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Comparison of two unit cold standby reliability models with three types of repair facilities
Microelectron. Reliab., Vol. 24, No. 1, pp. 35-49, 1984. Printed in Great Britain.
0026-2714/8453.00 + .00 © 1984 Pergamon Press Ltd.
COMPARISON OF TWO UNIT COLD STANDBY RELIABILITY MODELS WITH THREE TYPES OF REPAIR FACILITIES K. MURARIand VmRA GOYAL Department of Mathematics, University of Roorkee, Roorkee-247667, India
(Receivedfor publication 2 Au#ust 1983) Abstract--This paper develops three models for cold standby redundant systems, consisting of two identical units. These models are different in the sense that different types of repairmen are employed. In model 1, the repairman is always with the system. In model 2, he comes immediatelyat the failure of a unit, while in model 3, he takes some random time in reaching the system. Profit is evaluated in each case. Comparison of these profits is done in two parts. Part 1 considers the comparison of different models. In part 2 comparison is done by taking different criteria for evaluating the profit for the same model. Computer programs for these comparisons are also given (in the Appendix).
Before going back, the repairman repairs all the units which fail during his stay at the system. We employ the method of regenerative processes and semi-Markov processes to analyse each model. A large number of system measures such as, the total up-time of the system, the total time for which the system is under repair, the total number of (ordinary and immediate) visits by the repairman, the total number of repairs for a visit of the repairman, are found in steady state. Also mean time to system failure is calculated in each model. Now the question arises which type of repairman we should use to maximize the profit obtained in a two unit cold standby system. This depends on the parameters used. Comparison of models is done in two parts. In part 1 different models are compared while in part 2, comparison in the same model is done. Let us first compare different models under various assumptions.
INTRODUCTION
In a reliability system, often we come across the situation when we want to maximize the profit. The profit of a system, besides other factors, depends largely upon the repairman employed. The repairman can be at hand always or we have to call him when required. Also the question arises of how one should pay the repairman. Broadly speaking, there can be two alternatives. Either we should pay on the basis of time consumed in doing the repairs or on the basis of total number of repairs done. We have tried to study these points here. This paper presents three models. Each model consists of two identical units. Initially one unit is operative and the other is kept as cold standby i.e. it does not fail while standing by. When the operative unit fails, the standby unit takes its place immediately. When both units are in failed condition, the system fails. Constant and same failure rate is assumed in the case of all the models. These models are different as different types of repairmen are employed there. In model 1, the repairman is always with the system. As soon as a unit fails, repair is done by the repairman. When no unit is in a failed condition, the repairman remains idle. This model has been discussed by Rau [1]. We reproduce this here and find out many other measures for the sake of comparison. In model 2, the repairman is not always with the system, but can be made available whenever needed. Thus the repairman comes immediately at the failure of a unit. After repairing all the units which fail during his stay at the system, he goes back. In model 3, as soon as a unit fails, we inform the repairman to arrive at the system for the repair of a unit. The repairman takes some time in reaching the system. We call this time the "arrival time" of the repairman. This arrival time is arbitrarily distributed. If one unit is waiting for the arrival of the repairman and in the meantime another unit fails, we call the repairman immediately, say by fast conveyance. It is assumed that in this case arrival time is negligible.
Comparison of model I and model 2 If (i) we do not pay any charges for the visit of the repairman in model 2, and (ii) cost for doing the repairs and the repair rates are same in model 1 and model 2, then, obviously, model 2 will be preferable to model 1, because in model 1 we have to pay some money for the time for which the repairman remains idle. But, in practice, we have to pay some money for every visit of the repairman. So in this paper, we assume cost for the visit of the repairman in model 2 and different costs for doing the repair and different repair rates for model 1 and model 2 and then try to compare the two models on the basis of profit obtained.
Comparison of model 1 and model 3 In similar fashion, if we assume that (i) no charges are paid for immediate and ordinary visits by the repairman in model 3, and 35
36
K. MURARIand VIBHAGOYAL
(ii) repair rates and cost for doing the repairs are the same in the case of model 1 and model 3, then model 1 will be preferable to model 3 because up-time in model 1 is larger than that of model 3. But, in day to day life, we have to pay some money to the repairman for visiting the system. In each model 3 in such a way that M T S F becomes the same in each case. Then it is found that the total up-time in men take different times for doing the repairs. Now, unless the profit in each model is compared, we cannot say which model is preferable. So the profit in each model is calculated. As we know, in model 3 repairman takes some time in reaching the system while in model 1 it is always available, both the total up-time and the total time for which the system is under repair reduce in model 3. M T S F also reduces in model 3. To make model 3 more profitable, we take repair rates of model 1 and model 3 in such a way that M T S F becomes the same in each case. Then it is found that the total up-time in model 3 becomes larger than that of model 1. Taking this fact into account, we have tried to compare model 1 and model 3 in this paper.
Comparison of model 2 and model 3 If we assume that (i) in model 3, charges for ordinary and immediate visits are the same as and equal to the charges for the visit of the repairman in model 2, and (ii) cost for doing the repairs and the repair rates are the same in model 2 and model 3 then, model 2 will be preferable to model 3 because here also, the up-time in model 2 will be larger than that of model 3. But, in practice, one has to pay a higher charge for an immediate visit than that of an ordinary visit. Thus in this paper we assume different repair rates and different costs for the visit by the repairman. Just as we compare model 1 and model 3, here also we equate the M T S F of two models and then compare these models on the basis of their profits. In part 2, profit is calculated, taking different modes of payment to the repairman in the same model and then these profits are compared. The repairman can be paid according to the total time for which he does the repair or on the basis of total number of repairs done during his one visit to the system. Finally, computer programs are given (in the Appendix) and comparisons are done.
(4) Repairs are perfect i.e. the repairman never does any damage to the units. (5) The system is down when both the units are non-operative. (6) Failure of a unit is detected immediately and perfectly. (7) All random variables are mutually independent. (8) Failure time of each unit is exponentially distributed. (9) Repair time of a unit in each model is different and is arbitrarily distributed.
NOTATIONS The following notations are common to all models: ). constant failure rate of a unit ei(t), Gi(t) pdf and cdf of repair time of a unit in model i, i = 1,2,3 w(t), W(t) pdf and cdf of arrival time for the repairman ~(t) cdf of first passage time from regenerative state i to a failed state Qij(t ) cdf of first passage time from regenerative state i to a regenerative state j or to a failed state j without visiting any other regenerative state in (0, t] Qi~.k(t) cdf of first passage time from regenerative state i to a regenerative state j or to a failed state j visiting state k only once in (0, t] AVe(t) probability that the system is in up state at instant t given that the system started from regenerative state i at t = 0 ARAt ) probability that the system is under repair at instant t given that system entered regenerative stateiat t = 0 (SC) stands for Stieltjes convolution i.e. for any A(t) and B(t)
A(t)(SC)B(t) = (LC)
B(t-u)dA(ul
stands for Laplace transform i.e. for anyf(t) and g(t)
f(t)(LC)g(t) = A i
;o
=l-g*(s)+g*(s+2j,
fo
f(t-u)g(u)du
i= 1,2,3
Notations for diagram
ASSUMPTIONS
operative cold standby ur under repair waiting for repair wr ur.k repair is continuing on the unit from state k © good state [] failed state regeneration point
The following assumptions are c o m m o n to all the three models.
MODEL 1
(1) The system consists of two identical units. (2) Both units are in operating condition at time t = 0. But at t = 0, one unit starts operating while other is kept as cold standby. (3) Standby is switched on to operative state in negligible time.
0
CS
Special assumptions (10) There is a single repairman, who is available always. (11) When both the units are in an operating condition, the repairman remains idle.
Reliability analysis State transition diagram
37
equation is AVo(s) = (ur, o)
(0, CS)
(ur.1, wr)
sA l + 2 [ 2 - g * ( s ) ] (s +,~)[sAi + 2g~*(s)]"
(1.6)
In steady state, up time is given by
Fig. 1. AVo = lim AVo(t) The transition between states as shown in the diagram can be explained as follows: Initially the system is in state 0, in which one unit is operative and other is cold standby. When this operative unit fails, system transits from state 0 to state 1. In this state one unit is under repair and the other unit is operative. If the operative unit remains operative till the completion of repair on the failed unit, it returns back to state 0. On the other hand, if this operative unit fails before the completion of repair on first unit, the system transits to state 2. In state 2, repair on one unit is continuing from state 1, while other unit is waiting for repair.
t~oc
= lim [sAV*(s)]. s~o
Using L'Hospital rule, we have AVo-
1
Cost analysis Let us suppose K o = revenue per unit up-time K l l = cost per unit time for engaging a full time repairman. If at instant t, profit function is Pi(t), then in steady state profit in model 1 is given by
Mean time to system failure Analysis dPo(t) = Ool(t)(SC)$l(t) q~,(t) = Q12(t)+Qlo(t)(SC)c~o(t ).
P1 = lim (1.1)
Qol (s)g~2 (s) ** ** . (s)(21o (s)
j
P1 = K o A V o - K l l Ko g*(2)--2g*'(o)
Kli.
(1.8)
(1.2)
Taking Laplace Stieltjes transform of equations (1) and solving for 4~t*(s), we have
- (2ol
t
Hence,
dQo 1(t) = 2 e - l t dt dQio(t ) = e - i t g t ( t ) d t dQll.2(t ) = gl(t)[1 - e -it] dt dQ12(t) = 2 e - i t 61(t)dt.
1-7/Pl(t)/.
,~L
Where these transition probabilities are:
~b**(s) = 1
(1.7)
g*(2)-2g*'(o)
(1.3)
Making use of the fact
MODEL
2
Special assumptions (12) The repairman is not always with us, but can be called immediately at the failure of a unit. (13) After repairing all the units which fail during his stay in the system, the repairman returns back.
Q*~'(0) = 1
State transition diagram
Q~2 (o)+ g l o (o) = 1
it can be shown that ~b**(0) = 1, which implies that ~b0(t) is a proper distribution. Now, the MTSF, given that the system started at the beginning of state 0, is
(o, cs)
(ur, o)
(ur,1, wr)
Fig. 2.
1 - ~bo** (s)
T O = lim s~o
-
The transition between states can be explained as follows:
S
Using L'Hospital rule and value of ~b**(s) from (1.3) and equations (1.2) we get TO -
2-o*(~) 2(1 -g~'(2))
(1.4)
Availability analysis AVo(t ) = e - l t + 2e -~t (LC)AVI(t) AVI(t) = e - l t 6 1 ( t ) + e - l t gl(t)(LC)AVo(t).
(1.5)
The Laplace transform of the solution of the above
Initially the system is in state 0, in which one unit is operative and other is kept as cold standby. When this operative unit fails, standby takes its place immediately. At the failure of the operative unit we call a repairman immediately. Thus now the system is in state 1 where one unit is under repair and the other is operative. If the failed unit gets repaired before the failure of operative unit, system returns back to state 0. If this operative unit fails before the completion of repair on the other unit, system goes to state 2. In
38
K. MURARI and VIBHA GOYAL
state 2, one unit is under repair continued from state I, and other unit is waiting for repair. Mean time to system failure Equations are same as in model 1. But transition probabilities are dQol(t ) = ite-~tdt
Then, NVo(t) = Qol(t)(SC)[1 + NVI(t)] NVI(t ) = Qlo(t)(SC)NVo(t)+Q112(t)(SC)NVl(t). (2.9) Taking Laplace Stieltjes transform of above equations and solving for NV*(s), we get
dQ10(t) = e-;tg2(t)dt
NV*(s) -
dQ1L2(t ) = g z ( t ) [ 1 - e - a r ] d t dQ12(t) = ).e at(S'z(t)dt.
(2.1)
Proceeding in the same way as in model 1, we have M T S F as 2-g*(2) TO (2.2)
NV°= t~:lim[~]
Let (2.3)
In steady state, up-time is given by
RVi(t) = expected number of repairs for a visit by the repairman in (0, t], given that system just entered state i at t = 0. RVo(t ) = Ooltt)(SC)RVl(t) RV~(t) = Q~o(t). 1 +Q~Lz(t)(SC)[1 + RV~(t)].
itg*(s)
AVo-
RV*(s) - s(s + ).)A 2 .
s~o
1
(2.5)
g~()~) -- itg*'(o)
R V o = lim [RVo(t)] = lim [sRV~'(s)] t~oc
ARo(t) = ite-~t(LC)ARl(t)
RV° - g~'(it)'
(2.6)
Laplace transform of the solution of the above equations is
itO~(s) s i s % + 2.9* (s)]
(2.7)
In the long run, fraction of time for which system is under repair is given by A R o = lira [ARo(t)] = lim [sAR*(s)] s~0
- ito~'(0)
g*(it)-itg*'(0)
(2.14)
Cost analysis We have already described the two methods by which profits can be evaluated. For comparison of different models, we consider the profit P~, in steady state, which besides other factors depends on the time consumed in doing the repairs, i.e. P~ = K o A V o - K 2 1 A R o - K22NVo where Ko = revenue per unit up-time, K21 = cost per unit time for which system is under repair, K22 = cost per visit by the repairman.
t~oc
AR o -
s~o
1
ARI(t ) = e-at G2(t)+e-at gz(t)(LC)ARo(t )
AR*(s) -
(2.13)
In the long run,
Busy period'analysis of repairman
+92(011 - e - a t ] ( L C ) A R x ( t ) + (~2(t)[1 _ e at].
