A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM WITH CORRELATED FAILURE TIME AND REPAIR MACHINE FAILURE Текст научной статьи по специальности «Медицинские технологии»

Alka Chaudhary , Shivali Sharma and Anika Sharma RT&A, No 4 (76)

A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM Volume 18, December 2023

A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM WITH CORRELATED FAILURE TIME AND REPAIR MACHINE FAILURE

Alka Chaudhary, Shivali Sharma and Anika Sharma

Principal, KLPG College, Meerut, 250002 Department of Statistics, Meerut College, Meerut, 250004 Ch. Charan singh university, Meerut, 250002 alkachaudhary527@gmail.com , sharma14shivali@gmail.com , ash27sharma@gmail.com

Abstract

The paper deals with a system composed of two-non identical units (unit-1 and unit-2). Initially, unit-1 is operative and unit-2 is kept in cold standby. The cold standby unit can't fail in its standby mode. Each unit of the system has two possible modes: Normal (N) and total failure (F). When the unit-1 fails the cold standby (unit-2) becomes operative instantaneously with the help of a perfect and instantaneous switching device. A single repairman is always available with the system to repair a failed unit and failed RM. Unit-1 gets priority in operation and repair over unit-2. However, the RM gets priority in repair over any of the units. The RM machine is good initially and can't fail unless it becomes operative. The system failure occurs when both the units are in total failure mode. The joint distribution of failure and repair times for each unit is taken bivariate exponential distribution. Each repaired unit works as good as new. Using regenerative point technique, various important measures of system effectiveness have been obtained.

Keywords: Transition probabilities, mean sojourn time, bi-vairate exponential distribution, reliability, MTSF, availability, expected busy period of repairman, net expected profit.

1. Introduction

Two units standby system models have been investigated by a large number of authors including A. Kumar, D. Pawar and S.C. Malik [11], P. Chaudhary, A. Sharma and R.Gupta [3], P. Chaudhary and A. Sharma [2], N.Kumar and N. Nandal [12], P. Gupta and P. Vinodiya [9], R. Gupta and P.Bhardwaj [5], R. Gupta and A. Tyagi [8], N. Kumar, S.C. Malik and N. Nandal [10], P. Chaudhary and S. Masih [1], P. Chaudhary and L. Tyagi[4] by using the concepts of warm standby with common cause failure and human error, correlated failure and repairs, two types of repairmen, two priority units warm standby with preparation for repair, two unit priority standby with repair, two unit cold standby with two operating modes.

In the analysis of above system models it has been assumed that a failed unit is always repairable manually and after repair the unit becomes as good as new. There are many situations where a repair machine (RM) is needed to repair a failed unit and the RM may also fail during the

repair of a failed unit .In this case the RM is first taken up for repair and the failed unit waits for getting repair.

For example, in case of nuclear reactors, marine equipment etc. the robots are used for the repair of such type of systems. It is evident that a robot being a Machine, may fail while performing its intended task. In this case the repairman will repair the RM first and then begins the repair of the failed unit.

Keeping above fact in view, the present chapter deals with the analysis of a two non-identical units cold standby system model with constant failure and general repair rates assuming that the first unit gets priority in operation and repair both. The RM may also fail during the repair of a unit. The failure rate of RM is taken as constant and its repair rate as general.

The objective of the present paper is to provide the analysis of a two non-identical unit standby system with correlated failure time and repair machine failure. The joint distributions of failure and repair times of each unit are taken to be bivariate exponential distribution with p.d.f. of the type-

f(x,y) = a1ß1(1-r1)e"a-x-ß-yIo(^a1ß1r1xy)

; 1 = 1,2; x, y, a 1, ß 1 > 0;0 < r < 1

Where, I0 = £

k = 0

(z/2)

2k

(k!)'

is the modified Bessel function of type-I and order zero. Gupta et al. [6] and Gupta and Shivakar [7] have analyzed some of two unit redundant system models by taking the joint distribution of failure and repair as bivariate exponential having the above form of pdf.

Using regenerative point technique, the following measures of system effectiveness are obtained-

i. Transition probabilities and mean sojourn times in various states.

ii. Reliability and Mean time to system failure (MTSF).

iii. Point-wise and steady-state availabilities of the system as well as expected up time of the system during time interval (0, t).

iv. Expected busy period of repairman in the repair of unit-1 and unit-2 during time interval (0, t).

v. Net expected profit earned by the system in time interval (0, t).

2. System description and assumptions

1. The system consists of two non-identical units (unit-1 and unit-2). Initially, unit-1 is operative and unit-2 is kept in cold standby. The cold standby unit can't fail in its standby mode.

