~ ) Pergamon Microelectron. Reliab., Vol. 37, No. 5, pp. 851 856, 1997

Copyright © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0026-2714/97 $17.00 + .00 PII: S0026-2714(96)00104-7

TECHNICAL NOTE

ANALYSIS OF A SYSTEM HAVING SUPER-PRIORITY, PRIORITY AND ORDINARY UNITS WITH ARBITRARY DISTRIBUTIONS

RAKESH GUPTA, RAMKISHAN and RITU GOEL Department of Statistics, Ch. Charan Singh University, Meerut, India

(Receit~ed for publication 21 May 1996)

Abstract--This paper deals with the analysis of a three non-identical unit cold standby system model. A single repairman is available to repair a failed unit. The priority in respect of operation and repair is being given to the units in order. All failure and repair time distributions are assumed to be general having different p.d.f.'s. Several measures of system effectiveness are obtained by using a regenerative point technique. Copyright © 1997 Elsevier Science Ltd.

INTRODUCTION

Two-unit priority standby system models have been studied extensively in the past by many authors including Nakagawa and Osaki [ l l , Goel et al. [2] and Gupta et al. [3, 4]. However, the cases of n-unit redundant system models have been seen in the literature but not much attention has been paid to this study because of the complexity in the equations and not getting the results in closed form. Gopalan [5] studied a single server n-unit warm standby system model with constant failure rate and arbitrary repair time distribution. The n-unit cold standby system model with arbitrary failure and repair time distribution has been analysed by Goel et al. [6] but they could not get the results in closed form. However, they have obtained the same for n = 2 and n = 3 in the particular cases.

Gupta et al. [7] have studied the cost-benefit analysis of a single server three-unit redundant system with inspection, delayed replacement and two types of repair. Failure time distributions are taken as negative exponentials whereas all the other distribu- tions are assumed arbitrary. More recently Gupta and Bansal I-8] have analysed the cost function of a three-unit standby system subject to random shocks and linearly increasing failure rates. The shock time and repair time distributions are assumed to be negative-exponential.

In the present study we analyse a system model consisting of three non-identical units (super-priority, priority and ordinary). Initially the super-priority (sp) unit is operative and the other two are kept in cold standby. Each unit has two modes, normal (N) and total failure (F). A single repairman is available to repair a failed unit. The switching device to put the standby unit into operation is perfect and

851

instantaneous. The preference in operation and repair is being given to the super-priority (sp) unit over the other units. However the preference in operation and repair is also given to priority (p) unit over the ordinary (o) unit. The failure and repair time distributions are assumed to be arbitrary. The discontinued operation or repair of a unit is supposed to be a pre-emptive repeat type. Using a regenerative point technique the following characteristics have been obtained:

(i) Reliability of the system and MTSF. (ii) Pointwise and steady-state availabilities. (iii) Expected up time of the system and expected

busy period of the repairman in (0, t]. (iv) Net expected profit during an interval (0, t]

and in a steady-state.

The MTSF and profit function have also been studied through graphs in respect of various parameters when all the distributions are assumed to be negative exponentials.

E

FI('), f~(.) i = 1,2,3

Gi(' ), gi(' ) i = 1 , 2 , 3

ml, nl

Symbols for

No/Ns Fr/Fw

NOMENCLATURE

Set of regenerative states (So - ST) set of non-regenerative states (So, $4, Sa, $6) c.d.f, and p.d.f, of failure time of sp, p and o-unit, respectively, for i = 1, 2 and 3 c.d.f, and p.d.f, of repair time of sp, p and o-unit, respectively, for i = 1, 2 and 3 mean failure and mean repair times of super-priority unit.

the states of the system

unit in N-mode and operative/standby unit in F-mode and under repair/waiting for repair.

Using these symbols, the possible states of the

852 Technical Note

G3(.)

So

Fl(.)

G j(.)

G2(.)

St

F2(.)

$2

=1(.)

Fl(.)

$3

F3(.) F r

Fw Fw

$4

Fig. 1. Transition diagram.

sTI

k.....,

G2(. ) /

I I

Fl(.)

o

w

S5

• Regenerative point

X Non-regenerative point

system and the transitions between them along with the failure/repair time c.d.fs are shown in Fig. I. The epochs of transitions from $2 to So, $1 to $3, $3 to $4, $5 to $6, $6 to So and ST to $4 are non-regenerative. In each state the first, second and third symbols are used, respectively, for the sp, p and o-units.

