= Solution
For the <Erlang loss formula>, assume a <Poisson process> of arrivals with rate $\lambda$, <independent> holding times with <exponential distribution> of mean $1/\beta$, unit capacity usage per accepted call, and rejection without waiting when all $C$ units are occupied. Here $C$ is a positive integer and the offered load is $a=\lambda/\beta$. The call population on $\{0,\ldots,C\}$ is a <birth-death process>; <detailed balance> gives
$$
\pi_k=\frac{a^k/k!}{\sum_{\ell=0}^C a^\ell/\ell!}.
$$
By <Poisson arrivals see time averages>, the fraction of arrivals rejected is the time probability of full occupancy. Therefore
$$
\boxed{E(a,C)=\frac{a^C/C!}{\sum_{k=0}^C a^k/k!},\qquad E(0,C)=0.}
$$
The paper's $\nu$ is the offered load, rather than necessarily an arrival rate. Exponential holding times suffice for this derivation; no unproved <insensitivity of loss networks> assumption is needed.
A <loss network> with <fixed routing> has capacities $C_j$, call types $r$, and a <link-route incidence matrix> $A$ with $A_{jr}=1$ exactly when a type-$r$ call requires link $j$. An accepted call holds one unit on every link of its route for its holding time; if any required link is full, the call is rejected everywhere. Take <independent> <Poisson processes> of calls, <independent> holding times, and route offered loads $\nu_r$. For the finite-variable formulation below, take $C_j\geq1$; a zero-capacity link and all routes using it may be removed first.
The <Erlang fixed point approximation> treats blocking at different links as <independent> and thins the traffic offered to link $j$ by acceptance at the other links. Its reduced offered load is
$$
a_j(B)=\sum_r A_{jr}\nu_r\prod_{i\ne j}(1-B_i)^{A_{ir}}
=(1-B_j)^{-1}\sum_r A_{jr}\nu_r\prod_i(1-B_i)^{A_{ir}}.
$$
The equality uses $A_{jr}\in\{0,1\}$: only routes with $A_{jr}=1$ contribute. Treating this reduced stream as Poisson leads to $B_j=E(a_j(B),C_j)$. <Independence> and Poisson thinning here are the approximation, not exact properties of a general <loss network>.
To construct the <convex potential for the Erlang fixed point>, write $e_C(a)=E(a,C)$ and $m_C(a)=a[1-e_C(a)]$, the <carried load of an Erlang loss resource>, equivalently the mean occupancy in the single-link <stationary distribution>. Differentiating its finite sums gives, for $a>0$,
$$
m_C'(a)=\frac{\operatorname{Var}_a(N)}{a}>0,\qquad
e_C'(a)=\frac{e_C(a)[C-m_C(a)]}{a}>0.
$$
The <variance> is positive because the finite occupancy law is nondegenerate for $a>0$. Furthermore $e_C$ maps $[0,\infty)$ increasingly onto $[0,1)$, while $m_C$ increases from zero to $C$. Set
$$
y_j=-\log(1-B_j),\qquad
\boxed{U(z,C)=e^{-z}e_C^{-1}(1-e^{-z})=m_C\bigl(e_C^{-1}(1-e^{-z})\bigr).}
$$
In particular $U(0,C)=0$, $U(\cdot,C)$ is <continuous> and strictly increasing, and $U(z,C)\to C$ as $z\to\infty$.
For
$$
F(y)=\sum_r\nu_r e^{-\sum_jA_{jr}y_j}+\sum_j\int_0^{y_j}U(z,C_j)\,dz,
$$
the first sum is <convex>, and each integrated strictly increasing function is <strictly convex>. Thus $F$ is a <strictly convex function> on $y\geq0$. Since $U$ tends to a positive limit on every coordinate, $F$ is a <coercive function> there; it has a unique minimizer. Its coordinate <derivative> is
$$
\partial_jF(y)=-e^{-y_j}a_j(y_{-j})+U(y_j,C_j),\qquad
a_j(y_{-j})=\sum_rA_{jr}\nu_r e^{-\sum_{i\ne j}A_{ir}y_i}.
$$
The <Erlang fixed point> equation is exactly $\partial_jF=0$. For an unused link, $a_j=0$ and the optimum is $y_j=0$; for a used link with positive offered load, the <derivative> at zero is negative and the optimum is positive. Hence every fixed point gives the unique minimizer, and the minimizer gives a fixed point. \b[The <Erlang fixed point> is unique under these fixed-routing assumptions.]
A convergent <cyclic substitution for the Erlang fixed point> updates one coordinate at a time, repeatedly cycling through all links:
$$
\boxed{B_j\leftarrow E\!\left(\sum_rA_{jr}\nu_r\prod_{i\ne j}(1-B_i)^{A_{ir}},C_j\right).}
$$
Always use the most recently available values of the other coordinates. Start, for example, with $B=0$. In $y$ coordinates this is exact <coordinate descent> for $F$: with other coordinates fixed, the displayed update is the unique coordinate minimizer.
Here is a convergence proof. All iterates stay in the initial compact <sublevel set> of the <coercive function> $F$, and their potential values decrease to a limit. Each single-coordinate update is <continuous>, since its reduced load is <continuous> and finite and $E(a,C)<1$. Thus a whole-sweep map $T$ is <continuous>. If a full-sweep subsequence converges to $z$, continuity gives $F(Tz)=F(z)$, because both consecutive potential values converge to the same limit. Strict coordinate convexity means equality can occur only when no coordinate changes during that sweep. Thus $z$ minimizes every coordinate, satisfies the nonnegative-orthant first-order conditions, and is the unique global minimizer of $F$. Every subsequential limit is therefore the same point, proving convergence of the full-sweep iterates and, by continuity of the updates, all intermediate iterates. This argument applies to sequential substitution; it does not assume simultaneous updates converge.
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