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PROFIT ANALYSIS OF REPAIRABLE COLD STANDBY SYSTEM SUBJECT TO REBOOT FACILITY UNDER REFRESHMENTS
!Ajay Kumar and 2Ashish Sharma
iSOET, Raffles University, Neemrana, Rajasthan 2Department of Pharmacy, Sushant University, Gurugram [email protected], [email protected]
Abstract
This paper relates to the reliability measures analysis of two identical unit system with reboot facility. Initially, one unit of the system is in operative mode and another unit is kept in cold standby mode. A technician is always available with the system to perform repairing and rebooting activities. Here, the system operative unit failed in safe mode and unsafe mode. During unsafe failure, repair activity cannot be done immediately but first rebooting is done to transform unsafe failure into safe failure, and then repair activity is performed as usual. Sometimes, the technician needs refreshments due to continuous work and provides better services after taking refreshments. The unit works like a new one after repair. The failure time of the unit in safe mode, unsafe mode and technician refreshment request time are assumed to be general while the repair time of the unit, rebooting delay time and technician refreshment time are taken as exponential. Reliability measures such as mean time to system failure, availability of the system, busy period of the repairman, the expected number of visits by the technician and profit values are calculated using tables.
Keywords: Availability, cold standby, regenerative point, rebooting, and refreshment.
I. Introduction
In daily life, there are many situations such as the breakdown of the unit that caused machine failure. One way to avoid loss and increase the reliability of the system is to use the cold standby facility. With the occurrence of the complexity of machines and advancements in industrial sectors or organizations, the focus is on increasing the reliability and profit of the industry. The prominent point is that the designs and layout of complex machines or equipment should be so that it increases the reliability of the system and always tries to minimize the shortcomings responsible for its downtrends. Hence, designing a reliable system has become an essential step in almost every sector. So, the concept of rebooting is used to transform the unit from unsafe failure to safe failure. Sometimes, a technician is tired and needs refreshment. After taking refreshment, the repairman provides better service, and after getting the repair, the unit works like a new one. Many researchers such as Zhang and Wang [15] described a different unit repairable cold standby system that gives seniority to the operative unit. Hsu et al. [4] explored the standby system having reboot delay, general repair, switching failure, and unreliable repair facility. Jyh-Bin et al. [6] analyzed various reliability measures of a repairable system having standby switching failures and facility of reboot delay.
Ajay Kumar and Ashish Sharma
PROFIT ANALYSIS OF REPAIRABLE COLD STANDBY SYSTEM Rl&A, No 2 (78) UNDER REFRESHMENTS SUBJECT TO REBOOT FACILITY_Volume 19, June, 2024
Dhall et al. [2] discussed the reliability of the similar unit stochastic approach under the repair and replacement of the failed unit subjected to inspection. Ke and Liu [7] examined the repairable system having a single server that identified the failed unit before repair and rebooting. Kumar and Goel [10] highlighted the two-unit cold standby redundant system subjected to inspection before repairing the failed unit and using the concept of preventive maintenance. Goel et al. [3] explained the performance of a cold standby redundant system with a server to inspect the failed unit before repair.
Temraz [14] evaluated the reliability measures for dependent system with load sharing and subject to degradation facility. Jain et al. [5] described the machine system as having online and standby units for system sustainability with server vacation, observing imperfect fault and its recovery using the reboot approach. Kumar and Jain [11] examined the reliability measures of a warm standby machine system having multiple components with recovery failure supported by the reboot process. Levitin et al. [12] evaluated the cold standby systems with elements exposed to shocks during operation and task transfers under preventive maintenance. Agrawal et al. [1] described the nature of the water treatment reverses osmosis plant using the regenerative point graphical technique. Sengar and Mangey [13] examined the complex manufacturing system subject to inspection facility using copula methodology. Kumar and Sharma [8] evaluated the availability and profit analysis of a repairable two unit cold standby system under refreshment using the regenerative point technique. Kumar et al. [9] explored the performance of two unit cold standby system under inspection and subject to refreshment facility.
II. System Assumptions
There are following system assumptions:
• The whole system has two identical units- first operative and second cold standby.
• The cold standby unit takes place when the operative unit stops functioning.
