Availability for a 23 : G subsystem and secondary subsystem subject to shut-off rules

Microelectron. Reliab., Vol. 32, No. 7, pp. 925-930, 1992. Printed in Great Britain.

0026--2714/9255.00 + .00 © 1992 Pergamon Press Ltd

AVAILABILITY FOR A 2/3:G SUBSYSTEM AND SECONDARY SUBSYSTEM SUBJECT TO SHUT-OFF RULES YOW-MOW CHENt and T. FUJISAWA Department of Communication and Systems Engineering, University of Electro-Communications, Japan and H. 0SAWA Tokoha Gakuen Hamamatsu University,Japan (Received for publication 12 February 1991) Al~raet--Two-unit series subject to shut-off rules have been treated before. For increasing system availability a new model is considered, which is composed of a 2/3 : G main subsystem and another secondary subsystem. If more than two units of the main subsystem are inoperable, then the main subsystem shuts off the secondary subsystem. For this model, the system availability is obtained by the technique of the Markov renewal process. Then we obtain the mean inter-occurrence shut-off time and the steady-state probability that the system is under shut-off.

INTRODUCTION The availability of a series system subject to shut-off rules has been discussed by many authors. The shut-off rule is referred to as a rule in which the failure of one unit may shut off another, but not vice versa. For example, failure of the power equipment may shut down a computer but not vice versa, or failure of a tyre in a car causes the car to stop but the engine may continue to operate. Availability for such systems subject to shut-off rules was first discussed by Hudes [1]. Khalil [2] and Cao [3] dealt with the n-unit series system subject to shut-off rules under the assumption that all lifetimes are independent. Osaki et aL [4] treated the two-unit series system whose lifetimes are non-independent. Khalil [3] and Osaki et al. [4] dealt with a similar model of a series system. In their model, when one main unit failed it would shut off the other unit. Then the system cannot be operating. In this paper, for increasing availability, a new model is considered. In this model, the system is composed of a 2 / 3 : G main subsystem and a secondary subsystem. When more than two units of the main subsystem fail, the main subsystem will shut off the secondary subsystem. This model is depicted in Fig. I. However this model has not been discussed.

(ii) Failure of the main subsystem shuts off operation of the secondary subsystem, but not vice versa. Moreover, we assume that the secondary subsystem never fails or deteriorates during the shutting off period. (iii) The main subsystem is composed of three identical units. The lifetimes of the three units follow an exponential distribution with failure rate 21 . The lifetime of the secondary subsystem follows an exponential distribution with failure rate 2 2. The shut-off period is excluded from the lifetime of the secondary subsystem. However, by the Markov property of the exponential distribution, we shall note that its residual lifetime has the same exponential distribution with mean 22 as the lifetime distribution of a new secondary subsystem. (iv) Two subsystems are reparable. The repair time distribution of any unit in the main subsystem is exponential with repair rate /~t. The repair time distribution of the secondary subsystem is also exponential with rate/z2. The repaired unit or subsystem is as new. (v) The lifetimes of the main subsystem and the secondary subsystem are independent. (vi) The reliability of switchover in the main subsystem is I.

ASSUMPTIONS Considering this system subject to shut-off rules the following assumptions are made.

kv Pl

(i) The system is composed of a main subsystem and a secondary subsystem. That is, failure of any subsystem initiates system shutdown.

kl, Pl kl, Pl

Secondary subsystem

Main subsystem

t"Present address: Box 840, Tunghai University, Taiwan, R.O.C.

Fig. 1. System subject to shut-off rules. 925

YOW-MOWCHENet al.

926 0

1

2

0

Main unit 1

0

1

3

2

0

1

Main unit 2 Main unit 3 Secondary S (continues)

4

1

3

5

3

2

3

,,e..%.~:.%% "

~%=- =

l

..... o

Operating

~

Being repaired

Shutting off

- -

Waiting for repair

Failure

Fig. 2. A realization for the model. (vii) There is a single repair facility and the repair discipline where the main subsystem has the pre-emptive priority rule is considered. Under these assumptions, this model is formulated by the Markov renewal process. The system is described by the following six states: (i)

state 0:

The main subsystem is operating with three units functioning, while the secondary subsystem is operating. The main subsystem is operating with two units functioning and the remaining one failed, while the secondary subsystem is operating. The main subsystem is operating with three units functioning, while the secondary subsystem is under repair. The main subsystem is operating with two of the three units functioning and the remaining one under repair, while the secondary subsystem fails and waits for repair. The main subsystem is in repair with one unit functioning, the other one under repair and the remaining one fails and waits for repair. Then the main subsystem shuts off the operation of the secondary subsystem. One of the unitsof the main subsystem is operating, another one is under repair,

(ii) state l:

(iii) state 2:

(iv) state 3:

(v) state 4:

(vi) state 5:

and the remaining one and the secondary subsystem are waiting for repair. The system states of this model are depicted in Fig. 2. From Fig. 2 a state transition diagram can be obtained as shown in Fig. 3. Referring to Figs 2 and 3 will help us to explain the stochastic behaviour of this model. NOTATION

Qu(t ) one-step transition probability from state i to state j in time t

Q~(s) the Laplace-Stieltjes transform of Q#(t), that is Q~ (s) = S~ e- s, dQq(t) the probability that the system is in state j at time t, given that it was in state i at time 0 P* (s) the Laplace-Stieltjes transform of Pu(t) Ho(t ) the first passage time distribution from the state i to state j H*(s) the Laplace-Stieltjes transform of Hu(t ) lu the mean passage time from state i to state j, that is

Po(t)

l0 = f / t

dHo(t)

G~(t)

the probability that the system leaves state i in time t G* (s) the Laplace--Stieltjes transform of G~(t) A the steady-state availability. ANALYSIS OF THE MODEL

kl

kl

®7®. ~1

~2

-®

P,I

Under the above assumptions and notation, we have the following mass function Qo(t) (i,j = O, 1, 2, 3, 4, 5) from state i to state j for t t> O:

k2

kl

kl ®_-----®------®

Qol(t) =

321 e-3~'~e-~2~dx [1 - e - ° ~

Fig. 3. A state transition diagram for the model.

= 2~.1 + 22

+ a')~]

Availability for a 2/3 : G subsystem

927 0

Q02( t ) "~ ~0 A2e-3alx e-X2x dx

Qs3(t)=Jolhe-u'~dx

=

1 - e -u',.

(1)

The above state transitions are the only state transitions. Let these state transition probabilities be zero, i.e.,

22 [1 - e - (3~ + a~)~] 321 + 22

Qo(t) ffi O. Qlo(t) m ~0 #l e-rex e-2hx e-;2x dx

For each state the following equation is true, i.e., 5

#1

_ e_(2a, + ).2+ Ul)/]

lim~.Q#(t)ffil

= 22t + 22 + #1 [1

Hence it is evident that this model can be analysed using Markov renewal processes.

fo )~2e-rex e-2~1x e - h x dx

Ql3(t)

(i = 0, 1, 2, 3,4, 5).

t~oo j . 0

Taking the Laplace-Stieltjes transform of (1) gives: :.2 = 2~.1 + ~.2 + #1 [1

+ ~ + I,,),] e - (2~'1

Q~I(s) ~.

321 , s +321 +22

Qf3(s) m

321 , s +321 +#2

Q*2(s) m

22 , s + 321 + )-2

Qro(S) =

#2 s + 321 + #2'

,

Q3.2(s) m

#1 s+2,~q+# I'

22 , s + 221 + 22 + #1

Q*s(s) =

221 , s + 221 + #1

QI4(t) = f~ 22~ e -~'~ e -~'~ e -~2~ dx 22

Q*o(S)

[1 -- e-(2~' +~2 +u,)t]

221 + 22 + #1

s +221 + 22 +#~

QJ'~(s)=

[1 - e-°al+~2)q

#2

Q32(t) = fO #1 e-U'x e - 2'hx dx

-

[1 --e -(2a~+ul)']

(2)

221 e -m~ e -2~'x dx

(3) (4)

(5)

where P * H denotes the convolution, i.e., a(t) * b(t) = S[ (t - #) d b ~ ) , and 6 0 denotes Kroneeker's delta. The availability of this system is the probability that the system is operating, i.e., that two subsystems are operating. Thus the availability of this system starting in space 0 at time 0 is the sum of Poo(t) and

22----2--1[1 - e -c2x' +.o,] 22~ + #1

~o

(i , j )

