Enhance Capacity Region of Multi‐Class IIoT Network by Applying Entanglement Assistance Protocol

Abstract

ABSTRACT

In the Industrial Internet of Things (IIoT) context, heterogeneous IIoT nodes need diverse performance requirements, including throughput and quality of service (QoS). These IIoT nodes transmit data over a common shared communication medium. The existing critical challenge arises in efficiently scheduling access to this shared medium among a large number of connected IIoT nodes. To address the challenge of random access in IIoT networks, the power of the entanglement‐assisted (EA) protocol was exploited to expand the capacity region boundaries of the shared communication medium, thereby enhancing the throughput and quality‐of‐service (QoS) requirements of the heterogeneous IIoT network. In the literature, IIoT networks are mainly categorised into two types: centralised and distributed. In this paper, we proposed two distinct models: (1) a centralised multi‐class IIoT network based on EA protocol and (2) a distributed multi‐class IIoT network based on EA protocol. Next, the authors analytically demonstrated that integrating the EA protocol into both proposed types of multi‐class IIoT networks significantly increases the capacity region boundaries compared to the classical reference model, namely slotted ALOHA (SA). Finally, the network performance boundaries were evaluated by analysing the throughput values for different network classes and varying numbers of IIoT nodes. The results demonstrate that, for both proposed models (1) and (2), the transmitted load generated by the IIoT nodes over the shared medium achieves dramatically higher throughput compared to the reference IIoT network based on SA.

Introduction

On the Internet of Things (IoT), many devices communicate directly with each other or via servers [1–4]. This network has large data volumes, requiring efficient administration to avoid congestion and cyber threats. These issues are critical, especially in industrial IoT (IIoT) [5–8]. IIoT consists of heterogeneous devices that require complex algorithms to manage traffic and security [9]. These devices' disparate priorities and needs are complicated by managing traffic load and using medium access control (MAC) to access shared transmission channels [8].

Despite its importance, few studies have addressed this topic [8, 10–12]. Among the most essential random-access protocols is ALOHA, which is used with IoT networks due to its simplicity of implementation [12]. The recent improvement to the slotted ALOHA (SA) protocol enhances the throughput and scalability, making it the best choice for IoT networks and dealing with vast amounts of data. One of the most significant improvements is the use of irregular repetition slotted ALOHA (IRSA) [8]. Still, SA, despite these improvements in throughput and collision reduction, is unsuitable for use with heterogeneous networks.

The authors in ref. [8] studied applied IRSA to enhance heterogeneous networks. The author focuses on the relationship between the throughput of a single-class homogeneous network and the capacity region of a multi-class network. They analytically show how users transmit their frames with high throughput in such a heterogeneous network.

Eren et al. [12] suggested an energy-efficient MAC protocol based on combined non-orthogonal multiple access (NOMA) power domain concepts with pure ALOHA. In this protocol, five IoT users access the channel simultaneously by superposition coding (SC). A Successive Interference Cancelation (SIC) separates these users' packets at the receiver. This technique increases throughput by 1.27 times compared to pure ALOHA.

In the last five years, due to the development of artificial intelligence and machine learning technologies and the need for reliable remote control provided for machines, especially after the COVID-19 pandemic 2020, these technologies were proposed to improve IIoT performance [13–16]. Liu et al. [17] proposed a deep-reinforcement Learning (DRL) approach to enhance Blockchain scalability and adjust the dynamic nature of IIoT systems.

Later, in 2022, Liu et al. [18] suggested reinforcement learning (RL) to optimise spectrum access in IIoT networks. This paper uses a Q-learning-based algorithm for dynamic spectrum access to improve the throughput of IoT nodes and reduce user interference.

Sun et al. [19] proposed a complex-valued power allocation network (AttCVNN) with cross-channel and in-channel attention mechanisms to enhance the performance of IIoT high-dense networks. The AttCVNN aims to improve the relationship between internetwork users and primary users, while cross-channels enhance relationships among intra-network users.

Wang et al. [20], suggested improving reliability and throughput in IIoT based on full-duplex relaying, power allocation, and rate adaptation. A brief comparison of the existing approaches based on system types, key techniques and findings, and challenges and limitations are given in Table 1.

TABLE 1 A brief comparison of the relevant methods used to enhance throughput in IIoT networks.

Research	System type	Key technique	Key finding	Challenges and limitations
Noori et al. [8]	Centralised and distributed	Irregular repetition slotted ALOHA (IRSA)	Maximise throughput based on capacity region optimisation	Potential for collision
Eren et al. [12]	Centralised	Non-orthogonal multiple access (NOMA)	Improve the throughput and reduce interferences	Complex resource allocation
M. Liu, Yu et al. [17]	Decentralisation	Blockchain with deep reinforcement learning (DRL)	Enhance throughput effectively with a dynamic adaptive IoT environment.	Scalability challenges due to complex blockchain management
X. Liu, Sun et al. [18]	Centralised	Reinforcement learning (RL)	Improve the throughput and reduce interference	Complex learning algorithm
. Sun et al. [19]	Centralised	Deep complex network	A significant improvement in the throughput of IIoT environments	Complexity in training deep learning model
M. Wang and et al. [20]	Centralised	Full-duplex relaying, power allocation and rate adaptation	A significant improvement in throughput and reliability of IIoT environments with low latency and high data rate	Issues in self-interference management

The number of IIoT users is continuously increasing. This growth necessitates the simultaneous transfer of more significant amounts of data, which requires greater network capacity. However, the limited conventional capacity means the network's overall capacity cannot exceed a certain threshold. As a result, when one user increases their data usage, it often comes at the expense of another user. This situation heightens the risk of losing or delaying critical data. Additionally, the recent developments in quantum computing and its ability to analyse large numbers using Shor's algorithm [21–24] at great speed pose a security threat to classical IoT networks that rely on encryption in this way [25].

