On Some Proposed Models for Traffic in Connecting Networks

Three stochastic models for traffic, forming a progression of decreasing simplicity, are discussed with a view to discerning in what ways the various assumptions they depend on affect the formula for blocking probability. These models are the probability linear graph (due to C. Y. Lee), the thermodynamic model, and a model based on Markov processes (both proposed by the author). Certain basic inadequacies of the models are described. Lee's model lacks a sufficiently broad assignment of probabilities to events of' interest, with the result that the blocking probability is improperly defined; at the same time it bases congestion formulas on network conditions never achieved in practice. The thermodynamic model deals only with genuine system states, but makes calling rates depend unrealistically on available paths. Neither the graph model as originally proposed, nor the thermodynamic model, can take into account routing procedures. The author's Markov model is free of these drawbacks, but at this price: in nearly all practical situations in which losses occur, it leads to hitherto insurmounted combinatorial and computational difficulties. To stress and illustrate the effect that routing has on loss, the blocking probability formulas of all three models are compared at low traffic: it ofteri turns out that when the first two models indicate that (with X = offered traffic) loss = 0(Xm), 0, an analysis based on routing shows that in fact loss = o(Xm), X --» 0.