Reliably Controlling Massive Traffic between a Sensor Network End Internet of Things Device Environment and a Hub Using Transmission Control Protocol Mechanisms

Abstract

The constant growth of Internet of Things traffic is ensured by the ongoing evolution of the hierarchy of all hardware links of sensor networks. At the same time, the implementation of the Edge computing ideology regulates the complexity of the “first-mile” section (from the sensors array to the peripheral server). Here, the authors suggest paying attention to the growing share of massive traffic from target sensors in the total traffic of the sensors array. This circumstance makes it expedient to introduce an additional link to the peripheral server for summarizing massive traffic from target sensors. The authors present a sensor network end IoT device (SNEIoTD), implemented grounded on a reliable and cheap Raspberry Pi computing platform, as such a link. The introduction of this SNEIoTD makes it possible to reduce the probability of information loss from the critical infrastructure of a smart city and increase the flexibility of controlling the massive traffic of the first mile. In this context, the urgent task is the reliable control of information transfer from the SNEIoTD environment to a hub, which the authors formalize based on Transmission Control Protocol (TCP). This article proposes a mathematical model of the interaction of the main mechanisms of the TCP in the form of a queuing system. As part of this model, a semi-Markov process of an information transfer with a unified speed is selected and its stationary distribution is analytically formalized. A computationally efficient information technology for determining the TCP Window Size is formulated, taking into account the interaction of TCP mechanisms in the process of massive traffic control. Using the example of TCP Westwood+ protocol modification, it is shown that the results of the application of information technology permit increases in the stability of data transfer under the circumstances of increasing Round-Trip Times.

1. Introduction and State-of-the-Art

1.1. Relevance of the Research

The continued expansion of Internet of Things (IoT) traffic [1,2,3] is facilitated by the ongoing enhancement of the hardware links within sensor networks. Concurrently, the adoption of Edge computing ideology [4,5,6] serves to manage the intricacies of the “first-mile” segment (from the sensor array to the peripheral server). It is noteworthy to highlight that the increasing proportion of mass traffic originating from specific target sensors, which, in certain cases, surpasses the cumulative traffic of the entire sensor array (for instance, a single 4k camera in a city’s video surveillance system generates more traffic than the entirety of the city’s smart traffic lights). This observation underscores the necessity of introducing an additional link to the peripheral server to aggregate substantial traffic from these target sensors. The authors of this article have previously investigated, in [7,8], the issue of the integration of a sensor network end IoT device (SNEIoTD), developed on the reliable and cost-effective Raspberry Pi computing platform, as a viable solution to address this need. The incorporation of such a SNEIoTD aims to diminish the likelihood of information loss within the critical infrastructure of a smart city and enhance the adaptability of controlling massive traffic at the initial stage. Consequently, a pressing challenge in this context is the effective control of information transfer from the SNEIoTD environment to a hub, a task that the authors formalize based on the TCP.

The endpoint in the studied system is a SNEIoTD, which aggregates information from sensors that generate massive traffic, and, as it accumulates, transmits it to the hub. The hub can be either a peripheral server or a cloud data storage itself. At the same time, an SNEIoTD is a significantly simpler computing device than a hub (which is why we mention Raspberry Pi in this context). It should be noted that the idea of using an SNEIoTD as an intermediate link was inspired by practice. While organizing the process of transferring information from 4k cameras of a city’s video surveillance system, we were faced with multiple cases of information loss due to the unreliability of the 5G information transmission channel. After we introduced an additional link (the “sensor network end IoT device”) into the “4k camera array”—“hub” system, we were able to solve the above problem (Figure 1).

Let us explain in more detail why in this research we focus on the transport layer of the Open Systems Interconnection (OSI) model and the TCP. As we mentioned above, sensor networks are designed to collect critical information (for example, information from cameras of a city’s video surveillance system). This information can characterize functioning industries, various processes in a smart city, and people’s preferences. In each case mentioned, the loss of collected information can have extremely significant negative consequences, even catastrophic. In the context of the above, the issue of reliability in transferring information from the elements of the sensor array to the final point of its placement comes to the forefront.

An OSI model’s transport layer is developed to guarantee reliable data transmission from a sender to a recipient [9,10,11]. However, the degree of its reliability may exhibit significant fluctuations. There exist numerous transport-layer protocols [12,13,14], from protocols that solely offer fundamental transport capabilities (such as data transfer functions without acknowledgement) to protocols that guarantee the delivery of multiple data packets to a destination in the proper sequence. Multiplexing numerous data streams establishes a data flow control mechanism and ensures the reliability of the received data. For instance, the User Datagram Protocol (UDP) [15,16] is confined to overseeing the integrity of data within a single datagram and does not preclude the chance of losing an entire packet, duplicating packets, or disturbing the order of received data packets. On the other hand, the TCP guarantees dependable and uninterrupted data transmission, preventing data loss or disruption of the sequential order, and can redistribute data by breaking down large data portions into smaller fragments and, conversely, consolidating fragments into a single packet. This circumstance, as well as the fact that over 90% of the world’s data traffic is controlled via TCP [17,18], became a decisive factor influencing the authors’ choice.

1.2. State-of-the-Art

In different wireless network structures, such as those found in the IoT, TCP retains its significance as a transport protocol owing to its dependable delivery characteristics and extensive prevalence on the Internet [19,20]. Among the various attributes of the TCP, congestion control stands out as a key feature that has garnered sustained attention over time, remaining the focus of numerous research studies.

Research on this topic typically concentrates on the following methods: active queue control, network feedback techniques, and alterations in a TCP’s algorithm, as well as different combinations of these methods [21,22,23,24]. This section provides a brief overview of studies that have tackled issues concerning TCP and the IoT.

