Methods for computing a system with instantaneous feedback and variable input stream intensity

We study the Markov model of a servicing system with one server and instantaneous feedback. After servicing is complete, a part of the calls, according to the Bernoulli scheme, either leave the system or immediately return to receive re-servicing, which requires a positive (random) server switching time. The intensity of the incoming stream depends on the state of the server, which can be in operating mode or in switching mode. We find the ergodicity condition for the corresponding two-dimensional Markov chain and propose three methods for studying it: the method of generating functions, the method of spectral expansion, and the method of space merging. We present the results of numerical experiments.