Reliability Models & Applications Leadership in Engineering Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis Reliability Models & Applications Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

Reliability Model Mathematical Model: a set of rules or equations which represent the behavior of a physical system or process Reliability Math Model: a mathematical relationship which describes the reliability of an item in terms of its components and their associated failure rates. Objective: to represent the real world by expressing physical or functional relationships in mathematical form Approaches: Analytical Simulation

Comparison of Reliability Modeling Approaches TYPE MODEL Analytical Monte Carlo Advantages Gives exact results (given the assumptions of the model). Once the model is developed, output will generally be rapidly obtained. It need not always be implemented on a computer; paper analyses may suffice. Very flexible. There is virtually no limit to the analysis. Empirical distributions can be handled. Can generally be easily extended and developed as required. Easily understood by non-mathematicians. Disadvantages Generally requires restrictive assumptions to make the problem trackable. Because of (A) it is less flexible than Monte Carlo. In particular, the scope for expending or developing a model may be limited. Model might only be understood by mathematicians. This may cause credibility problems if output conflicts with preconceived ideas of designers or management. Usually requires a computer; can require considerable computer time to achieve the required accuracy. Calculations can take much longer than analytical models. Solutions are not exact but depend on the number of repeated runs used to produce the output statistics. That is, all outputs are ‘estimates’.

Reliability Modeling Development Stages System Analysis Model output requirements Input data requirements Strike a balance between model complexity and accuracy Model Development Formulate the problem in mathematical terms Specify assumptions and ground rules Determine modeling approach; analytical or simulation Model Validation Use test cases to validate the mathematical formulation Perform sensitivity analysis to confirm realism Identify modeling constraints

Reliability Model Selection Model selection is basic to reliability modeling and analysis Selection criteria should be based on Physical laws Experience Statistical goodness of fit System configuration and complexity Type of analysis

Reliability Models: Time to Failure Models: The random variable T is time to (or between) failure. Number of Successes Model: The random variable X is the number of successes that occur in N trials. Number of Failures Model: The random variable Y is the number of failures that occur in a period of time, t.

Statistical Models Time to Failure Models: The Exponential Model The Weibull Model The Normal (or Gaussian) Model The Lognormal Model Discrete Event Models: The Binomial Model The Poisson Model

f (t) = The Exponential Model: Definition A random variable T is said to have the Exponential Distribution with parameters , where  > 0, if the failure density of T is: , for t  0 , elsewhere f (t) =

The Exponential Model: Remarks The Exponential Model is most often used in Reliability applications, partly because of mathematical convenience due to a constant failure rate. The Exponential Model is often referred to as the Constant Failure Rate Model. The Exponential Model is used during the ‘Useful Life’ period of an item’s life, i.e., after the ‘Infant Mortality’ period before Wearout begins. The Exponential Model is most often associated with electronic equipment.

Properties of the Exponential Model: Probability Distribution Function Reliability Function MTBF (Mean Time Between Failure)

Properties of the Exponential Model: Standard Deviation of Time to Failure: Failure Rate Cumulative Failure Rate P(T > t1 + t2 | T > t1) = P (T > t2), i.e., the Exponential Distribution is said to be without memory

The Exponential Model - Example θ=1,000 Hours Failure Density f(t) 0 1,000 2,000 3,000

The Exponential Model – Example (continued) θ=1,000 Hours R(t) Reliability Function 1.0 0.8 0.6 0.4 0.2 0.3679 0.1353 0.0498 0 1,000 2,000 3,000

f (t) = The Weibull Model: Definition A random variable T is said to have the Weibull Probability Distribution with parameters  and , where  > 0 and  > 0, if the failure density of T is: , for t  0 , elsewhere Remarks  is the Shape Parameter  is the Scale Parameter (Characteristic Life) f (t) =

The Weibull Model - Distributions: Probability Density Function f(t) 1.8 β=5.0 1.6 β=0.5 1.4 β=3.44 1.2 β=1.0 1.0 β=2.5 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 t t is in multiples of 

Properties of The Weibull Model: Probability Distribution Function , for t  0 Where F(t) is the Fraction of Units Failing in Time t Reliability Function

The Weibull Model - Distributions: Reliability Functions R(t) t t is in multiples of  β=5.0 β=1.0 β=0.5 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

The Weibull Model - Weibull Probability Paper (WPP): Derived from double logarithmic transformation of the Weibull Distribution Function. Of the form where Any straight line on WPP represents a Weibull Distribution with Slope =  & Intercept = - ln 

The Weibull Model - Weibull Probability Paper (WPP): Weibull Probability Paper links http://engr.smu.edu/~jerrells/courses/help/resources.html http://perso.easynet.fr/~philimar/graphpapeng.htm http://www.weibull.com/GPaper/index.htm

Use of Weibull Probability Paper:

Properties of the Weibull Model: p-th Percentile and, in particular

Properties of the Weibull Model: MTBF (Mean Time Between Failure) Standard Deviation

The Gamma Function  Values of the Gamma Function

Properties of the Weibull Model: Failure Rate Notice that h(t) is a decreasing function of t if  < 1 a constant if  = 1 an increasing function of t if  > 1

Properties of the Weibull Model: Cumulative Failure Rate The Instantaneous and Cumulative Failure Rates, h(t) and H(t), are straight lines on log-log paper. The Weibull Model with  = 1 reduces to the Exponential Model.

The Weibull Model - Distributions: Failure Rates h(t) t t is in multiples of  h(t) is in multiples of 1/  3 2 1 0 1.0 2.0 β=5 β=1 β=0.5

Properties of the Weibull Model: Conditional Probability of Surviving Time t2, given survival to time t1, where t1 < t2, , if  > 1 Mode - The value of time (age) that maximizes the failure density function.

The Weibull Model - Expected Time to First Failure: n identical items are put on test or into service under identical conditions and at age of zero time. The failure distribution of each item is Weibull with parameter  and . The expected time to first failure is: The expected time to second failure is:

Turbine Spacer Life Expectancy The Weibull Model Turbine Spacer Life Expectancy WEIBULL FIT - A 100 99 98 97 96 95 Based on 2 failures (Excludes 128 hr failure) New Spacers New Engine Build - Undisturbed Percent Surviving at 1 Flight Hours Based on 3 failures WEIBULL FIT - B 0 100 200 300 400 500 600 700 800 900 Age (Flight Hours)

The Weibull Model – Example 1 Time to failure of an item has a Weibull distribution with characteristic life  = 1000 hours. Formulate the reliability function, R(t), and the Failure Rate, h(t), as a function of time (age) and plot, for: (a)  = 0.5 (b)  = 1.0 (c)  = 1.5

The Weibull Model – Example 1 Solution

The Weibull Model – Example 1 Solution continued

The Weibull Model – Example 2 Time to failure of an item follows a Weibull distribution with  = 2 and  = 1000 hours. (a) What is the reliability, R(t), for t = 200 hours? (b) What is the hazard rate, h(t), (instantaneous failure rate) at that time? (c) What is the Mean Time To Failure?

The Weibull Model – Example 2 Solution failures per hour

The Weibull Model – Example 2 Solution continued From the Gamma Function table: