Abstract: In this talk we consider a stochastic point process $i + \xi_i$, where $i\in \mathbb{N}$ and the $\xi_i's$ are i.i.d exponential random variables with standard deviation $\sigma$. Some properties of this process are investigated. We then study a discrete time single server queueing system with this process as arrival process and deterministic unit service time. We obtain a functional equation of the bivariate probability generating function of the stationary distribution for the system. The functional equation is quite singular, does not admit simple solution. We find the solution of such equation on a subset of its set of definition. Finally we prove that the stationary distribution of the system decays super-exponentially fast in the quarter plane. The queueing model, motivated by air and railway traffic, has been proposed by Kendall and others some five decades ago, but no solution of it has been found so far. This is a joint work with Gianluca Guadagni, Carlo Lancia and Benedetto Scoppola.