2. Each unit of the system has two possible modes: Normal (N) and total failure (F).

3. When the unit-1 fails the cold standby (unit-2) becomes operative instantaneously with the help of a perfect and instantaneous switching device.

4. A single repairman is always available with the system to repair a failed unit and failed RM.

5. Unit-1 gets priority in operation and repair over unit-2. However, the RM gets priority in repair over any of the units.

6. The RM machine is good initially and can't fail unless it becomes operative.

7. The system failure occurs when both the units are in total failure mode.

8. The joint distribution of failure and repair times for each unit is taken bivariate exponential with density function given by ,

f(x,y) = a1P1(1-r1)e-a-x-p-yIo(^a|pir1xy)

; i = 1,2; x, y, a i, P i > 0;0 < r < 1

Where, I0 = £

(z/2)

2k

0 k=s (k!)2

Each repaired unit works as good as new.

3.1. Notations:

3. Notations and states of the system

E

f(x,y)

gi (x)

ki (yXi = x)

M-) .j (•)

Pj(-) (-)

Pijix (-) .p(jkx (-)

©

Set of regenerative states.

Random variable denoting the failure and repair time for unit-

1and unit-2 respectively ;( i=1, 2)

Joint probability density function of (X;, Yi);( i=1, 2)

= aipi(1 - i;)e-a'x-ß'yI0 (2yJa^i;xy)dx ; x,y, a;,ßi > 0; 0 < r- < 1

Where, Io(^aißirxy) = £

(aißii;xy)k

k=0

(k!)2

Marginal p.d.f. of X = x; (i=1, 2)

= ai(1 - ri)e-a"(1-r)x

Conditional p.d.f. of Yi given Xi = x; (i=1, 2)

-(ß,y+a,r,x)£ (aißirixy)j

= ßie-

£ (j'r

P.d.f. of transition time from state S1 to Sj and S1 to Sj via Sk. Steady-state transition probabilities from state S1 to Sj and S1 to Sj via Sk .

Steady-state transition probabilities from state S1 to Sj and S1 to Sj via S k when it is known that the unit has worked for time x before its failure.

Symbol for Laplace Transform i.e. gjj (s) = J estq1j (t)dt Symbol for Laplace Stieltjes Transform i.e. Q(s) = J e stdQ1j (t)t

Symbol for ordinary convolution i.e.

t

A(t)©B(t)=J A(u)B(t-u)du

0

waiting time of unit-1.

3.2. Symbols for the states of the system

N^o

N2s Flr,F2r

Unit-1and Unit-2 is in N-mode and operative. Unit-2 is in N-mode and kept into cold standby. Unit-1 and unit-2 is in F-mode and under repair.

9

0

F1w, F2w

RMg

g

Unit -1 and Unit -2 is in F-mode and waiting for repair. Repair machine is good. Repair machine is operative. Repair machine is failed and under repair. Considering the above symbols in view of the assumptions stated earlier, we have the following states of the system:

RM0 RMr

Up States

S0 -

Si -

Nio,N2s

RMg

Failed States

S3 -

S2 -

S5 -

Fir,N20

RMo ,

'Fiw,N20A

S4 -

1r 2w

RM0

F F

r1w' 2w

RMr

V

N

RM

r

10' F2w

S6 -

RMr ,

N10,F2w'

RMr ,

The transition diagram of the system model along with the transition rate or transition time c.d.f. is shown in Fig.1. The epochs of the transition into state S4 from S2 and S6 from S5 are nonregenerative.

Figure 1: Correlation Model

+The limits of integration are 0 to « whenever they are not mentioned.

Alka Chaudhary , Shivali Sharma and Anika Sharma RT&A, No 4 (76) A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM_Volume 18, December 2023

4. Transition Probabilities and Mean Sojourn Times

Let X(t) be the state of the system at epoch t, then {X(t);t > 0} constitutes a continuous parametric Markov-Chain with state space E = {S0toS6} .The various measures of system effectiveness are obtained in terms of steady-state transition probabilities and mean sojourn times in various states.

