Reliability Engineering and System Safety 44 (1994) 119-124 © 1994 Elsevier Science Limited

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Analysis of a two-unit standby system with fixed allowed down time and truncated

exponential lifetime distributions

Rakesh Gupta, Ritu Goel & Alka Chaudhary Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut-250005, India

(Received 26 April 1993; accepted 8 September 1993)

This paper deals with the profit analysis of a single-server two-unit (priority and ordinary) cold standby system model. The priority (p) unit has three modes--normal (N), partial failure (P) and total failure (F), whereas the ordinary (o) unit has only two modes--normal (N) and total failure (F). The system remains down for some fixed time To if during the repair of the o-unit, the p-unit fails totally. The distributions of time to failure for the p- and o-units from the N-mode are taken as truncated exponentials with different parameters, while the other time distributions are negative exponentials. By identifying the system at suitable regenerative epochs, the profit functions under two different policies are obtained.

NOTATION AND STATES OF THE SYSTEM

E Eo q,,(.) Qq(.)J

Pq

T,,

Set of regenerative states -So to $5 State of the system at t = 0

p.d.f, and c.d.f, of transition time from regenerative state Si to Sj Transition probability from regenerative state Si to Sj = Qq(~) Mean sojourn time in state Si

rl i~J, #2

© , ®

tr~ exp - [aq(t - a)] p.d.f, of time to failure of p-unit in N-mode (t -> a)

or 2 Failure rate of p-unit in N-mode fl exp[-f l ( t - b)] p.d.f, of time to failure of o-unit in

N-mode (t - b) Repair rate of o-unit in F-mode Repair rates of p-unit in P- and F-modes, respectively Symbols for Laplace and Laplace-Stieljes transforms Symbols for ordinary and Stieljes convolutions, i.e.

£ A( t ) (~B( t )= A ( u ) B ( t - u ) d u

£ A( t )@B( t )= d A ( u ) B ( t - u )

119

For the states of the system we define following symbols: D System in down state Fw, Fr Unit in F-mode and waiting for repair,

under repair No, Ns Unit in N-mode and operative, standby Po, Pr Unit in P-mode and operative, under

repair Using the above symbols, the possible states of the

system and the transitions between them, along with failure/repair rates are shown in Fig. 1. Notations other than those above may be seen from Ref. 1.

INTRODUCTION

Various authors (e.g. Refs 1 and 2) have analysed two-unit priority standby system models, assuming that the lifetimes of the units follow negative exponential distribution having the density of the form:

f( t) = Oe-°t; t>-O (1)

This distribution has been used very frequently in the field of reliability theory due to its 'lack of memory' property. This property makes the behaviour of the system independent from its past history at various

120 Rakesh Gupta, Ritu Goel, Alka Chaudhary

1

51

j-- ~ l

L

L _ _ _ ~ ,0 ,, /

II ' ' 2

0 UP STATE C~ DOWN SI-ATE ~] FAILED L;tA]Z

Fig. 1.. Transition diagram.

transition epochs and as such it simplifies the analysis of the system.

Generalization of the above distribution can be achieved by cutting off and ignoring a fixed part, say a, lying to the left side of the range of random variables and is known as truncated negative exponential distribution. This distribution has the density of the form:

/ 0 e -°u "); t ~ a f ( t ) = tO; t < a (2)

From eqn (2) it is obvious that the failure cannot occur before 'a' unit of time. So one can interpret 'a ' as the 'guarantee' time. The distribution given by eqn (2) is also a negative exponential distribution over the range a to oc, which does not satisfy the 'lack of memory' property. For a system having a lifetime density (eqn (2)), the probability of failure of the system prior to time 'a' is negligible. The reliability function and mean lifetime of such systems are given by

R(t) = { 1; t -< a (3) e ou-,,); t ~ a

Behaviours of these profit functions are also studied through graphs.