On the other hand, a unique quantum phenomenon called quantum entanglement provides an amazing capability for communication networks [26, 27]. Quantum entanglement occurs when particles, such as photons or atoms, become intertwined to the point that their quantum states are linked regardless of distance [28, 29].

Both quantum and conventional communication benefit from quantum entanglement [27, 29, 30]. In security, entanglement can detect unauthorised changes by studying the predicted correlation between entangled particles. If these correlations deviate from what they should be, it signals a potential problem, such as a system breakdown or, more seriously, an unwanted infiltration. This principle is a fundamental component of quantum key distribution (QKD) [26, 31–33]. Applying quantum private comparison differs from classical comparison by using quantum bits as a carrier of information based on Bell states. This protocol lets classical users compare their secret information without knowing it [34].

Pre-shared entanglement also increases the capacity of traditional communication lines [35]. It applies to improve performance and efficiency in centralised systems, entanglement-assisted distributed sensing [36], and distributed quantum computing [30, 37, 38]. Furthermore, the entanglement-assistance protocol enhances classical communication by strengthening error correction [39], improving authentication [40–42], and optimising coding through quantum-dense coding [43, 44].

Given the limitation of traditional capacity regions, and the enhancement added by quantum entanglement motivate us to suggest using an entanglement-assisted (EA) protocol to increase the network's overall capacity region. This improvement results from using the quantum MAC [45, 46] and the recent development of quantum technologies [22, 24, 46, 47]. Combining different types of information transfer provides a synchronisation formula between classical and quantum information using quantum entanglement, which makes the capacity areas larger due to its influence of quantum capacity [35].

Based on the above, we suggest a novel design for a hybrid network that combines classical and quantum components for the IIoT. This technique takes advantage of both quantum and classical technology to improve future network performance, accounting for ongoing advances in these disciplines and the expanding volume of data. A specific communication strategy based on quantum mechanics is established to define users for each class. We discuss two network types: centralised and distributed. Analytically, the network operates as one class at a time, with better throughput and more significant capacity than the traditional ALOHA protocols.

The proposed method holds significant promise for a range of real-world applications. Practically, it addresses critical challenges in automated and connected factories, such as security vulnerabilities and massive data transfer management. In the context of remote monitoring and control, our method offers an innovative solution by applying a quantum MAC protocol such as EA. This improves resource allocation management, increases total throughput due to wider capacity region, and reduces packet loss due to collision cancelation. Moreover, using quantum entanglement enables us to generate a quantum security approach by applying the QKD protocol. Eventually, integrating this method into present systems will improve data transfer management operations and drive better security.

The remainder of this paper is organised as follows: Section 2 explains some background on classical concepts and basics, such as IIoT architecture and the main differences between IoT and Capacity region. Quantum principles such as quantum entanglement and how it enhances network performance are presented in Section 3. Then, Section 4 explains the network connection and behaviour. Numerical analysis and comparison are given in Section 5. Finally, Section 6 concludes this paper.

Background and Preliminaries

Industrial Internet of Things

The trip started with the first industrial revolution, which used steam-driven power and Earth's resources, while the second one initiated the use of electricity and the third introduced computers. In the last decade, Industry 4.0 introduced AI and intelligent systems. In 2020, Industry 5.0 suggested concentrating on human-machine collaboration, based on Industry 4.0's IoT substructure, to meet future market requirements by 2030 [48, 49].

The manufacturing and industrial sectors use IIoT. The main strengths of IIoT are sensing, acting, connecting, and processing data, enabling more thoughtful decision-making, and limited human intervention. It improves (M2M) communication and power through progress in wireless networks, sensors, and digitisation, enhancing efficiency and functioning. Though IIoT and IoT have standard protocols and flexible designs, they vary in targets, architecture, and specific applications [50–52] as shown in Figure 1.

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IIoT is commonly used in smart grids, manufacturing, automation, safety, maintenance, transportation, and logistics [42, 52]. In our research, we concentrate on IIoT in automation and manufacturing, where intelligent machines and interconnected systems improve the performance of factories [5–7]. By analysing data and enabling automation, IIoT reduces human intervention, supports more thoughtful decision-making, and meets product quality standards.

Industrial Internet of Things Architecture

Diverse IIoT and IoT architectures employ multi-layered structures based on requirements, goals, and technologies [42, 53–55]. The ITU suggests a five-layer IoT pattern (sensing, accessing, networking, middleware, and application) [53, 55, 56]. Other popular models contain four-layer (sensing, network, service, and interface) and three-layer architectures (application, network, and participation) [53].

This article concentrates on a three-layer IIoT architecture, where edge layer handles sensors and controllers via local networks [42]. These regional networks connect extensive access networks via the platform layer. The enterprise layer integrates user applications and domain-specific services with the service network [53].

