The Song Rule as a Validator of Analytical Results-A Note Correcting System Reliability Results in a Review of the Literature

A great deal of previous work has studied methods to determine system reliability. That work has defined system reliability as the probability that the production output meets a predetermined demand for a network with many workstations, each of which has random capacity determined by a discrete probability distribution. It is noteworthy that the archival work presents numerous examples, wherein entities are discrete and indivisible, while the analysis is based upon continuous flow, much like a fluid, through the network. The inconsistency, inherent in mixing discrete-entity examples with continuous-flow analysis, can result in erroneous conclusions about the system reliability. The current paper presents a rigorous discrete-analysis analytical approach, which is called the Song rule (Stochastic "Output >= demand" Networks and their Generation). Based on all examples studied in this paper, the absolute difference ratios between the previous incorrect (as published) and the correct probabilities (from the Song rule) are all greater than 23%. The proposed Song rule analysis is verified to be correct using discrete-event simulation. In addition to providing rigorous analysis for the network problems under consideration, the Song rule is a useful tool for assessing the validity of any future proposed approach for other stochastic reliability problems.