The cover time of a Markov chain is the first time at which every state has been visited at least once. In this talk, we consider random walks that jump to stationarity every L steps; or at rate 1/L, where L is a given parameter that may diverge. We will show that the order of the expected cover time is the same in both setups and study when the cover time is concentrated. Aldous proved, in 1991, that for reversible Markov chains, the cover time is concentrated around its expectation if and only if the maximal expected hitting time is of strictly smaller order than the maximal expected cover time. We will give a similar concentration criterion and show that the concentration of the cover time is equivalent in both setups under certain conditions. Joint work with Omer Angel, Jonathan Hermon and Pietro Lavino.