An example of the Monte Carlo method and simulation.

Lecture

For a given system structure

1) find the Monte Carlo method P ^{* of the} probability of the system’s failure-free operation, knowing the probability of the failure-free operation of its elements

2) find the absolute error | PP ^* |, where P is the system reliability, calculated analytically. Perform at least 100 tests.

Note Monte-Carlo simulation on a computer, attach a test program in any programming language (you can also use Excel, Mathcad, Scilab Matlab, etc.)

Initial data

No. Var.	P (A)	P (B)	P (C)	P (D)	P (E)	P (F)	P (H)
four	0.5	0.1	0.2	0.9	0.3	0.8	0.3

An example of the Monte Carlo method and simulation.

Decision

1. find the Monte Carlo method for estimating P ^{* the} probability of system uptime

P _AB = P _A + P _B -P _A P _B = 0.5 + 0.1-0.5 * 0.1 = 0.6-0.05 = 0.55

P _CE = P _C + P _E -P _C P _E = 0.2 + 0.3-0.2 * 0.3 = 0.5-0.06 = 0.44

P _DF = P _D P _F = 0.9 * 0.8 = 0.72

P _DFH = P _DF + P _H -P _DF P _H = 0.72 + 0.3-0.72 * 0.3 = 0.804

P _ABCE = P _AB P _CE = 0.55 * 0.44 = 0.242

P _ABCEDFH = P _ABCE P _DFH = 0.242 * 0.804 = 0.1946

P _analytical = 0,1946

2) find the absolute error.

we first define the simulation probability using Excel

P (A) = IF ($ B4 <$ I $ 3; 1; 0)

P (B) = = IF ($ C4 <$ J $ 3; 1; 0)

P (C) = = IF ($ D4 <$ K $ 3; 1; 0)

P (D) = = IF ($ F4 <$ L $ 3; 1; 0)

P (E) = = IF ($ F4 <$ M $ 3; 1; 0)

P (F) = = IF ($ G4 <$ N $ 3; 1; 0)

P (H) = = IF ($ H4 <$ O $ 3; 1; 0)

P (ABCE) = IF (AND (OR (I4; J4); OR (K4; M4)); 1; 0)

P (DFH) = IF (OR (AND (L4; N4); O4); 1; 0)

P (System) = IF (AND (P4; Q4); 1; 0)

