Information Technology for Maximizing Energy Consumption for Useful Information Traffic in a Dense Wi-Fi 6/6E Ecosystem

Abstract

In Wi-Fi standards, a relatively narrow range of frequency spectrums is declared as working, on the operation of which additional restrictions are imposed in different countries. When creating dense wireless network ecosystems focused on massive information traffic, this circumstance causes significant interference even in the case of using Wi-Fi 6/6E-compatible equipment. An effective solution to this problem is the implementation of a centralized management mechanism for the relevant parameters of the target network ecosystem. The growing attention to ecology and rational use of electricity makes the problem of maximizing energy consumption for useful information traffic in a dense Wi-Fi 6/6E ecosystem an urgent task. Only the addressed information traffic between the transmitter and the target subscriber, which are subjects of the OFDMA technology and the MU-MIMO multiple access system (with an emphasis on the latter), is considered useful. To solve the problem, the authors formalized the Wi-Fi 6/6E ecosystem’s energy consumption model, which takes into account the specifics of OFDMA and MU-MIMO, the influence of the communication channel characteristics on the speed of target information transfer, and detailed energy consumption for maintaining the network infrastructure in a functional state. Based on the created model, the research problem is represented by the difference between two monotonic functions, relative to which the problem of optimization with restrictions is set. The process of solving this problem is presented in the form of information technology with a branch-and-bound hierarchy and a nested unconditional optimization problem. The results of simulated modelling in the MATLAB-NS3 environment showed a significant advantage of the authors’ approach. The energy power consumption by the Wi-Fi 6/6E ecosystem, the parameters of which were adjusted with the help of the authors’ information technology, decreased by more than four times.

1. Introduction

Let us start the article by mentioning the success story of one of the Mettis Aerospace Corporation’s projects. This is the case when the factory tired of buying tons of cables and decided to give wireless solutions a chance. Mettis Aerospace Corporation is a supplier of parts for Airbus, Boeing, and Rolls-Royce, so their production is serious. The company allocated one of its production facilities in the UK with an area of 11 hectares for a test stand. At the highest level, the task was standard: to provide end-to-end communication between all floors of the building, including the production part. The greatest attention was paid to the reliability of communication between the production lines and the central monitoring and management system. It was necessary to achieve minimum delays without wires with the possibility of traffic prioritization. As tests, they transmitted 4K video broadcasts from production over the network, collected data from IoT sensors, and checked the continuity of the equipment in video conferencing mode, while transferring heavy files in parallel. At the same time, tests were conducted on the use of augmented reality in large-scale production. To do this, each augmented reality device involved needed a stable video stream. Tests on previous generations of Wi-Fi failed, but with Wi-Fi 6, they managed to squeeze a stable 700 Mbps out of the equipment using 80 MHz channels. The average delay was 6 ms, which is considered a very decent result. The tests involved Cisco Catalyst 9100 series routers configured for maximum signal stability.

The driver of success in the mentioned example is the use of equipment that supports the Wi-Fi 6 (IEEE 802.11ax) standard adopted in 2019 [1,2,3]. In 2020, the Wi-Fi Alliance decided to enhance the effect of Wi-Fi 6 and introduced an extension of the current standard to Wi-Fi 6E [3,4]. The main feature of Wi-Fi 6E is to increase the upper limit of the available frequency range to 6 GHz. Note that this extension was needed to answer an extremely important question: how to distribute Wi-Fi in a spacious room where there are a lot of people? Remember how it used to be: you enter the stadium or open sky concert hall, go into the crowd and immediately remain without a cellular connection. The reason was that the cells could not cope with so many people wanting to connect. Over time, GSM stations learned to serve large flows of subscribers (although not without the temporary deployment of additional capacities), but the problem remained with Wi-Fi–it needed to be made more stable and more productive. IEEE engineers achieved the first requirement by implementing support for OFDMA technology and the second by improving the MIMO system to MU-MIMO.

Orthogonal Frequency Division Multiple Access (OFDMA) [5,6,7] is a multi-user version of the well-known Orthogonal Frequency Division Multiplexing (OFDM) technology. In the OFDM variant, the modulation simply provides a good and stable communication channel. Keeping this, OFDMA also cuts the channel into independent sections (the so-called Resource Units (RUs), which work without interfering with each other. The router manages the distribution of RUs. The introduction of OFDMA makes it possible to repeatedly increase the stability and speed of communication in conditions of dense client placement. It is now possible to send frames to multiple users at the same time, which was not possible in previous versions of the Wi-Fi standard. The question arises: at what size of RU will we obtain the maximum gain from using OFDMA? Generally speaking, OFDMA shows better results with small RUs [5,8].

Now let us comment on the previously mentioned Multi-User Multiple-Input, Multiple-Output (MU-MIMO) system [6,7,9,10,11]. This is a good reworking of the already well-known MIMO system (introduced at the time of Wave-2 802.11ac). Unlike the latter, MU-MIMO supports both downlink and uplink. Note that now examples of the practical application of the MU-MIMO system are quite rare. The low prevalence may be due to physical difficulties: clients have to be sufficiently separated in space. In conditions where subscribers move actively and chaotically and the density of the presence of devices consuming traffic reaches several pieces per square meter (for example, shopping centers), this is difficult to achieve. On the other hand, the narrow beamforming technology requires the introduction of additional overhead for already overloaded packets. So MU-MIMO in all its glory can show itself in conditions where a large capacity of the information channel is important.

