Quality Level Linear Models Electronic Systems

Reliability and efficiency of electronics systems (ES) are described in [1–6]. The continuous operational control main probabilities characteristics modeling techniques for multilevel electronics systems is analyzed in [1–3]. These are useful when separate independent parameters defect level probabilities distributions are set (known) for chosen (selected) control schematics place. Denied electronics systems streams goes back to production process for regeneration and electronics systems classification rules in different control levels are similar, when electronics system classification first and second type errors are not denied by different parameters (good is denied or bad is accepted as good). Offered to use approximated models instead of exact whole electronics system defect level probabilities density transformed models because of complicated process of integration. Models of control quality are described in [4, 5]. Here a method is offered for synthesis of stochastic distributions of defectivity levels of multiparametric ES with interindependent parameters. This synthesis can be performed in groups of parameters or for entire product according to known distributions of defectivity levels of separate parameters. For practical applications it is advisable to differentiate average defectivity levels of separate parameters according to selected defectivity level of entire product, when ratio between defectivity levels in separate groups is selected or according to needed dispersion of parameters (selected variation coefficient).

The continuous operational control main probabilities characteristics modeling techniques for multilevel electronics systems is analyzed in [1][2][3].These are useful when separate independent parameters defect level probabilities distributions are set (known) for chosen (selected) control schematics place.Denied electronics systems streams goes back to production process for regeneration and electronics systems classification rules in different control levels are similar, when electronics system classification first and second type errors are not denied by different parameters (good is denied or bad is accepted as good).Offered to use approximated models instead of exact whole electronics system defect level probabilities density transformed models because of complicated process of integration.
Models of control quality are described in [4,5].Here a method is offered for synthesis of stochastic distributions of defectivity levels of multiparametric ES with interindependent parameters.This synthesis can be performed in groups of parameters or for entire product according to known distributions of defectivity levels of separate parameters.For practical applications it is advisable to differentiate average defectivity levels of separate parameters according to selected defectivity level of entire product, when ratio between defectivity levels in separate groups is selected or according to needed dispersion of parameters (selected variation coefficient).

Initial models
Multiparameter ES product defect level nonlinear transformation models in continuous quality control, evaluating separate parameter and the whole product probability characteristics also first and second type errors, when products are classified, described in [1,6].Electronic systems [ES] quality level probabilistic models, expressed by separate parameters probabilistic characteristics, when defect levels by separate parameters are characterized using beta densities.
Electronic tool quality level directly transformed probabilistic characteristics models from separate parameters probabilistic characteristics transformations and controlled parameters nomenclature variation are described in [1].When ES are repaired immediately after control operation and returned for repeated control with localized repair operation [2] for fixed second type classification errors by different parameters.For analysis needs we will use defect ES probabilities by i-th parameter θi and good product probabilities ηi=1-θi characteristic models [2] repeatedly, when θi~Be(bi, ai), ηi~Be(bi, ai)beta laws with parameters ai, bi: -averages dispersions densities: when For all l -electronic tool parameter Dispersion by two parameters (i=1, 2)

Common linear transformation models
Analyzing defect level  i linear transformation to defect level  i (Fig. 1).Single-stage control K with localized repair operation R is characterized by second type error probability  Ri -in operation R [1][2][3].If  Ri =const.exists and ES repeats ("spins") through these operations, until all are recognized as good (first type errors probability  i 0), then both operations K and R are characterized in generalized probability  0i [1,2]   If control system is made of k serial stages, then for all system by i-th parameter we get (when every stage is After control K defected ES probability  i and good ES probability Averages and dispersions are: after control by [1] transforming to generalized beta densities For all ES probabilities  and  general digital characteristics after control process are By two parameters (i=1, 2)  is found analogically (connecting) like  2 .
If j-th parameter during control process, j(1-ℓ), is not checked, then  0j 1 and

Transformed densities approximations
In the same nonlinear transformation case [1], densities h() are more useful to approximate more simple models for engineering analysis.Using single-parameter model [2], we use generalized beta density h  () With maximum h M h  ( M ) valid formulas Distribution function H  () found using programmed methods.
Density values for both cases are shown in Table 1 and densities are graphically shown in Fig. 2 and Fig. 3. Fig.
Mathematical realizations (l+3), when  , Density values are shown in Table 2 and densities are graphically shown in Fig. 3.