Study of Stochastic Model of a Two Unit System with Inspection and Replacement Under Multi Failure Текст научной статьи по специальности «Медицинские технологии»

Study of Stochastic Model of a Two Unit System with Inspection and Replacement Under Multi Failure

Neha Sharma, J P Singh Joorel •

Department of Statistics University of Jammu, Jammu, J&K -180006, India E mail: [email protected]

Abstract

The present paper studies a two non-identical units system model arranged in parallel with inspection and preparation time for replacement under multi failures. Initially, first unit (A) is in operative mode and other unit (B) is kept as warm standby. The first unit is subjected to two types of failures, i.e. minor failure and major failure. On failure of the first unit, it will be sent for inspection to check the type of failure i.e. whether minor or major failure. If some minor failure is found, it will be repaired and on major failure, the unit will be replaced by the new unit. However, the system will take some preparation time for replacement. Further, the standby unit may also fail during the standby mode. There is a single repairman which is always available with the system. Different measures of reliability have been obtained to study the effectiveness of the system such as transition probabilities, mean time to system failure, availability, busy period of repairman and net profit incurred and various system parameters are analysed graphically.

Keywords: inspection, preparation time, replacement, minor failure, major failure.

1. Introduction

Reliability is considered as important characteristic for the system design and plays a vital role in the planning of system expansion, operation and maintenance. Quality of supply can be improved by reliability. To obtain useful results from system reliability assessments, reasonable values of component reliability parameters need to be used. However, the required accuracy of the reliability depends on the system design, its performance and the failure phenomenon of the system components. However, the components failure rates may vary with component, time and the environmental conditions. Therefore, it is sometimes not accurate to assign identical failure rate to all components of a particular type. Each component is treated as an individual with a unique failure rate. Many authors had worked in the reliability modelling field with different failure rates and disciplines. Rander, Kumar and Tuteja [8] have discussed a two unit cold standby system with major and minor failure and preparation time in case of major failure. El-Damcese and Temraz[7] carried out the analysis for a parallel repairable system with different failure modes and Chander, Chand and Singh[2] has studies stochastic analysis of an operating system with two types of inspection subject to degradation. Further Bhatti, Chitkara and Bhardwaj [1] studied the profit analysis of two unit cold standby system with two types of failure under inspection policy and discrete distribution and Dhankhar and Malik[5] analyse the cost-benefit analysis of a system reliability models with server failure during inspection and repair while Chib, Joorel and Sharma

[3,4] have worked on MTSF and profit analysis of a two unit warm standby system with inspection and they also worked on the analysis of a two non-identical unit cold standby system with partial and total failure and priority and El-Damcese and Sharma [6] investigated reliability and availability analysis of a repairable system with two type of failure.

The present paper studies a two non-identical units system model arranged in parallel with inspection and preparation time for replacement under multi failures. Initially, first unit (A) is in operative mode and other unit (B) is kept as warm standby. The first unit is subjected to two types of failures, i.e. minor failure and major failure. On failure of the first unit, it will be sent for inspection to check the type of failure i.e. whether minor or major failure. If some minor failure is found, it will be repaired and on major failure, the unit will be replaced by the new unit. However, the system will take some preparation time for replacement. Further, the standby unit may also fail during the standby mode. There is a single repairman which is always available with the system. Different measures of reliability have been obtained to study the effectiveness of the system such as transition probabilities, mean time to system failure, availability, busy period of repairman and net profit incurred and various system parameters are analysed graphically.

2. Assumptions

1. All the times associated with different events are random variables and independent.

2. Failure time distribution of both the units is exponential but with different parameters.

3. Inspection time distribution is also exponential.

4. Repair time distribution of both the units is taken as general but with different cdfs and replacement time distribution of first unit is also general.

5. On failure of both the units, the system will break down.

6. Switch over time is negligible.

3. Notations

a inspection rate for unit A

p failure rate of unit B

ax rate of minor / major failure in unit A with probability p and q

a2 rate of completion of preparation for replacement

hx(t)/Hx (t) p.d.f and c.d.f of repair time of unit A

h2 (t)/H2 (t) p.d.f and c.d.f of replacement time of unit A

g(t)/G(t) p.d.f and c.d.f of repair time of unit B

nI (•) c.d.f of time to system failure when S^eE.

