Invariant measure for continuous open chain of contours with discrete time

A dynamical system is studied. This system belongs to the class of contour networks introduced by A.P. Buslaev. The system is a version of the system called an open chain of contours. Continuous and discrete versions of the system were considered earlier. The system contains N contours. There is one adjacent contour for the utmost left and the utmost right contour, and there are two adjacent contours for any other contour. There is a common point of any two adjacent contours. This point is called a node. There is a cluster in each contour. For the continuous version, the cluster is a segment, moving with constant velocity if there is no delay. For the discrete version, the cluster is a group of adjacent particles. Delays are due to that two clusters may not move through the same node simultaneously. The main system characteristic, studied earlier, is the average velocity of clusters. In this paper, the continious and discrete version of the open chain of contours are considered. The following version of the system is also considered. Each cluster is a segment, and the cluster is shifted onto the distance α at any discrete moment if no delay occurs. We have obtained the limit distribution of the system states. We have also obtained the limit state distribution (invariant measure) for the open chain of contours with continuous state space and continuos time and for the open chain of contours with discrete state space and discrete time.