On a class of reflected AR(1) processes

In this paper, we study a reflected AR(1) process, i.e., a process $(Z_n)_n$ obeying the recursion $Z_{n+1}$ = max\{$aZ_n + X_n, 0$\}, with $(X_n)_n$ a sequence of i.i.d. random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case $X_n$ can be written as $Y_n - B_n$, with $(B_n)_n$ being a sequence of independent random variables which are all exp($\lambda$) distributed, and $(Y_n)_n$ i.i.d.; when $|a| <1$ we can also perform the corresponding stationary analysis. Extensions are possible to the case that $(B_n)_n$ are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a Normal random variable conditioned on being positive. Keywords: Reflected processes . queueing . scaling limits