In the publication [25], a comprehensive comparative analysis of TCP’s congestion control algorithms within the IoT context is outlined. This study assesses 14 different alternatives, delineating their respective benefits and drawbacks. The findings suggest that the primary challenges observed among the evaluated algorithms predominantly lie in issues related to data transfer rate and equity.

With the proliferation of intercommunicating devices, congestion inevitably rises. In the realm of the IoT, this escalation is particularly pronounced, driven by the continual appearance of new devices able to join an existing network. In such a dynamic ecosystem, effective congestion control becomes imperative to accommodate a diverse range of devices, each potentially employing a different variation of a transport protocol. This involves adapting data transfer speeds in response to network congestion, all while ensuring fairness and sustaining throughput across the network [25].

Typically, proposals for modifications align with a standard TCP [26,27] or involve particular improvements to the control algorithms of established versions. Within these alterations, protocols utilizing Round-Trip Time for measuring network congestion are categorized as proactive, while those relying on segment losses for congestion identification are deemed reactive. As noted by the authors of [28], even though reactive protocols effectively utilize the entire accessible data transfer capacity, they may not be well-suited for equipment operating with a limited data transfer capacity.

Conversely, pre-emptive protocols aim to minimize data segment losses and retransmissions but may fall short of leveraging the entire network capacity. In light of these considerations, the study in [28] modifies the TCP that can function in either pre-emptive or proactive modes based on the network environments. This protocol initially operates in pre-emptive mode before congestion escalation is identified, at which point it transitions to proactive mode. The hybrid variant demonstrated commendable performance in terms of data transfer speeds and equity in contrast with the New Reno, CUBIC, HTCP, and FAST-FIT variants.

Another recent study delving into modifications of transport protocols is outlined in [29]. CERL+ stands out as a version inspired by TCP Reno, with its primary aim being to optimize the data transfer rate while minimizing the adverse effects stemming from lost data segments. Their suggestion utilizes mean and minimum Round-Trip Time (RTT) estimates to discern connection bottleneck circumstances and identify instances of random data segment loss attributed to connection failures, a common occurrence in 802.11 networks. In comparison with various other TCP variants [12,13,14], including NewReno, mVeno, Westwood+, YeAh, CUBIC, and New Jersey+, the CERL+ suggestion exhibited a superior data transfer rate and lower data segment loss, mitigating bottleneck congestion across diverse wireless network structures.

By integrating a blend of the aforementioned methodologies, the authors in [30] introduce FAIR+, a synthesis of Active Queue Management (AQM) mechanisms, network reactions, and TCP adjustments. Their proposal demonstrated favourable outcomes in terms of latency, goodput, and fairness metrics within multi-hop wireless network scenarios, outperforming varied alternatives including NRT, OQS, and Westwood.

A concept structurally close to the system described in Section 1.1 was presented in [31,32]. Those papers introduce a system for device indoor localization that uses variations in the strength of the wireless signal. Their proposed system addresses logistics use cases in which it is imperative to achieve reliable end-to-end delivery, such as pharmaceutic delivery, delivery of confidential documents and court exhibits, and even food delivery, since vulnerabilities in these deliveries present a potential risk of terrorism or other attacks. This work proposes a concept based on a low-power and low-cost LoRaWAN system that utilizes a Machine Learning technique based on Neural Networks to achieve high accuracy in device indoor localization by measuring the signal strength of a beacon device. However, in these studies, LoRaWAN is used as a communication technology. This platform does not provide the speed characteristics necessary for our case. We also note that the use of neural networks as a model of the studied process requires significant preliminary work on collecting and summarizing training data.

In general, it should be noted that the authors of the aforementioned studies strive to pay attention to the description of the competition for resources for their information and communication systems between a large number of non-resource-intensive connections. This does not fit into the concept of using the SNEIoTD described in Section 2.1, which determines the relevance of this research.

1.3. Main Attributes of this Research

Next, we will present the main elements that determine the orientation, structure and novelty of the presented research.

An object of this research is an extensive process of controlling massive data transfer from the SNEIoTD environment to a hub as a result of the interaction of the main mechanisms of the TCP.

A subject of this research includes the theory and methods of mathematical modelling, probability theory, queuing theory, and recovery theory.

The goal of this research is articulated as follows: to analytically determine the unified speed of information transfer from the SNEIoTD environment to a hub controlled by the main mechanisms of TCP, taking into account the specific features of this process.

Our research objectives are:

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to formalize the parametric space for an adequate representation of the research object;

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to formalize the concepts that characterize the features of the interaction of the main mechanisms of the TCP (Slow Start, Additive-Increase/Multiplicative-Decrease) when controlling massive data transfer from the SNEIoTD environment to a hub discretely and continuously;

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to formalize the information technology for calculating the TCP Window Size of both mentioned mechanisms and the <ssthresh> parameter;

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to implement a numerical experiment, the results of which will justify the effectiveness of the created mathematical apparatus.

Now, let us state the main contribution of the research.

The IoT systems are designed for the collection of information. If an entire system of smart city traffic lights generates dozens of megabytes of traffic, and a high-quality video surveillance system alone generates gigabytes of traffic, ensuring its reliable transmission to the cloud using a wireless communication channel becomes a non-trivial task. This article proposes an idea to alleviate this constraint by introducing an additional hardware element into the classical “sensor network”—”cloud” structure (SNEIoTD, see Figure 1). To achieve maximum benefit from this proposal, the authors have suggested a mathematically founded TCP-based information technology for effective control of reliable transmission of massive traffic between a SNEIoTD and the edge of the cloud.

This article proposes a mathematical model of the interaction of the main mechanisms of the TCP (Slow Start, Additive-Increase/Multiplicative-Decrease) in the form of a queuing system, which describes the process of reliable control of massive traffic from the SNEIoTD environment to a hub. As part of this model, a semi-Markov process is selected that describes the interaction of these mechanisms, locked to the definition of such a characteristic parameter as the unified speed of information transfer. As a result of recurrent analyses of the aforementioned process, a stationary distribution of an embedded Markov chain is determined. This made it possible to analytically describe the stationary distribution of the original semi-Markov process. The aforementioned stationary distribution analytical representation became the basis for the formalization of an accurate, computationally efficient information technology for calculating the TCP Window Size of both the previously mentioned mechanisms and the <ssthresh> parameter.

