**3.1 Basic elements of OTES stochastic modeling**

According to [13, 14] we introduce composite elements (CE) as OTES with the following elements: (1) basic technical means (TM) and TM being part of serving equipment; (2) staff. For creation unique stochastic model of interoperable OTES processes it necessary to define the data set forecasting CLC indicators at given period of exploitation. This set of indicators includes: (1) coefficient of CE performance at planning for given period of exploitation; (2) level of professional and

medical readiness; (3) investment readiness. Analogously the technical resources corresponding values of professional, staff health and investment resources are defined. Informational-analytical tools (IAT) perform probabilistic analytical modeling of OTES technical means for after sales servicing (ASS).

Solving problems:


3. CP is at user repair (first level) in amount *X*3ð Þ*t* ;

*Basic graph of TM state, professional level and staff health state.*

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

5. CP is at routine maintenance in amount *X*5ð Þ*t* ;

state are given on **Figure 2**.

[14, 15].

**121**

**Figure 2.**

4.CP is written off part and utilized in amount *X*4ð Þ*t* ;

7. CP is in capital repairs at supplier in amount *X*7ð Þ*t* .

**3.3 Stochastic processes and equations for OTES-CALS**

using **Figure 3** for after sales maintenance support system.

are transition intensities from one state to another. Values U1 tj

6.CP is in factory repair (second level) at supplier in amount *X*6ð Þ*t* ;

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

Graphs edges describe CP transition direction at states changes. Parameters pijð Þt

filling process of the store by SP at time moments *tj, j* ¼ 1*,* 2*,* … for providing technical readiness of TM. In general case value and time moment of replenishment are random. These factors must be taken into consideration in LS for OTES-CALS model development. Basic graph of TM state, professional level of staff and health

The developed methodology is unique as for modeling TM and OTES-CALS dynamical staff potentials. Therefore, for constructing unique forecasting costs of staff potential the stochastic model it is necessary to apply developed integrated approach for description and modeling professional level personal health state

Let us consider basic elements of stochastic OTES-CALS modeling and analysis

reflect the discrete

In addition it is possible to give analogous list of problems for modeling staff and medical services.

#### **3.2 Structural schemes of state change streams**

State graph of TM, graph of equipment infrastructure and state staff graph are the basis of stochastic OTES-CALS model. For example, let us consider (**Figure 2**) basic CP state graph. This graph is constructed in accordance with basic LC processes for TM and OTES infrastructure for each CP. Being CP in any states that corresponds the definite costs of various resources and the total production value. This production value must be the object of monitoring and statistical data processing for estimation of probability characteristics (means, probabilistic moments, distributions, etc.). For each CP being the part of aggregate with the help special technologies are sequentially aggregated for final product (FP).

Main modeling stage of usage and service processes consists in probabilistic forecast of main indicators final values: sum of production costs and technical readiness level for OTES-CALS technical means maintenance at given time period. Thus, graph (**Figure 2**) must be supplemented additional graph for calculating integrals cost values at this time period according to [14] recommendations.

Vertex of basic graph maps CP current LC states in two level operational capability:


*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*

**Figure 2.** *Basic graph of TM state, professional level and staff health state.*


Graphs edges describe CP transition direction at states changes. Parameters pijð Þt are transition intensities from one state to another. Values U1 tj reflect the discrete filling process of the store by SP at time moments *tj, j* ¼ 1*,* 2*,* … for providing technical readiness of TM. In general case value and time moment of replenishment are random. These factors must be taken into consideration in LS for OTES-CALS model development. Basic graph of TM state, professional level of staff and health state are given on **Figure 2**.

The developed methodology is unique as for modeling TM and OTES-CALS dynamical staff potentials. Therefore, for constructing unique forecasting costs of staff potential the stochastic model it is necessary to apply developed integrated approach for description and modeling professional level personal health state [14, 15].