(2.12)
Taking L.S.T. of above equations and solving we get
= lim AVo(t)] = lim [sAV*(s)] t ~
(2.11)
Expected number of repairs Jbr a visit by the repairman
Taking Laplace transform of above equations and solving, we get sA 2 + ) ~ [ 2 - g~'(s)] AV*(s) = (s+ 2)[sA2 + itg~(s)]. (2.4)
AVo
=lim[s2NV*(s)].~o
itg* ( it ) . . nVo = a*(;.)- /g~'(O)
Availability analysis A Vo(t) = e -;.7+ it e - at (LC)A V~(t)
(2.10)
In steady state, number of visits per unit time is given by
it(1 -g*(it))
AVI(t ) = e-;-t G2(t)+e at g2(t)(LC)AVo(t ) +g2(t)[1-e-a~](LC)AVt(t).
2A2
s[-sA2 + it~*(s)]
(2.8)
Expected number of visits by the repairman Let NV~(t) = expected number of visits by the repairm a n in (0, t], given that system started from regenerative state i at t = 0.
We also find the profit p2, taking into account the total number of repairs done and the total up-time in steady state, i.e. p2 = KoAVo_K23RVo .NVo. where K o has the same meaning as above and K2a is the cost per repair, done by the repairman. Putting values of A Vo, A R o, N Vo, R Vo from equations
Reliability analysis (2.5), (2.8), (2.11) and (2.14) respectively, we get
P~-
Ko + '~g~"(0)K21 - J'g~ (J')K22
0*(~)-,~O*'(0)
(2.15)
and p2 __
Ko-2K23
g * ( 2 ) - 2g*'(O)"
(2.16)
MODEL 3
Special assumptions
39
repair on one unit, it becomes operative and the other goes under repair. Thus system transits to state 2. In state 2, one unit is under repair and the other is operative. If the operative unit does not fail till the completion of repair on the failed unit, the system returns back to state 0. If this operative unit fails before the completion of repair on the failed unit, the system transits to state 4. In state 4, repair is continued on one unit from state 2 and other unit is waiting for repair.
Mean time to system failure
(14) At the failure of a unit, we call for the repairman to arrive at the system. The repairman needs some time to arrive. This arrival time has general probability distribution. (15) While one unit is waiting for repair, the other unit also fails, the same repairman arrives immediately, but at higher cost. Here, the arrival time is assumed to be negligible. (16) The repairman returns back, after repairing all the units which fail during his stay at the system.
Analysis O0(t) = Qoa(t)(SC)q51(t) thl(t ) = Qla(t)+Q12(t)(SC)~b2(t) ~bE(t) = Q24(t)+Q20(t)(sC)¢o(t). Where transition probabilities are dQ01 (t) = 2 e-At dt dQ12(t ) = e-'Zt w(t)dt dQ13(t) = 2e -'~t lTV(t)dt dQ20(t ) = e-Ztga(t )dt dQ22.4(t ) = ga(t)[1 - e-at] dt dQE4(t ) = 2 e-~t t~a(t) dt
State transition diagram 9
dQ32(t) = ga(t)dt.
(o,cs)
~
o
(3.1)
1
(ur.2,wr)
Taking L.S.T. of above equations and solving for ~b**(s), we get
(ur,wr)
~bJ*(s) =
Fig. 3.
The transition between states, as shown in the diagram, can be explained as follows: Initially the system is in state 0, in which one unit is operative and the other unit is kept as cold standby. When the operative unit fails, standby takes its place immediately and also we call for the repairman to arrive at the system. Thus the system transits to state 1, where one unit is operative and other is waiting for repair. If the repairman comes before the failure of the operative unit, unit goes under repair and the system transits to state 2. But if the operative unit fails before the arrival of the repairman, then the system fails. So we call the same repairman immediately. Then one unit undergoes repair and the other is waiting for repair and the system is in state 3. After completion of
(3.2)
Q01 (s)[Q13 (s) + Qt2 (s)Q24 (s)] 1 - Qo, (s)Q, 2 (s)Q20 (s)
(3.3)
It can be shown that ~bJ*(0) = 1 which proves that it is a proper distribution. Here we make use of the fact Q~a*(0) = 1 ** 0 ) + Q 1** Q12( 3( 0 )=1 ** 0 ) + Q 2** Q20( 4( 0)= 1 Q~'~'(0) = 1. Now MTSF, given that the system started from the beginning of state 0 is given by 1 - ~b~*(s)
TO = l i m S~O
S
Using equations (3.2) and (3.3) and L'Hospital rule, we get 2 - w*(2)g*(2 )
T° = 211 - w*(2)g~(2)]"
(3.4)
Availability analysis AVo(t ) = e-at + 2 e - a t ( L C ) A V l ( t ) A V~ (t) = e -a~W(t) + e -~t w(t)
(LC)A V2(t) + 2 e -at 17V(t) ( L C ) A V3 (t) e-X~] (LC)A V2(t)
A V2(t) = e-~t ~3(t ) + e-2tg3(t)(LC)A Vo(t ) + g3(t) [1 AV3(t ) = g3(t)(LC)AV2(t).
-
-
(3.5)
40
K. MURARI and VIBHAGOYAL
Taking L.T. of above equations and solving for AV*(s), we get AVo*(S ) =
(S + ,~,)2A3 + 2(s + 2)A3w*(s + .~) + ,).[(s + 2)w*(s + 2) + 2w*(s + 2)0 ~'(s)]g* (s + 2) (s+2)[(s+2)2Aa_2[(s+2)w,(s+2)+2~,(s+2)O,~(s)]g~(s+2) ]
(3.6)
In steady state, expected up-time is given by 1 + ~*(2)g~(2) A V o - [2-w*()OJg'~(2)-20*'(O)[ 1 + ~,(2)g~,(2) ] .
(3.7)
Busy period analysis of the repairman ARo(t ) = 2 e -;~ (LC)ARI(t) A R 1(t) = e-;" w(t)(LC)AR2(t ) + 2 e-At W(t)(LC)ARa(t ) AR2(t ) = e -;" G3(t)+ 93(t) [1 - e -~t] (LC)AR2(t) + e - ~ Oa(t)(LC)ARo(t) + (~a (t) [1 - e -;.t] ARa(t ) = ga(t)(LC)AR2(t).
(3.8)
Taking L.T. of above equations and solving them for AR*(s), we get
2g~'(s) I-(s + ~)w* (s + 2) + 2~* (s + ;~)0~(s)] AR* (s) - s[(s + 2)2A3 - 2g*(s + 2) [(s + 2)w* (s + 2) + 2ff*(s + 2)0~'(s)] ] '
(3.9)
In the long run, expected time for which system is under repair, is given by - 207'(0 ) AR° = [2-w*(2)]g*(2)-20*'(O)[1 +~*(2)g~(2)]"
(3.10)
Expected number of ordinary visits by repairman OVi(t ) = expected number of ordinary visits by the repairman in (0,t], given that system just entered regenerative state i at t = 0. (By the term "ordinary" we mean, repairman comes by its own after placement of order.) OVo(t) = Qol(t)(SC)OVa(t) OVl(t ) = Q,2(t)(SC)[1 +OV2(t)] +Q13(t)(SC)OVa(t)
ov2(t)
= Q2o(t)(sc)ovo(t)+Q2~.,(t)(sc)ov:(t)
OVa(t ) = Qa2(t)(SC)OV2(t).
(3.11)
Taking L.S.T. of above equations and solving them for O V* (s), we get
2(s + 2)A3w*(s + 2) 0 V~(S) = S[(S ÷ 2)2A3 -- 2g~(S ÷ 2) ['(S÷ 2)W*(S ÷ 2) ÷ 2W*(S ÷ 2)g~(S)]] "
(3.12)
In the long run, expected number of ordinary visits per unit time, is
2w*(2)o';(2) OV° = [ 2 - w * ( 2 ) ] O * ( 2 ) - 20~'(O)[1+ w*(2)g*(2)]"
(3.13)
Expected number of immediate visits by the repairman IVy(t) = expected number of immediate visits by the repairman in (0, t], given that system just entered regenerative state i at t = 0. (By the term "immediate" we mean we call the repairman immediately, at the failure of both the units). IVo(t ) = IVl(t ) = IV2(t ) = IV3(t ) =
Qoa(t)(SC)IVI(t) Q12(t)(SC)IV2(t)+Qla(t)(SC)[1 +IVa(t)] Q20(t)(SC)IVo(t)+Q22.g(t)(SC)IV2(t) Qa2(t)(SC)IV2(t ).
(3.14)
Taking L.S.T. of the above equations and solving them we get
AEA31~*(S÷ 2) I V*(s) = s[(s + 2)2A3 - 2g~ (s + 2)[(s + 2)w*(s + 2) + 2ff*(s + 2)g~'(s)]]"
(3.15)
In steady state, expected number of immediate visits per unit time is given by 2~*(2)g~(2) IV° = [ 2 - w * ( 2 ) J g * ( 2 ) - 20*'(O)[1 +~*(2)g*0.)]'
(3.16)
Reliability analysis
41
Expected number of repairs for an ordinary visit of the repairman ROVi(t ) = expected n u m b e r of repairs for an ordinary visit of the repairman in (0, t], given that the system just entered regenerative state i at t = 0. ROVo(t ) = QoI(t)(SC)ROVI(t) ROVI(t ) = Q12(t)(SC)ROV2(t)+Qta(t)(SC)ROVa(t) ROV2(t ) = Q20(t) • 1 +Q22.4(t)(SC)[1 +ROV2(t)] g o v a ( t ) = Qa2(t)(SC)ROV2(t ).
(3.17~
Taking L.S.T. of above equations and solving them we get
ROV*(s) -
,~#~(s)[(s + ~ )w*(s + ;0 + ,~*(s + ~.)#~(s)] s(s+2)2A 3
(3.18)
In the long run, expected number of repairs for an ordinary visit is given by
R O V o = lim [ROVo(t)] = lim [sROV*(s)] s~O
1
R O V o = 9,(2).
(3.19)
Expected number of repairs for an immediate visit by the repairman RIV~(t) = expected n u m b e r of repairs for an immediate visit of the repairman in (0, tl, given that the system just entered regenerative state i at t = 0. RI Vo(t ) RIVI(t ) RIV2(t ) RIVa(t )
= = = =
Qox(t)(SC)RIVI(t) Q12(t)(SC)RIV2(t)+ Qls(t)(SC)RIVs(t) Q20(t)(SC)RIVo(t) + Q22.4(t)(SC)[-1 + RIV2(t)] Qs2(t)(SC)[1 + RIVz(t)].
(3.20)
Taking L.S.T. of above equations and solving them for RIV*(s), we obtain
RI V* (s) =
,~1-,~* (s + 2)g~(s)zX3 + r(s + ~)w*(s + ,~) + ,l~,(s + ~.)g~(s)] • [g~(s) - 0~'(s + ,~)]] s(s + 2)2A3
In the long run, expected number of repairs for an immediate visit is given by
R I V o = lira [RIVo(t)] t~oO
= lim [RIV*(s)] s~O
1 - w*(2)g*(2)
o*(4) Cost analysis Similar to model 2, here also, we find profits by the same two ways. Both the profits are found out in steady state. P~ is based upon the total up-time, the total time for which system is under repair and the total number of ordinary and immediate visits by the repairman, i.e.
P~ = KoA V o - K s l A R o - K s 2 O V o - K s s l V o where, Ko = Ks 1 = Ksz = Kss =
revenue per unit up-time cost per unit time for which system is under repair cost per ordinary visit of the repairman cost per immediate visit of the repairman.
P~ can be calculated on the basis of total number of repairs done and the total up-time. Hence,
p2 = K o A V ° - Ka,t(ROV ° . O V ° + R I V ° .iVo) where, K o has the same meaning as above and Ks,~ is the cost per repair, done by the repairman.
