Hello, I'm trying to model a bank's liabilities using a queue. Suppose a bank begins with a cash reserve of $M$. Depositors are a M/M/$\infty$ queue; they arrive with rate $\lambda$ and deposit 1 dollar in the bank. During service time, depositors are earning compound interest of $r$ on their deposits. Departures are at rate $\mu N$; if the depositor had a waiting time of $t$, they will withdraw $e^{rt}$ when they depart. How can I find the steady-state distribution of $M$ (if it exists)?

If this has already been done, I would greatly appreciate pointers to the literature.