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Availability analysis of a repairable standby human-machine system
~
t Pergamon
Microelectron. Reliab., Vol. 35, No. 11, pp. 1401-1413, 1995 Elsevier Science Ltd Printed in Great Britain 0026-2714/95 $9.50 + .00 0026-2714(95) 00038-0
AVAILABILITY ANALYSIS OF A REPAIRABLE STANDBY HUMAN-MACHINE SYSTEM
N1ANFU YANG and B.S. DHILLON Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, KIN 6N5
(Received for publication 29 December 1994)
Abstract
- - This paper presents a model representing a two units active and one unit on
standby human-machine system with general failed system repair time distribution. In addition, the model takes into consideration the occurrence of common-cause failures. The method of linear ordinary differential equation is presented to obtain general expressions for system steady state availability for failed system repair time distributions such as Gamma, Weibull, lognormal, exponential, and Rayleigh. Generalized expressions for system reliability, timedependent availability, mean time to failure, and system variance of time to failure are also presented. Selected plots are presented to demonstrate the impact of human error on system steady state availability, reliability, time-dependent availability, and mean time to failure.
Introduction Various research publications have discussed the reliability and availability of human-machine systems [1-5]. But most of these publications assume that the system repair times are exponentially distributed. In practice, repair times often are not exponential distributed. This paper presents a two parallel units active and one unit on standby human-machine system with general failed system repair time distributions. Furthermore, the system/unit may fail either due to a common-cause failure or a critical human error. The standby unit is activated only when one of the operating unit fails. At time t=0, the parallel units start operating with the remaining unit in its standby mode. As soon as one of the parallel operating unit fails, the standby unit is switched into operation. At least one unit must function normally for the system to operate successfully. The human errors and common-cause failures can occur from any of the operable states of the system. The system can fail either due to a critical human error, a common-cause failure, or normal hardware failures. The following assumptions are associated with the model: 1) Common-cause, human error, and hardware failure rates are constant. 2) All failures and human errors are statistically independent. 3) All system units are identical. 4) The failed system repair times arc arbitrarily distributed. 5) The occurrence of a common-cause failure or a critical human error causes the entire system to fail from any of its operable states. The repair process begins soon after a unit failure. 6) The repaired unit or system is as good as new. 7) The switching mechanism for the standby is considered automatic and instantaneous. 8) No further failure can occur when system has been down.
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Nianfu Yang and B. S. Dhillon
Notation The following symbols are associated with the model under study: i ith state of the system (see system transition diagram of Figure 1): i=0, two active plus the one standby unit are in perfect working condition; i=l, one operating unit failed due to a hardware failure, the standby unit is switched into operation; i=2, one unit active and no standby; i=3, the system failed due to hardware failures; i=4, the system failed due to a common-cause failure; i=5, the system failed due to a critical human error. s Laplace transform variable ~.~
Constant common-cause failure rate from state i to state 4; for i = 0, 1, 2
L~ r~
Constant critical human error rate from state i to state 5; for i = 0, 1, 2 Constant hardware failure rate of a unit in state i; i = 0, I, 2
~tj
Constant repair rate of a failed unit in state i; i = 1, 2
~tj(x)
Time-dependent system repair rate when the system is in state j and has an elapsed repair time of x; forj = 3, 4, 5 The probability that the system is in state i at time t; for i = 0, 1, 2 The probability that the failed system is in state j and has an elapsed repair time of x; for j = 3, 4, 5 The steady state probability that the system is in state i; i = 0, 1, 2 ..... 5 The Laplace transform of P~(t) The mean time to system repair that the failed system is in state j and has an elapsed repair time of x; forj = 3, 4, 5 Steady state availability of the system Steady state unavailability of the system Time-dependent system availability probability density function (pdf) of the system repair time when the system is in state j and has an elapsed repair time of x; for j = 3, 4, 5
P~(t) Pj(x,t) Pi P~(s) Ej(x)
AV~ UAV. AV(t)
Nj(x) Nj(s)
Laplace transform of Nj(x).
Descriptive Equationsof The System The differential equations associated with Figure 1 can be written as follows: dP o(t._.__~) = - ( r o + ~.~o+ Lho)Po(t) + p~PL(t) +S:P3(x, t)l.h (x)dx + dt
(1)
dP,(t) T = r°P°(t) - (~t, + r~ + ~ , + ~.ht)P~(t)
(2)
+
~t2P2(l )
dP2(t) = r~Pt(t) - (~t2 + r 2 + ~.~2+ L~2)P2(t) dt 0 P3(x,t) 0 P3(x,t) - + ~ =-~t3(x)P3(x,t) Ot Ox O P4(x,t) Ot
+
O P4(x,t) ~ Ox
=-~,(x)e,(x,t)
0 P-~(x, t) 0 Ps(x, t) - ~- = -laffx)P~(x,t) c3 t c3 x
(3) (4) (5) (6)
The associated boundary conditions are as follows: P3( O,t ) = r2P2( t ) P4(O,t) = L.oPo(t) + Z.c~Pt(t) + Z.c2P2(t) P.frO, t) = LhoPo(t) + khlPl(t) + ~.h2P2(t)
At time t = 0,Po(0) = I,P~(0) = P2(0) = O, Pffx,O) = P4(x,0) = Ps(x,0) = 0
(7) (8) (9)
Availability analysis
1403
,--1
rts(x) Figure 1.
The state transition diagram of the system
Steady-State Availability Analysis As time approaches infinity, Equations (1)-(6) reduce to Equations (10)-(13), respectively.
,,:o - ~,:, = f?,(x,t)~,(x)~ +~o;°,(x,,).,(x)a~ + f?,(x,o~ (x)~ top o - a , P
+ la,P2 = 0
(1 I)
~P~ -a2P2 = 0 0 P,(x)
c3x
(lO) (12)
= -~t,(x)P,.(x)
(.for i = 3,4,5)
Similarly, the boundary conditions become: P3(O)= r2P2
(13)
(14)
/4(0) = ~.coPo + ~.c.P~ + ~.c2P2 Ps(0) = ~,oPo + ks~P, + k,h2P2
(15) (16)
where a 0 = r 0 + k,co + k.ho a I = r~ + ~,c) + ~
+ ~tI
a 2 = r 2 + L:2 + ~ 2 + la2 a3 = aI + a2
Pi is the steady-state probability that the system is in state i, for i = 0, 1, 2, ..., 5, and
g
=f:Pi(x)dx
f o r i = 3, 4, 5
(17)
Also, 5
y~=1
(18)
i:0
Solving differential Equation (13), we get Pf(x)
P,(O)exp(-fo~(to)do)
( f o r i = 3,4,5)
(19)
Thus, from Equation (17) and (19), we have
= f [ P3(O )exp(-fo; 3(co)do))dx =
(20)
r2P2Es(x )
Similarly, P. = I ? . ( x ) d x
=I: = (~.coPo + ~1Pl + k~2P2) E4(x)
(21)
1404
Nianfu Yang and B. S. Dhillon
= J'o Ps(0)exp(40~5(o)do)dx
(22)
= (~.hoPo + )~hiP~+ ~.h2P2) Es(x) where
(23) (for i = 3, 4, 5) E3(x), E 4 x ) and E s ( x ) are the mean times to repair from state 3 to state 0, from state 4 to state 0, and from state 5 to state 0, respectively.
