Given the following dynamical equation for $X(t)$ as follows:
$X(t+1) = X(t) - \min\{X(t), M\} + Y(t)$,
or can write it as follows:
$X(t+1) = \max\{X(t) - M, 0\} + Y(t)$,
Assume the PDF of $Y(t)$ is given by $f_Y(y)$ and $\mathbb{E}[Y]<M$. Is it possible to obtain the closed-form expression (or certain bound) of the following two quantities:
1) Average queue length: $\lim_{T \rightarrow \infty} \frac{1}{T}\sum_{t=1}^T X(t)$
2) Average resource utilization ratio: $\lim_{T \rightarrow \infty} \frac{1}{T}\sum_{t=1}^T\frac{min\{X(t), M\}}{M}$