On high-level crossing for a class of discrete-time stochastic processes

The aim of this paper is to study the asymptotic behaviour of the first passage time for some discrete-time stochastic processes arising in applied probability. The paper is organized as follows. The systems' description is given in § 1 along with the main results. The integer-valued random walks with impenetrable (as well as reflecting) barrier at origin
$$ W_{k}=\max(0,W_{k-1}+X_{k}),\ k \geq 1,\ W_{0}=x $$
are treated in § 2. The main object of investigation is $N_{x,n}=\inf\{k\colon W_{k}=n\}$, the first overflow time in terms of inventory theory. The limit distribution of normalized random variable $\tau_{x,n}=N_{x,n}(\mathrm{E}N_{x,n})^{-1}$ is obtained for all the initial states $x$ and possible values of $\mathrm{E}X_{k}$ for the case of the three-valued i. i. d. random variables $X_{k}$ (demand and supply in batches of fixed volume). The domain of model's stability with respect to initial state and system's parameters is established as well. The influence of two-level control policy on system's behaviour is dealt with in § 3. It is proved, in particular, that $\tau_{x,n}$ is asymptotically exponential if $\mathrm{E}X_{k}<0$ in a sufficiently wide band in the neighbourhood of the absorbing boundary $n$. The directions of further investigations and various possibilities of application are given in § 4.