42
K. MURARI and VIBHA GOYAL
Putting values of AV0, ARo, OVo, 1Vo, ROVo, R I V o from equations (3.7), (3.10), (3.13), (3.16), (3.19) and (3.22), we get p~ = [1 + u?*():)g~(),)]K o + ),g~'(0)K31 - 2w*().)g~()OK32 - ,:~w*(2)g~(,~)K33 [2 - w* (),)]g* 0.) - ).g*'(O)[1 + u?*(2)g* (2)]
(3.23)
and p~ = [1 + v~*(2)g*(2)]K o - 211 - w*(iO~*()Og~(iQ]K34 E2-w*(2)]g*(2)-itg*'(O)[1 + ~,*(2)g*(2)]
(3.24)
COMPARISON
Here we compare the profit obtained in different models. This comparison is done in two parts. Part 1 considers the comparison of profits in different models while part 2 carries the comparison of profits in the same model, obtained by two methods. Part 1 Comparison of model I and model 2 F r o m (1.8) and (2.15), we have go
P'
g*(2)-).g*'(O)
K,t
p~ = K0 + 2g~"(0)K21 - ")~g~('~)K22 g~(2)-2g~'(0) A computer program is given in the appendix for the comparison of P1 and P~' (program 1 of Appendix). Let us assume that model 2 is better than model 1 by at least a quantity M D I P (i.e. minimum difference in profits). N o w we find out the conditions on the cost K22. For example, if we take 2=0.1,
u 1 = 1.0,
K 0 = 10,
K l l = 15,
u 2 =0.9. K21 = 16
and M D I P = 10 Pz' > P, + 10
i.e. and this will be so only when
K22 < 36.2.
Comparison of model I and model 3 F r o m (1.8) and (3.23), we have
PI -
P~=
go
g * ( 2 ) - 2g*'(0)
K11
[1 + ~*().)g*(2)]K 0 + 2g*'(0)K3, - 2w*(2)g*(2)K32 - 2&*(2)g~'(2)K33 [ 2 - w*(2)]g'~(2) -).g*'(O)[1 + ~*(2)g*(2)]
Let us take repair rate of model 3 and model 1 in such a way that mean time to system failure becomes same in both the models i.e. if
g3(t) = u 3 e . . . . gl(t) = u, e uLt
then take AU1
u3 = (2 + u , ) w * ( 2 ) - u , "
Then the total up-time of model 3 becomes larger than the total up-time of model 1. Assuming the above, a computer program is made to compare the profits in two models (program 2 of Appendix). For example, if we take ~=0.1,
u, = 1.0
Reliability analysis
43
parameters of waiting time distribution as m = 5, K o=41.0, and
p = 2.0
K l 1 = 12.0, Kax = 3 3 . 0 M D I P = 10.0
i.e. model 3 will be better than model I by at least 10 units if the following inequality holds. w*(~)K32 +w*(,4)K33 < 8.14. If K 3 3 - K 3 2 = 20 (say), then for the inequality P~ > P1 + 10 to hold K33 should be less than 27.17 and K32 should be less than 7.17 units.
Comparison of model 2 and model 3 From (2.15) and (3.23), we have
P~ = Ko -I-~.g~'(O)K21- 2g~'(~.)K22 o~(,l)-,~o*'(o)
p~ = I-1 + ~* (,~)g* (,~)]Ko + ,~g*'(0)K~ 1 - Zw* (;0g~'(,~)K32 -,~¢*(,~)g~'(,l)K33 [2 - w*(2)2o*(2) - 2g*'(0)E 1 + ~*(2)g~'(2)2 Taking again, the M T S F of models 2 and 3 as equal, a computer program is made to compare the above profits. (Program 3 of Appendix.) For example, if 2=0.1,
u2=l.0
m=5,
p=2.0
parameters of waiting time are
and K o = 31.0,
K21 = Kal = 11.0,
K22 = 32.0,
M D I P = 1.3.
So we want P31 > P21+ 1.3. This will happen only when K33 < 59.47.
Part 2 Comparisons in model 2 Profits, found by two methods are
p~
Ko+29~'(O)K21-
,~g~ (,~)K22
g~'(,~)-,lg~"(0) p22 -
Ko-2K23 g~(,~)-,~g*'(o)
Program 4 of appendix, compares the above profits. Here repair time is assumed to have Erlangian distribution. For example, if £=0.1 parameters of repair time are n=5,
U2
=
2.0,
K21 = 1.0,
K22 = 12.0,
M D I P = 3.0.
So we want P~ > p2 + 3; this will happen only when K23 > 41.96.
Comparisons in model 3 Profits, obtained by two methods are p~ = [1 + v~*(2)g*(2)]K o + 2g~"(0)K31-2w*O.)g'~(2)K32 -).O*(2)g~'(2)K3a I-2 - w*(2)]g*(2)- 2g~'(O)[1 + if* (2)g~ (2)3
44
K. MURAR!and VIBHAGOYAL
and
P~
[1 + O*(2)g~(2)]K
o - ~[-1 -
w*(2)~,*(;.)g*(;.)]K34
[2 - w*(2)]g~'()~) - 2g*'(0)[ 1 + ~* (;.)g* ()~)]
C o m p a r i s o n between P~ a n d P~ is d o n e in p r o g r a m 5 of the Appendix. F o r example if X=0.1,
u 3 = 1.0
p a r a m e t e r s of waiting time are m = 5,
p = 2.0,
K31 = ll.0,
i.e.
K3a = 32.0,
M D I P = 5.0
P~ > P~ + 5
which holds only w h e n w*(2)g32 +ff*(/.)g33 > 79.79. This inequality reduces to K33 > 89.31 K32 > 79.31 where we have assumed t h a t
K33 --K32 =
10.
REFERENCES
1. J. G. Rau, Optimization and Probability in Systems Engineering, Van Nostrand Reinhold Company, (1970). 2. M. N. Gopalan and R. Subramanyam Naidu, Stochastic
behaviour of a two-unit repairable system subject to inspection, Microelectron. Reliab. 22, 717-722 (1982). 3. K. Murari and V. Goyal, Reliability system with two types of repair facilities, Microelectron. Reliab. (in press). 4. G. S. Hura, Enumeration of success paths in a graph, Microelectron. Reliab. 22, 1033-1034 (1982).
Reliability analysis
45
APPENDIX
Five computer programs for the comparison of models are given here.
Performanceguide
Program details
1. 2. 3. 4.
1. Specific word length limitation--none 2. Additional relevant information--none
Computer used DEC-2050 Input medium: punched cards Output medium: printer Work data files required: DSK.
Program I MAI{~.
P2PI.FOR
001 ~.~ 002. :
C C C C C C C
003<~:
004 ~:~ 005,~ 00~. ~ 007 ~,~ 008~'~ 009~ 010~ L 011~, 012~,:
021,,, 022i, , 023,Ji 024r, ~ 025,~ ,~ 02b~; ,,
C 10 11
SUHPRf]GRAM5 CO21 CCO
8QR
SCALARS
AND
ll-JUh-83
OF M O D E L
PAGE 1
I
CCtl ARRAYS
C "*"
:!O E X P L I C I T
*Cll
*01
20 25
11:28
CALLED
*DP}~FIT 2 .F~;O93 7 21
[ NO ERRORS D E T E C T E D
DEFINITION
*RQUAN .FO001
3 10 22
TEMPORARIES
MAIN.
/KI
CC21(~,~,C)=A%(A÷C)~SOR(A.B) , OPEl,( U I ~ I ' P = i , DP~VICE: ' DSK ' , F I L E = P2Pl. DAT t ) R~Ai~( I , ~) ,FRATEt HI.U2 R~;AZ;(I,*).CO,CII,C21,DPRFIT P l,,Ti . D P R F I T DIFF~:CO#CC0 (FRATE, U1, U2) ÷ C 1 1 # C C I 1 (FRATE, UI, U 2 ) ' C 2 1 ' C C 2 1 ( IFF, ATE. U I.U2) Ah=I~PRFIT*S'JR ( F R A T E . U I ) * S O R ( F R A T E ~ U2) R~,I]A~'~=(DIFF-AL) / ( F R A T E * ~ Q R ( F R A T E , UI) #U2#~2) F ~JR < ' , F I 0 . 4 , / / ) FL~AT(LX~'C22< OR > ' , E 1 7 . I 0 , I ) Ci~O;;.": ( O~I ~,=1 ) ~TGP E~'I)
*CO I .F:~¢06 6
, F;,~n4
V.5A(621)
COi~'AkI,~Oq OF MODEl, 1 AND 2 FGA LE:FAII,I~RE RATE UI=[~IPAIR RATE OF M O D E L 1 U%=~:E?AIF, RATE OF M O D E L 2 D P F F I T = P R O F I T OF MODEL 2 - P R O F I T C I U : C O S T OF MQDEL I CCId:CBEFF'ICIENT OF THE COST CIJ S'.~F (A, ~) :A*A+A*B+B*~ CC~ (A f rlt C) =A,A* (A+B+C)* (C-B)
Oi=h~: 015-., ~
016',~.~ 017:: 018,~.. 019i. 020..~
FORTEAN
]
-
N%n
NOT
*DIFF *FRATE .F0012
4 11
REFERENCED
]
.FOOl! *C21 .FO007
$ I~
K. MURARIand VIBHAGOYAL
46
Program 2 MAIN.
P3PI.FOR
00100 00200 O0300 00400 00500 0060C 00700 008~0 0090O OlOOO 0110O 01200 0130C 01400 01500 0160o 0170O
C C C C
C
C C
ll-OUb'B3
11:54
PAGE I
COMPARISONS OF MODEL 3 AND 1 FRATE=FAILUR£ RAT~.UI=R£PAIR RATE OY MODEL I M AND RHO ARE PARAMETERS OF WAITING TIME DISTRIHUTION (M -STAGE ERLANGIAI~ CCIJ=COEf'FICIENT)OF THE COST CIJ DCOST=C33-C32
DPRFIT= PROFIT OF MODEL3 - PROFIT OF MODEL 1 W(AnC~D)=(C*D/(A+C*D))**C SOR(AfB)=A*A+A*B+B*B ~ERM(A.B.C.D)=~*~÷A*CA+~I*(WCA~C.DI)**2
C C 0 ( A . B . C , O ) = A B (A~B+W(A.C,D) ( S O R ( A , B ) - C A + B } $ V 2 ~ W ~ A , C , D ) ) ) CC11(A,B,C,D)=SGR(A,~)'TERM(A,B.C,D)
CC31CA,~,CeD)=CA+R)*"CAeC,D)*((A+B)*w(A,C,D)'B)~OR(A,B)
CC32(A,C,D)=W(A,C.D) CC33(A.C.D)=I.'W(A,C.D). OP~i~(U~IT=I.D£VICE='DSK ,FIhE= BOUND.DKT') READ(I,#),FRATE,UIeM,RHO READ(I,*),C0,CII,C31 READ(I.~),DPPFIT,DCOST FM=FLOAT(M) D[FF=CO~CCO(FRATE,Ut.FM.RHO)+Ci%*CCII(FRATE,UI,FM,~HO)-C 131*CC31(FRATE U 1 F M RHO)
0180C
0190~ 0200C 0210~ 02200 0250~ 0260~ 0270o 02800 029O0 030O0 0310~ o32oe 03300 0340O 03500
FORTRAN V.5A(621) / K I
AL=DPRFIT*SQR(~RA~E.61)*TERM(FRATE,UI.FM,RHO) RQUA~=(~)IKF-AL)/((FRATE*UI**2)*SQR(FRAT~,U%)) PRI~CTII.RQUAh
10
11 12
0360~
BC33=ROUAN+DCOST*w(FRATE,FM,RHO) BC32=~C33"DCOST PRINTI2,DCOST,BC33.BC32 FOR.~ATC///.IX,'PROFIT OF ~ODEL 3 - PRL)FIT OF MODEL I > O 1R <'.FIO.&~///) F O R ~ A T ( I X , ' ( C 3 3 - C 3 2 ) * W ( F R A T E . F M . R H O ) - C 3 3 < OR >~,E17.10,/) FOR~TAT(IX.'IF C 3 3 - C 3 2 = ' ~ F I O . 4 , 1 0 X , ~ T H ~ C33 < OR >',E17. 110.'AhD C32 < OR >'rE17110) CLO~E(U~IT=I,DEVICE: D~ ) STOP END
SUBPROGRAMS CALLED W
CCO
CC31
FLOAT.
