# A Multigraph-Defined Distribution Function in a Simulation Model of a Communication Network

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## Abstract

**:**

## 1. Introduction

- A new method of defining network traffic was proposed. The distribution function for creating a simulation model of a communication network was developed, based on the description of communication events and the values of the parameters they determined. The application of this method enabled us to solve the problem of describing the time of data generation and distribution in the communication networks.
- The application of multigraphs for the mathematical derivation of a more precise distribution function of data was proposed and compared with other methods in which the distribution function of data was approximated by the type of network traffic and by the time variation of the data.
- The application of multigraphs and their related matrices enabled multiple descriptions of network traffic in terms of events and communication parameters, which enabled their change in time to be mathematically represented as a function of the schedule. The new approach enabled a more accurate description of the network traffic in the design of a simulation model of the communication network and time-accurate results in the simulation.

## 2. Related Work

## 3. Data Exchange in the Communication Network

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) when the participants in communication establish their communication interaction, achieve mutual communication, and, at the same time, exchange certain types and amounts of data.

#### 3.1. The Data of Network Distribution over Time

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}). Additionally, depending on the function in the communication process, the minimum and maximum amount of data generated by the network element, Ei, for distribution in the network is calculated. The transfer of information to the ITCN requires encrypting of the communication channels. The amount of data for distribution in the network can be increased by the amount of digital code required for protecting information (reconstruction, encryption, error detection). The increase in the amount of data is realized in relation to the header size of the individual layers of the OSI network model. The choice of the access technique, the technology and transmission medium, the communication protocols, and the data packet size (MTU) affect the amount of data to be transferred by the telecommunication links in the ITCN. The total data payload for distribution by the network from the source to destination is determined by the steps in the procedure shown in Figure 2.

#### 3.2. Distribution Function for Variations in the Amount of Data

## 4. Description of the ITCN Network Distribution Using Multigraphs

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) when data are exchanged, a multigraph is formed. The formed multigraph is joined with the similarity matrix. The corresponding value of the distribution function for each moment of time is calculated by mathematical estimation of the similarity matrix associated with the multigraph. The use of all the calculated values for all moments of time in the communication interval ΔT = [t

_{0}, t

_{m}] enables the definition of the appropriate distribution function.

#### 4.1. Data Distribution Time Scheme between ITCN Network Elements

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}). The moments of time are set at the beginning of the time interval in which the application service is active between the network elements. The generation of communication information is enabled and transformed into the appropriate amount of digital data for distribution to the network element Ej is performed. The time scheme of communication interactions (Figure 3), as in [1,2], shows the flow of these activities from the operational procedures.

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) of activation are separated from the given time scheme. The amount of data Adt = (Adt

_{0}, Adt

_{1}, …, Adt

_{N−1}, Adt

_{N}, Adt

_{m}) generated at the moment of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) in the application service Srv is also defined. Examples of the separate individual timing schemes for Services S1 and S2 are displayed in Figure 4.

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}). For example, the service S1 in the network element E1 with a data quantity of Adt

_{0}= 15 kbps for distribution to the network element E2 at time t

_{0}is denoted as E1E2S1_Adt

_{0}.

#### 4.2. Multigraphs of Data Distribution in ITCN Network Traffic

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) are shown by presenting the network traffic as a multigraph (Figure 6a). The single-service multigraph (labeled SSMG_Srv_Adt) shows the amount of data Adt (kbps or Mbps) exchanged between the network elements Ei and Ej (i ≠ j) by the application service Srv at time t. A single edge between the nodes (simple graphs) Ei and Ej (i ≠ j) represents the communication interaction between these network elements, where the amount of data Adt are distributed through the application service Srv at time t. The creation of all the single-service multigraphs between the network elements Ei and Ej (i ≠ j) through the application service Srv for each moment of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) enables the presentation of data exchanged during the communication time interval ΔT = [t

_{0}, t

_{m}]. The total data exchanged between the nodes Ei and Ej (i ≠ j) through all application services, are Srv = (S1, S2, …, Sn) with time t representing the unification of all the single-service multigraphs formed previously into one multi-service multigraph (labeled MSMG_S1Sn_Adt), as shown in Figure 6b. The multi-service multigraph enables the definition of network traffic among the ITCN’s network elements at the observed moments of time t = (t

_{0}, t

_{1}...t

_{N−1}, t

_{N}, t

_{m}) and enables the application of graph sampling theory to perform predictions, as in [22].

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) enables the presentation of the exchange of all data through all application services during the communication time interval ΔT = [t

_{0}, t

_{m}]. A set of multi-service multigraphs allows one to define the total network traffic among the ITCN’s network elements during the communication time interval ΔT = [t

_{0}, t

_{m}].

#### 4.3. Matrix Associated with the ITCN Network Traffic Distribution Multigraph

_{SSMG_Srv_Adt}in Equation (1) with integer terms and a diagonal of zero, where n is the number of network elements Ei. The associated symmetric matrix is formed by using a timeline or a time plane of the communication interactions (Figure 5) or by using a single-service multigraph (Figure 6a), such that

_{ij}t is the amount of data distributed in the communication interactions between the nodes Ei and Ej (i ≠ j) with the application service Srv = (S1, S2, Sn) at the moment of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}). Figure 7 shows the single-service multigraph for data exchanged among the network elements E1 to E8 with the application service S1 at the moment of time t

_{0}, and its associated symmetric 8 × 8 matrix.

_{SSMG_Srv_Adt}defines the matrix at each moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}), as in Equations (2) and (3):

_{SSMG_Srv_Adt}enables one to define the function for the distribution of data in the network through the service Srv = (S1, S2, …, Sn) in the communication time interval ΔT = [t

_{0}, t

_{m}].

_{0}, t

_{m}] is defined by the minimum and maximum values of the amount of data distributed among the network elements of the ITCN:

_{MSMG_S1Sn_Adt}of data distribution shown in Equation (7) is formed. The value of the distribution function of the total amount of data distributed through all application services Srv at the moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) is defined by the associated symmetric matrix T

_{MSMG_S1Sn_Adt}.

_{ij}t is the total amount of data distributed in communication interactions between nodes Ei and Ej (i ≠ j) with all the activated application services Srv = (S1, S2, …, Sn) at the moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}), where:

_{MSMG_S1Sn_Adt}at each moment of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) defines a set of multi-service multigraphs. The set of associated symmetric matrices T

_{MSMG_S1Sn_Adt}enables the definition of the value of the distribution function of the total amount of data distributed through all the application services Srv during the communication time interval ΔT = [t

_{0}, t

_{m}], such that

_{SSMG_Srv_Adt}and all the sets of the associated multi-service matrices T

_{MSMG_S1Sn_Adt}.

## 5. Generating the Data Distribution Function in the ITCN by Sampling Multigraphs

_{SSMG}(t) = T

_{SSMG_Srv_Adt}belongs to the set of associated symmetric matrices related to the distribution of data between the network elements Ei and Ej (i ≠ j) with the application service Srv at the moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}), where ΣT is the number of matrices in the set. The distribution function q(T

_{SSMG}(t)) > 0 for the matrix T

_{SSMG}(t) defines the amount of data for distribution between the network elements Ei and Ej (i ≠ j) through the application service Srv at the moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}). The estimated value of the distribution function is:

_{SSMG}(t)) is determined with the test distribution function q(·) by sampling the T

_{SSMG}(t) matrix column by column (c

_{1}, c

_{2}, …, c

_{n}), using the method and procedure in [20] and [21]. Here, q(T

_{SSMG}(t)) is represented by:

_{i}) of the matrix (n × n), denoted d

^{(2)}, d

^{(3)}, …, d

^{(i)}, and the updated row margins of the (n−1) × (n−1) submatrix are determined for the matrix T

_{Ei}(t).

_{SSMG}(t) is repeated until all the columns (c

_{1}, c

_{2}, …, c

_{n}) have been sampled. The value of each margin of the row (d

_{i}) and the total margin of the matrix (M) is calculated according to the following:

_{SSMG}(t) matrix (t), the number of multigraphs |Σd| is calculated. Submatrices are formed by removing columns. For forming the submatrices, the number of multigraphs |Σd

^{(i)}| is calculated, which corresponds to the associated submatrix. Based on the asymptotic approximation given in [20] and [21], the expression for |Σd| and for |Σd

^{(i)}| is performed.

_{1}, c

_{2}, …, c

_{n}) of the T

_{SSMG}(t) matrix determines the marginal distribution function of each column p(c

_{i}) ∼ q(c

_{i}). The marginal distribution function represents the derived distribution function q(T

_{SSMG}(t)).

_{1}):

_{1}|c

_{2}), …, q(c

_{n}|c

_{n−1}, …, c

_{1}) are derived in the same way. The value of q(T

_{SSMG}(t)) is calculated from the obtained values. The procedure given in [19] evaluates the sampling efficiency of the matrix and the accuracy of the derived distribution function q(T

_{SSMG}(t)) in relation to the marginal distribution p(T

_{SSMG}(t)). The value of the standard estimation error μ and the difference between the obtained values of cv

^{2}is used to calculate the following expression:

_{i}calculated by the procedure given in [20] and [22] realizes the correction and adjustment of values between the derived distribution function q(T

_{SSMG}(t)) and the marginal distribution p(T

_{SSMG}(t)).

_{0}, t

_{m}]. The calculated values of the distribution function form a set of values of the distribution function q(T

_{SSMG}(t)). These values enable one to define the data distribution function pdF(Srv(t)) of the application service Srv in the communication time interval ΔT = [t

_{0}, tm]:

_{MSMG}(t))) define the data distribution function pdF(S1Sn(ΔT)) of all application services Srv in the communication time interval ΔT = [t

_{0}, t

_{m}]. The calculated values form a set of values {q(T

_{SSMG}(t))}. The use of values from the set of values {q(T

_{SSMG}(t))} thus formed enables the creation of graphs of the distribution function pdF(Srv(ΔT)). Graphically, the values are connected in the order of the moments of time t = (t

_{0}, t

_{1}, …, t

_{N−1}, t

_{N}, t

_{m}) to which the values refer.

_{0}, t

_{m}].

_{0}, t

_{m}]. Determining the similarity of the graphs of the function pdF(Srv(t)) to the graphs of the known distribution functions (exponential, Poisson, Normal (Gaussian), uniform, Weibull, etc.) given in [8,10,16,17] allows one to identify the derived distribution function pdF(Srv(t)). The identification of pdF(Srv(t)) as a known distribution function enables the selection of the existing distribution function in the OPNET simulation model and the application of the parameter values from the graphs (Figure 8). If no similarity is found, the use of software tools integrated into the OPNET software allows one to import graphics of the derived distribution function pdF(Srv(t)). This way, the distribution function can be used as a newly defined distribution for realization of the ITCN simulation model.

## 6. Conclusions and Further Research

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The basic concept of the mapping timeline of the network elements’ distribution of data in the OPNET simulation model.

**Figure 2.**The procedure for determining the amount of data to send to the network elements in the ITCN.

**Figure 3.**Timeline of activation and repetition of the network elements’ communication interactions and the network application services (Srv).

**Figure 4.**Timeline of activation and repetition of network application services S1 and S2 with amounts of generated data Adt.

**Figure 5.**Activation and repetition of the application services S1 to S4 and the communication interactions among network elements E1 to E8: (

**a**) timeline; (

**b**) time plane.

**Figure 6.**Data exchange multigraph among network elements E1 to E8 with the application services S1 (blue line) to S2 (red line) at time t

_{0}: (

**a**) single-service multigraph; (

**b**) multi-service multigraph.

**Figure 8.**The graph of the data distribution function pdF through the application service Srv in the communication time interval ΔT = [t

_{0}, t

_{m}].

Reference | Methods | Measurement Source | Statistical Description | Traffic | Illustrating | Application | Country | Year |
---|---|---|---|---|---|---|---|---|

[3] | Traffic self-similarity, the approximation function of traffic | The average daily traffic recorded | Pareto distribution | 2G, voice, HSDPA, | Function distribution graph | Simulating real network traffic | Russia | 2021 |

[4] | Nonlinear analysis of traffic measurements | A medium-sized LAN with 200 to 250 interconnected computers | Kolmogorov’s scheme for describing network traffic, log-normal distribution, Gaussian distribution | NetBEUI, TCP/IP | Function distribution graph | Realistic dynamical models of network traffic | Russia | 2004 |

[5] | Mathematical approximation | Traffic volume recorded by routers, ethernet traffic traces | Poisson’s probability distribution | Ethernet, MPEG4, TCP/IP, web, email, multimedia | Function distribution graph | Traffic modeling | USA, Texas | 2007 |

[6] | Self-similarity statistical analysis of network traffic measurements | Computer network in small company | Gaussian or power-law probability distributions | Web, HTTP, internet, email, SSL, IPv6 | Function distribution graph | Computer network traffic analysis | Poland | 2021 |

[7] | Statistical analysis | Academic, commercial and residential networks; data centers | Log-normal distribution, Gaussian distribution, Weibull distribution | Internet IPv4 | Function distribution graph | Predicting the proportion of time traffic, statistically predicted outcomes for the network | USA, Chicago | 2019 |

[8] | Introductory techniques for input modeling; graphical and statistical methods; mathematics | Sample statistics, the Kolmogorov–Smirnov test statistic, the discrete-event simulation, hypothetical arrival process, stochastic processes | Binomial, degenerate Normal, exponential, Bezier curve, independent binomial, bivariate exponential, Markov chain, Poisson process, nonhomogeneous Poisson process, Markov process | Discrete, continuous modeling arrivals | Histogram, function distribution graph | Input models available to simulation analysts | USA | 2001 |

[9] | Simulation modeling process | Describing the behaviors and interactions | Classical statistics right-triangular distribution, cumulative distribution function, uniform distribution | Discrete event systems | Simulating and modeling operations, distribution modeling | USA | 2007 | |

[16] | Traffic modeling | Core router of a university, backbone links trans-Pacific backbone link | Gaussian distribution | Gaussian traffic model | Q–Q plots, timescales | Network modeling | Netherlands, Denmark | 2013 |

[10] | Traffic analysis | Counting process, inter-arrival time process, discrete-time traffic, | Poisson, Pareto, Weibull, Markov, Markov chain, on–off model, interrupted Poisson | The traffic on the network | Mathematically, graphs | Traffic modeling, capacity planning the design of networks and services | ||

[17] | Traffic analysis | The University of Jordan’s network | Poisson traffic model, long-tail traffic models | Internet traffic | Daily traffic flow graph | Traffic model QoS | Jordan | 2019 |

[11] | Traffic analysis, mathematics | 1998 FIFA World Cup website | Poisson traffic model, Gaussian distribution | Internet traffic | Function distribution graph | Simulation model | Russia | 2017 |

[12] | Traffic analysis | Hubs of cities in Europe and America | Normal probabilistic | Internet traffic, web traffic | Traffic flow graph, probabilistic distribution | Simulation model | Ukraine | 2019 |

[13] | Traffic analysis | Computer network traffic | Multimedia, VoIP | Average daily computer network traffic | Network modeling | Ukraine | 2019 |

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**MDPI and ACS Style**

Miletic, S.; Pokrajac, I.; Pena-Pena, K.; Arce, G.R.; Mladenovic, V.
A Multigraph-Defined Distribution Function in a Simulation Model of a Communication Network. *Entropy* **2022**, *24*, 1294.
https://doi.org/10.3390/e24091294

**AMA Style**

Miletic S, Pokrajac I, Pena-Pena K, Arce GR, Mladenovic V.
A Multigraph-Defined Distribution Function in a Simulation Model of a Communication Network. *Entropy*. 2022; 24(9):1294.
https://doi.org/10.3390/e24091294

**Chicago/Turabian Style**

Miletic, Slobodan, Ivan Pokrajac, Karelia Pena-Pena, Gonzalo R. Arce, and Vladimir Mladenovic.
2022. "A Multigraph-Defined Distribution Function in a Simulation Model of a Communication Network" *Entropy* 24, no. 9: 1294.
https://doi.org/10.3390/e24091294