4. System functional stability management

In general, IS periodic control involves performing a number of standard procedures:


Choosing a strategy and the number of control/backup points helps manage the system's stability, integrity, and accessibility levels [12]. For example, considering the earlier procedures, one can define the system availability ratio (operational availability factor [13]):

$$R = \left(\frac{t}{\left(t + M\_{n\_r}'\right)}\right) \left(p + (1 - p)\left(P\_{n\_\epsilon}' + \left(1 - P\_{n\_\epsilon}'\right)P\_{n\_\mathbb{P}}'\right)\right) \tag{25}$$

where t is the task solution time, Mr nr is the mathematical expectation of the program operation delay time in case of nr being the backup points, p is the SW error-free performance (SW efficiency), Pe ne is the error prevention probability in case of ne environment control points, and P<sup>p</sup> np is the error detection probability in case of np SW control points.

In the above formula, p is the SW failure-free performance probability; error prevention probability—P<sup>e</sup> <sup>n</sup> <sup>=</sup> ð Þ ne <sup>þ</sup> <sup>1</sup> <sup>∙</sup>F^z; error detection probability—Pp <sup>n</sup> <sup>=</sup> np <sup>þ</sup> <sup>1</sup> � �∙Fz^; availability

$$\text{factor } R' = \left(\frac{t}{\left(t + M\_{n\_r}\right)}\right); M\_{n\_r}' = t + (n\_r + 1)M\_{\bar{z}}.$$

The constraining factor (Eq. (25)) is the SW dependability cost index defined as the cost of standard procedures:

$$\mathbb{C}\left(n\_{\varepsilon}, n\_{p}, n\_{r}\right) = \mathbb{C}\_{\varepsilon} \left. n\_{\varepsilon} + \mathbb{C}\_{p} \left. n\_{p} + \mathbb{C}\_{r} \left. n\_{r} \right| \right. \tag{26}$$

3.1. Find a set of search interval values

<sup>0</sup>}, where Ll

}, where n<sup>1</sup>

4. If R0 < Rrg, perform the following operations:

4.1. Find a set of search interval values

<sup>0</sup>}, where Ll

<sup>i</sup> � R0 - /(Ci L0

7. Find a set of search interval values

<sup>τ</sup>�1}, where Li

}, where n<sup>τ</sup>þ<sup>1</sup>

8. If R0 > Rrg, perform the following operations:

<sup>τ</sup> <sup>¼</sup> <sup>L</sup><sup>i</sup> <sup>τ</sup>þ<sup>1</sup>=<sup>2</sup> .

8.1. Obtain a set of possible values of the number of standard procedures

<sup>i</sup> � <sup>L</sup><sup>τ</sup> i =2 .

, where i is the index of the standard procedure conforming to the minimiza-

<sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>τ</sup>

8.2. Find another set of values of the number of standard procedures

5. Increase the iteration index

}, where n<sup>1</sup>

<sup>0</sup> = nl <sup>u</sup> - n<sup>l</sup> <sup>0</sup>; n<sup>l</sup>

i =2 

<sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup>

<sup>0</sup> = nl <sup>0</sup> - nl lw; nl

<sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup>

3.2. Obtain the set of possible values of the number of standard procedures

<sup>i</sup> � <sup>L</sup><sup>0</sup> i =2: 

)), where i,j,k E {e,p,r}.

4.2. Obtain three sets of possible values of the number of standard procedures

)), where i,j,k E {e,p,r}.

<sup>i</sup> <sup>þ</sup> <sup>L</sup><sup>0</sup> i =2 .

4.3. Find another set of values of the number of standard procedures

3.3. Find another set of values of the number of standard procedures

lw is the lower boundary <sup>n</sup><sup>l</sup>

, where i is the index of the standard procedure conforming to the minimization

<sup>u</sup> is the upper boundary nl

, where i is the index of the standard procedure conforming to the maximization

.

http://dx.doi.org/10.5772/intechopen.75232

227

Periodic Monitoring and Recovery of Resources in Information Systems

.