SCALARS AND ARRAYS • CO .FOOl6 *FRATE ,F0024
1 6 13 20
T~R~
VDPRFZT 2
.F0~12 7 ,8C33 14 .F~020 21
,R.o'F°°°3 4o33
, F 0 ~ 3 1 44 .FOO07 51
.F0026 45 .FO001 52
P3PI.FOR
.F 025
MAI~,
*RuUAN 3 .FO006 10 *M 15 , F 0 0 1 4 22 .F0021 ~ .FO000 34
*A6
.C+',sr,¢I I?.4
[ NO ERROS',q FJiT~:CT£t) ]
41
.F0023 46
F(]RT~Ah: V.SA(621)
T~MPORARI E.5 .Qx+~:O0 1.25
CCli
[ "*" m~O EXPLICIT DEFIr:ITIDN - "%" NOT REFERENCED
.Foo3o 28 ,o[
MAIN.
SOR
/i',I
*DIFF
.FO004 .F0032 ,FOOl0 .FOOLS *BC32 ~DCUST ,FOOl7
11-JU6-83
4
11 16 23 30 35 42 47
]
,F0022 *C31 .F0027 ,FO002 .r0oli ~Cll
5 12 17 24 3z 36 ~FM 43 .F0013 50
11854
P A G E 1-1
47
Reliability analysis
Program 3 MAIN. 001~
C
002;~
C
003¢. 004~. O05v=
C
C C
006~." 007'~
008~ 009~
010~ 011-~
II-,IUL-B3
II:26
PAG£ I
),C,i,)=A*(A+r~)*TKR:~(A,¢~,C,D)
C C 3 t (A, ~',C e D) = (A+~) *$,0R ( A , 5 ) * W ( A, C, b)* ( ( A + B ) * W (A, C, D,I'B) C C 3 2 (A, 3,C, D ) = A * B * * 2 * S O R (.A,h)* ~ (a.C, D) CPE-(d[~I'i)=ItL~EVICL'='D,.%K',FILE='P3P2.DAT') R,~:A,~( I , *) ,F'~ATr;, U2 z ~",~MC} ~,
015.:, 016~
017<:;; 018~
019,~ 020~,~ 021~> 022v,i
0236~, 026v~
9
02U~ 029.¢
II
027C::
/KI
C~h~PARISL),i OF ~'ILID~lh8 3 ,~141 ~, 2 CCIJ=C{);-:FrlCIKhT (,'F TH~S COST CId FRAT6=FAILURE HA.~E.,U2=RKPAIR RATR OF' :~'o.r)F:L2.M A~D R:i[) ARK PAWA,'~.TLR,S :)F ,~AITIhG TI~E DI~TRI~UTIO~J(ERL, ANGIAN) D ? R F I ~ = P R O F I T OF hOt)El, 3 - PROFIT OF xOV)EJ, 2 ~.(A e C, L]) = C C*,'~/(A+C~D) ) **C S l-~~, t A e £ ) =~\* A+ A ' b + 6 * ~, ~:]R:: (A, H,C, 7;)= 8 * B + A * ( A+F~)* (~ { A ~ C, D) )*'2 CC2[(A,
012~ 013:,~ 014~,~
024v.~ 0256~
r',~I~'rPg:i V . L A ( 6 2 1 )
P 3 P 2 . FLtR
F :,'=~L~A i'( :,: ) O I F F = C J ~ C C O ( F R A T E , U2, FM, RHO) + C 2 1 * C C 2 1 ( F R A T E , U2, FM, IKHO) +C 122~CC22(FEA'I'['~,[i2,FX,v~nO)'C31*CC31(F'RATIS,U2,FV',RHO)'C32* 2 C C 3 2 ( I" F(APF. ~ [)2.FN.RP,(~) AL=~;,P*EFI T*Sq)E ( F R A T E . U 2 ) *TER.~ ( F'RATE, U 2 , F ~ , RHO) ROOA.~= (~-,IFF-AL) / ( ( F W . A T E * U 2 * * 2 ) *[;OR [ ['~ATF., iJ2)* ( I .-w ( F R A T E 1. F.,,,~ i~itg) ) ) P R I ' : ~ 1 I.R~.U~f" F!3R,,,AT(IX, P~F)F'IT OF ,~(]I:EL 3 - PHUFIT OF ~(3DEL 2 > OR <'
I~F13.4,//)
FC~R:~,%T(IX, ' C 3 3 >
OIK <' ,El7.1(!,//)
C~,,~,<.E ( l! ',~I T= t )
~TOf E':,>
030~ 031~c
SUBPRnGRAMS CALbE~ W
CC21 SQR
COO CC22
SCALARS
AND
*CO .FG~26 .F0504 *C32
! 6 13 20
,F;020 25 .F0025 .FJO03 *F'M .FU013
32 37 44 .51
CC32 ARRAYS
[ "*"
*[>PRFIT, .F.,022 *C3~ ,o2
CC31
~C E X P L I C I T
*ROUAh .F0016 *FRATE ,FG034
33 4C 45 .Fi.:O~7 52
.FC015 *C22 .Fo027 .FOOQ1
oF~','~)I~ 2e, .F,~021 .~'.30(,0 .Fv(,33
MAIN.
76
,,~I;Ovl
77
[ NO ERRaR5 DF~T~CTF.D ]
]'VR~
DEFII;ITION
7. 7 14 21
T~..MP OR A~ I E,~; .(~O000
FbOAT.
o~0,)10
3 lu 15 22 27 34 41 ~:6 53
- "%"
NnT R E F E R E N C E D
]
*DIFF 4 , F O 0 1 2 11 *C21 16 oF'O030 23
.F0032 .EO006 ~M .F0024
5 12 17 24
*RHh .~0023
~ AL .FO017
~3 50
. o o2 3o . oo11 3s 42 47
. F o o 3 1 31 .Fooo5
K. MURARI a n d VIBHA GOYAL
48
Program 4
MAI~'~.
P 2 . F ~ R - F,*,RTb~c, *
001 : * 002" 003~;~ 0 0 4 "~ 005 ,,
C
007 008 009
C C C C
, ,~ -
0 1 0 :~ 0il 012 • 01,~ ' 014: ' 015-, 016, , 0 1 7 ~" 0 1 8 .., ~ 01q '~ 020 ~' 021 , L 022.
C
I0
V.SA(621)
/K)"
3<'-JU~.-83
II:31
PAGE I
FRA .'CK:F ~ ! iA,.o,~ RATE N ~D o2 ~R~: PARA~ETEk,~ t~F ~ K P A I R T I M £ ( E R L A N G I A t ~ ) RPRTID=~q~nV!T I~ P A R T I - P P O F I T Ih P A R T II CI,.I=CnST G (AF B F(:) = ~ g~*Cl ( A + ~ * C ) ) * * ~ ~ERQG(A,~,C)=.I./C " . ~OE' ( o > I ' F : I , D F V I C £ : ' D S K ,FIbF~: P 2 . D A T ' ) ~TAi,( i , ¢) , F R A T F r N ~,lJ ~. RZA!'( 't, * ) , C21 , C22, DPRFI!' F'~:F[,OAT ( :! ) VAhUF;:C22*(; ( F~ATI':, F~;, {J2 )'C'2 I*I)ER VG(FRATE,FN, U2) ~.I,:,',PRF 11'* (G (FIXATE, F'N, U2.)-FHAT~:*D~RVG (F P,ATE,FN, U2) )/FRATE ROll.~ '! : V :\~'.L:~+A G R ~ i I ; w : J = ~ i ~ l ; T RED OUAh T I'.I'¥ P R I , i T I ;,';~HFIT,~OLiA!~ T Y P ~.~ . , i ; P ~ ) F I T , R O U A i , F n R A rcL.Kt,pRc;FIT [ ~ RA~LT I - P R O F I T IN P A R T II > OR < ' , E l 1~: 4 5,',, ,;,,bY ~,:HE;'i C'2~ > r!R <~ , E ~ 7 . 1 0 ) St'OP
$Ut~p~OGRA>! .~ CALhE.) DERVG FbOAT.
(;
SCAbARS A~;D ARRAYS [ *DPRFIT 1 *FRATE 5 *FN 1[ .Y;> 02 t 6
"*"
*KNt~A . *C2J. *C2L,
~; E X P L I C I T D £ F I ' I T I { I N 2 '~ t2
TEMPCU'AKIH,
,6) ~OO 4 t MAI~q.
[
,C;'::~,,l
.,2
;~0 £RRO.RS P/'ri~c'UPrl[' ]
"%"
NOT R£EERENCED ]
*r,
3
%.F0004
.FO001
4
*u2 *VALUE
7 13
%,F0003 *AI,
,FO000 .FO005
10 15
14
Reliability analysis
49
Program 5 MAIN, 001~.,,) 002:,
~
0030 ' 004(, " 005:., ~ OOboO 08o:> 0 9 '~ . 0095. 010o,. 0105011"r. 0120 • 013~?,' 014(~7, 015c, 0160 0181,.: 019o'~ 0245: 0 2,~ 013C
~
o34,>::: 035',,'. 0 3 6 .,,, 037:):' 0376" 0378.' 038(;" 3 8 5 :, 41o., 0415:" 0440~ 0445~.' 051:)3 052~> 0560?
P3.FOR
FOP'TF,,,~
v.%/.(~21)
C C C C C C
CC!'P~I;JS<<.,%
]I
/~I
3,"-obl -~3
1~:32
C
I
, CL E.!. i:,
U 3 : b' F)) A .,'.I- )'~ T~: U ;,'[> Fh .~ I-'IEI, t,'L'],i'F,,% OF" ~AITI),('. T.T~[: ['I::,TF].P.b'I'IL)N LAi~r;I/~ ' ) DPFFIT=;'IKf;FII i:, P A W T II - PF,'C,FIT I~- PAI
C C
PAGE
(.EI~
~(A ,C,I~ )=C C*I'/ ( A*C*I> ) )**C DE:~.(~- ( t, ~ i", C , I.) : t : * P . + F ~ I A + A * A + h * ( A + b ) * ( 1 .-%,. ( A, C , D ) } CC35.C;',,5,C,D)=A*(A÷£,) CC32(A,~,,C,[)=A*~:*~*~(~,C,|~) -,'" C C 3 3 ( k , k., C, [~) =A*~*F.,) ( I . ' , L A ¢ C , [ } J CC3 ~(;,,",,C,5)=A*B'(A÷5",'(A,C,F,)*( I. -~'(A,C,I}) )~B) OPE:i(Ui.[T=I~F;EV~C~,='[)~b ' , F I b E = ' P 3 . L A I ' ) REAr; ( I, ' ) , F'KAq'E:tlj3, ~, ~ H[.~ REA;>(~ , * ) , C g I , ( : 3 4 , f ' P R F I T R~;AI )( ! , * ) , t;'C~,'.,T PRII~Tg,FF~FI',r F ~=!"I,[)AT ('~) D[ V " = C 3 1 * C C 3 ~ ( F'~
C
TIP:':12, OCt)CT, ~ C 3 3 , b C 3 2 F',:~',~,AT(IX,'PR~]FIT .If~ FAF~ 11 - P~UF].T IF }'AI~T I • OR <', IF~.2I///) FU~.~ r(I.Y,'C32*;;(~WAT~:,FP,,p~o)÷C33*(1 -~-(F~AT£,F'M,RHO))> 1 ":!" < ' , K t T . I L " . / ) r O ~ : A ! ' ( l . < , ' I r C 3 3 - C 3 2 = ' ~ B . P ~ 5 X , ' T I * E ~ C33> OR <),F~17.10, 15X.','~L[) C32> .'.IF' < ' , E 1 7 . 1 0 e / / ) Ch{~SE (U-, IT= 1 )
9
11
12
057,!~
STC, P " L'-, ~
SUBPRfJGRAw,o C A L L E D W
DENOM
CC31
FLCAT.
CC'3~
8CAI, ANS A~'D ARRAYS { " * " *DPRFIT .F0~20 :~FRATE
I
*R(,~i~,
6
.F''~I4
~.F EXPLICI'I bEFI~,ITI(~r, 2
7 1.¢
*C3q
*~.C33
1.3
*~:
.F(:(:IC." J('
20 24 31 36
13
*~1~
MAIN.
PJ.FOR
FORTR~t: V.LA(821)
~I 25 37 ~.7
TEMPORARIE3 .O,,;;~O 117 MAIN.
MR 24/I--D
[
.(~'.,OC;I 12C
r~L) ERRORS DETECTED ]
"%" r,N']" IxEF~;RE.NCEb ]
3
.F0.922 .Fo~;23 ,re-:oe ,~FM
.F)C;Ih F * A C ~ ' 2I 7 . F _25
-
*1 IF'f"
g.FLOI2 .FC. CI3 26 * FKHF. 33 .~t021 '-,C,
/KZ
.~(,c't'4
.~OC(;b FC'O,.7 */,l, .F(.,(315
30-dUN-83
4
11
16
22 27 3~ 41
11z32
.~0024
5
.~O02b
17
*C31
12
.FOC('2 23 .FUO03 30 *DCOST 35 %.F('.01|
PAGE I - 1
0026-2714/8453.00 + .00 © 1984 Pergamon Press Ltd.