Two main MAC protocols manage devices' communication channels without collision. First are contention-free protocols (CFP), which prevent collisions, such as TDMA, FDMA, and CDMA, but still, it is a struggle with high traffic loads [12, 55]. Another allows nodes to set transmission time independently, called contention-based protocols (CBP), such as carrier sense multiple access CSMA, time division multiple access (TDMA), and ALOHA. In CSMA, nodes listen to the channel before sending data, TDMA divides the channel into fixed time slots dedicated to each user to prevent a collision, and ALOHA sends packets and waits for an acknowledgement (ACK), retrying after a timeout if no ACK is received [12].

Considering different user classes, controlling user access to a shared transmission channel via MAC is the primary difficulty in IoT networks. Because of its simplicity, ALOHA—especially with improvements like IRSA—is ideally suited for the internet of Things. However, managing heterogeneous data is still challenging because a vector expresses throughput instead of a single numerical value.

Classical IIoT Multiclass Modelling

In this context, we examine a multi-class IIoT network whose users have similar criteria organised into different classes. These classes share a common communication medium. Since all users have data ready to transmit, the network is heterogeneous. Various factors affect the priority allocated to each class, resulting in differential access to common communication resources.

To explain this approach, consider an N-users network consisting of K-disjoint classes. Each class $${C}_{i}$$ has $${N}_{i}$$ users where $$i\mathit{\in }\left\{1,2,\text{\ldots }.,\mathrm{K}\right\}$$, and $$N={\sum }_{i\mathit{\in }\mathrm{\mathrm{K}}}{N}_{i}$$. This study focuses on throughput as a vital parameter, representing the successful data transmission rate. Throughput is a significant metric of Quality of Service (QoS) that plays a significant role in determining network performance across various user groups. By optimising throughput, the data transfer rate increases, which leads to improvements in other QoS factors like transmission delay and energy efficiency.

Multi-class networks use SA to coordinate data transfer over a shared medium. SA divides time into discrete time slots. Each time slot is set to fit the network's maximum packet length to reduce collision probability and time delay. A group of M-slots aggregate to form a frame. The ratio Ni/M must remain constant as the number of users increases. To maintain equilibrium, both N and M must continue to expand eternally.

Assume $${L}_{i}$$ out of $${N}_{i}$$ users in class $${C}_{i}$$ are active users and $$L=\sum \nolimits_{i=1}^{k}{L}_{i}$$ represents the total number of active users in the network. The traffic load for each class $${C}_{i}$$ is as follows: 1 $${G}_{i}=\frac{{L}_{i}}{M}$$ where $${G}_{i}$$ represents the traffic load for each class i and the total traffic load $$({G}_{t})$$ $${G}_{t}=\sum \nolimits_{i=1}^{k}{G}_{i}$$. In multi-class heterogeneous networks, the vector $$\boldsymbol{G}={\left[{G}_{i}\right]}_{i\mathit{\in }k}$$ uses to represent traffic load. For the given packet transmitted successfully by each class i, $${S}_{i}(\boldsymbol{G})$$ the throughput for each class $${\text{Th}}_{i}(\boldsymbol{G})$$ expressed as follows: 2 $${\text{Th}}_{i}(\boldsymbol{G})=\frac{{S}_{i}(\boldsymbol{G})\,}{M}$$

Then, the probability of losing packets in class $$i$$ is as follows: 3 $${P}_{i}=1-\frac{{\text{Th}}_{i}(\boldsymbol{G})}{{G}_{i}}$$

Optimal Dual Network

It is an analysis used to evaluate throughput in multi-class IIoT networks by modelling them as homogeneous networks with a uniform number of users to simplify the study of varying traffic and service needs across classes. As mentioned before, in multi-class networks, each class $${C}_{i}$$ has $${N}_{i}$$ users and $$N$$ represents the total number of users in the network. In dual network $${N}_{1}$$ represents the number of users in class 1, it is assumed to be equal to $$N$$ with the exact traffic requirement to simplify the multiclass network's capacity region analysis. Because the dual network does not attain the same throughput in terms of user degree distribution and traffic load, the optimal dual network $${N}_{1}^{\ast }$$ is chosen. This version determines the maximum potential throughput $${\text{Th}}^{\ast }$$ for optimal load $${G}^{\ast }$$ written as follows: 4 $${\text{Th}}^{\ast }=\text{Th}\left({G}^{\ast }\right)=\max \text{Th}(G)$$

Capacity Region of Classical Multiclass Networks

Due to different traffic load requirements by each class in multiclass networks, the capacity region characterised throughput boundaries which used to represent all throughput compositions ($${\text{Th}}_{1},{\text{Th}}_{2},\mathit{\text{\ldots }}.,{\text{Th}}_{k}$$). Throughput $${\text{Th}}_{i}$$, of class $$i$$ depends on both $${G}_{i\,}$$ and $${\Lambda }(\mathrm{x})$$, where $${\Lambda }(\mathrm{x})$$, represents user distribution degree in the network.

By applying the concept of dual network to obtain maximum throughput for $${C}_{1}$$, $${\text{Th}}_{1}={\text{Th}}^{\ast }=.$$ While $${\text{Th}}_{2}=0$$ which represents the maximum throughput for $${C}_{2}$$ this means only users of $${C}_{1}$$ are active, the same thing obtained for $${C}_{2}$$. According to the above, the pair ($${\text{Th}}_{1},{\text{Th}}_{2}$$) obtained depending on each class's active users and distribution degree, as shown in Figure 2 capacity region for classical single-class and multi-class networks (K = 2), where $${\text{Th}}_{1}+{\text{Th}}_{2}\mathit{\le }C$$.