MODEL DESCRIPTION

The system consists of two units (priority and ordinary). Initially, the priority (p) unit is operative and the ordinary (o) unit is kept as cold standby. The p-unit has three modes--normal (N), partial failure (P) and total failure (F), while the o-unit has only two modes--normal (N) and total failure (F). A single repairman is available to repair a unit failed partially or totally, and the repaired unit works as good as new. The p-unit gets preference for operation while it is in N-mode and for repair while it is in F-mode over the o-unit. The preference for operation is also given to o-unit in N-mode over the p-unit in P-mode. The repair of p-unit is not possible when it is operative. The system remains down for a minimum fixed time Td (down time), if, during the repair of o-unit, p-unit also fails totally. 3 The system failure occurs if during the down time the repair of p-unit is not completed. The failure time distributions of both the units from N-mode are truncated exponential with different parameters, while the failure time distribution of p-unit in N-mode and all the repair time distributions are negative exponential over the interval (0, ~).

A practical example that best suits our model is that of a room-heating system consisting of two units: blower and simple room heater. The blower may be taken as priority unit and the room heater as nonpriority or ordinary unit. The blower is said to work in normal (N) mode if it works with full efficiency, i.e. both the fan and heater function satisfactorily. It functions in partial failure (P) mode if only the fan fails. The ordinary room heater either operates with full efficiency in N-mode or fails completely. The room heater starts functioning instantaneously if the blower enters in the P-mode. If, during the repair of the room heater, the blower working in P-mode also fails totally, then the temperature of the room remains maintained for some fixed time, say T,, known as down time.

and

E(T) = a + 1/0 (4)

The purpose of the present paper is to analyse a two-unit standby system model, assuming that the lifetime distributions of the units are of the form of eqn (2). Using the regenerative point technique in the Markov renewal process, we obtain expressions for the mean up time of the system, expected busy period of the repairman, the expected number of repairs during (0, t] and the two different profit functions.

TRANSITION PROBABILITIES A N D SOJOURN TIMES

POl = P54 = 1

p , , , = l - / 3 e x p ( - f f , b ) / ( ~ . + / 3 ) , p~2 = l - p , , ,

PEt=q/ (~ /+ t r2) , P23--1--P21

P34 = 1 -- exp(-u2T,,), P~4 = exp(--/~2Td)

P42 = or1 exp(--qa)/(q + crl), ~0)_ t,4~ - 1 - P ~ 2

Analysis of a two-unit standby system 121

Using the formula:

W, = f P(T~ > t) dt

for the mean sojourn time in state S,, its values for various states are

tIZo = a + l / tel , Ux'/2 = 1/( r /+ 0(2)

W~ = [1 - e x p ( - # , b ) ] / g , + exp(-g,b) / (U, + fl)

qJ3 = [1 - exp(-/~zTd)]//~2, tIZ5 = 1/t~2

tIJ4 = [1 - e x p ( - r /a)]/r /+ exp ( - r /a) /(r /+ tel)

PROFIT FUNCTION ANALYSIS

In order to find the profit functions Pl(t) and P2(t) we first obtain the following.

(i) Mean up time of the system during (0, t]

Let us define A~(t) as the probability that the system is up or down at epoch t [ Eo=S~. Elementary probability arguments yield the following relations for i = 0 to 5:

Ai(t) = Mi(t) + ~ qq(t)@Aj(t) (5) J

where

F o r

1 ift<_a i=O; j = l ; Mo(t)= e x p [ - a l ( t - a ) ] ; ift>-a

i=1; j = 0 , 2 ;

Ml(t) = [ e x p ( - ~ l t ) ; if t < b [ e x p ( ~ , t ) exp[- f l ( t - b)' if t --> b

i = 2 ; j - - l , 3; M2(t)=exp[--(r /+teE)t ]

i = 3 ; j = 4 , 5 ; M~(t)=(exp(-p2t); if t T~ ( 0; ift>-Ta

i = 4 ; j = l , 2;

M4(t) = ~exp( - ~/t); if t -< a (exp(~r/ t) exp[-te~(t - a)]; if t -> a

i = 5 ; j=4 ; Ms( t )=0

Taking the Laplace transform of eqn (5) and simplifying for A~(s), we obtain:

A,~s) = N,(s)/Dt(s) (6)

where

N~(s) = (M~ + q~MO[1 - q23qa2(q34 + q35q54)]

+ [(M2 + q23M3 + q23(q34 d- q35q54)M4*]qolq12

and :# * * * Dl(s)=(1-qo,q,o)[1 * * * - q23q42(q34 + q3~qs4)]

- q'{2[q~, + q23q4,(q34 + q35q54)]

Now let A~(t), A~(t) and A~(t) be the respective probabilities that the system is up in N, P and D states at epoch t lEo= So. The separate values of these probabilities in terms of their L.T. can be obtained by the result in eqn (6) on taking M~= M~= O, M,T = M~= M~= M~= 0 and M~ = M~= M~ = M~ = O, respectively.

