# Layman’s abstract for paper on modeling and analysis of system reliability using phase‐type distribution closure properties

Every few days, we will be publishing layman’s abstracts of new articles from our prestigious portfolio of journals in statistics. The aim is to highlight the latest research to a broader audience in an accessible format.

The article featured today is from Applied Stochastic Models in Business and Industry, with the full article now available to read in issue 36:4 here.

Alkaff, A, Qomarudin, MN. Modeling and analysis of system reliability using phase‐type distribution closure properties. Appl Stochastic Models Bus Ind. 2020; 1– 22. doi: 10.1002/asmb.2509

Reliable systems are everyday life’s necessities: reliable products, reliable machines, reliable services, etc.; hence, reliability is an important consideration in systems design. Design for reliability involves modeling and analysis of a system as an attempt to estimate, predict, and optimize reliability measures of the system. It is important for system’s operationability and maintainability. The model represents the relationship between system reliability and its components reliability through the system structure.
Due to systems’ large structure involving many components, the analysis requires computer software. The available analytical methods in computerized system reliability analysis is for calculating the system reliability value from its components’ reliability values. However, it is not sufficient. Ideally, the analysis should mathematically express the system reliability function, which measures how reliability value decays over time, as a function of its components reliability functions. Once it is obtained, other reliability measures can be calculated easily. Unfortunately, to achieve this ideal objective using currently available formulas and methods requires tedious manual calculations or complicated computerized symbolic processing.

This paper proposes an exact analytical method readily implementable as computer programs for generating the system reliability function directly from its components reliability functions and the system structure using only matrix algebra. The method also can directly estimate how the system’s failure rate increases over time (hazard function) and predict how long the system is expected to last (mean time to failure with its confidence interval). The idea is assuming that components’ reliability functions have independent non-identical phase-type (PH) distribution and employs closure properties of such a distribution. This assumption does not prevent the method to be used for more popular distributions in reliability analysis, such as Weibull or lognormal, since all distributions eventually converge to PH distribution. In addition, PH distribution makes the method applicable for multi-state reliability analysis, i.e., may include states other than good and failed as commonly assumed.

The authors derived algorithms for all basic system structures and provide their application for a general system structure. For practical purposes, the algorithms and application are implemented as MATLAB programs whose source codes is available in the journal’s website. The results of this study are fundamental and simple to make them suitable as course materials on system reliability.