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For positive integer capacity $C$ and offered load $a>0$, let $m_C(a)$ be the carried load of an Erlang loss resource. Differentiating the upper-truncated Poisson distribution gives $m_C^{\prime }(a)=\mathrm{Var}(N)/a>0$. Differentiating its probability at $C$ gives the displayed derivative of the Erlang loss formula; it is positive because $m_C(a)<C$. Thus blocking increases continuously from zero to one, and the carried load increases from zero to $C$. These properties justify the inverse blocking coordinates in the convex potential for the Erlang fixed point.

For unit resource requirements, treat the resource-blocking events as approximately independent. Let $B_j$ be the approximate blocking probability at resource $j$. Screening class-$r$ traffic by every resource other than $j$ produces the reduced-load approximationThe modeled resource is excluded from its own screening product. For general integer requirements, the standard generalized approximation is $a_j=(1-B_j{)}^{-1}{\sum }_r{A}_{jr}{\alpha }_r{\prod }_k(1-B_k{)}^{A_{kr}}$; in the unit-requirement case this reduces to the displayed formula. The approximate route acceptance is ${\prod }_j(1-B_j{)}^{A_{jr}}$. The independence used here is an approximation to the correlated exact stationary distribution, not an extra property of that law.

Here is a proof of fixed-routing existence and uniqueness. Suppose first that every retained capacity $C_j$ is a positive integer. Links of zero capacity permanently reject their routes and can be removed. If $K$ has the upper-truncated Poisson distribution with parameter $a$, its mean isThe derivative follows by differentiating the normalized weights: $dP(K=k)/da=P(K=k)(k-m_C(a))/a$. The probability at the upper endpoint similarly givesThus $E(\cdot ,C)$ increases from zero to one, while $m_C$ increases from zero to $C$. This is the monotonicity of the Erlang blocking probability.

Set $y_j=-\log (1-B_j)$. For $y\ge 0$ defineHere the inverse is with respect to offered load, and the value at zero is zero. This function is the carried load evaluated at the inverse blocking coordinate, so it is continuous and strictly increasing, with limit $C_j$ as $y\to \infty$. Consider the convex potential for the Erlang fixed pointEach exponential of a linear function is convex. Each integral is strictly convex because its derivative $u_j$ is strictly increasing. Their sum is strictly convex on the nonnegative orthant. It is also coercive: for large $y_j$, $u_j(y_j)$ is bounded below by a positive constant, so the integral grows without bound; the other terms are nonnegative. Hence $F$ attains a unique minimum.

At a positive coordinate, the first-order condition isAt $y_j=0$, the derivative equals minus the right-hand side and is nonpositive. A boundary minimum can therefore occur there only when that load is zero, in which case the derivative is zero and the same equation holds. Since $1-B_j=e^{-y_j}$, division by this positive acceptance probability makes the first-order condition precisely the reduced-load equations above. Conversely every fixed point is such a minimum. Thus fixed routing has exactly one Erlang fixed point, including zero blocking at unused resources. The proof does not rely on a potentially nonconvergent simultaneous substitution algorithm.

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 iswhere the Erlang B formula isThe 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 $p_j=-\log q_j\ge 0$. 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 $C$. DefineEach 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 isDividing 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 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.