Solving the set of Equations (I 1), (12), (18) and (20)-(22), we get the following steady state probabilities: Po = b° (2,1) BA(2,1)
( 24 )
p~
b I (2,1) BA(2,1)
( 25 )
P2 = b2 (2,1) BA(2,1)
( 26 )
P3 = b3(2'l)E~(x)
( 27 )
BA(2,1) ( 28 )
P4 = b4(2'l)E4(x)
BA(2,1) ( 29 )
p, = b~(2,1)E~(x)
BA(2,1)
where bo(2,1) = ala2 - rl P.2 b1(2,1) = a2ro b2(2,1) = rorr b3(2,1) = ror~r2 2
b, (2,1) = ~ b. (2,1)~.. i0 2
b,(2,1) = if'b, (2,1)~.,i i,0 2
5
BA (2,1) = ~-'6, (2,1) + ~ bj (2,1)E, (x) i-o
J =3
The system steady state availability is ,4 v~(2,l) = Po + ~ +1'2 2
~-" b, (2,1)
(30)
BA(2,1) Similarly, the system steady state unavailability is given by UAV~(2,1) = P3 + P, + P, 5
~"bj (2,1)Ej (x) .~-
j=3
.
BA(2,1)
(3l)
Availability analysis
1405
System Steady State Availability Special Cases Case I If the system repair time x is repair time is given by
Ni(x)
Gamma distributed and probability density function (pdf) of the
F(I 3) exp(-lajx)
(32)
(forj = 3, 4, 5; x_>0, 13>0, ~j>0 ) where 13and ~j are two parameters of the Gamma distribution. Thus, the mean time to repair Ej(x) is
E/(x) =f: xNj(x)dx -- ~---
( 33 )
(forj = 3, 4, 5) Substituting Equation (33) into Equation (30), we get the following resulting system steady state availability for the Gamma repair time distribution: 2
)'~ b,.(2,1) i~o s A V~(2,1) = 2 ~ b , (2,1) + '~b~ (2,1) 13/ta) i :0
( 34 )
j=3
If13 is an integer F(13) = (13 -1)!
( 35 )
Thus the Gamma distribution becomes
Erlangian distribution.
For 13 = 1, the Gamma distribution reduces to exponential distribution, and Equation (34) is the system steady state availability for the exponential repair time distribution. C a s e II If the system repair time x is
Weibull distributed, the pdf of the repair time is expressed by Ns(x) = pj~ 13x~-~exp(-~ts~x ~ ) ( 36 )
(forj = 3, 4, 5; x_>_>0,13>0, pj>0 ) where 13 and ~tj are two parameters of the Weibull distribution. Thus, the mean time to repair Ej(x) is given by
Ej(x) =f[xNj(x)dx = F(I + 1/13)
( 37 )
Pj (forj = 3, 4, 5) Substituting Equation (37) into Equation (30), we get the following system steady state availability for the Weibull repair time distribution: 2
'~-~b,.(2,1)
/:o
( 38 )
A V= (2,1) = ~-'~b,(2,1) + )-~bj (2,1)F(I + 1/13)/Pu i o
./=3
C a s e III If the system repair time x is Rayleigh distributed and the time dependent failed system repair rate and pdfofthe system repair time, respectively, are defined by
~j(x) = ~:2x
( 39 )
0=3,4,5) and N j(x)
= laj 2 x exp(-~tj zx" / 2)
(for j =3, 4, 5, x~_>0,p.j>0 )
( 40 )
1406
Nianfu Yang and B. S. Dhillon Thus, the failed system mean time to repair Ej(x) is expressed by
l~
(41)
(forj = 3, 4, 5) Substituting Equation (41) into Equation (30), we get the following system steady state availability 2
~-" b, (2,1) ,=o
A V~ (2,1) = 2
( 42 )
Eb,(2,1)+ i=0
j=3
N If the system repair time x is lognormal distributed, the pdf of the repair time is defined by =
N j(x)
1 , - - expl (In x -~tj)2.1 x % 42~ [ 2o: 2 J
( 43 )
(forj = 3, 4, 5, and x_>0) where pj and oj are the distribution parameters (mean value and standard deviation of In x, respectively). Thus, the mean time to repair Ej(x) is given by
E:(x)= ~ : x N : ( x ) d x
= exp
+
( 44 )
(forj = 3, 4, 5)
Substituting Equation (44) into Equation (30) yields the following system steady state availability: 2
~b,(2,1) AVe,(2,1)= 2
,~0
(45)
y b,(2,1)+Eb,(2,1)exp(p, i~0
i:3
System Steady State AvailabilityNumerical Examples Setting ro =r t =O.O005/hr 3¢o = L~I = 0.00005 / hr
r2 =O.O008/hr L2 = 0.00002 / hr
~t~ =O.O01/hr Lht = 0.0002 / hr
P2 =O.O02/hr ~2 = 0.0001 / hr
in Equation (30) leads to the following:
AV,, =
5.82 [2.0Effx) + 2.835E4(x) + 3.17 Effx)] x 10 4 + 5.82 + 4.1 lEs(x)Lho
The plots of system steady state availability as a function of human error rate, Lho, are shown in Figures 2 and 3 for different values of 13 and for Gamma and Weibull distributed repair times, respectively, for specified values of parameters P3 = 0.001 /hr, la4 = tx~ = 0.0009 / h r . Figure 4 provides the results of the system steady state availability as a function of human error rate, Lho, for lognormally distributed failed system repair times (p~ =1.001, la 4
= p.~ =1.0009, ~3 =o4 =o~ = o ).
