerative up states S0, S1 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 A0 (t) = M0 (t) + ^01 (t) © A1 (t) + q03 (t) © A3 (t) A1 (t) = M1 (t) + q10 (t) © A0 (t) + q12 (t) © A2 (t)
+ qu .4 (t) © A1 (t) + q11 4(56)„ (t) © A1 (t)
A2 (t) = M2 (t) + q21 (t) © A1 (t) + q21 7(65)„ (t) © A1 (t)
A3 (t) = q31 (t) © A1 (t) + q31 8(65)„ (t) © A1 (t) (8)
where, M0(t) = e_At, M1(t) = e~(^+A2)tG(t) , M2(t) = e_liG(i) (9)
Using LT of the above relation (8), there exist
.. NA [A0 P1,0 +^2 P1,2] - A0 = lim DAT = 7-T-;-T-7 (10)
i^0 D1 L«0P1,0 + M1 +V2P1,2 +<«3P0,3P1,0]
(11)
. + + (a - b)(d + X)X +X/j]
•'• A) =
2 (0 + X) + (dtp + 2(0 + ^))(a - b)2(<9 + 2) +2(0 + ^ + /u))^(a - b)x - (e + /u)bx<p{e+x)
III. Busy Period of the Server
From the transition diagram, it is clear that the technician is busy at states S1, S2 and S3 . 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 Si at t = 0. Then the repetitive interface for Bj (t) is B0 (t) = q01 (t) © B1 (t) + q03 (t) © B3 (t)
B1 (t) = W1 (t) + q10 (t) © B0 (t) + q12 (t) © B2 (t)
+ qu .4 (t) © B1 (t) + q11 4(56)„ (t) © B1 (t)
B3 (t) = W2 + q21 (t) © B1 (t) + q n (t) © B1 (t)
21.7(65)
B3(t) = W3(t) + q31(t) © B1(t) + q31 8(65)n (t) © B1(t) (12)
where, W1(t) = G1(t)e~(X+M)' + G1(t) ) ©G1(t)e~M' ■
-Xn
W2(t) = Gl(t)®Xe /l'Fl(f)©fi())
W3(t) = G1(t)e~M) + G1(t)^e~Mt © f1(t)©G1(t)-
(13)
Using LT on relations (12) then we get
D NB W*(0) + W2*(0)jP1,2 + W3*(0)p0,3 P1,0
B0 = lim —B =-:-:-— (14)
,^0 D'l [^0P1,0 + -«1 + -«2P1,2 + -«3P0,3P1,0]
Xp\0if>+X(ip + 0)\
0(/>2(0 + X) + (G<¡> + X(0 + M))(a - b)X(0 + X) + (e<¡> + X(0 + $ + ß)')ß(a - b)X - (0 + n)bX<¡)(6 + X)
(15)
IV. Estimated number of visits made by the server
The transition diagram explores that the technician visits at states S1 and S2 • Let Nj (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 Sj at t = 0. Then the repetitive interface for Nj (t) is N0 (t) = Qoi(t) 0 [1 + Nj (t)] + Q03 (t) 0 [1 + N3 (t)] Ni(t) = Qio(t) ® No(t) + Qi2(t) ® N2(t)
+ QhA(J) ® N1 (t) + QnAmn (t) ® N1 (t)
N2(t) = Q21(t) ® N1(t) + Q21 7(65)n (t) ® N1(t)
N3 (t) = Q31 (t) ® N1 (t) + Q31 8{65)n (t) ® N1 (t)
Using LST of the above relations (16) then we get
Nv P1,0
(16)
N0 = Kn-
5D[
N 0
lM0 P1,0 +Á +M2 P12 +^33 P0,3 P!0]
(0 + X)
1.2
(0 + X) + (0tj> + X(0 + ¡u))(a - b)X(0 + X)
X(0 + 0 + M))M(a - b)X - (0 + n)bX<¡>(0 + X)
(17)
(18)
2
V. Profit Analysis
It is an integral part of reliability measures that tell customers and system developers whether the system is beneficial or not. The profit values depend upon the MTSF, availability of the system, busy period of server and extended number of visits. Then the profit function of the system is defined by
P = ToAo - TlB0 - T2N0 (19)
where, To = 1500 (Revenue per unit up-time)
T\ = 500 (Charge per unit for server busy period) T2 = 200 (Charge per visit made by the server)
4. Discussion
The transition diagram is used to calculate the system reliability measures like MTSF, availability of the system, busy period of the server, expected number of visits made by the server and profit values. It can be seen from the table 2 that the tendency of MTSF increases smoothly with respect to increments in refreshment rate (0); however, other parameters such as failure rate of unit (X=0.55), server refreshment request rate (|i=0.4), repair rate of unit (^=0.5), cold stand by unit working probability (a=0.8) and not working probability (b=0.2) have fixed values. It is clear that when failure rate (X) increases then MTSF declines. When technician refreshment rate (|i) enhances then MTSF also declines but when repair rate increases then MTSF enhances. Thus, the concept of refreshment is beneficial for the owner and technician. When MTSF enhances then system reliability also enhances.
