Analyses with Index Matrices Mathematical Models of Network Systems

Stela Todorova; Ivan Ivanov

doi:doi:10.11648/j.ajam.20251306.18

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| Peer-Reviewed

Analyses with Index Matrices Mathematical Models of Network Systems

Stela Todorova^*

, Ivan Ivanov

Published in American Journal of Applied Mathematics (Volume 13, Issue 6)

Received: 17 November 2025 Accepted: 1 December 2025 Published: 26 December 2025

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Abstract

The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 6)
DOI	10.11648/j.ajam.20251306.18
Page(s)	462-468
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Index Matrices, Network Model, DCN, Fat-Tree Network Model, Probability Theory, Prediction, Poisson Process Model, Broadcast

1. Introduction

Let

I

be a fixed set of indices and  be the set of the real numbers. Let the standard sets

K

and

L

satisfy the condition:

K, L \subset I

. Let over these sets, the standard set-theoretical operations be defined. We call “IM with real number elements” (R-IM) the object:

where

K = \{k_{1}, k_{2}, . . ., k_{m}\}

and

L = \{l_{1}, l_{2}, . . ., l_{n}\}

, and for

1 \leq i \leq m

, and

1 \leq j \leq n : a_{k_{i}, l_{j}} \in R

When set

R

is changed with set {0, 1}, we obtain a particular case of an IM with elements being real numbers, that we denote by (0, 1)-IM.

When we choose to work with matrices, elements of which are logical variables, propositions or predicates – let us call these IM “Logical IMs (L-IMs)”.

Let the set of all used functions be Ƒ. The research over of this document.

IMs with function-type of elements has two cases:

1) each function of set Ƒ has one argument and it is exactly x (i.e., it is not possible that one of the functions has argument x and another function has argument y) – let us mark the set of these;

2) each function of set Ƒ has one argument, but that argument might be different for the different functions or the different functions of set Ƒ have different numbers of arguments.

[1-3]

2. Application of Probability Models in Network Systems

2.1. Probability Models

Theoretical mathematics is a key evolution factor of artificial intelligence (AI). Nowadays, representing a smart system as a mathematical model helps to analyze any system under development and supports different case studies.

[8]

Scientists check and make researches with different models and methodologies objects of their research whether they meet certain criteria. Also, the analysis of data from a scientific experiment - be it a survey or laboratory measurement uses predicate logic.

[15]

The core concept in probability theory is that of a probability model. The first component of a probability model is a sample space, which is a set whose elements are called outcomes or sample points. We will consider probability models only in the case where the sample space is finite. The second component of a probability model is a class of events, which can be considered for now simply as the class of all subsets of the sample space. The third component is a probability measure.

[6]

2.2. Distribution Function and Probability Space

Let us consider a space (Ω, F, P). For each outcome of a probability experiment, we assign a number, in other words, we set a function ξ=ξ(ω), which domain is Ω, and the set of values belongs to the set of real numbers. The values that ξ(ω) takes are random.

[9]

Let Ω be a space of elementary events. The function ξ_j=ξ_j(ω) maps each point ω of Ω to a point x_j = ξ_j(ω) on the number line, where 1≤j≤n.

A function ξ_j=ξ_j(ω), defined on the space of elementary events Ω, is called a random variable if ξ_j^-1(A_i) ϵF for every Borel set, where 1≤j≤n and 1≤i≤m and.

A function F(x_j) = P(ξ_j < x_j), defined for every x_jϵ, is called the distribution function of the random variable ξ_j. To each real number x_j we assign the number F(x_j), equal to the quantity of probability mass located to the left of x_j, where 1≤j≤n.

[13]

2.3. Representation of z-images with Sequences of Discrete Values

Let the continuous function f(t) be given by its values

f_{l_{j}}

= f(l_jT), called discretes, at discrete times t_j = l_jT, where T is the discretization interval, where 1≤ j ≤ n.

We assume

e^{sT}

,(1)

where the z-image F(z) of the sequence f(l_jT) for l_j = 0, 1, 2, …, n is represented in the form

F(z) =

\sum_{l_{1}}^{\infty} f (l_{j} T) z^{- l_{j}}

,(2)

where 1≤ j ≤ n.

The correspondences between z-images and sequences of discrete values are written:

F(z) =ζ{f(

l_{j} T

)},(3)

l_{j} T

)=ζ^-1{F(z)},(4)

where 1≤ j ≤ n.