#### **3.3 Stochastic processes and equations for OTES-CALS**

Let us consider basic elements of stochastic OTES-CALS modeling and analysis using **Figure 3** for after sales maintenance support system.

medical readiness; (3) investment readiness. Analogously the technical resources corresponding values of professional, staff health and investment resources are defined. Informational-analytical tools (IAT) perform probabilistic analytical

5. modeling streams of TM written off CP after random fault when we have exceeding of fixed number of repairs and/or reaching given resource;

7. modeling of SP delivery processes from stores into exploitation system;

8. cost modeling of after sales CP supply at given period of TM exploitation

9. forming total cost model of TM after sale processes for whole CP list at given

In addition it is possible to give analogous list of problems for modeling staff and

State graph of TM, graph of equipment infrastructure and state staff graph are the basis of stochastic OTES-CALS model. For example, let us consider (**Figure 2**) basic CP state graph. This graph is constructed in accordance with basic LC processes for TM and OTES infrastructure for each CP. Being CP in any states that corresponds the definite costs of various resources and the total production value. This production value must be the object of monitoring and statistical data processing for estimation of probability characteristics (means, probabilistic moments, distributions, etc.). For each CP being the part of aggregate with the help

6.modeling spare parts (SP) accumulation processes on stores;

period of exploitation and providing given level of TM park.

special technologies are sequentially aggregated for final product (FP).

2. CP service able exploiting in aggregate in amount of *X*2ð Þ*t*

Main modeling stage of usage and service processes consists in probabilistic forecast of main indicators final values: sum of production costs and technical readiness level for OTES-CALS technical means maintenance at given time period. Thus, graph (**Figure 2**) must be supplemented additional graph for calculating integrals cost values at this time period according to [14] recommendations. Vertex of basic graph maps CP current LC states in two level operational

modeling of OTES technical means for after sales servicing (ASS).

2. modeling technical means and exploitation processes;

Solving problems:

1. initial data forming;

*Probability, Combinatorics and Control*

3. modeling streams of plan works;

according payments items;

**3.2 Structural schemes of state change streams**

1. CP is on the stock in amount *X*1ð Þ*t* ;

medical services.

capability:

**120**

4.modeling streams of non-plan works;

#### **Figure 3.**

*General state graph for after sales maintenance support system.*

Vertex of graph (**Figure 3**) corresponds 1*,* 2*,* …*, n* states, where of the same type resource be. Let current amount of resources be *X*1ð Þ*t ,* …*, Xn*ð Þ*t* ,

*X t*ðÞ¼ ½ � *X t*ð Þ<sup>1</sup> *;* …*;X t*ð Þ*<sup>n</sup> <sup>T</sup>*. Graphs edges corresponds transition of resources from state to state *h k*ð Þ *; h* ¼ 1*;* 2*;* …*; n; k* 6¼ *n* at random time moments forming Poisson streams transition events properly. Find some of resources states corresponding in queuing system for repeatedly recovery. General capacity is defined by the number of channels and being essentially nonlinear intensity<sup>4</sup> function depending on the amount of input resource units. This fact is mapped by *ρkh*ð Þ *X; t* . In general case this nonlinear function has vector argument.

Stochastic equations and corresponding algorithms of analytical modeling for mathematical expectation *m* ¼ *m t*ð Þ, covariance matrix *θ* ¼ *θ*ð Þ*t* and matrix of covariance functions *K t*ð Þ <sup>1</sup>*; t*<sup>2</sup> are as follows [5]:

$$d\mathcal{X} = \varrho(\mathcal{X}, t)dt + \int\_{\mathcal{R}\_0^\ell} c(\mathcal{X}, v, t)\mathcal{P}(dv, dt) = \varrho(\mathcal{X}, t)dt + \sum\_{k, h = 0}^n \int\_{\mathcal{R}\_0^\ell} \mathcal{S}\_{kh}^T(v\_{kh}, \mathcal{X}, t)\mathcal{P}(dv\_{kh}, dt), \; \mathcal{X}(t\_0) = \mathcal{X}\_0, \; t \ge 0$$

(1)

Equation (2) are approximate and valid at conditions of normal (Gaussian) approximation method. For raising the accuracy of analytical modeling and analysis methods of probabilistic distributions (moments, semi-invariants, coefficients of

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

The developed ordinary differential equations with initial conditions may be

Following [16] let us consider informational-analytical tools (IAT) for aircraft vehicle park modeling and control by technical-economic efficiency criteria after

According to contract supplier creates NDB according to standards DEFSTAN 00-600, S1000D, S2000 M and specifications S1000D, S2000 M. For IAT acceleration there are designed emulated DB (DBE) in the form of additional tables. Information from NDB automatically comes into DBE. These data characterize:

orthogonal expansions of densities) [2, 3] may be used.

used for basic risk problems of systems engineering [6].

sales maintenance products (ASMP).