In the steady state, the above three probabilities are

A~ = (1 -P23P42)(qJ,, + m,) +P23qJ4/O, (7)

AP=p,2W2/D, (8)

and

where

A ° =P23W3/D, (9)

D, = (1 - P23P42)(P,,,W,, + W 1)

+ P,2[tttz + P23(qt3 + Wo + P35qJs)]

The mean up times of the system in N, P and D states during (0, t] are given by

and

ttN(t) = A~'(u) du

Up(t)= Aeo'(u) du

lto(t) = A° ' (u) du

(10)

( 1 1 )

(12)

(ii) Expected busy period of the repairman in (0, t]

We define Bi(t) as the probability that the repairman is busy at epoch t lEo--S~. By probabilistic reasoning the recursive relations for i = 0 to 5 are as follows:

Bi(t) -- Mi(t) + ~ qq(t)@ Bj(t) (13) J

where the values of j corresponding to different values of i are the same as in Section (i) and Mo(t)=O, Ms(t) = exp(-/u2t). Taking the L.T. of eqn (13) and simplifying for B~[s) we have

B,~s) = N2(s)/Dt(s) (14)

where * :# * * * *

N2(s) = q,~',[1 - q23q42(q34 + q35q54)lM,

+ qmql2[Mz + q23{M3q35M5

+ (q34 + q35q54)M,}]

In the steady state, the probability that the repairman

122 Rakesh Gupta, Ritu Goel, Alka Chaudhary

will be busy is given by

Bo = N2/D1 (15)

where

N 2 = (1 - pz3P42)W, +p,z[W 2 +Pz3(W3 + p3sqJ5 + W,)]

Now the expected busy period of the repairman in (0, t] is

~h(t) = Bo(u) du (16)

so that

u~(s) = B,,*(s)/s

(iii) Expected number of repairs during (0, t]

Let us suppose repairs during relations are as

Nj(t) = 010(0

N2(/) = O2l(t)

N3(/) = Q34(/)

N4(/) = Q41(/)

that Ni(t) is the expected number of (0, t] ] Eo = Si. Here, the recurrence follows:

No(t) = 0o1(0 (~ Nl(t) (17)

@[6o + No(t)] + O12(t)ONff t ) (18)

®[62 + N,(/)] + Q2.~(t)ON3(t) (19)

(_)[61 + N4(t)] + O3s(t)@N~(t) (20)

® [62 + N,(/)] + Q42(/) (~) N2(t) (21)

Ns(t) = O54(t)@[61 + N4(/)] (22)

Taking the L.S.T. of above relations and simplifying for ~,(s) we obtain:

We are now in a position to obtain the two cost (profit) functions by the system, considering the characteristics obtained above. The net expected total profit incurred in (0, t] is

P,(t) = expected total revenue in (0, t]

- expected amount paid to

repairman during (0, t]

= Ko/.tN(t) + K,ttt,(t) + K2kto(t) - K3ktt,(/)

Similarly,

P2(t) = Ko#N(t) + K,#p(t) + Kz~to(t)

- K4N~P(t) -- KsN~F(t) - K6Ng(t)

The expected total profits per unit time in the steady state are given by

and

Pj = KoA~ + K,A~ + K 2 A ~ - KsBo

P2 = KoA~ + KIA~ + K2Ao ° - K4N~ P - KsN~ F - K6N'~

where Ko, K~ and K2 are the revenues per unit up time of the system when it works in N-, P- and D-modes, respectively. K3 is the amount paid per unit of time to repairman when he is busy. K4, Ks and K6 are the per unit repair costs of p-unit from P-mode, p-unit from F mode and o-unit, respectively.

G R A P H I C A L S T U D Y

where

~,(s) = Nffs)/Dz(s) (23)

N (s) = 001Q,I,[1 - 023042(034 + Q3sOs4)]

× 60 + Q01Q,z[Qz162 + 023

x {(034 -4- 035054)(61 04162)}]

and Dz(s) can be written simply on replacing q/~ in Ol(s) by Q,.

