Queueing for an infinite bus line and aging branching process

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 age-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 techniques developed combine a spine approach for multitype branching process in varying environment and geometric ergodicity along the spine to control the increments of the normalized process.