COMPARISON OF TWO UNIT COLD STANDBY RELIABILITY MODELS WITH THREE TYPES OF REPAIR FACILITIES K. MURARIand VmRA GOYAL Department of Mathematics, University of Roorkee, Roorkee-247667, India
(Receivedfor publication 2 Au#ust 1983) Abstract--This paper develops three models for cold standby redundant systems, consisting of two identical units. These models are different in the sense that different types of repairmen are employed. In model 1, the repairman is always with the system. In model 2, he comes immediatelyat the failure of a unit, while in model 3, he takes some random time in reaching the system. Profit is evaluated in each case. Comparison of these profits is done in two parts. Part 1 considers the comparison of different models. In part 2 comparison is done by taking different criteria for evaluating the profit for the same model. Computer programs for these comparisons are also given (in the Appendix).
Before going back, the repairman repairs all the units which fail during his stay at the system. We employ the method of regenerative processes and semi-Markov processes to analyse each model. A large number of system measures such as, the total up-time of the system, the total time for which the system is under repair, the total number of (ordinary and immediate) visits by the repairman, the total number of repairs for a visit of the repairman, are found in steady state. Also mean time to system failure is calculated in each model. Now the question arises which type of repairman we should use to maximize the profit obtained in a two unit cold standby system. This depends on the parameters used. Comparison of models is done in two parts. In part 1 different models are compared while in part 2, comparison in the same model is done. Let us first compare different models under various assumptions.
INTRODUCTION
In a reliability system, often we come across the situation when we want to maximize the profit. The profit of a system, besides other factors, depends largely upon the repairman employed. The repairman can be at hand always or we have to call him when required. Also the question arises of how one should pay the repairman. Broadly speaking, there can be two alternatives. Either we should pay on the basis of time consumed in doing the repairs or on the basis of total number of repairs done. We have tried to study these points here. This paper presents three models. Each model consists of two identical units. Initially one unit is operative and the other is kept as cold standby i.e. it does not fail while standing by. When the operative unit fails, the standby unit takes its place immediately. When both units are in failed condition, the system fails. Constant and same failure rate is assumed in the case of all the models. These models are different as different types of repairmen are employed there. In model 1, the repairman is always with the system. As soon as a unit fails, repair is done by the repairman. When no unit is in a failed condition, the repairman remains idle. This model has been discussed by Rau [1]. We reproduce this here and find out many other measures for the sake of comparison. In model 2, the repairman is not always with the system, but can be made available whenever needed. Thus the repairman comes immediately at the failure of a unit. After repairing all the units which fail during his stay at the system, he goes back. In model 3, as soon as a unit fails, we inform the repairman to arrive at the system for the repair of a unit. The repairman takes some time in reaching the system. We call this time the "arrival time" of the repairman. This arrival time is arbitrarily distributed. If one unit is waiting for the arrival of the repairman and in the meantime another unit fails, we call the repairman immediately, say by fast conveyance. It is assumed that in this case arrival time is negligible.
Comparison of model I and model 2 If (i) we do not pay any charges for the visit of the repairman in model 2, and (ii) cost for doing the repairs and the repair rates are same in model 1 and model 2, then, obviously, model 2 will be preferable to model 1, because in model 1 we have to pay some money for the time for which the repairman remains idle. But, in practice, we have to pay some money for every visit of the repairman. So in this paper, we assume cost for the visit of the repairman in model 2 and different costs for doing the repair and different repair rates for model 1 and model 2 and then try to compare the two models on the basis of profit obtained.
Comparison of model 1 and model 3 In similar fashion, if we assume that (i) no charges are paid for immediate and ordinary visits by the repairman in model 3, and 35
36
K. MURARIand VIBHAGOYAL
(ii) repair rates and cost for doing the repairs are the same in the case of model 1 and model 3, then model 1 will be preferable to model 3 because up-time in model 1 is larger than that of model 3. But, in day to day life, we have to pay some money to the repairman for visiting the system. In each model 3 in such a way that M T S F becomes the same in each case. Then it is found that the total up-time in men take different times for doing the repairs. Now, unless the profit in each model is compared, we cannot say which model is preferable. So the profit in each model is calculated. As we know, in model 3 repairman takes some time in reaching the system while in model 1 it is always available, both the total up-time and the total time for which the system is under repair reduce in model 3. M T S F also reduces in model 3. To make model 3 more profitable, we take repair rates of model 1 and model 3 in such a way that M T S F becomes the same in each case. Then it is found that the total up-time in model 3 becomes larger than that of model 1. Taking this fact into account, we have tried to compare model 1 and model 3 in this paper.
Comparison of model 2 and model 3 If we assume that (i) in model 3, charges for ordinary and immediate visits are the same as and equal to the charges for the visit of the repairman in model 2, and (ii) cost for doing the repairs and the repair rates are the same in model 2 and model 3 then, model 2 will be preferable to model 3 because here also, the up-time in model 2 will be larger than that of model 3. But, in practice, one has to pay a higher charge for an immediate visit than that of an ordinary visit. Thus in this paper we assume different repair rates and different costs for the visit by the repairman. Just as we compare model 1 and model 3, here also we equate the M T S F of two models and then compare these models on the basis of their profits. In part 2, profit is calculated, taking different modes of payment to the repairman in the same model and then these profits are compared. The repairman can be paid according to the total time for which he does the repair or on the basis of total number of repairs done during his one visit to the system. Finally, computer programs are given (in the Appendix) and comparisons are done.
(4) Repairs are perfect i.e. the repairman never does any damage to the units. (5) The system is down when both the units are non-operative. (6) Failure of a unit is detected immediately and perfectly. (7) All random variables are mutually independent. (8) Failure time of each unit is exponentially distributed. (9) Repair time of a unit in each model is different and is arbitrarily distributed.
NOTATIONS The following notations are common to all models: ). constant failure rate of a unit ei(t), Gi(t) pdf and cdf of repair time of a unit in model i, i = 1,2,3 w(t), W(t) pdf and cdf of arrival time for the repairman ~(t) cdf of first passage time from regenerative state i to a failed state Qij(t ) cdf of first passage time from regenerative state i to a regenerative state j or to a failed state j without visiting any other regenerative state in (0, t] Qi~.k(t) cdf of first passage time from regenerative state i to a regenerative state j or to a failed state j visiting state k only once in (0, t] AVe(t) probability that the system is in up state at instant t given that the system started from regenerative state i at t = 0 ARAt ) probability that the system is under repair at instant t given that system entered regenerative stateiat t = 0 (SC) stands for Stieltjes convolution i.e. for any A(t) and B(t)
A(t)(SC)B(t) = (LC)
B(t-u)dA(ul
stands for Laplace transform i.e. for anyf(t) and g(t)
f(t)(LC)g(t) = A i
;o
=l-g*(s)+g*(s+2j,
fo
f(t-u)g(u)du
i= 1,2,3
Notations for diagram
ASSUMPTIONS
operative cold standby ur under repair waiting for repair wr ur.k repair is continuing on the unit from state k © good state [] failed state regeneration point
The following assumptions are c o m m o n to all the three models.
MODEL 1
(1) The system consists of two identical units. (2) Both units are in operating condition at time t = 0. But at t = 0, one unit starts operating while other is kept as cold standby. (3) Standby is switched on to operative state in negligible time.
0
CS
Special assumptions (10) There is a single repairman, who is available always. (11) When both the units are in an operating condition, the repairman remains idle.
Reliability analysis State transition diagram
37
equation is AVo(s) = (ur, o)
(0, CS)
(ur.1, wr)
sA l + 2 [ 2 - g * ( s ) ] (s +,~)[sAi + 2g~*(s)]"
(1.6)
In steady state, up time is given by
Fig. 1. AVo = lim AVo(t) The transition between states as shown in the diagram can be explained as follows: Initially the system is in state 0, in which one unit is operative and other is cold standby. When this operative unit fails, system transits from state 0 to state 1. In this state one unit is under repair and the other unit is operative. If the operative unit remains operative till the completion of repair on the failed unit, it returns back to state 0. On the other hand, if this operative unit fails before the completion of repair on first unit, the system transits to state 2. In state 2, repair on one unit is continuing from state 1, while other unit is waiting for repair.
t~oc
= lim [sAV*(s)]. s~o
Using L'Hospital rule, we have AVo-
1
Cost analysis Let us suppose K o = revenue per unit up-time K l l = cost per unit time for engaging a full time repairman. If at instant t, profit function is Pi(t), then in steady state profit in model 1 is given by
Mean time to system failure Analysis dPo(t) = Ool(t)(SC)$l(t) q~,(t) = Q12(t)+Qlo(t)(SC)c~o(t ).
P1 = lim (1.1)
Qol (s)g~2 (s) ** ** . (s)(21o (s)
j
P1 = K o A V o - K l l Ko g*(2)--2g*'(o)
Kli.
(1.8)
(1.2)
Taking Laplace Stieltjes transform of equations (1) and solving for 4~t*(s), we have
- (2ol
t
Hence,
dQo 1(t) = 2 e - l t dt dQio(t ) = e - i t g t ( t ) d t dQll.2(t ) = gl(t)[1 - e -it] dt dQ12(t) = 2 e - i t 61(t)dt.
1-7/Pl(t)/.
,~L
Where these transition probabilities are:
~b**(s) = 1
(1.7)
g*(2)-2g*'(o)
(1.3)
Making use of the fact
MODEL
2
Special assumptions (12) The repairman is not always with us, but can be called immediately at the failure of a unit. (13) After repairing all the units which fail during his stay in the system, the repairman returns back.
Q*~'(0) = 1
State transition diagram
Q~2 (o)+ g l o (o) = 1
it can be shown that ~b**(0) = 1, which implies that ~b0(t) is a proper distribution. Now, the MTSF, given that the system started at the beginning of state 0, is
(o, cs)
(ur, o)
(ur,1, wr)
Fig. 2.
1 - ~bo** (s)
T O = lim s~o
-
The transition between states can be explained as follows:
S
Using L'Hospital rule and value of ~b**(s) from (1.3) and equations (1.2) we get TO -
2-o*(~) 2(1 -g~'(2))
(1.4)
Availability analysis AVo(t ) = e - l t + 2e -~t (LC)AVI(t) AVI(t) = e - l t 6 1 ( t ) + e - l t gl(t)(LC)AVo(t).
(1.5)
The Laplace transform of the solution of the above
Initially the system is in state 0, in which one unit is operative and other is kept as cold standby. When this operative unit fails, standby takes its place immediately. At the failure of the operative unit we call a repairman immediately. Thus now the system is in state 1 where one unit is under repair and the other is operative. If the failed unit gets repaired before the failure of operative unit, system returns back to state 0. If this operative unit fails before the completion of repair on the other unit, system goes to state 2. In
38
K. MURARI and VIBHA GOYAL
state 2, one unit is under repair continued from state I, and other unit is waiting for repair. Mean time to system failure Equations are same as in model 1. But transition probabilities are dQol(t ) = ite-~tdt
Then, NVo(t) = Qol(t)(SC)[1 + NVI(t)] NVI(t ) = Qlo(t)(SC)NVo(t)+Q112(t)(SC)NVl(t). (2.9) Taking Laplace Stieltjes transform of above equations and solving for NV*(s), we get
dQ10(t) = e-;tg2(t)dt
NV*(s) -
dQ1L2(t ) = g z ( t ) [ 1 - e - a r ] d t dQ12(t) = ).e at(S'z(t)dt.
(2.1)
Proceeding in the same way as in model 1, we have M T S F as 2-g*(2) TO (2.2)
NV°= t~:lim[~]
Let (2.3)
In steady state, up-time is given by
RVi(t) = expected number of repairs for a visit by the repairman in (0, t], given that system just entered state i at t = 0. RVo(t ) = Ooltt)(SC)RVl(t) RV~(t) = Q~o(t). 1 +Q~Lz(t)(SC)[1 + RV~(t)].
itg*(s)
AVo-
RV*(s) - s(s + ).)A 2 .
s~o
1
(2.5)
g~()~) -- itg*'(o)
R V o = lim [RVo(t)] = lim [sRV~'(s)] t~oc
ARo(t) = ite-~t(LC)ARl(t)
RV° - g~'(it)'
(2.6)
Laplace transform of the solution of the above equations is
itO~(s) s i s % + 2.9* (s)]
(2.7)
In the long run, fraction of time for which system is under repair is given by A R o = lira [ARo(t)] = lim [sAR*(s)] s~0
- ito~'(0)
g*(it)-itg*'(0)
(2.14)
Cost analysis We have already described the two methods by which profits can be evaluated. For comparison of different models, we consider the profit P~, in steady state, which besides other factors depends on the time consumed in doing the repairs, i.e. P~ = K o A V o - K 2 1 A R o - K22NVo where Ko = revenue per unit up-time, K21 = cost per unit time for which system is under repair, K22 = cost per visit by the repairman.
t~oc
AR o -
s~o
1
ARI(t ) = e-at G2(t)+e-at gz(t)(LC)ARo(t )
AR*(s) -
(2.13)
In the long run,
Busy period'analysis of repairman
+92(011 - e - a t ] ( L C ) A R x ( t ) + (~2(t)[1 _ e at].