[10, 11]

3. Representation of a Graph with an Index Matrix

Let a set of vertices U be given. Let's

E = {〈

u_{w}

u_{r}

〉 |

u_{w}

u_{r}

∈U}, where 1≤w≤|U|, 1≤r≤|U|.

Obviously, this is the set of all pairs of vertices of the set U.

Let H be an arbitrary subset of K. The ordered pair G = [U, H] is called "unoriented graph".

We will call a directed graph a graph which direction is determined for each of its edges. Formally, directed graphs are defined by

G^* = [U, H^*],

where U is the set of vertices of the graph and is the set of edges

H^* ⊆ {〈

u_{w}

u_{r}

〉 |

u_{w}

u_{r}

∈U ∧ (〈

u_{w}

u_{r}

〉 ∈ H^*∧ 〈

u_{r}

u_{w}

〉 ∈ H^*⟶〈

u_{w}

u_{r}

〉 ≠〈

u_{r}

u_{w}

〉), where 1≤w≤|U|, 1≤r≤|U|.

On every directed graph G^* = [U, H^*] a dimension matrix can be mapped |U|  |U|, called an incidence matrix.

If the vertices of the graph are u₁, u₂, u₃, …, u_n, where n = |U|, then we will ask that the w-th row and the r-th column correspond to a vertex u_w.

Then (w,r), will correspond to the pair of vertices u_w and u_r will have value 1 if from vertex u_w there is an edge to vertex u_r, or value 0 otherwise.

Let a directed graph be given G = [U, H^*]. Then its index incidence matrix will have the form (by analogy with the standard matrix of incidence)

IM(G) = [U, U, {

a_{u_{w}, u_{r}}

}],

where

a_{u_{w}, u_{r}}

\{\begin{matrix} 1, oriented arc (u_{w}, u_{r}) \in H^{*} \\ 0, ot h erwise, \end{matrix}

[3, 5, 12]

where 1≤w≤|U|, 1≤r≤|U|.

4. Representation with Index Matrices Fat-Tree Network Model

We consider the set of all connected servers designed as directed weighted graph where weights equal the link capacities. The graph is denoted by G^* = [U, H^*], where U represents the set of nodes (i.e., Fat-Tree servers) and H^* the set of links attaching each server to neighbors.

[4]

Then we can construct an index incidence matrix of Fat-Tree Network Model will have the form

IM(G) = [U, U, {

a_{u_{w}, u_{r}}

}],

where

a_{u_{w}, u_{r}}

\{\begin{matrix} 1, oriented arc (u_{w}, u_{r}) \in H^{*} \\ 0, ot h erwise \end{matrix}

and 1≤ w ≤ |U|, 1≤ r ≤ |U|.

We denote the link from u_w to u_r as

a_{u_{w}, u_{r}} \in H^{*}

, 1≤ w ≤ |U|, 1≤ r ≤ |U|. We denote the residual bandwidth of the link

a_{u_{w}, u_{r}}

C_{w_{r}}

(

a_{u_{w}, u_{r}}

) at the time instant

T_{l_{j}}

, where 1≤ i ≤ m, 1≤ w ≤ |U|, 1≤ r ≤ |U|.

4.1. Centralized Multicast path Computation within Fat-Tree DCN

In this proposal, we consider Constant Bit Rate (CBR) flows. Let with the index set Q = {q₁, q₂, … q₅} define different arrival times of packets. Let with the index set E = {e₁ = 6, e₂ = 12, e₃ = 18, e₄ = 24, e₅ = 30} define flows requests with a fixed band width. Let be defined a Poisson process with velocities λ₁ = 6 per second and λ₂ = 3 flows per 2 seconds.

Then can be calculated times duration of a Poisson process with a velocity λ₁ = 6 per second of transmitting of each flow with formula

q_i=

\frac{e_{i}}{λ_{f}}

,(5)

where 1≤i≤5 and substituting with the values of the elements of the index set E of flows with different band width, we can assign q₁ =

\frac{e_{i}}{λ_{1}}

\frac{6}{6}

=1, q₂=

\frac{12}{6}

=2, q₃=

\frac{18}{6}

=3, q₄=

\frac{24}{6}

=4, q₅=

\frac{30}{6}

=5. Because the max value of the arrival times of packets is q₅ and the duration of transmitting each flow is 1 second, then with the index set Y = {y₁ =1, y₂ =2, y₃ =3, y₄ =4, y₅ =5} will define times duration of transmitting of each flow. Let with the index L = {l₁ =1, l₂ =2, l₃ =3, l₄ =4, l₅ =5} denote intervals of 5 units, where each unit is calculated by the function Z(y_j) = l_j = y_j*6, where 1 ≤ j ≤ 5. Because the elements of the index set L are intervals, we can assign the index set to a probability measure.