Which technology is more important–OFDMA or MU-MIMO? This is a question about a whale fighting an elephant. The first technology increases the efficiency of using the frequency band, and the second improves the capacity of information transfer channels. The first focuses on reducing latency, while the second improves the speed of connections. OFDMA is good for small packets, while MU-MIMO is good for large ones. Both technologies should work, but if the network ecosystem is oriented towards a massive transfer, more attention should be paid to MU-MIMO.

A review of modern publications devoted to MIMO modelling [6,7,12,13,14] showed that scientists prefer to describe such systems at the physical level with the implementation of an electromagnetic approach to modelling a MIMO transmission channel. In most cases, the functioning of multiantenna communication systems is modelled using the equivalent representation of the latter in the form of multipoles [14,15,16,17]. In the basis of this approach, the transmitting part (signal generators, transmitting devices, matching devices), the receiving part (receiving devices, matching devices, payloads) and the communication channel (signal propagation medium, transmitting and receiving antennas) are represented by equivalent multipoles and are described by the corresponding parametric matrices. Thus, the receiving and transmitting parts of the simulated MIMO system, as well as the signal propagation medium, appear to be separate but interconnected multipoles. However, more often there are models [17,18,19], in which the receiving and transmitting modules are presented as multipoles, and the medium model (channel matrix) is determined based on the a priori information available (for example, by a Deterministic Ray Tracing method) or based on statistical estimates of the channel. That is, even though the existing physical models of the operation of MIMO systems are not limited to a specific antenna configuration, they are, one way or another, tied to a given channel model obtained either based on experimental studies of the intended working environment or the basis of randomly specified probabilistic characteristics of the signal propagation channel. We can state the actual drift from physical models to statistical ones when the physical component of the simulated system is reflected in the form of certain transfer functions, and the main attention is focused on the values of the characteristic parameters of the simulation object (i.e., the functioning of the MIMO system).

In support of the trend we have identified, the authors of [20] proposed to determine the efficiency of power consumption of a modern communication system through the ratio of the volume of data confirmed received by subscribers to the amount of energy spent on the transfer. At the same time, the authors of the mentioned study introduce a limitation on the amount of energy available to the transmitter and focus on various methods of encoding information, depending on the quality characteristics of the transmission channel. However, the authors of [12,21] immediately focus on the shortcomings of the above approach, in which it is most efficient to transmit information in the form of the smallest RUs at the lowest speed–after all, the mentioned model does not take into account that the energy power is spent not only on information transmission but and to keep the transmission system operational. We also note that the above studies analyze the case of “one transmitter–wireless communication channel–one subscriber”, which does not reflect the specifics of the functioning of the MIMO system. In [22], a generalization is made to the case of a MIMO system, but the authors replace the concepts by assuming that the sum of the power spent on maintaining each connection separately and the power consumed by the MIMO system are equivalent. There are also approaches [23,24] that are close to the definition of the so-called Global Energy Efficiency, as well as to solving various optimization problems that play around with the mentioned term. However, these works do not consider both the specifics of the Wi-Fi 6/6E platform, in general, and the interpretation of the communication ecosystem transmitter as a complex phenomenon, in particular.

Next, the authors of this article identified the main attributes of scientific research.

The object of research is the process of energy consumption by a dense Wi-Fi 6/6E ecosystem with massive information traffic.

The subject of research includes functional analysis, signal transmission theory, and experiment planning theory.

The goal of the research is applied modeling of such a phenomenon as useful information traffic in a dense Wi-Fi 6/6E ecosystem with massive information exchange.

We define the following as research tasks:

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analytically formalize the Wi-Fi 6/6E ecosystem’s energy consumption model, which takes into account the specifics of OFDMA and MU-MIMO, the influence of the communication channel characteristics on the speed of target information transfer, and details of energy consumption for the operation of the network infrastructure,

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present the goal of the research based on the created model in the form of a statement of the optimization problem with restrictions,

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present the process of solving the stated optimization problem in the form of information technology with a branch-and-bound hierarchy,

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justify the adequacy of the proposed mathematical apparatus and demonstrate its functionality in the context of the research goal.

The main contribution of the research is as follows. The growing attention to ecology and rational use of electricity makes the problem of maximizing energy consumption for useful information traffic in a dense Wi-Fi 6/6E ecosystem an urgent problem. Only the addressed information traffic between the transmitter and the target subscriber is considered useful. To solve the problem, the authors formalized the novel Wi-Fi 6/6E ecosystem’s energy consumption model, which takes into account both the specifics of OFDMA and MU-MIMO, the influence of the communication channel characteristics on the speed of target information transfer and detailed energy consumption for maintaining the network infrastructure in a functional state. Based on the created model, the problem of optimization with restrictions is set. The process of solving this problem is presented in the form of information technology for maximizing energy consumption for useful information traffic in a dense Wi-Fi 6/6E ecosystem with a branch-and-bound hierarchy and a nested unconditional optimization problem.

