Asymptotics for the Late Arrivals Problem

TL;DR

This paper analyzes a queueing system with deterministic arrivals delayed by i.i.d. exponential times, proving super-exponential decay of the equilibrium distribution and proposing an effective numerical approximation method.

Contribution

It provides the first solution to the classical 'late arrivals problem' by deriving the equilibrium distribution and its decay properties.

Findings

Equilibrium distribution decays super-exponentially in the quarter plane.

A functional equation for the generating function is solved on a subset of its domain.

A simple approximation scheme effectively computes the stationary distribution.

Abstract

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays $\xi_{i}$. The standard deviation $\sigma$ of the delay is finite, but its value is much larger than the deterministic unit service time. We describe the model as a bivariate Markov chain, we prove that it is ergodic and then we focus on the unique joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution of such equation on a subset of its set of definition. This solution allows us to prove that the equilibrium distribution of the Markov chain decays super-exponentially fast in the quarter plane. Finally, exploiting the latter result, we discuss the numerical computation of the stationary distribution, showing the effectiveness of a simple approximation scheme in a wide region of the parameters. The model, motivated by air and railway traffic, was proposed many decades ago by Kendall with the name of "late arrivals problem", but no solution has been found so far.