I asked a question here about solving a system related to an absorbing Markov chain. I now have a variation where there are $m$ types (of student, job seeker, etc) each of which applies to institutions in a different order, and the total accepted by the institutions has to respect capacities $\mathbf{c}$. So, whereas the original was of the form (after some manipulation): $$ \mathbf{p}^T [\mathbb{I} - \mathbf{T} + \mathbf{A}\mathbf{T}]^{-1}\mathbf{A} = \mathbf{c}^T $$ The new form is $$ \sum_{i=1}^m \mathbf{p}_i^T [\mathbb{I} - \mathbf{T}_i + \mathbf{A}\mathbf{T}_i]^{-1}\mathbf{A} = \mathbf{c}^T $$ or in matrix notation $$ \mathbf{p}^T [\mathbb{I} - \mathbf{T} + \mathbf{A}^*\mathbf{T}]^{-1}\mathbf{A}^*\mathbf{S} = \mathbf{c}^T $$ where $\mathbf{p} = (\mathbf{p}_1,\mathbf{p}_2,\dots,\mathbf{p}_m)^T$ is an $mn$-vector of initial state probabilities, $\mathbf{T} = \mathrm{diag}(\mathbf{T}_1,\mathbf{T}_2,\dots,\mathbf{T}_m)$ is an $mn \times mn$ block diagonal matrix of transition probabilities, with rows summing to 1. $\mathbf{c}$ is still an $n$-vector of capacities. $\mathbf{S} = (\mathbb{I},\mathbb{I},\dots,\mathbb{I})^T$ is the $mn \times n$ summing matrix.
$\mathbf{A}^* = \mathrm{diag}(\mathbf{A},\mathbf{A},\dots,\mathbf{A})$ is just a replication of $\mathbf{A}$ which is itself a diagonal matrix with diagonal entries $\mathbf{a} = (a_1,a_2,\dots,a_n)^T$. These diagonal entries, the acceptance probabilities, are the only unknowns in the system and the quantities of interest.
Whereas before I could take $[\mathbb{I} - \mathbf{T} + \mathbf{A}\mathbf{T}]^{-1}\mathbf{A}$ over to the other side, I now can't because of the summation, which makes it more difficult. I still have reason to believe there is a solution for $\mathbf{a}$, although I could be wrong.
Note you can do: $$ \mathbf{p}^T [\mathbb{I} - \mathbf{T} + \mathbf{A}^*\mathbf{T}]^{-1}\mathbf{S} = \mathbf{c}^T\mathbf{A}^{-1} $$ where the $\mathbf{A}$ that has been taken across is the small version. For what its worth.
I hope someone can help.
P.s. I will also accept answers that show that this is not solvable or not linear in $\mathbf{a}$.