Second-order conditions on the overflow traffic function from the Erlang-B system: a unified analysis

Abstract

This paper presents a unified treatment of the mathematical properties of the second-order derivatives of the overflow traffic function from an Erlang loss system, assuming the number of circuits to be a nonnegative real number. It is shown that the overflow traffic function $\widehat A\left( {a,x} \right)$ is strictly convex with respect to x (number of circuits) for x ≥ 0, taking the offered traffic, a, as a positive real parameter. It is also shown that $\widehat A\left( {a,x} \right)$ is a strictly convex function with respect to a, for all (a, x) ∈ ℝ⁺ × ℝ⁺. Following a similar process, it is shown that $\widehat A\left( {a,x} \right)$ is a strict submodular function in this domain and that the improvement function introduced by K. O. Moe [11] is strictly increasing in a. Finally, based on some particular cases and numerous numerical results, there is a conjecture that the function $\widehat A\left( {a,x} \right)$ is strictly jointly convex in areas of low blocking where the standard offered traffic is less than −1.