Rearrangeable triangular loss network

Three nodes have integer link capacities $C_{12},C_{13},C_{23}$. Calls between a node pair use either its direct link or the two-link alternative. With unrestricted instantaneous rearrangement of existing calls, counts $n_{12},n_{13},n_{23}$ are feasible exactly when $n_{12}+n_{13}\le C_{12}+C_{13}$, $n_{12}+n_{23}\le C_{12}+C_{23}$ and $n_{13}+n_{23}\le C_{13}+C_{23}$. Necessity follows by counting calls crossing each node cut. For sufficiency, the inequalities allow at most one count to exceed its direct-link capacity. Divert precisely that excess to the other two links; the same inequalities ensure capacity there. The counts thus have the same feasible set as a fixed routing loss network with one virtual resource at each node, capacity equal to the sum of its two incident physical-link capacities, and each call using its two endpoint resources. With independent Poisson processes for arrivals and independent routing-independent exponential distributions for holding times, this gives the usual product-form stationary distribution of a loss network and exact Poisson arrivals see time averages blocking probabilities.

The three cyclic versions of this inequality are necessary and sufficient for routing unit calls on a triangle with complete instantaneous rearrangement. Necessity counts endpoint calls through a node cut. For sufficiency, at most one call count can exceed its direct-link capacity, since any two such excesses would violate their common endpoint cut. Route that one excess along the two other links; the two relevant cut inequalities guarantee room there. This gives an exact reduction to a loss network with a virtual resource at each node.