Also, the plots of the system steady state availability as a function of Z.ho are shown in Figure 5 for Rayleigh distributed failed system repair times for specified values of parameters. Figure 6 shows plots of system steady state availability as a function of human error rate, Z,ho, for specified values of parameters la3 = 0.001 / hr, la4 = kt5 = 0.0009 / hr, for different system repair time distributions: exponential, Gamma, Rayleigh, Weibull and lognormal (la 3 = 1.001, ~t4 = P5 = 1.0009, o 3 = o 4 =cr~ = 6 = 3).
Availability analysis
1407
1.0 0.9 0.8 >, 0,7 <
0.6
N co 0.5 N 0.4 E ~, 0.3 0.2 0.1 0.0
0.0000
h
0.0002
0.0004
0.0006
0.0008
0.0010
Human Error Rate, ~,0
Figure 2.
System steady state availability vs. ~'~oplots for the system with constant human error rates and the Gamma failed system repair time distribution 1.0 ~=2
0.9 0.8 ~5
0.7
<: 0.6 co 0.5 .~. 0.4 ¢n E 0.3
'~ 0.2 0.1
0.0 0.0000
0.0002
0.0004
0.0006
0.0008
-' 0.0010
Human Error Rate, L,o
Figure 3.
System steady state availability vs. ~'hoplots for the system with constant human error rates and the Weibull failed system repair time distribution 1.000 0,999 0.998
0=0.3
0.997 <
O=t .t)
0.996
ca 0.995 N
0.994
E 0.993
a=l.5
¢n 0.992 0,991 0.990 0.0000
0.0002
0.0004
0.0006
0,0008
0.0010
Human Error Rate, Xho
Figure 4. /'IR31-11-C
System steady state availability vs. Xh0 plots for the system with constant human error rates and the lognormal failed system repair time distribution
1408
Nianfu Yang and B. S. Dhitlon 0.9 0.8; 0.7 0,6
V.~=0.0008 0.5 !
p.~=0.00(16
0.4
~=0.0004
Go
o3
0.3
~=0.0002
~3 = O.O01/hr
0.2
p,, = O.O009/hr 0.1 0.0 0.0000
0.0001
0.0002 0.0003 0.0004
0.0005
0.0006
Human Error Rate, ~,ho
Figure 5.
System steady state availability vs. ~'hoplots for the system with constant human error rates and the Rayleigh failed system repair time distribution
Time-Dependent System AvailabilityAnalysis Using Laplace transforms and the initial conditions in Equations (1) - (9), we get
sPo(s) : 1 - aoPo(s) + ~tPt(s) + f ~P~(x, s ) ~ ( x ) d x + ¢ U
(46)
÷I/ , ( x, s ).,(x (s + a t ) P l ( s ) - roPo(s) - ~t2P2(s) = 0 (s + a2)P2(s ) - rlP~(s ) = 0
( 47 ) ( 48 )
~P~(x'S)+sP~(x,s)+~t,(x)P,(x,s)=O
(for i = 3,4,5)
c~x
(49)
and the boundary conditions: P3(0,s) = r2P2(s) P , ( 0 , s ) = X,.oPo(s) + ~.~,e~(s) + ~.~2P2(s)
( 50 )
P f f 0 , s ) = LhoPo(s) + L~lPt(s)+ Lh2P2(s) 1.0 ~
_
_
i toOnormaldistribution(o = 3)
•
•
j
\
N
s, oonoo,i d=,bot,on
•
~ 0.5
0.4 f 03 ~ . 0.0000
Gammadistribution (.3 = 2}/
. . . 0.0002
0.0004
, 0.0006
~, 0.0008
= 0.0010
Human Error Rate, k.o
Figure 6.
The comparision of AVs, vs. Lh0 for the system with constant human error rates and different failed system repair time distributions
Availability analysis
1409
Solving differential equation (49), we get P,. (x, s) = P,. (0, s)e %xp(-Io~, (m)&o)
(for i = 3, 4, 5)
(5~)
Substituting Equations (50) and (51 ) in Equation (46), lead to [s
+
a o - k~oNa(s) - ~oN.~(s)]Po(s) -
[p., + k~,N,(s) - £hzNs(s)]P~(s) [rzN3(s ) + L2N,(s) - kh2Ns(s)]P2(s) = 1
(52)
where Ni(x) is pdf of repair time and Ni(s ) is the Laplace transform of Ni(x )
N,(s) =IoeXp(- x)N,(x)a N,(x) = ~t, (x) exp[-Io~, (co)aco]
(for i = 3, 4, 5)
Since we know
P,(s)
(53)
(for i = 3,4,5)
Thus, utilizing Equation (51), we get 1- N 3(s) P3(s) = t ~ P 2 ( s ) - -
(54)
S
P,(s) = [A.coPo(s) + kciP,(s) + L.2P2(s)] I - N , ( s )
(55)
S
Ps(s) = [knoPo(s) + kh,P~(s) + kh2 P2(s)] l - Ns(s)
(56)
S
where 1 - N~(s) - I o e - = exp(-Io;,(o~)do))dx
(for i = 3,4,5)
(57)
S
Solving Equations (47), (48), (52) and (54) - (56), we get the following Laplace transforms of state probabilities Po(s),P,(s) ..... and P~(s): Po(s) = b°(2'l) + (al +a2)s+s2 CA(2,1)
( 58 )
P~ (s) = b, (2,1) + ros CA(2,1)
( 59 )
P2(s) = b2(2'1) CA(2,1)
( 60 )
P3(s) = b3 (2'1)[I - N3(s)] s. CA(2,1)
( 61 )
& (s)
[1 - N, (s)][b, (2,1) +
le(Z,l).s + )~cos:]
s . CA (2,1)
p~(s) = .[l_Us(s)j[bs(2,1)+lh(Z,1).s+~hos2] . . r s. CA(2,1) where CA(2,1) = cal(2,1) + ca2(2,1), s + ca3(2,1), s 2 + s 3 5
cal(2,1) : y [1- N ,(s)]b,(2,1) j=3
ca2(2,1) = '~-~b, (2,1) +/c(2,1)[1 - N4 (s)] +/h (2,1)[1 - N 5(s)] i=0
ca 3(2,1) =/0(2,1) + kco[1 - N, (s)]+ Lh, [I - Ns(s) ] /c(2,1) = (a~ + a2)~.co + ro~.c~ lh(2,1) = (a I + a2)~.ho + ro?~h, 10(2,1) = a~ + a 2 + r o
( 62 ) (63)
1410
Nianfu Yang and B. S. Dhillon The Laplace transform of the system availability is : ~'b, (2,1) +/0(2,1). s + S 2 A V(s) = i o
( 64 )
CA(Z,1)
Taking the inverse Laplace transforms of Equation (64), we can obtain the time-dependent system availability.
