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Basics of Monte Carlo Simulations: practical exercises
Piero Baraldi
Politecnico di Milano, Energy Department
Phone: +39 02 2399 6345
Piero BARALDI
EXERCISE 1
Piero BARALDI
Exercise 1
Consider the Weibull distribution:
with
1. Sample N=400 values from
2. Show a Figure reporting the distribution of the
obtained samples
3. Verify whether the obtained distribution provides a
good approximation of the Weibull distribution
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Sampling Random Numbers from FX(x)
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MATLAB COMMAND
• rand(M,1) provides a column of M random numbers sampled
from an uniform distribution in the range [0,1)
• N = hist(Y) bins the elements of vector Y into 10 equally
spaced counters and returns the number of elements in
each counter. More options if you write ‘help hist’
Piero BARALDI
Example: Weibull Distribution
• Time-dependent hazard rate
cdf:
pdf:
• Sampling a failure time T (by the inverse transform)
1 t
TF t P T t e
1 t
Tf t dt P t T t dt t e dt
1 t
R TR F r F t e
1
1 1ln 1TT F R R
1 tt
Piero BARALDI
% Exercise 1
clear all; close all
% Weibull parameters
beta = 1;
alpha = 1.5;
% Sample N values from the Weibull distribution
N = 400; % number of samples
r = rand(N,1);
t = (-log(1-r)/beta).^(1/alpha); % inverse transform method
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% Verify the distribution
% Estimated pdf
delta_channel = 0.1;
channel_centers = [delta_channel/2:delta_channel:5-
delta_channel/2];
num_samples = hist(t,channel_centers);
pdf_est = num_samples/(N*delta_channel);
% Compute the analytic pdf
analytic_weibull = alpha*beta*channel_centers.^(alpha-
1).*exp(-beta*channel_centers.^alpha);
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% plot the pdf
figure % pdf
plot(channel_centers,analytic_weibull); hold on;
plot(channel_centers,pdf_est,'-sr'); grid
legend('Analytic pdf','Sampled values distribution pdf',0)
% plot the cdf
figure cdf_est = cumsum(pdf_est)*delta_channel;
plot(channel_centers,
cumsum(analytic_weibull)*delta_channel);hold on;
plot(channel_centers, cdf_est,'*r'); grid
legend('Analytic cdf','Sampled values distribution cdf',0)
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Analytic pdf
Sampled values distribution pdf
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Analytic cdf
Sampled values distribution cdf
Piero BARALDI
EXERCISE 2
Piero BARALDI
13Excercise 2
Write the pseudo code for the estimation of the instantaneous
unavailability of a continuously monitored component with constant
failure rate (𝜆) and repair rate (𝜇)
Hints:
▪ You can assume a mission time of 10^3
▪ You can compute the instantaneous availability at times:
0,1,2,3,…10^5
values
3· 10-3 h-1
m 25· 10-3 h-1
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14
%Initialize parameters
Tm=1000; %mission time;
M=10000; %number of trials;
lambda=3E-3;
mu=25E-3;
Dt=1; %bin length;
Time_axis=0:Dt:Tm;
counter_q=zeros(1,length(Time_axis));
Excercise 2 - Solution
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15
for i=1:M %simulation of M MC trials
%parameter initialization for each trial
t=0;
state=0; %state=0 = working;
%state=1 = failed;
while t<Tm
if state == 0 %system is working sampling of failure time
t=t-[log(1-rand)]/lambda; %failure time
failure_time=t;
state=1; % new state=failed
lower_b=min(find(Time_axis>=failure_time)); % first unavailability counter to be increased
else %system is failed then sampling of repair time
t=t-[log(1-rand)]/mu; %repair_time
state=0;
repair_time=t;
if t<Tm
upper_b=max(find(Time_axis < repair_time)); %last unavailability counter to be increased
else %repair ends after mission time
upper_b=length(Time_axis);
end
counter_q(lower_b:upper_b)= counter_q(lower_b:upper_b)+1; %increase all unavailability counter between lower_b and
%upper_b
end
end
end
Excercise 2 - Solution
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16
Unav_MC=counter_q(:)/M;
Unav_true=(lambda)/(lambda+mu)-(lambda/(lambda+mu))*exp(-
(lambda+mu).*Time_axis);
plot(Time_axis,Unav_true,'blue')
hold on
plot(Time_axis,Unav_MC,'red')
Excercise 2 - Solution
M=10000
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17Excercise 2 - Solution
M=100000
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EXERCISE 3
