  We consider a service system where agents (or, servers) are invited
on-demand. Customers arrive as a Poisson process and join a customer queue.
Customer service times are i.i.d. exponential. Agents' behavior is random in
two respects. First, they can be invited into the system exogenously, and join
the agent queue after a random time. Second, with some probability they rejoin
the agent queue after a service completion, and otherwise leave the system. The
objective is to design a real-time adaptive agent invitation scheme that keeps
both customer and agent queues/waiting-times small. We study an adaptive
scheme, which controls the number of pending agent invitations, based on
queue-state feedback.
  We study the system process fluid limits, in the asymptotic regime where the
customer arrival rate goes to infinity. The fluid limit trajectories have
complicated behavior -- there are two domains where they follow different ODEs,
and a "reflecting" boundary. 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 (with zero queues). We derive
sufficient local stability conditions for the fluid limits. We conjecture that,
for our model, 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 the local stability conditions do hold,
simulations show good overall performance of the scheme.
