Monte Carlo Simulation
Reliability Evaluation Techniques:
Analytical Technique
represent the system by a
mathematical model (usually
simplified for practical systems)
direct mathematical solution
short solution time
same results for the same
problem (greater but perhaps
unrealistic confidence to user)
Simulation Technique
simulate the actual process (using
random numbers) over the period of
interest
repeat simulation for a large number of
times until convergence criteria is met
Advantages:
can incorporate complex systems
(analytical approach simplification can be
unrealistic)
wide range of output parameters
including probability distributions
(analytical approach usually limited to
expected values)
Toss of a Coin - MCS
Probability of getting a head, P(Head) = 0.5
Actual Tossing process: Toss a coin repeatedly for a large number of trials.
Simulation of Tossing Process:
Generate a random number Ui (between 0 and 1) for trial i.
If Ui ≥ 0.5, we have a Head; otherwise a Tail
Generate a random number repeatedly to simulate a large number of trials.
1
MCS Methods
Random Simulation
Basic (time) intervals chosen randomly
Can be applied when events in one basic interval does not affect the
other basic intervals
Sequential Simulation
Basic (time) intervals in chronological order
Required when one basic interval has a significant effect on the next
interval
Can also provide frequency and duration indices
Random Simulation
1
A1= 0.8
2
A2= 0.6
- Random # (0 – 1) Generator
- Simulation Convergence
Trial
#
Component 1
simulation
Component 2
simulation
Rand #
State
Rand #
State
1
0.12
Up
0.35
Up
2
0.87
Down
0.21
3
0.95
Down
0.62
4
0.59
Up
0.18
Up
System
State
System
Availability
Up
1/1 = 1.00
Up
Down
1/2 = 0.50
Down
Down
1/3 = 0.33
Up
2/4 = 0.50
5
2
Sequential Simulation
1
Component 1:
λ1 = 1 f/yr
2
Evaluate the system reliability for an operating
time of 20 hours.
Up time = -
Component 2:
λ2= 5 f/yr
1
ln X
λ
5
4
3
# of Sim ulations
2
1
0
20
T im e (h)
Sequential Simulation
1
2
Component 1:
λ1 = 1 f/yr
r1 = 100 hr
Component 2:
λ2= 5 f/yr
r2 = 444 hr
Usys = U1 x U2 = 0.00228 = 20 hr/yr
1
ln X
λ
1
Down time = - ln X
µ
Up time = -
U=
total outage time
total simulation time
total # of failures
Frequency of Failure = total simulation time
Duration of Failure =
total outage time
total # of failures
3
Simulation Results
0.6
0.5
0.4
Probability
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
Unavailability (hr/yr)
4