TRANSITION PROBABILITIES AND SOJOURN TIMES

Let Qij(t) be the c.d.f, of transition time from state Si to Sj. The transition probability matrix (t.p.m.) is given by

P = (p~j) = ( Q , ( o o ) ) = O(oo)

with non-zero elements

PolP4s = 1, P lo= f F2(t) dG,

p,~ = f dG(v) f: f3(v- u)dF2(u)

P~3 = f d , ( v ) dff2(v), 3

p~=p'~'~=fG(v)fi'dF3(v-u)dF2(u)

P2o = P~2 °~ = f Fl(v) dG2(v) = P56,

P23 = f (~2(v) dFl(v) = Ps4

P3e = F3(v) dGl(v), P34- = PI3~ ---- f GI(v) df3(v)

pts~ = FI(v) dG3(v - u) dGz(u ) = p(56'°)

f ; ,(6~ dfl(v) (~3(v - u) dG2(u) /157 =

P60 = pt60~ = f Pl(V) dG3(v),

P74 = P ~ = f Gl(v) dF2(v),

P67 = f G3(v) dFl(v)

P76 = f F2(v) dGl(v) •

Technical Note 853

It can be easily verified that

PlO + ffl~ + P(I~( = p~13g 4}) = 1, p(2 O) + P23 = 1

P32 ÷ P34-( = p(45)) = 1, P54 + P ~ + Pt56'°) = 1

P~,7 +P~6 ° ) = 1, P76 + P ~ - = 1.

For simplicity, throughout the paper it has been assumed

ql~ ' = ql~ ' ' ' = qij and p!~'= pl~ ' ' ' = Pij.

Using the formula

T~ = f P( g > t) dt

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

T o = f F , ( t ) d t = m , , T t = TT=fG~(t)F2(.t)

7"2= Ts = f F~(t)G2(t) dt, T3 = f cJ,(t)ff3(t) dt

T4 = f Gi(t) dt = n,, T6 = f F,(t)G3(t) dt.

RELIABILITY AND MEAN TIME TO SYSTEM FAILURE

Let the r,v. U~ be the time to system failure when it starts functioning from state Si. Then reliability of the system is given by

Ri(t) = P[Ui > t].

By using the simple probabilistic arguments, we have the following recursive relations for i --- 0 to 3

R,(t) = Z~(t) + ~, q,j(t) © Rj(t) (1) J

where for

i = 0; j = 1; Zo(t) = F,(t)

i = l; j = 0 , 2 ;

Zl(t) = Fz(t)Gl(t) + qls(t) © d,(t)ff 3(t)

i = 2 ; j = 1,3;

Zz(t) = Fl(t)G2(t) + q2o(t) @ ffl(t)

i =- 3; j = 2; Z3(t ) = Gl(t)ff3(t ).

Taking the Laplace transform L.T. of eqn (1) and simplifying for R*(s), we get

R*(s) = N, (s)/Ol (s) (2)

where

= q23q32)Zo + q*l N(s) ( 1 - * * - * * * q12qzl • * * * * * * *

q o l q l 2 ( Z 2 + x (1 -q23qa2)Zx + q23Z3)

and

D(s) = (1 - - q * t q ~ ' o ) ( 1 - - q23q32) - - q12qzt'* *

Using the usual formula, the M T S F is given by

E(To) = Nt/D, (3)

where in terms of Zt = Tt + PlaTa and Z2 = T2 + P2o m' we have

N 1 = (1 -- PI2P21 - - P23P32)ml + ( 1 - - P23P32)Zl

÷ PlE(Z2 ÷ P23T3)

and D 1 = (1 - Plo)(1 - P23P32) - P12P21.

PROFIT FUNCTION ANALYSIS

In order to find the profit function P(t), we first obtain the following:

(i) mean-up time of the system during (0, t].

Let us define Ai(t) as the probabil i ty that the system is up at epoch t l E o = S i. Using the probabilistic concepts we have the following relations for i = 0 to 7:

Ai(t ) = Zi(t ) + ~, qij(t) © Aj(t), (4) J

where the values of j corresponding to the various values of i are as follows:

i = 0 ; j = l ; i = l , j = 0 , 1 , 5 ;

i = 2 , j = 1,3; i = 3 , j = 2 , 5 ;

i = 4 ; j = 5 ; i = 5 , j = 1 , 4 , 7 ;

i = 6 , j = 1,7; i = 7 , j = 5,6

and

Zs(t ) = Fl(t)G2(t ) + q56(t) QFl(t)G3(t)

+q50(t)©ff l( t)

Z6(t) = Fl(t)G3(t) + q60(t) © ffl(t),

ZT(t ) = F2(t)G(t ),

Taking the L.T. of relations (4) and solving the resulting set of algebraic equations, we can get A*(s), the pointwise availability of the system in terms of its Laplace transform for given E o = S o.

