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ECE313.Lecture28

# ECE313.Lecture28 - ECE 313 Probability with Engineering...

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Hazard Rates and System Lifetimes Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 39 Life and Death Previous studies of system reliability Very simple model System is working or has failed P{system is working} = p P{system has failed} = q = 1–p We study system reliability once again What is the probability that a system works for T time units (seconds/hours/days)? What is the average lifetime of a system?
ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 39 Lifetime (death time?) of a system A system is put into operation at time t = 0 Assumption: the system actually is in working condition at t = 0 At some later time, the system fails The time of failure cannot be predicted, and is modeled as a random variable X X denotes the age of the system at death X is the length of life for the system = lifetime of the system = time of death

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ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 39 X is a positive random variable Since the system is working at t = 0, its lifetime X can take on positive values only Model X as a continuous RV with pdf f(u) f(u) = 0 for u ≤ 0 E[ X ] is the average lifetime of the system E[ X ] is also called the mean time to failure (MTTF) mean time before failure (MTBF)
ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 39 How does MTBF make any sense? A system begins to operate at t = 0 As soon as it fails, it is replaced by a brand new system When the substitute fails, it is replaced by yet another new system, … and so on The experiment consists of measuring (or observing) the lifetime of a system Successive trials of the experiment provide independent observations of X

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ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 39 Long-term average ≈ expectation If N systems together provided service for a total of T hours under the “replace the system when it fails” policy, the observed long-term average of the N lifetimes is T/N Long-term average T/N ≈ expectation E[ X ] MTBF is defined to be E[ X ] If service has to be provided for T hours using systems whose MTBF is L, then, on average, expect to need T/L systems
ECE 313 - Lecture 28 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 39 What does f(u) look like? f(u) = 0 for u ≤ 0

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ECE313.Lecture28 - ECE 313 Probability with Engineering...

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