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This concept can be extended for K-class network shared resources, where the sum of throughputs across all K-classes is lesser than or equal to total capacity (C). 5 $$\sum\limits _{i=1}^{k}{\text{Th}}_{i}\mathit{\le }C$$

For any $${C}_{i}$$, specific traffic loads $${G}_{i\,}$$, and the throughput for each class $${\text{Th}}_{i}=\left({\text{Th}}_{1},{\text{Th}}_{2},\mathit{\text{\ldots }}.,{\text{Th}}_{k}\right)$$ represents all possible throughput combinations. Capacity region maximum limits are determined according to the different productivity groups of the different classes. Since the boundaries of the capacity region $$\mathbb{C}$$, are equitable to the maximum throughput of the classes, its limit does not exceed 1/e if slotted aloha is used. Another limitation is due to the Shannon capacity limit.

Quantum Information Theory

Quantum communications are powered by qubits, which vary from classical ones and show special benefits such as high speed. Qubits can remain connected over any distance by quantum entanglement, providing quantum protocols such as quantum teleportation and superdense coding. These capabilities support cryptography, distributed systems, and secure quantum cloud computing [23, 26, 28, 30].

Quantum Entanglement

The two entangled qubits, known as an entangled pair, cannot be split into independent states, resulting in genuine quantum correlations between nodes A and B that do not exist in classical systems. Bell states are the most common examples of entangled pairs. 6 $$\vert {{\Phi }}^{+}\rangle =\frac{\vert 00\rangle +\vert 11\rangle }{\sqrt{2}}\vert {{\Psi }}^{+}\rangle =\frac{\vert 01\rangle +\vert 10\rangle }{\sqrt{2}}\vert {{\Phi }}^{-}\rangle =\frac{\vert 00\rangle -\vert 11\rangle }{\sqrt{2}}\vert {{\Psi }}^{-}\rangle =\frac{\vert 01\rangle -\vert 10\rangle }{\sqrt{2}}$$

W-State

W state describes a quantum entangled state of three qubits or more; the equation gives it: 7 $$\vert W\rangle =\vert 001\rangle +\vert 010\rangle +\vert 100\rangle $$

It is possible to generalise the ًW-state for N qubits [30], such that one of the qubits is excited while the rest are in the ground state. 8 $$\vert {w}_{n}\rangle =\frac{1}{\sqrt{n}}\sum\limits _{i=1}^{N}\vert {2}^{(n-i)}\rangle $$

Equation (8), modified by ref. [30], to give priority to certain users ($${\varphi }_{n}$$) over other users therefore, the generalised form is introduced as follows: 9 $$\vert {\tilde{w}}_{n}\rangle =\sum\limits _{i=1}^{n}{\varphi }_{n,{2}^{(n-i)}}\vert {2}^{(n-i)}\rangle $$ Where i represents the node that can access shared medium and probability amplitude is given by the following: 10 $${P}_{n,i}={\vert {\varphi }_{n,{2}^{(n-i)}}\vert }^{2}$$ Where all probabilities summation must be equal to one.

Greenberger-Horne-Zeilinger (GHZ) Entangled States

Another special type of quantum entanglement called Greenberger-Horne-Zeilinger (GHZ) states in quantum networks. GHZ represents a specific type of quantum entanglement that involves three or more particles that are entangled and highly correlated to perform quantum communication operations, $$\vert \text{GHZ}\rangle =(\vert 000\rangle +\vert 111\rangle )$$, this concept can be extended into N-qubits: 11 $$\vert {\text{GHZ}}_{N}\rangle =\frac{1}{2}\left({\vert 0\rangle }^{\mathit{\bigotimes}N}+{\vert 1\rangle }^{\mathit{\bigotimes}N}\right)$$

In this section, we show how this special state improves multi-class capacity.

Quantum Entanglement Assistance Capacity $$({C}_{\text{EA}})$$

Entanglement enables higher classical information transmission due to applying the Holevo bound. It expresses the capacity region limits of classical information transmission over a quantum channel, which surpasses Shannon's classical capacity. 12 $${C}_{\text{EA}}=\underset{{\rho }_{\text{AB}}}{\max }I(A;B)$$ Where $$I(A;B)$$ represents the mutual quantum information for entangled states between transmitter A and receiver B. It calculates the amount of information that can be shared through quantum entanglement. $${\rho }_{AB}$$ is an arbitrarily shared quantum state, maximising each input state $$\rho $$ ensures the selection of one that produces the strongest correlation between A and B. $${C}_{\text{EA}},$$ represents the entanglement-assisted capacity, 13 $$\sum\limits _{i=1}^{k}{\text{Th}}_{i}\mathit{\le }{C}_{\text{EA}}$$

Since $${C}_{\text{EA}}\, > C$$, then the capacity region will be expanded, which will allow for more data handling simultaneously [12]. Equation (13) specifies how larger capacity regions due to entanglement assistance protocol affect throughput. This is due to varied factors; the first is due to higher quantum mutual information (allows sending two bits using one qubit in superdense coding), which is defined as follows: 14 $$I(A;B)=S\left({\sigma }_{A}\right)+S\left({\sigma }_{B}\right)-S\left({\sigma }_{\text{AB}}\right)$$ Where $$S\left({\sigma }_{A}\right)$$ is the entropy of A's input, $$S\left({\sigma }_{B}\right)$$ is the entropy of B's output, and $$S\left({\sigma }_{\text{AB}}\right)$$ is the entropy of a joint state.

