Fast Engset computation

TL;DR

This paper improves the computation of blocking probabilities in finite-source bufferless queues by introducing a globally convergent Newton's method and establishing a new inequality involving hypergeometric functions.

Contribution

It demonstrates the limitations of fixed point iteration and proposes a more reliable Newton's method for Engset formula computation, along with a new mathematical inequality.

Findings

Fixed point iteration can fail to converge.

Newton's method guarantees global convergence.

New Turán-type inequality for hypergeometric functions.

Abstract

The blocking probability of a finite-source bufferless queue is a fixed point of the Engset formula, for which we prove existence and uniqueness. Numerically, the literature suggests a fixed point iteration. We show that such an iteration can fail to converge and is dominated by a simple Newton's method, for which we prove a global convergence result. The analysis yields a new Tur\'an-type inequality involving hypergeometric functions, which is of independent interest.