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