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information technology for maximizing energy consumption for useful information traffic in a dense Wi-Fi 6/6E ecosystem,

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a simulation model of energy consumption by the Wi-Fi 6/6E ecosystem, which takes into account the specifics of OFDMA and MU-MIMO.

2. Materials and Methods

2.1. Statement of the Research

Let there be a set of $R$ Wi-Fi 6/6E routers of the same type, united in a network ecosystem with active OFDMA and MU-MIMO, focused on the consumption of massive traffic by $U$ subscribers. The $R⥂U$ information transfer is implemented in the format of active information channels “router transmitter”–“subscriber”. The router spends energy power ${\eta }_c^{-1}{\pi }_c$ to support the channel $c$, where ${\eta }_c^{-1}$ is the efficiency of the amplifier, and ${\pi }_c$ is the average resulting energy power of the information signal in the channel $c$. We also note that to support the router in an active state (without taking into account the power that is aimed at supporting information interaction) power is equal to ${\pi }_r$ what is spent. Accordingly, $R$ routers of the same type consume energy power $R{\pi }_r$ regardless of the traffic intensity in the network architecture. The concept of multiple access assumes that the target information signal from the transmitter $c$ can be received not only by subscriber $c$, but also by other subscribers. Let us determine the share of information traffic from the transmitter $k$ to the subscriber $l$ in the form of a coefficient $0\le {\alpha }_{kl}\le 1$, $k\in R$, $l\in U$. Let us determine the share of information traffic from transmitter $k$ to transmitter $l$ in the form of a coefficient $0\le {\beta }_{kl}\le 1$, $k,l\in R$. In the context of what has been said, the useful information traffic (that is, from the transmitter $c$ to the subscriber $c$) can be characterized as ${\alpha }_{cc}>0$, ${\beta }_{cc}=0$ $\forall c$.

Based on what has been said, the classical approach [12,20,21] to determine the total power $\mathrm{H}$ consumed by a network architecture with an active multiple access system to support useful information transfer is formalized by expression

$$\mathrm{H}={\displaystyle \sum _{c=1}^{RU}{\mathrm{H}}_c}={\displaystyle \sum _{c=1}^{RU}\frac{s_c}{{\pi }_c+{\pi }_r}},$$

(1)

where $c$ is the channel index, $s_c$ is the average speed of information transfer in channel $c$.

The advantage of expression (1) for calculating energy consumption for the support of the researched process is its simplicity. At the same time, approach (1) is characterized by several serious shortcomings. First, expression (1) is a simple sum of energy consumption without taking into account the “physics” of the process, which, in the context of Wi-Fi 6/6E, consists in the support of the investigated network architecture not only for the MU-MIMO system but also for the OFDMA technology. Secondly, in the conditions when ${\displaystyle \sum _c{\pi }_c}\le R{\pi }_r$, the value of function (1) will characterize to a greater extent not the use of energy for useful information transfer, but the expenditure of energy for the functioning of the system as such. This tendency will be even more aggravated if we consider the expression (1) as the objective function of the problem of maximizing the expenditure of energy power for the support of useful information transfer. Finally, function (1) does not reproduce the dynamic characteristics of information transfer. Summarizing what has been said, let us express the static consumption of useful energy through the ratio of the average capacity of information channels on the subscriber side to the average consumed energy, i.e.,:

$$1/\mathrm{H}\left({\displaystyle \sum _{u=1}^U\mathrm{H}\left(s_u,\gamma \right)}/U\right),$$

(2)

where $\mathrm{H}\left(s_u,\gamma \right)=\left\{\left(s_u^\left(1-\gamma \right)\right)/\left(1-\gamma \right)\forall 0\le \gamma <1;\log \left(s_u\right)\forall \gamma =1\right\}$, and $\gamma \in \left[0,1\right]$ is the balancing coefficient, the variation of which value within the established limits allows synchronously changing the value of the function (2) in the range from the arithmetic mean to the geometric mean of its value.

Let us rethink expression (1) taking into account function (2):

$$\stackrel{⌢}{\mathrm{H}}\left(\overrightarrow{s},\overrightarrow{\gamma }\right)=1/\left(\mathrm{H}\left(\frac{1}{U}{\displaystyle \sum _{u=1}^U\mathrm{H}\left(s_u,\gamma \right)}\right)\left(R{\pi }_r+{\displaystyle \sum _{u=1}^U{\eta }_u^{-1}{\pi }_u}\right)\right).$$

(3)

We detail expression (3) by expressing the rate of information transfer $s_k$ in the channel $k$ as a function of energy power ${\pi }_k$ and energy power of Nyquist noise ${\tau }_u$ for subscriber $u$:

$$s_k=f\left({\rho }_k\left(\overrightarrow{\pi },\overrightarrow{\tau }\right)\right)=f\left({\alpha }_{kk}{\pi }_k/\left({\tau }_k+{\displaystyle \sum _{l=1,l\ne k}^U{\alpha }_{kl}{\pi }_l}\right)\right),$$

(4)

where $k,l\in U$, and ${\rho }_u\left(\pi ,\tau \right)$ is the signal-to-noise ratio for the information channel of subscribers $u\in U$, expressed in terms of our research.