L<sup>0</sup> = { Li

N<sup>1</sup> <sup>i</sup> = {n<sup>1</sup> i , n<sup>0</sup> j , n<sup>0</sup> k

N1= N<sup>1</sup> i

L0= { Li

N1= N<sup>1</sup> i

condition:

maxðR N<sup>1</sup>

τ ¼ τ þ 1.

R<sup>τ</sup> ¼ R Nð Þ<sup>τ</sup> .

Lτ= { Li

N<sup>τ</sup>

6. Calculate the value R

<sup>τ</sup>, <sup>L</sup><sup>j</sup> <sup>τ</sup>�<sup>1</sup>, Lk

<sup>i</sup> = {n<sup>τ</sup>þ<sup>1</sup> i ,n<sup>τ</sup> j , n<sup>τ</sup> k

N<sup>τ</sup> ¼ N<sup>τ</sup>þ<sup>1</sup>

tion condition:

i

N<sup>0</sup> <sup>i</sup> = {n<sup>1</sup> i , n<sup>0</sup> j , n<sup>0</sup> k

condition:

minðR<sup>0</sup> - R(N<sup>1</sup>

<sup>0</sup>, <sup>L</sup><sup>j</sup> <sup>0</sup>, Lk

i )/(Ci L<sup>0</sup> i =2 

<sup>0</sup>, Lj <sup>0</sup>, <sup>L</sup><sup>k</sup>

where Cp is the cost of one SW error detection control event; Ce is the cost of one environment control event for error prevention; and Cr is the cost of setting one checkpoint.

The SW operation security management task comes down to optimizing the availability factor, with the constraining factor (Eq. (26)). The following two optimization tasks can be defined:

1. Direct task: Using partial redundancy of the number of standard procedures, ensure that the SW security index is at least equal to the specified index Rrg, with a minimum possible cost of standard procedures in general, that is

$$\min \left\{ \mathbb{C}(n\_{\epsilon}, n\_{p}, n\_{r}) \,|\, \mathbb{R}\left(n\_{\epsilon}, n\_{p}, n\_{r}\right) > R\_{r\xi} \right\} \tag{27}$$

2. Reverse task: Using partial redundancy of the number of standard procedures, ensure that the cost of all standard procedures does not exceed the specified value Crg, with a maximum possible SW reliability index, that is

$$\max\left\{\ R\left(n\_{\epsilon}, n\_{p}, n\_{r}\right) \mid \mathbb{C}\left(n\_{\epsilon}, n\_{p}, n\_{r}\right) < \mathbb{C}\_{\text{rg}}\right\}\tag{28}$$

#### 4.1. Direct optimization task

Analysis (Eq. (25)) showed that R is a nondifferentiable monotone increasing function that is strictly convex upward.

In order to solve optimization tasks, therefore, it is advisable to employ sequential search methods.

Let us assume the value of incremental difference ΔR ,n ð <sup>r</sup>)/ΔC to be an enumeration criterion. Let us determine an enumeration step in accordance with the dichotomy rule. In this case, the computational scheme for solving the direct task can be presented as follows:

1. Define a set of initial values of the number of standard procedures:

N<sup>o</sup> = {n<sup>0</sup> i , n<sup>0</sup> j , n<sup>0</sup> k }, where i,j,k E {e,p,r}.

2. Calculate the initial value R:

R<sup>0</sup> = R(No).

3. If R0 > Rrg, perform the following operations:

3.1. Find a set of search interval values

factor <sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup>

standard procedures:

<sup>t</sup>þM<sup>r</sup> nr ð Þ 

226 Probabilistic Modeling in System Engineering

; Mr

cost of standard procedures in general, that is

mum possible SW reliability index, that is

4.1. Direct optimization task

strictly convex upward.

methods.

