Carried load of an Erlang loss resource ($m_C(a)=a[1-E(a,C)]$)

For offered load $a$ and capacity $C\ge 1$, the carried load equals the mean occupancy:Differentiating the truncated-Poisson equilibrium weights gives $m_C^{\prime }(a)={\mathrm{Var}}_a(N)/a>0$ for $a>0$. The carried load increases from zero to $C$. This monotonicity supplies the strictly increasing term in the convex potential for the Erlang fixed point.