Some exact and approximate methods for large scale systems steady-state availability analysis.

摘要

System availability is the probability of the system being operable at instant t. Markov chains are a model used for system availability analysis. The exact analytical solution in terms of component failure rates and repair rates for steady-state system availability is complex to find solving the large numbers of simultaneous linear equations that result from the model. Although exact analytical solutions have been developed for series and parallel systems and for some other small size systems, they have not been developed for large scale general systems with n distinct components. Some methods for approximate analytical solutions have been suggested, but limitations on network types, over simplified states merge conditions and lack of predictions of approximation errors make these methods difficult to use. Markov state transition graphs can be classified as symmetric or asymmetric. A symmetric Markov graph has two-way transitions between each pair of communicating nodes. An asymmetric Markov graph has some pair(s) of communicating nodes with only one-way transitions. In this research, failure rates and repair rates are assumed to be component dependent only. Exact analytical solutions are developed for systems with symmetric Markov graphs. Pure series systems, pure parallel systems and general k out of n systems are examples of systems with symmetric Markov graphs. Instead of solving a large number of linear equations from the Markov model to find the steady-state system availability, it is shown that only algebraic operations on component failure rates and repair rates are necessary. In fact, for the above class of systems, the exact analytical solutions are relatively easy to obtain. Approximate analytical solutions for systems with asymmetric Markov graphs are also developed based on the exact solutions for the corresponding symmetric Markov graphs. The approximate solutions are shown to be close to the exact solutions for large scale and complex systems. Also, they are shown to be lower bounds for the exact solutions. Design principles to improve systems availability are derived from the analytical solutions for systems availability. Important components can be found easily with the iteration procedure and computer programs provided in this research.