Analysis of Triple-Unit System with Operational Priority

Jyotishree Ghosh1, D. Pawar1*, S.C. Malik2

department of Statistics, Amity Institute of Applied Sciences, Amity University, Noida - 201313, INDIA 2Department of Statistics, Maharshi Dayanand University, Rohtak - 124001, INDIA

ijyotishreeg@gmail.com 1*dpanwar75@yahoo.com 2sc malik@rediffmail.com

Abstract

Reliability of three non-identical unit system is analyzed for various measures. Initially, main unit is operational, one is warm standby and other is cold standby. Single repair facility is present with the system. Operational priority is given to main unit over standby units. Failure times of all the components are exponentially distributed whereas repair time follows Weibull distribution. All the random variables are statistically independent. Semi-Markov process and regenerative point technique are used to analyze mean time to system failure, availability, busy period and expected number of visits by the server. System model's profit is analyzed for arbitrary values and are shown graphically.

Keywords: Reliability, Non-Identical, Standby, Priority, Semi-Markov, Regenerative Point.

I. Introduction

Unwavering property is the greatest amount of wanted trait of a system. One and all wishes to rehearse exceptionally robust systems in everyday life. Understanding high demand of highly reliable systems, numerous researchers studied various systems under various set of assumptions. Osaki and Asakura [1] discussed two-unit standby redundant system with repair and preventive maintenance, Murari and Goyal [2] studied a system model with three types of repair facilities, Gopalan and Nagarwalla [3] analyzed two-identical-unit cold standby system with repair and preventive maintenance, Dhillon and Yang [4] and Dhillon [5] done reliability and availability analysis of standby systems with common-cause failures and human errors. Goel et al. [6] discussed two-unit cold standby system with preventive maintenance and replacement of the duplicate unit, Kadyan et al. [7] stochastically analyzed non-identical units reliability models with priority and different failure modes, Kishan and Jain [8] studied two non-identical unit standby system with repair, inspection and post-repair under classical and Bayesian viewpoints, Kumar and Saini [9] analyzed single-unit system with preventive maintenance and Weibull distribution for failure and repair activities, Malik and Upma [10] analyzed profit of non-identical units system under preventive maintenance and replacement, Kumar et al. [11][12] analyzed warm standby non-identical units system with single server subject to priority and without priority for operation, Rathee et al. [13] studied two-unit parallel system subject to priority of repair of units over replacement. Kumar et al.

Jyotishree Ghosh, D. Pawar, S.C. Malik RT&A, No 3 (69)

ANALYSIS OF TRIPLE-UNIT SYSTEM WITH OPERATIONAL PRIORITY Volume 17, September 2022 [14][15] analyzed profit of a warm standby non-identical unit system with single server performing in normal/ abnormal environment, also analyzed profit of the system with preventive maintenance, Ashok et al. [16] performed reliability analysis of a redundant system with 'FCFS' repair policy subject to weather conditions, Jain et al. [17] studied the reliability of a 1-out-of-2 system with standby and delayed service. Jain et al. [18] analyzed profit of a 1-out of 2 unit system with a standby unit and arrival time of server.

Every time, it is not feasible to meet the expenses of an identical unit in spare. Therefore, to keep the system operational, non-identical units might be taken as warm/ cold standby. When dealing with highly sensitive server systems in I.T. sector to keep data, we cannot rely on the systems of single unit or single unit in standby. In such cases, there is a need to keep two or more units in standby, therefore, priority is given to non-identical units in standby. Here, we developed and analyzed a reliability model of triple unit system.

II. System Assumptions

• The system comprises of three non-identical units.

• Initially, main unit (M) is operational, one unit (Ux) is warm standby and other unit (U2) is cold standby.

• Single repair facility is present with the system.

• Operational priority is given to the main unit over the standby units.

• All unit works as new after repair.

• Failure times of the components are exponentially distributed whereas repair time follows Weibull distribution.

III. Method

Expressions for numerous reliability measures including mean time to system failure (MTSF), availability, busy period and expected number of visits by the server are evaluated using semi-Markov process and RPT. Profit of the system is analyzed for arbitrary values and represented graphically.

Mo,U1o,U2o U1ws/U2cs Mur/MUR U1ur/U1UR U1wr/U1WR U2ur/U2UR U2wr/U2WR Ä,Ä1,Ä2 f(t),r(t),r1(t) F(t), R(t), R1OO qij(t)/Qij(t)

Mi(t) Wi(t)

IV. Notations and Transition Diagram

Unit M, U1; U2 are operative.

U1 is warm standby/ U2 is cold standby.

M is under repair/ continuous repair.

U1 is under repair/ continuous repair.

U1 is waiting/ continuously waiting for repair.

U2 is under repair/ continuous repair.

U2 is waiting/ continuously waiting for repair.

Failure rate of units M, U1, U2 respectively.

p.d.f. of repair time of units M, U1, U2 respectively.

c.d.f. of repair time of units M, U1, U2 respectively.

p.d.f./ c.d.f. of earliest passing time from regenerative state i to j without passing through any other regenerative state in (0, t].

Probability that the system is initially up in regenerative state i at time t. Probability that the server is busy in regenerative state i at time t,.

mij

®/©/©n */**

K0 Ki K2

Contribution to mean sojourn time (/¿¿) in regenerative state i when system

transits directly to state j. = and mt/ = J tdQLj (t) = —q*j (0).

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Symbol of Stieltjes/ Laplace/ Laplace 'n' times convolution.

Symbol of Laplace/ Laplace-Stieltjes transformation.

Fixed revenue/ unit operational time of the system.

Fixed cost/ unit busy period of the server.

Fixed cost/ visit by the server.

Profit of the system.

The possible transition states are exhibited in figure 1.

Figure 1: State Transition Diagram Up-state • Regenerative Point | | Doum-state

V. Analysis of Reliability Measures a) Transition Probabilities and Mean Sojourn Time

Simple probabilistic considerations for non-zero elements generate the expressions as

r OO

Pij = Qij(°°) = l Rtj (t)dt Jo

We have p01 =j^-,p03 = p10 = f*^), p12 = l~ft(A1),p23 = f*{X2), p25 = 1 - f*(A2), p30

Jyotishree Ghosh, D. Pawar, S.C. Malik RT&A, No 3 (69)

ANALYSIS OF TRIPLE-UNIT SYSTEM WITH OPERATIONAL PRIORITY Volume 17, September 2022 r'W, p34 = 1 " r*{X), p43 = f*(A2), p45 = 1 - f*(A2), p56 = p86 = p116 = 1, p67 = r*(A), p68 = 1 -

r*(T), p70 = r^A + AO, p79 = [1 " ^(A + AJ], p7,10 = [1 " (A + K), P97 = r*(A), P98 = 1 "

r* (A), p10,7 = /*(Ai), Pio.n = 1 " r^i) It can be easy to verify that

Poi + Pos = Pio + P12 = P23 + P25 = Pso + P34 = P43 + P45 = Pse = P67 + Pes = P70 + P79 + P7,10 = P86 = P97 + P98 = Pl0,7 + PlO.ll = Pll,6 = 1

Now, iii in the state St are |it = X, mt/ and mt/ = J tdQij (t) = -q*j (0)

to = ik= £[1 " to = I [! - -"4 = ^ [1 - /*(A2)], Ai7 = J^ [1 - ri*(A + Ai9 =

7[l-r*(A)],Ai10=f [!-/'№)]

b) Reliability and MTSF

Let cdf of earliest passage time from regenerative state i to a failed state is </>j(£)- The recursive

relations for (pi (t) are

Фо(0 = ÇoiCO© 4>i(t) + <2оз(0©Фз(0

ФЛО = QioCO © ФоСО + <21з.2(0©ФЗ(0 + Ç15.2CO

Фз(0 = @зо(0 © Фо(0 + <2з4(0©Ф4(0

Ф4(0 = Ç43CO© Фз(0 + Ç45CO •••(!)

Taking LST of (1) and solving for 4>**(s), we have

The reliability of the system can be obtained by taking inverse LT of (2). The MTSF is given by lims^0R*O). Thus,

MTSF = where

Do

N0 = (Aio + AiiPoiXl " P43P34) + + Ai4P34)(PoiPi3.2 + Роз) and D0 = (1 - р34Р4з)(1 " PoiPio) " P30CP01P13.2 +P03)

c) Steady State Availability

Let Ai (t) be the probability of the system to be operational at time T provided that the system arrived at regenerative state i at t = 0. The recursive relations for A^t) are

A0(t) = M0(t) + <?0i(0©^i(0 + q03(0©^3(0

¿i(0 = Mi(0 + <7io(0©^o(0 + q132(t)©A3(t) + fe17.2f5f6(t) + ^17.2,5(6,8)" CO] ©^7 (0

A3(t) = W3(0 + q30(t)©Mt) + q34(t)©A4(t)

A4(t) = M4(t) + q43(t)©A3(t) + [<747.5,6(0 + <747.5(6,8)" CO] ©¿7(0

i47(t) = M7(t) + q70(t)©A0(t) + <?79(0©^(0 + q7,10(t)©A10(t)

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AgÇt) = M9(t) + [q„(t) + qgH6,8)n(t)]©A7(t)

A10 (0 = M10(t) + [q10f7(t) + <7i0,7.ii,6(0 + qio,7.ii,(6,8)" (0]©^7(0 ••• (3)

where, M0(t) = e"w+/li)£, M^t) = e"Ai£F(t), M3(t) = e"A£F(t), M4(t) = e'^FCt), M7(t) = e-(A+A1)tfii(t); M9(t) = e"A£fl(t), M10(t) = e~Alt¥(T)

Taking LT of (3) and solving for i4*(s). The steady state availability is given by

^(00) = lims/T(0 = A = —, where

Nt = [p70{(M0 + MiPoiXl - P34P43) + (M3 + M4p34)(p03 + P01P13.2)} + D0(M7 + M9p79 + M10p7,10)], £>i = [P7o{(i"o + i"iPoi)(l - P34P43) + (("3 + ("4P34XP03 + P01P13.2)} + D0([i7 + [iqp7q + AiioP7,io)] and £>0 is already specified.

d) Busy Period Analysis

Let Bj(t) be the probability that the server is employed in restoring the unit at time T given that the system arrived at regenerative state i at t = 0. The recursive relations for Bi (t) are Bo(0 = <7oi(0©Bi(0 + <7оз(0©Вз(0

Bi(0 = WiCO + q10(t)©B0(t) + q132(t)©B3(t) + fa17.2f5f6(t) + i17,2,5(6,8)"(t)]©B7(t)

B3(t) = W3(t) + q30(t)©B0(t) + q34(t)©B4(t)

B4(0 = W4(t) + q43(t)©B3(t) + [<747.5,б(0 + <747.5(6,8)40]©B7(0

B7(t) = M/7(t) + q70(t)©B0(t) + q79(t)©B9(0 + q7,io(0©Bio(0

B9(t) = M/9(t) + [q97(t) + <797.C6,8)"(0]©B6(0

Bio (0 = ^lo (0 + [9ю,7 (0 + <?io,7.n,6 (0 + ^10,7.11,(6,8)" (0] ©B6 CO • • • (4)

where, W^t) = W3(t) = е~иН€), W4(t) = e"A2£F(t), W7(t) = е-(я+я1)£д1^); M/9(t) =

е"Я£Д(7), M/10(t) = e~Alt¥(T)

Taking LT of (4) and solving for B*(s). The busy period of the server can be obtained as

B(00) = lims£*(Y) = В = —, where

N2 = [P7o{WPoi)(l " Р34Р43) + (W3 + м/4р34)(р03 + P01P13.2)} + + м/9р79 + М/10р7Д0)] and D0,

D, are previously specified.

e) Expected Number of Visits by the Server

Let expected number of visits by the server in (0, t] is JVj(t), given that the system arrived at the regenerative state i at t = 0. The recursive relations for N^t) are ВД = <2oi(0©{i + ВД} + <2оз(0© {1 + N3(t)}

Wl(0 = <2lo(0©W0(0 + <2l3,(0©JV3(0 + [<2l7.2,5,6(0 + <2l7.2,5(6,8)40]©W7(t)

N3(t) = Q30(t)(s)N0(t) + Q34(t)®N4(t)

W4(0 = 043 (0 ©W3(0 + [047.5,6 (0 + 047.5(6,8)40] © N7 (t)

W7(0 = 07o(0©Wo(0 + <?79(0©ВД + 07,io(0©w10(t)

97 CO + 097.(6,8)" CO] ©Л?б(0 Wlo(0 = [0ю,7(0 + Oio,7.11,6(0 + 010,7.11,(6,8)40] ©tf6(0 ••• (5)

Taking LT of (5) and solving for N**(s). The expected number of visits by the server can be obtained as

N(oo) = \imsN"(s) = N = where N3 = p70( 1 - P34P43) and D1 is already specified.

VI. Profit Analysis

In steady state, system model's profit can be evaluated as P = K0A — Kj В — K2N

VII. Particular Case

Let f{t) = arjtr,~1e~atV, r(t) = a^t^e-"^, r±(t) = a^t^e-«^ ... (6)

are pdfs of Weibull distribution for repair time of units M, U, and U2 respectively. Where a, a1 and a2 are different scale parameters and rj is shape parameter.

On taking rj = 1 in (6), Weibull distribution becomes exponential distribution. The transition probabilities p01and p03 remains same whereas the remaining are

Vw = ~±'Vl2 =^Тя7'Р23 =ТТГ2'Рз° =^ТЯ7'Рз4 = ^fvP43 =~2'PiS =^T2'Ps6 =

P86 = Pll,6 = P™ = a2+X+X1 ' P79 = a2+X+X1 ' P?0 = a2+X+Xt ' P97 = ^TI' =

'Pl0-7 = 'Pl0-11 =

l

' I+I7

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- A* i:

1

Then iii are fi0

i

(a+A^V

„ , , r(l+i) r(l+i) r(l+i) r(l+i) Similarly, W1 =-W3 =-W4 =-W7 =-

(ii+Ai)1' (iii+Ai)1' (a+A2y> " '

1 r(l+i) r(l+i) r'

(a+A^V

i) For ?7 = 0.5, the

__(1+|) _ r(i+|) r(i+|) _ r(i^)

■ > 1^3 ~ lift4 ~ 1> №7 ~ 1> №9 — 1> 1^10 -

(iii+Ai)1' (ii+A2)'i (A+A1+ii2)'' (/l+ai)1'

and M(

r(l+i) 2L_ m =_

1 ; '"'4 — 1

r(l+i)

-,M/9 =-\

7) r(l+i)

'-T,w10 =—h

h=-t; ,M3=-■l-t,M4=-^

(a+Xtyi (iii+Ai)1' (ii+A2)'i

the repair time distribution reduces

As

2

= -—,Wg = -±

(A+A1+a2)'i (/l+ai)1' (a+zli)''

r(l+i) r(l+i) r(l+i)

:,M7=-=-\,M10 =-2t

- -- -= -~>'V' 10 = --

(ii+A2)'i (A+A1+a2)'i (/l+ai)1' («+¿1)''

reduces to /(t) = r(t) =

—¡=e~~

^ J = ui " 3 a result iii changes to ¿¿0

2 _ 2

_ 2 _ 2 _ 2 _ 2 _

-171?^ ~ (a+AO2'^3 ~ («i+Ai)2'^4 ~ C«+A2)2 ~ a+l1+a2)i ' to ~

' (iH-A^

Mn = ——, M, = —, M, =--—r, M4 = —, M7 =---r, Mq=-

0 A+At ' 1 (a+At)2 ' J (at+At)2 ' 4 (a+A2)2 ' 7 (A+Ai+ii;,)2 ' 9 ( ' (a+Ai)2

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and W, = —W3 = —W4 = —W7 =---r, Wg = —^—r, W„ = —

1 (a+Ai)2 ' J (iii+Ai)2 4 (a+A2)2 ' 7 (A+A1+a2)2 ' 9 (A+iii)2 10 (a+Ai)2

ii) For 7} = 1, the repair time distribution reduces to exponentials having pdf /(t) = ae~at, a1e~CClt, r±(t) = a2e~a2t

ii I

z 2

a^'too

Similarly,

2

M10 =

, r(t) =

As

_ i

/■¿10 —

changes to ^ = ^, ^ =, № = = = , ^ =

M° = 171? Ml = Ms = ^

_ i

a result ¡ii

i

a+A

Similarly, and w1 = -^—,W3 = , M/4 = —--

1 a+A1 A «l+Ai * a+A

: —— , M4 = —, M7 = ---, Mq = —— ,Mi n = -—

a1+A1 ' 4 a+A2 ' 7 A+At+a2 ' 9 A+ai ' 10 a+A1

—-r-, W7 = —--,Wg = -7——, W10 = -i-

a+A2 ' A+At+a2 9 A+at lu a+A1

VIII. Graphical Presentation

600000 500000 400000 T 300000

Mh

CD

b 200000

■A=0.04 Al=0.03 A2=0.001 «1=0.3 «2=0.2 -A=0.06, Al=0.03, A2=0.001, «1=0.3, «2=0.2 -A=0.04, Al=0.04, A2=0.001, «1=0.3, «2=0.2 -A=0.04, Al=0.03, A2=0.002, «1=0.3, «2=0.2 -A=0.04, Al=0.03, A2=0.001, «1=0.5, «2=0.2 A=0.04, Al=0.03, A2=0.001, «1=0.3, «2=0.4

0,4

0,5 0,6 0,7 0,8 0,9 1

1,1

1,2

1,3

Repair Rate (a) —> Figure 2: MTSF vs Repair rate (rj = 0.5)

Jyotishree Ghosh, D. Pawar, S.C. Malik RT&A, No 3 (69)

ANALYSIS OF TRIPLE-UNIT SYSTEM WITH OPERATIONAL PRIORITY Volume 17, September 2022 I A=0.04 Al=0.03 A2=0.001 «1=0.3 «2=0.2 —•—A=0.06, Al=0.03, A2=0.001, «1=0.3, «2=0.2 5qqqqq _*_A=0.04, Al=0.04, A2=0.001, «1=0.3, «2=0.2

—A=0.04, Al=0.03, A2=0.002, «1=0.3, «2=0.2

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0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3

Repair Rate (a) —>

Figure 3: MTSF vs Repair rate (r\ = 1)

•A=0.04 Al=0.03 A2=0.001 «1=0.3 «2=0.2 A=0.04, Al=0.04, A2=0.001, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.001, «1=0.5, «2=0.2

A=0.06, Al=0.03, A2=0.001, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.002, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.001, «1=0.3, «2=0.4

T

.■S 4750

MH

O

£ 4700

0,4

0,5

0,6

0,7 0,8 0,9 Repair Rate (a) —>

Figure 4: Profit vs Repair rate (rj = 0.5)

1

1,1

1,2

1,3

o

SH

Ph

■A=0.04 Al=0.03 A2=0.001 «1=0.3 «2=0.2 -A=0.04, Al=0.04, A2=0.001, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.001, «1=0.5, «2=0.2

A=0.06, Al=0.03, A2=0.001, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.002, «1=0.3, «2=0.2 A=0.04, Al=0.03, A2=0.001, «1=0.3, «2=0.4

0,4 0,5 0,6 0,7 0,8 0,9

Repair Rate (a) —>

Figure 5: Profit Vs Repair Rate (rj = 1)

1,1

1,2

1,3

IX. Conclusion

Figure 2 clearly indicate that for ^ = 0.5, MTSF is increasing with increasing repair rate of main unit (a) while figure 3 indicate comparatively less increment in MTSF for increasing a when ^ = 1. Therefore, we conclude that MTSF increases with increasing a. Figure 4 shows that for q = 0.5, system model's profit is increasing with increasing a whereas figure 5 shows relatively higher increase in profit for increasing a and a1 (restoration rate of warm standby unit) when q = 1. Hence, profit of the system increases with increasing a and a1 for constant q.

Present study concludes that the increasing restoration rate of the main unit increases MTSF whereas to make the system more profitable we should increase repair rate of both main as well as warm standby units.

References

[1] Osaki, Shunji and Asakura, Tatsuyuki (1970). A two-unit standby redundant system with repair and preventive maintenance. Journal of Applied Probability, 7(3):641-648.

[2] Murari, K. and Goyal, V. (1984). Comparison of two-unit cold standby reliability models with three types of repair facilities. Microelectronics Reliability, 24(1):35-49.

[3] Gopalan, M. N. and Nagarwalla, H. E. (1985). Cost-benefit analysis of a one-server two-unit cold standby system with repair and preventive maintenance. Microelectronics Reliability, 25(2):267-269.

[4] Dhillon, B.S. and Yang, N. (1992). Reliability and availability analysis of warm standby systems with common-cause failures and human errors. Microelectronics Reliability, 32(4):561-575.

[5] Dhillon, B.S. (1993). Reliability and availability analysis of a system with standby and common cause failures. Microelectronics Reliability, 33(9):1343-1349.

[6] Goel, L.R.; Mumtaz, S.Z. and Gupta, Rakesh (1996). A two-unit duplicating standby system with correlated failure-repair/ replacement times. Microelectronics Reliability, 36(4):517-523.

*i*Не можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

[7] Kadyan, M.S.; Chander, S. and Grewal, A.S. (2004). Stochastic analysis of non-identical units reliability models with priority and different modes of failure. Journal of Decision and Mathematical Sciences, 9(1-3):59-82.

[8] Kishan, Ram and Jain, Divya (2012). A two non-identical unit standby system model with repair, inspection and post-repair under classical and Bayesian viewpoints. Journal of Reliability and Statistical Studies, 5(2):85-103.

[9] Kumar, A. and Saini, M. (2014). Cost-benefit analysis of a single-unit system with preventive maintenance and Weibull distribution for failure and repair activities. Journal of Applied Mathematics, Statistics and Informatics, 10(2):5-19.

[10] Malik, S.C. and Upma (2016). Cost-benefit analysis of a system of non-identical units under preventive maintenance and replacement. Journal of Reliability and Statistical Studies, 9(2):17-27.

[11] Kumar, Ashok; Pawar, Dheeraj and Malik, S.C. (2018a). Profit analysis of a warm standby non-identical units system with single server subject to priority. International Journal on Future Revolution in Computer Sciences & Communication Engineering, 4(10):108-112.

[12] Kumar, Ashok; Pawar, Dheeraj and Malik, S.C. (2018b). Economic analysis of a warm standby system with single server. International Journal of Mathematics and Statistics Invention, 6(5):1-6.

[13] Rathee, Reetu; Pawar, D. and Malik, S.C. (2018). Reliability modelling and analysis of a parallel unit system with priority to repair over replacement subject to maximum operation and repair times. International Journal of Trend in Scientific Research and Development, 2(5):350-358.

[14] Kumar, Ashok; Pawar, Dheeraj and Malik, S.C. (2019a). Profit analysis of a warm standby non-identical unit system with single server performing in normal/abnormal environment. Life Cycle Reliability and Safety Engineering, 8(3):219-226.

[15] Kumar, Ashok; Pawar, Dheeraj and Malik, S.C. (2019b). Profit analysis of a warm standby non-identical unit system with single server subject to preventive maintenance. International Journal of Agriculture and Statistical Sciences, 15(1):261-269.

[16] Kumar, Ashok; Pawar, D. and Malik, S.C. (2020). Reliability analysis of a redundant system with 'FCFS' repair policy subject to weather conditions. International Journal of Advanced Science and Technology, 29(3):7568-7578.

[17] Jain, Pooja; Pawar, D. and Malik, S.C. (2020). Reliability measures of a 1-out-of-2 system with standby and delayed service. International Journal of Mechanical and Production Engineering Research and Development, 10(3):12725-12732.

[18] Jain, Pooja; Pawar, D. and Malik, S.C. (2022). Profit analysis of a 1-out of 2 unit system with a standby unit and arrival time of server. International Journal of Agricultural and Statistical Sciences, 18(1):29-33.