EMP5169 Midterm 2005 . 2
EMP5169 Midterm 2000 . 6
EMP5169 Midterm 2001 . 8
-1-
EMP5169 Midterm 2005
1. Prove using the Markov method that a systems steady state unavailability,
UAVss, is given by:
UAVss
where
is the constant failure rate of the system

Q1
1. Hazard Rate Function:
Failure rate is a special case of hazard rate that changes with time
(t )
f (t )
R(t )
(t )
1 dR(t )
R(t ) dt
2. Accumulate probability function
t
F (t )
F(t)
t
f ( x)dx f(x)dx
0
: Accumulate probability function
f (x) : f

Z
X
C
D or CD
CD
Y
X or Y
0
X (Y Z ) XY XZ
X YZ ( X Y )( X Z )
X X X
XX X
X
( XY )
X * 1 ( XY )
X (1 Y )
X
X ( XY )
( XX )Y
XY
0
P( x) 1
n
P ( x1
x2
. x n ) 1
1 P ( xi )
i 1
P( x1
P ( x1
x2
P( S ) 1
. x n )
P( S )
P ( x1 x 2 .x n )
P ( x1 ) P ( x 2 ) . P

1
(100)
16
6.25%
1
(100)
15
Time t=0
t (hours)
hc (cumulative hazard)
ln(t) (hours)
6.67%
L
f (t1 ; ) f (t 2 ; ). f (t m ; )
log e L
0
e
t1
L
e
L
m
e
L
m
e
t
e
f (t ; )
t2
t3
e
. e
tm
t1 t 2 t3 . t m
m
i 1
ti
m
log e L
m log e ( )
ti
i 1
log e L
m
m
ti
0

0
1
2
k
k
R s tan d
by
i 0
(t )
t
0
i
(t )dt e
i!
(t )
k
R s tan d
t
by (t )
i 0
MTTF
0
R s tan d
k
MTTF
MTTF
0
i 0
k 1
0
by
i
dt
t
e
i!
(t )dt
( t)i e
i!
t
dt
t
0
( t ) dt
1
2
k
m
m
Rk / m (t )
i k
m
i
m i
m!
(m i )!i!
(R F )m
m 3
Rseries
R parallel
R2 /