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This model is due to Stein and Dattero (1984). 34 TAXONOMY FOR WEIBULL MODELS Model V-3 The distribution function is given by " FðtÞ ¼ 1 À exp À t X # " rðiÞ ¼ 1 À exp À i¼1 t X # cibÀ1 t ¼ 0; 1; 2; . . ð2:54Þ i¼1 This model is due to Padgett and Spurrier (1985). 8 TYPE VI MODELS: MULTIVARIATE WEIBULL MODELS This group of models is multivariate extensions of the univariate case so that the distribution function is given by an n-dimensional distribution function Fðt1 ; t2 ; . . ; tn Þ. The models can be grouped into two groups based on basis of the extension.

Propose four different models (involving a Weibull and a non-Weibull distribution) belonging to Type III(a)-3. Give the details of the model structure. 5. 4 with Type III(a)-3 replaced by Type III(b)-3. 6. 4 with Type III(a)-3 replaced by Type III(c)-3. 7. 5? If so, indicate the possible relationship between parameters of the model and the relevant factors. 1 Taxonomy for Weibull Models. 8. Describe three real-world problems where one or more variables can be modeled by a Type V model. 9. For many components the degradation and failure is a function of age and usage.

The survivor function is given by " ðtÞ ¼ G  ! y0 t 0 aij aij exp À yij ti þ r i¼1 j¼1 r X k Y t>0 ð2:68Þ where t0 ¼ maxðt1 ; . . ; tr Þ. 9 TYPE VII MODELS: STOCHASTIC POINT PROCESS MODELS These are stochastic point process models with some link to the standard Weibull model. 1 Weibull Intensity Function Models Model VII(a)-1: Power Law Process This is a point process model with the intensity function given by lðtÞ ¼ b tbÀ1 ab ð2:69Þ This is identical to the hazard function for the standard Weibull model.

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