, 1

P01 =1 P21 =

(9 + a2(1 -12))

P43 = J dG (t) = 1 P65 = (Q + ',-"

J (Q + a1(1 - r))

P101X =Ja2 (1 -r2)e-a2(1-r2)te-QtdK1(t|x) = K1[Q + a 2(1 - r2)|x] = k*[Q + a 2(1 - r2)|x],

P121 x =QJe^+^^dK (t | x)

= QJ e-[Q+a2(1-r2)]tdt [1 - Kj (t|x )]

- k*[Q + a2(1 - r2)|x])

P131 x =Ja2 (1 - r2 ) e-a2 (1-r )te-QtdtK1 (t|x )

= a2 (1 - r2 )J e-[Q+a (1-r2)] tdt [1 - K1 (t|x )]

= (e+^ )(1 - k*[Q + a2^ - r2)|x])

P341 x =JQe-QtdtK1 (t|x )

= JQe-Qtdt [1 - K1 (t | x)] = 1 - k* (Q|x), P351 x =J e-QtdK1 (t|x ) = K1 [Q | x ] = k* (Q|x)

P501x =Je-0tdK2 (t|x)

= K 2 [Q+a1 (1 - r1) | x ] = k*2 (Q + a1 (1 -r1 )|x)

P531 x = Ja (1 - r) e-a (1-r1 )te-QtdtK2 (t|x)

= ax (1 - % )J e-[Q+a1 (1-r1)] tdt [1 - K2 (t|x)]

^ora!^) )(1 - k2[Q+ai(1 - ri)|x])

P56|x =QJ e-[6+a-(1-r1)] ^ [1 - K2 (t|x)]

= (e^R) )(1 - k2[Q+a1(1 - 0|x])

p243)=J(1 _ e^O-rJt ) dG (t )

= 1 _-

ö+a2 (1 _ r2 )'

p64)=j(1 _ e^1^) dG (t) = 1 --

1

8 + a1 (1 _ r1)

It can be easily verified that

p01 = p45|x = 1 '

,(4) =

P12 + P13 = 1,

,(6) =1

P34 = 1

P20 | x + P25| x = 1 P50 | x + Ps2| x = 1 (1-5)

From the conditional steady state transition probabilities, the unconditional steady state transition probabilities can be obtained by using the result-

Pj =J P1j | x g(x) dx

Thus,

P10 =

P12 =

P13 = P34 =

P35 = P50 =

P53 = P56 =■

P1 (1 _ 11)

[a2(1 _ 12) + P1 (1 _ 11) + 8]

8

8 + a2 (1 _ r2)

a2 (1 _ r )

1 _-

P1 (1 _ O

[a2(1 _ 12) +P1 (1 _ O + 8]

8 + a2 (1 _ r2)

1 —

P1 (1 _ H)

[a2(1_ 12) + P1C1 _ 11) +8]

1 _-

P1 (1 _ 0 [P1C1 _ O + 8] P1 (1 _ 0 [P1(1 _ 11) + 8]

P2 (1 _ r2) [8 + P2 (1 _ r2 ) + a1 (1 _ r1)]

a1 (1 _ r1)

8 + a1 (1 _ r1)

1 —

P2 (1 _ 12)

8

8 + a1 (1 _ r1) Thus, we have

P01 = P43 =1,

P50 + P53 + P56 =1

1 _

K(1 _ O + P2(1 _ r2) + 8]

P2(1 _ r2) [a1(1 _ rj + P2(1 _ r2) + 8]

P10 + P12 + P13 =1 P21 + P23 =1

P34 + P35 =1 P65 + P643) = 1

(6-11)

1

5. Mean Sojourn Time

The mean sojourn time in state S; is defined as the expected time taken by the system in state S; before transiting into any other state. If random variable U; denotes the sojourn time in state S; then,

= j P [U; > t ]dt

The mean sojourn times in various states are as follows-

Alka Chaudhary , Shivali Sharma and Anika Sharma RT&A, No 4 (76) A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM_Volume 18, December 2023

1 1

Vo = —-t , V =-

ai(1 - ri) 0+02(1-r2)+Pi(1-ri)

1 1

w2 =-, w3 =-

e + a2(1 - r2) 0 + Pi(1-r)

v4 = i G (t) dt = 1, y5 =-1-

4 0+01(1-ii)+p2 (1 -12)

1

^6 =e—(1—)

0 + 01(1 -1) (12-18)

6. Analysis of Characteristics

6.1. Reliability and MTSF

Let Rj (t) be the probability that the system operates during (0, t) given that at t=0 system starts from S; e E . To obtain it we assume the failed states S3 and S4 as absorbing. By simple probabilistic arguments, the value of R0 (t) in terms of its Laplace Transform (L.T.) is given by

1 - q56q65 )[(1 - Q12Q21) Zo + qo1Z1 + qoxq12Z2

R0 (s ) = -

(1)

(**w ** * * \ 1 - q56Q65 )(1- Q12Q21 - QoxQxo)

We have omitted the argument's from q*j(s) and Z*(s) for brevity. Z*(s); i = 0, 1, 2, 5, 6 are the