Table 2: MTSF vs. Refreshment Rate
0 X=0.55, |i=0.4 X=0.65, |i=0.6 ^=0.7
1 ^=0.5, a=0.8
b=0.2
0.1 3.8613371 3.218144 3.775294 4.0647311
0.2 3.8894957 3.235483 3.806505 4.1010786
0.3 3.9129156 3.250148 3.833119 4.1312067
0.4 3.9327001 3.262713 3.856081 4.1565858
0.5 3.9496349 3.273599 3.876094 4.1782569
0.6 3.9642941 3.283122 3.893693 4.1969772
0.7 3.9771076 3.291522 3.90929 4.2133109
0.8 3.9884034 3.298987 3.923207 4.2276872
0.9 3.998436 3.305664 3.935703 4.240438
1 4.007406 3.311673 3.946984 4.2518242
The availability of the redundant system is affected by the refreshment rate (0), repair rate unit failure rate (X) and server refreshment request rate (|i). Table 3 explores the availability of the system and its value increase corresponding to increments in refreshment rate (0) when the system's other parameters X=0.55, |i=0.4, ^=0.5, a=0.8, b=0.2 possess constant values. When the failure rate of unit changes (X=0.55 to 0.65) then the availability of system declines. Also, when the technician request rate changes (|i=0.4 to 0.6) then the system's availability declines but when the repair rate of unit changes (^=0.5 to 0.7) then the availability of the system enhances.
Table 3: Availability vs. Refreshment Rate
0 X=0.55, |i=0.4 X=0.65, |i=0.6 ^=0.7
1 ^=0.5, a=0.8
b=0.2
0.1 0.214076 0.185918 0.190417 0.25028
0.2 0.319498 0.281347 0.293845 0.364639
0.3 0.381465 0.338725 0.358019 0.427923
0.4 0.421888 0.376681 0.401314 0.467056
0.5 0.450148 0.403461 0.432266 0.493101
0.6 0.470912 0.423258 0.455361 0.511375
0.7 0.48675 0.438421 0.473168 0.524719
0.8 0.499191 0.450366 0.487264 0.534774
0.9 0.509198 0.45999 0.498663 0.542545
0.1 0.517404 0.46789 0.508046 0.548679
It is an important part of the system that tells the customers about the performance of the product that it is beneficial or not. So, the cold standby redundant system is used to enhance the system's profit. It is evident from table 4 that the system uses constant parameters such that X=0.55, |i=0.4, ^=0.5, a=0.8, b=0.2 and the trend of profit values enhanced with respect to increments in refreshment rate (0). When the failure rate of unit (X) changes from 0.55 to 0.65 then the profit of system declines. Also, when the technician request rate (|i) changes from 0.4 to 0.6 then profit values decline but when the repair rate of unit changes from 0.5 to 0.7 then the profit of the system enhances.
