On the lifetime comparison of series systems with random number of components

In this paper, we stochastically compare Harris family distributions having random tilt parameter with Harris family distributions having fixed tilt parameter. In reliability context, this applies to stochastic comparison of lifetime of a series system with random number of components with another similar but mixed system. Our comparison tools are various types of orderings. In addition, we obtain an upper bound for maximum error in evaluating the reliability function. We also present two bounds for a Harris family survival function conditioned on its random tilt parameter, which are useful in distinguishing failure probability of a component after a time t. Several previous findings, regarding Marshall-Olkin family follow as spacial cases of our results.