Then Z(y_j) has only a finite or countable number of possible sample values, say y₁, y₂, y₃, y₄, y₅ the probability Z_j { Z_j (y_j) = y_i} of each sample value y_i is called the probability mass function at y_i and denoted by p_Z(Y_i); such a random variable is called discrete for 1≤i≤n. The ‘staircase function,’ (the CDF of a discrete random variable) is having a jump of magnitude p_Z(y_i) at each sample value y_i and stays constant between the possible sample values for 1≤i≤n. As a result, the discrete random variables can be specified by the probability mass function and the cumulative distribution function.

Because the velocity of a Poisson process is assigned to λ₁ = 6 per second, then we can define the derivative(probability density) of Y at y_i as the time duration of transmitting each flow and denoted by Z(y_j); for δ>0 sufficiently small if there is a function Z(y_j) δ, such that for each y_i ∈

R

, that defines the existence of a cumulative distribution function satisfies

\int_{- \infty}^{| Y |} Z (y_{i}) d (y_{i - 1} - y_{i}

), where 1 ≤ i ≤ |Y-1|.

There are 5 complete results of the experiment to be modeled, so we can define 5 outcomes or sample points in a probability model.

The possible outcomes (the space of possible results) are 1, 2, 3, 4 or 5 that are mutually exclusive and collectively constitute. In order to define the finest grain result of the probability outcome, will define singleton event ω that contains no proper subsets.

With the restriction to the max value of elements of the index set L, we can assign to non-negative value to each outcome. Then, let with the index set K = {k₁ = “F(x₁)≤6”, k₂ = “F(x₂)≤12”, k₃ = “F(x₃)≤18”, k₄ = “F(x₄)≤24”, k₅ = “F(x₅)≤30”} denote predicates.

Then, according to the definition that a random variable (rv), we can assign the function

F_{k_{i}}

(

y_{l_{j}}

) as a function Z from the sample space Ω of a probability model to the set of real numbers

R

, where 1≤i≤5, 1≤j≤5. First,

F_{k_{i}}

(

y_{l_{j}}

) might be undefined or infinite for a subset of Ω that has 0 probability, where where 1≤i≤5, 1≤j≤5. Second, the mapping

Z_{k_{i}}

(

ω_{k_{i}}

) must have the property that {ω∈ Ω:

F_{k_{i}}

(

y_{l_{j}}

) ≤ e_i} is an event for each k_i ∈

R

, where where 1≤i≤5, 1≤j≤5. Third, every finite set of random variables

F_{k_{1}}

,...,

F_{k_{5}}

has the property that for each

y_{l_{j 1}}

∈

R

,...,

y_{l_{5}}

∈

R

,{ ω:

F_{k_{1}}

(ω) ≤ e₁,...,

F_{k_{5}}

(ω) ≤ e₅} is an event.

Then we can construct an Index matrix A=[K, L, {

a_{k_{i}, l_{j}}

}] of probability trials, where

a_{k_{1}, l_{1}}

= 1,

a_{k_{1}, l_{2}}

= 0,

a_{k_{1}, l_{3}}

= 0,

a_{k_{1}, l_{4}}

= 0,

a_{k_{1}, l_{5}}

= 0,

a_{k_{2}, l_{1}}

= 1,

a_{k_{2}, l_{2}}

= 1,

a_{k_{2}, l_{3}}

= 0,

a_{k_{2}, l_{4}}

= 0,

a_{k_{2}, l_{5}}

= 0,

a_{k_{3}, l_{1}}

= 1,

a_{k_{3}, l_{2}}

= 1,

a_{k_{3}, l_{3}}

= 1,

a_{k_{3}, l_{4}}

= 0,

a_{k_{3}, l_{5}}

= 0,

a_{k_{4}, l_{1}}

= 1,

a_{k_{4}, l_{2}}

= 1,

a_{k_{4}, l_{3}}

= 1,

a_{k_{4}, l_{4}}

= 1,

a_{k_{4}, l_{5}}

= 0,

a_{k_{5}, l_{1}}

= 1,

a_{k_{5}, l_{2}}

= 1,

a_{k_{5}, l_{3}}

= 1,

a_{k_{5}, l_{4}}

= 1,

a_{k_{5}, l_{5}}

= 1 and 1≤i≤5, 1≤j≤5.