Another reason is that the quantum channels $$(\mathcal{N})$$ are described as completely positive trace-preserving (CPTP) maps. This property is fundamental when dealing with multiple processes or working together, especially with channels without memory. Each channel is modelled as a map, each with an interlacing capacity given by Equation (12). One of the marked consequences of modelling channels as CPTP maps is additive classical capacity in the entanglement-assisted scenario. This means the capacity for one channel use is computed by Equation (12) and then scaled for multiple uses (n), avoiding the need for complex regularisation limits [57, 58]; entanglement assistance additive capacity is given by the following: 15 $${C}_{\text{EA}}({\mathcal{N}}^{\mathit{\bigotimes}n})=n{C}_{\text{EA}}(\mathcal{N})$$

Furthermore, the overall capacity achieved is higher than the sum of individual ones for some quantum channels. This resulted in the reality $${C}_{\text{EA}}=C+Q+\mathit{{\increment}}$$, where Q, represents quantum channel capacity, and $$\mathit{{\increment}}$$ is pre-shared entanglement gain. This means $${C}_{\text{EA}}$$ can achieve more capacity than the summation of classical and quantum capacities due to the superadditivity that appears when using quantum channels and tensor products [59, 60]. However, superadditivity only works in cases where the term entropy is not saturated.

Hybrid Quantum-Classical Multi-Class System Design

In this section, we suggest and analyse a hybrid quantum–classical multi-class IIoT network. It is based on special enhancement provided by EA protocol to enhance the network capacity region mentioned before. This capacity region extension, due to pre-shared entanglement, also maximises the throughput of the network because of collision cancelation, which can significantly reduce latency and maximise the energy efficiency compared to SA, primarily by reducing the number of retransmissions. Two approaches are discussed: centralised and distributed.

Centralised IIoT Networks

In this subsection, we first study the effect of shared entanglement on both capacity region and throughput as a function of load for a centralised network. Each node is coordinated by the central unit and connected to this unit by two channels: quantum channel (fibre base) and wireless classical channel.

Hybrid Quantum-Classical Multi-Class Centralised IIoT Network

The network shown in Figure 3 has N-nodes aggregated into K-disjoint classes $${C}_{i}$$. Each node consists of two parts classical node (CN) representing the IIoT node and a Quantum part (Q), which performs quantum measurement (QM) and entanglement generation (EG) operation and is connected to server (S) via quantum and classical channels. Quantum channel is used for entanglement sharing to coordinate classical transmission, and classical channels are uplink channels that are portioned into time slots (M). Its length (t) depends on the class requirement for information transmission between nodes, and S. Nodes cannot communicate with each other in such a system. They only communicate with the S.

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The Quantum Entanglement Assistance (QEA) Method

First, we consider a centralised network with two classes, each containing active nodes equal to L_i. Each node, by using its Q-part, shares an entangled pair $$\vert {{\Phi }}^{+}\rangle $$ with S, only active Node share $$\vert 11\rangle $$. In the second step, S measure his qubit on a computational basis {$$\vert 0\rangle ;\vert 1\rangle $$}, he obtains the same values, regarded as tokens. According to the received state, S selects active users and then coordinates their access by generating a special type of $$\vert {w}_{L}\rangle $$ which is described in Equation (8), where only certain nodes have a higher access probability. The coordination process is done through the generation power of two in binary form, resulting in a binary vector randomly distributed to active nodes. Only a node that receives bits equal to one can send information into S. Finally, each node is accessed in time slots without collision, and the network works as a single class at each time.

To derive the mathematical analysis of a capacity region for the aforementioned network, assume that $${\text{Th}}_{1}$$ and $${\text{Th}}_{2}$$ represent the throughput of nodes per class1 and class2, respectively, and that $$\text{Th}$$₁* and $$\text{Th}$$₂* are the maximum possible throughput achieved by each. To consider the EA effect on the capacity region, an entanglement-enhanced capacity factor is introduced for user1 and user2, η₁ and η₂, respectively. 16 $$\frac{{\text{Th}}_{1}}{{\eta }_{1}{\text{Th}}_{1}^{\ast }}+\frac{{\text{Th}}_{2}}{{\eta }_{2}{\text{Th}}_{2}^{\ast }}\mathit{\le }1$$

To reflect the effect of traffic load (G), let us introduce α as a fraction of the traffic load for class1, then $${G}_{1}=\alpha G$$ and $${G}_{2}=(1-\alpha )\,G$$, where $$0\le \alpha \le 1$$.

