All results subjected to a confidence analysis follow the following calculations. Let:
X: random variable (maximum end-to-end delay). µ: Average of X
σ2: Variance of X
Xi: sample of X obtained during it OPNET simulation (using different seed) n: No. of OPNET simulations
x: Sample mean s2: Sample variance 𝑥 =!! !! ! !!! (1) 𝑠! = 1 𝑛 − 1 𝑋! − 𝑥 ! ! !!! (2)
OPNET Network Modeler requires a ‘seed’ value. The value is used to generate different random number generator equations. The equations help to simulate the behavior of non-deterministic aspects. If the distribution of a random variable is unknown, the distribution of its sample mean will approach a normal distribution, as the number of samples increases based on the Central Limit Theorem (CLT). The sample mean also approaches the ensemble mean and the variance of the sample mean is a scaled version of the ensemble mean (mean of x = µ = mean of X and variance of x= 2 = where σ2 = variance of X [32, 49].
x
σ n
2
63
z: is a normal random variable (mean= 0 & variance = 1).
𝑃 −𝑧! < 𝑧 < 𝑧! = 𝛼 (4)
𝑃 𝑥 − 𝜇
𝜎! < 𝑧! = 𝛼
(5)
By using 33 simulations, n > 30 and hence the sample standard deviation s can be used instead of σ as it is difficult to find . The Normal distribution will be used and zα is calculated for a confidence level α = 95%.
n x σ
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