On a class of one-dimensional random walks

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.