• A technician is always available to repair the failed unit.
• The failed unit behaves like a new one after repair.
• When the unit fails in unsafe mode then the reboot process is used to convert it to safe mode.
• Refreshment is offered to the technician to enhance his efficiency.
• Repair time, refreshment time and reboot delay time are exponentially distributed whereas times for failure of the unit in safe mode and unsafe mode and technician refreshment request are general.
III. System Notations
There are following system notations:
R Collection of regenerative states (i = 0,1,2,3,9)
O / O(g)/ Cs The system unit is operative and in normal mode / suitable good condition
mode /cold standby mode a / b The probability that the cold standby unit is working/ not working
A / Ay / jU The constant failure rate of the unit of the system in safe mode/rate with unit
goes to unsafe mode/ rate by which the repairman needs refreshment gy (t) / Gy (t) PDF/ CDF of the repair time of the unit
fl (t) / Fy (t) PDF/ CDF of refreshments time that restores freshness to the technician
hy (t) / Hy (t) PDF/ CDF of reboot delay time
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 system has up states as well as down states and these individual states are described in 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) and other is failed & waiting for repair (FWR) continuously from the previous state.
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 & waiting for repair (Fwr) and the other is failed & waiting for repair (FWR) continuously from the previous state.
58 This down state has two units under refreshment facility continuously from the prior state (SUT) in unsafe mode of failure of unit such that operative unit is failed under unsafe mode F(uns) and the other is failed and waiting for repair (FWR) continuously from the previous state.
S9 This down state has two units such that the operative unit is failed under unsafe mode F(uns) and the other is in good condition.
S10 This down state has two units such that the operative unit is failed under unsafe mode F(uns) and the other is failed under repair (FWR) continuously from the previous state.
Figure 1: State Transition Diagram
V. Transition Probabilities
The transition probabilities are calculated using fi(t) = ве ~в 1, g i( t) = ф e ~фг, hi( t) = $e 1 Aa
P03 =
P01 = Pl2 = Р21 =
A + \a' M
Ab Aa
' p09 = „ . „ ' р10 =
Ф
ф + A + A + M A
A + Aa
' Pl4 =
A + Aa
A
P28 =
ф + A + A +M A
Pi,w =
ф + A + A +M A
в
в+A+A в+A+A
p54 = p61 = p76 = p87 = p91 = P 100,4 = 1 It is smoothly verified that
P01 + P03 + P09 = 1 P41 + P45 = 1
, P31 = P41 =
ф + A + A + M
Ф
' P21 =
в + A + A
ф + M
, P35 = P45 =
M
ф + M
(1)
A0 + Pl2 + Pl4 + P1,10 = A0 + Pl2 + -P1I.4 + ^11(45)" + ^11.(10,4) + ^1L,0(45)"
.10(45)"
= 1
P21 + P27 + P28 = P21 + P21.(76) + ^21.8(76) = 1, P31 + P35 = P31 + P
31.(54)"
= 1
(3)
VI. Mean Sojourn Time
In the cold standby redundant system, / represents the mean sojourn time. Mathematically, time
<x>
consumed by a system in a particular state is, / = Zmj j = JP(T > t)dt. Then
j 0
/0 = m01 + m03 + m09 = JP(T > t)dt = A-—
0 A + Ai&
1 1
/ = m10 + m12 + m14 + m1,10 = , . „ . „ , .. , /2 = m21 + m27 + m28 =
j + X + X + j
1 1
j3 = m31 + m35 = --, j4 = m41 + m45 =
1
( + j
1 , (0 + j)
j = f = J10 =~ , f = m31 + m31.(54)" = (
f1 = mX0 + m12 + m11.4 + m11(45)" + m11.(10,4) + m1L
10(45)"
[dj(£ + \) + ^ + f)(X + X)]
d(Ç(( + X + Xx + j)
j2 = m21 + m21.(76) + m21.8(76)
[(f + X)+#(X + X )(( + ?)]
