Queuing for an infinite bus line and aging branching process

TL;DR

This paper analyzes a queueing system with Poisson arrivals on an infinite bus line, exploring the asymptotic behavior of buses and their coalescence using branching process techniques.

Contribution

It introduces a novel connection between queueing dynamics and aged structured branching processes with immigration in varying environments.

Findings

Identifies three regimes of bus behavior, with two leading to almost sure coalescence.

Establishes a Kesten-Stigum type theorem for the branching process normalization.

Develops new techniques combining spine approach and geometric ergodicity for analysis.

Abstract

We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is less than d\_i stations before, where d\_i is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually coalesce almost surely by coupling arguments. Three regimes appear, two of which leading to a.s. coalescing of the buses.The approach relies on a connection with aged structured branching processes with immigration and varying environment. We need to prove a Kesten Stigum type theorem, i.e. the a.s. convergence of the successive size of the branching process normalized by its mean. The technics developed combines a spine approach for multitype branching process in varying environment and geometric ergodicity along the spine to control the increments of the normalized process.