Since 17a $${\text{Th}}_{1}=\min \left({G}_{1}{\text{Th}}_{1}^{\ast },{\eta }_{1}{\text{Th}}_{1}^{\ast }\right)$$ 17b $${\text{Th}}_{2}=\min \left({G}_{2}{\text{Th}}_{2}^{\ast },{\eta }_{2}{\text{Th}}_{2}^{\ast }\right)$$

Substituting $${G}_{1}=\alpha G\,\text{and}\,{G}_{2}=(1-\alpha )G$$ 18a $${\text{Th}}_{1}=\min \left(\alpha G{\text{Th}}_{1}^{\ast },{\eta }_{1}{\text{Th}}_{1}^{\ast }\right)$$ 18b $${\text{Th}}_{2}=\min \left((1-\alpha )G{\text{Th}}_{2}^{\ast },{\eta }_{2}{\text{Th}}_{2}^{\ast }\right)$$

For readily extrapolated, we demonstrate the above procedure into more significant systems consisting of K-classes. The capacity region of K-classes is represented as the shutout of the convex hull of all $$\text{Th}(G)=(\text{Th}\left({G}_{1}\right),\text{Th}\left({G}_{2}\right)\mathit{\text{\ldots }},\left(\text{Th}\left({G}_{k}\right)\right)$$ satisfying: 19 $${\text{Th}}_{i}=\min \left({G}_{i}{\text{Th}}_{i}^{\ast },{\eta }_{i}{\text{Th}}_{i}^{\ast }\right)$$

Since we are applying the EA protocol described above, only one user is active in class C_i, and all other classes are silent. The load vector of the K-classes network is $$\left[0,0,\mathit{\text{\ldots }},{G}_{i},\mathit{\text{\ldots }},0,0\right]$$. If $${G}_{i}\mathit{\le }{G}^{\ast }=\frac{{N}_{i}}{M}$$, then 20 $${\text{Th}}_{i}(G)=\min \left(\frac{{N}_{i}}{M}{\text{Th}}_{i}^{\ast },{\eta }_{i}{\text{Th}}_{i}^{\ast }\right)$$

And total throughput constraints equal to $$\sum \nolimits_{i=0}^{k}\frac{\min (\frac{{N}_{i}}{M}{\text{Th}}_{i}^{\ast },{\eta }_{i}{Th}_{i}^{\ast })}{{\eta }_{i}{\text{Th}}_{i}^{\ast }}\mathit{\le }1$$, which simplified to: 21 $$\sum\limits _{i=0}^{k}\left(\frac{{N}_{i}}{M{\eta }_{i}},1\right)\mathit{\le }1$$

According to load distribution and $$\frac{{N}_{i}}{M}$$ ratio the throughput of each C_i as a function of traffic load written as a combined relationship: 22 $${\text{Th}}_{i}=\left\{\begin{array}{@{}c@{}}\frac{{N}_{i}}{M}{\text{Th}}_{i}^{\ast },if\,{N}_{i}\mathit{\le }{M\eta }_{i}\\ {\eta }_{i}{\text{Th}}_{i}^{\ast },if\,{N}_{i} > {M\eta }_{i}\end{array}\right.$$

And capacity region is equal to the sum of all classes' throughput: 23 $$\sum\limits _{i=1}^{k}{\left(\frac{{\text{Th}}_{i}}{{\eta }_{i}{Th}_{i}^{\ast }}\right)}^{2}\mathit{\le }1$$

In this case, the total throughput is bounded by total system capacity with EA C_EA. Still, it is impossible to virtualise directly as the K-dimensional capacity region as performed for K = 2. The two-dimensional capacity region is applied for K-classes, and then the statistical method is used to expand the quantum EA effect on other users.

Distributed Multi-Class IIoT Network

In the previous subsection, we discussed the performance of EA when using a centralised approach. In this subsection, we discuss the impact of EA when using a distributed approach. Distributed IIoT networks ensure no need for centralised decision-making, making the network more flexible for use in applications such as IoT, where no central point of failure exists. Still, it suffers from security and complex management problems.

Distributed Classical-Quantum Multi-Class IIoT Network

The proposed distributed IIoT classical-quantum network contains N non-homogeneous Nodes grouped into classes C_k, each class has N_i homogenous Node connected through quantum and classical channels. A quantum channel is an optical fibre, and a classical channel is a wireless channel. Nodes' distance from a neighbour is fixed equal to $$l$$. Stated differently, every Node in the network is situated at a distinct distance from every other node, indicating that the distance between any two nodes is not constant and does not exceed 1 Km. Assume each Node block consists of a classical node (CN) representing the IIoT node and quantum (Q), which consists of entanglement generator EG and quantum state measurement (M). EG generates entangled pairs to coordinate communication between Nodes and M to measure the received entangled pair. Figure 4a,b shows the described distributed IIoT classical-quantum network.

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S coordinates communication in the previous network (centralised) model, but this cannot be applied in the distributed model. In distributed networks, each node makes its own decision. Each Node communicates with its neighbours, and there is only one step at each step in most cases.

Consider two Nodes systems: the node wants to start communication, sharing an entangled pair $$\vert {{\Phi }}^{+}\rangle $$, equal to $$\vert 11\rangle $$ to another node (who wants to share information with it). The pre-shared entangled state coordinates communication between nodes before any transmission process, which does not carry any information. If the second node is BUSY (communicates with another node), it will wait until the IDLE state and then ACCEPT the entangled state. After the entangled pair equal to $$\,\mathit{\vert }11\rangle $$ shared by EG successfully, and both CNs had JIONT communication. By applying the EA protocol, CNs share information and disconnect at the session's end.

