 Solution

For fixed routing, let each route $r$ require one unit on every resource in its fixed set, let $C_j\ge 1$ 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 giveswhere the Erlang loss formula isScreening 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 $C\ge 1$ 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 givesConsequently $e_C$ increases continuously from $0$ to $1$, and $m_C$ increases from $0$ to $C$. Put $y_j=-\log (1-B_j)$ andEach $g_j$ is continuous and strictly increasing, tending to $C_j$. Consider the convex potential for the Erlang fixed pointThe 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 $y^{\ast }$.

Its derivative isFor 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 ${\partial }_{j}F=0$. 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 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 satisfyDefine the right side minus $B$ as $G(B)$. The stable Erlang loss formula recursion $b_0=1$, $b_k=a{b}_{k-1}/(k+a{b}_{k-1})$ givesAll 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 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.