  We study a system, where a random flow of customers is served by servers
(called agents) invited on-demand. Each invited agent arrives into the system
after a random time; after each service completion, an agent returns to the
system or leaves it with some fixed probabilities. Customers and/or agents may
be impatient, that is, while waiting in queue, they leave the system at a
certain rate (which may be zero). We consider the queue-length-based feedback
scheme, which controls the number of pending agent invitations, depending on
the customer and agent queue lengths and their changes. The basic objective is
to minimize both customer and agent waiting times.
  We establish the system process fluid limits in the asymptotic regime where
the customer arrival rate goes to infinity. We use the machinery of switched
linear systems and common quadratic Lyapunov functions to approach the
stability of fluid limits at the desired equilibrium point, and derive a
variety of sufficient local stability conditions. For our model, we conjecture
that local stability is in fact sufficient for global stability of fluid
limits; the validity of this conjecture is supported by numerical and
simulation experiments. When local stability conditions do hold, simulations
show good overall performance of the scheme.