((Ö + X+X)
ö + X+X
1
"7- f = f = ^ , f =T
j + j Ö j
(4)
VII. Reliability Measures Evaluations
I. Mean Time to System Failure (MTSF)
Let the cumulative distribution function of the first elapsed time be (j (t) from the regenerative state Si to the failed state of the system. Treating the failed states as an absorbing state then the repetitive interface for pj (t) being
Pc(t) = QUO+Qb(0+Qd(t) ® p(t)
P (t) = 01,10 (t) + 014(t) + 012 (t) ® (2 (t) + 010 (t) ® P0 (t)
P2(t) = 028(t) + 027(t) + 021(t) ® P1(t) (5)
Taking LST on the above equation (5) then get
(s) = 1"^0 (s)
R* (s)=-
Now, system reliability is accessed by using the inverse LT on equation (6) such that
1 -P^(s) = (1 - P01 P10)/2 + P21/1 + P21P10/0
MTSF = lim-
(1 - P01 P10 - P12 P21)
(6)
(7)
s
s
Ajay Kumar and Ashish Sharma
PROFIT ANALYSIS OF REPAIRABLE COLD STANDBY SYSTEM Rl&A, No 2 (78) UNDER REFRESHMENTS SUBJECT TO REBOOT FACILITY_Volume 19, June, 2024
II. Availability of the system
From the transition diagram, the system is available at the regenerative up states So , Sj 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) = qo9(t) © MO + 9o3(t) © A3(t) + qoj(t) © Ai(t) + Mo(t)
Aj (t) = qn (t) © A2 (t) + qw (t) © Ao (t) +
[qu.4(t) + qUX45f (t) + qu.(io,4)(t) + qUA0(45)n (t)] © a,(t)+m,(t) A2(t) = [q2i(t) + q2i.(76)(t) + q2i^ (t)] © A(t)+M2(t)
A3(t) = [q3i(t) + q3L{54f (t)] © A,(t)
Mt) = qgi(t) © Aj(t) (8)
(9)
Where, M 0 (t) = e-(A+Aia)t, Mx (t) = G(t) e -0-+W, M 2 (t) = F (t) e-(A+Ai)t
Using LT of the above relation (8), then get
* N „
M") = limsA0(s) = D (!0)
where, Na = [^0 Pl0 + Vl + V2 Pl2]
and D' = [(^0 + ¡¿,P03 - V9P09)P10 + + ¿2Pi2]
III. Busy Period of the Server
From the transition diagram, it is clear that the technician is busy at states Sj, 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 Sj at t = 0. Then the repetitive interface for Bj (t) is
B0(t) = q09(t) ©B9(t) + q03(t) ©B3(t) + q0j(t) © Bj(t)
Bj (t) = qj2 (t) © B2 (t) + qj0 (t) © B0 (t) +
[qii. 4(t) + qll( {45)n (t) + qii. (WA)(t) + qu A0(45f (t)] © bx (t) + Wx (t)
B2 (t) = [q2i (t) + q2i.(76) (t) + +q2i.^ (t)] © Bj (t) + W2 (t) B3 (t) = [q3i (t) + q3l 54A)n (t)] © Bj (t) + W3 (t)
B9(f) = q9j(t) © Bj(t) + W9(f) (11)
where, W1 (t) = G1 (ty(Ä+M)t + G1 (t)Äe-(Ä+M)t © Gx (t)e-Mt +
W2(t) = [{A^^F^ Ae-^F^)®^)}^)} © Gj(t) W3(t) = G1 (t)e-Mt + G1 (t)^e Mt © f,(t)© G,(t)e-Mt +.....
and W9(t) = Hl(t)
Using LT on the above relations (11) then get BR = lims^0 sB*(s) = N^r
Where, Nb = W*(0) + W2*(0)pj2 + Ws'(0)P03 - W*(0)P09)pw (12)
and D' is formerly declared.
IV. Estimated number of visits made by the server
The transition diagram explores that the technician visits at states Sj and S2 . Let N (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 S;- at t = 0. Then the repetitive interface for N (t) is
Vo(t)=Q»(0 ®[1+V9(0]+Qo3(t) ®[1+V3(t)]+Qoi(t) ®[1+Vjct)]
Vi(t) = 012 (t) ® V2(t) + Q10 (t) ® Vo(t)
+ [011.4 (t) + 011(45)„ (t) + 011.(10,4) (t) + 0„.1o(45)n (t)] ® V (t) V2 (t) = [021 (t) + 021.(76) (t) + 021.8(76) (t)] ® V (t) V3(t) = [031(t) + 03L(54)„ (t)] ® Vx(t)
V9 (t) = 091 (t) ®V1(t)
Using LST on the above relations (13), then get
Vo(W) = limsVo**(s)
V = D Where V„ = Ao
and D' is formerly declared.
