A tandem queue with a gate mechanism

Inspired by a problem regarding cable access networks, we consider a two station tandem queue with Poisson arrivals. At station 1 we operate a gate mechanism, leading to batch arrivals at station 2. Upon arrival at station 1, customers join a queue in front of a gate. Whenever all customers present at the service area of station 1 have received service, the gate before as well as a gate behind the service facility open. Customers leave the service area and enter station 2 (as a batch), while all customers waiting at the gate in front of station 1 are admitted into the service area. For station 1 we analyse the batch size and the time between two successive gate openings, as well as waiting and sojourn times of individual customers for different service disciplines. For station 2, we investigate waiting times of batch customers, where we allow that service times may depend on the size of the batch and also on the interarrival time. In the analysis we use Wiener–Hopf factorization techniques for Markov modulated random walks.