Fixed routing

Under fixed routing, each call type always requests the same collection of resources. The link-route incidence matrix records those requirements. This differs from alternative routing, in which a rejected direct request can try another route.

For a loss network with feasible occupancies $An\le C$, its stationary distribution is proportional to ${\prod }_r{\alpha }_r^{n_r}/n_r!$. It is the law of independent Poisson random variables conditioned on the resource constraints. With exponential holding times, detailed balance for a continuous-time Markov chain proves the formula directly.

Let $Z(C)$ sum the unnormalized occupancy weights ${\prod }_r{\rho }_r^{n_r}/n_r!$ over feasible counts. The states admitting another type-$r$ call are exactly those feasible under capacities $C-A_r$. Poisson arrivals see time averages therefore gives the displayed blocking probability. Take $Z$ to be zero when the reduced capacities make its feasible set empty.

For positive arrival rates and capacities with every route nonempty, maximizing $H_{\nu }(x)$ over $x\ge 0$, $Ax\le C$ has a unique positive optimizer $x^{\ast }$. The Karush-Kuhn-Tucker conditions give $x_r^{\ast }={\nu }_r{e}^{-(A^{T}z{)}_r}$ for nonnegative link prices with complementary slackness. Under proportional scaling by $N$, the Stirling formula gives a uniform logarithmic weight approximation $\log w_N(n)=N{H}_{\nu }(n/N)+O(\log N)$. Strict concavity and compactness give an objective gap away from $x^{\ast }$, while only polynomially many feasible states exist. Thus the stationary distribution has $n/N\to x^{\ast }$ in probability, with exponentially small probabilities outside any fixed neighborhood of $x^{\ast }$. In particular $\mathbb{E}[n]=N{x}^{\ast }+o(N)$.

Substituting the optimizer relation ${\nu }_r=x_r^{\ast }{e}^{(A^{T}z{)}_r}$ into the product-form stationary distribution of a loss network proves the displayed exact identity. The independent Poisson random variables give a reference law centered at the optimized occupancies, while the constraint and exponential factor account for saturated resources. An unconstrained Gaussian approximation can miss these boundary effects.

For the standard fixed routing loss network with independent Poisson processes, the stationary occupancy law depends on independent holding-time distributions only through their means. The occupancy process alone need not be a Markov process when those holding times lack the memoryless property.