V. Particular Cases
(13)
(14)
Suppose that f (t) = 6e~6t, g 1(t) = 4e , h1 (t) = _it
P11.4 =
¿4
¿j
-Pn.(1o,4)
^21.(76)
-, P n _-
(4 + x + X + j)(4 + j) 11(45)n (4 + x + X + j)(4 + j) ¿4 ¿J
-, p
(4 + x + X+j)(4 + j) muOC«)" (4 + x + x+^)(4 + ^) x _ ¿1 _ f
^21.8(76) _77T" 7" 7T , -P3i(54)n _
(6 + X + X)
(6 + x + x) 3L(54) (4+j)
Also, Mo _
1
_Vo , M1 _■
1
_jf, M2 _-
(¿ + Xa) (4 + J + X + X) (6 + 1 + 1)
W1(t), W2(t)=jf, W3(t)_f, W9(t)_j
MTSF _
[(X + ¿a){(4 + X + X + j) + 6} - + 6)] [(6 + ¿ + ¿){(X + ¿a)(4 + X + X + f) - ¿a4} + j6(X + ¿a)]
_64;[4(6 + ¿ + ¿) + (¿ + ¿a){6 + ¿ + ¿ + f)]
Ao _ : :
A + A2
where, A _[4{6 + X + \){6<4^ + (6+ jufAb^-¿a64}]
"64(^ + x1)(6 + x + x1 + j) + ^(x + x){(6 + j)(6 + x + x) + 64j(4 + 6)}
"{¿b^(6 + j) -¿1a64}(6 + X + X1)" + 6^(1 + ¿1a )(6 + X + ¿1 + j)
(15)
(16)
A2 _ (¿ + ¿a)
4
Bo _
A1 + A2
(17)
1
Ajay Kumar and Ashish Sharma
PROFIT ANALYSIS OF REPAIRABLE COLD STANDBY SYSTEM Rl&A, No 2 (78) UNDER REFRESHMENTS SUBJECT TO REBOOT FACILITY_Volume 19, June, 2024
where Aj and Aj are defined above.
V = 0^2^(X + Xa )(0 + X + X)
0 a + A2
VI. Profit Analysis
Using reliability parameters, the profit (P) of the system during the time interval (0,t] is
P = T A - TBR - TV (19)
Where, T0 = 1000 (Price tag per unit uptime)
T = 500 (Cost per unit time for technician Busy) T2 = 100 (Charge per visit by the technician)
VIII. Discussion
Generally, cold standby redundancy is used to enhance the system performance and sometimes refreshment is offered to the technician to enhance his efficiency.
The system performance is calculated with reliability measures such as MTSF, availability of the system and profit values. Table 2 shows the increasing trend of MTSF with respect to refreshment rate 0, keeping the values of other parameters X=0.3, X1=0.2, |=0.4, ^=0.3, ^=0.2 are failure rate of the unit in safe mode, unsafe mode, refreshment request rate, repair rate of unit, rebooting delay rate respectively, and these are taken constantly for simplicity. When X changing from 0.3 to 0.4, X1 changing from 0.2 to 0.3, | varying from 0.4 to 0.5 then MTSF declined.
The table reveals that as the rate of repair ^ changes from 0.3 to 0.4, and reboot delay rate £, changes from 0.2 to 0.3 then MTSF enhances.
