A polling system whose stability region depends on a whole distribution   of service times
  We present an example of a single-server polling system with two queues and an adaptive service policy where the stability region depends on the expected values of all the primitives and also on a certain exponential moment of the service-time distribution in one of the queues. The latter parameter can not be determined, in general, in terms of any finite number of power moments. It follows that the fluid approximation approach may not be an appropriate tool for the stability study of this model. 
Natalia Chernova, Sergey Foss and Bara Kim
math.PR