You are asking the queue $Q\to (Q+Y,0)^+$ to be ergodic, where $Y$ is your $X-C$, and the stationary distribution of this queue to be integrable. Ergodicity requires that $E(Y)<0$, i.e. $\mu < p$. Integrability holds as soon as $Y$ (or, equivalently, $X$) is square integrable.

Amongst many other places, you might want to check example I.5.7 of Applied Probability and Queues by Søren Asmussen. (Are you sure this is not HW?)

You are asking the queue $Q\to (Q+Y)^+$ to be ergodic, where $Y$ is your $X-C$, and the stationary distribution of this queue to be integrable. Ergodicity requires that $E(Y)<0$, i.e. $\mu < p$. Integrability holds as soon as $Y$ (or, equivalently, $X$) is square integrable.

Amongst many other places, you might want to check example I.5.7 of Applied Probability and Queues by Søren Asmussen. (Are you sure this is not HW?)