Steady-State Probabilities for a Queue with a General Service Distribution and State-Dependent Arrivals

We consider a queueing system with a general service distribution having a possibility of feedback. The customers belong to the same class, and the queueing discipline is first-in, first-out. There is only one server in the station. The facility is rendered Markovian by means of fictitious stages. The input flow depends on the state of the station. It is shown that the equilibrium probabilities can be simply expressed by means of a matrix product. Two particular cases are studied.