L. T. of

Z0 (t) = e-a1(1-r1)t

Z1 (t) = J e-(0+O2 (1-12 ))tK1 (11 x) dt, Z2 (t) = J e-a2 (1-» G (t) dt

Z (t ) = e-(e+a1 (1-r1 ))t , -a(1-r )t_

Z5 (t) =e ' Z6 (t) = Jea1 (1r1 }tG(t) dt (2-6)

Taking the Inverse Laplace Transform of (1), we can get the reliability of the system when system initially starts from stateS0 . The MTSF is given by,

E (T ) fR (t) R*() (1 - P56P65 )[(1 - P12P21 )^0 +^1 + P12^2 ] (7)

E(To ) = J Ro (t) = !imR0 (s) =-j.---^-n---(7)

J (1 - P56P65 )[1 - P12P21 - P10 J

6.2. Availability Analysis

Let Aj (t) be the probability that the system is up at epoch t, when initially it starts operation from state S; e E. Using the regenerative point technique and the tools of Laplace transform, one can obtain the value of A0 (t) in terms of its Laplace transforms i.e. A0 (s) given as follows-

A0 (•) = £$ (8)

Where,

N1 (s ) = (i - Z0 + q0iZ;]+ Z^i

* * * * (6 )* qis + qi3q34q45qs2

+Z5qoi

* (4)* * * *

qisq2s + q^q34q45

J1 ^ and

/ s. (4)* (6)* *** ***** (6)* * * * (4)* *****

D1 (s) =1 - q2s qv52 - qoxqisqso- qo1qi3q2oq34q45qv52 - qotq^qW - qoxqi3q34q45q5o (9)

Alka Chaudhary , Shivali Sharma and Anika Sharma RT&A, No 4 (76) A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM_Volume 18, December 2023

Where, Z, (t), i=0,1,2,5,6 are same as given in section 6.1.

The steady-state availability of the system is given by

A0 = lim A0 (t) = lim s AO (s) (10)

ti<» si0

We observe that Di (0) = 0

Therefore, by using L. Hospital's rule the steady state availability is given by N1 (s) N,

A0 = lim-^^ = —1- (11)

0 s-0D, (s) D, V '

Where,

N1(0) = (1 - p(25)p(562)) [V 0 + Vl ] + V2 f1" P13P50 ] + V5 fl" P12P20 ]

and

D1 = (v0 + Vl ) (1 - P(245)P562) ) + V3P13 (1 - P245)P562) ) + n1 (1 - Pl3P20P562) ) + n2 (1 - Pl(P(0 ) (12)

The expected up time of the system in interval (0, t) is given by

t

^up (t ) = j A0 (U) dU

0

*

* , , A0 (s)

So that, ^ (s) = (13)

6.3. Busy Period Analysis

Let B, (t) and B2 (t) be the respective probabilities that the repairman is busy in the repair of unit-

1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair at epoch t, when initially the system starts operation from state S; e E . Using the regenerative point technique and the tools of L. T., one can obtain the values of above two probabilities in terms of their L. T. i.e. B,* (s) and Bf* (s) as follows-

B,*(s) = ^ B?*(s) = ^

Di(s) Di(s) (14-15)

Where,

** / * * \ / * * \ * * / * (4)* * \ * * / * * \ / * (4)* * \

n2 (s)=Ziq0i (1-q(6q65)(1 -q34q43)+q3(q43 (^63 + fe) + Z3q0i (1 -q(6q65)(^23 + qi3) (16)

and

/s. * * / * » (4)*\„»

N3 (s) = -q0lQ35 (q,3 + ^23 ) Z5

and D, (s) is same as defined by the expression (9) of section 6.2.

The steady state results for the above two probabilities are given by Where,

B0 = lim s B0* (s) = N2 \ D, and B2 = sim s B2* (s) = N3 \ D, (17-18)

N2 (0) = Vl (1 - P56P65 )(1 - P34 )+ P35 (P(6P64)+ P53 ) +V3 (1 - P56P65 )(Pl2P24)) + Pl3 ) (19)

N4 (0) = -P35 (P13 + PkPZ^^) V5 (20)

and D, is same as given in the expression (12) of section 6.2.

The expected busy period in repair of unit-1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair during time interval (0, t) are respectively given by-

(t ) = { B0 (u ) du

So that,

) =

Bq* (s)

and

and

^b (t ) = { b2 (u) du

^*(s ) =

Bq* (s)

(21-22)

o

o

s

s

7. Profit Function Analysis

The net expected total cost incurred in time interval (0, t) is given by

P (t) = Expected total revenue in (0, t) - Expected cost of repair in (0, t)

= K0H-Up (t)-K^b (t)-(t) (23)

Where, K0 is the revenue per- unit up time by the system during its operation. Kj and K2 are the amounts paid to the repairman per-unit of time when the system is busy in repair of unit-1 failed due first repair with priority of unit-1 and unit-2 failed due to second repair respectively. The expected total profit incurred in unit interval of time is P = K0A0 - Kxb0 - K2B2

8. Particular Case

Let, G(t) = e-M

In view of above, the changed values of transition probabilities and mean sojourn times.

