A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering

Routing mechanisms for stochastic networks are often designed to produce state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the limiting process to a lower-dimensional subset of its full state space. In a fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding mode control that forces the fluid trajectories to a lower-dimensional sliding manifold. Within deterministic dynamical systems theory, it is well known that sliding-mode controls can cause the system to chatter back and forth along the sliding manifold due to delays in activation of the control. For the prelimit stochastic system, chattering implies fluid-scaled fluctuations that are larger than typical stochastic fluctuations. In this paper we show that chattering can occur in the fluid limit of a controlled stochastic network when inappropriate control parameters are used. The model has two large service pools operating under the fixed-queue-ratio with activation and release thresholds (FQR-ART) overload control which we proposed in a recent paper. The FQR-ART control is designed to produce asymptotic SSC by automatically activating sharing (sending some customers from one class to the other service pool) once an overload occurs. We have previously shown that this control can be effective, even if the service rates are less for the other shared customers, if the control parameters are chosen properly. We now show that, if the control parameters are not chosen properly, then delays in activating and releasing the control can cause chattering with large oscillations in the fluid limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even when the arrival rates are smaller than the potential total service rate in the system, a phenomenon referred to as congestion collapse. We show that the fluid limit can be a bi-stable switching system possessing a unique nontrivial periodic equilibrium, in addition to a unique stationary point.