Let us assume a distributed network consisting of two classes with N-nodes. These Nodes use an EA protocol to coordinate their communication. Each node from the class that wants to share information with another node or more from the class needs to coordinate communication by entanglement sharing first. The second step is for the Node on the receiver side to apply measurement to their entangled pair. In the third step, the quantum layer controls the classical layer to obtain optimal performance based on the second step; finally, classical Nodes perform the classical communication. Each class has aggregated throughput based on its nodes; if Th₁ represents the throughput of class1 and Th₂ is the throughput of class2. The mathematical expression for the capacity region of two-class distributed IoT networks is: 24 $$\frac{{\text{Th}}_{1}}{{\eta }_{1}{\text{Th}}_{1}^{\ast }}+\frac{{\text{Th}}_{2}}{{\eta }_{2}{\text{Th}}_{2}^{\ast }}\mathit{\le }1$$

The above-described two-class distributed network procedure extended to a K-class network for plausible discussion. To present this, assume a K-class network consisting of N-nodes; each node $$i$$ has a set of neighbouring nodes equal to $${\mathcal{N}}_{j}$$. Each neighbour $$j$$ has throughput equal to $${\text{Th}}_{j}$$ and maximum total throughput equal to $${\text{Th}}_{j}$$*, with entanglement enhancement factor to communicate with its neighbours $$i$$ is $${\eta }_{j,i,c}$$.

Since each node belongs to a specific class, the effective capacity of each node $$i$$ must consider the overall throughput requirements along with the entanglement improvement factor of its neighbours for each class. We need the sum of the normalised throughput of each neighbour j. Where each neighbour j normalised throughput is equal to $$\frac{{\text{Th}}_{j,c}}{{\eta }_{j,i,c}{\text{Th}}_{j,c}^{\ast }}$$, then the total capacity region for node i in the K-class network is expressed as follows: 25 $$\sum\limits _{c=1}^{k}\sum\limits _{j\mathit{\in }{\varkappa }_{i}\mathit{\cap }c}\frac{{\text{Th}}_{j,c}}{{\eta }_{j,i,c}{\text{Th}}_{j,c}^{\ast }}\mathit{\le }1$$

Numerical Analysis and Discussion

In this section, we analyse the capacity region and throughput of EA for different numbers of class networks and load for centralised and distributed categories. We use Python's Matplotlib library to create visualisations of the data. In connection with the system performance evaluation system, SA is in our scope of focus since it was the earliest to work towards improving the efficiency of communication networks by reducing the chance of collision. Up to the present time, we use SA as a suitable instance of evaluation. The goal was to study how a multi-class system would act using a novel quantum protocol. We study K-classes' effect with N-Nodes and M-slot time for each approach.

Centralised Network

Initially, the network contains two classes with uniform access probabilities capacity region shown in Figure 5. For the same network, throughput with different numbers of nodes (N) and timeslots (M) is shown in Figure 6.

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Figure 6 shows good performance for SA at low load states, while it is inefficient when N becomes larger because of contention. While EA performs better for high N, it degrades when M increases but performs better than SA. Figure 7 shows how entanglement factor η and the number of slots affect total throughput when η increases EA increased and vice versa.

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To check the EA performance on the k-class network, we apply it to a network consisting of five classes; each class has a different η and Th*, as shown in Figure 8.

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To assess the scalability of the EA-base centralised approach, Figure 9 shows the network performance as the number of IIoT nodes increased up to 5000. The results of the EA model are compared against SA, considering multiple time slot values (M = 100, 200, 300, 400, 500).

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Distributed Multi-Class Network

Networks mostly used for IoT networks are distributed ones. Unlike centralised networks, they have not had a single point of failure. Notwithstanding this benefit, it has a drawback because no central location regulates the communication process; instead, the nodes must connect. This problem deteriorates with increased nodes and classes in such networks. This part proves how EA can enhance the capacity region and throughput of multi-class distributed networks over classical methods.

Initially, the network contains two classes with uniform access probabilities. Figure 10 shows the capacity region for (K = 2) for SA and EA methods. SA performs better than EA for low load, but as load increases, the EA performs better. This means the throughput increases as the traffic load increases for EA, as shown in Figure 11. Figure 11 shows that as the load increases, the collision probability increases when SA protocol is used, especially if the ratio of N/M is larger than one.

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In distributed networks, the ratio N/M has a more profound effect, especially on classical performance. When M is less than N, it means G is less than one, leading to a decrease in throughput due to idle slots; conversely, if N is much larger than M, the throughput decreases. Such a problem is enhanced when quantum EA is used as shown in Figure 12. This effect becomes more assertive when a multi-class network (K > 2) is used.

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Using EA improves throughput for multi-class networks compared to classical methods but has not reached a satisfactory level as shown in Figure 13. This depends on factors such as entanglement distribution and node management.

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Discussion

In this part, we outline the main findings when we applied EA protocol into centralised and distributed IIoT multi-class networks. Then, we introduce the main differences between the suggested quantum protocol and classical ones illustrated in the “Introduction” section. Later, we insert a comparison table illustrating differences between EA and SA protocols based on the obtained results presented in Figures 5–13. Lastly, we review the proposed protocol's limitations and suggest future work to address them.