Ai(t) Pr [starting fromS^ , the system is up at time t].

Bi(t) Pr [Repairman is busy at time t\£'0 = SieE].

Vt (t) Expected number of visits by repairman in (0,t]. Mean sojourn time in state S^ eE.

Following Symbols are used to study the proposed model:

Ao/Bo unit A/B is operative

Ar/Br unit A/B is under repair

AI unit A is under inspection and

Apr/Ar unit A is under replacement or preparation for replacement

Bws unit B is in warm standby mode

Bwr unit B is waiting for repair

Different possible states of the system are described and shown by Fig.: 1

Up states Down states

$o: (Ao,Bws^ S2: (Ar,B0) S4-: (Ar,B0) S6 = (Aj,Bwr) S8: (APR,Bwr)

Si: (Ai,B0) S3: (Apr,B0) (A0,Br) S7: (Ar,Bwr) S9 (AR,Bwr)

Fig. : 1 Transition Diagram

^^ : Operative S

X: Noil-Regenerative

4. Transition Probabilities and Mean Sojourn Time

If T1, T2T3 ... denote the epochs at which the system enter any state and Xn denotes the state visited at point Tn+, i.e. just after the transition at Tn then the transient and steady state transition probabilities are defined as Q^OO = P[Xn+1 = - Tn < t|Xn = i] and Pu = lim Qn(t)

* J t^m J

respectively. The following steady state transition probabilities of the system are obtained:

P01

Po5 z Pl2 = Pl3 =

Pl6 =

a

' a+p

P

a+p ai + P

q<*i <*i+P P

<*i+P

Pii =

Pvi ai+P

PlB = ai+p

P20 = K(p)

p27 = i-hm

p725 = i-hm

P34 P3B

Pl9 = ■

"2

a2 + P P

a2 + P

P

a2 + P

P40 = K(p)

p49 = 1-h'2(fi)

PÏs = 1-K№ Pso = gXa) P56 = 1-g*(a) P67 = Pl P6B = Rl

P75 = Ps9 = P95 = 1 It may be noted that 'ZjPij = 1, for all possible values of i

(1) (2)

Further, if T denotes the sojourn time in state St then mean sojourn time is defined as the time of stay in state St before transiting to any other state and is denoted by ^ . Following are the expressions for mean sojourn time:

Vo Vl

a+p

l

^i + P

V3

<X2 + P

V4=l[1-h2(p)]

vL2 = j[1-hlm

V6

l «i

V9 = SrH2(U)dU

V7 = foHl(u)du

Vs = l[1 - al(a)]

Ve = —

a2

(33)

l

l

Neha Sharma, J P Singh Joorel

STUDY OF STOCHASTIC MODEL OF A TWO UNIT SYSTEM WITH I4RT&A' N° ?ni9 INSPECTION AND REPLACEMENT UNDER MULTI FAILURE_o ume , eptem er

5. Mean Time To System Failure

The distribution function of time to system failure is obtained by considering the failed states as absorbing state and the time taken by the system to reach in the failed state for the first time is known as time to system failure and is denoted by and ^ (t) denotes its expected value which is known as the mean time to system failure.

The following result for mean time to system failure is obtained by using Laplace transformation.

a[(a2 + P)P+paiP(9+a){1-hi(P)}(a2 + P) + qai{P+a2{1-hi(P)}}(e+a)] + P(a2 + P){1-g*(P)}(a1 + P)

(4)

6. Availability Analysis

(t) is defined as the probability that a system will be in operational service during a scheduled operating period i.e., probability that the system is up at epoch 't' given that initially it starts from state 5; without transiting to any non-regenerative state. By using simple probabilistic concepts, recurrence relations among At (t)'s are obtained and on solving those equations by using the

Laplace- transformation following results hold:

_ JV2CS)

where,

N2O) _ [M0 + ^Ml + 01*2^2 + ^1*3^3* + ^M^Kl - 956967975 - ^MOG^) +

m*r * i * / * (7)* 1 * * (9)* 1 * (8)* * . (6)* * * . (6)* *

Ms[qo5 + qoi(qi2q2s) + qi3q34q4s) + qi3q39) q95 + ql8) q^95 + ql7) q75

A>0) _ (1 - 9*6967975 - 956968989995)[1 - 90l(92292O + 91*39*4940)] - 9*o [9o5 + 9oi (q52q2(57) + ql3q3(98)q95 + q1l6)ql9q*95 + 9l76)975)]

