transform - X is non-negative real value The LT of X is 9...

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CDA6530: Performance Models of Computers and Networks Review of Transform Theory
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2 Why using transform? Make analysis easier Two transforms for probability Non-negative integer r.v. Z-transform (or called probability generating function (pgf)) Non-negative, real valued r.v. Laplace transform (LT)
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3 Z-transform Definition: G X (Z) is Z-transform for r.v. X Example: X is geometric r.v., p k = (1-p)p k For pz<1
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4 Poisson distr., p k = λ k e - λ /k!
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5 Benefit Thus Convolution: X, Y independent with pdfs G X (z) and G Y (z), Let U=X+Y
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6 Solution of M/M/1 Using Transform Multiplying by z i , using ρ = λ / μ , and summing over i
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7 E [ N ]= dG N ( z ) dz | z =1 = 1 ρ (1 ρ z ) 2 ρ | z =1 = ρ 1 ρ = 1 μ λ
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8 Laplace Transform R.v. X has pdf f X (x)
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Unformatted text preview: X is non-negative, real value The LT of X is: 9 Example X: exp. Distr. f X (x)= λ e-λ x Moments: 10 Convolution X 1 , X 2 , , X n are independent rvs with If Y=X 1 +X 2 + +X n If Y is n-th Erlang, F ∗ X 1 ( s ) ,F ∗ X 2 ( s ) , · · · ,F ∗ X n ( s ) F ∗ Y ( s ) = F ∗ X 1 ( s ) · F ∗ X 2 ( s ) · · · F ∗ X n ( s ) F ∗ Y ( s ) = ( λ λ + s ) n 11 Z-transform and LT X 1 , X 2 , , X N are i.i.d. r.v with LT N is r.v. with pgf G N (z) Y= X 1 +X 2 + +X N If X i is discrete r.v. with G X (Z), then F ∗ X ( s ) G Y ( z ) = G N ( G X ( z ))...
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This note was uploaded on 01/14/2012 for the course CDA 6530 taught by Professor Zou during the Fall '11 term at University of Central Florida.

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transform - X is non-negative real value The LT of X is 9...

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