The PH/PH/1 multi-threshold queue

We consider a PH/PH/1 queue in which a threshold policy determines the stage of the system. The arrival and service processes follow a Phase-Type (PH) distribution depending on the stage of the system. Each stage has both a lower and an upper threshold at which the stage of the system changes, and a new stage is chosen according to a prescribed distribution. The PH/PH/1 multi-threshold queue is a Quasi-Birth-and-Death process with a tri-diagonal block structured boundary state which we model as a Level Dependent Quasi-Birth-and-Death process. An efficient algorithm is presented to obtain the stationary queue length vectors using Matrix Analytic methods.