(5)

(6)

Steady state availability of the system starting from state s0 is obtained as follows:

¿o _ lim ¿0 (») _ lim s^0 (5) _ (7)

A _

[(ai + P)(a2 + P)P + a(ai + P)(a2 + P)P+Pai(ai + P)a(a2 + P){1—h2(P)}+qaiP(a+P) + qaia2(a+P){1—h2(P)}

g*(a)a+{1-g*(a)}P[(ai + P)(a2 + P) + a{paiP(a2 + P){1-h!(P)}(a+P) + qaia2P{1-h2(P)}(a+P) + qPai(a+P)

+ (a+P)(a2 + P)P}]62 a2_

g*(a)[(a1 + P)(a2 + P)Paa282 + a(a2 + P)aPa282+pa:L(a2 + P)a{1-h*(P)}a282a+qaa:L82aP + qaa:L82a282 a{1-h1(P)}] + [{1-g*(a)}a+p][(a+P)P(ai + P)(a2 + P)a-a{p(a2 + P)h1(a) ai + qPaia2h2(a){1-g*(a)}]82a2 + /Hl(u)[apl{1-g•(a)}(al + P)(a2 + P)(a+P)-a{palh2(P)(a2 + P) + qala2Ph2(P)}+ag•(a)Pplala(a2 + P) + aqi{1-g*(a)}[(ai + P)(a2 + P)(a+P)-paaig*(a)(a2 + P) + qPaia2{1-h2(P)}+a2qiP(a2 + P)] + a2PJHi(u) [ql(al + P){1-g2(a)}(a2 + P)(a+P)-a{pal(a2 + P)g6(a)h2(P) + qala2P}+a{1-g*(a)}{qPal + Ppl(a2 + P)}]

(8)

7. Busy Period Analysis

(t) is the probability that the repairman is busy due to repair of the unit at an instant 't' given that system entered state S; at t=0. Now we will determine these probabilities. To illustrate the calculations we consider B0(t) and similar arguments may be employed for other probabilities.

0W D2(S)

N3(5) _ q01(M1* + 91*2^* + 9Íз984M4* + ql^X + ?1f Ml + ?1f q^Xl - 9l6967

q*75 - 95696l 969 995) + (^5* + 956^6* + q^l^l + 95696l9l9^9)[9c)1(9i29257) +

* * *(9) . * *(l) * . *(6) * * . *(6) * N . * -i

q13q34q45 + q13q39 q95 + qn qi9q95 + q17 q75) + qo5]

A>0) _ (1 - - 95696l 969 995)[1 - 901(912920 + 9i39°494o)] - 95*o[905 + 9°)1

STUDY OF STOCHASTIC MODEL OF A TWO UNIT SYSTEM WITH I4RT&A' N° ^mo INSPECTION AND REPLACEMENT UNDER MULTI FAILURE_° ume , eptem er

(q*i2q2(s7) + q*i3q3(98)q*95 + qî(86)q*89q*95 + 9Î76)97s)] (i°)

Therefore, busy period analysis for repairman is given by:

B0(0)=® (11)

a[{(ai + P)P(a2 + P)a2 + (a+P)(a2 + P)pa2ai{1-h1(P)}+(a+P){1-h2(P)}qaia|}+aiPa2(a+P)JH2(u)q+qiP

(a+P)(a2 + P) + qiP(ai + P)(a2 + P) J H2(u)(a+P)]g7(a)a+[a[{aia202 + a2a02 + aaiqi02 + aqi}+(ai + P) (a2 + P)(a+P)][a{paiP02(a2 + P){1-h1(P)}+qPaia2a2{1-h2(P)}+qPai + qiP(a2 + P)a2+PiP(a2 + P)a2]

