Explicit Stationary Comparisons for Join-the-Shortest-Queue Approximations

We compare the stationary occupancy law of a finite join-the-shortest-queue system with its finite-buffer and sampled-routing approximations. For \(N\) unit-rate exponential servers, Poisson arrival rate \(0<\lambda<N\) and \(\rho=\lambda/N\), regeneration gives \(H=((1-\rho)^{-N}-1)/\lambda\). With capacity \(b\) and blocking probability \(B_b\), the total variation error is at most \(\lambda H(bN+1)B_b\). Explicit upper and lower bounds prove sharp order \(\lambda^{bN+1}\) as \(\lambda\) tends to zero for fixed \(N,b\). For shortest-of-\(k\) routing we obtain explicit total variation bounds for both sampling conventions. These estimates answer the two comparison requests in AIM Problem 1.35 for every bounded occupancy observable. Constants can be loose or trivial at high load or large \(N\); no sharp many-server approximation is asserted.