If we define N~P(t), N~F(t) and N~(t), respectively, as the expected number of repairs of p-unit in P-mode, of p-unit in F-mode and of o-unit during (O, t] l Eo= S,, then the values of these three probabilities in terms of their L.S.T. can be obtained by the result in eqn (23) on putting (6o = 1, 6~ = 62 = 0), (61 = 1, 6o = 62 = 0) and (62 = 1, 6o = 6, = 0).

In the steady state the expected number of repairs of p-unit in p-mode, of p-unit in F-mode and of o-unit per unit of time [Eo = Si are given by

N~ P = (1 - P23P42)P.,/D1

N~F=p12P23/D1 and N~=pl2(p21 +P23P41)/DI

For a more concrete study of profit functions (P, and P2), we plot these characteristics with respect to (w.r.t.) tel (for different values of a) and #2 (for different values of Td) in Figs 2 and 3, respectively. In both the figures, each smooth curve shows the behaviour of profit P1, and the dotted curves indicate the behaviour of profit/ '2.

Figure 2 represents the behaviour of the profit functions P1 and P2 w.r.t, tel for different values of a (=0, 10 and 20), whereas all the other parameters are kept constant. From the figure it is clear that P~ and P2 decrease as tr~ increases. Also, with an increase in the value of 'a ' both P~ and P2 tend to increase. Further, it is important to note that P2 is better than P~ for all values of trl.

In Fig. 3, each curve represents the graph between both the profit functions and/~z for different values of Td (=0, 50 and 100), whereas all the other parameters are kept constant. The curves clearly indicate that Pl and P2 tend to increase with an increment in the value of/~2- Also, as Td increases, both Pt and P2 increase. Here also, the figure indicates that P2 is better than P~ for each and every value of ~t2.

Analysis of a two-unit standby system 123

230 i, K 0 :150 , K 1 :100 , K 2 : 5 0 , K 3 : 75, K 4 = 3 0 , K 5 = 6 0 , K 6 : 2 5 ,

;~, )ut 1 : '005, 0 :01 , h. : '1 , cc 2:.01 , ,U 2=.05, T d =10, b =15

21o ",','>, p, ": "' . . . . . . 1:'2

19C

~, a = 20 17(]

a 10 ~,~ " " " - " - - . _ _ . _ . . _ . . _ _ . . _ _ ", _","\ . . . . . . . . . a : 0

15o ' \ \ ,

13U , ~ ,

x x x xx " - - . . . .

110 ",, " - . " . . . . . . . . . . a = 20

°° f 0.6, ' o .~3 ' o.o'5 ' o . & " o.6.~ ' o.~'~ " o. f i ' o.~;

Fig. 2. Behaviour of cost functions with respect to aq for different values of a.

19C

I?C

15C

130

II0

90

70

K 0 = 1 5 0 , K 1=100, K 2 ; 50 , K 3 = 75 , K / . : 3 0 , K 5 . 6 0 , K 6 = 2 5 ,

,u 1=.005, I'Y =01 , h =.1 , ~ 1 =05 ,Q:2=.01 , b =15 , a .10 .. T d = 100

- - PI ,..-" T d =1173 ---, P2 ......~ .- . . Td:50

. Td: 0

0-61 " 0.63 0.05 0-07 0.03 0.1] 0-13 045 #2

Fig. 3. Behaviour of cost functions with respect to #2 for different values of Td.

124 Rakesh Gupta, Ritu Goel, Alka Chaudhary

ACKNOWLEDGEMENTS REFERENCES

The authors are grateful to Professor L. R. Goel, Head, Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut (India), for his constant encouragement and for providing necessary research facilities in the department. The third author also thanks the CSIR, New Delhi, for the award of a Senior Research Fellowship.

1. Gupta, R. & Bansal, S., Profit analysis of a two-unit priority standby system subject to degradation. Int. J. System Sci., 22(1) (1991) 61-72.

2. Gupta, R. & Goel, L. R., Profit analysis of a two-unit priority standby system with administrative delay in repair. Int. J. System Sci., 20(9) (1989) 1703-13.

3. Gupta, S. M., Jaiswal, N. K. & Goel, L. R., Stochastic behaviour of a two-unit cold standby system with three modes and allowed down time. Microelectron. and Reliab., 23(2) (1983) 333-6.