FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 1, PAGES 81-107

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

E.V.Bulinskaya

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 Nx,n = inf {k: Wk = n}, the first overflow time in terms of inventory theory. The limit distribution of normalized random variable $\tau_{x,n} = N_{x,n} (PN_{x,n})^{-1}$ is obtained for all the initial states x and possible values of PXk for the case of the three-valued i. i. d. random variables Xk (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 PXk<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.

All articles are published in Russian.

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