= Solution
First consider unit requirements, $A_{jr}\in\{0,1\}$. The <Erlang loss formula> for a resource with capacity $C$ and <offered traffic> $a$ is
$$
E(a,C)=\frac{a^C/C!}{\sum_{k=0}^C a^k/k!}.
$$
The <Erlang fixed point approximation> assumes that resources block independently and that the traffic retained after screening at other resources can be treated as a <Poisson process>. If $B_j$ is the approximate blocking probability and $q_j=1-B_j$, the <reduced-load approximation> gives
$$
\boxed{a_j=\sum_{r:A_{jr}=1}\alpha_r\prod_{k\ne j}q_k^{A_{kr}},\qquad
B_j=E(a_j,C_j),\qquad
L_r\approx1-\prod_jq_j^{A_{jr}}.}
$$
The link whose load is being calculated is excluded from the screening product. Otherwise one would confuse its offered load with its carried load.
For existence, these equations define a continuous map from $[0,1]^J$ into itself. The <Brouwer fixed-point theorem> gives a <fixed point>. Assume $C_j\geq1$; a zero-capacity resource forces rejection of every call needing it and can be removed together with those call types.
For uniqueness, let $K$ have probabilities proportional to $a^k/k!$, $0\leq k\leq C$, and write $m_C(a)=\mathbb E K$. Direct differentiation gives
$$
m_C(a)=a\bigl(1-E(a,C)\bigr),\qquad
m_C'(a)=\frac{\operatorname{Var}(K)}a>0,\qquad
\frac{\partial E(a,C)}{\partial a}
=\frac{E(a,C)}a\bigl(C-m_C(a)\bigr)>0.
$$
These identities hold for $a>0$, and $m_C(0)=0$. Also $E(a,C)$ increases from zero to one and $m_C(a)$ increases from zero to $C$.
Put $p_j=-\log q_j$. Define $a_C(p)$ by $E(a_C(p),C)=1-e^{-p}$, and define $h_C(p)=m_C(a_C(p))$. This is a continuous, strictly increasing function on $[0,\infty)$, starting at zero and tending to $C$. Multiplying each <Erlang fixed point approximation> load equation by $q_j$ transforms it into
$$
h_{C_j}(p_j)=\sum_r A_{jr}\alpha_r\exp\left(-\sum_kA_{kr}p_k\right).
$$
These are exactly the zero-<gradient> conditions of
$$
\Phi(p)=\sum_j\int_0^{p_j}h_{C_j}(u)\,du
+\sum_r\alpha_r\exp\left(-\sum_jA_{jr}p_j\right),\qquad p\geq0.
$$
Each integral is a <strictly convex function>; the exponential terms are <convex functions>. Thus $\Phi$ is a <strictly convex function>. It is also a <coercive function>, since $h_{C_j}(p)\to C_j>0$, so it has a unique minimizer. At $p_j=0$, its partial derivative is negative if some positive-traffic route uses $j$, so that coordinate of the minimizer is positive. If no route uses $j$, the unique minimizing coordinate is zero. The minimizer therefore satisfies the equations in every coordinate. Conversely any <fixed point> has $B_j<1$, finite $p$, and these zero-<gradient> equations, so must equal that unique minimizer. Hence \b[the <Erlang fixed point approximation> exists and is unique for <fixed routing>].
For integer requirements, the common generalized <Erlang fixed point approximation> uses
$$
a_j=\sum_{r:A_{jr}>0}A_{jr}\alpha_rq_j^{A_{jr}-1}\prod_{k\ne j}q_k^{A_{kr}}.
$$
Multiplication by $q_j$ gives the same equations for $\Phi$, so the existence and uniqueness argument also covers this generalized approximation. It remains an approximation, rather than the exact blocking law for a call requesting several units.
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