= Proportional scaling limit of the Erlang loss formula
{title2=$E(N\nu,NC)\to\max\{0,1-C/\nu\}$}
Scale the offered load and capacity together. Reversing the denominator of the <Erlang loss formula> gives
$$
E(N\nu,NC)^{-1}=\sum_{j=0}^{NC}\prod_{\ell=0}^{j-1}\frac{NC-\ell}{N\nu}.
$$
For each fixed $j$, the summand tends to $(C/\nu)^j$. If $\nu>C$, the <dominated convergence theorem> applies with the summable <geometric series> bound $(C/\nu)^j$. Otherwise each fixed finite partial sum has limiting value at least its number of terms, so the reciprocal diverges. Taking reciprocals proves the formula, including the critical case $\nu=C$.
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