On the Capacity of Disjointly Shared Networks

Multi-access broadcast channels (such as ETHERNET or Fasnet [5], [6]) have the feature that a message is broadcast to all users, so that only one user can transmit on the channel at a time. The utilization of such a channel would be increased if it were possible for more than one user to occupy the channel at the same time. We consider the performance of a hypothetical channel having the property that users can simultaneously transmit without interference on physically disjoint parts of the channel. We call a channel (or network) having this property a disjointly shared channel. The problem motivating this study is: how much extra capacity does a disjointly shared channel have compared to a multi-access broadcast channel? Here we measure capacity in terms of the number of pairs of users simultaneously using the channel. We model a disjointly shared channel schematically as a graph, as in Figure 1. The vertices of the graph represent sites of origin or destination of transmissions and the edges are the channel transmission facilities. We model the disjointly shared property of the channel by supposing that when two users are transmitting to each other they seize all vertices and edges on a path connecting the two users, and no other users have access to these vertices or edges during their transmission. (Another model, which we do not consider, supposes that the two users seize the edges on a path connecting the two users, but other user pairs do have access to the vertices on the path. This alternate model corresponds to a situation where the vertices can perform a switching function.) If the graph is a tree, as it is in both examples in Figure 1, the choice of connecting path is unique, but it need not be in other network topologies, such as the graph in Figure 2.