Basically I am attempting doing load balancing between two machines and I have equations but no clear road to a solution.
First, we know that $C_{tot} = C_1 + C_2$ and $C_{tot}$ is known. $C$ stands for capacity in this case.
And $C_n = A_nx_n+B_n$ where $A_n$ and $B_n$ are known and $n \in \{1,2\}$.
Also that $\frac{C_1}{C_2} = \frac{\ln \left( \frac{D-x_2}{E-x_2} \right)}{\ln \left( \frac{D-x_1}{E-x_1} \right)}$ where $D$ and $E$ are known.
I have a guess for $x_1$ as well. I tried simply using $x_1$ and $C_{tot}$, combining equations $1$ and $2$ and finding a formula for $x_2$ in terms of $x_1$, but it gives incorrect answers since it is not properly balanced
It almost looks like there should be a simplification that can be done to the last equation, but I have not found one. I am looking for a direct calculation for the ratio of the capacity, or some method of iterating to a solution.
Write $L_1=\ln\left(\frac{D-x_1}{E-x_1}\right)$ and $L_2=\ln\left(\frac{D-x_2}{E-x_2}\right)$. If we start with an initial guess for $x_1$ and $x_2$, then we obtain corresponding guesses for $L_1$ and $L_2$ from these equations. Now, we have a linear system of equations in $C_1$ and $C_2$: \begin{align*} C_1+C_2 &= C_{tot}\\ L_1C_1-L_2C_2 &= 0 \end{align*} which has solution $C_1=\frac{C_{tot}L_2}{L_1+L_2}$, $C_2=\frac{C_{tot}L_1}{L_1+L_2}$. Solving for $x_n$ in $C_n=A_nx_n+B_n$, we obtain a new guess for $x_1$ and $x_2$, providing an iterative approach to solving the problem.