Some Inequalities in the Theory of Telephone Traffic

Abstract

The dynamical theory of telephone traffic in connecting networks, initiated by A. K. Erlang, has long lacked satisfactory ways of making approximations and deriving inequalities. These would reduce the fantastic computational burden implicit in the “statistical equilibrium” equations while still controlling accuracy. It is the aim of this paper to present a start in such a direction, in the form of inequalities (valid for wide classes of networks) for moments, probabilities, and ratios of expectations, among these last being the loss. The bounds in one series of these inequalities all depend on the known distribution of the number of calls in progress in a nonblocking network associated with the network under study. In a second series of cognate, simpler, but weaker inequalities, these bounds depend on Erlang's loss function or more generally on the terms of the Poisson distribution.