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Two-parameter electronic devices quality models/Daugiaparametriu gaminiu kokybes lygio tiesines transformacijos modeliai.

Introduction

Multiparameter mechatronics 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 [1-4]. Electronic device [ED] quality level probabilistic models, expressed through separate parameters probabilistic characteristics, when defect levels by separate parameters are characterized using beta densities described in [5-8]. ED quality level directly transformed probabilistic characteristics models, dependent from different parameter probabilistic characteristics transformations and controlled parameters nomenclature variation, when all condemned ED are repaired immediately after control operation and returned for repeated control with localized repair operation, for fixed second type classification errors by different parameters, has been designed. Linear transformed defect level models, for different parameters [9-12] are found and used for control system functioning modeling.

Initial models

For further analysis, we use defected ED probabilities by i-th parameter [[theta].sub.i] and also good ED probabilities [[eta].sub.i] = 1 - [[theta].sub.i] characteristic models [11-13] used once more, when [[theta].sub.i] ~Be([b.sub.i], [a.sub.i]), [[eta].sub.i]~Be([b.sub.i], [a.sub.i])--beta laws with parameters [a.sub.i], [b.sub.i],

where E[[theta].sub.i] = [[mu].sub.i] = [a.sub.i]/[a.sub.i] + [b.sub.i], E[[eta].sub.i] = 1 - [[mu].sub.i] = [[bar.[mu]].sub.i] - averages,

V[[theta].sub.i] = [[sigma].sup.2.sub.i] = [[mu].sub.i][[bar.[mu]].sub.i]/[a.sub.i] + [b.sub.i] + 1 = V[[eta].sub.i] - dispersion s,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[[phi].sub.i]([[eta].sub.i]) = [g.sub.i](1 (1 - [[eta].sub.i]) - densities,

when

B([a.sub.i], [b.sub.i]) = [GAMMA]([a.sub.i])[GAMMA])[b.sub.i])/[GAMMA]([a.sub.i] + [b.sub.i]) - beta functions,

[GAMMA](z) - gama function.

For all l - parametric ED [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Dispersion by two parameters (i=1, 2) V[[theta].sub.12] = [[sigma].sup.2.sub.12] =[[sigma].sup.2.sub.1] [[bar.[mu]].sup.2.sub.1] + [[sigma].sup.2.sub.1] [[sigma].sup.2.sub.2].

Then link [[sigma].sup.2.sub.12] with [[sigma].sup.2.sub.34] or [[sigma].sup.2.sub.12] with [[sigma].sup.2.sub.3] etc, until [[theta].sup.2] by all l parameters.

Distribution functions [PHI]([eta]), G([theta]) and densities [phi]([eta]), g([theta]) for all electronic device--by [13-15].

Common linear transformation models

Analyzing defect level [[theta].sub.i] linear transformation to defect level [[tau].sub.i] (Fig. 1). Single-stage control K with localized repair operation R is characterized by second type error probability [[beta].sub.Ri]--in operation R.

[FIGURE 1 OMITTED]

If [[beta].sub.Ri] = const. exists and ED repeats ("spins") through these operations, until all are recognized as good (first type errors probability [[alpha].sub.i]=0), then both operations K and R are characterized in generalized probability [[beta].sub.0i] [7]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

If control system is made of k serial stages, then for all system by i-th parameter we get (when every stage is characterized [[beta].sub.0i](j))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

After control K defected ED probability [[tau].sub.i] and good ED probability [[zeta].sub.i]=1-[[tau].sub.i] by i-th parameter are equal [2]

[[tau].sub.i][[beta].sub.0i][[theta].sub.i], [[zeta].sub.i] = [[bar.[beta]].sub.i] + [[beta].sub.0i][[eta].sub.i], [[bar.[beta]].sub.i], = 1 - [[beta].sub.0i]. (3)

Averages and dispersions are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Beta densities [g.sub.i]([[theta].sub.i]), [[phi].sub.i]([[eta].sub.i]) after control by [13-15] transforming to generalized beta densities [h.sub.i] ([[tau].sub.i]) and [[??].sub.i]([[zeta].sub.i]) with [[tau].sub.i], [[zeta].sub.i] variation interval [[tau].sub.i] [member of] (0, [[beta].sub.0i]) and [[zeta].sub.i] [member of]([[bar.[beta]].sub.i], 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

For all ED probabilities [tau] and [zeta] general digital characteristics after control process are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

By two parameters (i=1,2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Later [[sigma].sup.2.sub.[tau]] is found analogically (connecting) like [[sigma].sup.2].

