Cluster Motion in a Two-Contour System with Priority Rule for Conflict Resolution

A deterministic dynamical system representing the contour network is considered. The number of contours is two. At each contour there is a segment moving with a constant velocity which is called the cluster, because in the discrete variant of the system it corresponds to the cluster of particles, that is, to the group of particles occupying the adjacent cells and moving simultaneously. The lengths of contours and the lengths of clusters are prescribed. There is a common point named a node. Clusters cannot pass a node simultaneously. A cluster stops and waits for the node to empty if this cluster comes to the node at the instance when another cluster passes through the node. If clusters come to a node simultaneously, then precedence is given to the cluster considered the priority cluster (the priority rule of conflict resolution). The theorems on the average speed of cluster motion are proved taking delays in different types of the system’s behavior into account. It is established that the average speed of motion of each cluster in the system is independent of the cluster position at the initial time instance in contrast to the analogous system with another rule of conflict resolution considered previously, where such dependence appears in the general case. The possible practical interpretation of the studied system is given. The presented system is referred to as the class of dynamical networks introduced and investigated by A.P. Buslaev. The results may be applied to solve questions on the automatization of motion of a continuous mass, simulate the motion of transport, and other areas.