Given $C_1$, $C_2$ and $\lambda_i$ for $1 \le i \le N$, I am looking for unknown $\tau_1$ and $\tau_2$ in the following system:
\begin{array}{c,c} \sum_{i=1}^N P_{i}(\tau_1,\tau_2) = C_1 & (1)\\ \\ \sum_{i=1}^N Q_{i}(\tau_1,\tau_2) = C_2 & (2)\\ \\ \end{array} in which
\begin{array}{c,c} P_{i}(\tau_1,\tau_2) = \Big(Q_{i}(\tau_1,\tau_2)\big(1 - exp(-\lambda_i\tau_1)\big)\Big)/\Big(exp(-\lambda_i\tau_1) + Q_{i}(\tau_1,\tau_2)\big(1-exp(-\lambda_i\tau_1)\big)\Big) & (3)\\ \\ Q_{i}(\tau_1,\tau_2) = 1 - exp\Big(-\lambda_i\big(1-P_{i}(\tau_1,\tau_2)\big)\tau_2\Big) & (4)\\ \\ \lambda_i > \lambda_j \quad \forall i,j \quad i < j \quad 1 \le i,j \le N & (5) \end{array} Since $Q_{i}(\tau_1,\tau_2)$ and $P_{i}(\tau_1,\tau_2)$ are interdependent, I guess a fixed-point iterative procedure is required.
Could someone please help me with a suggestion or hint to help me find $\tau_1$ and $\tau_2$?
Thanks in advance.
To make things simpler, consider that you have to solve $$F(x,y)=0$$ $$G(x,y)=0$$ and you need "good" starting values for $x_0$ and $y_0$.
Consider $$\Phi(x,y)=F^2(x,y)+G^2(x,y)$$ which would be $0$ at solution.
So, make a $2D$ grid computing $\Phi(x,y)$ for different values of $x,y$; look for which pair you get the minimum value of $\Phi(x,y)$ and start from this point.