I have the following equation for the delay in a queue\begin{align} d(f)=\frac{c(1-f)^2}{2(1-a)}+\frac{\lambda b}{2f(f-a)}\end{align} where $0\le f\le 1;\quad c,a=\lambda\tau, \ b=\tau^2$ or $\tau^{(2)}$ depending upon the queueing model, are the parameters of the system. I have two such queues and I have to solve the following equation $$\lambda_1d_1'(f_1)=\lambda_2d_2'(f_2)$$ where $'$ signifies differentiation with respect to $f$ and we have the relation between $f_1,f_2$ $$f_1+f_2=1-\delta,\quad 0<\delta<1$$ $\delta $ is given. I have reduced the equation to the following equivalent form (assuming $b=\tau^2$)$$\frac{2c}{(1-a_1)(1-a_2)}\left[a_1(1-a_2)(f_1-1)-a_2(1-a_1)(f_2-1)-a_1+a_2\right]\\ = \tau\left[\frac{a_2^3(2f_2-a_2)}{f_2^2(f_2-a_2)^2}-\frac{a_1^3(2f_1-a_1)}{f_1^2(f_1-a_1)^2}\right]$$ with $f_1+f_2=1-\delta$
But now I am stuck, because though I can get numerical values for $f_1,f_2$, I want to study the dependence of $f_1,f_2$ upon the various parameters $c,\lambda, \tau$.
I don't know how to proceed from here. Is some kind of approximation going to help me? Please give me some suggestions regarding how to get a feel for the dependence of $f_1,f_2$ upon the parameters, in some intelligent way, except extensive numerical study.