If j-th parameter during control process, j [member of] (1-l), is not checked, then [[beta].sub.0j]=1 and [[tau].sub.j] [equivalent to] [[theta].sub.j], [h.sub.j]([[tau].sub.j]) [equivalent to] [g.sub.j]([[tau].sub.j]). Stochastic value [tau], [zeta] distribution functions H([tau]), [??]([zeta]) and densities h([tau]), [??]([zeta]) are analyzed farther.

Electronic device, when l = 2. Stochastic values [zeta] and [tau] with error probabilities [[beta].sub.01], [[beta].sub.02] varies in interval

[zeta] [member of] ([[bar.[beta]].sub.0],1), [tau] [member of] (0, [[beta].sub.0]); (8)

here [[bar.[beta]].sub.0] = [[bar.[beta]].sub.1] [[bar.[beta]].sub.2], [[beta].sub.0] = 1 - [[bar.[beta]].sub.0], [[bar.[beta]].sub.i] = 1 - [[beta]].sub.0i], i = 1, 2.

Value [zeta] = [[zeta].sub.1] [[zeta].sub.2] possible values range [zeta] [member of] ([[bar.[beta]].sub.1] [[bar.[beta]].sub.2],1) in coordinate system ([[zeta].sub.1], [[zeta].sub.2]) is a rectangular, restricted with lines [[zeta].sub.1] = [[zeta].sub.2] = 1, [[zeta].sub.1] = [[bar.[beta]].sub.1], [[zeta].sub.2] = [[beta].sub.2] (Fig. 2). On the tops of the rectangular [zeta] gets values [[bar.[beta]].sub.1][[bar.[beta]].sub.2], [[bar.[beta]].sub.1], [[bar.[beta]].sub.2] and 1. When [[bar.[beta]].sub.1] [not equal to] [[bar.[beta]].sub.2], indicated [zeta] values on rectangular tops divide the whole [zeta] variation interval to 3 partial intervals dependent on transformation coefficient ratio [[beta].sub.01]/ [[beta].sub.02]. When [[beta].sub.01]<[[beta].sub.02] ([[bar.[beta]].sub.1] > [[bar.[beta]].sub.2]), we get [[bar.[beta]].sub.1] [[bar.[beta]].sub.2] [less than or equal to] [zeta] [less than or equal to] [[bar.[beta]].sub.2], [[bar.[beta]].sub.2] [less than or equal to] [zeta] [less than or equal to] [[bar.[beta]].sub.1], [[bar.[beta]].sub.1] [less than or equal to] [zeta] [less than or equal to] 1.

In partial intervals, when [[beta].sub.02]< [[beta].sub.01], [[bar.[beta]].sub.1] alternates with [[bar.[beta]].sub.2]. In particular case, when [[beta].sub.01] = [[beta].sub.02] = [[beta].sub.0i], we get two partial intervals [[bar.[beta]].sup.2.sub.i] [less than or equal to] [zeta] [less than or equal to] [[bar.[beta]].sub.i], [[bar.[beta]].sub.i] [less than or equal to] [zeta] [less than or equal to] 1.

[FIGURE 2 OMITTED]

In particular partial intervals, function [PHI]([zeta]) models are different, dependent on integrate interval (Fig. 2) Functions [??]([zeta]) are expressed using two-dimensional defined integrals.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Densities [??]([zeta]) are expressed using single-dimensional integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Here [[??].sub.i]([[zeta].sub.t])- as (5), and integration range depends on [[beta].sub.01]/[[beta].sub.02]. Using [y.sub.01] [less than or equal to] [[zeta].sub.i] [less than or equal to] [y.sub.11], [y.sub.02] [less than or equal to] [[zeta].sub.2] [less than or equal to] [y.sub.12] we get:

1 case: [[beta].sub.01] = [[beta].sub.02] = [[beta].sub.0i], [[bar.[beta]].sub.1] = [[bar.[beta]].sub.2] = [[bar.[beta]].sub.i].