N<sup>o</sup> = {n<sup>0</sup> i , n<sup>0</sup> j , n<sup>0</sup> k

R<sup>0</sup> = R(No).

2. Calculate the initial value R:

nr ¼ t þ ð Þ nr þ 1 Mz^.

C ne; np; nr

control event for error prevention; and Cr is the cost of setting one checkpoint.

min C ne; np; nr

max R ne; np; nr

The constraining factor (Eq. (25)) is the SW dependability cost index defined as the cost of

where Cp is the cost of one SW error detection control event; Ce is the cost of one environment

The SW operation security management task comes down to optimizing the availability factor, with the constraining factor (Eq. (26)). The following two optimization tasks can be defined: 1. Direct task: Using partial redundancy of the number of standard procedures, ensure that the SW security index is at least equal to the specified index Rrg, with a minimum possible

<sup>j</sup> R ne; np; nr

2. Reverse task: Using partial redundancy of the number of standard procedures, ensure that the cost of all standard procedures does not exceed the specified value Crg, with a maxi-

<sup>j</sup> <sup>C</sup> ne; np; nr

Analysis (Eq. (25)) showed that R is a nondifferentiable monotone increasing function that is

In order to solve optimization tasks, therefore, it is advisable to employ sequential search

Let us assume the value of incremental difference ΔR ,n ð <sup>r</sup>)/ΔC to be an enumeration criterion. Let us determine an enumeration step in accordance with the dichotomy rule. In this case, the

computational scheme for solving the direct task can be presented as follows:

1. Define a set of initial values of the number of standard procedures:

}, where i,j,k E {e,p,r}.

3. If R0 > Rrg, perform the following operations:

> Rrg

< Crg

(27)

(28)

<sup>¼</sup> Ce ne <sup>þ</sup> Cp np <sup>þ</sup> Cr nr (26)

L<sup>0</sup> = { Li <sup>0</sup>, Lj <sup>0</sup>, <sup>L</sup><sup>k</sup> <sup>0</sup>}, where Ll <sup>0</sup> = nl <sup>0</sup> - nl lw; nl lw is the lower boundary <sup>n</sup><sup>l</sup> .

3.2. Obtain the set of possible values of the number of standard procedures

N<sup>1</sup> <sup>i</sup> = {n<sup>1</sup> i , n<sup>0</sup> j , n<sup>0</sup> k }, where n<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup> <sup>i</sup> � <sup>L</sup><sup>0</sup> i =2: 

3.3. Find another set of values of the number of standard procedures

N1= N<sup>1</sup> i , where i is the index of the standard procedure conforming to the minimization condition:

minðR<sup>0</sup> - R(N<sup>1</sup> i )/(Ci L<sup>0</sup> i =2 )), where i,j,k E {e,p,r}.

4. If R0 < Rrg, perform the following operations:

4.1. Find a set of search interval values

L0= { Li <sup>0</sup>, <sup>L</sup><sup>j</sup> <sup>0</sup>, Lk <sup>0</sup>}, where Ll <sup>0</sup> = nl <sup>u</sup> - n<sup>l</sup> <sup>0</sup>; n<sup>l</sup> <sup>u</sup> is the upper boundary nl .

4.2. Obtain three sets of possible values of the number of standard procedures

N<sup>0</sup> <sup>i</sup> = {n<sup>1</sup> i , n<sup>0</sup> j , n<sup>0</sup> k }, where n<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup> <sup>i</sup> <sup>þ</sup> <sup>L</sup><sup>0</sup> i =2 .

4.3. Find another set of values of the number of standard procedures

N1= N<sup>1</sup> i , where i is the index of the standard procedure conforming to the maximization condition:

maxðR N<sup>1</sup> <sup>i</sup> � R0 - /(Ci L0 i =2 )), where i,j,k E {e,p,r}.

5. Increase the iteration index

$$
\tau = \tau + 1.
$$

6. Calculate the value R

R<sup>τ</sup> ¼ R Nð Þ<sup>τ</sup> .

