Second Order Conditions on the Overflow Traffic from the Erlang-B System

This paper presents in a unified manner mathematical properties of the second order derivatives of the overflow traffic from an Erlang loss system, assuming the number of circuits to be a nonnegative real value. It is shown that the overflow traffic function Â(a, x) is strictly convex with respect to x (number of circuits), with x ≥ 0, taking the offered traffic, a, as a positive real parameter. The convexity (in the wider sense) has been proved by A.A. Jagers and Erik A. Van Doorn [8]. Using a similar procedure to the one used by those authors it is shown that Â(a, x) is a strictly convex function with respect to a, for all (a, x) ∈ IR+ × IR+ — a well known result for the case of x being a positive integer, due to C. Palm [10, pp.180–181]. These two results are obtained by determining the sign of the second order derivatives Â′′ aa(a, x) and  ′′ xx(a, x) for (a, x) ∈ IR+× IR+. In the same manner it is proved that the rectangular derivatives Â′′ ax(a, x) and  ′′ xa(a, x) are negative for all (a, x) ∈ IR+× IR + 0 . Finally, a first approach to the analysis of the strict joint convexity of Â(a, x) in some open convex subdomain of IR+ × IR+, is discussed. Finally, based on some particular cases and extensive computational results it is conjectured that the function Â(a, x) is strictly jointly convex in areas of low blocking where the standard offered traffic is less than −1.