With a certain rate $R$ balls fall into a box. There is no limit to the number of balls the box can hold, but each ball has a rate $\gamma$ to leave the box and when two balls hit each other they leave both the box. The probability that two balls hit each other is $\beta$.
One can build a markov chain to describe this process. The rate matrix of this problem is the following pentadiagonal matrix:
$$ P = \begin{pmatrix} -R & R & 0 & 0 & \dots & 0 & \dots \\ w_{1} & q_1 & R & 0 & \dots & \vdots & \dots \\ v_2 & w_{2} & q_2 & R & \dots & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots & R & \dots \\ 0 & \dots & 0 & v_i & w_{i} & q_i & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots\\ \end{pmatrix} $$ where $$\begin{cases} v_i=\beta i (i-1) \\ w_i=\gamma i \\ q_i = -R-v_i-w_i \end{cases} $$
To find the stationary state I have to solve: $$ \bar{x}P = 0 $$ that corresponds to this system:
\begin{equation} \begin{cases} -R x_0 + w_1 x_1 + v_2 x_2 = 0 \\ R x_0 -(v_1 + w_1 + R) x_1 + w_2 x_2 + v_3 x_3 = 0\\ \vdots \\ R x_{i-1} -(v_i + w_i + R) x_i + w_{i+1} x_{i+1} + v_{i+2} x_{i+2} = 0\\ \vdots \\ \end{cases} \label{system_stat_stat} \end{equation}
there exist a closed (not recursive) analytical solution of it? if yes, how can I get it?