Optimization of Velocity Mode in Buslaev Two-Contour Networks via Competition Resolution Rules

In computer networks based on the principle of packet switching, the important transmitting function is to maintain packet queues and suppress congestion. Therefore, the problems of optimal control of the communication networks are relevant. For example, there are  users, and no more than a demand of one user can be served simultaneously. This paper considers a discrete dynamical system with two contours and two common points of the contours called the nodes . There are n cells and  particles, located in the cells. At any discrete moment the particles of each contour occupy neighboring cells and form a cluster. The nodes divide each contour into two parts of length  and  (non-symmetrical system). The particles move in accordance with rule of the elementary cellular automaton 240 in the Wolfram classification. Delays in the particle movement are due to that more than one particle cannot move through the node simultaneously. A competition (conflict) occurs when two clusters come to the same node simultaneously. We have proved that the spectrum of velocities contains no more than two values for any fixed n , d and l .  We have found an optimal rule which minimizes the average velocity of clusters. One of the competition clusters passes through the node first in accordance with a given competition rule. Two competition resolutions rules are introduced. The rules are called input priority and output priority resolution rules. These rules are Markovian, i.e., they takes into account only the present state of the system.  For each set of parameters n, d and l , one of these two rules is optimal, i.e., this rule maximizes the average velocity of clusters. These rules are compared with the left-priority resolution rule, which was considered earlier.  We have proved that the spectrum of velocities contains no more two values for any fixed n, l, and d . We have proved that the input priority rule is optimal if , and the output priority rule is optimal if .