Considering a queuing system with three nodes and a single server (in a not necessary very realistic regime) results in an interesting billiard-type problem on the sides of equilateral triangle. The detailed study of the trajectories reveals that in almost all situations there are at most 4 attractor orbits for which the order of visits to the sides of the triangle eventually becomes periodic, while the periods may be arbitrary long. At the same time there is an uncountable set of parameters, yet of measure zero, for which the trajectories will be chaotic. Not surprisingly, the stochastic system approximated by the billiard trajectories exhibits exactly the same properties.