In the following sections of the article, three main sections are highlighted. Section 2 introduces the parametric space that characterizes the interaction of TCP mechanisms in the process of traffic control from the SNEIoTD environment to a hub. The distribution function of the stochastic parameter TCP Window Size for the studied process was investigated. Elements of the research on the dynamic properties of the TCP Window Size stochastic parameter over time are also formalized. Section 3 is devoted to simulation modelling to demonstrate the information technology formulated in Section 2.2. Section 3 also provides empirical results on the example of the TCP Westwood+ protocol, which proves the functionality of the proposed mathematical apparatus. In Section 4, generalized conclusions and directions for further investigation are outlined.

2. Materials and Methods

2.1. Statement of the Research

As mentioned above, in our research we will focus on the analytical description of the influence of the settings of the Slow Start (SlS) and Additive-Increase/Multiplicative-Decrease (AI/MD) mechanisms integrated in TCP on the value of the TCP Window Size controlled parameter $S_w$ for the target connection between the SNEIoTD environment, which generates massive, mission-critical traffic, and a hub. The parameter $S_w$ defines the size of the data block that the SNEIoTD environment can send to a hub without confirmation of receipt from the latter. The interaction of the SlS and AI/MD mechanisms is determined by a controlled parameter, $T_{sls}$, known as the Slow Start Threshold [21,22,23,24], and feedback signals $s=\left\{s^{+},s^{-},s^o\right\}$ from a hub, which will be mentioned later in the paper. The parameter known as Round-Trip Time [21,22,23,24], which characterizes the time required to send a block of data to the hub and receive confirmation of its receipt, is denoted by $T_{rt}$.

After each confirmation from the hub, the SlS mechanism increases the value of the parameter $S_w$ by one. If such a trend is maintained, the value of the parameter $S_w$ doubles in a time not less than $T_{rt}$. In turn, after each confirmation from the hub, the AI/MD mechanism increases the value of the parameter $S_w$ by $1/S_w$. The value of such an increase is presented in integer form: $⌊1/S_w⌋$, therefore (while maintaining the trend of receiving confirmations from a hub) after each subsequent time interval, not less than $T_{rt}$, the value $S_w$ will increase by one. The inherent TCP Three-Way Handshake logic [17,18] assumes the existence of three types of reverse communication signals (confirmation signals from a hub):

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data block received in full: $s^{+}$,

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data block received with skips: $s^{-}$,

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reverse signal not received (communication channel overload): $s^o$.

The initiation of an information interaction between the SNEIoTD environment and a hub begins with setting the initial value of the parameter $T_{sls}$. Further, after each SNEIoTD reception of the signal $!s^{+}$, the value of the parameter $T_{sls}$ is redefined: $T_{sls}:=\max \left(2,⌊S_w/2⌋\right)$. The characteristics of the information interaction between the SNEIoTD and a hub are regulated with the SlS mechanism until the SNEIoTD environment receives the signal $s^{-}$ or the value of the parameter $S_w$ reaches the value of $T_{sls}$. The moment of detection of certain facts regulates the transfer of connection control to the AI/MD mechanism. Upon receiving the signal $s^{-}$, the AI/MD mechanism continues to work: $S_w:=\max \left(2,⌊S_w/2⌋\right)$. Returning control of the connection to the SlS mechanism is possible only upon receipt of the SNEIoTD signal $s^o$. At the same time, the SlS mechanism resets the value $S_w$ to one: $S_w:=1$, and regulates reattempts to send the data block to a hub after a time equal to $T_{rt}$ has passed. If the SNEIoTD does not receive a signal $s^{+}$ in response, the SlS mechanism authorizes another attempt to send the data block to a hub after a time interval equal to $T_{rt}:=2T_{rt}$. The value $S_w$ remains unchanged: $S_w=1$. These described actions are repeated until the SNEIoTD receives the signal $s^{+}$.

We will choose a theoretical basis for an analytical description of the process of information interactions between the SNEIoTD environment and a hub, taking into account the described logic of control. The authors propose a formalization of the researched process based on the queuing theory [7,8] with a controlled input flow. To serve the latter, we assume one device with an infinite queue, a reverse connection, and a deterministic service of duration $t_m=1/T_{cc}$, where $T_{cc}$ is the actual capacity of the route between the SNEIoTD and a hub. The set of data blocks represents a population of incoming requests. The control of the flow of incoming requests is regulated by the above-described logic of application of the SlS and AI/MD mechanisms built into the TCP. Let us assume that at the time of receiving the signal $s$ from the population of incoming service requests, $n>0$ blocks of data are transmitted, where the value $n$ depends on the parameter $S_w$ and several data blocks being sent to a hub without confirming the fact of their receipt. Accepted incoming requests are served in order of arrival (FIFO).

We denote with $p_n^{+}$ the probability of obtaining a series of signals $s^{+}$ of length $n$. We denote with $p_n^{+-}$ the probability of obtaining an arbitrary series of signals $s^{+}$, $s^{-}$ of length $n$. We denote with $p_n^o$ the probability of obtaining a signal $s^o$ for a series of data blocks of length $n$ sent to a hub at $S_w=n$ (if the studied process is controlled via the AI/MD mechanism) or $S_w\ge n$ (if the studied process is controlled via the SlS mechanism).

