A Two Unit Parallel System with Inspection, Repair and Post Repair

A large number of papers including (Goel and Singh, 1985), (Gupta and Singh, 1985) with two types of repair have widely been studied in literature. They analysed a single unit multi-component system model with two types of repair (minor and overhaul) where the decision about the type of repair was taken by inspection. The purpose of this paper is to analyses a two-unit active (parallel) redundant system. Both the units of the system have two modes – Normal (N) and total failure (F). The system breaks down when both the units enter into F-mode whenever a unit fails it first inspected by the repairman who decides the needed type of repair (Type I or Type II) and then accordingly the repair of the failed unit is started. On completion of the repair, the unit is finally checked and re-repaired if required by the repairman. The failure and repair times of a unit are assumed to be independent and uncorrelated random variables. The distribution of time to failure of a unit is taken to be exponential. By using regenerative point technique, the following economic measures of system effectiveness are obtained – (i) Reliability of the system and MTSF. (ii) Point wise and steady-state availabilities of the system. (iii) Expected up time of the system duringt,0. (iv) Expected busy period of the repairman in inspection, in type-I repair, in type-II repair and in cost repair duringt,0. (v) Net expected profit incurred by the system during t,0 and in steady state.

0N = Unit in N (normal) mode and operative. 1F = Unit in F (Failure) mode and under inspection. 1rF = Unit in F-mode and under repair of type-I. 2rF = Unit in F-mode and under repair of type-II. PrF = Unit in F-mode and under post repair. 1WF = Unit in F-mode and waiting for inspection.  = Constant failure rate of an operating unit. P = Probability that a failed unit needs type I repair after inspection. q = Probability that a failed unit needs type II repair after inspection.

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.2.1,gg = p.d.f. of time to repair of type I and type II. .2.1,GG = C.d.f. of time to repair of type I and type II respectively. ..,Kk = p.d.f. and c.d.f. of inspection time of a failed unit. ..,Hh = p.d.f. and c.d.f. of post repair time of a repaired unit.

The transition diagram along with the possible transitions between the states is shown in fig. 1. The epochs of the entrance from states 1s to 24,ss to 5s 3s to 6s and 7s to 8s are non-regenrative.

The transition probability matrix (t.p.m.) is given by QQpPijij If the system transits from 0s to 1s then

eduetQ22

00112

udKePtQut

012

udKeqtQut

013

duuKetQut

014

1027

1025

2037

2036

045

tKqtQ46

11058

22068

uHdetQut

070

duuHetQut

078

081

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ijm as the mean sojourn time by the system in state is. dttqttQtmijijij

2

tdKetpmt12 tdKetqmt13 dttKetmt14 dttGetmt125

tGdetmt127

tGdetmt237

dttGetmt236 PtKdtpm45 qm46 158nm 268nm tdHetmt70 dttHetmt78 mm81 tKdetpmt1415 tKdetqmt1416

tGdetmt15281

tGdetmt26381

tHdetmt1871 Availability analysis tAAt00lim

*00lim



sNss2

20lim

Reliability and MISF.



1

1*0

*7*37*13*27*12*01*3*13*01*2*12*01*1*0*01zqqqqqzqqzqqzqzsN *70*37*13*27*12*0111qqqqqqsD

*

00lim



0

0

1 1

N

.

Let tBi1 be the probability that the repairman is busy in inspection of a failed unit at time t. tBctqtB110110)( tBctqtBctqtBctqtBctqtKtB16416154151313121211 tBctqtBctqtB18528172712 tBctqtBctqtB18638173713 tBctqtB185815

tBctqtBctqtB11871107017 tBctqtB118118

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tP = expected total revenue in t,0 – expected total

tKtKtKtKtKpbbbbup42312110 0K is the revenue per unit up time. 1K is the cost of inspection per-unit of time. 2K is the cost of type-I repair per unit of time. 3K is the cost of type-II repair per-unit of time and 4K is the per-unit of time cost of post repair.

up00

b100

1

b100

1

b200

2

duuBtPtPb00

Then

PBKBKBKBKAKP0420310210100

All the repair time distribution are also negative exponential. tetK1 tetG111 tetG212 tetH1 So that





ssK~



1

11~

 



2

22~

 





ssH~

Then

 

PP12 



qP13

 

1

127



125



2

237



236



 

 



 

PP415



 

qP416

11 121



231



111



221



17 

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In fig. 2, we observe that MTSF decreases uniformly as  increases. Moreover, it increases with the increase in . The similar trends w.r.t.  and  are observed for the case of profit in fig. 3 but here the trends are almost linear.

Chung, Who Kee (1990). ‘A reliability analysis of a K-out of – N:G redundant system with common – cause failure and critical human error.’ Microeletorn – Reliab, 30, p. 237. Goel, L. R. R. Gupta and S. K. Singh (1985). ‘Profit analysis of a cold standby system with two repair distribution’, Microelectron. Reliab, 25, pp. 467-475. Goel, L.R. Rakesh Gupta and S.K. Singh (1985). ‘Cost analysis of a two-unit cold standby system with two types to operation and repair’, Microelectron. Reliab, 25(1), pp. 71-75. Mittal, S. K. & Surbhi Gupta et. al. (2006). Analytical Behaviour of a parallel redundant complex system involving Environmental Failure under had of line Repair Discipline. Acta Cinecia India. Vol XXX, II M, No I. Wang K. H. (2004). A survey of Maintenance Polices of Deteriorating System, Euro. J. Res. Vol 139 (3).

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