A service system with randomly behaving on-demand agents

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 uid limits, in the asymp- Totic regime where the customer arrival rate goes to infinity. We use the machinery of switched linear systems and com- mon quadratic Lyapunov functions to derive suffcient condi- Tions for the local stability of uid limits at the desired equi-librium point (with zero queues). We conjecture that, for our model, local stability is in fact suffcient for global stability of uid 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.