(2.12)
Taking L.S.T. of above equations and solving we get
= lim AVo(t)] = lim [sAV*(s)] t ~
(2.11)
Expected number of repairs Jbr a visit by the repairman
Taking Laplace transform of above equations and solving, we get sA 2 + ) ~ [ 2 - g~'(s)] AV*(s) = (s+ 2)[sA2 + itg~(s)]. (2.4)
AVo
=lim[s2NV*(s)].~o
itg* ( it ) . . nVo = a*(;.)- /g~'(O)
Availability analysis A Vo(t) = e -;.7+ it e - at (LC)A V~(t)
(2.10)
In steady state, number of visits per unit time is given by
it(1 -g*(it))
AVI(t ) = e-;-t G2(t)+e at g2(t)(LC)AVo(t ) +g2(t)[1-e-a~](LC)AVt(t).
2A2
s[-sA2 + it~*(s)]
(2.8)
Expected number of visits by the repairman Let NV~(t) = expected number of visits by the repairm a n in (0, t], given that system started from regenerative state i at t = 0.
We also find the profit p2, taking into account the total number of repairs done and the total up-time in steady state, i.e. p2 = KoAVo_K23RVo .NVo. where K o has the same meaning as above and K2a is the cost per repair, done by the repairman. Putting values of A Vo, A R o, N Vo, R Vo from equations
Reliability analysis (2.5), (2.8), (2.11) and (2.14) respectively, we get
P~-
Ko + '~g~"(0)K21 - J'g~ (J')K22
0*(~)-,~O*'(0)
(2.15)
and p2 __
Ko-2K23
g * ( 2 ) - 2g*'(O)"
(2.16)
MODEL 3
Special assumptions
39
repair on one unit, it becomes operative and the other goes under repair. Thus system transits to state 2. In state 2, one unit is under repair and the other is operative. If the operative unit does not fail till the completion of repair on the failed unit, the system returns back to state 0. If this operative unit fails before the completion of repair on the failed unit, the system transits to state 4. In state 4, repair is continued on one unit from state 2 and other unit is waiting for repair.
Mean time to system failure
(14) At the failure of a unit, we call for the repairman to arrive at the system. The repairman needs some time to arrive. This arrival time has general probability distribution. (15) While one unit is waiting for repair, the other unit also fails, the same repairman arrives immediately, but at higher cost. Here, the arrival time is assumed to be negligible. (16) The repairman returns back, after repairing all the units which fail during his stay at the system.
Analysis O0(t) = Qoa(t)(SC)q51(t) thl(t ) = Qla(t)+Q12(t)(SC)~b2(t) ~bE(t) = Q24(t)+Q20(t)(sC)¢o(t). Where transition probabilities are dQ01 (t) = 2 e-At dt dQ12(t ) = e-'Zt w(t)dt dQ13(t) = 2e -'~t lTV(t)dt dQ20(t ) = e-Ztga(t )dt dQ22.4(t ) = ga(t)[1 - e-at] dt dQE4(t ) = 2 e-~t t~a(t) dt
State transition diagram 9
dQ32(t) = ga(t)dt.
(o,cs)
~
o
(3.1)
1
(ur.2,wr)
Taking L.S.T. of above equations and solving for ~b**(s), we get
(ur,wr)
~bJ*(s) =
Fig. 3.
The transition between states, as shown in the diagram, can be explained as follows: Initially the system is in state 0, in which one unit is operative and the other unit is kept as cold standby. When the operative unit fails, standby takes its place immediately and also we call for the repairman to arrive at the system. Thus the system transits to state 1, where one unit is operative and other is waiting for repair. If the repairman comes before the failure of the operative unit, unit goes under repair and the system transits to state 2. But if the operative unit fails before the arrival of the repairman, then the system fails. So we call the same repairman immediately. Then one unit undergoes repair and the other is waiting for repair and the system is in state 3. After completion of
(3.2)
Q01 (s)[Q13 (s) + Qt2 (s)Q24 (s)] 1 - Qo, (s)Q, 2 (s)Q20 (s)
(3.3)
It can be shown that ~bJ*(0) = 1 which proves that it is a proper distribution. Here we make use of the fact Q~a*(0) = 1 ** 0 ) + Q 1** Q12( 3( 0 )=1 ** 0 ) + Q 2** Q20( 4( 0)= 1 Q~'~'(0) = 1. Now MTSF, given that the system started from the beginning of state 0 is given by 1 - ~b~*(s)
TO = l i m S~O
S
Using equations (3.2) and (3.3) and L'Hospital rule, we get 2 - w*(2)g*(2 )
T° = 211 - w*(2)g~(2)]"
(3.4)
Availability analysis AVo(t ) = e-at + 2 e - a t ( L C ) A V l ( t ) A V~ (t) = e -a~W(t) + e -~t w(t)
(LC)A V2(t) + 2 e -at 17V(t) ( L C ) A V3 (t) e-X~] (LC)A V2(t)
A V2(t) = e-~t ~3(t ) + e-2tg3(t)(LC)A Vo(t ) + g3(t) [1 AV3(t ) = g3(t)(LC)AV2(t).
-
-
(3.5)
40
K. MURARI and VIBHAGOYAL
Taking L.T. of above equations and solving for AV*(s), we get AVo*(S ) =
(S + ,~,)2A3 + 2(s + 2)A3w*(s + .~) + ,).[(s + 2)w*(s + 2) + 2w*(s + 2)0 ~'(s)]g* (s + 2) (s+2)[(s+2)2Aa_2[(s+2)w,(s+2)+2~,(s+2)O,~(s)]g~(s+2) ]
(3.6)
In steady state, expected up-time is given by 1 + ~*(2)g~(2) A V o - [2-w*()OJg'~(2)-20*'(O)[ 1 + ~,(2)g~,(2) ] .
(3.7)
Busy period analysis of the repairman ARo(t ) = 2 e -;~ (LC)ARI(t) A R 1(t) = e-;" w(t)(LC)AR2(t ) + 2 e-At W(t)(LC)ARa(t ) AR2(t ) = e -;" G3(t)+ 93(t) [1 - e -~t] (LC)AR2(t) + e - ~ Oa(t)(LC)ARo(t) + (~a (t) [1 - e -;.t] ARa(t ) = ga(t)(LC)AR2(t).
(3.8)
Taking L.T. of above equations and solving them for AR*(s), we get
2g~'(s) I-(s + ~)w* (s + 2) + 2~* (s + ;~)0~(s)] AR* (s) - s[(s + 2)2A3 - 2g*(s + 2) [(s + 2)w* (s + 2) + 2ff*(s + 2)0~'(s)] ] '
(3.9)
In the long run, expected time for which system is under repair, is given by - 207'(0 ) AR° = [2-w*(2)]g*(2)-20*'(O)[1 +~*(2)g~(2)]"
(3.10)
Expected number of ordinary visits by repairman OVi(t ) = expected number of ordinary visits by the repairman in (0,t], given that system just entered regenerative state i at t = 0. (By the term "ordinary" we mean, repairman comes by its own after placement of order.) OVo(t) = Qol(t)(SC)OVa(t) OVl(t ) = Q,2(t)(SC)[1 +OV2(t)] +Q13(t)(SC)OVa(t)
ov2(t)
= Q2o(t)(sc)ovo(t)+Q2~.,(t)(sc)ov:(t)
OVa(t ) = Qa2(t)(SC)OV2(t).
(3.11)
Taking L.S.T. of above equations and solving them for O V* (s), we get
2(s + 2)A3w*(s + 2) 0 V~(S) = S[(S ÷ 2)2A3 -- 2g~(S ÷ 2) ['(S÷ 2)W*(S ÷ 2) ÷ 2W*(S ÷ 2)g~(S)]] "
(3.12)
In the long run, expected number of ordinary visits per unit time, is
2w*(2)o';(2) OV° = [ 2 - w * ( 2 ) ] O * ( 2 ) - 20~'(O)[1+ w*(2)g*(2)]"
(3.13)
Expected number of immediate visits by the repairman IVy(t) = expected number of immediate visits by the repairman in (0, t], given that system just entered regenerative state i at t = 0. (By the term "immediate" we mean we call the repairman immediately, at the failure of both the units). IVo(t ) = IVl(t ) = IV2(t ) = IV3(t ) =
Qoa(t)(SC)IVI(t) Q12(t)(SC)IV2(t)+Qla(t)(SC)[1 +IVa(t)] Q20(t)(SC)IVo(t)+Q22.g(t)(SC)IV2(t) Qa2(t)(SC)IV2(t ).
(3.14)
Taking L.S.T. of the above equations and solving them we get
AEA31~*(S÷ 2) I V*(s) = s[(s + 2)2A3 - 2g~ (s + 2)[(s + 2)w*(s + 2) + 2ff*(s + 2)g~'(s)]]"
(3.15)
In steady state, expected number of immediate visits per unit time is given by 2~*(2)g~(2) IV° = [ 2 - w * ( 2 ) J g * ( 2 ) - 20*'(O)[1 +~*(2)g*0.)]'
(3.16)
Reliability analysis
41
Expected number of repairs for an ordinary visit of the repairman ROVi(t ) = expected n u m b e r of repairs for an ordinary visit of the repairman in (0, t], given that the system just entered regenerative state i at t = 0. ROVo(t ) = QoI(t)(SC)ROVI(t) ROVI(t ) = Q12(t)(SC)ROV2(t)+Qta(t)(SC)ROVa(t) ROV2(t ) = Q20(t) • 1 +Q22.4(t)(SC)[1 +ROV2(t)] g o v a ( t ) = Qa2(t)(SC)ROV2(t ).
(3.17~
Taking L.S.T. of above equations and solving them we get
ROV*(s) -
,~#~(s)[(s + ~ )w*(s + ;0 + ,~*(s + ~.)#~(s)] s(s+2)2A 3
(3.18)
In the long run, expected number of repairs for an ordinary visit is given by
R O V o = lim [ROVo(t)] = lim [sROV*(s)] s~O
1
R O V o = 9,(2).
(3.19)
Expected number of repairs for an immediate visit by the repairman RIV~(t) = expected n u m b e r of repairs for an immediate visit of the repairman in (0, tl, given that the system just entered regenerative state i at t = 0. RI Vo(t ) RIVI(t ) RIV2(t ) RIVa(t )
= = = =
Qox(t)(SC)RIVI(t) Q12(t)(SC)RIV2(t)+ Qls(t)(SC)RIVs(t) Q20(t)(SC)RIVo(t) + Q22.4(t)(SC)[-1 + RIV2(t)] Qs2(t)(SC)[1 + RIVz(t)].
(3.20)
Taking L.S.T. of above equations and solving them for RIV*(s), we obtain
RI V* (s) =
,~1-,~* (s + 2)g~(s)zX3 + r(s + ~)w*(s + ,~) + ,l~,(s + ~.)g~(s)] • [g~(s) - 0~'(s + ,~)]] s(s + 2)2A3
In the long run, expected number of repairs for an immediate visit is given by
R I V o = lira [RIVo(t)] t~oO
= lim [RIV*(s)] s~O
1 - w*(2)g*(2)
o*(4) Cost analysis Similar to model 2, here also, we find profits by the same two ways. Both the profits are found out in steady state. P~ is based upon the total up-time, the total time for which system is under repair and the total number of ordinary and immediate visits by the repairman, i.e.
P~ = KoA V o - K s l A R o - K s 2 O V o - K s s l V o where, Ko = Ks 1 = Ksz = Kss =
revenue per unit up-time cost per unit time for which system is under repair cost per ordinary visit of the repairman cost per immediate visit of the repairman.
P~ can be calculated on the basis of total number of repairs done and the total up-time. Hence,
p2 = K o A V ° - Ka,t(ROV ° . O V ° + R I V ° .iVo) where, K o has the same meaning as above and Ks,~ is the cost per repair, done by the repairman.
