  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.