In Figure 1 is illustrated the transmission of flows in time.

Download: Download full-size image

Figure 1. Transition of flows in time.

With the elements of the index set L, we illustrate that we performing exactly 5 trials of the same experiment (i.e., repeat the same idealized experiment exactly 5 times). With the these 5-tuples of sample points, will be calculated the probabilities of each outcome and the prediction of occurrence of an event in the future. The results are important to make decision for the state of a machine.

[14]

is given an example of fault detection in machine using the programme product of index matrices.

The Cartesian product is the sample space of the 5-repetition model.

Ω^|K|= {(

ω_{k_{1}}, ω_{k_{2}}

ω_{k_{3}}

ω_{k_{4}}

ω_{k_{5}}

ω_{k_{i}}

∈Ω for eachi, 1≤i≤5},(6)

i.e., the set of all 5-tuples for which each of the 5 components of the 5-tuple is an element of the original sample space Ω. Since each sample point in the 5-repetition model is an 5-tuple of points from the original Ω, it follows that an event in the n-repetition model is a subset of Ω^|^K^|, i.e., a collection of 5-tuples (

ω_{k_{1}}, ω_{k_{2}}

ω_{k_{3}}

ω_{k_{4}}

ω_{k_{5}}

), where each

w_{k_{i}}

is a sample point 5 from Ω, where 1≤i≤5. This class of events in Ω^|^K^| should include each event of the form {(

A_{k_{1}}

A_{k_{2}}

A_{k_{3}}

A_{k_{4}}

A_{k_{5}}

)}, where {(

A_{k_{1}}

A_{k_{2}}

A_{k_{3}}

A_{k_{4}}

A_{k_{5}}

)} denotes the collection of 5-tuples (

ω_{k_{1}}, ω_{k_{2}}

ω_{k_{3}}

ω_{k_{4}}

ω_{k_{5}}

) where

ω_{k_{i}}

∈

A_{k_{i}}

for 1≤i≤5. The set of events (for 5-repetitions) must also be extended to be closed under complementation and countable unions and intersections.

In order to create the probability model in the experiment, there are defined 5-trials that are statistically independent. Moreover, we assume that for each extended event {(

A_{k_{1}}



A_{k_{2}}



A_{k_{3}}



A_{k_{4}}



A_{k_{5}}

)} contained in Ω^|5|, we have

P{(

A_{k_{1}}



A_{k_{2}}



A_{k_{3}}



A_{k_{4}}



A_{k_{5}}

)} =

\prod_{k_{i} = 1}^{5} P_{k_{i}} {A_{k_{i}}}

,(7)

where

P_{k_{i}} {A_{k_{i}}}

is the probability of event

A_{k_{i}}

in the original model for 1≤i≤5. Each events in each trial are independent of those of any other trials of the extended independent 5-repetition model. From Figure 1, we can see that the element k₁ of the index set K and element l₁ of the index set L satisfies the first condition, and for the other indexes of the index set K of columns and the index set L of rows in progressive order, the count of elements that satisfies the conditions are growing in linearly, so we can conclude that for less time less flows will be transmitted. Let with the index set 0 < S₁ < S₂ < S₃< S₄< S₅the sequence of increasing random variables that represent an arrival process.

Definition 3. A renewal process is an arrival process for which the sequence of interarrival times is a sequence of IID random variables.

Definition 4. A Poisson process is a renewal process in which the interarrival intervals have an exponential distribution function; i.e., for some real λ> 0, each

X_{k_{i}}

has the density

f_X(x) = λexp(-λ

x_{i}

)(8)

for

x_{i}

≥0 and for 1≤i≤5.

The parameter

λ_{1}

is called the rate of the process. We shall see later that for any interval of size t,

λ_{i}

is the expected number of arrivals in that interval. Thus is called the arrival rate of the process.