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Exercise 3
• Consider the following system
• Transition rates (in arbitrary time-units):
Failure: λA = 0.001; λB = 0.002; λC = 0.005;
Repair: μA = 0.1; μB = 0.15; μC = 0.05;
• Estimate the reliability of the system at Tmiss = 500
A
CB
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Components’ times of transition between states are exponentially distributed
Arrival
Initial1 2 3
1(nominal) 0 ,C)(BA )(31BA
2 (failed) m ,C)(BA 0 )(32
BA
3 (failed) )(13BA
)(23
BA 0
Indirect Monte Carlo
A
CB
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• The rate of transition of the system out of its current configuration (1,1, 1) is:
• We are now in the position of sampling the first system transitiontime t1, by applying the inverse transform method:
where Rt ~ U[0,1)
CBA 1,1,1
)1ln(1
1,1,101 tRtt
Analog Monte Carlo Trial
Sampling the time of transition
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• Assuming that t1 < TM (otherwise we would proceed to the
successive trial), we now need to determine which component
has undergone the transition
• The probabilities of components A, B, C undergoing a transition
out of their initial nominal states 1, given that a transition occurs
at time t1, are:
• Thus, we can apply the inverse transform method to the discrete
distribution
1,1,11,1,11,1,1 , ,
CBA
RC U[0,1)
Analog Monte Carlo Trial
Sampling the kind of Transition (1)
100
Rc
C 1,1,1
A
1,1,1
B
1,1,1
C
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• As a result of this first transition, at t1 the system is
operating in configuration (1,2,1).
• The simulation now proceeds to sampling the next
transition time t2 with the updated transition rate
m CBA 1,2,1
Analog Monte Carlo Trial
Sampling the kind of Transition (2)
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24
• When the system enters a failed, tallies are
appropriately collected for the unreliability estimates;
• After performing a large number of trials M, we can
obtain estimates of the system unreliability by simply
dividing by M, the number of MC simulation ended in a
failure configuration
Unreliability and Unavailability Estimation
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% Reliability of Simple Systems (A\\(B+C))
% System parameters
lambda_A = 0.001; mu_A = 0.1;
lambda_B = 0.002; mu_B = 0.15;
lambda_C = 0.005; mu_C = 0.05;
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% 2 = Failure; 1 = Working
Trans_A = [0 lambda_A; mu_A 0];
Trans_B = [0 lambda_B; mu_B 0];
Trans_C = [0 lambda_C; mu_C 0];
failed_states = [2 1 2; 2 2 1; 2 2 2];
initial_state = [1 1 1];
% Mission time
Tmiss = 500;
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% MC simulation parameters
M = 1e3;
% MC cycle
unrel_counter = zeros(1,N);
for n = 1:M % Main Monte Carlo cycle
unrel_flag = 0; % 0 if no failures before Tmiss, 1 otherwise
t = 0;
current_state = initial_state;
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while (t < Tmiss)
% check if the system is in a failure configuration
for jj = 1:3 % for each one of the 3 failure states:
if sum(current_state == failed_states(jj,1:3)) == 3
unrel_flag = 1;
end
end
% find the system transition rate (Lambda_sys)
if unrel_flag ~= 1
lambda_A_out = sum(Trans_A(current_state(1),:));
lambda_B_out = sum(Trans_B(current_state(2),:));
lambda_C_out = sum(Trans_C(current_state(3),:));
lambda_sys = lambda_A_out + lambda_B_out +
lambda_C_out;
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% Sample transition time (t_trans)
t_trans = -1/lambda_sys*log(rand);
t = t+t_trans;
if t < Tmiss % if the transition time falls within Tmiss Sample the kind
% of transition
r = rand;
sum_l = cumsum([lambda_A_out lambda_B_out
lambda_C_out])/lambda_sys;
comp = min(find(sum_l>r)); % Component that makes the
transition
%Change the state of the component that makes the transition
old_st = current_state(comp);
current_state(comp) = 3-current_state(comp);
end
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else %(unreliability flag is 1)
break % break the while loop
end %(end of the if of the unreliability flag)
end %(of the while loop)
unrel_counter(n) = unrel_flag;
end % end of the N simulation
% Estimate the reliability by the MC samples
rel_MC = 1-mean(unrel_counter);
var_MC = var(unrel_counter)/M;