1 - G,*(s) & ( s ) = 1 - H~ (s)

fo'

Q41 (t) = f ~ # t e - U : d x = l - e

s +#~

Pu(t) = Pu(t) * Hu(t ) + 6u[1 - G,(t)]

221 + #1

Q35(t) =

#l

Q,*,(s) = ~ ,

#1 . s+#t

P~(s) = P~(s)H,y(s)

#1

,

Equations (2) are summarized in Table 1. All the Laplace-Stieltjes transforms Q~(s) have been derived. The desired availability can then be obtained by applying the theory of Markov renewal processes, i.e., the following theorem [5] is used. Theorem. For t >t 0 and s > 0, we have

Q20(t) ~" fo #2 e - 3,~x e - mx dx #:

2).1

Q*(s)--

3 2 ~ [1 - e -Oh +re)t] 321 + #2

- 32~

#1 s+221+).2+#1

Q*3(s) m

Q23(t) = fo 321 e - 3a,x e-U2x dx _

=

-u:

/'01(0.

Table 1. Transition probabilities Y i

0

1

2

3

4

5

0 1 2 3 4 5

0 Q~o(s) Q~o(S) 0 0 o

Q*, (s) 0 0 0 Q~t (s) o

O.~2(s) 0 0 Q~2(s) 0 o

0 Q~3(s) Q~3(s) 0 0 Q~3(s)

0

0

Q*,(s)

0

0 0 0 o

0 Q~s(s) 0 o

Yow-Mow C~mN et al.

928

According to the above theorem, we obtain their Laplace-Stieltjes transforms as follows: e*(s)

1 - G*(s)

=

1 - n~o(S) 1 -G*(s)

P*I (s) = l

H * 1(S)

-

It is found that

H*o(s) = C~(s)Q~o(s) + C~(s)Q~'3(s)H~o(S). (12)

(6)

Concerning H~o(s), there are two situations:

(7)

(i) the process moves from state 2 to state 0 in one step; (ii) the process moves from state 2 to state 3 and then eventually returns to state 0.

and

Hence H~o(S) is written as

(8)

P*) (s) = P*I ( s ) ' H*j (s)

H~o(S) = Qgo(S) + Q~3(s)H*(s).

where G*(s) and G*(s) can be easily obtained as follows:

H*o(S) in equations (12) and (13) is the LaplaceStieltjes transform of the duration that the process moves from state 2 to state 0 after visiting states 3, 2 and 5 many times. Hence H*o(S) is given by

G~(s) = Q*I (s) + Q~2(s) G*(s)=Q*o(s)+Q*3(s)+Q*(s).

(13)

(9)

n~o(S)=m~=o~o(mn ){Q23(s)Q32($)}{Q35(s)Q53(s)}Q32(s)Q20(S) oo

=

-+-n

,

,

m

•

*

n

*

*

Q~2(s)Q~o(S){1 _Q23(s)Q32(s)_Q35(s)Q53(s) } 1.

First we consider H~o(S). H*(s) is the LaplaceStieltjes transform of the distribution of recurrence time for state 0. It can be separated into the following two situations: (i) the process moves from state 0 to state 1 in one step and then eventually returns to state 0; (ii) the process moves from state 0 to state 2 and then eventually returns to state 0.

(14)

Based on the above results, the mean first passage times are given as follows:

1 3 ° = - l (i sm) {} ~do 0H*° = 1//a2 + (2)-1 + Ul)(3)'1 + #2)/(/a~#2).

(15)

From (12) and (13) we have ) . 1 ( 1 + 1 )

110 = ~

In situation (i) the Laplace-Stieltjes transform of transition probability is

~

2).1 + ).2 +/~1

1

-I- . ).5

+2).i+25+#1

Q* (s) "H*o(s).

130

(16)

A2+/.tl

and In situation (ii) the Laplace-Stieltjes transform of transition probability is

/2o = (1 + 3).j/30)/(321 + #2). Moreover, from (10), we have

Q*2(s)" H*o(S).

loo=(l + 3211jo+ 2212o)/(32j + 22).

Hence we have

H~o(S) = Q*I (s)H*o(s) + Q*2(s)H*o(s).

(10)

Let C'4 (s) be the Laplace-Stieltjes transform of the duration that the process visits states 1 and 4 many times, i.e.,

H~',(s) = Q ~(s)Q*l (s) + Q*o(s)H*, (s) + Q ?3(s)H*(s)H*, (s).

