Performance Measures Of A Two Non-Identical Unit System Model With Repair And Replacement Policies Текст научной статьи по специальности «Медицинские технологии»

Performance Measures Of A Two Non-Identical Unit System Model With Repair And Replacement Policies

Urvashi Gupta, Pawan Kumar

Department of Statistics University of Jammu Jammu-180006 Email: [email protected]

Abstract

The present paper deals with two non-identical units A and B, both are in operative mode. If the unit A fails then it is taken up for preparation of repair before entering into repair mode and the unit-B gives a signal for repair before going into failure mode. If the unit gets repaired then it becomes operative otherwise it is replaced by the new one. A single repairman is always available with the system to repair the failed units and the priority in repair is always given to the unit-A. The failure time distributions of both the units are taken as exponential and the repair time distributions are taken as general. Using regenerative point technique the various characteristics of the system effectiveness have been obtained such as Transition Probabilities and Mean Sojourn times, Mean time to system failure (MTSF), Availability of the system, Busy Period of repairman, Expected number of Replacement, Expected profit incurred.

Keyword: Preparation, Signal, Repair and Replacement.

1.1 INTRODUCTION

Several researchers have considered and studied numerous reliability system models having identical units. In view of their growing use in modern technology the study of reliability characteristics of different stochastic models have attracted the attention of the researchers in the field of reliability theory and system engineering. To help system designers and operational managers, various researchers including [1,2,3] in the field of reliability theory have analysed two unit system models with two types of repairs, replacement policy, signal concept etc. They obtained various economic measures of system effectiveness by using regenerative point technique. The common assumption which is taken in most of these models is that a single repairman is always available with the system to repair the failed unit and after the repair the unit becomes as good as new. But in many practical situations, it is not possible that a single repairman perform the whole process of repair particularly in case of complicated unit/machine. Goel [1] analyzed that the multi standby, multi failure mode system model with repair and replacement policy

and there are various authors who have carried out study on repair and replacement policies.

In the present paper, we study a two non-identical units system model. The units are named as A and B and are taken to be in operative mode. If the unit A fails then goes for preparation for repair before entering into repair. Unit-B while in operation gives a signal for its repair before going in to failure mode and if it gets repaired it starts its functioning in usual manner otherwise it is replaced by the new one. A single repairman is always available with the system to repair the failed units and the priority in repair is always given to the unit-A The failure time distributions of both the units are taken as exponential and the repair time distribution is taken as general. All random variable are statistically independent.

Using semi- Markov process and regenerative point technique the expressions for the following important performance measures of the system have been derived in steady state

1. Transition Probabilities and mean Sojourn times.

2. Mean time to system failure (MTSF).

3. Availability of the system.

4. Busy period of repairman.

5. Expected number of replacement of the unit.

6. Net expected profit earned by the system during the interval (0,t) and in steady state.

1.2 MODEL DESCRIPTION AND ASSUMPTIONS

1. The system comprises of two non-identical units A and B initially both are in operative mode.

2. Upon the failure of unit A, it will go for preparation for repair before taken up for repair.

3. Unit-B while in operation gives a signal for its repair before going in to failure mode and if it is not repaired in a stipulated time it is replaced by the new one.

4. A single repairman is always available with the system to repair and replace the failed units and the priority in repair is always given to the unit A over unit B

5. The failures of the units are independent and the failure time distributions of the units are taken as Exponential.

6. The repair time distributions of the units are taken as general.

1.3 NOTATIONS AND STATES OF THE SYSTEM

We define the following symbols for generating the various states of the system.

A10,B20 : Unit A and unit B are in operative mode.

A1r/ A1P : Unit A under repair/ preparation for repair.

B2sr/ B2srw : Unit B in operative mode and gives signal for repair/waiting of signal for repair.

B2r/ B2rw : Unit B under repair/waiting for repair.

