Classical Perturbation Theory for Monte Carlo Studies of System Reliability

A variational principle for a Markov system allows the derivation of perturbation theory for models of system reliability, with prospects of extension to generalized Markov processes of a wide nature. It is envisaged that Monte Carlo or stochastic simulation will supply the trial functions for such a treatment, which obviates the standard difficulties of direct analog Monte Carlo perturbation studies. The development is given in the specific mode for first- and second-order theory, using an example with known analytical solutions. The adjoint equation is identified with the importance function and a discussion given as to how both the forward and backward (adjoint) fields can be obtained from a single Monte Carlo study, with similar interpretations for the additional functions required by second-order theory. Generalized Markov models with age-dependence are identified as coming into the scope of this perturbation theory.