Optimal allocation problem in a hybrid queueing system

A controllable hybrid queueing system consists of a system controller and two nodes in tandem of the type $M/M/n_i$. Customers arrive to the controller who allocates them between the nodes. After service completion at node 2 the controller can allocate the customer waiting at node 1 to node 2. With probability $p$ after a service completion at node 1 a failure occurs. In this case the customer from node 1 joins node 2. With complement probability $1-p$ the service completion at node 1 is successful. For the given cost structure we formulate an optimal allocation problem to minimize the long-run average cost per unit of time. Using dynamic-programming approach we show the existents of thresholds which divides the state-space into two contiguous regions where the optimal decision is to allocate the customers to node 1 or to node 2. Some monotonicity properties of the dynamic-programming value function are established.