= Solution
A <loss network> models calls that require several resources simultaneously and are rejected if any required capacity is unavailable; rejected calls do not queue. For <fixed routing>, let $A_{jr}\in\mathbb Z_{\geq0}$ be the number of units of resource $j$ used by a call of type $r$, and let $C_j$ be its capacity. Independent <Poisson processes> supply type-$r$ calls at rate $\nu_r$, with independent holding times having an <exponential distribution> of mean $1/\mu_r$. Write $\alpha_r=\nu_r/\mu_r$ for the <offered traffic>, and
$$
\mathcal N(C)=\{n\in\mathbb Z_{\geq0}^R:An\leq C\}.
$$
Assume finitely many resources and call types, with every call using a resource, so this set is finite. The occupancy <continuous-time Markov chain> has rates
$$
q(n,n+e_r)=\nu_r\,\mathbf1_{\{n+e_r\in\mathcal N(C)\}},\qquad
q(n,n-e_r)=\mu_r n_r.
$$
Its <stationary distribution> is
$$
\boxed{\pi(n)=\frac1{Z(C)}\prod_{r\in R}\frac{\alpha_r^{n_r}}{n_r!},\quad n\in\mathcal N(C),\qquad
Z(C)=\sum_{n\in\mathcal N(C)}\prod_r\frac{\alpha_r^{n_r}}{n_r!}.}
$$
Indeed, for each feasible upward transition,
$$
\pi(n)\nu_r=\pi(n+e_r)\mu_r(n_r+1),
$$
which is <detailed balance for a continuous-time Markov chain>. Thus this is a <reversible Markov chain>. With positive arrival rates it is irreducible on $\mathcal N(C)$: departures reach the empty state, and any feasible state can be assembled by arrivals. Its <stationary distribution> is consequently unique.
The formula is a <product-form stationary distribution of a loss network>: equivalently, independent <Poisson random variables> of means $\alpha_r$ conditioned on $An\leq C$. The conditioning couples the occupancies, so the resources are generally not independent. By <Poisson arrivals see time averages>, the acceptance probability for a type-$r$ arrival is
$$
\boxed{1-L_r=\frac{Z(C-A_{\cdot r})}{Z(C)},}
$$
because the prearrival state must leave $A_{jr}$ free units at each resource. Set the numerator to zero if its capacity vector has a negative entry. This exact formula is often expensive to evaluate, which motivates the <Erlang fixed point approximation>.
The same occupancy <stationary distribution> extends to independent general holding-time distributions with these means by <insensitivity of loss networks>; the exponential assumption above makes the occupancy process itself a <continuous-time Markov chain> and permits the direct <detailed balance> proof.
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