The steady-state availability of the system is given by

A o = lim sAo(s) = N2/D2, say (5)

where in terms of

Z5 = 7"2 + P56T6 q" psomt and Z6 = T6 + Pore1

854 Technical Note

we have

N2 = [(1 - PlzP21 - Pz3P32)m1 + (1 - p23P32)Z1

+ P12(Z2 + P23T3)][(1 - P54)(1 - P67P76)

-- PsvP75] + (1 -- m1)[PlEP23P35

÷ Pl 5( 1 -- P23P32)] [(1 -- P67P76)(P51 + Zs)

÷ P57(P76Z6 ÷ TI ÷ P61P76)]

and

D2 = [(ploml + n0(1 - P23P32)

+ p12(ml + P23nl)] [(1 - p54)(1 - P67P76)

- PsvPvs] + [(P54nl + mr)(1 - - P 6 7 P 7 6 )

+ P57(Pv6ml + hi)]

× [(1 -- Plo)(1 -- P23P32) -- P12P21].

The expected up time of the system is given by

p.p(t) = t ' Ao(u) du 30

so that

#u*(s) = A*(s)/s. (6)

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

We define Bi(t) as the probability that the repairman is busy at epoch t iE o = Sv Here the governing relations for i = 0 to 7 are as follows

Bi(t) = Wi(t) + E qiJ(t) @ Bj(t) (7) Y

where, the values of j corresponding to different values of i are the same as in Section (i) and

Wl(t ) = Gl(t)F2(t ) + qla(t) @ Gl(t)Fa(t)

+ q, , ( t ) © Gl(t)

W2(t ) = Fl(t)G2(t )

W3(t) = Ga(t)Fa(t) + q34(t) © (~t(t),

W4(t ) = G,(t)

Ws(t ) = Fl(t)G2(t ) + qs6(t) © Fl(t)G3(t),

W6(t ) = Fl(t)Ga(t )

Wv(t ) = G,(t)F2(t ) + qv4(t) © G,(t).

Taking the L.T. of relations (7) and solving the resulting set of algebraic equations, we can get B'd(s), the L.T. of Bo(t).

In the long run, the fraction of time for which the system is under repair is given by

B 0 = lira sB'~(s) = N3/D 2 (8) s~O

where in terms of

W1 = T1 + P t 3 T 3 + p l , n l , W3= Ta+P34nl

and

W7 = T1 + P74nl

we have

N 3 = [ (1 - - p23P32)W1 + p 1 2 ( T 2 + P 2 3 W 3 ) ]

× [(I - p54)(1 - - P 6 7 P 7 6 ) - - P57P75]

÷ [P12P23P35 + P15(1 - P 2 3 P 3 2 ) ]

× [ (1 - - P67P76)(Psl + p s 4 n l + Zs)

÷ P57(P76T6 ÷ W7 ÷ P 6 1 P 7 6 ) ] .

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

#b(t) = I ' Bo(t) dt 30

so that

tt*(s) = B~(s)/s. (9)

We are now in the position to obtain the profit function 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]

= Kol~,p(t ) - K1/~b(t ). (10)

The expected total profit per-unit time in the steady-state is given by

P(t) P = lim - - = KoA o - K~Bo (11)

t~c~ t

where K o is the revenue per-unit up time by the system and K1 is the amount paid per-unit of time to the repairman when he is busy.

PARTICULAR CASE

When the failure and repair time distributions of the unit are negative exponentials i.e.

El(t) = 1 - e - ' ' ' , F2(t ) = 1 -- e -'2',

F3(t ) = 1 - e - '~ '

and

Gl(t) = 1 - e -a~', G2(t) = 1 - e -a2',

G3(t ) = 1 - e -a3'.