g _ +(ai + P)P(a2 + P)]_

0 g*(a)[(a1 + P)(a2 + P)Paa202 + a(a2 + P)aPa202+pa1(a2 + P)a{1-h*(P)}a202a+qaa102aP+qaa192a202 a{1-hi(P)}] + [{1-g'(a)}a+p][(a+P)P(ai + P)(a2 + P)a-a{p(a2 + P)h1(a) ai + q|3aia2h7(a){1-g*(a)}]02a2 + JHi(u)[api{1-g*(a)}(ai + P)(a2 + P)(a+P)-a{paih2(P)(a2 + P) + qaia2Ph2(P)}+ag*(a)Ppiaia(a2 + P) + aqi{1-g*(a)}[(ai + P)(a2 + P)(a+P)-paaig*(a)(a2 + P) + qPaia2{1-h7(P)}+a2qiP(a2 + P)] + a2PJHi(u) [qi(ai + P){1-g2(a)}(a2 + P)(a+P)-a{pai(a2 + P)g2(a)h2(P) + qaia2P}+a{1-g*(a)}{qPai + Ppi(a2 + P)}]

(12)

8. Expected Number of Visits by Repairman

(t) is the expected number of visits by the repairman to the system to repair the failed unit, when the system initially starts from regenerative state S;. By probabilistic reasoning the recurrence relations for ^(t) are obtained and solving those relations by using Laplace transformation, we have,

70*(s) = W4(s)

D2(S)

N4(s) = [(q*oi + q*os) + q*oi(qi3q*34 + q1(86)q*89)][1 - q^q^^s - q56q*68q89q*95] + 9s*6 * * r * i * * *(7) i * * *(9) i * »(8) * . »(8) * * . »(6) * q67q75[qos + qoi(qi2q25 +qi3q34q45 + q«q39 q95 + q89q95 + q17 q75)]

A>(s) = (1 - 9*6967975 - 9569»89*9 995)[1 - 9o1(9*29*o + 9i*39»494o)] - 95o[9o5 + 9oi

(q»2q2(57) + ql3q3(98)q95 + q^^q^s + 9176)97s)] (14)

In steady state, the number of times the repairman visits the system is given by:

7o* = lims7o(s) (15)

o s^o ow o'(o) v '

[(ai+P)(a+P)(a2+P)+aaia2q+qiP(a+P)(a2+P)l+qi{i-^*(a))[P(ai+P)(a2+P)+a{(ai+P)(0i+P)—Pai

g2(«)}-q"i"2h2(P)]fl*(«)_

g*(a)[(a1 + P)(a2 + P)Paa202 + a(a2 + P)aPa202+pa1(a2 + P)a{1-h*(P)}a202a+qaa:L02aP+qaa:L02a202 a{1-hi(P)}] + [{1-g*(a)}a+p][(a+P)P(ai + P)(a2 + P)a-a{p(a2 + P)h1(a) ai + qPaia2h*(a){1-g*(a)}]02a2 + JHi(u)[api{1-g*(a)}(ai + P)(a2 + P)(a+P)-a{paih2(P)(a2 + P) + qaia2Ph2(P)}+ag*(a)Ppiaia(a2 + P) + aql{1-g*(a)}[(al + P)(a2 + P)(a+P)-paalg*(a)(a2 + P) + qPala2{1-h6(P)}+a2qlP(a2 + P)] + a2PJHl(u) [qi(ai + P){1-g2(a)}(a2 + P)(a+P)-a{pai(a2 + P)g2(a)h2(P) + qaia2P}+a{1-g*(a)}{qPai + Ppi(a2 + P)}]

(13)

vn =

(16)

9. Profit Analysis

The profit in steady state generated by proposed model may be obtained as follows:

The expected profits incurred in (0,t] = expected total revenue in (0,t] - expected total repair in (0,t]

-expected cost of visit by repairman in (0,t]

Therefore, profit analysis of the system can be written as:

?1 = Ko^o - ^o - ^2^0

where,

Ko = revenue per unit up time of the system, K1 = Cost per unit time for which the repair is busy K2 = Cost per unit visits by the repairman

The expressions for Ao, Bo and Vo are given by equations (8), (12)and (16) respectively.