All further initiatives regarding the formalization and research of the relevant queuing system are based on the following initial postulates:

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the probabilities $p=\left\{p_n^{+},p_n^{+-},p_n^o\right\}$ are determined exclusively by an existing value of the parameter $S_w$ and do not depend on the previous values of the parameters $S_w$, $T_{sls}$;

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for the probabilities $p$, the equality $p_n^{+}+p_n^{+-}+p_n^o=1$ is fulfilled. At the same time, when $n=1$, the values of probabilities $p$ represent the functioning of the SlS mechanism, and when $n>1$ the values of probabilities $p$ represent the functioning of the AI/MD mechanism;

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SNEIoTD always has relevant data for sending to a hub;

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the initiating value $S_w^{\max }$ is at the same time a deterministic upper limit of the value of the parameter $S_w$ for any moment of implementation of the studied process;

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parameter $T_{rt}$ is a stochastic value with distribution function $T_{rt}\left(x\right)$.

A visual interpretation of the above-described logic of interaction between the SlS and AI/MD mechanisms is presented in Figure 2.

2.2. Research of the Distribution Function of the Stochastic Parameter TCP Window Size for the Studied Process

In Section 2.1, we noted that the controlled parameter $S_w$ is significant for the target connection between the SNEIoTD environment and a hub. Let us investigate the stochastic nature of this parameter, taking into account that the studied process is controlled via SlS & AI/MD mechanisms, the interaction between which is characterized by the controlled parameter $T_{sls}$.

Let us introduce the vector $\overrightarrow{F}=\left(S_w,T_{sls},\alpha \right)$, where $\alpha$ is a binary parameter whose value $\alpha =0$ corresponds to the SlS mechanism, and $\alpha =1$ corresponds to the AI/MD mechanism. Next, we will investigate the stochastic process ${\left\{\overrightarrow{F}\left(t\right)\right\}}_{t>0}$. Let the value of the parameter $S_w$ be measured at discrete moments represented by the sequence ${\left\{t_i\right\}}_{i>0}$. Subject to the postulates defined in Section 2.1, the sequence ${\left\{{\overrightarrow{F}}_k=\overrightarrow{F}\left(t_k\right)\right\}}_{k>0}$ ordered by the value of the argument is the aperiodic, irreducible Markov chain with a bounded set of states and with a stationary distribution $q_{\overrightarrow{F}}$. Considering that $T_{sls}:=⌊S_w/2⌋$ after each fact of receiving a signal $s^{-}$ or $s^o$, then (under the condition $\exists n:$ $p_n^{+-}>0$ or $p_n^o>0$) all $\overrightarrow{F}$ of the inequalities $T_{sls}>S_w^{\max }$, $q_{\overrightarrow{F}}=0$ are true, because all such states will be irreversible. Let us introduce the notation $q_{\overrightarrow{F}}=q_{T_{sls},S_w}^{\alpha }$. In this case, starting from the logic of interactions of SlS and AI/MD mechanisms described in the previous section, we can write: $q_{T_{sls},S_w}^0=0$ $\forall S_w\ge T_{sls}$, $q_{T_{sls},S_w}^1=0$ $\forall S_w<T_{sls}$.

We accept that $L=⌊S_w^{\max }/2⌋$. Taking into account the introduced notations, the stationary distribution of the stochastic parameter $S_w$ for SlS and AI/MD mechanisms will be described by the expressions $q_i^0={\displaystyle \sum _{n=i+1}^L{q}_{n,i}^0}$, $q_i^1={\displaystyle \sum _{n=2}^{\min \left(i,L\right)}{q}_{n,i}^1}$, respectively. For all $2<n\le ⌊L/2⌋$, the probability $q_{\overrightarrow{F}}$ satisfies the following equations:

$$q_{n,n}^1=p_{2n}^{+-}{\displaystyle \sum _{i=2}^{2n}{q}_{i,2n}^1}+p_{2n+1}^{+-}{\displaystyle \sum _{i=2}^{2n+1}{q}_{i,2n+1}^1}+p_1^{+}{q}_{n,n-1}^0+p_1^{+-}\left({\displaystyle \sum _{i=2n+1}^L{q}_{i,2n}^0}+{\displaystyle \sum _{i=2n+2}^L{q}_{i,2n+1}^0}\right),$$

(1)

$$q_{n,1}^0=p_{2n}^o{\displaystyle \sum _{i=2}^{2n}{q}_{i,2n}^1}+p_{2n+1}^o{\displaystyle \sum _{i=2}^{2n+1}{q}_{i,2n+1}^1}+p_1^o\left({\displaystyle \sum _{i=2n+1}^L{q}_{i,2n}^0}+{\displaystyle \sum _{i=2n+2}^L{q}_{i,2n+1}^0}\right).$$

(2)

Equations (1) and (2) match with the first quarter of the index space and characterize the system’s transitions between the SlS and AI/MD mechanisms. At the same time, Equation (1) characterizes the transitions for the AI/MD mechanism after the arrival of the signal $s^{-}$ and the transitions from the SlS mechanism to the AI/MD mechanism, which can occur if, as a result of the operation of the SlS mechanism, $S_w$ becomes equal to $T_{sls}$, or if the SlS mechanism detects a pass in the data sequence. Equation (2) is interpreted so that a transition to a state $q_{n,1}^{)}$ can take place from any other state of a Markov chain if the SNEIoTD receives a signal $s^o$.

