Stochastic comparisons of random processes, with applications in reliability

Abstract

The usual definition of stochastic comparison of random vectors is extended to stochastic comparison of random processes. Conditions are stated under which {X (t), t ≧ 0} stochastically larger than {Y (t), t ≧ 0} implies that for increasing functionals f.Applications are made to reliability problems, yielding stochastic comparisons for systems of independently operating machines assuming exponential failure and exponential repair. From these stochastic comparisons we may then deduce similar stochastic comparisons for functionals of practical importance in reliability applications, such as the total machine up-time, the first time that the number of functioning machines drops below a specified number, the total time during which at least a specified number of machines are functioning, etc.