1.1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

1.2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2 case: [[beta].sub.01] < [[beta].sub.02], [[bar.[beta]].sub.1] > [[bar.[beta]].sub.2].

2.1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

2.2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

2.3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For further analysis we use only the case when [a.sub.1]=[a.sub.2]=[a.sub.i], [b.sub.1]=[b.sub.2]=[b.sub.i], so the case when [[beta].sub.02]<[[beta].sub.01] equals to the 2nd case. Analyzing the most simple [g.sub.i]([[theta].sub.i]) case [9-15]:

[a.sub.i]=[b.sub.i]=1, [g.sub.i]([[theta].sub.i])=1, g([theta])=-ln(1-[theta]).

We get: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using densities models

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In partial case, when [[beta].sub.02]=1 (the second parameter is not checked), insert [[bar.[beta]].sub.2] = 0 to common models (with [[beta].sub.02]<[[beta].sub.2]) and find h([tau]/[[beta].sub.02] = 1), [tau] [member of] (0,1).

Transformed densities approximations

In the same nonlinear transformation case [1-4], densities h([tau]) are more useful to approximate more simple models for engineering analysis. Using single-parameter model [9-15], we use generalized beta density [h.sub.[sigma]]([tau])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (11)

here

[a.sup.*] = [[mu].sub.[tau]]/[[beta].sub.0] [[mu].sub.[tau]]([[beta].sub.0] - [[mu].sub.[tau]]/[[beta].sup.2.sub.[tau]] - 1], [b.sup.*] = ([[beta].sub.0]/[[mu].sub.[tau]] - 1) [a.sup.*]

and mode

[[tau].sub.M] = [[beta].sub.0] [a.sup.*] - 1/[a.sup.*] + [b.sup.*] - 2.

With maximum [h.sub.M[sigma]] = [h.sub.[sigma]]([[tau].sub.M]) valid formulas:

[[tau].sub.[tau]] = [[beta].sub.0] [a.sup.*]/[a.sup.*] + [b.sup.*], [[sigma].sup.2.sub.[tau]] = [[beta].sub.0] = ([[mu].sub.[tau]] x [b.sup.*]/[a.sub.*] + [b.sup.*])([a.sub.*] + [b.sup.*] + 1). (12)

Distribution function [H.sub.[sigma]]([tau]) found using programmed methods.

In partial case, when one of the parameters is not checked, [h.sub.[sigma]]([tau]) becomes beta density, because [tau][member of](0,1).

Mathematical realizations, l = 2. [a.sub.i] = [b.sub.i] = 1.

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Density values for both cases are shown in table 1 and densities are graphically shown in Fig. 3 and Fig. 4.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusions

1. It is advisable to use beta densities for different parameters defect level characterization in control system modeling, when linear transformation, also as in nonlinear transformation scheme, is used.

2. Linear transformed defect levels densities by different parameters are more simple than in case of linear transformation, but for all device, especially when parameters are increasing, it becomes more complicated because of integral intervals disjunction to separate intervals where density models become different.

3. If one of the parameters is not checked in the control system, approximated density [h.sub.[sigma]]([tau]) from approximated beta density becomes beta density, because [tau] variation interval becomes 0 [less than or equal to] [tau] [less than or equal to] 1, when non-checked parameters increase, density [h.sub.[sigma]]([tau]) and also h([tau]) diverges into initial density g([theta]).

4. It is offered to use this modeling technique for engineering projection of control systems with localized repair places.

Received 2010 04 02

References

[1.] Eidukas D., Kalnius R. Models Quality of Electronics Products // Electronics and Electrical Engineering.--Kaunas: Technologija, 2010--No. 2(98).--P. 29-34.

[2.] Eidukas D., Kalnius R. Models of Quality of Mechatronic Products // Electronics and Electrical Engineering.--Kaunas: Technologija, 2009.--No. 7(95)--P. 59-62.

[3.] Kalnius R., Eidukas D., Reability Model Quality of Informations Systems // Electronics and Electrical Engineering.--Kaunas: Technologija, 2009.--No. 4(92).--P. 13-18.

[4.] Eidukas D., Kalnius R. Multiparameter Electronics Systems Control Probablistic Evaluation // Second International Conference on Advances in Circuits, Electronics and Microelectronics.--October 11-16, 2009.--Silema, Malta.