In the signal transmission theory [25], the function $f\left({\rho }_k\left(\overrightarrow{\pi },\overrightarrow{\tau }\right)\right)$ is defined for a wide range of pulse-code sequences. In the context of expression (4), we will further interpret the function (3) as $\stackrel{⌢}{\mathrm{H}}\left(\overrightarrow{\pi },\overrightarrow{\tau },\overrightarrow{\gamma }\right)\equiv \stackrel{⌢}{\mathrm{H}}\left(\overrightarrow{\pi }\right)$, where $\pi$ is a vector of controlled parameters and $\gamma$ is a balancing constant determined by the researcher.

So, the problem of maximizing energy consumption for useful information transfer in the Wi-Fi 6/6E ecosystem of routers of the same type with an emphasis on MU-MIMO operation and taking into account OFDMA technology is formalized by an objective function and a system of constraints of the form

$$\begin{array}{c}\stackrel{⌢}{\mathrm{H}}\left(\pi \right)\to \max ,\\ \left\{\begin{array}{l}{\beta }_{kl}{\pi }_l\le {\stackrel{⌢}{T}}_k^{\left(R\right)}\forall k,{\pi }_l>0,\\ {\alpha }_{kl}{\pi }_l\le {\stackrel{⌢}{T}}_k^{\left(U\right)}\forall k,{\pi }_l>0,\\ 0\le {\pi }_k\le {\stackrel{⌢}{T}}_k^{\left(\pi \right)}\forall k,\end{array}\right.\end{array}$$

(5)

where $k,l\in U$.

The first limitation regulates the “transmitter”–“transmitter” synchronization and characterizes the connectivity of the network architecture at the specified minimum threshold values of energy consumption for the support of information signals ${\stackrel{⌢}{T}}_k^{\left(R\right)}$. The second limitation is focused on OFDMA and regulates the “subscriber”–“transmitter” synchronization, which is typical for establishing a Wi-Fi connection, at the given minimum threshold values of energy consumption for the support of information signals ${\stackrel{⌢}{T}}_k^{\left(U\right)}$. The third limitation is focused on MU-MIMO and limits the power of the information transfer signal to the corresponding threshold value ${\stackrel{⌢}{T}}_k^{\left(\pi \right)}$.

2.2. The Markov Concept of the Energy Efficiency Assessment of the Edge Computing Infrastructure Peripheral Server Functioning over Time

The set optimization problem (5) explicitly does not take into account the function (4). Let us remove this restriction by solving the system of linear equations $s_k=f\left({\rho }_k\left(\overrightarrow{\pi },\overrightarrow{\tau }\right)\right)$.

Let us present the logarithm of function (3) as the difference between two monotonic functions of the argument $\overrightarrow{s}$:

$$\log \stackrel{⌢}{\mathrm{H}}\left(\overrightarrow{s}\right)=\varphi \left(\overrightarrow{s}\right)-\phi \left(\overrightarrow{s}\right),$$

(6)

$$\varphi \left(\overrightarrow{s}\right)=\log \mathrm{H}\left(U/{\displaystyle \sum _{c=1}^U\mathrm{H}\left(s_c\right)}\right),$$

(7)

$$\phi \left(\overrightarrow{s}\right)=\log \left(R{\pi }_r+{\displaystyle \sum _{c=1}^U{\eta }_c^{-1}{\pi }_c\left({\rho }_c\left(s_c\right)\right)}\right).$$

(8)

Let us examine the functions (7) and (8). The first of them is a monotonic function of the argument $\overrightarrow{s}$. Let us focus on function (8). The classical signal transmission theory [25] says: if the function of the signal-to-noise ratio is ${{\rho }^{\prime }}_c\ge {\rho }_c\forall c\in U$, then the inequality ${{\pi }^{\prime }}_c\ge {\pi }_c\forall c\in U$ is satisfied for the corresponding power characteristics. At the same time, if ${s^{\prime }}_c\ge s_c\forall c\in U$, then ${{\rho }^{\prime }}_c\ge {\rho }_c\forall c\in U$. Following this sequence of statements, we can conclude that the function $\phi \left(\overrightarrow{s}\right)$ is also a monotonic function of its argument.

If function (6) is formed by the difference between two identical (monotone) functions, then classical optimization and mathematical programming methods [26,27] can be used to find its extremum. For this, it is enough to introduce the variable $\phi \in \left[-\exp \left({\mathrm{\Phi }}_{\max }\right),-\exp \left({\mathrm{\Phi }}_{\min }\right)\right]$ and take this circumstance into account by formulating the objective function based on (6):

$$\underset{s,\phi }{\max }\varphi \left(\overrightarrow{s}\right)-\log \left(-\phi \right)\forall \exp \left(\phi \left(\overrightarrow{s}\right)\right)+\phi \le 0.$$

(9)

The objective function (9) is a monotonic function of the arguments $\left\{\overrightarrow{s},\phi \right\}$. At the beginning of Section 2.2, we focused on the fact that there is a linear dependence between the elements of the vectors $\overrightarrow{s}$, $\overrightarrow{\pi }$. Therefore, all constraints of the optimization problem (9) will be monotone functions of the form $f\left(\overrightarrow{s},\phi \right)\le 0$. Thus, the objective function (9) is a replacement for the entity of the same type in the formulation of the optimization problem (5). The controlled variables in this version of the optimization problem will be $\left\{\overrightarrow{s},\phi \right\}$, unlike the original (5), where the controlled variables are $\left\{\overrightarrow{\pi }\right\}$.

We formalize the process of static solution of the problem of maximizing energy consumption for useful information transfer in the Wi-Fi 6/6E ecosystem with an active MU-MIMO system and OFDMA technology. Let us present the solution of such an optimization problem with the objective function (9) and constraints (5) based on the branch-and-bound hierarchy [26]. With this approach, the original optimization problem is represented by a tree of subproblems, each of which describes a rectangular fragment with an upper face ${\left(\overrightarrow{s},\phi \right)}^{+}$ and a lower face ${\left(\overrightarrow{s},\phi \right)}^{-}$ in the parametric space $\left(\overrightarrow{s},\phi \right)$, respectively.

The information technology for obtaining the optimal solution for the objective function (9) will consist of a hierarchical sequence of the following stages:

1. Formalization stage. The value of the desired solution error $\delta$ is set. The creation of a stack of subtasks is initiated, into which the first instance of a subtask describing a rectangular area with a lower face $\left(\overrightarrow{s}\ne 0,\phi =-\exp \left({\mathrm{\Phi }}_{\max }\right)\right)$ and an upper face $\left(\overrightarrow{s}=\left\{r_c=f\left({\alpha }_{cc}{\stackrel{⌢}{\pi }}_c/u_c\right)\right\},c=\overline{1,U};\phi =-\exp \left({\mathrm{\Phi }}_{\min }\right)\right)$ is entered;

2. Selection stage. We implement a variant of the First-In-First-Out method, therefore, at this stage, the last subtask recorded in the stack is selected as active for the next stage;

3. Branching stage. We find the longest face of the rectangle corresponding to the active subproblem and split the latter in half along this face, forming two new subproblems. Note that the parametric space $\left\{\overrightarrow{s},\phi \right\}$ is formed by parameters of a different nature ($\overrightarrow{s}$ is the vector of transfer speed values in information channels, and $\phi$ is the energy used to support them). To bring the values of these controlled parameters to an approximately common order, it is permissible to apply the normalization procedure to the established parameters of the solved optimization problem;

4. Solution stage. For each pair of subproblems obtained at the previous stage, within the corresponding rectangles, there are approximate solutions by the chosen zero-order unconditional optimization method (this means, for example, the Dichotomy method, the Golden Section method, the Fibonacci method. By default, the authors recommend choosing the second method). To implement the selected method, auxiliary variables $\overrightarrow{a}={\left(\overrightarrow{s},\phi \right)}^{-}$, $\overrightarrow{b}={\left(\overrightarrow{s},\phi \right)}^{+}$ are initiated. Next, the vector $\overrightarrow{d}=0.62\overrightarrow{a}+0.38\overrightarrow{b}$ is calculated and it is checked whether the values of the elements of this vector satisfy the constraint (5) and the additional condition (9). If the checks were successful, the vector $\overrightarrow{d}$ replaces the vector $\overrightarrow{a}$, otherwise, the vector $\overrightarrow{b}$. In this way, the domain of admissible solutions is reduced. The Golden Section method completes its work after finding a vector ${\overrightarrow{a}}^{\ast }=\left({\overrightarrow{s}}_{{\overrightarrow{a}}^{\ast }},{\phi }_{{\overrightarrow{a}}^{\ast }}\right)$ for which the value of the objective function $\varphi \left({\overrightarrow{s}}_{{\overrightarrow{a}}^{\ast }}\right)-\log \left(-{\phi }_{{\overrightarrow{a}}^{\ast }}\right)$ exceeds the previously recorded suboptimal value of this function.

The logic of using the vector $\overrightarrow{b}$ to estimate the upper bound for the objective function (9) repeats what was described above about the vector $\overrightarrow{a}$. The difference is that if the found estimate for the upper bound of the objective function (9) exceeds the previously fixed value of this characteristic by less than the value of $\delta$, then the found values of the controlled variables are fixed, the current subtask is removed from the stack, and the technology moves to the next stage (Finalization). Otherwise, the found values of the controlled variables are fixed, the stack is supplemented with the found subtasks, and the technology proceeds to the implementation of Stage 2.

5. Finalization stage. The stack is checked for current subtasks. The presence of such when activating admission to the current stage indicates the incompatibility of restrictions (5) or non-fulfilment of the additional condition (9). If the stack is empty, then the technology is considered successfully implemented, and the found suboptimal values are further considered optimal, found with an error of $\delta$.

We recall that the solution obtained using the above-described technology is interpreted by us as static, which is typical for optimization problems that are solved for values of controlled parameters of the model fixed at a certain moment.

It should be noted that even if stationary terminals act as subscribers, the technology of finding the optimal speed scheme for implementing information transactions in the studied Wi-Fi 6/6E ecosystem should be repeated. The obvious reason for this is the need for periodic synchronization of communication between the subscriber and the router, neglecting which can lead to the blocking of the corresponding channel. If the studied ecosystem is open to a large number of mobile subscribers, then it will be necessary to implement the proposed technology to support the optimal parameters of information transactions much more often (it was this circumstance that prompted the authors to reduce the optimization problem (9) to an unconditional optimization problem (see Stage 4 of information technology), because the proposed technology is simple enough for software implementation in the system environment of a centralized management controller (for example, Cisco Catalyst 9800 series) without reducing the functionality of the latter. However, let us try to express the recommended frequency of restarting the proposed technology as a function of the amount of data transmitted by the information channels of the network ecosystem.

Let it be established that during the censored period of operation of the studied network ecosystem, a sufficiently large amount of data, equal to $I={\displaystyle \sum _{c=1}^U{I}_c}$, was transmitted by all information channels, for which the energy capacity $\mathrm{\Pi }$ was spent. Let us express the objective function (6) taking into account these data:

$$\log \stackrel{⌢}{\mathrm{H}}=\log \mathrm{H}\left(U/{\displaystyle \sum _{c=1}^{U}U\left(I_c\right)}\right)-\log \mathrm{\Pi }.$$

(10)

It is logical to connect the periodicity of the implementation of the proposed technology with the increase in the value of the objective function (10). A natural way to evaluate the latter is to transit to the derivative of the function (10), whose analytical form will be obtained taking into account expression (2) $\forall 0\le \gamma \le 1$ as

$${\left(\log \stackrel{⌢}{\mathrm{H}}\right)}^{\prime }=\left({\displaystyle \sum _{c=1}^U\frac{{I^{\prime }}_c}{I_i^{\gamma }}}\right)/\left({\displaystyle \sum _{c=1}^U{I}_c^{1-\gamma }}\right)-\frac{{\mathrm{\Pi }}^{\prime }}{\mathrm{\Pi }}.$$

(11)

Considering that ${I^{\prime }}_c=s_c\forall c\in U$, ${\mathrm{\Pi }}^{\prime }=-\phi$, expression (11) will acquire its completed form:

$${\left(\log \stackrel{⌢}{\mathrm{H}}\right)}^{\prime }=\left(\log \stackrel{⌢}{\mathrm{H}}\left(\overrightarrow{s},\phi \right)\right)=\left({\displaystyle \sum _{c=1}^U\frac{s_c}{I_i^{\gamma }}}\right)/\left({\displaystyle \sum _{c=1}^U{I}_c^{1-\gamma }}\right)+\frac{\phi }{\mathrm{\Pi }}$$

(12)

Function (12), like function (9), is a monotone function of the arguments $\left(\overrightarrow{s},\phi \right)$. Therefore, to maximize the function (12), we can use the proposed technology, summarized in Stages 1–5. At the same time, the objective function based on expression (12) is supplemented by an additional condition (9) and a system of constraints (5).

3. Results

The information technology presented in Section 2 has an obvious applied orientation. However, this circumstance does not make simulation modelling impractical. Let us formulate the initial conditions for such a study. Let a dense Wi-Fi 6/6E ecosystem be deployed within a square perimeter with an area of ${10}^4$ square meters. In total, 140 stationary subscribers’ terminals are evenly distributed in this area (this scenario simulates a typical corporate workspace, in which the use of mobile gadgets is prohibited in the corporate security and information protection policy).

The number of routers is a controlled parameter, the value of which varies in the range of $\left[1,30\right]$ instances. The order of placement of routers within the specified perimeter is uniform and symmetrical relative to its center. The specified conditions are included in the scheme described in the study [28]. In this work, for the initial conditions formulated above, the information signal fading function at a distance $L$ from the router is formalized as

$$L\left(s\right)=40.1+20\left(\lg \left(f_c/2.4\right)+\lg \left(\min \left(s,10\right)\right)\right)+{{\rm I}}_{s>10}35\lg \left(0,1s\right)$$

(13)

where $f_c$ is the frequency of the wireless connection (for Wi-Fi 6/6E: $f_c\in \left[1,6\right]$ GHz) and ${{\rm I}}_{s>10}=\left\{1\forall s>10;0\forall 0\le s<10\right\}$ is an indicator function.

The authors information technology for the specified conditions will be implemented with the value of the balancing coefficient $\gamma \in \left[0,1\right]$ (see expression (2)). We also provide a list of established parameters, the values of which affect the simulation results:

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the height at which the router is mounted: $2.7$ m;

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the height of placement of the subscriber’s terminal: $1.0$ m;

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$MAX\left({\stackrel{⌢}{\pi }}_c\right)\forall c\in U=40$ mW;

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noise spectral power density ${\rho }_c$, $c\in U$: −175 dBpHz;

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amplifier noise spectral power density ${\tau }_k$, $k\in R$: 8 dB;

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${\eta }_k^{-1}=0.1$, $\forall k\in R$;

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channel width: $80$ MHz;

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subscriber’s adapter sensitivity: $-96$ dBm;

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router’s power consumption in standby mode ${\pi }_r$: 850 mW.

Before the implementation of the authors’ information technology, it is necessary to evaluate the relationship between the information transfer speed $s_c$ and the signal-to-noise ratio ${\rho }_c$ for the instance of the channel $c\in U$. We obtained the following dependency for the MutFed speed adaptation algorithm in the MATLAB-NS3 environment (project site on GitHub: https://github.com/vkrepo/MATLAB-NS3 (accessed on 27 July 2023)). The obtained empirical dependence $s_c=f\left({\rho }_c\right)$ in the step function form is presented in Figure 1.

