Predicting the Reliability of Systems An important application

Predicting the Reliability of Systems • An important application of probability is reliability analysis used to predict an overall system’s reliability • It is convenient to define relaiability in terms of survival probability, using the notation Ri(t) = Pr[component i survives beyond time t]

Predicting the Reliability of Systems • The reliability of the system is then a function of the reliabilities of its components: Rs(t) = f{R 1(t), R 2(t), …} • There are two primary system forms: One applies when the components are arranged in parallel and another when they joined in a series.

Predicting the Reliability of Systems • More complex cases involve modular subsystems as building blocks. • We can portray the logic of systems schematically in similar fashion to electrical circuits.

Systems with Series Components • A series system perform satisfactorily as long as all components are fully functional. • Under the series logic, the system will survive past time t only if its components do so. • System reliability, therefore, is equal to the product of the component reliabilities: Rs(t) = R 1(t)(R 2(t)…. . Rn(t) The above follows directly from the multiplication law for independent events.

Systems with Parallel Components • A parallel system is one that performs as long as any one of its components remains operational. • Failure of the system occurs only if all components fail. • The probability of system failure is therefore the product of those failure probabilities [each being 1 minus the respective survival probability]: Pr[system failure at or before t] = [1 -R 1(t)][1 R 2(t)]…. [1 -Rn 9 t)]

Increasing System Reliability • A design issue in systems is how to increase system reliability. One way this might be accomplished is by raising the reliability of individual components (perhaps by substituting better materials, such as gold wiring instead of copper).

Complex Modular Systems • Individual components may be aggregated into subsystems according to their interrelationships. • To find the system reliability, we must first establish separate reliabilities for each subsystems.

Problems from text book page#186

Chapter 7 This chapter focuses on the uncertain quantities that may arise from a random experiment. Because the particular level of a variable is uncertain and subject to chance, these quantities are referred to as random variables.

Random Variable as a Function • The number of defective castings in a sample and the number of test failures of electronic components are examples of random variables.