EEL6507sp09L17

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EEL6507 Spring 2009, Lecture 17, Monday 2009/02/16 (Notes created by Kyungyong Lee)

  • Method of stages
  • non-exponential service, arrival(not discussed in this class)

Laplace Transform

  • Definition of Laplcae transform
B^*(s) = \int\limits_{0}^{\infty}e^{-sx}b(x)dx
  • Comparing Laplace transform and Z-transform

Z-transform :

P(z) = \sum_{k=0}^{\infty} Z^kP(k)
P(1) = \sum_{k=0}^{\infty} P_k = 1
P'(1) = \sum_{k=0}^{\infty} kP_k = <K>
P''(1) = \sum_{k=0}^{\infty} k(k-1)P_k = <K^2> - <K>

Laplace transfrom :

B^*(0) = \int\limits_{0}^{\infty}e^{-0x}b(x)dx = 1
B^{*'}(0) = \int\limits_{0}^{\infty}-xe^{-0x}b(x)dx = -<X>_b
\begin{alignat}{2}...\end{alignat}
\begin{alignat}{2}B^{*(n)}(0) = (-1)^n<X^n>_b\end{alignat}

Multiplication of Laplace transform

\begin{alignat}{2} B_1^*(s)B_2^*(s) & = \int\limits_{0}^{\infty}e^{-sx}b_1(x)dx\int\limits_{0}^{\infty}e^{-sy}b_2(y)dy\\ & = \int\limits_{0}^{\infty}\int\limits_{0}^{\infty}e^{-s(x+y)}b_1(x)b_2(y)dxdy\\ & Let \ \ y^' = x+y \Rightarrow\ dy^'=dy , y=y'-x\\ & = \int\limits_{0}^{\infty}\int\limits_{x}^{\infty}e^{-sy^'}b_1(x)b_2(y^'-x)dy^'dx\\ & = \int\limits_{0}^{\infty}\int\limits_{0}^{y^'}e^{-sy^'}b_1(x)b_2(y^'-x)dxdy^'\\ & = \int\limits_{0}^{\infty}e^{-sy^'}(\int\limits_{0}^{y^'}b_1(x)b_2(y^'-x)dx)dy^' \end{alignat}

If we let

b_3(y^')=\int\limits_{0}^{y^'}b_1(x)b_2(y^'-x)dx

Then,

B_1^*(s)B_2^*(s) = B_3^*(s)
  • What is \begin{alignat}{2}b_3(y^')\end{alignat}

\begin{align} 1. non negative\\ 2. normalized\\ \end{align} \begin{align} \Rightarrow\ b_3\ is\ a\ pdf\ of\ distribution\ :\ X+Y \geq Y^' \\ \end{align}

Lecture Voice Recording

lecture 16 recording

lecture 17 recording

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