This paper deals with a stochastic stability concept for discrete-time Markovian jump linear systems. The random jump parameter is associated to changes between the system operation modes due to failures or repairs, which can be well described by an underlying finite-state Markov chain. In the model studied, a fixed number of failures or repairs is accepted, after which, the system is brought to a halt for maintenance or for replacement. The usual concepts of stochastic stability are related to pure infinite horizon problems, and are not appropriate in this scenario. A new stability concept is introduced, named stochastic τ-stability that is tailored to the present setting. Necessary and sufficient conditions to ensure the stochastic τ-stability are provided, and the almost sure stability concept associated with this class of processes is also addressed. The paper also develop an equivalence among second order concepts that parallels the results for infinite horizon problems.1 334339Ji, Y., Chizeck, H.J., Jump linear quadratic Gaussian control: Steady-state solution and testable conditions (1990) Control Theory and Advanced Technology, 6 (3), pp. 289-319Ji, Y., Chizeck, H.J., Loparo, K.A., Stability and control of discrete-time jump linear systems (1991) Control Theory Advanced Technology, 7, pp. 247-270Costa, O.L.V., Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters (1993) Journal of Mathematical Analysis and Applications, 179, pp. 154-178Fang, Y., A new general sufficient condition for almost sure stability of jump linear systems (1997) IEEE Transactions on Automatic Control, 42 (3), pp. 378-382Li, Z.G., Soh, Y.C., Wen, C.Y., Sufficient conditions for almost sure stability of jump linear systems (2000) IEEE Transactions on Automatic Control, 45 (7), pp. 1325-1329Zhou, K., Doyle, J.C., Glover, K., (1996) Robust and Optimal Control, , Prentice-HallKozin, F., A survey of stability of stochastic systems (1969) Automatica, 5, pp. 95-112Fang, Y., Loparo, K.A., Feng, X., Almost sure and δ-moment stability of jump linear systems (1994) International Journal of Control, 59 (5), pp. 1281-1307