A system encounters $m = 12$ missing experts during decode, with PCIe bandwidth $B_P = 32\text{ GB/s}$ and host bandwidth $B_H = 128\text{ GB/s}$. Based on the optimal execution policy, how many missing experts should be routed to the concurrent PCIe cache-fill branch?

In the runtime implementation of the $q^*$ execution policy, a minimum of at least one cache fill is retained to ensure continuous cache warming.

Under the $q^*$ execution policy, what behavior does the system dynamically degenerate to as the host bandwidth approaches PCIe bandwidth ($B_H \to B_P$)?

Explain the mathematical derivation used to determine the optimal expert cache-fill count $q^*$ for serving $m$ missing experts of size $S$. Include the expressions for both concurrent branches, how they are equated, and the resulting closed-form policy.