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

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

We should find the best $W >0$ when $F( \cdot )$ 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. $

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

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

We should find analytically the optimal $W >0$ which maximize the first equation subject to the second equation, where $F( \cdot )$ is comulative distribution function (CDF), and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $0<\pi_0, \pi_1<1$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume that, for example, $L_0$ and $L_1$ may have Erlang or exponential distribution.

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

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

We should find the best $W$ when $\xi$ is constant, $F( \cdot )$ 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 $F_{L_1}$ such as below:

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

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

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

We should find the best $W >0$ when $F( \cdot )$ 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. $

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

$W$ 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, $F_{L_0}$ and $F_{L_1}$ 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}

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

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

We should find the best $W$ when $\xi$ is constant, $F( \cdot )$ 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 $F_{L_1}$ such as below:

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