Nonstationary blocking in telephone traffic

Blocking is considered for an N-trunk group of exponential servers with Poisson-offered load whose rate parameter varies with time. The infinite trunk case is solved by means of a rapidly convergent series of Poisson-Charlier polynomials. This solution is used to obtain practical approximations of blocking probability, transition probabilities, and recovery function for general time-variable offered load in the finite trunk-group case. An integral equation is derived satisfied by the blocking probability in the general case. In the situation of constant offered load, two additional methods are derived for providing easily computable approximations; one based on the integral equation, the other based on an approximate inversion of the Laplace transform. To aid in the latter approximation, bounds on the roots of Poisson-Charlier polynomials are obtained; in particular, an approximation is obtained for the dominant root. The inversion of the integral equation is studied with the purpose of providing the basis for future investigations of errors of approximation. Curves are provided for a number of examples permitting comparison of exact and approximate solutions.