B2R/B2Rw : Unit B under replacement/waiting for replacement

b) NOTATIONS:

E : Set of regenerative states = {S0, Sx, S2, S3, S4, S5, S6, S8}

E : Set of non — regenerative states = {S7, S9, S10, SX1}

ax : Failure rate of unit - A

a2 : Repair rate of unit — B

в1 : Parameter for signal

в2 : Repair rate of unit — A

в3 : Repalcement rate of unit — B

H1 : cdf of repair time of unit — B

H2 : cdf of repair time of unit — A

G1 : cdf of replacement time of unit — B

TRANSITION DIAGRAM

О : Operative State

□ : Down-State

Fig.1.1

X

: Non-Regenerative

1.4 TRANSITION PROBABLITIES

Let An denotes the state visited at epoch Tn+ just after the transition at Tn, where T1, T2 .... represents the regenerative epochs, then{An, Tn} constitute a Markov-Renewal process with state space E and

Ql/(0 = P[*n+1 = У, ^n+1 — t |^'n = i]

is the semi Markov kernel over E.

Then the transition probability matrix of the embedded Markov chain is

P = Pij = Qij(OT) = Q(ot)

We obtain the following direct steady-state transition probabilities:

Poi

= «,/

e-(ai+Pi)udu =

ai

Similarly,

p02 = P24 =

Pi

(ai + ei) ’ ai [1

P13

(ai + Pi)

P2

(«2+«1) p30 = H2(Pi)<

ai

(P2 + P1)' Hi(a2 + ai)],

P20 = H1(a2 + ai)

p25 =

ii

a2

[1-Hl(a2 + ai)]

P46 =

в2

p57

a2+P27

[l-HKai + Рз)],

P80 = G2(ai), p11,8 = 1

(a2+ai)

P50 = H2(ai + рз)

Рз r-l TT*

p58

вз+ai

p8,10 = 1 G*(a1)

ai+вз

p62 = H*(a2), p79 = p95 = p10,11

The indirect transition probability may be obtained as follows: в1Р2 f _-(^2+a2)v

[1 — H*(ai + вз)]

(4) P1P2 f

p‘6 = свТ-О)! e

P1P2

-(P2+Pi)VdU

(в2 + a2)(P2 + Pi)

Similarly,

p'4-7) = 1 +

P2a2

P1P2

(Pi — a2)(p2 + Pi) (p2 — a2)(p2 + a2)

„(6,9) ,, PiH2(a2) I a2H2(Pi)

p35 = 1 - ' + '

(7)

p49) =

(Pi-a2) (Pi-a2)'

(6)

P32) =

Pi

a2

a2+P2

It can be easily verified that

p0i + p02 = 1,

(9)

p65) = 1

(Pi-a2)

H*(a2)

[H*(a2) — H*(Pi)]

P30 + p365,9) + p362 = 1

P62 + P695 = 1,

P13 + pi97) + pi? = 1

P46 + p4? = 1, p80 + p8,10 = 1,

P20 + P24 + P25 = 1 P50 + P57 + P58 = 1

p79 = p95 = pi0,ii = pii,8 = 1

A) MEAN SOJOURN TIMES

The mean sojourn time in state Si denoted by Rj is defined as the expected time taken by the system in state Si before transiting to any other state. To obtain mean sojourn time Rj, in state Sj, we observe that as long as the system is in state Sj, there is no transition from Sj to any other state. If Tj denotes the sojourn time in state Sj then mean sojourn time Rj in state

Si is:

Rj = E[Tj] = / P(Tj > t)dt

Therefore,

i

Ro = a^ '

i

R3 = — H*(Pi),

r6 = “ H*(a2),

a2

R9 = R11 = / H2 (t)dt

_ 1

Ri = p!+p?

_ 1

R4 a2 + 02 ,

R8=-^—G*(ai)

ai

1

R2=-r—H*(ai)

ai

1

R5=e^ — H*(e3 + a1)

1

R7 = R10 = ^

1.5 ANALYSIS OF RELIABILITY

Let Tj be the random variable denoting time to system failure when system starts up from state Sj £ £j , then the reliability of the system is given by

Ri(t) = P№ > t]

To obtain Ri(t), we consider failed states as absorbing states.

