Stochastic Reservoir Calculations

Prabhu (1958) obtained the stationary distribution of storage level $Z_{t}$ in a reservoir of finite volume $v$, given an inflow $X_{t}$ and an outflow $Y_{t}$. Time $t$ is assumed to be discrete, $X_{t} \sim$ Gamma$(p,\mu)$ are independent and $p$ is a positive integer. The mean inflow is $p/\mu$; the target outflow is $m$ (constant). We attempt to clarify intricate details, often omitted in the literature, by working through several examples. Of special interest are the probabilities of depletion ($Z_{t}=0$) and spillage ($Z_{t}=v$). For prescribed {$v,p,\mu$}, what value of $m$ minimizes both of these?