In a Jackson network, organized as a tree rooted at queue r, several customers are queued at time 0 and there is no new customer arrival. The service time of each customer in queue i is geometrically distributed with rate $p_i$. Service times are independent and service can be preempted. Each customer arrives at parent queue after it is served in queue i. The objective is to find a scheduling policy that maximizes the expected number of customers in queue r at time T. Note queue r is never active and once a customer arrives at r, it stays.
The problem falls in the arena of stochastic scheduling in Markov Decision Process. However, there are two constraints in my problem: no two queues sharing a parent can be active at the same time; plus, a queue and its parent queue cannot be active together. Can anyone please give some hint (reference books, papers, etc.) on how this or similar problem is approached in the literature? Is there some optimal result, like $\mu c$-rule, for this problem? Thanks in advance.