Ergodic and stability theorems for random walks in the strip and their applications

Let $\{N_n,\tau_n^e,\tau_n^s;\,1\le n<\infty\}$ be a stationary sequence of positive random variables, $\xi_n=\tau_n^s-\tau_n^e$. In this paper ergodic and stability theorems are obtained for the sequences $\{w_{n+k};\,k\ge 0\}$ as $n\to\infty$, which are defined by the recurrent equations of two types. The equations of the first type have the form
$$\begin{align*} &w_{n+1}=\max(0,w_n+y_n),\qquad n\ge 1,\\ &\text{where}\ y_n= \begin{cases} \xi_n,&\text{if}\ w_n\le N_n,\\ -\tau_n^e,&\text{if}\ w_n> N_n. \end{cases} \end{align*}$$
The equations of the second type are the following:
$$w_{n+1}=\min\{N_{n+1},\max(0,w_n+\xi_n)\},\qquad n\ge 1.$$
The applications to the queueing theory are considered.