A polling model with threshold switching

We consider a single-server two-queue Markovian polling system with the following special feature. If the server is serving the infinite-buffer queue Q2 and the single-buffer queue Q1 is empty, then it stays at Q2 until it has become empty; but if a customer joins an empty Q1, then the server only stays at Q2 as long as that queue has at least N customers (the threshold). If that customer joins Q1 while Q2 has less than N customers, then service at Q2 is preempted and the server instantaneously switches to Q1. Arrivals to Q1 when it is occupied are blocked and lost. This threshold discipline contrasts with the classical multi-queue polling model, where switching instants are typically determined by the length of the queue being served. We (i) derive explicit expressions for the joint queue length distribution; (ii) analyze the busy period distribution by employing an original approach that uses taboo states; and (iii) determine the sojourn time distribution for customers in both queues.