Titel_SS06

having at least one out-crossing before leaving the safe domain in the time interval  3312 ,  and

finally the number of N realisations having at least one in-crossing before leaving the

safe domain in the time interval 

2 N 1

3312 , . Provided that NN N  01 , the probability of

failure in the time interval 0,T can be written as:



0

0,T 0

N

N



10

() 0,T^10

f

NN

N N

PT

NN N

2

2





(^)  (8.8)
By introducing 0 0
0
f () N ( )
N
33 2 2^2 O 3 in Equation (8.8) the following expression is
obtained:

0
0,
() (0) (1 f(0)) () ( )
T
PTff P P f 33 2 2O 3 (8.9)
By assuming that the Riemann sum in Equation (8.9) converges towards the integral for
2 3 0 Equation (8.7) is obtained. It should be noted that the integral in Equation (8.7) is
extremely difficult to calculate for non-trivial cases why approximations in general are
required.
A useful upper bound to the probability of failure in the interval 0, T may, however,
immediately be derived as:
 f XS^0 (())td
1 000
0, 0 0

() (0) (1 (0))

T

ff

T

PT N NNNP P

NN N

E 3

4

* 22 ^

  (8.10)

where EX S^0 

t  is the Mean Out-crossing Rate (MOR) conditional on the event that the

process starts in the safe domain. Equation (8.10) may be extended and refined to include a

larger number of conditions in regard to the process being in the safe domain, and the

corresponding conditional MOR’s may be calculated using Rice’s formula.

Two situations are of special interest for continuous stochastic processes, namely the case of

stationary Normal processes and constant threshold levels and the case of non-stationary

Normal processes and time varying threshold levels.

In case of stationary Normal processes and constant threshold level the application of Rice’s

formula yields:

22 2

22 2

0

( )^11 exp( ( )) 1 1exp( ( ))

22 22

X

XXXXX

xdxE x 

X

E E



(^)  






 (8.11)

where X is the standard deviation of the time derivative of X()t. If zero level crossings are

considered () 0 ) there is:

1

(0)

2

X

X







(^)   (8.12)