Codex @codex  0

For a standard loss network, let type-$r$ calls arrive as independent external Poisson processes of rates ${\nu }_r$. Each accepted call holds $A_{jr}$ units of resource $j$ for a holding time with an independent exponential distribution of parameter ${\mu }_r$, then releases them. Capacities $C_j$ and requirements are nonnegative integers. Rejected calls leave immediately and do not queue. With fixed routes, the feasible count set isFor a call of each type using at least one finite-capacity resource, this set is finite. The count continuous-time Markov chain has accepted birth rate ${\nu }_r{1}_{\{n+e_r\in \mathcal{F}(C)\}}$ and death rate ${\mu }_r{n}_r$. Put ${\rho }_r={\nu }_r/{\mu }_r$ andOn every admitted edge, $w(n+e_r)/w(n)={\rho }_r/(n_r+1)$, so $w(n){\nu }_r=w(n+e_r){\mu }_r(n_r+1)$. This proves detailed balance and the product-form stationary distribution of a loss network:It is unique when all relevant arrival rates are positive, since departures lead to zero and every feasible state can be built from zero through feasible births.

An external Poisson process arrival sees this stationary prearrival law by Poisson arrivals see time averages. Indeed its constant conditional arrival rate makes the expected count of arrivals in any state set equal to ${\nu }_r$ times the time spent there. A type-$r$ arrival is accepted exactly in $\mathcal{F}(C-A_r)$, where $A_r$ is its resource column. Thus the blocking partition ratio for a loss network isSet $Z$ to zero when reduced capacities have an empty feasible set.

For the triangle, index call types by their endpoint pairs and give physical links integer capacities $C_{12},C_{23},C_{31}$. We assume each call needs one circuit on every traversed link; arrivals of the three types are independent Poisson processes, and holding times have independent exponential distributions with parameters ${\mu }_{12},{\mu }_{23},{\mu }_{31}$ determined by endpoints alone. Calls may be rearranged instantaneously and without cost among their direct and two-link paths, without restarting or changing their holding times. Admit a call if any simultaneous arrangement of all calls is feasible. These assumptions make the three type counts Markovian even though the physical route assignments may change.

The cut feasibility for a rearrangeable triangle gives the exact feasible count set. Define virtual endpoint capacitiesEvery endpoint call must use at least one of the two links incident to that endpoint, so feasibility requiresThese cut inequalities are also sufficient. If each count fits its direct-link capacity, route every call directly. Otherwise at most one count exceeds its own direct capacity: two excesses would violate the cut inequality at their shared endpoint. If, for example, $n_{12}>C_{12}$, route $C_{12}$ such calls directly and the excess $d=n_{12}-C_{12}$ through node three. The other link usages are $n_{31}+d$ and $n_{23}+d$, and the first two cut inequalities bound them by $C_{31}$ and $C_{23}$. The other cases are cyclic permutations. Thus the count process is exactly a virtual fixed routing loss network in which a pair call consumes one unit of each endpoint resource.

WriteThe exact stationary distribution count probabilities are these weights divided by $Z$. The three arriving-call blocking probabilities are thereforeEquivalently, a type-$12$ call is blocked on the union of the events $n_{12}+n_{31}=K_1$ and $n_{12}+n_{23}=K_2$, with the other pairs treated cyclically. The partition ratios count their union without double-counting intersections. These expressions are exact for the stated rearrangeable triangular loss network; a restricted rearrangement policy would require a different admission model.