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edited Dec 18 2010 at 9:36

Venous007
11●2

$\max{\pi_0 W_{opt}=\arg {\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }$ \)}$

subject to $\textrm{s.t.}\quad \quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$ \

We should find analytically the best optimal $W >0$ when which maximize the first equation subject to the second equation, where $F( \cdot )$ is probability comulative distribution function , (CDF), and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $\pi_0, 0<\pi_0, \pi_1>0$ pi_1<1$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume that, for example, $F_{L_0}$ and $F_{L_1}$ such as below ($\lambda$ L_0$ and $\mu$ are constant:$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0.$ $F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0L_1$ may have Erlang or exponential distribution.$

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edited Nov 13 2010 at 13:13

Venous007
11●2

$\max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }$ \

$\textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$ \

We should find the best $W$ W >0$ when $\xi$ is constant, $F( \cdot )$ is probability distribution function, and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $\pi_0, \pi_1>0$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume, for example, $F_{L_0}$ and $F_{L_1}$ such as below ($\lambda$ and $\mu$ are constant:

$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0.$ $F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0. $

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edited Nov 13 2010 at 12:56

Venous007
11●2

$$\textrm{Problem}\qquad\qquad \begin{cases}\max{\pi_0 \max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }, \\ $ \textrm{s.t.}\quad

$\textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi'' <\xi$ \end{cases} $$

We should find the best $W$ when $\xi$ is the variableconstant, and $F( \cdot )$ is probability distribution function, and $L_0$ and $L_1$ are random variables. Also, $\pi_0, \pi_1>0$ and $\pi_0 + \pi_1 =1$. Further, if needed, we can assume, for example, $F_{L_0}$ and $F_{L_1}$ can be such as below:

\begin{eqnarray} F_{L_0}(\alpha)

$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0. \label{PDF0}\ F_{L_1}(\alpha) 0.$ $F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0. \label{PDF1} \end{eqnarray}$

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edited Nov 8 2010 at 13:30

Andrey Rekalo
13.9k●1●32●68

\begin{eqnarray}\label{Problem no error} \textrm{Problem}\hspace{-.1in}&:&\left{ \begin{array}{ll}

$$\textrm{Problem}\qquad\qquad \max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0 ^W begin{cases}\max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }, \\ \textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi''

                  \end{array}
                \right.
\end{eqnarray}end{cases}
$$

$W$ is the variable, and $F( \cdot )$ is probability distribution function, $L_0$ and $L_1$ are random variables. Also, $\pi_0, \pi_1>0$ and $\pi_0 + \pi_1 =1$. Further, for example, $F_L_0$ F_{L_0}$ and $F_L_1$ F_{L_1}$ can be such as below:

\begin{eqnarray}
  F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0. \label{PDF0}\
  F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0.  \label{PDF1}
\end{eqnarray}

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asked Nov 8 2010 at 12:07

Venous007
11●2

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One-Variable Optimization Problem