# A M/M/$\infty$ queue of depositors with compound interest

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.

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I must be missing something. Where is the bank's money coming from? Each depositor, upon leaving, is taking away more than they gave. So the bank's reserve should decrease approximately linearly in time, eventually becoming negative. The only increase is due to customers who have arrived but not yet left; but since the queue is stable, the number of such customers at any time is bounded in distribution. –  James Martin May 28 '11 at 14:13
Yes, it will go negative. I would like to have an asset process that is basically the mirror image of this one (borrowers arrive in a queue; they take money out when they arrive, then put back more money when they depart) to balance it out, but I'd like to understand one queue before trying to solve two together. –  Ronaldo Carpio May 28 '11 at 17:35
OK - but you asked for the "steady-state distribution of M". If the reserve has a negative drift in this way, it will converge to $-\infty$; there will not be a steady-state distribution. –  James Martin May 29 '11 at 1:00