This paper considers a consecutive k-out-of-n:F system when the failure of a component in the system induces higher failure rate of the preceding survivor. The reliability, mean time to failure(MTTF), and average failure number of a consecutive k-out-of-n:F system are obtained, when the failure of a component increases the failure rate of the survivor which is working just before the failed component. Then the optimal number of consecutive failed components to minimize this long run average cost rate can be obtained. An example is considered to calculate the reliability, MTTF and average failure number of the system. And two procedures that find the optimal number of consecutive failed components are studied. Then, various cases of system parameters are also studied.