X is a random number from 0 to 1

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R
n	X	X	X	X	X	X	X	A	B	C	D	E	F	H	ABCE	Dfh	System
								0.5	0.1	0.2	0.9	0.3	0.8	0.3
one	0.52	0.38	0.25	0.90	0.51	0.13	0.68	0	0	0	one	0	one	0	0	one	0
2	0.60	0.97	0.59	0.43	0.85	0.04	0.30	0	0	0	one	0	one	0	0	one	0
3	0,69	0.79	0.35	0.05	0.37	0.44	0.82	0	0	0	one	0	one	0	0	one	0
four	0.73	0.56	0.87	0.08	0.92	0.24	0.33	0	0	0	0	0	one	0	0	0	0
five	0.85	0.38	0.84	0.61	0.07	0.68	0.71	0	0	0	one	one	one	0	0	one	0
6	0.36	0.79	0.66	0.31	0.12	0.27	0.90	one	0	0	one	one	one	0	one	one	one
7	0.63	0.24	0.46	0.64	0.74	0.93	0.17	0	0	0	one	0	0	one	0	one	0
eight	0.54	0.29	0.57	0.05	0.09	0.14	0.01	0	0	0	one	one	one	one	0	one	0
9	0.05	0.20	0.87	0.01	0.88	0.30	0.24	one	0	0	one	0	one	one	0	one	0
ten	0.30	0.99	0.59	0.59	0.76	0.50	0.37	one	0	0	one	0	one	0	0	one	0
eleven	0.85	0.07	0.37	0.25	0.39	0.31	0.46	0	one	0	one	0	one	0	0	one	0
12	0.45	0.05	0.87	0.92	0.19	0.34	0.86	one	one	0	one	one	one	0	one	one	one
13	0.46	0.66	0.53	0.47	0.75	0.11	0.36	one	0	0	one	0	one	0	0	one	0
14	0.22	0.02	0.63	0.39	0.33	0.51	0.19	one	one	0	one	0	one	one	0	one	0
15	0.16	0.72	0.81	0.45	0.89	0.91	0.62	one	0	0	one	0	0	0	0	0	0
sixteen	0.25	0.50	0.15	0.59	0.50	0.45	0.18	one	0	one	one	0	one	one	one	one	one
17	0.53	0.11	0.33	0.52	0.83	0.45	0.56	0	0	0	one	0	one	0	0	one	0
18	0.65	0.96	0.58	0.58	0.47	0.58	0,69	0	0	0	one	0	one	0	0	one	0
nineteen	0.58	0.70	0.61	0.31	0.31	0.20	0.51	0	0	0	one	0	one	0	0	one	0
20	0.28	0.99	0.08	0.47	0.44	0.73	0.94	one	0	one	one	0	one	0	one	one	one
21	0.62	0.32	0.70	0.19	0.07	0.36	0.88	0	0	0	one	one	one	0	0	one	0
22	0.09	0.21	0.27	0.31	0.93	0.84	0.75	one	0	0	0	0	0	0	0	0	0
23	0.54	0.45	0.26	0.71	0.56	0.07	0.99	0	0	0	one	0	one	0	0	one	0
24	1.00	0.90	0.21	0.56	0.33	0.79	0.31	0	0	0	one	0	one	0	0	one	0
25	0.15	0.16	0.19	0.94	0.62	0.94	0.22	one	0	one	one	0	0	one	one	one	one
26	1.00	0.18	0.28	0.80	0.91	0.03	0.79	0	0	0	0	0	one	0	0	0	0
27	0.13	0.86	0.35	0.04	0.20	0.05	0.93	one	0	0	one	one	one	0	one	one	one
28	0.32	0.36	0.60	0.41	0.96	0.56	0.30	one	0	0	0	0	one	0	0	0	0
29	0.80	0.45	0.49	0.54	0.58	0.25	0.79	0	0	0	one	0	one	0	0	one	0
thirty	0.68	0.36	0.31	0.80	0.27	0.96	0.57	0	0	0	one	one	0	0	0	0	0
31	0.10	0.49	0.34	0.80	0.89	0.02	0.91	one	0	0	one	0	one	0	0	one	0
32	0.92	0.83	0.57	0.29	0.38	0.08	0.53	0	0	0	one	0	one	0	0	one	0
33	0.14	0.11	0.81	0.38	0.77	0.30	0.88	one	0	0	one	0	one	0	0	one	0
34	0.22	0.73	0.39	0.45	0.83	0.13	0.43	one	0	0	one	0	one	0	0	one	0