Table 2: MTSF vs. Refreshment Rate
0 X=0.3, X1=0.2 X=0.4 X1=0.3 |=0.5 <^=0.4 ^=0.3 | 1=0.4, ^=0.3 ^=0.2 a=0.8 b=0.2
0.1 1.49505 1.40182 1.38431 1.42983 1.45287 1.50588
0.2 1.52616 1.42729 1.40848 1.45878 1.48644 1.53438
0.3 1.55598 1.45179 1.4316 1.48666 1.51862 1.56687
0.4 1.58457 1.47538 1.45372 1.51351 1.54951 1.59368
0.5 1.61202 1.4981 1.47492 1.53948 1.57917 1.62598
0.6 1.63839 1.51475 1.49524 1.56438 1.60768 1.64788
0.7 1.66375 1.54115 1.51475 1.58849 1.6351 1.67466
0.8 1.68814 1..5615 1.53349 1.61178 1.6615 1.69586
0.9 1.71164 1.58118 1.5515 1.63429 1.68694 1.72568
1 1.73427 1.60021 1.56883 1.65605 1.71145 1.74655
The availability of the redundant system is also affected by the refreshment and reboot facilities. 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=0.3, X1=0.2, |=0.4, ^=0.3, ^=0.2 possess constant values. When the failure rate of a unit in safe mode changes (X=0.3 to 0.4), unsafe mode
changes (Xi=0.2 to 0.3) then the availability of system declines.
Also, when the technician request rate changes (|=0.4 to 0.6) then the system's availability declines but when the repair rate of unit changes (^=0.5 to 0.7), reboot rate of unit changes (^=0.2 to 0.3) then the availability of the system enhances.
Table 3: Availability vs. Refreshment Rate
0 X=0.3, Xx=0.2 X=0.4 Xx=0.3 |=0.5 ^=0.4 ^=0.3 | 1=0.4, ^=0.3 ^=0.2 a=0.8 b=0.2
0.1 0.11217 0.10168 0.0973 0.10666 0.1181 0.11682
0.2 0.156 0.14251 0.13534 0.14863 0.16336 0.16537
0.3 0.19261 0.17718 0.16728 0.18385 0.20078 0.20742
0.4 0.22302 0.20641 0.19401 0.21326 0.23162 024353
0.5 0.24825 0.23098 0.21639 0.23776 0.25703 0.2744
0.6 0.26922 0.25162 0.23516 0.25819 0.27804 0.30074
0.7 0.28668 0.26899 0.25094 0.27525 0.29546 0.32323
0.8 0.30128 0.28362 0.26427 0.28955 0.30998 0.34246
0.9 0.31354 0.296 0.27557 0.30157 0.32214 0.35894
1 0.32389 0.3065 0.28525 0.31172 0.33237 0.3731
It is evident from table 4 that the system uses constant parameters such that X=0.3, X1=0.2, |=0.4, ^=0.3, Ç=0.2and the trend of profit values enhanced with respect to increments in refreshment rate 0. When the failure rate of a unit X in safe mode changes from 0.3 to 0.4 and unsafe mode changes from 0.2 to 0.3 then the profit of the system decreases.
Also, when the technician request rate | changes from 0.4 to 0.5 then profit values decline but when the repair rate of unit ^ changes from 0.5 to 0.7 and reboot rate changes from ^=0.2 to 0.3 then the profit value enhances.
Table 4: Profit vs. Refreshment Rate
0 X=0.3, Xi=0.2 X=0.4 Xi=0.3 | =0.5 ^=0.4 ^=0.3
1 1=0.4, ^=0.3
^=0.2 a=0.8
b=0.2
0.1 46.91405 41.65736 39.76882 43.94127 50.09497 50.82808
0.2 66.50537 59.48843 55.46915 62.40709 70.57429 74.43605
0.3 83.49143 75.18573 68.73058 78.49941 88.13762 96.1274
0.4 98.10396 88.87597 79.89434 92.41253 103.1117 115.7624
0.5 110.635 100.7582 89.2898 104.399 115.8591 133.3744
0.6 121.3797 111.0531 97.21094 114.7191 126.7243 149.0903
0.7 130.6097 119.9755 103.9094 123.616 136.0128 163.0801
0.8 138.5628 127.7217 109.5955 131.3055 143.9851 175.5265
0.9 145.4418 134.4642 114.4424 137.9734 150.859 186.6077
1 151.4167 140.3518 118.5933 143.7773 156.8144 196.4892
IX. Conclusion
The refreshment and reboot approach plays a vital role in system configuration and its
functioning. These features enhance the capacity of the technician and performance of the system.
Tables explore the increasing trends of MTSF, availability and profit values of the system using
reboot and refreshment facilities.
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