10. Particular cases

As we have assumed that the repair time and replacement time distribution is general, so firstly we convert it into exponential with parameters 9, 0l and 02. We have assumed G(t) = 0e-et, Hl(t) = 9le-0it, H2(t) = 92e-02t

So under these assumptions the expressions for different transitions with their mean sojourn time, MTSF, availability and profit function are obtained as:

Transition probabilities

_ Ai _ p

P20 = a P27

p4s

p

02 + jS

Pso

0i+0 0

0 + a

P25 =

Ps6

0i+0 a

e+a

02

P40 62+^

P95 = 1

P49

02 + ^

Mean Sojourn Time

ll

V2 =

0i+jS

V4 =

02 + ^

Vs =

0 + a

V7 = ^7

Mean time to system failure

a[(0i + P)(a2 + P)(e2 + P)+paiP(e+a)(a2 + P)(e2 + P) + qai{(02 + P) + a2P}(e+a)(e2 + P)] + aP(ei + P)(a2 + P)

MTSF

_(ai + P)(62 + P)

(ai + p)(0i + p)(a2 + p)(e2 + p)(e+a)(p + a)-a{pai0i(a2 + p)(e2 + p) + qaia2e2(ei + p)(e+a)}-pe(ai + p) (9i + P)(a2 + P)(92 + P)

Availability

[(ai + P)(a2 + P)(ei + P)(02 + P) + a(ai + P)(ei + P)(02 + P)(a2 + P)P+pai(ai + P)a(e2 + P)(a+P) + qai(a+P) (0i + P)(02 + P)P+qaia2(a+P)a]e+[(ai + P)(a2 + P)(ei + P)(e2 + P) + a{paiP(a2 + P)(e2 + P)(a+P) + qaia2 P(a+P)(9i + P) + qPai(a+P)(9i + P)(92 + P) + (a+P)(9i + P)(92 + P)(a2 + P)P}]_

A =

a2992[(ai + P)(9i + P)(92 + P)(a2 + P) + a(a2 + P)(9i + P)(92 + P)92a+pai(a2 + P)(92 + P) + (92 + P)q(9i + P) aa292a+(9i + p)qaai92a292a] + [2a+p][(a+p)p(ai + p)(a2 + p)(9i + p)(92 + p)-a{p(92 + p)(a2 + p) ai + qPaia2(9i + P))]92a2 + [api(ai + P)(9i + P)(92 + P)(a2 + P)(a+P)-a{pai(a2 + P)9i(92 + P) + (92 + P)qai a2P}+aPpiaia9(a2 + P)(9i + P)(92 + P) + aqi[(ai + P)(a2 + P)(a+P)(9i + P)(92 + P)(9+a)-paai9i(a2 + P) (92 + P) + qPaia2(9i + P) + a2qiP(a2 + P)(9i + P)(92 + P)] + a2P[qi(ai + P)(a2 + P)(9i + P)(92 + P)(a+P)-a{pai 9i(a2 + P) + qaia2P(9i + P)}+a{qPai + Ppi(a2 + P)}]

P

P

l

1

1

Busy period

a[(ai + P)P(a2 + P)(9i + P)(92 + P)a2 + (a+P)(92 + P)(a2 + P)pa2ai + (9i + P)(a+P)qaia2 + ai(9i + P)(92 + P)P a2(a+P)q+qiP(a+P)(a2 + P)(92 + P) + qiP(ai + P)(9i + P)(92 + P)(a2 + P)(a+P)]9+[{aia292 + a2a92 + aaiqi 92 + aqi}(9i + P)(92 + P)(ai + P)(a2 + P)(a+P)][aa2{paiP92(a2 + P)(92 + P) + qPaia2a2(9i + P) + qPai(9i + P) _ (92 + P) + qiP(a2 + P)(9i + P)(92 + P)a2+PiP(9i + P)(92 + P)(a2 + P)a292]

Bo

a2992[(ai + P)(9i + P)(92 + P)(a2 + P) + a(a2 + P)(9i + P)(92 + P)92a+pai(a2 + P)(92 + P) + (92 + P)q(9i + P) aa292a+(9i + p)qaai92a292a] + [2a+p][(a+p)p(ai + p)(a2 + p)(9i + p)(92 + p)-a{p(92 + p)(a2 + p) ai + qPaia2(9i + P))]92a2 + [api(ai + P)(9i + P)(92 + P)(a2 + P)(a+P)-a{pai(a2 + P)9i(92 + P) + (92 + P)qai a2P}+aPpiaia9(a2 + P)(9i + P)(92 + P) + aqi[(ai + P)(a2 + P)(a+P)(9i + P)(92 + P)(9+a)-paai9i(a2 + P) (92 + P) + qPaia2(9i + P) + a2qiP(a2 + P)(9i + P)(92 + P)] + a2P[qi(ai + P)(a2 + P)(9i + P)(92 + P)(a+P)-a{pai 9i(a2 + P) + qaia2P(9i + P)}+a{qPai + Ppi(a2 + P)}]