Time-Dependent System Availability Special Cases Substituting the Laplace transforms of the pdf of the system repair times, N3(s), N4(s), and Ns(s), in Equation (64) and then substituting the same applicable data of the earlier system steady state availability numerical examples into the resulting expression (also setting p,3=0.001/hr and p,4=~ts=0.0009/hr and taking inverse Laplace transform), we get the system time-dependent availability. For exponential failed system repair time distribution and ~,~o= 0.0002 / hr we get A V(t) = 0.765733 + 0.0115416 exp(-0.00355207t) +
0.00738703 exp(-0.00096036t) + 2.1631 sin(0.000186153t) exp(-0.00140379t) + 0.215338[cos(0.0001861530 exp(-0.00140379t) 7.54103 sin(0.000186153 0 exp(-0.00140379t)] For different values of Z,ho, the system time-dependent availability plots are shown in Figure 7. For Gamma distributed failed system repair time, the plots of the time-dependent system availability for varying human error rate, ~,hO,are shown in Figures 8 (13 = 2) and 9 (13 = 3).
System Reliability and MTTF With and Without Repair Setting ~3(x) = ~t4(x) = ~5(x) = 0 in Figure 1, the system of differential equations becomes: dPo(t) = - a o P o ( t ) + ~uP~(t) dt
( 65 )
dP,( t ) _ roPo( t ) - a~Pt( t ) + ~t2P2( t ) dt
( 66 )
1.00 0.95 0.90 0.85
0.60
~-ho = 0.0002
~, 0.75 L.0 = 0.0004 0.70
0.65 0.60
1000
2000
3000
4000
5000
Time, t
Figure 7.
Time-dependent system availability plots for the system with constant human error rates and failed system repair rates
Availability analysis
1411
1.0
o.91
5" 0.8
0.7
iiiiii.
o~
0.6
0.5 1000
2000
3000
4000
5000
Time, t
Figure 8.
Time-dependent system availability plots for the system with constant human error rates and the Gamma failed system repair time distributions ( 13= 2) ( 67 )
dPz(t) = r~P~(t)- a2P2(t) dt dP3( t ) = r2Pfft) dt
( 68 )
dP4(t) = 3.coPo(t)+ k.~Pl(t) + Xc2P2(t) dt
( 69 )
d P f f t ) = 3.hoPo(t) + 3.h~P,(t) + Lh2P2(t) dt
( 70 )
At time t = 0, P o ( 0 ) = 1 and P~(0) = 0 for i = 1,2,3,4,5. Using Laplace transforms in Equations (65) - (70), and solving the resulting set of equations, we can get the Laplace transforms of the state probabilities, Po(s), P~(s) ..... Ps(s). Thus, the Laplace transform of the system reliability with repair is 2
~ b , (2,1) + 10(2,1). s + s 2 R(s) = i=o
(71)
DA(2,1) 1.0 0.9 0.8 0.7 E 0.6
0.5 0.4 0.3 1000
2000
3000
4000
Time, t
Figure 9.
Time-dependent system availability plots for the system with constant human error rates and the Gamma failed system repair time distributions ([3 = 3)
1412
Nianfu Yang and B. S. Dhillon where DA(2,1) = dal(2,1) + da2(2,1), s + da3(2,1), s 2 + s 3 dal(2,1) = aoror1 - a2r&h da2(2,1) = ao(a ~ + a2 ) + rot~ - r&h da3(2,1) = a o + a I + a 2 System reliability with repair can be obtained by inverting Equation (71) e'~'(s°s~ - s o s , - s ~ s , +s2 2)
e " ( s ~ s b -s°s~ -,%s~ +sL 2) R(t)
+
-
(s, - s 2)(s, - s3)
-
(s 2 - s,)(s~
+
- s3)
(72)
e ~Jt ( s = s b - s=s 3 - s o s 3 + s~ 2)
(s~ - s~)(s~
- s,)
where s, and sb are the roots of the numerator of Equation (71), and sl, s2 and s3 are the real and unique roots of the denominator of that Equation. The system mean time to failure (MTTF) with repair is given by 2
Zb,(2,1) M T T F = lim R ( s ) = i=0
,~o
( 73 )
dal(2,1)
The variance of time to failure with repair of the system is expressed by tr -' = -21im R ' ( s ) - ( M T T F ) 2 s~o
2da2(2,1)~-" b~(2,1) - 2-10(2,1) • dal(2,1) -
b, (2,1)
( 74 )
i=0
[dal(2,1)] 2 Plots of the system reliability with repair are shown in Figure 10 for same applicable data of
the system steady state availability numerical examples. Figure 11 shows the plots of system MTTF with repair. Setting I.tl = ~ = 0 in Equations (72) - (74), we can get the system reliability, MTTF and the variance of time to failure without repair, respectively.
1.0 L"°=° ~'.o= 0.0002
~'J 0.8
0.6 ,-r" 0.4
0.2
0.0 2000
4000
6000
8000
10000
Time, t
Figure 10.
System reliability with repair plots for different human error rates
Availability analysis
1413
10000
8000
/~"
o
~'~o= 000005
6000
}4000
2000
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
Human Error Rate, Z.~
Figure 11.
System mean time to failure with repair plots for different human error rates
References 1. Abbas, B.S. and Kuo, Way, "Stochastic Effectiveness Models for Human-Machine Systems," 1EEE Transactions on S.vstems. Man. and Cybernetics, Vol. 20, No. 4, 1990, pp.826-834. 2. Dhillon, B.S., Rayapati, S.N., "Analysis of Redundant Systems With Human Errors," Proc. of the Annual Reliability and Maintainability Symposium, Jan. 1985, pp. 315-321. 3. Dhillon, B.S., Yang, N., "Stochastic Analysis of Standby Systems With Common-Cause Failures and Human Errors," Microelectron. Reliab., Vol. 32, No. 12, 1992, pp. 16991712. 4. Goel, L.R., Sharma, G.C. and Gupta, P., "Stochastic Analysis of a Man-machine System With Critical Human Error," Microelectron. Reliab., Vol. 25, No. 4, 1985, pp. 669-674. 5. Lee, K. W., Higgins, J.J. and Tillman, F.A., "Stochastic Modelling of Human-Performance Reliability," IEEE Transactions on Reliabili.ty, Vol, 37, No. 5, 1988.