The equations for $q_{n,n}^1$, $q_{n,1}^0$ for indices $⌊L/2⌋<n\le L$ are obtained as a result of the reduction of Equations (1) and (2), with the exclusion of the aforementioned irreversible states:

$$q_{n,n}^1=p_1^{+}{q}_{k,k-1}^0+p_{2n}^{+-}{\displaystyle \sum _{i=2}^{2n}{q}_{i,2n}^1}+p_{2n+1}^{+-}{\displaystyle \sum _{i=2}^{2n+1}{q}_{i,2n+1}^1},$$

(3)

$$q_{n,1}^0=p_{2n}^o{\displaystyle \sum _{i=2}^{2n}{q}_{i,2n}^1}+p_{2n+1}^o{\displaystyle \sum _{i=2}^{2n+1}{q}_{i,2n+1}^1}.$$

(4)

Finally, let us characterize the traffic control on the SNEIoTD side under the conditions of the arrival of signals $s^{+}$, which leads to an increase in the value of the parameter $S_w$. For all, $2<n\le S_w^{\max }$ we write:

$$q_{n,i}^1=p_{i-1}^{+}{q}_{n,i-1}^1,n<i;$$

(5)

$$q_{n,S_w^{\max }}^1=p_{S_w^{\max }-1}^{+}{q}_{n,S_w^{\max }-1}^1+p_{S_w^{\max }}^{+}{q}_{n,S_w^{\max }}^1.$$

(6)

After the value of the parameter $S_w$ reaches $S_w^{\max }$, its growth stops, even if new signals $s^{+}$ arrive. For the SlS mechanism, this means that

$$q_{n,i}^0=p_1^{+}{q}_{n,i-1}^0,n<L,n>i.$$

(7)

Based on Equation (7), we write:

$$q_{n,i}^0={\left(p_1^{+}\right)}^{i-1}{q}_{n,1}^0.$$

(8)

Let us select implementations of Equation (5) for which the condition $L<i<S_w^{\max }$ is fulfilled and sum them for $\forall n=\overline{2,L}$:

$$q_i^1={\displaystyle \sum _{n=2}^L{q}_{n,i}^1}=p_{i-1}^{+}{\displaystyle \sum _{n=2}^L{q}_{n,i-1}^1}=p_{i-1}^{+}{q}_{i-1}^1.$$

(9)

Based on Equation (9), we write: $q_i^1=H_i{q}_L^1$ $\forall i>L$, where $H_i={\displaystyle \prod _{k=L}^{i-1}{p}_k^{+}}$. Note that $q_i^1={\displaystyle \sum _{n=2}^{\min \left(i,L\right)}{q}_{n,i}^1}$. If $i>L$, then $q_{i,i}^1=0$; otherwise, $q_{i,i}^1>0$. Accordingly, the system of Equations (1)–(7) can be reduced to the form of a system of equations concerning the variables $q_{n,1}^0$ and $q_{n,n}^1$.

Based on the value of the probabilities $q_i^0$ and $q_i^1$, we can analytically express the dynamics of the value of the parameter $S_w$ for AI/MD and SlS mechanisms, respectively.

We characterize the dynamics of the value of the TCP Window Size parameter for the case when the information interaction between a SNEIoTD and a hub is controlled via the AI/MD mechanism through probabilities $q_i^1$. For $i<L$, we write: $q_{n,i}^1=p_{i-1}^{+}{q}_{n,i-1}^1$, $n=\overline{2,i-1}$, $i=\overline{n+1,L}$. Let us sum the obtained equations in terms of $n$:

$${\displaystyle \sum _{k=2}^{i-1}{q}_{k,i}^1}=p_{i-1}^{+}{\displaystyle \sum _{k=2}^{i-1}{q}_{k,i-1}^1}.$$

(10)

Based on the previously introduced notation, Equation (10), we can write $q_i^1-q_{i,i}^1=p_{i-1}^{+}{q}_{i-1}^q$ and, in turn:

$$q_i^1=H_i{q}_L,\forall i>J,$$

(11)

$$q_{i-1}^1=\left(q_i^1-q_{i,i}^1\right)/p_i^{+},\forall i\le J.$$

(12)

Now, let us characterize the dynamics of the value of the TCP Window Size parameter for the case in which the information interaction between the SNEIoTD and a hub is controlled via the SlS mechanism through probabilities $q_i^0$. This mechanism functions as long as the condition $S_w<T_{sls}$ is fulfilled. In terms of $q_i^0$, this means that

$$q_i^0={\displaystyle \sum _{k=i+1}^L{q}_{k,i}^0}={\left(p_1^{+}\right)}^{i-1}{\displaystyle \sum _{k=i+1}^L{q}_{k,1}^0}.$$

(13)

Let us reduce the value of the iterator in Equation (13) by one: $q_{i-1}^0={\displaystyle \sum _{k=i}^L{q}_{k,i-1}^o}={\left(p_1^{+}\right)}^{i-2}{\displaystyle \sum _{k=i}^L{q}_{k,1}^0}$, from which $q_{i-1}^0={\left(p_1^{+}\right)}^{i-2}\left({\left(p_1^{+}\right)}^{1-i}{q}_i^0+q_{i,1}^0\right)$. Let us present the last equation in its completed form:

$$q_{i-1}^0={\left(p_n^{+}\right)}^{-1}{q}_i^0+{\left(p_1^{+}\right)}^{i-2}{q}_{i,1}^0.$$

(14)

Therefore, Equations (11), (12) and (14) characterize the dynamics of the value of the TCP Window Size parameter for AI/MD and SlS mechanisms (in terms of probabilities $q_i^0$ and $q_i^1$, respectively).

Let us summarize the process of solving the system of Equations (1)–(4) in the context of the obtained recurrent expressions (8) and (9). As a result of the substitution, we find:

$$q_{n,i}^{\alpha }=q_L^1{C}_{n,i}^{\alpha },$$

(15)

where $\forall ⌊L/2⌋\le n\le L$: $C_{k,1}^0=p_{2n}^o{H}_{2n}+p_{2n+1}^o{H}_{2n+1}$, $C_{n,n}^1=p_{2n}^{+-}{H}_{2n}+p_{2n+1}^{+-}{H}_{2n+1}+{\left(p_1^{+}\right)}^{n-1}{C}_{n,1}^0$; and $\forall 2<n<⌊L/2⌋$: $C_{n,1}^0=p_1^o\left(B_{2n}+B_{2n+1}\right)+p_{2n}^o{A}_{2n}+p_{2n+1}^o{A}_{2n+1}$, $C_{n,n}^1=p_{2n}^{+-}{A}_{2n}+p_{2n+1}^{+-}{A}_{2n+1}+{\left(p_1^{+}\right)}^{n-1}{B}_{n,1}+p_1^{+-}\left(B_{2n}+B_{2n+1}\right)$, where $A_i=q_i^1/q_L^1$, $i<L$; $A_{n,i}=q_{n,i}^1/q_L$, $B_{n,i}=q_{n,i}^0/q_L$, $B_i=q_i^0/q_L^1$.

