Alternative routing

Alternative routing lets a call try another resource path when its preferred route is blocked. Overflow onto longer paths can create feedback and multiple solutions of the Erlang fixed point approximation, even though an exact finite irreducible occupancy Markov chain has a unique stationary distribution.

A symmetric triangle with direct calls and two-link overflow routes has Erlang fixed point approximation equation $B=E(\lambda [1+2B(1-B)],C)$. For $C=1000$ and $\lambda =950$, this has at least three solutions, certified by the intermediate value theorem using signs at $B=0,0.01,0.1,0.4$.

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.

On a three-node loss network, each pair offers direct calls of load $v$ to its link of capacity $C$. If blocked, the call tries the two-link path through the remaining node. The symmetric reduced-load approximation gives a link its direct load and two overflow contributions $vB(1-B)$, hence the displayed equation. For $C=1000$, $v=940$, its residual has alternating signs at $B=0,0.01,0.10,0.25$, giving at least three distinct roots. This multiplicity belongs to the Erlang fixed point approximation, not to the exact stationary law of a finite reachable Markov chain.