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chap8_p5 - Probability and Statistics with Reliability...

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Copyright © 2003 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 8 (Part 5) :Continuous Time Markov Chains Reliability Modeling Dept. of Electrical & Computer engineering Duke University Email: [email protected] URL: www.ee.duke.edu/~kst
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Copyright © 2003 by K.S. Trivedi 2 Outline of This Part of Chapter 8 Software Reliability Growth Models Hardware Reliability Models A Safety Model A Security Model A Real-Time System Model
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Copyright © 2003 by K.S. Trivedi 3 Software Reliability Growth Models
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Copyright © 2003 by K.S. Trivedi 4 Failure data is collected during testing Calibrate a reliability growth model using failure data; this model is then used for prediction Many SRGMs exist – NHPP Jelinski Moranda We revisit the above models which we studied in Chapter 5, studying them now as examples of CTMCs. Software Reliability Growth Models
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Copyright © 2003 by K.S. Trivedi 5 0 1 2 ....... Poisson Process The Poisson process,{ N(t) | t 0 }, is a homogeneous CTMC (pure birth type) with state diagram shown below Since failure intensity is time independent, it cannot capture reliability growth. Hence we resort NHPP. λ λ λ
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Copyright © 2003 by K.S. Trivedi 6 Example –Software Reliability Growth Model (NHPP) Consider a Nonhomogenous Poisson process (NHPP) proposed by Goel and Okumoto, as a model of software reliability growth during the testing phase. Note that the Markov property is satisfied and it is an example of a non- homogeneous CTMC Assume that the number of failures N ( t ) occurring in time interval (0 , t ] has a time-dependent failure intensity λ (t). Expected number of software failures experienced (and equated to the number of faults found and fixed) by time t: = = t dx x t N E t m 0 ) ( )] ( [ ) ( λ
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Copyright © 2003 by K.S. Trivedi 7 Example –Software Reliability Growth Model (NHPP) (Contd.) Using previous equation the instantaneous failure intensity can be rewritten by This implies that failure intensity is proportional to expected no. of undetected faults at ‘t’ Many commonly used NHPP software reliability growth models are obtained by choosing different failure intensities λ ( t ), e.g. Goel-Okumoto, Musa-Okumoto model etc.
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Copyright © 2003 by K.S. Trivedi 8 Nature of the failure occurrence rate per fault and the corresponding NHPP model Constant : Goel-Okumoto model Increasing : S-shaped model Generalized Goel-Okumoto model Decreasing : Generalized Goel-Okumoto model Increasing/Decreasing : Log-logistic model Software Reliability Growth Model Finite failure NHPP models
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Copyright © 2003 by K.S. Trivedi 9 Example- Jelinski Moranda Model This model is based on the following assumptions: The number of faults introduced initially into the software is fixed, say, n .
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