noindent This paper studies a one-dimensional Markov chain ${X_n,n=0,1,dots$ that satisfies the recurrence relation $X_n = max(0, X_{n-1 + eta_n^{(m) )$ if $X_{n-1 =m leq a$; for $X_{n-1 > a$ it satisfies the same relation with $eta_n^{(m)$ replaced by $xi_n$. Here ${ eta_n^{(m) $ and ${ xi_n $ are independent sequences of independent, integer-valued random variables. The limiting distribution of $X_n$ is determined, using Wiener-Hopf factorization. It requires solving a set of $a+1$ linear equations. The asymptotic behaviour of the limiting distribution is also described. Various applications to queueing models are discussed.

Markov processes (msc 60Jxx), Queueing theory (msc 60K25)
CWI
Department of Operations Research, Statistics, and System Theory [BS]
Combinatorial Optimization and Algorithmics

Boxma, O.J, & Lotov, V.I. (1995). On a class of one-dimensional random walks. Department of Operations Research, Statistics, and System Theory [BS]. CWI.