Measuring availability indexes with small samples for component and network reliability using the Sahinoglu-Libby probability model

With the advances in pervasive computing and wireless networks, the quantitative measurements of component and network availability have become a challenging task, especially in the event of often encountered insufficient failure and repair data. It is well recognized that the Forced Outage Ratio (FOR) of an embedded hardware component is defined as the failure rate divided by the sum of the failure and the repair rates; or FOR is the operating time divided by the total exposure time. However, it is also well documented that FOR is not a constant but is a random variable. The probability density function (pdf) of the FOR is the Sahinoglu-Libby (SL) probability model, named after the originators if certain underlying assumptions hold. The SL pdf is the generalized three-parameter Beta distribution (G3B). The failure and repair rates are taken to be the generalized Gamma variables where the corresponding shape and scale parameters, respectively, are not identical. The SL model is shown to default to that of a standard two-parameter Beta pdf when the shape parameters are identical. Decision Theoretic (Bayesian) solutions are employed to compute small-sample Bayesian estimators by using informative and noninformative priors for the component failure and repair rates with respect to three definitions of loss functions. These estimators for component availability are then propagated to calculate the network expected input-output or source-target (s-t) availability for four different fundamental networks given as examples. The proposed method is superior to using a deterministic way of estimating availability simply by dividing total up-time by exposure time. Various examples will illustrate the validity of this technique to avoid over- or underestimation of availability when only small samples or insufficient data exist for the historical lifecycles of components and networks.