Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.