= Solution
For <fixed routing>, let each route $r$ require one unit on every resource in its fixed set, let $C_j\geq1$ be integer resource capacities, and let $\alpha_r$ be the <offered load>, arrival rate times mean holding time. The <Erlang fixed point approximation> treats distinct resource availability events as independent. If $B_j$ is the approximate blocking probability at resource $j$, the <reduced-load approximation> gives
$$
a_j(B)=\sum_{r:j\in r}\alpha_r\prod_{k\in r\setminus\{j\}}(1-B_k),\qquad B_j=E(a_j(B),C_j),
$$
where the <Erlang loss formula> is
$$
E(a,C)=\frac{a^C/C!}{\sum_{k=0}^Ca^k/k!}.
$$
Screening excludes resource $j$ itself: it is the load offered to that resource, not merely the load ultimately admitted there. Approximate route acceptance is $\prod_{j\in r}(1-B_j)$. This specifies the approximation, not an assertion of exact independence in the actual <loss network>.
Here is a constructive variational proof of both existence and uniqueness. For $C\geq1$ write $e_C(a)=E(a,C)$ and $m_C(a)=a[1-e_C(a)]$, the <carried load of an Erlang loss resource>. Under its upper-truncated Poisson occupancy law, differentiating the finite sums gives
$$
m_C'(a)=\frac{\operatorname{Var}_a(K)}a>0,\qquad e_C'(a)=\frac{e_C(a)}a[C-m_C(a)]>0\qquad(a>0).
$$
Consequently $e_C$ increases continuously from $0$ to $1$, and $m_C$ increases from $0$ to $C$. Put $y_j=-\log(1-B_j)$ and
$$
g_j(y)=e^{-y}e_{C_j}^{-1}(1-e^{-y})=m_{C_j}\bigl(e_{C_j}^{-1}(1-e^{-y})\bigr),\qquad g_j(0)=0.
$$
Each $g_j$ is continuous and strictly increasing, tending to $C_j$. Consider the <convex potential for the Erlang fixed point>
$$
F(y)=\sum_r\alpha_r\exp\left(-\sum_{j\in r}y_j\right)+\sum_j\int_0^{y_j}g_j(z)\,dz,\qquad y\geq0.
$$
The first sum is <convex>, and each integral is <strictly convex> because its derivative $g_j$ strictly increases. Hence $F$ is <strictly convex>. Also each integral grows at least linearly for sufficiently large $y_j$, since $g_j(y_j)\to C_j>0$. Thus $F$ is <coercive>, its bounded sublevel sets are compact, and it attains a unique minimum $y^*$.
Its derivative is
$$
\partial_jF=g_j(y_j)-\sum_{r:j\in r}\alpha_r\exp\left(-\sum_{k\in r}y_k\right)=g_j(y_j)-e^{-y_j}a_j(B).
$$
For a resource used by some positive-load route, this derivative is negative at $y_j=0$ for every finite choice of the other coordinates, so its minimizing coordinate is positive and satisfies $\partial_jF=0$. Cancelling $e^{-y_j}$ gives $e_{C_j}^{-1}(B_j)=a_j(B)$, exactly the required fixed-point equation. An unused resource minimizes its integral at $y_j=0$, giving $B_j=0=E(0,C_j)$. Conversely any fixed point has $B_j<1$, since the total offered load is finite; its logarithmic coordinates satisfy these same minimizing conditions. Strict <convexity> therefore proves \b[one and only one fixed-routing Erlang fixed point]. Zero-capacity resources can be removed with their blocked routes, whose acceptance probabilities are already zero.
For a concrete <alternative routing> counterexample, take three nodes forming a triangle, each link of capacity $1000$, with <offered traffic> $950$ for each endpoint pair. Try the direct link first; if it is blocked, try the other two links together. In a symmetric <reduced-load approximation>, a link has its direct load plus two overflow streams, each of load $950B(1-B)$: the preferred link is blocked with probability $B$, and the other link of the alternative path is available with probability $1-B$. The link itself is excluded from screening. Thus a common link blocking probability must satisfy
$$
B=E\bigl(950[1+2B(1-B)],1000\bigr).
$$
Define the right side minus $B$ as $G(B)$. The stable <Erlang loss formula> recursion $b_0=1$, $b_k=ab_{k-1}/(k+ab_{k-1})$ gives
$$
\begin{aligned}
G(0)&\simeq0.0036493>0,&G(0.01)&\simeq-0.0009231<0,\\
G(0.1)&\simeq0.0144598>0,&G(0.4)&\simeq-0.1095146<0.
\end{aligned}
$$
All four arguments are rational, so the same recursion can certify these signs with rational arithmetic, without trusting rounded values. Continuity and the <intermediate value theorem> give \b[at least three distinct symmetric fixed points], one in each intervening interval. This is nonuniqueness of the approximation; the exact finite irreducible <loss network> still has a unique <stationary distribution>.
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