Table 4: Profit vs. Refreshment Rate
0 X=0.55, |i=0.4 X=0.65, |i=0.6 ^=0.7
1 ^=0.5, a=0.8 b=0.2
0.1 2046.417 1588.805 1690.491 2692.9
0.2 3640.771 3027.839 3256.171 4428.411
0.3 4576.626 3891.418 4226.372 5389.1
0.4 5186.276 4461.578 4880.064 5983.344
0.5 5611.935 4863.085 5346.797 6378.977
0.6 5924.286 5159.353 5694.6 6656.653
0.7 6162.259 5385.877 5962.46 6859.482
0.8 6348.978 5563.998 6174.235 7012.365
0.9 6498.993 5707.276 6345.293 7130.568
1 6621.895 5824.711 6485.948 7223.9
7. Conclusion
The results of the study show that providing refreshments to the server during the job generally enhances his efficiency which is crucial for any repairable system. From the above discussion, the MTSF, availability and profit values of the system increase with respect to increments in refreshment rate as well as repair rate but the reliability values decline when server refreshment request rate and failure rate of unit are enhanced. It is clear from tables that the server has to override his emotions and try to satisfy the customers. The idea of refreshment is used by corporate sectors, industries, cybercafés, education, university systems, etc.
8. Future Scope
Refreshment to the server plays an essential role in the water-boosting station system where one unit is operative and another is kept on cold standby.
References
[1] Agrawal, A., Garg, D., Kumar, A. and Kumar, R. (2021). Performance analysis of the water treatment reverses osmosis plant. Reliability: Theory & Applications, 16(3): 16-25.
[2] Barak, M. S., Garg, R. and Kumar, A. (2021). Reliability measures analysis of a milk plant using RPGT. Life Cycle Reliability and Safety Engineering, 10: 295-302.
[3] Bao, Y. and Mays, L. W. (1990). Model for water distribution system reliability. Journal of Hydraulic Engineering, 116: 1119-1137.
[4] Chaudhary, P. and Sharma, A. (2022). A two non-identical unit parallel system with priority in repair and correlated lifetimes. Reliability Theory and Application,17(67): 113-122.
[5] 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, 181-190.
[6] Diaz, V. G., Gomez, J. F., Lopez, M., Crespo, A., and de Leon, P. M. (2009). Warranty cost models State-of-Art: A practical review to the framework of warranty cost management. ESREL, 2051-2059.
[7] Garg, D. and Garg, R. (2022). Reliability analysis and profit optimization of briquette machine by considering neglected faults. Reliability Theory and Application, 17(67): 210-222.
[8] Gnedenko B. and Igor A.U. (1995). Probabilistic Reliability Engineering. John Wiley and Sons.
[9] Jia, H., Peng, R., Yang, L., Wu, T., Liu, D., and Li, Y. (2022). Reliability evaluation of demand based warm standby systems with capacity storage, Reliability Engineerings System Safety, 218: 108-132.
[10] 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.
[11] Kumar, A., Baweja, S. and Barak, M. (2015). Stochastic behavior of a cold standby system with maximum repair time. Decision Science Letters, 4(4): 569-578.
[12] Kumar, J. and Goel, M. (2016). Availability and profit analysis of a two-unit cold standby system for general distribution. Cogent Mathematics, 3(1): 1262937.
[13] Levitin, G., Finkelstein, M., and Xiang, Y. (2020). Optimal preventive replacement for cold standby systems with elements exposed to shocks during operation and task transfers. IEEE Transactions on Systems, Man and Cybernetics: Systems, 10: 2168-2216.
[14] Mahmoud, M. A. W. and Moshref, M. E. (2009). On a two-unit cold standby system considering hardware, human error failures and preventive maintenance. Math. Comput. Model, 51: 736-745.
[15] 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.
[16] Sengar S. and Mangey R. (2022). Reliability and performance analysis of a complex manufacturing system with inspection facility using copula methodology. Reliability: Theory & Applications, 4(71): 494-508.
[17] Subramanian, R. (1978). Availability of two-unit system with preventive maintenance and one repair facility. IEEE Transactions on Reliability, 27(2): 171-172.
[18] Temraz, N. S. Y. (2018). Availability and reliability analysis for dependent system with load sharing and degradation facility. International Journal of Systems Science and Applied Mathematics, 3(1): 10-15.
[19] 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.
