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Let $G_N(B)=E(20N[1+2B(1-B)],21N)-B$. The proportional scaling limit of the Erlang loss formula makes its limits at $B=1/100$ and $B=1/8$ respectively $-1/100$ and $7/520$. Since $G_N(0)>0$ and $G_N(1)<0$, sufficiently large $N$ give alternating signs at these four points. The intermediate value theorem supplies three distinct zeros in the intervening intervals. These are multiple solutions of the Erlang fixed point approximation, not multiple stationary laws of an exact finite irreducible Markov chain.

Start from the Erlang loss formula and reverse the order in its denominator:For each fixed $j$, the product tends to $(C/\nu {)}^j$. If $\nu >C$, it is bounded by this same summable geometric series. Extend the summand by zero for $j>NC$ and apply the dominated convergence theorem with counting measure to obtainIf $\nu \le C$, retain the first $K+1$ summands. Their limiting sum is at least $K+1$, so the reciprocal tends to infinity as $K$ can be arbitrarily large. This includes $\nu =C$ without a separate normal approximation. ThereforeHere $C$ and $N$ are positive integers, as appropriate for circuit counts; taking $\lfloor NC\rfloor$ gives the same limit for a positive real capacity scale. This is the proportional scaling limit of the Erlang loss formula.

The alternative routing term feeds additional traffic back into the link. We can prove the existence of three solutions rather than relying on an iteration that happens to find different answers. Choose ${\nu }_N=20N$, $C_N=21N$, and define the continuous residualBy the proportional scaling limit of the Erlang loss formula, at each fixed $B$,At $B=1/100$, the multiplier is $1.0198<21/20$, so $G_{\infty }(1/100)=-1/100$. At $B=1/8$ it is $39/32$, givingAlso $G_N(0)>0$ and $G_N(1)<0$ for every $N$, since a finite-capacity Erlang loss formula gives a blocking probability strictly between zero and one. For all sufficiently large $N$, the signs at $0,1/100,1/8,1$ therefore alternate. Applying the intermediate value theorem in the three disjoint intervening intervals proves at least three distinct fixed points. This supplies a fluid scaling proof of multiple Erlang fixed points.

This multiplicity concerns the Erlang fixed point approximation. It does not imply several equilibrium laws for an exact finite irreducible Markov chain, which has a unique stationary distribution.