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Probability mass function (pmf) of discrete random variable X

Probability mass function (pmf) of discrete random variable X. Expected value of X. Z-transform. Expected value of X. The z -transform of the sum of independent random variables is the product of the individual z -transforms of these variables :. f.

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Probability mass function (pmf) of discrete random variable X

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  1. Probability mass function (pmf) of discrete random variable X Expected value of X

  2. Z-transform Expected value of X

  3. Thez-transform of the sum of independent random variables is the product of the individualz-transforms of these variables:

  4. f The Universal Generating Function Xi Composition Operator Arbitrary math. object f(X1,…,Xn) E(F(X1,…,Xn))=U'(1)

  5. pmf of function Y=f(X1,…,Xn) Composition operator pmf of n random variables f(X1,…,Xn)

  6. f Properties of composition operators Associative Commutative Recursive

  7. min min min f Properties of composition operators Collection of the like terms pizx+pjzx=(pi+pj)zx (0.2z0+0.3z4+0.5z7) (0.2z0+0.8z5) =0.04z0+0.06z0+0.1z0+0.16z0+0.24z4+0.4z5 =0.36z0+0.24z4+0.4z5 (0.2z0+0.3z4+0.5z7) (0.2z0+0.8z5) (0.2z0+0.3z4+0.4z7+0.1z8) 3X2=6 6X4=24 3X4=12 3

  8. Functions in composition operators … Series systems Processing speed Identical elements:

  9. Functions in composition operators Series systems Transmission capacity … Identical elements:

  10. D E(max(w-G,0)) Series systems Performance measures w R E 0 Transmission capacity: Processing speed:

  11. Functions in composition operators Parallel systems Flow dispersion Transmission capacity No flow dispersion Flow dispersion n identical elements:

  12. D E(max(w-G,0)) Flow transmission parallel systems Performance measures w R E

  13. Functions in composition operators Parallel systems Work sharing Processing speed No work sharing No work sharing n identical elements:

  14. D E(max(w-G,0)) Task processing parallel systems Performance measures No work sharing w R E

  15. fser fser fser fpar fser fpar Series-parallel systems Generalized RBD method Usystem(z)

  16. UGF + Markov chain techniques 1 A B 3 2 1 3 2 Element 1 2 1 g12=1.5 g11=0 Element 3 3 2 1 g32=1.8 g31=0.0 g33=4.0 Element 2 1 2 g22=2.0 g21=0

  17. … 1 1.5, 2, 4 3.5 2 4 0, 2, 4 1.5, 2, 1.8 2 1.8 3 1.5, 0, 4 1.5 5 6 8 0, 2, 1.8 1.5, 2, 0 0, 0, 4 1.8 0 0 7 1.5, 0, 1.8 1.5 10 9 0, 2, 0 0, 0, 1.8 0 0 11 1.5, 0, 0 0 12 0, 0, 0 0 Pure Markov approach System state-space diagram System of 12 differential equations

  18. + min u3(z) 3 equations 2 equations UGF + Markov approach 2 equations Element 1 u1(z) 2 1 g12=1.5 g11=0 Element 3 U=u1(z) u2(z) u3(z) 3 2 1 g32=1.8 g31=0.0 g33=4.0 Element 2 1 2 g22=2.0 g21=0 u2(z)

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