The recursive relations among Ri(t) can be developed on the basis of probabilistic arguments. Taking their Laplace Transform and solving the resultant set of equations for R° (s), we get

p* („) _ Ni(s)

R°(S) = D,(s)

(1.5.1)

Where

Ni(s) = [(1 - q*24q4eq*62)(Z* + q°pZ* + q^q*^*)]

+

+

q°i (q1sq(362* + ql4^) + q°2] (z* + q24Z4 + q25z* + q25q*58Z5) q°iq146)* + q24q46 (q°iqi3q(3<2)* + q°2)] z*

^1(s) = [(1 — q24q46q62)(1 — q°1q13q3°)]

— [(q13q32 + qi6 q62) q°1 — q°2] (q2° + q5° + q25q58q8°)

Taking the Inverse Laplace Transform of (1.5.1), one gets the reliability of the system. To get MTSF, we use the well known formula E(T°) = / R°(t)dt = lim R° (s) = Ni(0)/Di(0)

s^°

where,

Ni(0) = [(1 - P24P46P62)(b° + glP°1 + R3P°lPl3)]

+

+

p°1 (Pl3P3<2) + Pl6)P62) + p°2] (b2 + P24b4 + P2Sb5 + P25P58b8)

(4) (6)

P°lPl6) + P24P46 (P°lPl3P32) + P°2

D1(0) = [(1 p24p46p62)(1 p°1p13p3°)]

(6) (4)

)]

b6

- [(Pl3P362 + P1'6)P62)P°1 - P°2] (P2° + P5° + P25P58P8°) Since, we have q*j(0) = pij and limZ* (s) = / Zi(t)dt = g

1.6 AVAILABILITY ANALYSIS

Let Ai(t) be the probability that the system is available at epoch t, when it initially starts from Si e E. Using the regenerative point technique and the tools of L.T., one can obtain the value of above probabilities in terms of their L.T. i.eA”*(s).Solving the resultant set of equations and simplifying for A° (s), we have

A3*(s) = N2(s)/D2(s)

N2(s) = q8°(1 — q57)(1 — q24q46q62)[Z° + q°1Z1] + q°1[q13q8°(1 — q57)(1

+ q8°(i - qs7)(Z2 + q24(Z4 + q46Z6)) [q°lql3q(32) + q°2]

+ {(q8°Z5 + qs8Z8) [(q695)q46 + q45) q24 + q25]} (q°lq(362) + q°2) + q°l(q8°z5 + qs8Z8)(i - q24q46q62) [q(36s9) + q1497)

(1.6.1)

q24q46q62)Z3]

(4)

+ q°lql6

q8°(1 - q57) [(Z6 + q62Z2 + q24q62Z4)

+ (q8°z5 + qs8Z8) {q^s + q62q25 + q25q45}

(1.6.2)

and

D2(s) — q80(1 — q57)(1 — q24q46q62) — {q01 [q13q80(1 — q57)(1 — q24q46q62) (l — q32^)]} — q01q13q80q32q80(1 — q57)(1 — q24q46q62) q01q80q1g7'>(1 — q57)(1 — q24q46q62)

q01q80qi6 (1 — q57)(1 — q24q46q62) — q02q80(1 — q57)(1 — q24q46q62)

(1.6.3)

The steady state availability is given by

Ao — lim Ao(t) — lim sA0(s) —

t^OT S^0 d2 (0)

As we know that, qij (t) is the pdf of the time of transition from state Si to Sj and qij (t)dt is the probability of transition from state Si to Sj during the interval (t, t + dt), thus limZrOO — /Zi(t)dt — g and q-j(s) — q-j(0) — pij , we get Therefore,

N2(0) — P80(1 - P57)(1 - P24P46P62)[q0 + P01q1] + P01[Pl3P80(1 - P57)(1 -P24P46P62)q3] + P80(1 - P57)(q2 + P24(q4 + P46q6)) [pOiPi3p3'2) + P02] + {(P80q5 +

P58q8) [(p615)P46 + P4^))P24 + P25]} (p0ip3<2) + P02) + P01(P80q5 + P58q8)(1 -

P24P46P62) [p3s9) + P14,7)] + P01P16 [P80(1 - P57) [(q6 + P62q2 + P24P62q4) +

(P80q5 + P58q8) {p695 + P62P25 + P25P4^}]]

(1.6.4)