42
K. MURARI and VIBHA GOYAL
Putting values of AV0, ARo, OVo, 1Vo, ROVo, R I V o from equations (3.7), (3.10), (3.13), (3.16), (3.19) and (3.22), we get p~ = [1 + u?*():)g~(),)]K o + ),g~'(0)K31 - 2w*().)g~()OK32 - ,:~w*(2)g~(,~)K33 [2 - w* (),)]g* 0.) - ).g*'(O)[1 + u?*(2)g* (2)]
(3.23)
and p~ = [1 + v~*(2)g*(2)]K o - 211 - w*(iO~*()Og~(iQ]K34 E2-w*(2)]g*(2)-itg*'(O)[1 + ~,*(2)g*(2)]
(3.24)
COMPARISON
Here we compare the profit obtained in different models. This comparison is done in two parts. Part 1 considers the comparison of profits in different models while part 2 carries the comparison of profits in the same model, obtained by two methods. Part 1 Comparison of model I and model 2 F r o m (1.8) and (2.15), we have go
P'
g*(2)-).g*'(O)
K,t
p~ = K0 + 2g~"(0)K21 - ")~g~('~)K22 g~(2)-2g~'(0) A computer program is given in the appendix for the comparison of P1 and P~' (program 1 of Appendix). Let us assume that model 2 is better than model 1 by at least a quantity M D I P (i.e. minimum difference in profits). N o w we find out the conditions on the cost K22. For example, if we take 2=0.1,
u 1 = 1.0,
K 0 = 10,
K l l = 15,
u 2 =0.9. K21 = 16
and M D I P = 10 Pz' > P, + 10
i.e. and this will be so only when
K22 < 36.2.
Comparison of model I and model 3 F r o m (1.8) and (3.23), we have
PI -
P~=
go
g * ( 2 ) - 2g*'(0)
K11
[1 + ~*().)g*(2)]K 0 + 2g*'(0)K3, - 2w*(2)g*(2)K32 - 2&*(2)g~'(2)K33 [ 2 - w*(2)]g'~(2) -).g*'(O)[1 + ~*(2)g*(2)]
Let us take repair rate of model 3 and model 1 in such a way that mean time to system failure becomes same in both the models i.e. if
g3(t) = u 3 e . . . . gl(t) = u, e uLt
then take AU1
u3 = (2 + u , ) w * ( 2 ) - u , "
Then the total up-time of model 3 becomes larger than the total up-time of model 1. Assuming the above, a computer program is made to compare the profits in two models (program 2 of Appendix). For example, if we take ~=0.1,
u, = 1.0
Reliability analysis
43
parameters of waiting time distribution as m = 5, K o=41.0, and
p = 2.0
K l 1 = 12.0, Kax = 3 3 . 0 M D I P = 10.0
i.e. model 3 will be better than model I by at least 10 units if the following inequality holds. w*(~)K32 +w*(,4)K33 < 8.14. If K 3 3 - K 3 2 = 20 (say), then for the inequality P~ > P1 + 10 to hold K33 should be less than 27.17 and K32 should be less than 7.17 units.
Comparison of model 2 and model 3 From (2.15) and (3.23), we have
P~ = Ko -I-~.g~'(O)K21- 2g~'(~.)K22 o~(,l)-,~o*'(o)
p~ = I-1 + ~* (,~)g* (,~)]Ko + ,~g*'(0)K~ 1 - Zw* (;0g~'(,~)K32 -,~¢*(,~)g~'(,l)K33 [2 - w*(2)2o*(2) - 2g*'(0)E 1 + ~*(2)g~'(2)2 Taking again, the M T S F of models 2 and 3 as equal, a computer program is made to compare the above profits. (Program 3 of Appendix.) For example, if 2=0.1,
u2=l.0
m=5,
p=2.0
parameters of waiting time are
and K o = 31.0,
K21 = Kal = 11.0,
K22 = 32.0,
M D I P = 1.3.
So we want P31 > P21+ 1.3. This will happen only when K33 < 59.47.
Part 2 Comparisons in model 2 Profits, found by two methods are
p~
Ko+29~'(O)K21-
,~g~ (,~)K22
g~'(,~)-,lg~"(0) p22 -
Ko-2K23 g~(,~)-,~g*'(o)
Program 4 of appendix, compares the above profits. Here repair time is assumed to have Erlangian distribution. For example, if £=0.1 parameters of repair time are n=5,
U2
=
2.0,
K21 = 1.0,
K22 = 12.0,
M D I P = 3.0.
So we want P~ > p2 + 3; this will happen only when K23 > 41.96.
Comparisons in model 3 Profits, obtained by two methods are p~ = [1 + v~*(2)g*(2)]K o + 2g~"(0)K31-2w*O.)g'~(2)K32 -).O*(2)g~'(2)K3a I-2 - w*(2)]g*(2)- 2g~'(O)[1 + if* (2)g~ (2)3
44
K. MURAR!and VIBHAGOYAL
and
P~
[1 + O*(2)g~(2)]K
o - ~[-1 -
w*(2)~,*(;.)g*(;.)]K34
[2 - w*(2)]g~'()~) - 2g*'(0)[ 1 + ~* (;.)g* ()~)]
C o m p a r i s o n between P~ a n d P~ is d o n e in p r o g r a m 5 of the Appendix. F o r example if X=0.1,
u 3 = 1.0
p a r a m e t e r s of waiting time are m = 5,
p = 2.0,
K31 = ll.0,
i.e.
K3a = 32.0,
M D I P = 5.0
P~ > P~ + 5
which holds only w h e n w*(2)g32 +ff*(/.)g33 > 79.79. This inequality reduces to K33 > 89.31 K32 > 79.31 where we have assumed t h a t
K33 --K32 =
10.
REFERENCES
1. J. G. Rau, Optimization and Probability in Systems Engineering, Van Nostrand Reinhold Company, (1970). 2. M. N. Gopalan and R. Subramanyam Naidu, Stochastic
behaviour of a two-unit repairable system subject to inspection, Microelectron. Reliab. 22, 717-722 (1982). 3. K. Murari and V. Goyal, Reliability system with two types of repair facilities, Microelectron. Reliab. (in press). 4. G. S. Hura, Enumeration of success paths in a graph, Microelectron. Reliab. 22, 1033-1034 (1982).
Reliability analysis
45
APPENDIX
Five computer programs for the comparison of models are given here.
Performanceguide
Program details
1. 2. 3. 4.
1. Specific word length limitation--none 2. Additional relevant information--none
Computer used DEC-2050 Input medium: punched cards Output medium: printer Work data files required: DSK.
Program I MAI{~.
P2PI.FOR
001 ~.~ 002. :
C C C C C C C
003<~:
004 ~:~ 005,~ 00~. ~ 007 ~,~ 008~'~ 009~ 010~ L 011~, 012~,:
021,,, 022i, , 023,Ji 024r, ~ 025,~ ,~ 02b~; ,,
C 10 11
SUHPRf]GRAM5 CO21 CCO
8QR
SCALARS
AND
ll-JUh-83
OF M O D E L
PAGE 1
I
CCtl ARRAYS
C "*"
:!O E X P L I C I T
*Cll
*01
20 25
11:28
CALLED
*DP}~FIT 2 .F~;O93 7 21
[ NO ERRORS D E T E C T E D
DEFINITION
*RQUAN .FO001
3 10 22
TEMPORARIES
MAIN.
/KI
CC21(~,~,C)=A%(A÷C)~SOR(A.B) , OPEl,( U I ~ I ' P = i , DP~VICE: ' DSK ' , F I L E = P2Pl. DAT t ) R~Ai~( I , ~) ,FRATEt HI.U2 R~;AZ;(I,*).CO,CII,C21,DPRFIT P l,,Ti . D P R F I T DIFF~:CO#CC0 (FRATE, U1, U2) ÷ C 1 1 # C C I 1 (FRATE, UI, U 2 ) ' C 2 1 ' C C 2 1 ( IFF, ATE. U I.U2) Ah=I~PRFIT*S'JR ( F R A T E . U I ) * S O R ( F R A T E ~ U2) R~,I]A~'~=(DIFF-AL) / ( F R A T E * ~ Q R ( F R A T E , UI) #U2#~2) F
*CO I .F:~¢06 6
, F;,~n4
V.5A(621)
COi~'AkI,~Oq OF MODEl, 1 AND 2 FGA LE:FAII,I~RE RATE UI=[~IPAIR RATE OF M O D E L 1 U%=~:E?AIF, RATE OF M O D E L 2 D P F F I T = P R O F I T OF MODEL 2 - P R O F I T C I U : C O S T OF MQDEL I CCId:CBEFF'ICIENT OF THE COST CIJ S'.~F (A, ~) :A*A+A*B+B*~ CC~ (A f rlt C) =A,A* (A+B+C)* (C-B)
Oi=h~: 015-., ~
016',~.~ 017:: 018,~.. 019i. 020..~
FORTEAN
]
-
N%n
NOT
*DIFF *FRATE .F0012
4 11
REFERENCED
]
.FOOl! *C21 .FO007
$ I~
K. MURARIand VIBHAGOYAL
46
Program 2 MAIN.
P3PI.FOR
00100 00200 O0300 00400 00500 0060C 00700 008~0 0090O OlOOO 0110O 01200 0130C 01400 01500 0160o 0170O
C C C C
C
C C
ll-OUb'B3
11:54
PAGE I
COMPARISONS OF MODEL 3 AND 1 FRATE=FAILUR£ RAT~.UI=R£PAIR RATE OY MODEL I M AND RHO ARE PARAMETERS OF WAITING TIME DISTRIHUTION (M -STAGE ERLANGIAI~ CCIJ=COEf'FICIENT)OF THE COST CIJ DCOST=C33-C32
DPRFIT= PROFIT OF MODEL3 - PROFIT OF MODEL 1 W(AnC~D)=(C*D/(A+C*D))**C SOR(AfB)=A*A+A*B+B*B ~ERM(A.B.C.D)=~*~÷A*CA+~I*(WCA~C.DI)**2
C C 0 ( A . B . C , O ) = A B (A~B+W(A.C,D) ( S O R ( A , B ) - C A + B } $ V 2 ~ W ~ A , C , D ) ) ) CC11(A,B,C,D)=SGR(A,~)'TERM(A,B.C,D)
CC31CA,~,CeD)=CA+R)*"CAeC,D)*((A+B)*w(A,C,D)'B)~OR(A,B)
CC32(A,C,D)=W(A,C.D) CC33(A.C.D)=I.'W(A,C.D). OP~i~(U~IT=I.D£VICE='DSK ,FIhE= BOUND.DKT') READ(I,#),FRATE,UIeM,RHO READ(I,*),C0,CII,C31 READ(I.~),DPPFIT,DCOST FM=FLOAT(M) D[FF=CO~CCO(FRATE,Ut.FM.RHO)+Ci%*CCII(FRATE,UI,FM,~HO)-C 131*CC31(FRATE U 1 F M RHO)
0180C
0190~ 0200C 0210~ 02200 0250~ 0260~ 0270o 02800 029O0 030O0 0310~ o32oe 03300 0340O 03500
FORTRAN V.5A(621) / K I
AL=DPRFIT*SQR(~RA~E.61)*TERM(FRATE,UI.FM,RHO) RQUA~=(~)IKF-AL)/((FRATE*UI**2)*SQR(FRAT~,U%)) PRI~CTII.RQUAh
10
11 12
0360~
BC33=ROUAN+DCOST*w(FRATE,FM,RHO) BC32=~C33"DCOST PRINTI2,DCOST,BC33.BC32 FOR.~ATC///.IX,'PROFIT OF ~ODEL 3 - PRL)FIT OF MODEL I > O 1R <'.FIO.&~///) F O R ~ A T ( I X , ' ( C 3 3 - C 3 2 ) * W ( F R A T E . F M . R H O ) - C 3 3 < OR >~,E17.10,/) FOR~TAT(IX.'IF C 3 3 - C 3 2 = ' ~ F I O . 4 , 1 0 X , ~ T H ~ C33 < OR >',E17. 110.'AhD C32 < OR >'rE17110) CLO~E(U~IT=I,DEVICE: D~ ) STOP END
SUBPROGRAMS CALLED W
CCO
CC31
FLOAT.
SCALARS AND ARRAYS • CO .FOOl6 *FRATE ,F0024
1 6 13 20
T~R~
VDPRFZT 2
.F0~12 7 ,8C33 14 .F~020 21
,R.o'F°°°3 4o33
, F 0 ~ 3 1 44 .FOO07 51
.F0026 45 .FO001 52
P3PI.FOR
.F 025
MAI~,
*RuUAN 3 .FO006 10 *M 15 , F 0 0 1 4 22 .F0021 ~ .FO000 34
*A6
.C+',sr,¢I I?.4
[ NO ERROS',q FJiT~:CT£t) ]
41
.F0023 46
F(]RT~Ah: V.SA(621)
T~MPORARI E.5 .Qx+~:O0 1.25
CCli
[ "*" m~O EXPLICIT DEFIr:ITIDN - "%" NOT REFERENCED
.Foo3o 28 ,o[
MAIN.