Then in results (3), (5) and (8) we have the following changes:

Pxo = fll/(fll + a2), P12 = flx~2/(fll + ct2)(fll + ct3)

P13 = ~2/(~1 + ~2),

P14 = Pl5 = ct2~ta/(fll + ~2)(fll + ~t3)

Techn ica l N o t e 855

u.,

70

60

50

40

30

20

10

I I I I 0 0.05 0.15 0.25 0.35

= 0.75, ot 3 = 0.95 ~2 = 0.15, 1~3 = 0.25

- o - . - - o . . . . o ~51 = 0.75 - e ' - - . e ~ o [|1 = 0 .50

~|1 = 0.25

I I O.45 0.55

Fig. 2. B e h a v i o u r o f M T S F w.r.t, ct 1 for different values o f f i r

250 -

2 0 0 -

1,° I lOOk-

50

O -'---- O ....,...

O --..... O " " e ' ' ° ~ e ~ o J31 = 0.75

~ , ct I = 0.75, ~t 3 = 0.95 " ~ 132 - 0.15, Ig = 0.25

• ° ~ o ~ o ~

• ~ ° ~ ° ~ • ~ • ~ ° ~ ° - " " ' e 13,

I I 1 1 I I 0.05 0.15 0.25 0.35 0.45 0.55

= 0 .50

= 0 . 2 5

Fig. 3. B e h a v i o u r o f prof i t w.r.t. % for different values of fl,.

MR 37/54

856 Technical Note

P20 = P21 = f12/(°~l + f12) = Ps6,

P23 = :Xl/(~Xl + f12) : P54

P32 = / J l / (~3 q'- i l l ) , P34 : P35 = 0~3/(0~3 -}- i l l )

P50 = P51 = fl2fl3/(OC1 + f12)(~l + f13)

ps~ = ~ 3 2 / ( ~ + 32)(~ + 3~),

P60 = /961 = f13/(~l + f13) P67 = 0~1/(~i + f13),

P74 : P75 = 0C2/(0~Z At- i l l ) ' ]976 : fll/(0~2 + i l l )

To : 1/~1, T~ = T~ = 1/(~2 + ~ ) ,

T2 = T5 = 1 / ( ~ l "~ f12), T3 = 1/(/J1 -{- ~3),

T4 = 1 / i l l , T6 -~- l / ( ~ l + f13)"

STUDY OF SYSTEM BEHAVIOUR THROUGH GRAPHS

For the graphical study of M T S F i.e. E(To) and profit function-P in the above discussed particular case, we plot these characteristics w.r.t. ~ (failure rate of super priority unit) alternatively by changing one parameter fll (repair rate of super priority unit) and keeping the other fixed. The curves so obtained are shown in Figs 2 and 3, respectively.

From the curves of Fig. 2 we conclude that the M T S F decreases uniformly as :q increases. The decrease is rapid initially and tends to vanish as ~ becomes large. Also M T S F is higher for the higher values o f /~r

The curves of Fig. 3 clearly indicate that the profit-P decreases uniformly as the ~ increases. This decrease is rapid for small values of ill, gradually becomes slow and finally vanishes when fll becomes large. Further, as fll increases the profit also increases irrespective of other parameters.

Acknowledgement--The first author is thankful to UGC, New Delhi for the award of a Major Research Project.

REFERENCES

1. Nakagawa, T. and Osalii, S., Stochastic behaviour of a two-unit standby system with imperfect switchover. IEEE Trans. Reliab., 1975, R-24(2), 193-196.

2. Goel, L. R., Gupta, R. and Singh, S. K., Cost analysis of a two-unit priority standby system with imperfect and arbitrary distributions, Microelectron. Reliab., 1985, 25, 65-69.

3. Gupta, R. and Bansal, S., Profit analysis of a two-unit priority standby system subject to degradation. Int. J. Systems Sci., 1991, 22, 61-72.

4. Gupta, R., Chaudhary, A. and Goel, R., Profit analysis of a two-unit priority standby system subject to degradation and random shocks. Microelectron. Reliab., 1993, 33, 1073-1079.

5. Gopalan, M. N., Probabilistic analysis of a single server n-unit system with (n - 1) warm standbys. Oper. Res., 1975, 23, 591-595.

6. Goel, L. R. and Gupta, R., Reliability analysis of multi-unit cold standby system with two operating modes. Microelectron. Reliab., 1983, 23, 1045-1050.

7. Gupta, R., Bajaj, C. P. and Singh, S. K., Cost-benefit analysis of a single k-server three-unit redundant system with inspection, delayed replacement and two types of repair. Microelectron. Reliab., 1986, 26, 247-253.