Probability Density Estimation for Composite Systems Reliability Indices Using Cross-Entropy Based Continuous Time Markov Chain Simulation

How to calculate the probability density function (PDF) of reliability indices (RIs) in the importance sampling (IS) based sequential simulation is an intractable problem not yet solved. To fill this gap, this article proposes a creative PDF estimation method by combining the IS of continuous time Markov chain (CTMC) simulation with the kernel density estimation (KDE) technique. Firstly, the cross entropy based IS of CTMC (CE-CTMC) is introduced, and then the analytic parameter updating rules of the CTMC IS-PDF are given. Secondly, using the CTMC path samples generated from the CTMC IS-PDF the PDF estimation of RI is achieved based on a creative modification of the classic KDE. The theoretical proof of the improved PDF estimation is given. Besides, to take full advantage of the samples drawn in the pre-simulation, a combined estimation method for the expectation and PDF estimation of RI is put forward, which leads to a further improvement in the accuracy and efficiency of CE-CTMC. The proposed method is verified by the comparable results using the IEEE-RTS79 and IEEE-RTS96.