We can prove the functionality of the authors’ information technology only in comparison with a competitor. This was the selected scheme called OFB (implementation of an instance of Wi-Fi 6/6E ecosystem based on Cisco Catalyst 9115, 9120, 9130 series routers interacting with a centralized Cisco Catalyst 9800 series controller through a Cisco Catalyst 9100 series Embedded Wireless Controller. An instance of the network ecosystem was implemented with controlled parameters, the values of which were set by immersion, that is, “Out of The Box”). This competitor was opposed by the AIT scheme (optimization of communication parameters for $MAC$-subscribers through Embedded Wireless Controller methods. Connection parameters were determined centrally using the authors’ information technology, software implemented in the information environment of the Cisco Catalyst 9800 series controller).

For both competing schemes, the information exchange between sets of “subscribers” and “routers” was implemented according to a single, defined scenario with a linearly increasing intensity of information transfer. An additional source of reducing energy consumption in a dense Wi-Fi 6/6E ecosystem (with a defined schedule of information transactions) is turning off routers that will not serve any subscriber during the time interval defined in the schedule (Adaptive Power mechanism option). The implementation of such a power-saving mechanism for the access control schemes described above is characterized by adding the symbol “+” after the identifier of the corresponding scheme, that is, OFB+, AIT+.

For a defined Wi-Fi 6/6E ecosystem and a single schedule of information transactions, we will estimate the dependence of the average arithmetic capacity $M=\frac{1}{U}{\displaystyle \sum _{c=1}^U{I}_c}$ of the information channel $c\in U$ on the number of routers $R=\overline{1,30}$ and the selected version of the access speed control scheme from the set $\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}$. In graphic form, the dependence $M=f\left(R=\overline{1,30};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ is presented in Figure 2.

For a defined Wi-Fi 6/6E ecosystem and a single schedule of information transactions, we will estimate the dependence of the average geometric capacity $G={\left({\displaystyle \prod _{c=1}^U{I}_c}\right)}^{\frac{1}{U}}$ of the information channel $c\in U$ on the number of routers $R=\overline{1,30}$ and the selected version of the access speed control scheme from the set $\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}$. In graphic form, the dependence $G=f\left(R=\overline{1,30};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ is presented in Figure 3.

For a defined Wi-Fi 6/6E ecosystem and a single schedule of information transactions, we will estimate the dependence of the power $\mathrm{\Pi }$ consumed by routers $R=\overline{1,30}$ to support information transfer in the volume $I={\displaystyle \sum _{c=1}^U{I}_c}$, on the selected version of the access speed control scheme from the set $\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}$. In graphic form, the dependence $\mathrm{\Pi }=f\left(R=\overline{1,30};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ is presented in Figure 4.

For a defined Wi-Fi 6/6E ecosystem and a single schedule of information transactions, we calculate the dependence of energy consumption $\mathrm{H}$ consumed by routers $R=\overline{1,30}$ for useful information transfer from the volume $I$ on the selected version of the access speed control scheme from the set $\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}$ using expression (11). In graphic form, the dependence $\mathrm{H}=f\left(R=\overline{1,30};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ is presented in Figure 5.

It is also interesting to evaluate the dependence of the characteristics $M$, $G$ and $1/\mathrm{H}$ on the number of active subscribers $U=\overline{10,140}$ and the selected version of the access speed control scheme from the set $\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}$. In graphic form, the dependences $M=f\left(U=\overline{10,140};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$, $G=f\left(U=\overline{10,140};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ and $1/\mathrm{H}=f\left(U=\overline{10,140};\left\{OF{B}_{-}^{+},AI{T}_{-}^{+}\right\}\right)$ are presented in Figure 6, Figure 7 and Figure 8, respectively.

As shown in Figure 6 and Figure 7, dependencies are obtained for a defined Wi-Fi 6/6E ecosystem, a single schedule of information transactions, and the number of routers $R=24$.

4. Discussion

Let us start the discussion with the results presented in Figure 2 and Figure 3, which, in general, characterize the dependence of the average arithmetic $M$ (Figure 2) and average geometric $G$ (Figure 3) transfer speed in the information channel on the number of routers $R$ in a dense Wi-Fi 6/6E ecosystem. It should be noted that the information technology presented in the article is focused on maximizing energy consumption for useful information traffic and not on achieving the maximization of the average speed of the information connection in the network ecosystem. It is this circumstance that explains the fact that $\left\{M,G\left(OF{B}_{-}^{+}\right)\right\}>\left\{M,G\left(AI{T}_{-}^{+}\right)\right\}\forall R>1$. Only at $R=1$ does this trend break because the entire channel resource of the ecosystem is centrally distributed by a single router. Under such conditions, energy consumption for useful traffic decreases with an increase in the average speed in the information channel, which, however, is limited by other factors. In general, the growing trend of the graphs in Figure 2 and Figure 3 can be explained by the fact that multiple access algorithms coordinate information transactions so that the subscriber connects to the router closest to him. Decreasing this distance encourages the activation of faster signaling code tables to service the connection. We also note that with the increase in the number of routers in the network ecosystem, the probability of frame collisions caused by the appearance of hidden stations increases. This is what can explain the minimum of graphs $M,G\left(AI{T}_{-}^{+}\right)$ $R=5$. However, $R>5$ these graphs grow steadily, which confirms the functionality of the AIT scheme.

