The optimal competition resolution rule for a controlled binary chain

A dynamical system that belongs to the class introduced by A. P. Buslaev is investigated. The system contains N contours. There are two cells and one particle on each contour. For each contour there is one common point, called a node, with each of the neighboring nodes. In the deterministic version of the system, at any discrete moment, each particle moves to another cell if there is no delay. The delays are due to the fact that two particles cannot pass through the node at the same time. If two particles tend to cross the same node, then only one particle moves in accordance with a given rule of competition resolution. In the stochastic version the particle tends to move in a state corresponding to the state of the deterministic system in which the particle is moving. This attempt is implemented in the corresponding system with a probability of 1-ε, where ε - is a small value. A rule for resolving competition, called the long cluster rule, is obtained, such that this rule puts the system in such a state that all particles move without delay at the present moment and in the future (the state of free movement), and the system gets into a state of motion in the shortest possible time. The average number of vi displacements of a particle of the i-th contour per unit of time is called the average velocity of this particle, i=1,…,N. For the stochastic version of the system, the following is established under the assumption that N=3. For the long rule, the average particle velocities v1=v2=v3=1-2ε+o(ε) (ε→0). For the left-priority rule, according to which, in competition, the particle of the contour with the lower number has priority, the average particle velocity v1=v2=v3=67+o(ε√).