= Solution
The exact normalizing sum can be expensive on a large <loss network>. The <Erlang fixed point approximation> replaces joint resource acceptance by a product of marginal acceptances. Here consider unit requirements $A_{jr}\in\{0,1\}$, positive integer capacities and finitely many fixed routes. Let $B_j$ be resource blocking and $q_j=1-B_j$. A route-$r$ call contributes to the <offered traffic> at resource $j$ after surviving the other resources on its route. Thus the <reduced-load approximation> is
$$
\boxed{v_j=\sum_{r:j\in r}\rho_r\prod_{k\in r\setminus\{j\}}q_k,\qquad
B_j=E(C_j,v_j)},
$$
where the <Erlang B formula> is
$$
E(C,v)=\frac{v^C/C!}{\sum_{m=0}^{C}v^m/m!}.
$$
The estimated route acceptance is $\prod_{j\in r}q_j$. This is an <independence> approximation, not an alternative exact factorization of the stationary law in part (i).
Uniqueness follows from a <convex potential for the Erlang fixed point>. For a single resource, the <carried load of an Erlang loss resource> is $m_C(v)=v[1-E(C,v)]$. It increases strictly from zero to $C$ as $v$ increases: differentiating the <expected value> of its <upper-truncated Poisson distribution> with respect to $\log v$ gives the strictly positive occupancy <variance>. Blocking also increases strictly, as is evident on dividing the Erlang denominator by its final term.
Use $p_j=-\log q_j\geq0$. Let $v_C(p)$ be the unique offered load with $1-E(C,v_C(p))=e^{-p}$, and set $g_C(p)=v_C(p)e^{-p}=m_C(v_C(p))$, with $g_C(0)=0$. This function is strictly increasing and tends to $C$. Define
$$
F(p)=\sum_r\rho_r e^{-\sum_{j\in r}p_j}
+\sum_j\int_0^{p_j}g_{C_j}(u)\,du.
$$
Each exponential term is a <convex function>, and each <integral> is a <strictly convex function> because its <derivative> is strictly increasing. Therefore $F$ is a <strictly convex function>. It is also a <coercive function> on the nonnegative orthant: an unbounded coordinate makes its <integral> grow asymptotically linearly with positive slope $C_j$. A unique minimizer exists.
If resource $j$ carries some positive offered route, its inward <derivative> at $p_j=0$ is negative, so its minimizing coordinate is positive. At such a coordinate the first-order equation is
$$
g_{C_j}(p_j)=\sum_{r:j\in r}\rho_r e^{-\sum_{k\in r}p_k}.
$$
Dividing by $q_j$ gives precisely $v_j=\sum_{r:j\in r}\rho_r\prod_{k\in r\setminus\{j\}}q_k$. A resource with no positive offered route uniquely has $p_j=0$, hence $B_j=0$. Thus the minimizer and the fixed point coincide, proving \b[existence and uniqueness of the <Erlang fixed point> for fixed unit-resource routing]. This does not by itself guarantee convergence of every simultaneous substitution algorithm; the uniqueness claim concerns the solution of the equations.
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