In this note, we study a new repair replacement model for a deteriorating system, in which the successive survival times of the system form a geometric process and are stochastically non-increasing, whereas the consecutive repair times after failure also constitute a geometric process but are stochastically non-decreasing. Two kinds of replacement policy are considered, one based on the working age of the system and the other one determined by the number of failures. The explicit expressions of the long-run average costs per unit time under these two kinds of policy are calculated.

We have recently discussed the replacement problem of a parallel system in a random environment. This paper extends the same replacement problem for the following two cases which are more plausible: (i) The probability that an operating component fails by the j th shock depends on the number j. (ii) The replacement cost is a linear function of failed components. The expected cost of the above model is obtained. A numerical example is finally presented when the probability of failure time of a component is a negative binomial distribution.

Råde considered a parallel system with n components in a random environment. The shocks cause the components to fail with certain probabilities. This paper considers the replacement policy for the above system in which it is exchanged before failure if the total number of failed components is more than k, and it is replaced if all of n components have failed. An optimum number of k which will minimize the expected cost is obtained. A numerical example is finally presented.