Expected number of repairs

0[(ai + P)(a+P)(a2 + P) + aaia2q+qiP(a+P)(a2 + P)](9i + P)(92 + P) + qia[P(ai + P)(9i + P)(92 + P)(a2 + P) + a{(ai + P)(9i + P)-pai9i}-qaia292]_

vn =

0 a2992[(ai + P)(9i + P)(92 + P)(a2 + P) + a(a2 + P)(9i + P)(92 + P)92a+pai(a2 + P)(92 + P) + (92 + P)q(9i + P) aa292a+(9i + P)qaai92a292a] + [2a+p][(a+P)P(ai + P)(a2 + P)(9i + P)(92 + P)-a{p(92 + P)(a2 + P) ai + qPaia2(9i + P))]92a2 + [api(ai + P)(9i + P)(92 + P)(a2 + P)(a+P)-a{pai(a2 + P)9i(92 + P) + (92 + P)qai a2P}+aPpiaia9(a2 + P)(9i + P)(92 + P) + aqi[(ai + P)(a2 + P)(a+P)(9i + P)(92 + P)(9+a)-paai9i(a2 + P) (92 + P) + qPaia2(9i + P) + a2qiP(a2 + P)(9i + P)(92 + P)] + a2P[qi(ai + P)(a2 + P)(9i + P)(92 + P)(a+P)-a{pai 9i(a2 + P) + qaia2P(9i + P)}+a{qPai + Ppi(a2 + P)}]

11. Graphical Study of the System Model

In order to have a graphical analysis of the above discussed model, we graphed these characteristics i.e., MTSF, availability and profit function. Firstly we have obtained the values of MTSF, availability and profit function with respect to failure and repair rates using C++ language and then we have plotted those values using STATISTICA. Firstly graphs are plotted for MTSF,

Availability and Profit with respect to failure rate a1 for different values of repair rate 01,02 & 8 keeping all other parameters constant as a = 0.5, a2 = 0.25, p = 0.35, ko = 1000,k1 = 300, k2 = 200,p = 0.5, q = 0.5, p1 = 0.5, q1 = 0.5

Behaviour of MTSF w.r. t Failure rate r;, for different values of repair rates

4.0 3.9 3.8 3.7 3.6

Hh

00 3.5

H

2 3.4 3.3 3.2

3.1 3.0

! ! i _ 0t= 0.20,02 \ = 0.30, 9 = 0.02

i _ Bi = 0.30,0-, = 0.40, a = 0.06

__9i - 0.40,02 = 0.50, 9 = 0.1

h— _ 6>x = 0.50, = 0.60, B = 0.14

-

01 23456789 10 11

ai -

Fig, 2

Behaviour of availability w. r. t failure rate a! for different values of repair rates

>-

H d 0.6

rn

<

►J 0.5

<

>

< 0.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ft I _

Fig, 3

Behaviour of profit w. r. t failure rate c^ for different values of repair rates

£200

E

o

pi 150

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a! _

Fig.: 4

0.9

0.8

0.7

0.3

0.2

50

From Fig 2, 3 & 4, we have observed that MTSF, availability and profit function respectively, decreases with the increase in the failure rate of the system and these characteristics shows an increase, as we increase the repair rate of the system. Therefore, we can conclude here that the expected lifetime of the system can be increased by providing the proper repair facility to the system, as regular repair of the units improves the reliability and effectiveness of the system.

References

1. Bhatti. J, Chitkara, A and Bhardwaj, N. (2011). Profit analysis of two unit cold standby system with two types of failure under inspection policy and discrete distribution, International Journal of Scientific & Engineering Research, Vol. 2(12), pp.2229-5518.

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