35	0.85	0.29	0.83	0.24	0.46	0.01	0.64	0	0	0	one	0	one	0	0	one	0
36	0.35	0.83	0.78	0.35	0.41	0.98	0.27	one	0	0	one	0	0	one	0	one	0
37	0.20	0.04	0.47	0.83	0.29	0.78	0.21	one	one	0	one	one	one	one	one	one	one
38	0.71	0.63	0.99	0.26	0.27	0.41	0.19	0	0	0	one	one	one	one	0	one	0
39	0.42	0.60	0.67	0.17	0.48	0.71	0,69	one	0	0	one	0	one	0	0	one	0
40	0.44	0.58	0.50	0.80	0.35	0.34	0.30	one	0	0	one	0	one	one	0	one	0
41	0.48	0.34	0.01	0.93	0.98	0.18	0.44	one	0	one	0	0	one	0	one	0	0
42	0.48	0,69	0.71	0.25	0.70	0.45	0.12	one	0	0	one	0	one	one	0	one	0
43	0.27	0.83	0.35	0.86	0.76	0.89	0.60	one	0	0	one	0	0	0	0	0	0
44	0.61	0.60	0.35	0.31	0.59	0.27	0.25	0	0	0	one	0	one	one	0	one	0
45	0.61	0.70	0.54	0.13	0.24	0.36	0.86	0	0	0	one	one	one	0	0	one	0
46	0.70	0.85	0.53	0.18	0.14	0.11	0.70	0	0	0	one	one	one	0	0	one	0
47	0.57	0.35	0.19	0.48	0.38	0.64	0.25	0	0	one	one	0	one	one	0	one	0
48	0.94	0.02	0.88	0.45	0.59	0.07	0.47	0	one	0	one	0	one	0	0	one	0
49	0.86	0.89	0.20	0.15	0.40	0.47	0.22	0	0	one	one	0	one	one	0	one	0
50	0.95	0.28	0.71	0.59	0.45	0.28	0.97	0	0	0	one	0	one	0	0	one	0
51	0.24	0.14	0.41	0.49	0.32	0.00	0.75	one	0	0	one	0	one	0	0	one	0
52	0.58	0.00	0.13	0.39	0.92	0.96	0.00	0	one	one	0	0	0	one	one	one	one
53	0.35	0.62	0.98	0.83	0.25	0.33	0.65	one	0	0	one	one	one	0	one	one	one
54	0.35	0.79	0.88	0.73	0.24	0.51	0.46	one	0	0	one	one	one	0	one	one	one
55	0.03	0.99	0.35	0.34	0.50	0.76	0.63	one	0	0	one	0	one	0	0	one	0
56	0.34	0.22	0.26	0.94	0.76	0.81	0.22	one	0	0	one	0	0	one	0	one	0
57	0.73	0.01	0.96	0.77	0.38	0.84	0.93	0	one	0	one	0	0	0	0	0	0
58	0.39	0.73	0.84	0.95	0.55	0.29	0.24	one	0	0	one	0	one	one	0	one	0
59	0.33	0.39	0.64	0.84	0.89	0.28	0.25	one	0	0	one	0	one	one	0	one	0
60	0.27	0.51	0.63	0.38	0.77	0.61	0.01	one	0	0	one	0	one	one	0	one	0
61	0.38	0.91	0.76	0.04	0.87	0.36	0.90	one	0	0	one	0	one	0	0	one	0
62	0.93	0.94	0.56	0.66	0.41	0.98	0.72	0	0	0	one	0	0	0	0	0	0
63	0.39	0.73	0.71	0.91	0.73	0.51	0.84	one	0	0	one	0	one	0	0	one	0
64	0.60	0.08	0.72	0.09	0.72	0.58	0.79	0	one	0	one	0	one	0	0	one	0
65	0.19	0.61	0.05	0.89	0.46	0.13	0.57	one	0	one	one	0	one	0	one	one	one
66	0.89	0.18	0.34	0.08	0.42	0.35	0.23	0	0	0	one	0	one	one	0	one	0
67	0.17	0.38	0.50	0.03	0.59	0.45	0.94	one	0	0	one	0	one	0	0	one	0
68	0.59	0.40	0.42	0.95	0.12	0.99	0.07	0	0	0	one	one	0	one	0	one	0
69	0.56	0.54	0.07	0.54	0.82	0.65	0.57	0	0	one	one	0	one	0	0	one	0
70	0.25	0.08	0.94	0.38	0.49	0.42	0.48	one	one	0	one	0	one	0	0	one	0
71	0.50	0.14	0.76	0.79	0.17	0.75	0.97	one	0	0	one	one	one	0	one	one	one
