My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair $(i,j), i\neq j$ we'll have an observation $k_{ij}$ of poissoin distribution with parameter $\lambda=a_i/b_j$.
So, I want to estimate the parameters, is it correct if I use the likelihood function? $$L=\prod_{(i,j),i\neq j} \frac{(a_i/b_j)^{k_{ij}} e^{-a_i/b_j}}{k_{ij}!}$$

If yes, and if I define $\sum 1/b_i = 1$ (since the all the parameters are redundant modulo multiplication) then I get $$a_i \sum_{j\neq i}k_{ij} = 1-1/b_i$$ $$b_j \sum_{i\neq j}k_{ij} = \sum_{i\neq j} a_i$$

Thank you in advance.