On a Class of One-Dimensional Random Walks

#### O.J. Boxma, V.I. Lotov

1996, v.2, №2, 349-362

ABSTRACT

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.

Keywords: random walk,Wiener-Hopf factorization,queues