72	0.86	0.47	0.15	0.58	0.92	0.45	0.19	0	0	one	0	0	one	one	0	one	0
73	0.95	0.64	0.88	0.95	0.64	0.35	0.76	0	0	0	one	0	one	0	0	one	0
74	0.33	0.35	0.10	0.58	0.14	0.36	0.59	one	0	one	one	one	one	0	one	one	one
75	0.20	0.65	0.72	0.56	0.54	0.25	0.74	one	0	0	one	0	one	0	0	one	0
76	0.13	0.77	0.41	0.01	0.99	0.95	0.25	one	0	0	0	0	0	one	0	one	0
77	0.10	0.45	0.87	0.15	0.75	0.02	0.15	one	0	0	one	0	one	one	0	one	0
78	0.15	0.04	0.94	0.14	0.54	0.58	0,69	one	one	0	one	0	one	0	0	one	0
79	0.95	0.66	0.99	0.35	0.21	0.30	0.42	0	0	0	one	one	one	0	0	one	0
80	0.99	0.79	0.42	0.36	0.70	0.83	0.51	0	0	0	one	0	0	0	0	0	0
81	0.59	0.74	0.19	0.49	0.50	0.11	0.29	0	0	one	one	0	one	one	0	one	0
82	0.99	0.64	0.40	0.24	0.20	0.89	0.17	0	0	0	one	one	0	one	0	one	0
83	0.19	0.43	0.71	0.78	0.19	0.89	0.53	one	0	0	one	one	0	0	one	0	0
84	0.71	0,69	0.87	0.33	0.14	0.85	0.48	0	0	0	one	one	0	0	0	0	0
85	0.00	0.56	0.05	0.21	0.67	0.38	0.51	one	0	one	one	0	one	0	one	one	one
86	0.53	0.81	0.26	0.72	0.63	0.84	0.70	0	0	0	one	0	0	0	0	0	0
87	0.18	0.05	0.91	0.12	0.17	0.16	0.84	one	one	0	one	one	one	0	one	one	one
88	0.04	0.30	0.46	0.19	0.77	0.58	0.46	one	0	0	one	0	one	0	0	one	0
89	0.70	0.71	0.43	0.20	0.48	0.36	0.27	0	0	0	one	0	one	one	0	one	0
90	0.49	0.88	0.64	0.30	0.14	0.72	0.73	one	0	0	one	one	one	0	one	one	one
91	0.68	0.33	0.41	0.12	0.79	0.25	0.74	0	0	0	one	0	one	0	0	one	0
92	0.64	0.99	0.32	0.39	0.59	0.98	0.88	0	0	0	one	0	0	0	0	0	0
93	0.29	0.13	0.04	0.37	0.41	0.96	0.27	one	0	one	one	0	0	one	one	one	one
94	0.66	0.55	0.76	0.44	0.16	0.04	0.27	0	0	0	one	one	one	one	0	one	0
95	0.71	0.08	0.12	0.88	1.00	0.07	0.59	0	one	one	0	0	one	0	one	0	0
96	0.55	0.72	0.87	0.90	0.08	0.64	0.81	0	0	0	one	one	one	0	0	one	0
97	0.21	0.78	0.01	0.02	0,69	0.39	0.27	one	0	one	one	0	one	one	one	one	one
98	0.93	0.92	0.40	0.00	0.93	0.48	0.71	0	0	0	0	0	one	0	0	0	0
99	0.63	0.76	0.19	0.32	0.62	0.31	0.56	0	0	one	one	0	one	0	0	one	0
##	0.28	0.43	0.82	0.25	0.46	0.41	0.47	one	0	0	one	0	one	0	0	one	0

																	18
																0.18

absolute error of failure-free operation of the system calculated by the Montecarlo method and the method of imitation modeling

| PP * | = | 0,1946-0.18 | = 0.01

Comments

To leave a comment

If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.

To reply

Comment

To confirm that you are not a bot, answer:

Name

Email(not published)

Vote

An example of the Monte Carlo method and simulation.

Comments

To leave a comment

System modeling

Terms: System modeling