t Pergamon
Microelectron. Reliab., Vol. 35, No. 11, pp. 1401-1413, 1995 Elsevier Science Ltd Printed in Great Britain 0026-2714/95 $9.50 + .00 0026-2714(95) 00038-0
AVAILABILITY ANALYSIS OF A REPAIRABLE STANDBY HUMAN-MACHINE SYSTEM
N1ANFU YANG and B.S. DHILLON Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, KIN 6N5
(Received for publication 29 December 1994)
Abstract
- - This paper presents a model representing a two units active and one unit on
standby human-machine system with general failed system repair time distribution. In addition, the model takes into consideration the occurrence of common-cause failures. The method of linear ordinary differential equation is presented to obtain general expressions for system steady state availability for failed system repair time distributions such as Gamma, Weibull, lognormal, exponential, and Rayleigh. Generalized expressions for system reliability, timedependent availability, mean time to failure, and system variance of time to failure are also presented. Selected plots are presented to demonstrate the impact of human error on system steady state availability, reliability, time-dependent availability, and mean time to failure.
Introduction Various research publications have discussed the reliability and availability of human-machine systems [1-5]. But most of these publications assume that the system repair times are exponentially distributed. In practice, repair times often are not exponential distributed. This paper presents a two parallel units active and one unit on standby human-machine system with general failed system repair time distributions. Furthermore, the system/unit may fail either due to a common-cause failure or a critical human error. The standby unit is activated only when one of the operating unit fails. At time t=0, the parallel units start operating with the remaining unit in its standby mode. As soon as one of the parallel operating unit fails, the standby unit is switched into operation. At least one unit must function normally for the system to operate successfully. The human errors and common-cause failures can occur from any of the operable states of the system. The system can fail either due to a critical human error, a common-cause failure, or normal hardware failures. The following assumptions are associated with the model: 1) Common-cause, human error, and hardware failure rates are constant. 2) All failures and human errors are statistically independent. 3) All system units are identical. 4) The failed system repair times arc arbitrarily distributed. 5) The occurrence of a common-cause failure or a critical human error causes the entire system to fail from any of its operable states. The repair process begins soon after a unit failure. 6) The repaired unit or system is as good as new. 7) The switching mechanism for the standby is considered automatic and instantaneous. 8) No further failure can occur when system has been down.
1401
1402
Nianfu Yang and B. S. Dhillon
Notation The following symbols are associated with the model under study: i ith state of the system (see system transition diagram of Figure 1): i=0, two active plus the one standby unit are in perfect working condition; i=l, one operating unit failed due to a hardware failure, the standby unit is switched into operation; i=2, one unit active and no standby; i=3, the system failed due to hardware failures; i=4, the system failed due to a common-cause failure; i=5, the system failed due to a critical human error. s Laplace transform variable ~.~
Constant common-cause failure rate from state i to state 4; for i = 0, 1, 2
L~ r~
Constant critical human error rate from state i to state 5; for i = 0, 1, 2 Constant hardware failure rate of a unit in state i; i = 0, I, 2
~tj
Constant repair rate of a failed unit in state i; i = 1, 2
~tj(x)
Time-dependent system repair rate when the system is in state j and has an elapsed repair time of x; forj = 3, 4, 5 The probability that the system is in state i at time t; for i = 0, 1, 2 The probability that the failed system is in state j and has an elapsed repair time of x; for j = 3, 4, 5 The steady state probability that the system is in state i; i = 0, 1, 2 ..... 5 The Laplace transform of P~(t) The mean time to system repair that the failed system is in state j and has an elapsed repair time of x; forj = 3, 4, 5 Steady state availability of the system Steady state unavailability of the system Time-dependent system availability probability density function (pdf) of the system repair time when the system is in state j and has an elapsed repair time of x; for j = 3, 4, 5
P~(t) Pj(x,t) Pi P~(s) Ej(x)
AV~ UAV. AV(t)
Nj(x) Nj(s)
Laplace transform of Nj(x).
Descriptive Equationsof The System The differential equations associated with Figure 1 can be written as follows: dP o(t._.__~) = - ( r o + ~.~o+ Lho)Po(t) + p~PL(t) +S:P3(x, t)l.h (x)dx + dt
(1)
dP,(t) T = r°P°(t) - (~t, + r~ + ~ , + ~.ht)P~(t)
(2)
+
~t2P2(l )
dP2(t) = r~Pt(t) - (~t2 + r 2 + ~.~2+ L~2)P2(t) dt 0 P3(x,t) 0 P3(x,t) - + ~ =-~t3(x)P3(x,t) Ot Ox O P4(x,t) Ot
+
O P4(x,t) ~ Ox
=-~,(x)e,(x,t)
0 P-~(x, t) 0 Ps(x, t) - ~- = -laffx)P~(x,t) c3 t c3 x
(3) (4) (5) (6)
The associated boundary conditions are as follows: P3( O,t ) = r2P2( t ) P4(O,t) = L.oPo(t) + Z.c~Pt(t) + Z.c2P2(t) P.frO, t) = LhoPo(t) + khlPl(t) + ~.h2P2(t)
At time t = 0,Po(0) = I,P~(0) = P2(0) = O, Pffx,O) = P4(x,0) = Ps(x,0) = 0
(7) (8) (9)
Availability analysis
1403
,--1
rts(x) Figure 1.
The state transition diagram of the system
Steady-State Availability Analysis As time approaches infinity, Equations (1)-(6) reduce to Equations (10)-(13), respectively.
,,:o - ~,:, = f?,(x,t)~,(x)~ +~o;°,(x,,).,(x)a~ + f?,(x,o~ (x)~ top o - a , P
+ la,P2 = 0
(1 I)
~P~ -a2P2 = 0 0 P,(x)
c3x
(lO) (12)
= -~t,(x)P,.(x)
(.for i = 3,4,5)
Similarly, the boundary conditions become: P3(O)= r2P2
(13)
(14)
/4(0) = ~.coPo + ~.c.P~ + ~.c2P2 Ps(0) = ~,oPo + ks~P, + k,h2P2
(15) (16)
where a 0 = r 0 + k,co + k.ho a I = r~ + ~,c) + ~
+ ~tI
a 2 = r 2 + L:2 + ~ 2 + la2 a3 = aI + a2
Pi is the steady-state probability that the system is in state i, for i = 0, 1, 2, ..., 5, and
g
=f:Pi(x)dx
f o r i = 3, 4, 5
(17)
Also, 5
y~=1
(18)
i:0
Solving differential Equation (13), we get Pf(x)
P,(O)exp(-fo~(to)do)
( f o r i = 3,4,5)
(19)
Thus, from Equation (17) and (19), we have
= f [ P3(O )exp(-fo; 3(co)do))dx =
(20)
r2P2Es(x )
Similarly, P. = I ? . ( x ) d x
=I: = (~.coPo + ~1Pl + k~2P2) E4(x)
(21)
1404
Nianfu Yang and B. S. Dhillon
= J'o Ps(0)exp(40~5(o)do)dx
(22)
= (~.hoPo + )~hiP~+ ~.h2P2) Es(x) where
(23) (for i = 3, 4, 5) E3(x), E 4 x ) and E s ( x ) are the mean times to repair from state 3 to state 0, from state 4 to state 0, and from state 5 to state 0, respectively.
