Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Quantitative Evaluation MethodsRandom Variable X–a function that assigns a real number X(s) to each sample point s in sample space S–e.g. coin toss, number of heads in a sequence of 3 tosses » s hhh hht hth htt thh tht tth ttt X(s) 3 2 2 1 2 1 1 0 –X is a random variable taking on values in the set 1Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Quantitative Evaluation MethodsCumulative Distribution Function (cdf)–The cdf of a random variable X is defined as the probability of the eventlim ( )xXF x→∞= 1lim ( )xXF x→−∞= 02Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Quantitative Evaluation MethodsProbability Density Function (pdf) –The pdf of a random variable is the derivation of – Since is a non-decreasing function,3Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Quantitative Evaluation MethodsExpectation of a random variable–in order to completely describe the behavior of a random variable, an entire function, namely the cdf or pdf, must be given–however, sometime we are just interested in parameters that summarize informationi.e. mean time to failure = expected lifetime of the system4Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Reliability R(t)R(t) = probability that system is working at time t, and any time before that => [0,t] X = random variable representing life of systemLet 5Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Reliability R(t)instantaneous rate at which components arefailing6Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5 div byto get this is called hazard function hazard rate failure rate functionwhich is the normalized failure rate(2)(1)7Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5using (1) in (2), i.e.expressed in terms of Reliability only withwe get8Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5expressed in term of unreliability Q(t)Result often used:9Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Bathtub CurveInfant mortality phase–burn-in to bypass infant mortalityUseful life periodWear-out phase–exchange before wear-out phaseTherefore one may assume constant failure rate function z(t), i.e z(t) = λ10Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Bathtub CurveFailure Rate FunctionTime tInfantMortalityPhaseWear-outPhaseUseful Life PeriodConstantFailureRate z(t) = λJohnson 1989, page 17311Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5assuming constant z(t)solving the differential equation we get R(t) Q(t)t t12Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5solving 13Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Lecture 6Mean Time to Failure (MTTF)Expected lifetime€ E[X] = xf ( x)dx−∞∞∫€ MTTF = tf (t)dt−∞∞∫Mean Time to Failurewhere f(t) is the failure density function€ f (t) =dQ(t)dt=d(1− R(t))dtPage: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Lecture 6Mean Time to Failure (MTTF)€ udv∫= uv − vdu∫€ MTTF = tQ(t)dtdt0∞∫= − tR(t)dtdt0∞∫= −tR(t) + R(t)dt∫[ ]0∞= R(t)dt0∞∫Now, we can rewriteto get€ d(1− R(t))dt= −dR(t)dtand use integration by partsu and v are both functions of t(recall)Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Lecture 6Mean Time to Failure (MTTF)Thus the expected lifetime isPage: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Reliability of Series SystemAny one component failure causes system failureReliability Block Diagram (RBD)...1 2 n17Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Reliability of Series SystemthusMean time to failure of series system:Thus the MTTF of the series system is much smaller thanthe MTTF of its componentssystem is weakerthan weakestcomponent18Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5Reliability of Parallel SystemAll components must fail to cause system failureReliability Block Diagram (RBD) –assume mutual independence...12n19Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5 X is lifetime of the systemn componentsAssuming all components have exponential distributionwith parameter λ20Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 5 from previous pageProduct law of unreliabilityTrivedi 1982, Page