Let us summarize the material of this section by formulating the information technology for calculating the probabilistic characteristics of TCP Window Size for the process of information interaction between the SNEIoTD environment and a hub controlled via AI/MD and SlS mechanisms. The input parameters are the values of probabilities $p_n^{+}$, $p_n^{+-}$, and $p_n^o$ (see Section 2.1), and the value of the parameter $S_w^{\max }$ (the maximum allowable sliding window size). The output parameters are the calculated values of the probabilities $q_i^0$, $q_{i,1}^0$, $q_i^1$, and $q_{i,1}^1$. Information technology includes the following stages:

Stage 1. For $k=\overline{L,S_w^{\max }-1}$, we calculate $A_{k+1}=p_{k-1}^{+}{A}_k$, $A_L:=1$ and take $B_k:=1$.

Stage 2. We calculate $A_{S_w^{\max }}=A_{S_w^{\max }-1}/\left(1-p_{S_w^{\max }}^{+}\right)$.

Stage 3. For each $k=\overline{L,3}$, we calculate

$$\begin{gathered} C_{i,1}^0=p_{2i}^o{A}_{2i}+p_{2i+1}^o{A}_{2i+1}+p_1^o\left(B_{2i}+B_{2i+1}\right)I\left(i\right), \\ {B}_i=C_{i+1,1}^0{\left(p_1^{+}\right)}^{i-1}+B_{i+1}/p_1^{+}, \\ {C}_{i,i}^1=p_{2i}^{+-}{A}_{2i}+p_{2i+1}^{+-}{A}_{2i+1}+{\left(p_1^{+}\right)}^{i-1}{C}_{i,1}^0+p_1^{+-}\left(B_{2i}+B_{2i+1}\right)I\left(i\right), \\ {A}_{i-1}=\left(A_i-C_{i,i}^1\right)/p_i^{+}, \end{gathered}$$

consecutively, considering at each iteration that $I\left(i\right)=\left\{\begin{array}{l}0\forall i>⌊L/2⌋,\\ 1\forall i\le ⌊L/2⌋.\end{array}\right.$

Stage 4. We calculate $q_L=1/{\displaystyle \sum _{k=1}^{S_w^{\max }}\left(B_k+A_k\right)}$.

Stage 5. We accept that $q_i^0:=B_i{q}_L$, $q_{i,1}^0=C_{i,1}^0{q}_L$, $q_i^1=A_i{q}_L$, and $q_{i,1}^1=C_{i,1}^1{q}_L$.

The information technology presented above is characterized by a linear computational complexity $O\left(S_w^{\max }\right)$. This makes it possible to calculate $3S_w^{\max }$ variables.

2.3. Elements of the Research of a TCP Window Size Dynamic Properties in Continuous Time

Let us examine the stochastic process ${\left\{\overrightarrow{F}\left(t\right)\right\}}_{t>0}$ in continuous time. For this process, a Markov chain ${\left\{{\overrightarrow{F}}_k\right\}}_{k>0}$ is needed; accordingly, the process ${\left\{\overrightarrow{F}\left(t\right)\right\}}_{t>0}$ is semi-Markov. Let $\left\{t_i\right\}$ be a sequence of moments during which the value of the TCP Window Size parameter changed. For all $T_{sls}$ values, we introduce the notation ${\gamma }_{S_w}^{\alpha }=\Phi \left[t_{i+1}{-}_i{t}_i\left|{\overrightarrow{F}}_i=\left(T_{sls},S_w,\alpha \right)\right.\right]$.

If $T_{rt}$ has a finite, non-zero mathematical expectation, then the stochastic process ${\left\{\overrightarrow{F}\left(t\right)\right\}}_{t>0}$ is ergodic and its distribution converges to the stationary one:

$$P_{\overrightarrow{F}}=\underset{t\to \infty }{\lim }{P}_{\overrightarrow{F}}\left(t\right)={\gamma }_{S_w}^{\alpha }{q}_{T_{sls},S_w}^{\alpha }/{\displaystyle \sum _{S_w=1}^{S_w^{\max }}{\gamma }_{S_w}^{\alpha }{\displaystyle \sum _{j=2}^L{q}_{j,S_w}^{\alpha }}}.$$

(16)

The statement enclosed in Equation (16) is derived as per an ergodic theorem [33] for semi-Markov processes. It is important to mention that $t_{i+1}-t_i>0$, $t_{i+1}-t_i\ge S_w/T_{cc}$; $t_{i+1}-t_i=T_{rt}$ if $T_{rt}>S_w/T_{cc}$. Accordingly, the stochastic process ${\left\{\overrightarrow{F}\left(t\right)\right\}}_{t>0}$ is regular, so $0<{\gamma }_{S_w}^{\alpha }<\infty$ if $\Phi \left[T_{rt}\right]$ is finite. To establish the stationary distribution (16), it is enough to calculate ${\gamma }_{S_w}^{\alpha }$.