[6]

Analogically, can be calculated times duration a Poisson process with a velocity λ₂ = 3 per 2 seconds of transmitting of each flow with formula (1), where 1≤i≤5 and substituting with the values of the elements of the index set E of flows with different band width, we can assign q₁ =

\frac{e_{i}}{λ_{1}}

\frac{6}{3}

=2, q₂=

\frac{12}{3}

=4, q₃=

\frac{18}{3}

=6, q₄=

\frac{24}{3}

=8, q₅=

\frac{30}{3}

=10. Because the max value of the arrival times of packets is q₅ and the duration of transmitting each flow is 2 seconds,then the max duration of transmitting each flow will be q₅2 with the index set X = {x₁ =1, x₂ =2, x₃ =3, x₄ =4, x₅ =5, x₆ =6, x₇ =7, x₈ =8, x₉ =9, x₁₀ =10} will define times duration of transmitting of each flow. Let with the index V = {v₁ =2, v₂ =4, v₃ =6, v₄ =8, v₅ =10, v₆ =12} denote intervals of 6 units, where each unit is calculated by the function F(x_j) = l_j = x_j*3, where 1 ≤ j ≤ 6.

Let with the index set Y = {y₁ = “F(x₁)≤6”, y₂ = “F(x₂)≤12”, y₃ = “F(x₃)≤18”} denote predicates.

Then we can construct an Index matrix B=[Y, V, {

b_{y_{i}, v_{j}}

}] of probability trials, where

b_{y_{1}, v_{1}}

= 1, b_y1,_v2= 0, b_y1,_v3= 0, b_y1,_v4= 0, b_y1,_v5= 0, b_y1,_v6= 0, b_y2,_v1= 1, b_y2,_v2= 1, b_y2,_v3= 0, b_y2,_v4= 1, b_y2,_v5= 0, b_y2,_v6= 0, b_y3,_v1= 1, b_y3,_v2= 1, b_y3,_v3= 1, b_y3,_v4= 0, b_y3,_v5= 0, b_y3,_v6= 0 and 1 ≤ i≤ 3, 1 ≤ j ≤ 6

In Figure 2 is illustrated the transmission of flows in time.

Download: Download full-size image

Figure 2. Transition of flows in time.

For the two experiments, we have given bigger value of the velocities to less value of time for transmitting flows from one node to another in a network system, and we can conclude that for less value of

λ

, requires more time for transmitting flows from one node to another in a network system.

Each flow

F_{k_{i}}

requests a fixed band width equals to

B_{k_{i}, l_{j}}

\frac{V_{k_{i}}}{T_{l_{j}}}

,(9)

where the volume of transferred bits is denoted by

V_{k_{i}}

, then we can calculate each band width for a each flow, 1≤ i ≤ m, 1≤ j ≤ n. It follows a random uniform distribution within duration denoted by

T_{l_{j}}

interval which follows the random exponential distribution, 1≤ j ≤ n. We model the arrival rate of

F_{k_{i}}

as a Poisson process with density λ_f, 1≤ i ≤ m.

In order to maximize the QoS satisfaction in the network, our objective is to calculate the optimal multicast tree for each

F_{k_{i}}

by allocating the requested bandwidth, where 1≤ i ≤ m. Then, the SDN controller exploits i) our optimization algorithm (SDN application) proposal and ii) OpenFlow rules to respectively calculate and install the multicast routing tree within Fat-Tree DCN.

To be more specific, for flow

F_{k_{i}}

from source u_w to the set of destinations D, we search for a multicast tree that maximizes the residual bandwidth with the fewest possible number of links, 1≤ i ≤ m, 1≤ w ≤ |U|. Then, we consider the binary variable

x_{u_{w}, u_{r}}

equals to 1 when the link

a_{u_{w}, u_{r}}

is selected for the tree path allocation and equals to 0 otherwise, where 1≤ w ≤ |U|, 1≤ r ≤ |U|.

[4]

We denote

y_{u_{w}, u_{r}}

as an auxiliary variable quantifying the residual capacity of the link, expressed as:

y_{u_{w}, u_{r}}

C_{w_{r}}

(

a_{u_{w}, u_{r}}

B_{k_{i}}

x_{u_{w}, u_{r}}

(10)

where

C_{w_{r}}

(

a_{u_{w}, u_{r}}

) should be greater than the requested bandwidth

B_{p_{r}}

and represents the capacity of all links

a_{u_{w}, u_{r}}

. Note that only links

a_{u_{w}, u_{r}}

with sufficient capacity

C_{w_{r}}

(

a_{u_{w}, u_{r}}

) in terms of bandwidth can be considered to build the multicast routing tree in the system, where 1≤ w ≤ |U|, 1≤ r ≤ |U|.

