Summary: For \(\{X(t), t \in G_\delta\}\) a centered Gaussian process with variance \(\sigma^2\) and stationary increments on a discrete grid \(G_\delta = \{0, \delta, 2 \delta, \dots\}\), where \(\delta > 0\), we investigate the stationary reflected process \[ Q_{\delta, X}(t) = \sup_{s \in [t, \infty) \cap G_\delta} (X(s) - X(t) - c(s - t)), \; t \in G_\delta \] with \(c > 0\). We derive the exact asymptotics of \(\mathbb{P} \Bigg(\sup_{t \in [0, T] \cap G_\delta} Q_{\delta, X} (t) > u \Bigg)\) and \(\mathbb{P} \bigg(\inf\limits_{t \in [0, T] \cap G_\delta} Q_{\delta, X}(t) > u \bigg)\), as \(u \to \infty\), with \(T > 0\). It appears that \(\varphi = \lim_{u \to \infty} \frac{\sigma^2(u)}{u}\) determines the asymptotics, leading to three qualitatively different scenarios: \(\varphi = 0, \varphi \in (0, \infty)\) and \(\varphi = \infty\).