D2(0) — p80(1 - p57)(1 - p24p46p62) - {p01 [p13p80(1 - p57)(1 - p24p46p62) (l - p32)]}

- P01P13P80P362)P80(1 - P57) - PoiP80p1^9^)(1 - P57)(1 - P24P46P62)

- p01p80p16(1 - p57)(1 - p24p46p62) - p02p80(1 - p57)(1 - p24p46p62)

The steady state probability that the system will be up in the long run is given by

Л0 — lim ^o(t) — lim sAo(s) — limN2(s) lim5-^T

t^OT s^o s^0 d2(S) s^0 s^0^2(s)

as s^ 0, D2(s) becomes zero.

Therefore, by L' Hospital's rule,A0 becomes A0 — N2(0)/D2(0) (1.6.5)

where,

D2(0) — g0{P80(1 - P57)(1 - P24P46P62)} + MP01P80(1 - P57)(1 - P24P46P62)} + q2 {P80(1 - P57)P01 [P13P362) + P1^P62] + P80(1 - P57)P02} + q3{P01P80P13(1 - P57)(1 -

P24P46P62)} + q4 {P80(1 - P57)P24 [P01P1?P62 + P01P13P3<2) + P02]} + (q5 + q7 + q8 + q9 + g10 + q11)(1) + q6 {P02P80(1 - P57)P24P46 + P01P80(1 - P57)P24P46P32)} (1.6.6)

Using the results (1.6.4) and (1.6.6) in (1.6.5), we get the expressions for A0.

The expected up (operative) time of the system during (0, t] is given by

gUp(t) — I A0 (u)du 0

So that,

A*0(s) gUp(s) — —-

1.7 BUSY PERIOD OF REPAIRMAN

Let Bi(t) be the probability that the repairman is busy in the repair of failed unit at epoch t, when the system initially starts operation from state Si e E. Developing the recursive

relations amongBi(t),s and solving the resultant set of equations and simplifying forBl(s), we have

BS(s) = Ns(s)/D2(s) (1.7.1)

where

N3(s) = [q*01(1 - Я24Я46Я62) [я1зЯ(359)* + q695* + q(197)*]

+ q0i(1 - q2o- q*24q46q*62) (q(3<2)* + q!6)*q62)

q02(1 — q20 — q24q46q62)] {q80M5 + q8,10q58Ml1 + q80q57q79M2}

+

+

{q0iq80(i — q*57)(i — q*24q*46q*62) [q^Ml + qffMi] + q0iqi4)*M;}

+ q*80(1 — q*57) [m2 + q*24 (q*46M6 + q479)9Ml)] {q0i (qi3q(362* + q146)9q*62)

+ q02}

In the long run, the expected fraction of time for which the expert server is busy in the repair of failed unit is given by

B0 = lim B0 (t) = limBl (s) = N(0) = N3 (1.7.2)

t^OT s^0 D2(0) u2

(0) = [p01(1 — P24P46P62) [pi3p365,9) + p695 + Pl49’7)] + P01(1 — P20 — P24P46P62) (p3<2') + p1t)P62) + P02(1 — P20 — P24P46P62)] {P80q5 + P8,10P58q11 + P80PS7P79q9}

N

{P01P80(1 — P57)(1 — P24P46P62) [Pl3q3 + p116)^] + P0lPS'7)q9} + P80(1 — P57) [q2 +

P24 (P46q6 + P49)q9)] {P01 (P13P32) + p1?P62) + P02}

(1.7.3)

and D2(s) is same as given by (1.6.6).

Thus using (1.7.3) and (1.6.6) in (1.7.2), we get the expression for B0.

