Optimal multi-attempt missions with cumulative effect

Many real-world systems utilize operation aborting to enhance their survivability. Specifically, an operation can be aborted when a risk of a system's loss during the rest of a mission becomes too high. In this case, a rescue procedure is initiated to save the system. Missions with a possibility of multiple attempts can occur in different real-world scenarios when accomplishing a mission is important, whereas the cost-related and the time-wise restrictions are not dominant. The system can start the new attempt if it was saved by the rescue procedures after aborting or completing the previous attempts. In this paper, we assume that the probability of achieving a mission goal is a function of the number of successfully completed attempts. The attempt success probability as well as the rescue success probability depend on the number of shocks experienced by the system during the corresponding mission phases. The probabilistic model for the multi-attempt case is suggested and the tradeoff between the overall mission success probability and the system loss probability is discussed. The corresponding optimization problems are formulated. For the considered illustrative example, the detailed analysis is performed that shows specifically that the optimal number of attempts under the risk avert policy can be obtained along with the optimal abort policy.