7. Find a set of search interval values

Lτ= { Li <sup>τ</sup>, <sup>L</sup><sup>j</sup> <sup>τ</sup>�<sup>1</sup>, Lk <sup>τ</sup>�1}, where Li <sup>τ</sup> <sup>¼</sup> <sup>L</sup><sup>i</sup> <sup>τ</sup>þ<sup>1</sup>=<sup>2</sup> .

8. If R0 > Rrg, perform the following operations:

8.1. Obtain a set of possible values of the number of standard procedures

N<sup>τ</sup> <sup>i</sup> = {n<sup>τ</sup>þ<sup>1</sup> i ,n<sup>τ</sup> j , n<sup>τ</sup> k }, where n<sup>τ</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>τ</sup> <sup>i</sup> � <sup>L</sup><sup>τ</sup> i =2 .

8.2. Find another set of values of the number of standard procedures

N<sup>τ</sup> ¼ N<sup>τ</sup>þ<sup>1</sup> i , where i is the index of the standard procedure conforming to the minimization condition:

minðR0 - R(N<sup>τ</sup>þ<sup>1</sup> i )/(Ci L<sup>i</sup> <sup>τ</sup>=<sup>2</sup> � � � �)), where i,j,k E {e,p,r}.

If N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>�1, withdraw from the procedure.

9. Otherwise (R<sup>τ</sup> < Rrg), perform the following operations:

9.1. Obtain the set of possible values of the number of standard procedures:

$$\mathbf{N}\_{\tau+1}{}^{i} = \{n\_{\tau+1}{}^{i}, n\_{\tau}{}^{j}, n\_{\tau}{}^{k}\}, \text{ where } n\_{\tau+1}{}^{i} = n\_{\tau}{}^{i} + ||L\_{\tau}{}^{i}/2||.$$

9.2. Find another set of values of the number of standard procedures

N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>þ<sup>1</sup> i , where i is the index of the standard procedure conforming to the maximization condition:

R Nð Þ¼ <sup>τ</sup>þ<sup>1</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>

and the initial value n2 <sup>¼</sup> <sup>n</sup><sup>2</sup>

R Nð Þ¼ <sup>τ</sup>�<sup>1</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>

n ¼ Crq=C.

5. Conclusion

restricted Bernoulli's flows [11].

<sup>τ</sup> þ 1Þ,

τ:

7. If R<sup>τ</sup>�<sup>1</sup> ¼ max Rð Þ <sup>τ</sup>�<sup>1</sup>;Rτ; R<sup>τ</sup>þ<sup>1</sup> , let τ = �1 and proceed to item 4.

value in accordance with a distribution law, and so on.

stochastic external factors on the system operation process.

5. If R<sup>τ</sup> ¼ max Rð Þ <sup>τ</sup>�<sup>1</sup>;Rτ;R<sup>τ</sup>þ<sup>1</sup> , withdraw from the computation scheme;

4. Calculate the maximum value R by directed enumeration n2 at the fixed value n<sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>

6. If R<sup>τ</sup>þ<sup>1</sup> ¼ max Rð Þ <sup>τ</sup>�<sup>1</sup>;Rτ; R<sup>τ</sup>þ<sup>1</sup> , let τ ¼ τ þ 1, perform item 3 and proceed to item 5;

In practice, there may be a task of calculating indices not for the SW functional stability (dependability) system in general but for a part thereof (checkpoint or error prevention/detection mechanisms). This means a transition from multidimensional to unidimensional task interpretation, which helps substantially simplify computational procedures. Thus, solving a partial reverse optimization task boils down to a single calculation of a specific index when

When solving a direct task, the effectiveness of the computation scheme can be additionally improved by adjusting the variable change interval, for example, by defining the next variable