[4]

4.2. Representation of the Non-stationary Poisson Process Model with a Fuzzy Index Matrix

Suppose that the network system consists of n nodes, which will be indexed as

u_{w}

= 1,..., |U| the rest of this paper. Let with the index set K = {k₁ = 1, k₂ = 2, k₃ =3, k₄ = 4, k₅ = 5} denote number of nodes of the network system. Let with the index set L = {l₁ = 1, l₂ = 2, l₃ = 3} denote a time interval in seconds. The arriving flow

F_{k_{i}}

(t) of the

u_{w}

-th node at the

l_{j}

-th time interval is assumed to behave as a non-stationary Poisson process

P (

F_{k_{i}}

(

t_{l_{j}}

)) =

\frac{e^{- λ_{k_{i}} (t_{l_{j}})} λ_{k_{i}} {{(t}_{l_{j}})}^{| U |}}{| U |!}

(11)

where, P denotes the abbreviation of the probability.

Moreover, the varying process of the arriving rates

λ_{k_{i}}

(

t_{l_{1}}

) = 9,

λ_{k_{i}}

(

t_{l_{2}}

) = 6,

λ_{k_{i}}

(

t_{l_{3}}

) = 3 are assumed to change according to a network wide finite-state deterministic or stochastic process, where 1≤ i ≤ 5. The system will enter one state during a certain time interval τ and then enter another state, which as a result changes the arriving rate as well as the arriving flux of the Poisson process.

[7]

Also, we can construct a Fuzzy Index matrix of values of probabilities of a transient process of flows requests with a fixed band width C=

[K, L, \{c_{k_{i}, l_{j}}\}]

, where

c_{k_{1}, l_{1}}

⟨

0.001,0.999

⟩

c_{k_{1}, l_{2}}

⟨

0.015,0.985

⟩

c_{k_{1}, l_{3}}

⟨

0.149,0.851

⟩

c_{k_{2}, l_{1}}

⟨

0.005,0.995

⟩

c_{k_{2}, l_{2}}

⟨

0.045,0.955

⟩

c_{k_{2}, l_{3}}

⟨

0.224,0.776

⟩

c_{k_{3}, l_{1}}

⟨

0.015,0.985

⟩

c_{k_{3}, l_{2}}

⟨

0.089,0.911

⟩

c_{k_{3}, l_{3}}

⟨

0.224,0.776

⟩

c_{k_{4}, l_{1}}

⟨

0.034,0.966

⟩

c_{k_{4}, l_{2}}

⟨

0.134,0.866

⟩

c_{k_{4}, l_{3}}

⟨

0.168,0.832

⟩

c_{k_{5}, l_{1}}

⟨

0.061,0.939

⟩

c_{k_{5}, l_{2}}

⟨

0.161,0.839

⟩

c_{k_{5}, l_{3}}

⟨

0.101,0.899

⟩

where the elements

c_{k_{i}, l_{j}} = ⟨ μ_{k_{i}, l_{j}}, 1 - μ_{k_{i}, l_{j}} ⟩

are fuzzy pairs, where

μ_{k_{1}, l_{j}} + 1 - μ_{k_{1}, l_{j}}

≤ 1 and

μ_{k_{i}, l_{j}}

is calculated with the formula:

F_{k_{i}}

(

z_{l_{j}}

) =

\frac{e^{- λ_{k_{i}} (t_{l_{j}})} λ_{k_{i}} {{(t}_{l_{j}})}^{| U |}}{| U |!}

, where 1≤ i ≤ 5, 1≤ j ≤ 3.

In Figure 3 is illustrated non-stationary Poisson process of arriving flows of nodes in a network system in duration of 1 second, 2 seconds and 3 seconds.

Download: Download full-size image

Figure 3. Non-stationary Poisson process of arriving flows of nodes in a network system.

5. Conclusions

In this article the authors analyze with Index matrices atraffic in network with different scenarios in duration of time and number of flows the load in the Poisson distribution of the network. Moreover, with the non-stationary Poisson process model, represented with Fuzzy Index matrices, can be predicted the probabilities of a transmission of flows in nodes of a network system. As a result, the obtained data from the calculations according to different variants of cases can be programmatically implemented in machine learning of DCN.

