User-optimal state-dependent routeing in parallel tandem queues with loss

We consider a system of parallel, finite tandem queues with loss. Each tandem queue consists of two single-server queues in series, with capacities C-1 and C-2 and exponential service times with rates mu(1) land mu(2) for the first and second queues, respectively. Customers that arrive at a queue that is full are lost. Customers arriving at the system can choose which tandem queue to enter. We show that, for customers choosing a queue to maximise the probability of their reaching the destination (or minimise their individual loss probability), it will sometimes be optimal to choose queues with more customers already present and/or with greater residual service requirements (where preceding customers are further from their final destination).