[5.] Eidukas D., Kalnius R., Vaisvila A. Probability Distribution Transformation in Continuous Information Systems Control // ITI 2007: proceedings of the 29th international conference on Information Technology Interfaces, June 25-28, 2007, Dubrovnik, Croatia / University of Zagreb. University Computing Centre. Zagreb : University of Zagreb, 2007.--P. 609-614.

[6.] Kalnius R., Vaisvila A., Eidukas D. Probability Distribution Transformation in Continuous Production Control // Electronics and Electrical Engineering.--Kaunas: Technologija, 2006.--No. 4(68).--P. 29-34.

[7.] Levitin G. The Universal Generating Function in Reliability Analysis and Optimization.--Springer.--2005.--442 p.

[8.] Brogliato B., Lozano R., Maschke B., Egeland O. Dissipative Systems Analysis and Control Theory and Applications 2nd ed.--Springer.--2006.--576 p.

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[10.] Eidukas D., Kalnius R. Stochastic Models of Quality in Continuous Information Systems // Proceedings of 30 International Conference on Information Technology Interfaces ITI 2008, June 23-26, Cavtat/Dubrovnik, Croatia.--P. 703-708.

[11.] Eidukas D. Stochastic Quality Electronics Systems // Electronics and Electrical Engineering.--Kaunas: Technologija, 2008.--No. 5(85).--P. 41-14.

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[13.] Kruopis J., Vaisvila A., Kalnius R. Mechatronikos gaminiu kokybe. Vilnius: Vilniaus universiteto leidykla, 2005.--518 p.

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D. Eidukas, R. Kalnius

Cathedral of Electronics Engineering, Kaunas University of Technology, Studentu str. 50, LT-51368 Kaunas, Lithuania, phone: +370 37 351389, e-mail: danielius.eidukas@ktu.lt
Table 1. Densities: l = 2

Var.  [[beta].sub.0i]      Density values

1.1   [[beta].sub.01]=     [tau]               0,05    0,15    0,25
      [[beta].sub.02]=3/8  h([tau])           0,365    1,156   2,046
                           [h.sub.[sigma]]    0,240    1,256   2,273
                             ([tau])

1.2   [[beta].sub.01]=1/4  h([tau])           0,410    1,300   2,302
      [[beta].sub.02]=1/2  [h.sub.[sigma]]    0,400    1,345   2,088
                             ([tau])

1.3   [[beta].sub.02]=1    [tau]               0,05     0,1     0,2
      [[beta].sub.01]=1/4  h([tau])           0,205    0,421   0,893
                           [h.sub.[sigma]]    0,376    0,511   0,778
                             ([tau])

Var.  [[beta].sub.0i]      Density values

1.1   [[beta].sub.01]=     [tau]              0,375    0,45     0,5
      [[beta].sub.02]=3/8  h([tau])           3,342    2,433   1,755
                           [h.sub.[sigma]]    2,783    2,438   1,893
                             ([tau])

1.2   [[beta].sub.01]=1/4  h([tau])           2,302    2,302   2,302
      [[beta].sub.02]=1/2  [h.sub.[sigma]]    2,465    2,293   1,971
                             ([tau])

1.3   [[beta].sub.02]=1    [tau]               0,25     0,5     0,9
      [[beta].sub.01]=1/4  h([tau])           1,151    1,151   1,151
                           [h.sub.[sigma]]    0,868    1,172   1,173
                             ([tau])

Var.  [[beta].sub.0i]      Density values

1.1   [[beta].sub.01]=     [tau]               0,55     0,6
      [[beta].sub.02]=3/8  h([tau])           1,006    0,169
                           [h.sub.[sigma]]    1,102    0,152
                             ([tau])

1.2   [[beta].sub.01]=1/4  h([tau])           1,1459   0,516
      [[beta].sub.02]=1/2  [h.sub.[sigma]]    1,445    0,640
                             ([tau])

1.3   [[beta].sub.02]=1    [tau]              0,999      1
      [[beta].sub.01]=1/4  h([tau])           1,151    1,151
                           [h.sub.[sigma]]    0,495      0
                             ([tau])
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Title Annotation:ELECTRONICS/ELEKTRONIKA
Author:Eidukas, D.; Kalnius, R.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:4EXLT
Date:Sep 1, 2010
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