Two-Dimensional Moran Model

In this paper, we consider a two-dimensional Moran model of random walks consisting of two queues evolving in parallel which can (at each unit of time) either increase by one or reset to 0. We analyze the joint law of their final altitude and prove that the asymptotic distribution of each component is a shifted geometric distribution and we analyze the maximum of these two components, also giving closed forms for the mean and variance.