EMIS 7300 SYSTEMS ANALYSIS METHODS FALL 2005 Dr. John Lipp Copyright © 2005 Dr. John Lipp

Reliability Definitions Reliability is the probability that an item will successfully perform its intended function In general, reliability is a function of time, R(t). Un-reliability is denoted Q(t) = 1 – R(t). Reliability / un-reliability are related to the CDF R(t) = P(working up to time t) = 1 – F(t). Q(t) = P(failure before time t) = F(t). Note that R(t) + Q(t) = 1. EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Reliability Definitions (cont.) The probability of failure in a given time interval, t1 to t2, can be expressed in terms of either reliability or unreliability functions, i.e., P(t1 < T < t2) = R(t1) - R(t2) = Q(t2) - Q(t1) EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Reliability Definitions (cont.) The mean time to failure (MTTF) is the preferred term when a device is not repairable The term mean time between failure (MTBF) is preferred when a device can be repaired EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Hazard Rate The hazard rate is the instantaneous probability that a part that has survived to time t will suddenly fail Typically, h(t) is high during infancy and old age “Burn-in” “Old age” 1 EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Hazard Rate (cont.) The instantaneous failure rate, h(t), has the following properties: h(t)  0 , t  0 EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Series Reliability Independent, series connected devices with reliabilities r1, r2, r3, etc. have an overall reliability of r = r1 r2 r3 … rN. Device 1: r1 Device 2: r2 Device 3: r3 Device N: rN … Device: r = EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Parallel Reliability Device 1: q1 Device 2: q2 Device 3: q3 Device N: qN … = Device: q Independent, parallel connected devices with un-reliabilities q1, q2, q3, etc. have overall un-reliability q = q1 q2 q3 … qN. Equivalent to r = 1 – (1 – r1)(1 – r2)(1 – r3) … (1 – rN). EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Exponential Reliability The most basic model of reliability is the situation where the failure rate is constant over time. The result is an exponential model, R(t) = e-lt Q(t) = 1 – e-lt h(t) = l MTTF / MTBF = 1/l Applies when the failure mechanism is simple . Recall that the exponential distribution is memoryless The Exponential Model is most often associated with electronic equipment. EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Exponential Probability Paper -log R(t) = lt, that is, a logarithm paper should show a straight line with exponential reliability. Sample correlation coefficient r quality of linear fit ln 0.01 -log R(t) slope = l ln 0.1 ln 1.0 time EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Exponential Parameter Estimation The parameter l can be estimated with or confidence intervals computed via EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Half-life of radioactive material  exponential Print / make transparency EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Series Exponential Reliability A series connection of N identical, exponentially reliable components has an overall exponential reliability EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Non-constant Failure Rates Increasing Failure Rate ln 0.01 -log R(t) ln 0.1 Decreasing Failure Rate ln 1.0 time EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Weibull Reliability When the failure rate isn’t constant with respect to time, the distribution that fits the data is usually the Weibull Reliability Hazard Rate When b < 1 the failure rate is increasing. When b = 1 the failure rate is constant. When b > 1 the failure rate is increasing. MTTF/MTBF Gamma Function EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Weibull Reliability (cont.) Weibull is the first choice reliability model Models “weakest link in the chain” (series). 20-30 points usually required to discredit. Note: includes exponential (b = 1).  is the shape parameter l is the scale parameter EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Weibull PDF f(t) t t is in multiples of  1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 β=0.5 β=5.0 β=3.44 β=2.5 β=1.0 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 EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Weibull Probability Paper Probability paper for the Weibull is based on the logarithm of the logarithm of the inverse reliability That is, Y = log(failure time) X = log(log(1 / (1 – Median Rank (Y)))) Fit Y = mX + b using the least squares method (AKA simple linear regression). Estimate of l = e-b and b = m-1. EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

The Log-Normal Distribution A log-normal tends to be a model of reliability when A large number of failures have to occur, that is, an effective model is that of a large, parallel system, the failure tends to have a log-normal distribution. Non-linear increases in the failure rate of the components. The log-normal appears concave down on Weibull paper. The Lognormal Model is often used as the failure distribution for mechanical items and for the distribution of repair times. If T ~ LN(,), then Y = lnT ~ N(,). EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp

Copyright  2005 Dr. John Lipp Homework Probability and Statistics for Reliability: An Introduction http://quanterion.com/ReliabilityQues/V4N2.html EMIS 7300 Fall 2005 Copyright  2005 Dr. John Lipp