# An Approach for Mathematical Modeling and Investigation of Computer Processes at a Macro Level

## Abstract

**:**

## 1. Introduction

_{G}. In both cases, the organization of the processes at each of the two levels is subject to probabilistic parameters and conditions in the computer environment [8]. Formally, from the point of view of formalization, any investigation of processes at the macro level and communications between them is subject to a defined procedure. In this aspect, a generalized scheme of a procedure for analysis and optimization of complex-structured objects during the modelling and algorithms is proposed in [9]. Numerical methods and algorithms for finding the optimal solution are used. Analysis of the features of the management process is carried out to increase the efficiency of using these components of the analyzed system.

## 2. Materials and Methods

_{O}with another object-model Ω

_{M}through which the properties or behavior in certain situations of the original are studied by experimenting with the model [22].

_{O}can be an arbitrary system or process that may not actually exist. Nevertheless, its system properties can be described by finite sets, such as S

_{O}—system parameters characterizing the internal state of the real system, its structure, and functioning; Y

_{O}—quantitative characteristics of system parameters, describing mainly resultant behavioral features that are important in the interaction with other systems; and X

_{O}—external actions influencing the behavior of system parameters. In this way, a formal representation of the object-original as a class of finite discrete sets Ω

_{O}= {S

_{O}, Y

_{O}, X

_{O}} can be made. The peculiarity is that the main sets in the formed class may be too large, which will require the selection of adequate subsets. In this reason, when studying a system, a subset {y

_{o}} ∊ Y

_{O}is usually chosen to be analyzed under the influence of external factors {x

_{o}} ∊ X

_{O}, and each individual characteristic y

_{oi}(i = 1 ÷ K, where K is the total number of elements in the subset of individual characteristics) depends on some subset {s

_{o}} ∊ S

_{O}of system parameters (usually the influence of the other parameters is neglected). This subset of selected system parameters determines the spice of the system, and the characteristics are the data describing its organization and behavior according to the goal of the research. In this sense, each object of study should be considered as a complex of two related parts—static (independent of time) and dynamic (system parameters and characteristics that depend on time).

_{M}must reproduce with sufficient accuracy the real object under study, being in accordance with the selected goal of the concrete research. The model should comply with the selected conditions for the analysis of the behavior of the original system or process. This determines the need to coordinate the subsets of the class Ω

_{O}with appropriate components used for realization of Ω

_{M}. In the concrete case, this coordination requires correct transformation of the selected subset in mathematically presented subsets for the model realization. As a result, the following components can be defined: {s

_{m}} ∊ S

_{M}—parameters of the object-model, {y

_{m}} ∊ Y

_{M}—characteristics of the object-model, and {x

_{m}} ∊ X

_{M}—external factors of influence for the object-model. The replacement Ω

_{O}→Ω

_{M}is admissible if the determined model characteristics {y

_{m}} ∊ Y

_{M}sufficiently reflect the respective quantitative characteristics {y

_{o}} ∊ Y

_{O}, defined in the modelling process. In this sense, the modelling is a replacement of a real functional dependence {y

_{o}} = Φ

_{O}[{s

_{o}}, {x

_{o}}, T

_{o}], describing the behavior of the original object in time, with a corresponding equivalent dependence {y

_{m}} = Φ

_{M}[{s

_{m}}, {x

_{m}}, T

_{m}], where usually the model time T

_{m}is related to the real time T

_{o}by scaling. The main requirement in modelling is to find such an analytical dependence that describes with sufficient accuracy the behavior of the original in terms of the objectives of the study.

_{O}) to the actual development of a mathematical model (Ω

_{M}). The phase of mathematical description is the basic stage in the investigation because it must build a sufficiently adequate model of the real object (process or system). This requires an appropriate mathematical environment to be selected (formal, deterministic, probabilistic, empirical, etc.). The third phase of program realization transforms the mathematical description into a suitable software environment and creates the final result of the modelling, which will be subjected to experiments. A large number of program languages (universal and specialized) can be used to develop the program model, but the choice must take into account the type of mathematical description (Ω

_{M}) and the nature of the model investigation—deterministic, probabilistic, simulation, statistical, or heterogeneous. A good opportunity is provided by the APL2 language and the TryAPL2 operating environment, which is relevant to the deterministic model investigation of computer process organization presented in the following sections.

_{1}and number of tasks N

_{1}) and k calls to external memory (intensity f

_{2}and number of tasks N

_{2}). It is accepted that N = N

_{1}+ N

_{2}= const; the task falls in the processor to (k + 1); and the bandwidth p coincides with the intensity for falling f

_{0}into a passive state, which allows for the presentation of the following simple analytical dependences:

_{1}(N

_{1}), which is based on the following: (a) there is zero intensity for the CPU (Central Processor Unit) load when it does not process a task (N

_{1}= 0), for example, due to an infinitely long time for servicing the tasks in state S

_{2}; (b) the service is performed only in one processor and if number of tasks is N

_{1}≥ 1, the analytical dependence f

_{1}(N

_{1}≥ 1) = const will be saturated and limited to maximum bandwidth (k + 1)/t; (c) to solve the situation 0 < N

_{1}< 1, it is necessary to know f

_{2}(N

_{2}), which is usually a nonlinear dependence.

_{1}(N

_{1}) and p, requiring the presentation of of the controllable model parameters (T—fixed processing time for processor, K—number of external memory access). To construct a functional dependence, successive experiments are performed according to a randomized factor plan, for example at values N = 5; T = 10; K = 2, 3, 4, 5. The results of execution of the function at a fixed value for K = 4 and different values (0; 0.2; 0.4; 0.6; 0.8; 1; 2; 4) of the argument N1 are graphically presented in Figure 2c.

## 3. Mathematical Formalization and Model Construction

_{j}, which can be considered as components of a global functional algorithm A

_{G}. Theoretically, the complete algorithm A

_{G}can be described logically if the individual algorithms A

_{j}and the conditions for their activation at a specific input information flow are known. This undermines its formal description by a directed graph representing a graph scheme of algorithm (GSA), in which the matrix of connections {c

_{ij}} describes the existence of an information connection A

_{i}→A

_{j}between two individual algorithms. In an investigation of the computer processes organization, it is possible to apply both approaches—stochastic (if the relationships are defined as probabilities) or deterministic (the matrix of relationships is Boolean).

#### 3.1. Mathematical Formalization

_{O}is made on the basis of the requirement of the phase [2] of the general technological procedure presented in Figure 1. In the current case, the object Ω

_{O}of model investigation is a global algorithm A

_{G}of exemplary computer processing presented as a GSA (Figure 3a). It is a generalized structure of communicating program modules, each of which can be activated by another, depending on the development of the generalized process. In practice, the interactions between different modules and the possible activations of a concrete sequence of them is a probabilistic process, but the task determined for the research allows a deterministic approach to be applied. In this case, the goal is to determine the number and lengths of all possible paths representing the possible realizations of information processing.

_{ij}} is applied, where l

_{ij}= 1 (if there is a connection) and l

_{ij}= 0 (in the absence of a connection), as seen in Figure 3b.

_{1},…,A

_{n}}, which are executed in different sequences with possible input/output interactions. This allows for the determination of a tree of relations between sequential processes in GSA on the basis of the consequence matrix L—l

_{ij}:A

_{i}→A

_{j}(i,j ∊ {1,2,…,n}), as shown in Figure 3b. Usually, the investigation is connected with determining the reachability of a given final task (algorithm) from one or several initial tasks. For this purpose, a system of algebraic equations is compiled, describing the presence of edges a

_{ij}between individual algorithms A

_{i}and A

_{j}:

_{i}→A

_{j}} ⇒ A

_{i}< A

_{j}; FOR ∀ A

_{i}, A

_{j}(i ≠ j).

- All initial nodes (without predecessors) are numbered first and included in layer (1).
- A node is numbered and included in the current layer if all its predecessors are already numbered (included in previous determined layers).
- Nodes to be numbered are successors of already numbered nodes.

#### 3.2. Deterministic Mathematical Description

_{M}on the basis of the following steps.

_{Aj}, describing the successors of each node, fulfilling the condition LT[i,j] ≠ 0 (j = 1 ÷ n), shown in Figure 4a. This can be realized by constructions LT←L & N←ρL [1;] and LT←⌀⊃L.

_{1}= ∑ V

_{Aj}(j = 1 ÷ n; n = 8) from LT:

_{1}[j] = LT[j,1] + LT[j,2] + ... + LT[j,n]; for j = 1 ÷ n.

_{1}[j] = 0 ⇒ A

_{j}∊ Layer(1), node A

_{j}must be included in the layer (1), which is marked in the matrix of layers AL. In this case, it is only V

_{1}[1] = 0, which determines that the algorithm A

_{1}must be included in Layer(1) (see column AL

_{1}in Figure 4b).

_{2}on the basis of vector V

_{1}and row j of LT:

_{2}[j] = 0 determine algorithms included in the Layer(2)—in our case, algorithms A

_{2}and A

_{7}, column AL

_{2}in Figure 4b.

_{q}(q = 3, 4, ...) to determine algorithms included in the Layer(q):

_{q}[j] = 0 (j= 1 ÷ n).

#### 3.3. Possibility for Application of Formalization in Process Dispatching

_{1}, …, t

_{n}} for realization of processes A

_{i}(i = 1 ÷ n) is known. If the system of resources has a heterogeneous nature, it is possible for a given process A

_{i}to occupy several devices in succession, staying in each for different times {t

_{i1}, t

_{i2}, …, t

_{ik}}. Then, t

_{i}= ∑t

_{ik}and the vector T can be modified in a matrix T* = {t

_{ij}} with n = |A| rows and m = |S| columns. This will allow for the formalization of the process of creating a dispatching plan for the analyzed processes in the system, which can be presented as a ordered discrete structure p = <S, A, G, T, F>, where S is a set of system resources with |S| = m, A is a final discrete partially ordered set of processes in the environment S with |A| = n, G(A,L) is a directed graph for describing GSA of the processes from set A, T = {t

_{1}, …, t

_{n}} presents the vector of execution times for all processes in the environment, and F determines a formal criteria (strategy) for process dispatching.

_{1}(t), …, d

_{m}(t)} defined in the interval (0, τ) and accepting integer values from the set of indices (1, 2, …, n) of A. Thus, the elements d

_{i}(t) = j (1 ≤ i ≤ m; 1 ≤ j ≤ n) of the function D will represent the occupation of a resource S

_{i}in the moment t during the execution of process A

_{j}.

## 4. Program Realization and Experimental Results

#### 4.1. Program Model Realization

_{M}is presented in Figure 5a. The results of the model execution are shown in Figure 5b and include ✓ the Boolean matrix L, describing the initial graph-scheme of the algorithm; ✓ the transposed matrix LT; and ✓ the calculated vectors V[q] (q = 1,2, ...) and the matrix of the layers AL (Table 1), which corresponds to the ordered graph-scheme OGSA (Figure 6). Each of the four defined layers includes procedures that can be performed independently of each other.

_{1}, ..., S

_{m}} and equal labor intensity of the processes forming the set A = {A

_{1}, …, A

_{n}}. This means that the mathematical expectation of the times for the realization of the processes were the same, i.e., E[t(A

_{i})] ≡ E[t

_{i}] = const, which allowed for the application of a binary graph to represent the GSA and a fixed time vector T = {t

_{i}= τ/i = 1 ÷ n}. In this case, the formed ordered scheme from Figure 6 determined six paths, as the maximum length was 4, which allowed for the definition of a maximum parallel plan with a minimum execution time for this set of eight processes (minimum number of layers). In this case, Table 1 can be transformed into an optimal parallel plan, requiring three independent processor nodes for realization, as shown in Table 2 (u is the total time for parallel form realization).

_{1}, …, A

_{14}} = {1,…, 14}, whose GSA is described below by the existing arcs <A

_{i}→A

_{j}> ≡ “i-j”:

1-2, 1-3, 1-4; | 4-7, 4-10; | 8-11; |

2-5, 2-8; | 6-9; | 9-11; |

3-2, 3-6, 3-12; | 7-9, 7-12; | 10-9; 10-13; 10-14. |

^{1}; {3, 4}

^{2}; {2, 6, 7, 10}

^{3}; {5, 8, 9, 12, 13, 14}

^{4}; {11}

^{5}}

#### 4.2. Experimental Result Discussion and Examples for Application

_{5}= a

_{25}·A

_{2}+ a

_{85}·A

_{8}= a

_{25}·[a

_{12}·A

_{1}] + a

_{85}·[a

_{28}·A

_{2}] = a

_{25}·a

_{12}·A

_{1}+ a

_{85}·a

_{28}·a

_{12}·A

_{1}

_{1}→A

_{2}→A

_{5}> and <A

_{1}→A

_{2}→A

_{8}→A

_{5}>. These two paths were equivalent—{S

_{2}, S

_{3}}, and another couple of equivalent paths were {S

_{4}, S

_{5}}. The presence of equivalent paths is marked by more than one “1” in the columns for the final tasks in the matrix SA (Figure 7b).

_{i}) ≡ t

_{i}= const for separate resources S

_{j}∊ S. Communication times for exchange between processors can be ignored if there is shared memory. The weights of the nodes can be transferred along the outgoing arcs of the GSA, which allows for the use of the procedure for finding a path in a graph and defining the optimal plan to relate to the maximum path and the corresponding critical time in the study of parallel planning (maximum path length in GSA; Figure 7c). Weight or binary graphs can be applied, depending on the specific environment for the realization of the parallel processes.

_{i}(i = 1 ÷ 7): {1

^{(4)}; 2

^{(2)}}, {3

^{(2)}}, {4

^{(5)}; 5

^{(4)}, 6

^{(8)}}, {7

^{(5)}}, where the notation is i

^{(ti)}. The maximum processor environment for the implementation of the plan was determined by the maximum power of the layer in the ordered GSA (in this case, three processors).

- Relative average resource load factor: $\eta =\frac{{\displaystyle \sum {\tau}^{\prime}}}{{\displaystyle \sum \tau}}=\frac{{\displaystyle \sum {\tau}^{\prime}}}{m.u}$;
- Relative weight of inefficient work (stay): $\chi =u\left(\frac{{\displaystyle \sum {\tau}^{\u2033}}}{m}\right)$;

## 5. Conclusions

_{i}) for execution, which in the formalization can be presented by scalar weights of the graph nodes. In this respect, the further research will be directed to an extension of the model investigation to application of a probabilistic approach, which is typical for computer processes. This will be well supported by the capabilities of the software environment TryAPL2 for presentation and executions of stochastic processes.

## Funding

## Conflicts of Interest

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**Figure 2.**A simple example for analytical model investigation by using APL2—(

**a**) program code, (

**b**) model execution, (

**c**) results interpretation.

**Figure 3.**Mathematical formalization of the investigated object Ω

_{O}(

**a**) graph scheme of algorithm (GSA), (

**b**) matrix of connections (conceptual model).

**Figure 8.**Graph interpretation of the dispatching plan based on results obtained from GGSA and PATH execution.

Layer 1 | Layer 2 | Layer 3 | Layer 4 |
---|---|---|---|

A_{1} = f(0) | A_{2} = a_{12}·A_{1} | A_{3} = a_{23}·A_{2} | A_{5} = a_{25}·A_{2} + a_{85}·A_{8} |

A_{7} = a_{17}·A_{1} | A_{4} = a_{24}·A_{2} | A_{6} = a_{26}·A_{2} + a_{36}·A_{3} | |

A_{8} = a_{28}·A_{2} |

Time Resource | 1 | 2 | 3 | 4 |
---|---|---|---|---|

S_{1} | A_{1} | A_{2} | A_{8} | A_{5} |

S_{2} | - | A_{7} | A_{3} | A_{6} |

S_{3} | - | - | A_{4} | - |

u |

Layer Path | I | II | III | IV | V | Clock Path | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | – | 2 | 5 | – | 1 | 1 | 2 | 5 | – | – |

2 | 1 | – | 2 | 8 | 11 | 2 | 1 | 2 | 8 | 11 | – |

3 | 1 | 3 | 2 | 5 | – | 3 | 1 | 3 | 2 | 5 | – |

4 | 1 | 3 | 2 | 8 | 11 | 4 | 1 | 3 | 2 | 8 | 11 |

5 | 1 | 3 | 6 | 9 | 11 | 5 | 1 | 3 | 6 | 9 | 11 |

6 | 1 | 3 | – | 12 | – | 6 | 1 | 3 | 12 | – | – |

7 | 1 | 4 | 7 | 9 | 11 | 7 | 1 | 4 | 7 | 9 | 11 |

8 | 1 | 4 | 7 | 12 | – | 8 | 1 | 4 | 7 | 12 | – |

9 | 1 | 4 | 10 | 9 | 11 | 9 | 1 | 4 | 10 | 9 | 11 |

10 | 1 | 4 | 10 | 13 | – | 10 | 1 | 4 | 10 | 13 | – |

11 | 1 | 4 | 10 | 14 | – | 11 | 1 | 4 | 10 | 14 | – |

D1(t) | D3(t) | |||||||||||

S_{1} | 1 | 3 | 2 | 8 | 11 | S_{1} | 1 | 3 | 2 | 6 | 5 | 14 |

S_{2} | – | 4 | 6 | 9 | – | S_{2} | – | 4 | 10 | 8 | 12 | 11 |

S_{3} | – | – | 7 | 5 | – | S_{3} | – | – | 7 | 9 | 13 | – |

S_{4} | – | – | 10 | 12 | – | |||||||

S_{5} | – | – | – | 13 | – | |||||||

S_{6} | – | – | – | 14 | – | |||||||

u = 5τ | u = 6τ | |||||||||||

D2(t) | D4(t) | |||||||||||

S_{1} | 1 | 3 | 2 | 8 | 11 | S_{1} | 1 | 3 | 2 | 6 | 5 | 14 |

S_{2} | – | 4 | 6 | 9 | 13 | S_{2} | – | 4 | 10 | 8 | 12 | 11 |

S_{3} | – | – | 7 | 5 | 14 | S_{3} | – | – | 7 | 9 | 13 | – |

S_{4} | – | – | 10 | 12 | – | |||||||

u = 5τ | u = 6τ | |||||||||||

D5(t) | ||||||||||||

S_{1} | 1 | 3 | 2 | 7 | 8 | 5 | 13 | 11 | 1 | |||

S_{2} | – | 4 | 6 | 10 | 9 | 12 | 14 | – | – | |||

u = 8τ |

Plan | m | u | η | χ | σ_{1} = u·η·χ | σ_{2} = m·σ_{1} | σ_{3} = (u·χ)/η |
---|---|---|---|---|---|---|---|

D1(t) | 6 | 5 | 0.466 | 13.33 | 31.06 | 186.36 | 143.026 |

D2(t) | 4 | 5 | 0.7 | 7.5 | 26.25 | 105 | 53.57 |

D3(t) | 3 | 6 | 0.777 | 8 | 37.296 | 111.88 | 61.776 |

D4(t) | 3 | 6 | 0.777 | 8 | 37.296 | 111.88 | 61.776 |

D5(t) | 2 | 8 | 0.875 | 8 | 56 | 112 | 73.143 |

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Romansky, R. An Approach for Mathematical Modeling and Investigation of Computer Processes at a Macro Level. *Mathematics* **2020**, *8*, 1838.
https://doi.org/10.3390/math8101838

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Romansky R. An Approach for Mathematical Modeling and Investigation of Computer Processes at a Macro Level. *Mathematics*. 2020; 8(10):1838.
https://doi.org/10.3390/math8101838

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Romansky, Radi. 2020. "An Approach for Mathematical Modeling and Investigation of Computer Processes at a Macro Level" *Mathematics* 8, no. 10: 1838.
https://doi.org/10.3390/math8101838