Two-queue polling models with a patient server

We consider two-queue polling models with the special feature that a timer mechanism is employed at Q(1): whenever the server polls Q(1) and finds it empty, it activates a timer and remains dormant, waiting for the first arrival. If such an arrival occurs before the timer expires, a busy period starts in accordance with Q(1)'s service discipline. However, if the timer is shorter than the interarrival time to Q(1), the server does not wait any more and switches back to Q(2). We consider three configurations: (i) Q(1) is controlled by the 1-limited protocol while Q(2) is served exhaustively, (ii) Q(1) employs the exhaustive regime while Q(2) follows the 1-limited procedure, and (iii) both queues are served exhaustively. In all cases, we assume Poisson arrivals and allow general service and switchover time distributions. Our main results include the queue length distributions at polling instants, the waiting time distributions and the distribution of the total workload in the system.