SOR
/i',I
*DIFF
.FO004 .F0032 ,FOOl0 .FOOLS *BC32 ~DCUST ,FOOl7
11-JU6-83
4
11 16 23 30 35 42 47
]
,F0022 *C31 .F0027 ,FO002 .r0oli ~Cll
5 12 17 24 3z 36 ~FM 43 .F0013 50
11854
P A G E 1-1
47
Reliability analysis
Program 3 MAIN. 001~
C
002;~
C
003¢. 004~. O05v=
C
C C
006~." 007'~
008~ 009~
010~ 011-~
II-,IUL-B3
II:26
PAG£ I
),C,i,)=A*(A+r~)*TKR:~(A,¢~,C,D)
C C 3 t (A, ~',C e D) = (A+~) *$,0R ( A , 5 ) * W ( A, C, b)* ( ( A + B ) * W (A, C, D,I'B) C C 3 2 (A, 3,C, D ) = A * B * * 2 * S O R (.A,h)* ~ (a.C, D) CPE-(d[~I'i)=ItL~EVICL'='D,.%K',FILE='P3P2.DAT') R,~:A,~( I , *) ,F'~ATr;, U2 z ~",~MC} ~,
015.:, 016~
017<:;; 018~
019,~ 020~,~ 021~> 022v,i
0236~, 026v~
9
02U~ 029.¢
II
027C::
/KI
C~h~PARISL),i OF ~'ILID~lh8 3 ,~141 ~, 2 CCIJ=C{);-:FrlCIKhT (,'F TH~S COST CId FRAT6=FAILURE HA.~E.,U2=RKPAIR RATR OF' :~'o.r)F:L2.M A~D R:i[) ARK PAWA,'~.TLR,S :)F ,~AITIhG TI~E DI~TRI~UTIO~J(ERL, ANGIAN) D ? R F I ~ = P R O F I T OF hOt)El, 3 - PROFIT OF xOV)EJ, 2 ~.(A e C, L]) = C C*,'~/(A+C~D) ) **C S l-~~, t A e £ ) =~\* A+ A ' b + 6 * ~, ~:]R:: (A, H,C, 7;)= 8 * B + A * ( A+F~)* (~ { A ~ C, D) )*'2 CC2[(A,
012~ 013:,~ 014~,~
024v.~ 0256~
r',~I~'rPg:i V . L A ( 6 2 1 )
P 3 P 2 . FLtR
F :,'=~L~A i'( :,: ) O I F F = C J ~ C C O ( F R A T E , U2, FM, RHO) + C 2 1 * C C 2 1 ( F R A T E , U2, FM, IKHO) +C 122~CC22(FEA'I'['~,[i2,FX,v~nO)'C31*CC31(F'RATIS,U2,FV',RHO)'C32* 2 C C 3 2 ( I" F(APF. ~ [)2.FN.RP,(~) AL=~;,P*EFI T*Sq)E ( F R A T E . U 2 ) *TER.~ ( F'RATE, U 2 , F ~ , RHO) ROOA.~= (~-,IFF-AL) / ( ( F W . A T E * U 2 * * 2 ) *[;OR [ ['~ATF., iJ2)* ( I .-w ( F R A T E 1. F.,,,~ i~itg) ) ) P R I ' : ~ 1 I.R~.U~f" F!3R,,,AT(IX, P~F)F'IT OF ,~(]I:EL 3 - PHUFIT OF ~(3DEL 2 > OR <'
I~F13.4,//)
FC~R:~,%T(IX, ' C 3 3 >
OIK <' ,El7.1(!,//)
C~,,~,<.E ( l! ',~I T= t )
~TOf E':,>
030~ 031~c
SUBPRnGRAMS CALbE~ W
CC21 SQR
COO CC22
SCALARS
AND
*CO .FG~26 .F0504 *C32
! 6 13 20
,F;020 25 .F0025 .FJO03 *F'M .FU013
32 37 44 .51
CC32 ARRAYS
[ "*"
*[>PRFIT, .F.,022 *C3~ ,o2
CC31
~C E X P L I C I T
*ROUAh .F0016 *FRATE ,FG034
33 4C 45 .Fi.:O~7 52
.FC015 *C22 .Fo027 .FOOQ1
oF~','~)I~ 2e, .F,~021 .~'.30(,0 .Fv(,33
MAIN.
76
,,~I;Ovl
77
[ NO ERRaR5 DF~T~CTF.D ]
]'VR~
DEFII;ITION
7. 7 14 21
T~..MP OR A~ I E,~; .(~O000
FbOAT.
o~0,)10
3 lu 15 22 27 34 41 ~:6 53
- "%"
NnT R E F E R E N C E D
]
*DIFF 4 , F O 0 1 2 11 *C21 16 oF'O030 23
.F0032 .EO006 ~M .F0024
5 12 17 24
*RHh .~0023
~ AL .FO017
~3 50
. o o2 3o . oo11 3s 42 47
. F o o 3 1 31 .Fooo5
K. MURARI a n d VIBHA GOYAL
48
Program 4
MAI~'~.
P 2 . F ~ R - F,*,RTb~c, *
001 : * 002" 003~;~ 0 0 4 "~ 005 ,,
C
007 008 009
C C C C
, ,~ -
0 1 0 :~ 0il 012 • 01,~ ' 014: ' 015-, 016, , 0 1 7 ~" 0 1 8 .., ~ 01q '~ 020 ~' 021 , L 022.
C
I0
V.SA(621)
/K)"
3<'-JU~.-83
II:31
PAGE I
FRA .'CK:F ~ ! iA,.o,~ RATE N ~D o2 ~R~: PARA~ETEk,~ t~F ~ K P A I R T I M £ ( E R L A N G I A t ~ ) RPRTID=~q~nV!T I~ P A R T I - P P O F I T Ih P A R T II CI,.I=CnST G (AF B F(:) = ~ g~*Cl ( A + ~ * C ) ) * * ~ ~ERQG(A,~,C)=.I./C " . ~OE' ( o > I ' F : I , D F V I C £ : ' D S K ,FIbF~: P 2 . D A T ' ) ~TAi,( i , ¢) , F R A T F r N ~,lJ ~. RZA!'( 't, * ) , C21 , C22, DPRFI!' F'~:F[,OAT ( :! ) VAhUF;:C22*(; ( F~ATI':, F~;, {J2 )'C'2 I*I)ER VG(FRATE,FN, U2) ~.I,:,',PRF 11'* (G (FIXATE, F'N, U2.)-FHAT~:*D~RVG (F P,ATE,FN, U2) )/FRATE ROll.~ '! : V :\~'.L:~+A G R ~ i I ; w : J = ~ i ~ l ; T RED OUAh T I'.I'¥ P R I , i T I ;,';~HFIT,~OLiA!~ T Y P ~.~ . , i ; P ~ ) F I T , R O U A i , F n R A rcL.Kt,pRc;FIT [ ~ RA~LT I - P R O F I T IN P A R T II > OR < ' , E l 1~: 4 5,',, ,;,,bY ~,:HE;'i C'2~ > r!R <~ , E ~ 7 . 1 0 ) St'OP
$Ut~p~OGRA>! .~ CALhE.) DERVG FbOAT.
(;
SCAbARS A~;D ARRAYS [ *DPRFIT 1 *FRATE 5 *FN 1[ .Y;> 02 t 6
"*"
*KNt~A . *C2J. *C2L,
~; E X P L I C I T D £ F I ' I T I { I N 2 '~ t2
TEMPCU'AKIH,
,6) ~OO 4 t MAI~q.
[
,C;'::~,,l
.,2
;~0 £RRO.RS P/'ri~c'UPrl[' ]
"%"
NOT R£EERENCED ]
*r,
3
%.F0004
.FO001
4
*u2 *VALUE
7 13
%,F0003 *AI,
,FO000 .FO005
10 15
14
Reliability analysis
49
Program 5 MAIN, 001~.,,) 002:,
~
0030 ' 004(, " 005:., ~ OOboO 08o:> 0 9 '~ . 0095. 010o,. 0105011"r. 0120 • 013~?,' 014(~7, 015c, 0160 0181,.: 019o'~ 0245: 0 2,~ 013C
~
o34,>::: 035',,'. 0 3 6 .,,, 037:):' 0376" 0378.' 038(;" 3 8 5 :, 41o., 0415:" 0440~ 0445~.' 051:)3 052~> 0560?
P3.FOR
FOP'TF,,,~
v.%/.(~21)
C C C C C C
CC!'P~I;JS<<.,%
]I
/~I
3,"-obl -~3
1~:32
C
I
, CL E.!. i:,
U 3 : b' F)) A .,'.I- )'~ T~: U ;,'[> Fh .~ I-'IEI, t,'L'],i'F,,% OF" ~AITI),('. T.T~[: ['I::,TF].P.b'I'IL)N LAi~r;I/~ ' ) DPFFIT=;'IKf;FII i:, P A W T II - PF,'C,FIT I~- PAI
C C
PAGE
(.EI~
~(A ,C,I~ )=C C*I'/ ( A*C*I> ) )**C DE:~.(~- ( t, ~ i", C , I.) : t : * P . + F ~ I A + A * A + h * ( A + b ) * ( 1 .-%,. ( A, C , D ) } CC35.C;',,5,C,D)=A*(A÷£,) CC32(A,~,,C,[)=A*~:*~*~(~,C,|~) -,'" C C 3 3 ( k , k., C, [~) =A*~*F.,) ( I . ' , L A ¢ C , [ } J CC3 ~(;,,",,C,5)=A*B'(A÷5",'(A,C,F,)*( I. -~'(A,C,I}) )~B) OPE:i(Ui.[T=I~F;EV~C~,='[)~b ' , F I b E = ' P 3 . L A I ' ) REAr; ( I, ' ) , F'KAq'E:tlj3, ~, ~ H[.~ REA;>(~ , * ) , C g I , ( : 3 4 , f ' P R F I T R~;AI )( ! , * ) , t;'C~,'.,T PRII~Tg,FF~FI',r F ~=!"I,[)AT ('~) D[ V " = C 3 1 * C C 3 ~ ( F'~
C
TIP:':12, OCt)CT, ~ C 3 3 , b C 3 2 F',:~',~,AT(IX,'PR~]FIT .If~ FAF~ 11 - P~UF].T IF }'AI~T I • OR <', IF~.2I///) FU~.~ r(I.Y,'C32*;;(~WAT~:,FP,,p~o)÷C33*(1 -~-(F~AT£,F'M,RHO))> 1 ":!" < ' , K t T . I L " . / ) r O ~ : A ! ' ( l . < , ' I r C 3 3 - C 3 2 = ' ~ B . P ~ 5 X , ' T I * E ~ C33> OR <),F~17.10, 15X.','~L[) C32> .'.IF' < ' , E 1 7 . 1 0 e / / ) Ch{~SE (U-, IT= 1 )
9
11
12
057,!~
STC, P " L'-, ~
SUBPRfJGRAw,o C A L L E D W
DENOM
CC31
FLCAT.
CC'3~
8CAI, ANS A~'D ARRAYS { " * " *DPRFIT .F0~20 :~FRATE
I
*R(,~i~,
6
.F''~I4
~.F EXPLICI'I bEFI~,ITI(~r, 2
7 1.¢
*C3q
*~.C33
1.3
*~:
.F(:(:IC." J('
20 24 31 36
13
*~1~
MAIN.
PJ.FOR
FORTR~t: V.LA(821)
~I 25 37 ~.7
TEMPORARIE3 .O,,;;~O 117 MAIN.
MR 24/I--D
[
.(~'.,OC;I 12C
r~L) ERRORS DETECTED ]
"%" r,N']" IxEF~;RE.NCEb ]
3
.F0.922 .Fo~;23 ,re-:oe ,~FM
.F)C;Ih F * A C ~ ' 2I 7 . F _25
-
*1 IF'f"
g.FLOI2 .FC. CI3 26 * FKHF. 33 .~t021 '-,C,
/KZ
.~(,c't'4
.~OC(;b FC'O,.7 */,l, .F(.,(315
30-dUN-83
4
11
16
22 27 3~ 41
11z32
.~0024
5
.~O02b
17
*C31
12
.FOC('2 23 .FUO03 30 *DCOST 35 %.F('.01|
PAGE I - 1