Solving the set of Equations (I 1), (12), (18) and (20)-(22), we get the following steady state probabilities: Po = b° (2,1) BA(2,1)
( 24 )
p~
b I (2,1) BA(2,1)
( 25 )
P2 = b2 (2,1) BA(2,1)
( 26 )
P3 = b3(2'l)E~(x)
( 27 )
BA(2,1) ( 28 )
P4 = b4(2'l)E4(x)
BA(2,1) ( 29 )
p, = b~(2,1)E~(x)
BA(2,1)
where bo(2,1) = ala2 - rl P.2 b1(2,1) = a2ro b2(2,1) = rorr b3(2,1) = ror~r2 2
b, (2,1) = ~ b. (2,1)~.. i0 2
b,(2,1) = if'b, (2,1)~.,i i,0 2
5
BA (2,1) = ~-'6, (2,1) + ~ bj (2,1)E, (x) i-o
J =3
The system steady state availability is ,4 v~(2,l) = Po + ~ +1'2 2
~-" b, (2,1)
(30)
BA(2,1) Similarly, the system steady state unavailability is given by UAV~(2,1) = P3 + P, + P, 5
~"bj (2,1)Ej (x) .~-
j=3
.
BA(2,1)
(3l)
Availability analysis
1405
System Steady State Availability Special Cases Case I If the system repair time x is repair time is given by
Ni(x)
Gamma distributed and probability density function (pdf) of the
F(I 3) exp(-lajx)
(32)
(forj = 3, 4, 5; x_>0, 13>0, ~j>0 ) where 13and ~j are two parameters of the Gamma distribution. Thus, the mean time to repair Ej(x) is
E/(x) =f: xNj(x)dx -- ~---
( 33 )
(forj = 3, 4, 5) Substituting Equation (33) into Equation (30), we get the following resulting system steady state availability for the Gamma repair time distribution: 2
)'~ b,.(2,1) i~o s A V~(2,1) = 2 ~ b , (2,1) + '~b~ (2,1) 13/ta) i :0
( 34 )
j=3
If13 is an integer F(13) = (13 -1)!
( 35 )
Thus the Gamma distribution becomes
Erlangian distribution.
For 13 = 1, the Gamma distribution reduces to exponential distribution, and Equation (34) is the system steady state availability for the exponential repair time distribution. C a s e II If the system repair time x is
Weibull distributed, the pdf of the repair time is expressed by Ns(x) = pj~ 13x~-~exp(-~ts~x ~ ) ( 36 )
(forj = 3, 4, 5; x_>_>0,13>0, pj>0 ) where 13 and ~tj are two parameters of the Weibull distribution. Thus, the mean time to repair Ej(x) is given by
Ej(x) =f[xNj(x)dx = F(I + 1/13)
( 37 )
Pj (forj = 3, 4, 5) Substituting Equation (37) into Equation (30), we get the following system steady state availability for the Weibull repair time distribution: 2
'~-~b,.(2,1)
/:o
( 38 )
A V= (2,1) = ~-'~b,(2,1) + )-~bj (2,1)F(I + 1/13)/Pu i o
./=3
C a s e III If the system repair time x is Rayleigh distributed and the time dependent failed system repair rate and pdfofthe system repair time, respectively, are defined by
~j(x) = ~:2x
( 39 )
0=3,4,5) and N j(x)
= laj 2 x exp(-~tj zx" / 2)
(for j =3, 4, 5, x~_>0,p.j>0 )
( 40 )
1406
Nianfu Yang and B. S. Dhillon Thus, the failed system mean time to repair Ej(x) is expressed by
l~
(41)
(forj = 3, 4, 5) Substituting Equation (41) into Equation (30), we get the following system steady state availability 2
~-" b, (2,1) ,=o
A V~ (2,1) = 2
( 42 )
Eb,(2,1)+ i=0
j=3
N If the system repair time x is lognormal distributed, the pdf of the repair time is defined by =
N j(x)
1 , - - expl (In x -~tj)2.1 x % 42~ [ 2o: 2 J
( 43 )
(forj = 3, 4, 5, and x_>0) where pj and oj are the distribution parameters (mean value and standard deviation of In x, respectively). Thus, the mean time to repair Ej(x) is given by
E:(x)= ~ : x N : ( x ) d x
= exp
+
( 44 )
(forj = 3, 4, 5)
Substituting Equation (44) into Equation (30) yields the following system steady state availability: 2
~b,(2,1) AVe,(2,1)= 2
,~0
(45)
y b,(2,1)+Eb,(2,1)exp(p, i~0
i:3
System Steady State AvailabilityNumerical Examples Setting ro =r t =O.O005/hr 3¢o = L~I = 0.00005 / hr
r2 =O.O008/hr L2 = 0.00002 / hr
~t~ =O.O01/hr Lht = 0.0002 / hr
P2 =O.O02/hr ~2 = 0.0001 / hr
in Equation (30) leads to the following:
AV,, =
5.82 [2.0Effx) + 2.835E4(x) + 3.17 Effx)] x 10 4 + 5.82 + 4.1 lEs(x)Lho
The plots of system steady state availability as a function of human error rate, Lho, are shown in Figures 2 and 3 for different values of 13 and for Gamma and Weibull distributed repair times, respectively, for specified values of parameters P3 = 0.001 /hr, la4 = tx~ = 0.0009 / h r . Figure 4 provides the results of the system steady state availability as a function of human error rate, Lho, for lognormally distributed failed system repair times (p~ =1.001, la 4
= p.~ =1.0009, ~3 =o4 =o~ = o ).
