A “thermodynamic” theory of traffic in connecting networks

Two new theoretical models for representing random traffic in connecting networks are presented. The first is called the “thermodynamic” model and is studied in detail. The second model is formulated in an effort to take methods of routing into account and to meet certain drawbacks of the “thermodynamic” model in describing customer behavior; since it is more realistic than the first, it leads to results that are vastly more complicated and must be described in another paper. The “thermodynamic” model is worth considering for four reasons: (1) It is faithful to the structure of real connecting systems. Indeed it is an improvement over many previous models in that it only considers physically accessible states of the connecting network, while the latter suffer the drawback that a large fraction of the network states on which calculation is based are physically meaningless, being unreachable under normal operation. (2) It gives rise to a relatively simple theory in which explicit calculations are possible. (3) The “thermodynamic” model provides a good simple description of traffic in the interior of a large communications network. (4) It has an analogy to statistical mechanics which permits us to be guided by the latter theory as we try to use the model to understand the properties of large-scale connecting systems. The two models to be described differ in only one respect. In the first (the “thermodynamic”) model, the system moves from a state x to a state y that has one more call in progress, at a rate λ the effective calling-rate per idle inlet-outlet pair is thus proportional to the number of paths usable in x from that inlet to that outlet. In the second model, the calling-rate per idle inlet-outlet pair is set at λ, and is then spread over the paths usable in x from that inlet to that outlet in accordance with some routing rule. This provides a mathematical description of routing, and avoids the unwelcome feature that a single customer's calling-rate depends on the state of the network.