Abstract
<p>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 <em>nodes</em>. There are <em>n</em> 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 <em>n , d</em> and <em>l</em>. 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 <em>n, d</em> and <em>l</em>, 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 <em>n, l,</em> and <em>d</em>. We have proved that the input priority rule is optimal if , and the output priority rule is optimal if .</p>
Highlights
A class of dynamical systems introduced by A.P
We have proved that the input priority rule is optimal in the sense of a given criterion if the length of cluster is not greater than the smaller distance between the nodes, and the input priority rule is optimal in the sense of a given criterion if the length of cluster is greater than the smaller distance between the nodes
If a competition at the node 2 occurs, the system occurs in a state of collapse in the case of input priority rule, and the system results in a spectral cycle with the average velocity n v1 = v2 = 2(l + d) in the case of the left-priority rule or the output priority rule
Summary
A class of dynamical systems introduced by A.P. Buslaev can be used for modeling the work of information and communication systems. The present paper studies closed chains of contours such that the rules can be chosen in these systems for resolution of particle competitions, which occur when two particles come to the node simultaneously. In [9]– [11], conditions for the self-organization and collapse (from a moment, no particle moves) Another type of movement in the contour network was introduced in [12]. In the continuous version of contour network, the clusters are segments moving with constant velocity if there is no delay. Networks with regular structures periodic structure were considered (in particular, closed and open chains of contours, [13] – [15]), two contour systems, [16]— [19], systems with one common node — the flower, [20]. We have proved that the input priority rule is optimal in the sense of a given criterion if the length of cluster is not greater than the smaller distance between the nodes, and the input priority rule is optimal in the sense of a given criterion if the length of cluster is greater than the smaller distance between the nodes
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