Abbreviations

IM	Index Matrix
L-IM	Logical IM
DCN	Data Center Network

Author Contributions

Stela Todorova: Formal Analysis, Data curation, Methodology

Ivan Ivanov: Conceptualization, Investigation, Resources

Data Availability Statement

The data supporting the outcome of this research work has been reported in this manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Atanassov, K. Index Matrices: Towards an Augmented Matrix Calculus. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10945-9
[2]	Todorova, S. Overview of publications on index matrices. (2022-2023) Yearbook of Section "Informatics". Union of Scientists in Bulgaria. pp. 32-62. Tom XII (Volume XII).
[3]	Atanassov, K. Bureva V. Short course on discrete structures. Avangard Prima. Sofia. 2018.
[4]	Roua Touihri. optimizing routing and resource allocation in sdn-based server only data center networks for private cloud architectures. Hardware Architecture [cs.AR]. Université Paris-Est, 2021. English. NNT: 2021PESC0061.
[5]	Todorova, S. Research of the index matrices and their applications. University "Prof. A. Zlatarov" (AU). Faculty of Technical Sciences. Academic degree: Doctor (PhD). 30.09.2024. Supervisor: Assoc. Prof. Veselina Kuncheva Bureva, PhD; Assoc. Prof. Nora Angelova Angelova, PhD. Reviewers: Acad. Prof. Krassimir Todorov Atanassov, DSc DSc Assoc. Prof. Velin Stoqanov Andonov, PhD. Burgas. 113 pages.
[6]	Robert G. Gallager. Stochastic Processes: Theory for Applications. Cambridge University Press. Kindle Edition. 2013.
[7]	Yu-Dong, C., Li, L., Yi, Z., & Jian-Ming, H. Fluctuations and pseudo long range dependence in network flows: a non-stationary Poisson process model. Chinese Physics B, 18(4), 1373. 2009.
[8]	El, Fawal, A. H., Mansour, A., & Nasser, A. Markov-Modulated Poisson Process Modeling for Machine-to-Machine Heterogeneous Traffic. Applied Sciences, 14(18), 8561. 2024. https://doi.org/10.3390/app14188561
[9]	Todorova, S. Representation with Index Matrices Discrete Random Variables. Am J Appl Math. 13(1): 57-63.2025. https://doi.org/10.11648/j.ajam.20251301.14
[10]	Farhi, S. Nenov, G. Kuyumdzhiev, T. Practical circuits with switched capacitors (SC - circuits). State Publishing House "Technika". Sofia. 1987.
[11]	Todorova, S Analysis of Charging and Discharging of Capacitor with Fuzzy Index Matrices. Applied Matrix Theory and Principal Component Analysis in the Digital Era [Working Title]. IntechOpen. 2025. Available at: http://dx.doi.org/10.5772/intechopen.1009996
[12]	Todorova, S. Autoreferat of Research of the index matrices and their applications. University "Prof. A. Zlatarov" (AU). Faculty of Technical Sciences. Academic degree: Doctor (PhD). 30.09.2024. Supervisor: Assoc. Prof. Veselina Kuncheva Bureva, PhD; Assoc. Prof. Nora Angelova Angelova, PhD. Reviewers: Acad. Prof. Krassimir Todorov Atanassov, DSc DSc Assoc. Prof. Velin Stoqanov Andonov, PhD. Burgas. 113 pages.
[13]	Todorova, S. Representation with an Index Matrix, Which Elements are Derivative Functions, Values Read by Arduino Uno and Photoresistor. Tenth Conference on Lighting (Lighting). Sozopol. Bulgaria. 2025, pp. 1-5, https://doi.org/10.1109/Lighting64836.2025.11081781
[14]	Todorova S., Bureva V. Angelova N. Proykov M. Programme Product for Index Matrices. Book of Abstracts of International Symposium on Bioinformatics and Biomedicine 8-10 October 2020, Burgas, Bulgaria. 2020.
[15]	S. Todorova. Method of predicate calculus applying a membership classification rule to create an indexed matrix of belonging. Electronic scientific journal Scientific Atlas. 2021.