If the process of information interaction between the SNEIoTD environment and a hub is controlled through the AI/MD mechanism, the value of the TCP Window Size parameter changes when the number of sent data blocks matches a value of $S_w$ at the current time of sending data to a hub. The described procedure for sending data at a fixed TCP Window Size takes $S_w{t}_0$. If at the end of this time interval, the reverse communication signal from a hub is not received, the SNEIoTD waits for it, subsequently updating the value $S_w$. The waiting interval is equal to $T_{rt}$; therefore,

$${\gamma }_{S_w}^1={\displaystyle \underset{S_w{t}_0}{\overset{\infty }{\int }}xd{T}_{rt}\left(x\right)}+S_w{t}_0{T}_{rt}\left(S_w{t}_0\right),$$

(17)

where $t_0=1/T_{cc}$.

If the process of information interaction between the SNEIoTD environment and a hub is controlled through the SlS mechanism, then the value of the TCP Window Size parameter is doubled in the period $T_{rt}$. Accordingly, the value of the parameter $S_w$ manages to change several times during one implementation $T_{rt}$. A direct description of this variant of the studied process of information interaction within the framework of this research is not rational. It is more productive to use the verified approximate scheme of the structure $T_{rt}$ described in [34]: ${\gamma }_{2S_w}^0={\displaystyle \underset{2S_w{t}_0}{\overset{\infty }{\int }}\left(x-2\left(S_w-1\right)t_0\right)d{T}_{rt}\left(x\right)}+t_0{T}_{rt}\left(2S_w{t}_0\right)$. In all other cases, ${\gamma }_i^0=t_0$, except for ${\gamma }_1^0$, the value of which is regulated by the postponement procedure. As we noted above, in the absence of a signal $s^{+}$, the delay procedure involves doubling the waiting period $k$ times in a row. Taking this into account, we write:

$${\gamma }_1^0=\left(p_1^{+}{\displaystyle \sum _{i=1}^{k-1}i{\left(1-p_1^{+}\right)}^{i-1}}+k{\left(1-p_1^{+}\right)}^{k-1}\right)\Phi \left[T_{rt}\right]$$

(18)

To calculate the characteristics in (16)–(18) based on the $P_{\overrightarrow{F}}\left(t\right)$ distribution, we suggest using the method described in [35].

3. Results and Discussion

We will demonstrate the functionality of the mathematical apparatus presented in the previous section by applying simulation modelling theory providences. Our focus will be on the stationary distributions $q_i=q_i^0+q_i^1$, $i=\overline{1,S_w^{\max }}$, determined using the information technology formulated in Section 2.2, and their first two moments (mathematical expectation and variance).

In our calculations, we used the initial values synthesized according to the rules generalized in the form of the corresponding sets. Set 1 is formed on the basis that the facts of receiving SNEIoTD signals $s^{-}$ (which indicate that a hub received data blocks with errors) are independent and characterized by the probability $p_{s^{-}}$. At the same time, it was assumed that the probability $p_{s^o}$ of receiving a communication channel overload signal $s^o$ when sending one data block did not change with increases in the value of the parameter $S_w$ for the AI/MD mechanism. Accordingly, in the parametric space $\left\{p^{+},p^{+-},p^o\right\}$, Set 1 was characterized by the following analytical expressions: $p_n^{+}={\left(1-p_{s^{-}}\right)}^n\left(1-p_{s^o}\right)$, $p_n^{+-}=\left(1-{\left(1-p_{s^{-}}\right)}^k\right)\left(1-p_{s^o}\right)$, $p_n^o=p_{s^o}$. Set 2 differed from Set 1 in that, when forming it, we considered the facts of receiving signals $s^o$ from the SNEIoTD environment to be independent. Accordingly, we analytically characterized Set 2 as follows: $p_n^{+}={\left(1-p_{s^{-}}\right)}^n{\left(1-p_{s^o}\right)}^n$, $p_n^{+-}=\left(1-{\left(1-p_{s^{-}}\right)}^k\right){\left(1-p_{s^o}\right)}^n$, $p_n^o=1-{\left(1-p_{s^o}\right)}^n$. The specific ranges of variation in the values of the controlled parameters $S_w$, $p_{s^{-}}$, and $p_{s^o}$ were chosen to ensure the visibility and interpretability of the obtained results.

We determined the visual form of the dependence of $q_i=f\left(S_w,\left\{\mathrm{Set}1,\mathrm{Set}2\right\}\right)$ on $p_{s^o}=10\times {10}^{-3}$, $p_{s^{-}}=\left\{0.5\times {10}^{-3},1.5\times {10}^{-3},10\times {10}^{-3}\right\}$. These calculated results are presented in Figure 3 and Figure 4.

Graphs from Figure 3 and Figure 4 demonstrate that if the probability $p_{s^o}$ of receiving a signal $s^o$ is low, then this characteristic parameter does not have a noticeable effect on the shape of the stationary distribution $q_i$ for the entire investigated range of values of the parameter $S_w$. The form of the obtained stationary distributions for different values of the probability $p_{s^{-}}$ corresponds to expectations and indicates that, for the established initial conditions, the information interaction between the SNEIoTD environment and a hub is almost constantly under the control of the AI/MD mechanism.

The transition from Set 1 to Set 2 does not change the nature of the distributions $q_i$, absorbing only the amplitude values of the latter. However, we investigated the sensitivity of the $q_i$ distributions to increases in the value of $p_{s^o}$ by increasing the value of this parameter to the level of $p_{s^o}=25\times {10}^{-3}$. We then repeated this calculation of dependences for $p_{s^o}=25\times {10}^{-3}$, $p_{s^{-}}=\left\{0.5\times {10}^{-3},1.5\times {10}^{-3},10\times {10}^{-3}\right\}$. The obtained results are visualized in Figure 5 and Figure 6.

