Dependent Competing Failure Processes in Reliability Systems.

This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H1, between H1 and H2, and above H2, respectively. The system's failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ-shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ-shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ-shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation.

On an unbiased and consistent estimator for mutation rates.

Spontaneous mutations are stochastic events. The mutation rate, defined as mutations per genome per replication, is generally very low, and it is widely accepted that spontaneous mutations occur at defined, but different, rates in bacteriophage and in bacterial, insect, and mammalian cells. The calculation of mutation rates has proved to be a significant problem. Mutation rates can be calculated by following mutant accumulation during growth or from the distribution of mutants obtained in parallel cultures. As Luria and Delbrück described in 1943, the number of mutants in parallel populations of bacterial cells varies widely depending on when a spontaneous mutation occurs during growth of the culture. Since 1943, many mathematical refinements to estimating rates, called estimators, have been described to facilitate determination of the mutation rate from the distribution or frequency of mutants detected following growth of parallel cultures. We present a rigorous mathematical solution to the mutation rate problem using an unbiased and consistent estimator. Using this estimator we demonstrate experimentally that mutation rates can be easily calculated by determining mutant accumulation, that is, from the number of mutants measured in two successive generations. Moreover, to verify the consistency of our estimator we conduct a series of simulation trials that show a surprisingly rapid convergence to the targeted mutation rate (reached between 25th and 30th generations).