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Todorova, S., Ivanov, I. (2025). Analyses with Index Matrices Mathematical Models of Network Systems. American Journal of Applied Mathematics, 13(6), 462-468. https://doi.org/10.11648/j.ajam.20251306.18

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Todorova, S.; Ivanov, I. Analyses with Index Matrices Mathematical Models of Network Systems. Am. J. Appl. Math. 2025, 13(6), 462-468. doi: 10.11648/j.ajam.20251306.18

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Todorova S, Ivanov I. Analyses with Index Matrices Mathematical Models of Network Systems. Am J Appl Math. 2025;13(6):462-468. doi: 10.11648/j.ajam.20251306.18

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@article{10.11648/j.ajam.20251306.18,
  author = {Stela Todorova and Ivan Ivanov},
  title = {Analyses with Index Matrices Mathematical Models of Network Systems},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {462-468},
  doi = {10.11648/j.ajam.20251306.18},
  url = {https://doi.org/10.11648/j.ajam.20251306.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.18},
  abstract = {The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.},
 year = {2025}
}

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TY  - JOUR
T1  - Analyses with Index Matrices Mathematical Models of Network Systems
AU  - Stela Todorova
AU  - Ivan Ivanov
Y1  - 2025/12/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.18
DO  - 10.11648/j.ajam.20251306.18
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 462
EP  - 468
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.18
AB  - The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.
VL  - 13
IS  - 6
ER  -

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Author Information

Stela Todorova

Department of Mathematics, Informatics and Physics, Burgas State University “Prof. Dr. Assen Zlatarov”, Burgas, Bulgaria

Biography: Stela Todorova is a Doctor of Technical sciences graduated at the University "Prof. Dr. Asen Zlatarov", Department “CST”, Faculty of Technical Sciences with the topic of her dissertation "Research of the index matrices and their applications". She graduated High School of Natural Sciences and Mathematics “Acad. Nikola Obreshkov”, Bachelor's degree in Informatics and Computer Science and Master's degree in Information Security at BFU, Burgas. Her current research interests are in the field of index matrices, intuitionistic fuzzy sets and etc.

Research Fields: Index Matrices, Fuzzy logic

Contact Email

http://orcid.org/0000-0002-0418-5853
Ivan Ivanov

Faculty of Technicks and Technologies, Trakia University, Stara Zagora, Bulgaria

Biography: Ivan Ivanov is a doctoral student at Trakia University – Faculty of Engineering and Technology, Department of Electrical Engineering, Electronics and Automation. His dissertation is on the topic of “Optimization of packets in the control plane of SDN networks using machine learning”. He holds a Master’s degree in Precision Mechanical Engineering and Instrumentation from the Technical University – Gabrovo (2000) and a Master’s degree in Automation and Computer Systems from the Faculty of Technicks and Technologies - Yambol (2016). He has participated in international scientific conferences, and his research interests are focused on analysis and optimization of SDN network protocols and application of machine learning for network traffic management.

Research Fields: analysis and optimization of SDN network protocols, application of machine learning for network traffic management

http://orcid.org/0000-0002-2074-8089

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Todorova, S., Ivanov, I. (2025). Analyses with Index Matrices Mathematical Models of Network Systems. American Journal of Applied Mathematics, 13(6), 462-468. https://doi.org/10.11648/j.ajam.20251306.18

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ACS Style

Todorova, S.; Ivanov, I. Analyses with Index Matrices Mathematical Models of Network Systems. Am. J. Appl. Math. 2025, 13(6), 462-468. doi: 10.11648/j.ajam.20251306.18

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AMA Style

Todorova S, Ivanov I. Analyses with Index Matrices Mathematical Models of Network Systems. Am J Appl Math. 2025;13(6):462-468. doi: 10.11648/j.ajam.20251306.18

Copy | Download

@article{10.11648/j.ajam.20251306.18,
  author = {Stela Todorova and Ivan Ivanov},
  title = {Analyses with Index Matrices Mathematical Models of Network Systems},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {462-468},
  doi = {10.11648/j.ajam.20251306.18},
  url = {https://doi.org/10.11648/j.ajam.20251306.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.18},
  abstract = {The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Analyses with Index Matrices Mathematical Models of Network Systems
AU  - Stela Todorova
AU  - Ivan Ivanov
Y1  - 2025/12/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.18
DO  - 10.11648/j.ajam.20251306.18
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 462
EP  - 468
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.18
AB  - The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.
VL  - 13
IS  - 6
ER  -

Copy | Download