Also, the plots of the system steady state availability as a function of Z.ho are shown in Figure 5 for Rayleigh distributed failed system repair times for specified values of parameters. Figure 6 shows plots of system steady state availability as a function of human error rate, Z,ho, for specified values of parameters la3 = 0.001 / hr, la4 = kt5 = 0.0009 / hr, for different system repair time distributions: exponential, Gamma, Rayleigh, Weibull and lognormal (la 3 = 1.001, ~t4 = P5 = 1.0009, o 3 = o 4 =cr~ = 6 = 3).
Availability analysis
1407
1.0 0.9 0.8 >, 0,7 <
0.6
N co 0.5 N 0.4 E ~, 0.3 0.2 0.1 0.0
0.0000
h
0.0002
0.0004
0.0006
0.0008
0.0010
Human Error Rate, ~,0
Figure 2.
System steady state availability vs. ~'~oplots for the system with constant human error rates and the Gamma failed system repair time distribution 1.0 ~=2
0.9 0.8 ~5
0.7
<: 0.6 co 0.5 .~. 0.4 ¢n E 0.3
'~ 0.2 0.1
0.0 0.0000
0.0002
0.0004
0.0006
0.0008
-' 0.0010
Human Error Rate, L,o
Figure 3.
System steady state availability vs. ~'hoplots for the system with constant human error rates and the Weibull failed system repair time distribution 1.000 0,999 0.998
0=0.3
0.997 <
O=t .t)
0.996
ca 0.995 N
0.994
E 0.993
a=l.5
¢n 0.992 0,991 0.990 0.0000
0.0002
0.0004
0.0006
0,0008
0.0010
Human Error Rate, Xho
Figure 4. /'IR31-11-C
System steady state availability vs. Xh0 plots for the system with constant human error rates and the lognormal failed system repair time distribution
1408
Nianfu Yang and B. S. Dhitlon 0.9 0.8; 0.7 0,6
V.~=0.0008 0.5 !
p.~=0.00(16
0.4
~=0.0004
Go
o3
0.3
~=0.0002
~3 = O.O01/hr
0.2
p,, = O.O009/hr 0.1 0.0 0.0000
0.0001
0.0002 0.0003 0.0004
0.0005
0.0006
Human Error Rate, ~,ho
Figure 5.
System steady state availability vs. ~'hoplots for the system with constant human error rates and the Rayleigh failed system repair time distribution
Time-Dependent System AvailabilityAnalysis Using Laplace transforms and the initial conditions in Equations (1) - (9), we get
sPo(s) : 1 - aoPo(s) + ~tPt(s) + f ~P~(x, s ) ~ ( x ) d x + ¢ U
(46)
÷I/ , ( x, s ).,(x (s + a t ) P l ( s ) - roPo(s) - ~t2P2(s) = 0 (s + a2)P2(s ) - rlP~(s ) = 0
( 47 ) ( 48 )
~P~(x'S)+sP~(x,s)+~t,(x)P,(x,s)=O
(for i = 3,4,5)
c~x
(49)
and the boundary conditions: P3(0,s) = r2P2(s) P , ( 0 , s ) = X,.oPo(s) + ~.~,e~(s) + ~.~2P2(s)
( 50 )
P f f 0 , s ) = LhoPo(s) + L~lPt(s)+ Lh2P2(s) 1.0 ~
_
_
i toOnormaldistribution(o = 3)
•
•
j
\
N
s, oonoo,i d=,bot,on
•
~ 0.5
0.4 f 03 ~ . 0.0000
Gammadistribution (.3 = 2}/
. . . 0.0002
0.0004
, 0.0006
~, 0.0008
= 0.0010
Human Error Rate, k.o
Figure 6.
The comparision of AVs, vs. Lh0 for the system with constant human error rates and different failed system repair time distributions
Availability analysis
1409
Solving differential equation (49), we get P,. (x, s) = P,. (0, s)e %xp(-Io~, (m)&o)
(for i = 3, 4, 5)
(5~)
Substituting Equations (50) and (51 ) in Equation (46), lead to [s
+
a o - k~oNa(s) - ~oN.~(s)]Po(s) -
[p., + k~,N,(s) - £hzNs(s)]P~(s) [rzN3(s ) + L2N,(s) - kh2Ns(s)]P2(s) = 1
(52)
where Ni(x) is pdf of repair time and Ni(s ) is the Laplace transform of Ni(x )
N,(s) =IoeXp(- x)N,(x)a N,(x) = ~t, (x) exp[-Io~, (co)aco]
(for i = 3, 4, 5)
Since we know
P,(s)
(53)
(for i = 3,4,5)
Thus, utilizing Equation (51), we get 1- N 3(s) P3(s) = t ~ P 2 ( s ) - -
(54)
S
P,(s) = [A.coPo(s) + kciP,(s) + L.2P2(s)] I - N , ( s )
(55)
S
Ps(s) = [knoPo(s) + kh,P~(s) + kh2 P2(s)] l - Ns(s)
(56)
S
where 1 - N~(s) - I o e - = exp(-Io;,(o~)do))dx
(for i = 3,4,5)
(57)
S
Solving Equations (47), (48), (52) and (54) - (56), we get the following Laplace transforms of state probabilities Po(s),P,(s) ..... and P~(s): Po(s) = b°(2'l) + (al +a2)s+s2 CA(2,1)
( 58 )
P~ (s) = b, (2,1) + ros CA(2,1)
( 59 )
P2(s) = b2(2'1) CA(2,1)
( 60 )
P3(s) = b3 (2'1)[I - N3(s)] s. CA(2,1)
( 61 )
& (s)
[1 - N, (s)][b, (2,1) +
le(Z,l).s + )~cos:]
s . CA (2,1)
p~(s) = .[l_Us(s)j[bs(2,1)+lh(Z,1).s+~hos2] . . r s. CA(2,1) where CA(2,1) = cal(2,1) + ca2(2,1), s + ca3(2,1), s 2 + s 3 5
cal(2,1) : y [1- N ,(s)]b,(2,1) j=3
ca2(2,1) = '~-~b, (2,1) +/c(2,1)[1 - N4 (s)] +/h (2,1)[1 - N 5(s)] i=0
ca 3(2,1) =/0(2,1) + kco[1 - N, (s)]+ Lh, [I - Ns(s) ] /c(2,1) = (a~ + a2)~.co + ro~.c~ lh(2,1) = (a I + a2)~.ho + ro?~h, 10(2,1) = a~ + a 2 + r o
( 62 ) (63)
1410
Nianfu Yang and B. S. Dhillon The Laplace transform of the system availability is : ~'b, (2,1) +/0(2,1). s + S 2 A V(s) = i o
( 64 )
CA(Z,1)
Taking the inverse Laplace transforms of Equation (64), we can obtain the time-dependent system availability.
