  We consider a service system where agents are invited on-demand. Customers
arrive exogenously as a Poisson process and join a customer queue upon arrival
if no agent is available. Agents decide to accept or decline invitations after
some exponentially distributed random time, and join an agent queue upon
invitation acceptance if no customer is waiting. A customer and an agent are
matched in the order of customer arrival and agent invitation acceptance under
the non-idling condition, and will leave the system simultaneously once matched
(service times are irrelevant here).
  We consider a feedback-based adaptive agent invitation scheme, which controls
the number of pending agent invitations, depending on the customer and/or agent
queue lengths and their changes. The system process has two components -- `the
difference between agent and customer queues' and `the number of pending
invitations', and is a countable continuous-time Markov chain.
  For the case when the customer arrival rate is constant, we establish fluid
and diffusion limits, in the asymptotic regime where the customer arrival rate
goes to infinity, while the agent response rate is fixed. We prove the process
stability and fluid-scale limit interchange, which in particular imply that
that both customer and agent waiting times in steady-state vanish in the
asymptotic limit. To do this we develop a novel (multi-scale) Lyapunov drift
argument; it is required because the process has non-trivial behavior on the
state space boundary. When the customer arrival rate is time-varying, we
present a fluid limit for the processes in the same asymptotic regime.
Simulation experiments are conducted to show good performance of the invitation
scheme and accuracy of fluid limit approximations.