(19)

Here H0*l(s) is

H*l (s) = ~ {Q*2(s)H~o(S)}"" Q*, (s)

n=O

= Q*I (s) ~/{1 - Q*2(s)H~o(s)}.

(20)

From (19) and (20), we have 10t = (1 + 22/2o)/32 l and

C*4(s) = ~ {Q*(s)Q*(s)}" n~0

-- {1 -- Q*(s)Q*l(s)} -1.

(18)

Similarly, we consider H*t(s). H*l(s) is the Laplace-Stieltjes transform of the distribution of recurrence time for state 1. Here we have

We now consider H*o(s) and H*o(s) in equation (I0). Concerning H*o(s), two situations are considered: (i) after visiting states 1 and 4 many times, the process returns to state 0 in one step; (ii) after visiting states 1 and 4 many times; the process visits state 3 and eventually returns to state 0.

(17)

(11)

Ill = {1 +221//Zl + ().2+ #l )101+).2130}/(2).1 + ).2+/~1 ). (21)

929

Availability for a 2/3 : G subsystem Finally we obtain the steady state (limiting) availability A by rHospital's rule and Tauberian's theorem, i.e.,

Hence we derive the mean shut-off time, given that the system starts state O, as /o4 = 10t -~

A = lim {Poo(t)+Pol(t)} t~oo

2)-i + '~.2+/~t Itt 22t

1 /~t"

(26)

Moreover, the study-state probability that the system is under shut-off is given by

= lira {Q'*2(s) + Q'~2(s)}/loo + lim {Q']'o(S) s~O s~O

1 22 t Po4 =/al'--~ =/at (2).1 + 25 + ~ t ) l , "

+ Q'~(3(s) + Q'*4(s)}/IH

(27)

= 1/(1 + 32~blo + )-212o)+ 1/{1 + 221/ta t (22)

"1L(22 "Jr-#t)/Ol "{- 22130 }.

As we know, state 4 indicates that the main subsystem shuts off the operation of the secondary subsystem. Therefore, since the recurrence time of state 4 represents the duration between the subsequent times when shut-off occurs we call it the inter-occurrence shut-off time. In order to derive its distribution we need to investigate H*(s). For the recurrence of state 4, the system starts in this state, transfers to state 1 in one step, repeats the recurrence of state l many times without visiting state 4 and eventually returns to state 4. Hence we get that

H~(s) = Q*,(s) ~, {Q*o(s)H*l(s) n~O

+ O "3(s)n*o (s)n~ (s)}"Q*,(s) = Q*,(s) ~. {H*l(s) - Q*,(s)Q*l(s)}"O*((s ) n~O

Q *4(s)Q*l (s) 1 -- H*l(s) + Q'~4(s)Q~l(s)'

(23)

Thus the mean inter-occurrence shut-off time is given by 221 +/~l + ;~2 /44 = lH (24) 22~ where lH is in (21). We next consider H~(s) in a similar manner as H * ( s ) and find that

H~(s) = H~',(s) ~ {H~'l(s ) - Q*4(s)Q'~l(s)}"Q~,(s) n=O

H*,(s)Q':,(s)

(25)

1 - H* 1(s) + Q*4(s)Q* l (s)"

NUMERICAL EXAMPLES AND CONCLUSIONS From the above formulations, we can conclude the process of computation as follows. First we compute/3o. Then we compute/~0,120,100, 10~and In in sequence. Finally, we can get availability A and steady probability P04 that the system is under shut-off. Here we have a system which is composed of a main subsystem and a secondary subsystem. The main subsystem has 2/3 : G power facilities. The main subsystem has 2/3:G power facilities. If more than two units of the main subsystem are inoperable, then the main subsystem shuts off the secondary subsystem. The secondary subsystem is a computer. Failure of the power facilities may shut off the computer but not vice versa. By the property of an exponential distribution, we know that l/R1 and 1/22 indicate the mean lifetimes of power facility and computer. Moreover, l/pl and l//h indicate their mean repair times. Now we give the actual performance data to demonstrate the computation of availability A. (l) Suppose that 1//~-- 10 (h), 1//~ 2 -~ 10 (h) and l/~a ffi 1000 (h) are known, and 1/21 varies from 1000 to 10,000 (h), then the availability increases from 0.9895 to 0.9900. This means that the mean lifetime of supply equipment increases as the availability of the system increases. Table 2 and Fig. 4 give the results of this case. We use the above data to demonstrate the computation of the steady-state probability P04 that the system is under shut-off. (2) Suppose that l//~ = l0 (h), 1//~2= 10 (h) and 1/22 -~ 1000 (h) are known, and 1/21 varies from 1000 to 10,000 (h), then the steady-state probability that the system is under shut-off decreases from 0.000571