Time-Dependent System Availability Special Cases Substituting the Laplace transforms of the pdf of the system repair times, N3(s), N4(s), and Ns(s), in Equation (64) and then substituting the same applicable data of the earlier system steady state availability numerical examples into the resulting expression (also setting p,3=0.001/hr and p,4=~ts=0.0009/hr and taking inverse Laplace transform), we get the system time-dependent availability. For exponential failed system repair time distribution and ~,~o= 0.0002 / hr we get A V(t) = 0.765733 + 0.0115416 exp(-0.00355207t) +
0.00738703 exp(-0.00096036t) + 2.1631 sin(0.000186153t) exp(-0.00140379t) + 0.215338[cos(0.0001861530 exp(-0.00140379t) 7.54103 sin(0.000186153 0 exp(-0.00140379t)] For different values of Z,ho, the system time-dependent availability plots are shown in Figure 7. For Gamma distributed failed system repair time, the plots of the time-dependent system availability for varying human error rate, ~,hO,are shown in Figures 8 (13 = 2) and 9 (13 = 3).
System Reliability and MTTF With and Without Repair Setting ~3(x) = ~t4(x) = ~5(x) = 0 in Figure 1, the system of differential equations becomes: dPo(t) = - a o P o ( t ) + ~uP~(t) dt
( 65 )
dP,( t ) _ roPo( t ) - a~Pt( t ) + ~t2P2( t ) dt
( 66 )
1.00 0.95 0.90 0.85
0.60
~-ho = 0.0002
~, 0.75 L.0 = 0.0004 0.70
0.65 0.60
1000
2000
3000
4000
5000
Time, t
Figure 7.
Time-dependent system availability plots for the system with constant human error rates and failed system repair rates
Availability analysis
1411
1.0
o.91
5" 0.8
0.7
iiiiii.
o~
0.6
0.5 1000
2000
3000
4000
5000
Time, t
Figure 8.
Time-dependent system availability plots for the system with constant human error rates and the Gamma failed system repair time distributions ( 13= 2) ( 67 )
dPz(t) = r~P~(t)- a2P2(t) dt dP3( t ) = r2Pfft) dt
( 68 )
dP4(t) = 3.coPo(t)+ k.~Pl(t) + Xc2P2(t) dt
( 69 )
d P f f t ) = 3.hoPo(t) + 3.h~P,(t) + Lh2P2(t) dt
( 70 )
At time t = 0, P o ( 0 ) = 1 and P~(0) = 0 for i = 1,2,3,4,5. Using Laplace transforms in Equations (65) - (70), and solving the resulting set of equations, we can get the Laplace transforms of the state probabilities, Po(s), P~(s) ..... Ps(s). Thus, the Laplace transform of the system reliability with repair is 2
~ b , (2,1) + 10(2,1). s + s 2 R(s) = i=o
(71)
DA(2,1) 1.0 0.9 0.8 0.7 E 0.6
0.5 0.4 0.3 1000
2000
3000
4000
Time, t
Figure 9.
Time-dependent system availability plots for the system with constant human error rates and the Gamma failed system repair time distributions ([3 = 3)
1412
Nianfu Yang and B. S. Dhillon where DA(2,1) = dal(2,1) + da2(2,1), s + da3(2,1), s 2 + s 3 dal(2,1) = aoror1 - a2r&h da2(2,1) = ao(a ~ + a2 ) + rot~ - r&h da3(2,1) = a o + a I + a 2 System reliability with repair can be obtained by inverting Equation (71) e'~'(s°s~ - s o s , - s ~ s , +s2 2)
e " ( s ~ s b -s°s~ -,%s~ +sL 2) R(t)
+
-
(s, - s 2)(s, - s3)
-
(s 2 - s,)(s~
+
- s3)
(72)
e ~Jt ( s = s b - s=s 3 - s o s 3 + s~ 2)
(s~ - s~)(s~
- s,)
where s, and sb are the roots of the numerator of Equation (71), and sl, s2 and s3 are the real and unique roots of the denominator of that Equation. The system mean time to failure (MTTF) with repair is given by 2
Zb,(2,1) M T T F = lim R ( s ) = i=0
,~o
( 73 )
dal(2,1)
The variance of time to failure with repair of the system is expressed by tr -' = -21im R ' ( s ) - ( M T T F ) 2 s~o
2da2(2,1)~-" b~(2,1) - 2-10(2,1) • dal(2,1) -
b, (2,1)
( 74 )
i=0
[dal(2,1)] 2 Plots of the system reliability with repair are shown in Figure 10 for same applicable data of
the system steady state availability numerical examples. Figure 11 shows the plots of system MTTF with repair. Setting I.tl = ~ = 0 in Equations (72) - (74), we can get the system reliability, MTTF and the variance of time to failure without repair, respectively.
1.0 L"°=° ~'.o= 0.0002
~'J 0.8
0.6 ,-r" 0.4
0.2
0.0 2000
4000
6000
8000
10000
Time, t
Figure 10.
System reliability with repair plots for different human error rates
Availability analysis
1413
10000
8000
/~"
o
~'~o= 000005
6000
}4000
2000
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
Human Error Rate, Z.~
Figure 11.
System mean time to failure with repair plots for different human error rates
References 1. Abbas, B.S. and Kuo, Way, "Stochastic Effectiveness Models for Human-Machine Systems," 1EEE Transactions on S.vstems. Man. and Cybernetics, Vol. 20, No. 4, 1990, pp.826-834. 2. Dhillon, B.S., Rayapati, S.N., "Analysis of Redundant Systems With Human Errors," Proc. of the Annual Reliability and Maintainability Symposium, Jan. 1985, pp. 315-321. 3. Dhillon, B.S., Yang, N., "Stochastic Analysis of Standby Systems With Common-Cause Failures and Human Errors," Microelectron. Reliab., Vol. 32, No. 12, 1992, pp. 16991712. 4. Goel, L.R., Sharma, G.C. and Gupta, P., "Stochastic Analysis of a Man-machine System With Critical Human Error," Microelectron. Reliab., Vol. 25, No. 4, 1985, pp. 669-674. 5. Lee, K. W., Higgins, J.J. and Tillman, F.A., "Stochastic Modelling of Human-Performance Reliability," IEEE Transactions on Reliabili.ty, Vol, 37, No. 5, 1988.