In this paper, we study the so-calledN-model,which consists of two pools of servers. Pool 1 containsN1serverswhile pool 2 consists of one server, which can be inactive timeto time. The 1st pool is fed by a Poisson input of the 1st classcustomers. Provided all servers of pool 1 are busy and the queuesize exceeds a given threshold, a class-1 customer, with a givenprobability, jumps to server of pool 2, if this server is active.Under exponential assumptions, the number of customers inthe 1st pool is a Markov birth-death process. The periods ofactivity/inactivity constitute initially a ”transient” process, whichconverges to a stationary regime. In this research we deducestationary distribution of this Markov process in an explicit form.Moreover, we find conditions, when the mentioned convergenceof the alternating process of activity/inactivity of the server inthe 2nd pool implies convergence of the birth-death process,describing the 1st pool, to stationary regime. Moreover, thisconvergence is demonstrated by simulation.