Table 2. Data for A

'~1 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

'~'2 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

~l 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100

/~2 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.I00

Lu0 779.277 635.075 536.411 464.656 410.122 367.274 332.720 304.263 280.422 260.157

L01 10.100 10.100 10.101 10.102 10.102 10.103 10.104 10.105 10.106 10.107

L~ 10.030 10.060 10.091 10.121 10.152 10.182 10.213 10.244 10.275 10.306

L30 L01 L, 20.050 3366.767 3370.212 20.100 1683.434 1686.893 20.151 1122.323 1125.795 20.201 841.767 845.253 2 0 . 2 5 2 673.434 676.933 2 0 . 3 0 2 561.212 564.724 20.353 481.578 484.578 2 0 . 4 0 4 420.935 424.472 2 0 . 4 5 5 374.176 377.726 2 0 . 5 0 6 336.769 340.331

A 0.9900 0.9900 0.9899 0.9899 0.9898 0.9897 0.9897 0.9896 0.9896 0.9895

YOW-MOWCHEN et al.

930 0.9901

0.0G0571

0.99 0.000463

0.9899 0.000367

0.9898 0.9897 0.9896

0,000014.5

0.000003

0.9895 I I I I I I I I I 0.9894 o.oool o.otm o.oo=o.ooo, o.oooe o.oooe 0.0007 o.oooeo.oooso.oolo

0 , ~ 0.ooo2o.oo03o.0oo4o.ooo50 . ~ o.o0o70 , ~ 0 . ~ 0.~0

(M)

Fig. 4. Availability of A.

Fig. 5. Probability of Po4. Table 3. Data for P04

21 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

22 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

/~l 0.100 0.100 0.I00 0.100 0.100 0.100 0.100 0.100 0.100 0.100

#2 0.I00 o. 100 0.100 0.I00 0.100 0.100 0.100 0.100 0.100 0.I00

to 0.000006. This means that the mean lifetime of supply equipment increases as the steady-state probability that the system is under shut-off decreases. The results of this case are shown in Table 3 and Fig. 5. In this paper we first analysed the case of the identical components of the main 2/3: G redundancy subsystem. The case of non-identical components, the priority o f the repair method and other redundancy models with shut-off rules, etc. will be treated as further problems.

L. /-~ 1705327.52 1708684.28 427627.27 429300.70 190634.59 191746.92 107558.39 108390.16 69047.12 69710.55 48095.62 48646.83 35443.42 35914.47 27219.27 27630.21 21572.34 21936.52 17527,07 17853.84

2. 3.

4.

5. REFERENCES

1. E. S. Hudcs, Availability theory for systems whose components are subject to various shut-off rules. Ph.D.

6.

Po4 0.000006 0.000023 0,000052 0.000093 0.000145 0.000208 0.000282 0.000367 0.000464 0.000571

dissertation, Department of Statistics, University of California, Berkeley, U.S.A. (1979). Z, S. Khalil, Availability of series systems with various shut-off rules. IEEE Trans. Reliab. R-34, 187-189 (1985). J. Cao, Availability and failure frequency of two-unit series with shut-off rule. In Reliability Theory andApplication, Proc. China-Japan Reliability Syrup. (Edited by S. Osaki and J. Cao), pp 1-13. World Scientific, Singapore (1987). S. Osaki, S. Yamada and J. Hishitani, Availability theory of two-unit nonindependent series systems subject to shut-off rules. Reliab. Engng Syst. Safety 25, 33-42 (1989). R. Pyke, Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 12431259 (1961). R. E. Barlow and F. Proschan, Statistical Theory'

of Reliability and Life Testing--Probability Models. Holt, Rinehart & Winston, New York (1975).