• forecasting processes block for ASMP;

• catalog of codified items supply.

IAT modular includes:

• operative DB;

**Table 1.**

*Forming the structure matrix S.*

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

*3.4.1 Normative DB (NDB)*

**123**

**3.4 Modeling and analysis of aircraft vehicles park life cycle**

• normative data base (DB) of passported aggregates;

• interrepair resource of final MP (FMP) and CP;

• mean duration of capital repair (CR) of FMP and CP;

• optimization block of delivery programs (annual application);

$$\begin{aligned} \dot{m} &= \mathbf{M}[\rho(\mathbf{X}, t) + \mathbb{S}\rho], \ m(t\_0) = m\_0, \\ \dot{\theta} &= \mathbf{M}\{ [\rho(\mathbf{X}, t) + \mathbb{S}\rho] \mathbf{X}^{0T} + \mathbf{X}^0 \left[ \rho^T(\mathbf{X}, t) + \rho^T \mathbf{S}^T \right] + \mathbf{S} \operatorname{diag}(\rho) \mathbf{S}^T \}, \ \theta(t\_0) = \theta\_0, \\ \frac{\partial \mathcal{K}(t\_1, t\_2)}{\partial t\_2} &= \mathbf{M}\{ \mathbf{X}\_1^0 \left[ \rho^T(\mathbf{X}\_2, t\_2) + \rho^T(\mathbf{X}\_2, t\_2) \mathbf{S}^T \right], \ \mathcal{K}(t\_1, t\_1) = \theta(t\_1). \end{aligned} \tag{2}$$

Hence M is symbol of mathematical expectation; *P* is symbol of probabilistic measure; *<sup>X</sup>* and *<sup>X</sup>*<sup>0</sup> are noncentered and centered state vectors; *<sup>φ</sup> Xt* ð Þ *; <sup>t</sup>* is in general vector nonlinear function reflecting current value of OTES-CALS efficiency criterion; *S v*ð Þ structure matrix of Poisson streams of resources (production) with values *<sup>v</sup>* according to state graph; *<sup>S</sup>и*ð Þ¼ *<sup>v</sup> suk*1ð Þ *vuk* …*sukng* ð Þ *vuk* h i is *<sup>u</sup>*-row of matrix *S v*ð Þ; *<sup>ρ</sup>* is the intensity vector.

Forming the structure matrix *S* ¼ ½ � *Skh* for OTES-CALS (**Figure 2**) is shown in **Table 1**.

*Column 1*. Transition direction in system graph. Total amount of rows is equal to amount transitions—*m*; amount of columns is equal to states amounts—*n* (graph vertex); two servicing columns: first and *n* þ 2.

*Column 2—n* þ 1. Values of intensities of transition—*ρkh*.

So columns from 2 till *n* þ 1 presents *m* � *n* matrix *S* and column under number ð Þ *n* þ 2 is *m* dimensional vector with intensity vector *ρ*.

<sup>4</sup> Stream of random events intensity is being mean number of events per time unit.

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*


Vertex of graph (**Figure 3**) corresponds 1*,* 2*,* …*, n* states, where of the same type

*X t*ðÞ¼ ½ � *X t*ð Þ<sup>1</sup> *;* …*;X t*ð Þ*<sup>n</sup> <sup>T</sup>*. Graphs edges corresponds transition of resources from state to state *h k*ð Þ *; h* ¼ 1*;* 2*;* …*; n; k* 6¼ *n* at random time moments forming Poisson streams transition events properly. Find some of resources states corresponding in queuing system for repeatedly recovery. General capacity is defined by the number of channels and being essentially nonlinear intensity<sup>4</sup> function depending on the amount of input resource units. This fact is mapped by *ρkh*ð Þ *X; t* . In general case this

Stochastic equations and corresponding algorithms of analytical modeling for mathematical expectation *m* ¼ *m t*ð Þ, covariance matrix *θ* ¼ *θ*ð Þ*t* and matrix of

