Systems prone to faults are often equipped with a controller whose aim consists in restricting the behaviour of the system in order to perform a diagnosis. Such a task is called active diagnosis. However to avoid that the controller degrades the system in view of diagnosis, a second objective in terms of quality of service is usually assigned to the controller. In the framework of stochastic systems, a possible specification, called safe active diagnosis requires that the probability of correctness of the infinite (random) run is non null. We introduce and study here two alternative specifications that are in many contexts more realistic. The notion of (γ,v)-fault freeness associates with each run a value depending on the discounted length of its correct prefix where the discounting factor is γ. The controller has to ensure that the average of this value is above the threshold v. The notion of α-resiliency requires that asymptotically, at every time step, a proportion greater than α of correct runs remain correct. From a semantic point of view, we determine the equivalences and (non) implications between the three notions of degradations both for finite and infinite systems. From an algorithmic point of view, we establish the border between decidability and undecidability of the diagnosability problems. Furthermore in the positive case, we exhibit their precise complexity and propose a synthesis of the controller which may require an infinite memory.