The expected busy period of repairman during the time interval (0,t] is given by

gb(t) = f B0 (u)du 0

So that Rb(s) =

B0(s)

1.8 EXPECTED NUMBER OF REPLACEMENTS

Let Vrp(t) be the expected number of replacements by the server in (0,t] given that the system entered the regenerative state Si at t=0. Framing the relations among Vrp(t),taking L.S.T and solving for V0(s), we get

Vrp(s) = f(Sr (1.8.1)

where,

N/>(s) = Q01 [q12Q21 + Q25Q21(Q78Q89 + Q79)Q57 Q90 + Q13Q34 (Q46 Q69 + Q48 Q89 + Q49) + Q13Q37( Q78Q89 + Q79)Q90]

and D2 (s) can be obtained on replacing q^s by in 1.6.6

In steady-state per-unit of time expected number of replacement by server is given

VrP = limV0p(t) = lim VqP (s)=N^ = NZ (1.8.2)

0 t s^0 0 w D2(0) D2 V ’

Where

N4P = Р01 [P12P21 + Pl2P25(P78P89 + P79)P57P90 + Pl3P34 (P46p69 + P89p4? + p45 ) +

s

Pl3P357)(P78P89 + P79)P9o] (1.8.3)

Thus using (1.8.3) and (1.6.6) in (1.8.2), we get the expression for Vp.

1.9 PROFIT FUNCTION ANALYSIS

The net profit incurred during (0,t) is given by

P(t) = Expected total revenue in (0,t]- Expected total expenditure in (0,t]

= KoRupOO - K^boO - K2^nP(t)

Where K0 is the revenue per unit up time by the system, and K1 repair cost per unit of time in repairing the failed unit by repairman and K2 is per unit replacement cost of the failed unit.

Also,

Rup(t) = f Ao(u)du 0

So that, RUp(t) = ^7^

In the similar way Rb(t), nhP(t)can be defined.

Now the expected profit per unit of time in steady state is given by P(t)

P = lim-----= lims2P*(s)

t^OT t s^0

= KoAo - K1B0 - K2V1

1.10 CONCLUSION

To study the behavior of MTSF and profit function through graphs w.r.t various parameters, curves are plotted for these characteristics w.r.t failure parameter a1 in Fig.2.1 and Fig.2.2 respectively for three different values of repair rate p2 = (0.20,0.50,0.60) whereas other parameters are kept fixed as a2 = 0.03, p1 = 0.25, p>3 = 0.20, h1 = 0.30, h2 =

0.02, g1 = 0.03.

Fig.2.1 represents variation in MTSF for varying values of failure parameter a1 for three different values of repair rate p2 . The graph shows decrease in MTSF with the increase in failure rate and an increase with the increase in repair rate. The curves also indicates that for the same value of failure rate, MTSF is higher for higher values of repair rate .So we conclude that the repair facility has a better understanding of failure phenomenon resulting in longer lifetime of the system.

Fig.2.2 represents the variation pattern in profit function w.r.t. varying values of failure parameter a1 for three different values of repair rate p2, it is observed from graph that profit decreases with the increase in failure rate a1 and increases with increase in repair rate p2 irrespective of other parameters. The curve also indicates that for the same value of repair rate, profit is lower for higher values of failure rate and decrease in both MTSF and profit function is almost exponential.

Hence, it can be concluded that the expected life of the system can be increased by decreasing failure rate and increasing repair rate of the unit which in turn will improve the reliability and hence the effectiveness of the system.

PROFIT FUNCTION

Behavior of MTSF wrt a± for different values of/?2

«1

FIG 2.1

Behavior of Profit Function wrt a.± for different values of

FIG 2.2

References

1. Chander, S (2007) : MTSF and Profit evaluation of an electric transformer with inspection for on line repair and replacement; Journal of Indian Soc. Stat. Operation Research, Vol. 28(1-4), pp. 33-43

2. Goel, L.R and Gupta, R. (1983) : A multi standby, multi failure mode system with repair and replacement policy, Microelectron reliability, Vol. 28(5), 805-808.

3. Gupta, R., C.K. Goel and A. Tomer (2010); A two dissimilar unit parallel system with administrative delay in repair and correlated lifetimes, International Transaction in Mathematical Sciences and Computer, Vol.3, pp.103-112.

4. Kumar et.al (2012) : Reliability analysis of a two non-identical unit system with repair and replacement having correlated failures and repairs, Journal of Informatics and Mathematical Sciences, Vol.4(3), 339-350.

5. Singh, M. and Chander, S. (2005) : Stochastic analysis of reliability models of an electric transformer and generator with priority and replacement, Jr. Decision and Mathematical Sciences, Vol.10, No.1-3, pp. 79-100.