*<sup>θ</sup>* <sup>¼</sup> <sup>M</sup> ½ � *<sup>φ</sup>*ð Þþ *<sup>X</sup>; <sup>t</sup> <sup>S</sup><sup>ρ</sup> <sup>X</sup>*<sup>0</sup>*<sup>T</sup>* <sup>þ</sup> *<sup>X</sup>*<sup>0</sup> *<sup>φ</sup><sup>T</sup>*ð Þþ *<sup>X</sup>; <sup>t</sup> <sup>ρ</sup>TS<sup>Т</sup>* � � <sup>þ</sup> *<sup>S</sup>* diagð Þ*<sup>ρ</sup> <sup>S</sup><sup>Т</sup>* � �*, <sup>θ</sup>*ð Þ¼ *<sup>t</sup>*<sup>0</sup> *<sup>θ</sup>*0*,*

<sup>1</sup> *<sup>φ</sup><sup>T</sup>*ð Þþ *<sup>X</sup>*2*; <sup>t</sup>*<sup>2</sup> *<sup>ρ</sup><sup>T</sup>*ð Þ *<sup>X</sup>*2*; <sup>t</sup>*<sup>2</sup> *<sup>S</sup><sup>Т</sup>* � �*,K t*ð Þ¼ <sup>1</sup>*; <sup>t</sup>*<sup>1</sup> *<sup>θ</sup>*ð Þ *<sup>t</sup>*<sup>1</sup> *:*

Hence M is symbol of mathematical expectation; *P* is symbol of probabilistic measure; *<sup>X</sup>* and *<sup>X</sup>*<sup>0</sup> are noncentered and centered state vectors; *<sup>φ</sup> Xt* ð Þ *; <sup>t</sup>* is in general vector nonlinear function reflecting current value of OTES-CALS efficiency criterion; *S v*ð Þ structure matrix of Poisson streams of resources (production) with values

h i

Forming the structure matrix *S* ¼ ½ � *Skh* for OTES-CALS (**Figure 2**) is shown in

*Column 1*. Transition direction in system graph. Total amount of rows is equal to amount transitions—*m*; amount of columns is equal to states amounts—*n* (graph

So columns from 2 till *n* þ 1 presents *m* � *n* matrix *S* and column under number

*k, <sup>h</sup>*¼<sup>0</sup>

ð

*ST*

*kh*ð Þ *vkh;X; t P dv* ð Þ *kh; dt ,Xt*ð Þ¼ <sup>0</sup> *X*0*,*

is *u*-row of matrix *S v*ð Þ; *ρ* is

(1)

(2)

*Rq* 0

resource be. Let current amount of resources be *X*1ð Þ*t ,* …*, Xn*ð Þ*t* ,

*c X*ð Þ *; <sup>v</sup>; <sup>t</sup> P dv* ð Þ¼ *; dt <sup>φ</sup>*ð Þ *<sup>X</sup>; <sup>t</sup> dt* <sup>þ</sup> <sup>X</sup>*<sup>n</sup>*

nonlinear function has vector argument.

*m*\_ ¼ M½ � *φ*ð Þþ *X; t Sρ , mt*ð Þ¼ <sup>0</sup> *m*0*,*

*v* according to state graph; *Sи*ð Þ¼ *v suk*1ð Þ *vuk* …*sukng* ð Þ *vuk*

*Column 2—n* þ 1. Values of intensities of transition—*ρkh*.

<sup>4</sup> Stream of random events intensity is being mean number of events per time unit.

ð Þ *n* þ 2 is *m* dimensional vector with intensity vector *ρ*.

vertex); two servicing columns: first and *n* þ 2.

ð

*Rq* 0

<sup>¼</sup> <sup>M</sup>f*X*<sup>0</sup>

*dX* ¼ *φ*ð Þ *X; t dt* þ

*<sup>∂</sup> K t*ð Þ <sup>1</sup>*; <sup>t</sup>*<sup>2</sup> *∂t*2

the intensity vector.

**Table 1**.

**122**

\_

**Figure 3.**

covariance functions *K t*ð Þ <sup>1</sup>*; t*<sup>2</sup> are as follows [5]:

*General state graph for after sales maintenance support system.*

*Probability, Combinatorics and Control*

Equation (2) are approximate and valid at conditions of normal (Gaussian) approximation method. For raising the accuracy of analytical modeling and analysis methods of probabilistic distributions (moments, semi-invariants, coefficients of orthogonal expansions of densities) [2, 3] may be used.

The developed ordinary differential equations with initial conditions may be used for basic risk problems of systems engineering [6].
