Asymptotics for the late arrivals problem

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays \(\xi _{i}\). We describe the model as a bivariate Markov chain, prove its ergodicity and study the joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution on a subset of its domain. This solution allows us to prove that the equilibrium distribution of the chain decays super-exponentially fast in the quarter plane. We exploit the latter result and discuss the numerical computation of the solution through a simple yet effective approximation scheme in a wide region of the parameters. Finally, we compare the features of this queueing model with the standard M / D / 1 system, showing that the congestion turns out to be very different when the traffic intensity is close to 1.