You cannot be the first in everything–you have to be the first in what is important. This truth of life best characterizes the graphs presented in Figure 4 (which describes the total energy consumption of the network ecosystem) and Figure 5 (which describes the expenditure of energy on useful information traffic) and the triumph of the proprietary information technology-based AIT scheme over the competing OFB scheme. It can be seen that in metric $\mathrm{\Pi }$, the AIT scheme outperforms the OFB scheme by two times, and the AIT+ scheme outperforms the OFB+ scheme by almost four times. Recall that “+” means activation of the router’s shutdown function if, according to the schedule of information transactions, it will not serve any subscriber for the corresponding time interval. Interestingly, the “turning point” in the character of those shown in Figure 4 and Figure 5 dependencies have the same value $R=5$ of the argument. This can be explained by the fact that precisely with a smaller number of routers, the authors’ information technology could not ensure an even distribution of the channel resource between active routers for the corresponding number and location of subscribers (see exclusion in Stage 5 of information technology). The dominance of the AIT and AIT+ schemes over competitors is also evident in the $\mathrm{H}$ metric (Figure 5). The only thing that saves the competitor is the reduction of energy consumption by turning off inactive routers (OFB+ graph in Figure 5), but this possibility is provided exclusively by the availability of a reliable schedule of information transactions, which, in the real conditions of operation of the network ecosystem, is more the exception than the rule.

Finally, let us comment on Figure 2, Figure 3, Figure 4 and Figure 5 in the complex. Experiments have shown that the use of the authors’ information technology (schemes AIT, AIT+) leads to a decrease in the average arithmetic and average geometric speed of the information connection in the investigated network ecosystem by one and a half times compared to the competing schemes OFB, OFB+. Instead, the experiments showed that the application of the authors’ information technology allows the investigated network ecosystem to spend more than four times less energy (compared to OFB, OFB+) to transmit the same amount of information. If we talk about the ratio of energy consumption to support useful information traffic (Figure 5), the advantage of the AIT and AIT+ schemes is even more convincing. Thus, the goal of the research can be considered achieved, and the functionality of the authors’ information technology can be considered proven.

Let us draw the attention of respected readers to the graph $\mathrm{\Pi }={\displaystyle \sum _R{\pi }_r}$ from Figure 4. This line characterizes the minimum possible power consumed by the investigated network ecosystem, all of which routers are in standby mode, not supporting any information transaction. Comparing the rest of the graphs in Figure 4 with this line, it is possible to state, firstly, the reliability of the measurements for the OFB and AIT schemes, the rates of which are above the Σπr curve, and secondly, that the author’s AIT+ scheme (with an active function of disabling inactive routers) allows the ecosystem to perform its functional purpose while consuming even less power than the corresponding number of routers solely to support its active state. This trend becomes more apparent as the number of routers in the ecosystem grows. This circumstance allows us to predict a promising direction of further research, namely, the creation of information technology for the formation of an optimal plan of network connections for the target Wi-Fi 6/6E ecosystem.

As shown in Figure 6, Figure 7 and Figure 8, the results only confirm the conclusions and trends formulated by us above. In addition, let us pay attention to the fact that increasing the number of subscribers from 10 to 100 led to a decrease in the average connection speed in the network ecosystem by more than seven times with a constant number of routers equal to 24. This trend should be taken into account if the connection speed is a critical parameter for the network ecosystem being created. We also draw readers’ attention to Figure 8, which shows the increasing energy consumption in Joules for the transmission of 1 Mbit to the target subscriber with the increase in the total number of the latter. The rapid growth of all graphs in this figure is the best confirmation of the relevance of our research topic.

5. Conclusions

The growing attention to ecology and rational use of electricity makes the problem of maximizing energy consumption for useful information traffic in a dense Wi-Fi 6/6E ecosystem an urgent task. Only the addressed information traffic between the transmitter and the target subscriber, which are subjects of the OFDMA technology and the MU-MIMO multiple access system (with an emphasis on the latter), is considered useful.

To solve the problem, the authors formalized the novel Wi-Fi 6/6E ecosystem’s energy consumption model, which takes into account both the specifics of OFDMA and MU-MIMO technologies, the influence of the communication channel characteristics on the speed of target information transfer, and detailed energy consumption for maintaining the network infrastructure in a functional state.

Based on the created model, the research problem is represented by the difference between two monotonic functions, relative to which the problem of optimization with restrictions is set. The process of solving this problem is presented in the form of information technology with a branch-and-bound hierarchy and a nested unconditional optimization problem. The results of simulated modelling in the MATLAB-NS3 environment showed a significant advantage of the author’s approach.

The authors believe that a promising direction of further research will be the creation of information technology for the energy-efficient network connections plan formation for the target Wi-Fi 6/6E ecosystem. At the same time, the authors plan to take into account the experience of such known studies as [29,30,31,32].