Using Metrics in the Throughput Analysis and Synthesis of Undirected Graphs

Abstract

The usual representations of the communication environment are graphs with certain properties. Like reliability, throughput is one of the most important characteristics of such graphs. Metric tasks often arise when simplifying complex and practically important problems on graphs. A traditional metric, such as the usual shortest paths, forms the basis of the traditional throughput index. In this case, the metric is used to obtain the distribution of multi-colour flows in graphs more complex than trees. To achieve better results than when using ordinary shortest paths, one can use the Euclidian metric. If one starts with the Kleinrock formula for the average packet delay, then the Euclidian (quadratic) metric allows one to practically refuse multiple distributions over the shortest paths with variable edge lengths in the cut saturation procedures. The same Euclidian metric describes the distribution of the flow of any colour in an arbitrary graph as the best approximation to the ideal distribution in a complete graph in the sense of quadratic deviation. Such independence of the result from the Kleinrock formula demonstrates the effectiveness of linear metric models in the throughput analysis and synthesis of graphs. The Euclidian metric also allows you to introduce the throughput index of an arbitrary graph into these tasks in the form of an abstract measure. Therefore in such tasks one can completely disregard the distribution of the flows. Theoretical results are illustrated by an example of graph synthesis.