I encounter a problem in which I need to characterize the asymptotic behaviour of a sequence.

$\{s_{n,k}\}$ is a stationary probability distribution, i.e., $\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}}=1}$, and satisfies

$s_{0,1}=\frac{\alpha}{\delta+\beta}$,

$s_{n,k}=0$ for $n<0$ or $k<0$,

and $(n\mu+k\beta+\delta)s_{n,k}=(k+1)\beta s_{n-1,k+1}+n\mu s_{n,k-1}$,

where $\alpha, \delta, \beta, \mu$ are constants.

What I want to know is the asymptotic behaviour of $\sum_{n=m}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}}}$ when $m \rightarrow \infty$

Initially I considered the generating function $F(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ which satisfies the PDE $\delta F=\beta(x-y)F_y+\mu x(y-1)F_x+\alpha y$, but I didn't find anything useful since this PDE doesn't seem solvable.

Can anyone suggest a way to do it or suggest references? This problem arises in an economic model for firms (you can think of $n$ as size and $k$ as the number of vacancies, and thus the recursive relation describes how firm size grows). It's like a generalization of the birth-death process, but is more complicated.

Thank you for your help!