To protect users from early failures, manufacturers often use burn-in to screen out weak units and meanwhile providing warranty to customers. Some performance-based burn-in models have been proposed in the literature. However, relations between these indices are not clear. In this study, we look into this problem and reveal the internal relations between these indices. More specifically, we focus on the probability of failure within the warranty period, mean number of failures within the warranty period and the percentile residual life that maximize the warranty period given a specified proportion of field failure. It is found that there are some dual relations between these indices. An illustrative example is used to demonstrate our results.Keywords -Probability of failure, warranty, percentile of the residual life

Failures of complex systems may be various types. Natural failure due to degradation is one main mechanism. Other factors such as traumatic failure caused by random shocks from external environment are also critical to systems' operation. This paper proposes a new model to study complex systems that experience both degradation and shocks which are dependent. System reliability is analyzed and maintenance actions including periodic inspection, preventive maintenance and replacement are adopted to minimize the long-run maintenance cost rate. Sensitivity analysis is also included.