  Consider a system of $n$ parallel server pools where tasks arrive as a
time-varying Poisson process. The system aims at balancing the load by using an
inner control loop with an admission threshold to assign incoming tasks to
server pools; as an outer control loop, a learning scheme adjusts this
threshold over time in steps of $\Delta$ units, to keep it aligned with the
time-varying overall load. If the fluctuations in the normalized load are
smaller than $\Delta$, then we prove that the threshold settles for all large
enough $n$ and balances the load when $\Delta = 1$. Our model captures a
tradeoff between optimality and stability, since for higher $\Delta$ the degree
of balance decreases, but the threshold remains constant under larger load
fluctuations. The analysis of this model is mathematically challenging,
particularly since the learning scheme relies on subtle variations in the
occupancy state of the system which vanish on the fluid scale; the methodology
developed in this paper overcomes this hurdle by leveraging the tractability of
the specific system dynamics. Strong approximations are used to prove certain
dynamical properties which are then used to characterize the behavior of the
system, without relying on a traditional fluid-limit analysis.
