A queueing model for local traffics to join a main stream under a leading space condition

We consider congestion of traffics that are randomly produced in a bounded area. Those traffics start at local lines which are connected to a main line, and they have a common destination located at the end of the main line. We assume that a stream on the main line runs with a constant speed, while local traffics can join the main stream only if a certain free space condition is satisfied. So, there may arise queues at junctions where the local lines meet the main line. Our primary interest is to see congestion at the junctions. To this end, we formulate this model as a discrete time queueing process, and compute the stationary distributions of the queue lengths at the junctions, provided stability conditions are satisfied. In particular, we give inductive formulas to compute the mean queue lengths. This model may be applied to road traffics and synchronous data transfers in telecommunication networks. From the theoretical point of view, the model may be considered as an extension of priority queues. Some numerical examples are presented as well.