Thus, in this section, we have considered stochastic and deterministic models of SW periodic monitoring and backup, which allow for time and computational/data resource constraints. Representing monitoring and backup points as a restricted Bernoulli's flow helps obtain random time intervals with the preset number thereof and, accordingly, allow for the effect of

Comparative analysis of stochastic and deterministic models showed the former's effectiveness with a small number of control and backup points. Therefore, when managing IS stability by numerical methods, it is possible to identify preferred models (stochastic, deterministic, or combined) which enhance IS functional stability. This gives an effect akin to introducing structure redundancy, that is, a special type of redundancy (stochastic), the use of which is unlikely to result in higher costs. The application of stochastic models in engineering systems can be facilitated by using a random-impulse generator that forms random-

A similar approach was taken as a basis to solve the problem of efficiency assessment of the diagnostic mechanism for data array failures. Apart from the IS resource control and backup domain, the above-stated results can be of use in assessing the cost-effectiveness of control measures and mechanisms being implemented in various engineering and management systems. For better use of stochastic models, it is possible to use a random pulse generator (e.g., [14]).

<sup>τ</sup> � 1Þ, where N<sup>τ</sup>�<sup>1</sup> meets the normalization requirement;

Periodic Monitoring and Recovery of Resources in Information Systems

http://dx.doi.org/10.5772/intechopen.75232

<sup>τ</sup> � 1

229

where N<sup>τ</sup>þ<sup>1</sup> meets the normalization requirement;

maxðR N<sup>τ</sup>þ<sup>1</sup> <sup>i</sup> � <sup>R</sup><sup>0</sup> � �=(Ci Li <sup>τ</sup>=<sup>2</sup> � � � �)), where i,j,k E {e,p,r}.

9.3. If N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>�1, record the value R<sup>τ</sup>þ<sup>1</sup> = R (N<sup>τ</sup>�1) and withdraw from the procedure.

10. Proceed to item 5.

The period of this computation scheme can be reduced as follows:


#### 4.2. Reverse optimization task

The reverse task can be solved using the branch-and-bound procedure. In this case, the computation scheme will be as follows:

1. Specify a cost-ordered set N of initial values of the number of standard procedures

Nτ= {n<sup>τ</sup> 1,n<sup>τ</sup> <sup>2</sup>, n<sup>τ</sup> 3}, C<sup>1</sup> ≥C<sup>2</sup> ≥C<sup>3</sup> ,

which meets the normalization requirement:

0 ≤Crq - P<sup>3</sup> <sup>i</sup>¼<sup>1</sup> <sup>n</sup> <sup>C</sup><sup>i</sup> � � <sup>≤</sup> Ci ; i ¼ 1; 3,

where τ is the ramification index;

2. Calculate the maximum value R by directed enumeration n2at the fixed value n1 <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>r</sup> and the initial value n<sup>2</sup> <sup>¼</sup> n2 r :

R Nð Þ¼ <sup>τ</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> τÞ,

where N<sup>τ</sup> meets the normalization requirement;

3. Calculate the maximum value R by directed enumeration n2at the fixed value n1 <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>τ</sup> þ 1 and the initial value n2 <sup>¼</sup> <sup>n</sup><sup>2</sup> τ:

R Nð Þ¼ <sup>τ</sup>þ<sup>1</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>τ</sup> þ 1Þ,

minðR0 - R(N<sup>τ</sup>þ<sup>1</sup>

228 Probabilistic Modeling in System Engineering

<sup>i</sup> = {n<sup>τ</sup>þ<sup>1</sup> i ,n<sup>τ</sup> j , n<sup>τ</sup>

i

<sup>i</sup> � <sup>R</sup><sup>0</sup> � �=(Ci Li

N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>þ<sup>1</sup>

maxðR N<sup>τ</sup>þ<sup>1</sup>

10. Proceed to item 5.

4.2. Reverse optimization task

Nτ= {n<sup>τ</sup>

0 ≤Crq -

1,n<sup>τ</sup> <sup>2</sup>, n<sup>τ</sup>

P<sup>3</sup>

the initial value n<sup>2</sup> <sup>¼</sup> n2

R Nð Þ¼ <sup>τ</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>

and the initial value n2 <sup>¼</sup> <sup>n</sup><sup>2</sup>

computation scheme will be as follows:

<sup>i</sup>¼<sup>1</sup> <sup>n</sup> <sup>C</sup><sup>i</sup> � � <sup>≤</sup> Ci

where τ is the ramification index;

3}, C<sup>1</sup> ≥C<sup>2</sup> ≥C<sup>3</sup>

which meets the normalization requirement:

r :

τÞ, where N<sup>τ</sup> meets the normalization requirement;

τ:

,

; i ¼ 1; 3,

zation condition:

N<sup>τ</sup>þ<sup>1</sup>

i )/(Ci L<sup>i</sup> <sup>τ</sup>=<sup>2</sup> � � �

If N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>�1, withdraw from the procedure.

9. Otherwise (R<sup>τ</sup> < Rrg), perform the following operations:

<sup>k</sup>}, where <sup>n</sup><sup>τ</sup>þ<sup>1</sup>

<sup>τ</sup>=<sup>2</sup> � � �

�)), where i,j,k E {e,p,r}.

9.1. Obtain the set of possible values of the number of standard procedures:

<sup>i</sup> <sup>þ</sup> <sup>L</sup><sup>τ</sup> i =2 � � � �:

�)), where i,j,k E {e,p,r}.

9.3. If N<sup>τ</sup>þ<sup>1</sup> ¼ N<sup>τ</sup>�1, record the value R<sup>τ</sup>þ<sup>1</sup> = R (N<sup>τ</sup>�1) and withdraw from the procedure.

• by specifying the effective initial values, for example, by using personnel's experience

• by reducing the calculation of standard procedure indices to their calculation only as per deterministic models. This is acceptable with a great number of standard procedures (more than 5–20) when stochastic models are less effective than deterministic ones.

The reverse task can be solved using the branch-and-bound procedure. In this case, the

1. Specify a cost-ordered set N of initial values of the number of standard procedures

2. Calculate the maximum value R by directed enumeration n2at the fixed value n1 <sup>¼</sup> <sup>n</sup><sup>1</sup>

3. Calculate the maximum value R by directed enumeration n2at the fixed value n1 <sup>¼</sup> <sup>n</sup><sup>1</sup>

<sup>r</sup> and

<sup>τ</sup> þ 1

, where i is the index of the standard procedure conforming to the maximi-

<sup>i</sup> <sup>¼</sup> <sup>n</sup><sup>τ</sup>

9.2. Find another set of values of the number of standard procedures

The period of this computation scheme can be reduced as follows:

(knowledge) or statistically accumulative tables;

where N<sup>τ</sup>þ<sup>1</sup> meets the normalization requirement;

4. Calculate the maximum value R by directed enumeration n2 at the fixed value n<sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>τ</sup> � 1 and the initial value n2 <sup>¼</sup> <sup>n</sup><sup>2</sup> τ:

R Nð Þ¼ <sup>τ</sup>�<sup>1</sup> maxð<sup>R</sup> <sup>j</sup> <sup>n</sup><sup>1</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>τ</sup> � 1Þ, where N<sup>τ</sup>�<sup>1</sup> meets the normalization requirement;


In practice, there may be a task of calculating indices not for the SW functional stability (dependability) system in general but for a part thereof (checkpoint or error prevention/detection mechanisms). This means a transition from multidimensional to unidimensional task interpretation, which helps substantially simplify computational procedures. Thus, solving a partial reverse optimization task boils down to a single calculation of a specific index when n ¼ Crq=C.

When solving a direct task, the effectiveness of the computation scheme can be additionally improved by adjusting the variable change interval, for example, by defining the next variable value in accordance with a distribution law, and so on.
