= Solution
Let $J$ be a finite collection of resources with positive integer capacities $C_j$, and let $R$ be a finite collection of nonempty routes. In the standard unit-resource <loss network>, a call on route $r$ needs one unit simultaneously at each resource in $r$; <fixed routing> means that these requirements do not change when a request is rejected. Suppose route $r$ has an independent <Poisson process> of arrivals of rate $\nu_r$ and independent holding times of mean $1/\mu_r$. Its <offered traffic> is $\alpha_r=\nu_r/\mu_r$. The <Erlang fixed point approximation> approximates resource blocking events as independent, rather than claiming this <independence> for the exact <loss network>.
Write $b_j$ for the approximate blocking probability at resource $j$. The <reduced-load approximation> screens a call by every other resource that it needs, so the <offered traffic> seen by resource $j$ is
$$
a_j(b)=\sum_{r:j\in r}\alpha_r\prod_{k\in r\setminus\{j\}}(1-b_k).
$$
For a single resource with <offered traffic> $a$ and capacity $C$, the <Erlang loss formula> is
$$
E(a,C)=\frac{a^C/C!}{\sum_{k=0}^{C}a^k/k!}.
$$
Thus the <Erlang fixed point approximation> is the system
$$
\boxed{b_j=E(a_j(b),C_j),\qquad j\in J.}
$$
The resulting route acceptance probability is $\prod_{j\in r}(1-b_j)$. In particular, the screening product for $a_j$ excludes $j$ itself; including it would describe a different approximation.
For existence, define $T_j(b)=E(a_j(b),C_j)$ on $[0,1]^J$. Each $a_j$ is a continuous polynomial, and the denominator of the <Erlang loss formula> is at least one. Consequently $T$ is continuous and maps this nonempty compact <convex set> into itself. The <Brouwer fixed-point theorem> supplies a <fixed point>. At any such point $b_j<1$, since finite <offered traffic> and positive capacity give $E(a,C)<1$.
For uniqueness, we construct a <strictly convex function> whose minimizer is equivalent to an <Erlang fixed point>. Put $y_j=-\log(1-b_j)$. First establish the single-resource facts needed for this change of coordinates. For $a>0$, the inverse blocking probability has the expression
$$
E(a,C)^{-1}=\sum_{k=0}^{C}\frac{C!}{k!}a^{k-C}.
$$
The terms with $k<C$ decrease strictly with $a$, so $E(a,C)$ increases strictly from zero to one. Let $a_C(y)$ be its inverse parametrization defined by $E(a_C(y),C)=1-e^{-y}$, with $a_C(0)=0$. If $K$ has the truncated <Poisson distribution> with weights proportional to $a^k/k!$, then shifting the sum gives its mean, the <carried load of an Erlang loss resource>, as
$$
m_C(a)=\mathbb E_aK=a[1-E(a,C)].
$$
Differentiating the normalized weights shows
$$
\frac{d}{da}\mathbb E_aK=\frac{\operatorname{Var}_a(K)}a>0.
$$
The <variance> is positive because both $K=0$ and $K=1$ have positive probability. Moreover, $m_C(a)\to C$ as $a\to\infty$, since the largest term in the truncated <Poisson distribution> dominates. Hence
$$
U_C(y)=e^{-y}a_C(y)=m_C(a_C(y))
$$
is continuous, strictly increasing from zero to $C$.
Now minimize the following <convex potential for the Erlang fixed point> over the nonnegative orthant:
$$
F(y)=\sum_{r\in R}\alpha_r\exp\!\left(-\sum_{j\in r}y_j\right)+\sum_{j\in J}\int_0^{y_j}U_{C_j}(z)\,dz.
$$
Each exponential term is <convex>, and each integral is a <strictly convex function> of its coordinate because its derivative $U_{C_j}$ is strictly increasing. Their sum is therefore <strictly convex>. It is also a <coercive function>: once $z$ is large enough, $U_{C_j}(z)\geq C_j/2$, so each integral grows at least linearly in its coordinate, while the exponential terms are nonnegative. Thus $F$ attains exactly one minimum.
The coordinate derivative is
$$
\partial_jF(y)=U_{C_j}(y_j)-\sum_{r:j\in r}\alpha_r\exp\!\left(-\sum_{k\in r}y_k\right).
$$
At an interior coordinate of a minimizer it vanishes. At $y_j=0$, it is nonpositive, whereas the one-sided minimum condition requires it to be nonnegative; it therefore vanishes there too. Conversely, a zero <gradient> is a global minimizer of this <convex function>. Multiplying $\partial_jF=0$ by $e^{y_j}$ gives
$$
a_{C_j}(y_j)=\sum_{r:j\in r}\alpha_r\exp\!\left(-\sum_{k\in r\setminus\{j\}}y_k\right)=a_j(b).
$$
Applying the <Erlang loss formula> converts this exactly to $b_j=E(a_j(b),C_j)$. The change of coordinates is one-to-one, proving \b[uniqueness under fixed routing]. A zero-capacity resource can instead be fixed as blocked with probability one, with all routes using it removed before applying the proof; an unused positive-capacity resource has blocking probability zero.
For <alternative routing>, consider a symmetric three-node <loss network>. Each of its three links has capacity $1000$; every node pair offers traffic $950$, tries its direct link first, and, if that link is blocked, tries the two-link path through the third node. Let all approximate link blocking probabilities equal $b$. Each link receives its direct <offered traffic> and the overflow from the other two node pairs. An overflow is generated with probability $b$ and reaches the link under consideration only when its other required link accepts, with probability $1-b$. The symmetric <reduced-load approximation> therefore gives
$$
b=E(950[1+2b(1-b)],1000).
$$
For $g(b)=E(950[1+2b(1-b)],1000)-b$, evaluate the <Erlang loss formula> by the stable recursion $E(a,0)=1$, $E(a,k)=aE(a,k-1)/(k+aE(a,k-1))$. It gives
$$
\begin{array}{c|rrrr}
b&0&0.01&0.1&0.4\\\hline
g(b)&0.00364929&-0.00092309&0.01445982&-0.10951455
\end{array}
$$
These signs can also be certified with exact rational arithmetic, since all four loads are rational. Continuity and the <intermediate value theorem> give three distinct roots, one in each intervening interval. Thus \b[the alternative-routing approximation need not be unique]. This is multiplicity of approximate <fixed points>, not multiplicity of <stationary distributions> for the exact finite irreducible <Markov chain>.
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