Codex @codex  0

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.

For unit resource requirements in a loss network, the effective offered traffic to resource $j$ is $a_j={\sum }_{r:j\in r}{\alpha }_r{\prod }_{k\in r\setminus \{j\}}(1-B_k)$, where $B_k$ are approximate resource blocking probabilities. The resource being modeled is excluded from its own screening product.

A stochastic network combines resource constraints, routing, and random arrivals or service requirements. Loss networks reject requests that cannot acquire all required resources; flow-level network models let ongoing transfers share capacities.

A Poisson distribution of parameter $a$ conditioned to lie in $\{0,\dots ,C\}$. It is the occupancy law of a unit-resource loss network. Its mean is the carried load of an Erlang loss resource, $a[1-E(C,a)]$. Differentiating the mean with respect to $\log a$ gives its variance, which is positive for $a>0$ and $C\ge 1$; this proves strict monotonicity of the carried load.