On the Erlang loss function

Abstract

We present various properties of the Erlang loss function

$$B(x,a)=\Bigl( a \int_0^\infty e^{-at} (1+t)^x \,dt \Bigr) ^{-1} \quad{(x\geq 0,\ a>0)}.$$

Among other results, we prove:

• (1) The function $x\mapsto B(x,a)^{\lambda}$ is convex on $[0,\infty)$ for every $a>0$ if and only if $\lambda\leq 0$ or $\lambda \geq 1$.

• (2) The function $x\mapsto ({1-B(1/x,a)})^{-1}$ is strictly convex on $(0,\infty)$. This leads to the functional inequality

  $$\frac{2}{1-B( H(x,y) ,a)}< \frac{1}{1-B(x,a)} +\frac{1}{1-B(y,a)}\quad{(x,y>0,\ x\neq y)}, $$

  where $H(x,y)=2xy/(x+y)$ denotes the harmonic mean of x and y.

• (3) Let $a>0$. The inequality $B(x,a) + B(1/x,a)\leq 1$ holds for all $x>0$ if and only if $a\leq 1$.