Modeling and simulation in engineering
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Uniform numbers testing; Ho: random versus Ha: not random for 250,000 runs. Ho is not rejected. After 40 cycles × 250K = 10,000K = 10,000,000 simulations there is still no reject Ho = Random Sequence. This may signal still no rejection of random sequence from the earlier safe threshold: 50K simulations for a JAVA coding uniform random number generator. Important Note: In this figure, buttons indicate: No of values = 250,000 (simulation runs), DF = 6 (Section on Generic Theory, by Knuth's Technique10,11), Significance level (Type‐I error) = 5%, Total Runs: 41,606 ×1 + 51,836 ×2 + 23,059 ×3 + 6583 ×4 + 1482 ×5 + 290 ×6.093 (average for >6) = 250,000, where bold numbers from 1 to >6 are calculated run sizes by Knuth's method. χ2 calculated = 7.57 <χ2 critical value = 12.59. Do NOT reject Ho: random sequence.
Uniform numbers testing; Ho: random versus Ha: not random for 100,000 runs. Ho is NOT rejected. After 50 cycles × 100K = 5000K = 5,000,000 simulations, there is still no reject Ho = random sequence. This may signal still no rejection of random sequence from the earlier safe threshold: 50K for a JAVA coding uniform random number generator.
Uniform numbers testing; Ho: random versus Ha: not random for 50,000 runs. Ho is NOT rejected. After 60 cycles × 50K = 3000K = 3,000,000 simulations there is still no reject Ho = random sequence. This may signal a cut‐off point of no rejection of random sequence from this point on. Safe threshold may be 50K for JAVA coding uniform random number generator.
Uniform numbers testing; Ho: random versus Ha: not random for 10,000 runs. Ho is rejected. On the average, one out of 25 cycles of 10,000 = 250,000 simulations will end up rejecting Ho: random.
Uniform numbers testing; Ho: random versus Ha: not random for 10000 runs. Ho is NOT rejected.
Uniform numbers testing; Ho: random versus Ha: not random for 5000 runs. Ho is rejected. On the average, one out of 10 cycles of 5000 = 50,000 simulations will end up rejecting Ho: random.
Uniform numbers testing; Ho: random versus Ha: not random for 5000 runs. Ho is NOT rejected.
Computer modeling and interplay between experiments, simulation, and theory.25
The Monte Carlo (MC) simulation results of the 2 × 2 × 2 security meter sampling design.
Uniform numbers testing; Ho: random versus Ha: not random for 500 runs. Ho is rejected. On the average, one out of 40 cycles of 500 runs = 20,000 simulations will end up rejecting Ho: random.
Uniform numbers testing; Ho: random versus Ha: not random for 500 runs. Ho is not rejected.
Monte Carlo simulations for the cyber server ($8000 asset) example with inputs in Table 8.
Simplest 2 × 2 × 2 tree diagram for two threats and for two vulnerabilities in a cyber‐risk scenario.
Complex network of seven units with input data, where source: s = 1 and target: t = 7.
Relationships for distributions in statistical simulation where α1 = α2 or α1≠α2, and L = (β1/β2) for SL (α, β, L). (Dashed arrows indicate ‐→∝ Reprinted with permission from Ref 20 Copyright 2007, Wiley & Sons, Inc)
Similar to Figure 12 but with Median (M), first and third quartiles as location measures for n = 100,000 simulation runs plotted for UP (r), DER (d), and DOWN (q).
P(DER) Cumulative reliability plot with 10,000 Monte Carlo simulation runs.
Given the input table on the l.h.s. column, the p.d.f.s of the three states are plotted for UP (r), DER (d), and DOWN (q) for a 90% confidence level showing mode (m), mean (E) with upper & lower confidence as centrality measures for n = 100,000 simulation runs.
The input data in Table 1, and simulation results in Tables 2–4 and Figures 5–7 display the cumulative reliability plots of the three states for UP (r), DER (d), and DOWN (q).
P(DOWN) Cumulative reliability plot with 10,000 Monte Carlo simulation runs.
P(UP) Cumulative reliability plot with 10,000 Monte Carlo simulation runs.
A sample illustration of feasible transitions from Figure 3 implemented to subsections of Three‐State Sahinoglu Probability Model of Production Units (Monte Carlo Simulation).
Three‐state Markov diagram of a repairable hardware unit with UP, DOWN and DER states.
Given the input table, the p.d.f. of the two‐state SL is plotted for UP (r) and DOWN (q) for 90% confidence analytically showing mode (m), mean (E) with upper & lower confidence bounds.