Properties of Lossy Communication Nets

Abstract

A communication net is considered in which the edges are lossy in the sense that flow through the edge is attenuated. Such a model describes many power or information transmission systems. The classical problem for such a net is that of maximizing the flow from a source to a sink. The solution to the problem for the lossless case is well known; the lossy case has been considered only recently by Fujisawa. In determining properties of the lossy communication nets, a saturated edge is defined as one in which the edge flow is equal to the capacity of that edge. The concept of the saturated cut set is introduced to obtain relationships that must be satisfied by the terminal capacities. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{t}_{pq}</tex> is the source terminal capacity (maximum flow at the source vertex to receive maximum flow at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> ) and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\underline{t}_{pq}</tex> is the sink terminal capacity (maximum flow that can be received at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> when the maximum flow is sent from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> ) then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\underline{t}_{ij} \geq \underline{t}_{kj}</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\overerline{t}_{ij} \geq \overline{t}_{ik}</tex> for any vertices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i,j,</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> . The property analogous to the S-submatrix property in the lossless case is also obtained. All results given reduce to known results when the loss of every edge becomes zero. This includes the maximum flow-minimum cut theory of Ford and Fulkerson, which is applicable only in the lossless case.