From the findings presented in Figure 5 and Figure 6, our results show that increasing the value of the parameter $p_{s^o}$ by 2.5 times led to a noticeable increase in the frequency of observation of low TCP Window Size values. At certain ratios of the values of parameters $p_{s^{-}}$, $p_{s^o}$ distributions $q_i$ degenerate into the form of monotonically decreasing functions. The latter means that the information interaction between the SNEIoTD environment and a hub is almost constantly under the control of the SlS mechanism, which is accompanied by the degradation of the bandwidth of the target connection. In Figure 6, this effect is more pronounced, because the values from Set 2 take into account the fact that the probability $p_{s^o}$ increases with the increase in the value of $S_w$; accordingly, the dominance of the SlS mechanism begins to be observed at lower values of the parameter $S_w$, and as the value of this parameter increases, it ceases to depend on the value of the probability $p_{s^{-}}$. To emphasize the conclusions made here, we visualized the three-dimensional dependencies $q_i=f\left(S_w,p_{s^o},p_{s^{-}}=2.5\times {10}^{-3}\right)$ for Set 1 (Figure 7) and Set 2 (Figure 8).

The results shown in Figure 7 demonstrate the presence of an extremum of the function $q_i=f\left(S_w,p_{s^o}\right)$. Moreover, the value of the argument $p_{s^o}$ affects only the value of the function at the extreme points, but not the positioning of these points in the dimension $S_w$. This fact echoes the results of the analysis of Figure 3. In general, it can be concluded that for initial data synthesized within the limits defined by Set 1, the mathematical apparatus proposed in Section 2.2 allows for the determination of the optimal TCP Window Size value, in particular, $S_w=18$. Figure 8 demonstrates that for small values of $S_w$ (in the range of $\left[2,30\right]$), the target connection between the SNEIoTD environment and a hub, characterized by the output data synthesized within the limits defined by Set 2, is under the control of the SlS mechanism. At values of $S_w>30$, the control initiative of the target connection passes to the AI/MD mechanism. This fact is fully confirmed by the logic of the TCP, which we focused on in Section 2.1.

Now let us examine how the moments of the first (mathematical expectation) and second (variance) orders for the TCP Window Size parameter depend on the values of probabilities $p_{s^{-}}$ and $p_{s^o}$. The data for this calculation were obtained in the context of the rules summarized in Set 1. This calculation’s results are presented in Figure 9 and Figure 10.

Figure 9 shows that the mathematical expectation of the parameter $S_w$ is a monotonically decreasing function, the value of which is affected by the argument $p_{s^{-}}$ significantly more than the argument $p_{s^o}$. This circumstance can be explained by the fact that (for Set 1) the equality of $q_i^0=0$ is fulfilled for all $i>L$, that is, with a sufficiently large value of $S_w$, the probability of activation of the SlS mechanism due to the arrival of the signal $s^o$ is unlikely. From Figure 10, it can be seen that the variance of the parameter $S_w$ is characterized by the presence of extreme values in the range of $\left[0,1\right]$ for both arguments, $p_{s^{-}}$ and $p_{s^o}$. However, the tendency for the greater influence of the parameter $p_{s^{-}}$ compared to the parameter $p_{s^o}$ on the value of the resulting function persists here. We also emphasize that the variance in the TCP Window Size can be a monotonic function because it is determined not only by the values of the parameters $p_{s^{-}}$, $p_{s^o}$, and $S_w^{\max }$, but also by the variance of the parameter $T_{rt}$.

We completed the experimental phase of this research by proving the positive effect of using the proposed mathematical apparatus. In the previous part of this research, published in [7], the authors compared the modified version of the TCP Westwood+ protocol with analogues such as TCP Westwood+ (without modification), TCP NewReno, TCP Vegas, BIC TCP, and TCP Veno. The object of this research was the process of information interaction between the SNEIoTD and a hub with a communication channel implemented based on the 5G platform. The modification consisted of the implementation of the scheme for controlling the Window Size parameter proposed by the authors. This research complements the possibilities presented in [7] due to the information technology formulated in Section 2.2, locked to control the <ssthresh> parameter of the SlS mechanism. Figure 11 demonstrates the effect of the introduction of the authors’ information technology on an example of modification of the mentioned TCP Westwood+ parameter. The remaining parameters of the experiment are identical to those described in [7].

From the findings presented in Figure 11, our results show that the use of the authors’ information technology for controlling the <ssthresh> parameter, according to the example of using the Westwood+ TCP to control the information interaction between a SNEIoTD environment and a hub, has a positive effect that increases with the increase in the value of the parameter $T_{rt}$. In particular, it was possible to increase the stability of data transfer in situations where the signal $s^o$ is 80% received (see the graphs shown in Figure 11 for the values $T_{rt}>9$ of this argument).

4. Conclusions and Future Work

This article proposes a mathematical model of the interaction of the main mechanisms of the TCP (Slow Start, Additive-Increase/Multiplicative-Decrease) in the form of a queuing system, which describes the process of reliably controlling massive traffic from the SNEIoTD environment to a hub. As part of our model, a semi-Markov process is selected that describes the interaction of these mechanisms, locked to the definition of such a characteristic parameter as the unified speed of information transfer. As a result of recurrent analyses of the aforementioned process, a stationary distribution of an embedded Markov chain was determined. This made it possible to analytically describe the stationary distribution of the original semi-Markov process. An analytical representation of the above-mentioned stationary distribution became the basis for our formalization of an accurate, computationally efficient information technology for calculating the TCP Window Size of both the studied mechanisms and the <ssthresh> parameter.

The conducted simulation made it possible to determine a number of important regularities in the distributed process of controlling data transfer from the SNEIoTD environment to a hub in the context of using the TCP. Using the example of TCP Westwood+ protocol modification, it has been shown that the results of applying the information technology proposed in Section 2.2 make it possible to increase the stability of targeted data transfer in conditions of an increasing Round-Trip Time.

As a promising direction for further research, the authors suggest the expansion of the mathematical apparatus presented in Section 2 according to the analytical description of the optimizing mechanisms specific to different versions of the TCP.