  Consider a system of identical server pools where tasks with exponentially
distributed service times arrive as a time-inhomogenenous Poisson process. An
admission threshold is used in an inner control loop to assign incoming tasks
to server pools while, in an outer control loop, a learning scheme adjusts this
threshold over time to keep it aligned with the unknown offered load of the
system. In a many-server regime, we prove that the learning scheme reaches an
equilibrium along intervals of time where the normalized offered load per
server pool is suitably bounded, and that this results in a balanced
distribution of the load. Furthermore, we establish a similar result when tasks
with Coxian distributed service times arrive at a constant rate and the
threshold is adjusted using only the total number of tasks in the system. The
novel proof technique developed in this paper, which differs from a traditional
fluid limit analysis, allows to handle rapid variations of the first learning
scheme, triggered by excursions of the occupancy process that have vanishing
size. Moreover, our approach allows to characterize the asymptotic behavior of
the system with Coxian distributed service times without relying on a fluid
limit of a detailed state descriptor.