J J

P21 -T,-T > P« =

^ + a2 (l - r2)' 65 ^ + al (l -r,)

p(4) = l__1__pM= l - 1

P23 1 , , /, \ ' F63 1

^+a2 (l - r2 )' ^ + al (l - r,)

1 l

^2 = 7---T, =-

X + a2(l - r2) X + al(l - rl)

9. Graphical Study of Behaviour and Conclusions

For a more clear view of the behaviour of system characteristics with respect to the various parameters involved, we plot curves for MTSF and profit function in Fig. 2 and Fig. 3 w.r.t. a1=for three different values of correlation coefficient pt=0.15, 0.45, 0.85 and two different values of repair parameter r2 =0.8, 0.9 while the other parameters are 0 = 0.9,= p2= =0.4, r, =0.7, a2= =0.8, X=0.9. It is

clearly observed from Fig. 2 that MTSF increases uniformly as the value of a1=and r2increase and it decrease with the increase in a1=. Further, to achieve MTSF at least 80 units we conclude for smooth curves that the values of a1=must be less than 0.1, 0.13 and 0.4 respectively for pt=0.15, 0.45, 0.85 when r2 =0.8. Whereas from dotted curves we conclude that the values of a1=must be less than 0.1, 0.11 and 0.22 for pt=0.15, 0.45, 0.85 when r2 =0.9.

Similarly, Fig.3 reveals the variations in profit (P) with respect to afor three different values of Pj =0.15, 0.45, 0.75 and two different values of r2 =0.01, 0.03, when the values of other parameters 0 = 0.8,=p2==0.98, r, =0.3, a2==0.6, X=0.8, K0=95, K1=250 and K2=225. Here also the same

Alka Chaudhary , Shivali Sharma and Anika Sharma RT&A, No 4 (76) A TWO NON IDENTICAL UNITS COLD STANDBY SYSTEM_Volume 18, December 2023

trends in respect of a , P1 and r2 are observed in case of MTSF. Moreover, we conclude from the smooth curves that the system is profitable only if a is less than 0.1, 0.19 and 0.59 respectively for P1 =0.15, 0.45, 0.75 when r2 =0.01. From dotted curves, we conclude that the system is profitable only if a is less than 0.1, 0.25 and 0.32 respectively for P1 =0.15, 0.45, 0.75 when r2 =0.03.

ai

Figure 2: Behaviour of MTSF with respect to a1, p1 and r2

— p1=0.15,r2=0.01 —•— p1=0.15,r2=0.03

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[5] Gupta, R. and Bhardwaj, P. (2019). A Discrete Parametric Markov-Chain Model of a two-unit cold standby system with appearance and disappearance of repairman. Reliability: Theory & Applications, 14(1(52)):13-22.

[6] Gupta, R. and kumar, K. (2008). A two unit complex system with correlated failure and repair times. PureandappliedMathekaScience LXVII,(1-2):23-34.

[7] Gupta, R. and Shivakar. (2010). Cost-Benefit analysis of a two unit parallel system with correlated failure and repair times. IAPQRTransactions, (35(2)):117-140.

[8] Gupta, R. and Tyagi, A. (2019). A Discrete Parametric Markov-Chain Model of a two-unit cold standby system with repair efficiency depending on environment. Reliability: Theory & Applications, 14(1(52)):23-33.

[9] Gupta, P. and Vinodiya, P. (2018). Analysis of reliability of a two non-identical units cold standby repairable system has two types of failure.ComputerScienceandEngineering, 8(3):343-349.

[10] Kumar, N., Malik, S.C. and Nandal, N. (2022). Stochastic analysis of a repairable system of non-identical units with priority and conditional failure of repairman. Reliability: Theory & Applications, 17(1(67)):123-133.

[11] Kumar, A., Panwar, D. and Malik, S.C. (2019). Profit analysis of a warm standby non-identical units system with single server subject to preventive maintenance. Agriculture and Statistical Science, 15(1):261-269.

[12] Kumar, N. and Nandal, N. (2020). Stochastic modeling of a system of a two non-identical units with priority for operation and repair to main unit subject to conditional failure of repairman. Statistics and Reliability Engineering.(1):114-122.