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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.