Asymptotics for the late arrivals problem

Abstract

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays ξ 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. Keywords Late arrivals • Exponentially delayed arrivals • Pre-scheduled random arrivals • Queues with correlated arrivals • Bivariate generating function Partially supported by PRIN 2012 "Problemi matematici in teoria cinetica ed applicazioni".

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