As IIoT networks expanded, ensuring efficient communication became increasingly challenging. Due to collision, classical transmission organisation protocols, such as SA, CSMA, IRSA, etc, suffer from packet loss due to collision. Due to limited resources and the capacity constraints in the region, increasing throughput for one user leads to a decrease for another.

In contrast, our suggested quantum-assisted protocol provides better resource allocation due to capacity region enhancement caused by entanglement enhancement factor ($$\eta $$). In quantum communication, each class's throughput is saturated at ($${\eta }_{k}{\text{Th}}_{k}^{\ast }$$), which gives a higher limit for maximum throughput than the classical ones. Furthermore, this quantum protocol provides better transmission organisation and collision-free communication. Our proposed method leverages entanglement sharing to enhance network scalability with a minimised effect on throughput than classical techniques.

To assess the productivity of our method, we conducted a mathematical analysis evaluating capacity region and throughput across centralised and distributed IIoT networks with different loads and visualised data by Python's Matplotlib library. The key findings of the EA protocol are that it has a larger capacity region than SA, throughput is higher for both EA approaches than SA, and total network throughput is more significant than throughput summation for each class. Unlike classical methods that struggle with resource constraints, EA protocol maintains higher performance efficiency even as the number of connected IoT nodes increased beyond one hundred devices for the centralised approach and fifty for the distributed one with significantly high throughput. Additionally, due to the entanglement enhancement factor ($$\eta $$) the network's overall throughput is higher than the throughput summation for each class, as shown in Figure 8. This factor added additional gain to the overall throughput due to capacity region enhancement $$\sum \nolimits_{i=1}^{k}{\text{Th}}_{i}\mathit{\le }{C}_{\text{EA}}=\sum \nolimits_{i=1}^{k}{\text{Th}}_{i}\mathit{\le }{\eta }_{i}{\text{Th}}_{i}^{\ast }$$.

Furthermore, in both centralised and distributed networks' approaches, the ratio N/M affects throughput value but has a more profound effect in the distributed one, especially on classical performance. When $$(\mathrm{M} > \mathrm{N})$$ means $$(\mathrm{G}< 1)$$, the throughput decreases due to idle slots; conversely, if ($$\mathrm{N}\gg \mathrm{M})$$, the throughput decreases due to the collision in the classical protocol and waiting time in the EA protocol; still, quantum protocol performs better than the classical one when $$(\mathrm{G} > 1)$$ as shown in Figures 6 and 7.

Moreover, our novel quantum protocol addresses issues of classical MAC protocols. Compared with TDMA, the EA-based protocol solves the problem of unused slots. Due to using Equation (8), EA protocol allows the assignment of slots based on real-time traffic demand. Additionally, it solves the CSMA collision problem due to more than one node sending simultaneously after sensing an idle channel, which causes performance degradation due to collisions and backoffs. EA offers collision cancelation and load optimisation with the required number of slots in each transmission period; the system operates efficiently even when the number of slots is lower than the number of users. This addresses the IIRAS [8] challenge, which requires more slots than users to mitigate collisions in heavy traffic situations, negatively impacting SIC efficiency—conversely, low traffic results in bandwidth wastage. Further, it does not suffer from limitations introduced in other classical protocols, such as decoding complexity in the NOMA protocol [12], scalability compared to methods based on blockchain [17], and needs for complex training node algorithms such as those used in refs. [18, 19], due to the entanglement sharing management.

Based on our analytic results illustrated in Subsections 5.1 and 5.2 for centralised and distributed IoT networks, Table 2 summarises the differences across varied performance aspects such as throughput, collision handling, class fairness, Resource Allocation, and complexity between our protocol and SA classical method.

TABLE 2 Comparison between EA and SA protocols for multi-class IoT networks with varied performance aspects.

Aspect	Centralised network with EA	Distributed network with EA	Centralised network with SA	Distributed network with SA
Throughput	High, its saturation depends on η Thi*	Adequate depends on the contention	Its maximum value is 0.36	Low depends on contention
Collision handling	Collision-free	Minimal	Frequent collisions at high load	Severe collisions at high load
Classes fairness	High, centrally managed fairness	Adequate fairness based on local coordination	Based on traffic distribution	Incredibly low, based on load
Resource allocation	Optimal, centrally coordinated	Imperfect, based on distributed decisions	No central coordination, random access	Random access, Unmanaged allocation
Complexity	High, needs central coordination	Low	Simple, no need for a control mechanism	Simple but inefficient

To evaluate the scalability of the EA protocol, we analyse the throughput of the centralised approach performance with the number of nodes increased into thousands as shown in Figure 9. The results show that the EA protocol performs efficiently, but its efficiency degrades when the number of nodes exceeds one thousand. The reason for such degradation is the limited number of slots due to limited bandwidth, which causes throughput saturation, but EA still has a higher throughput than SA.

Although our work focuses on the efficiency and scalability of IIoT networks by implementing the EA protocol, it also supports security features against classical and quantum cyberattacks. Entanglement-based techniques ensure secure communication channels immune to classical eavesdropping strategies such as Man-in-the-Middle (MITM) due to the non-cloning theorem (which states that unknown quantum states cannot be copied). Additionally, the ability to apply entangled-based QKD prevents attacks by detecting attempts to measure or copy quantum information that crash the system. Future research can discuss how EA-based protocols enhance resilience versus cyber threats in IIoT networks.

One of the main obstacles faced by real-world entangled quantum networks is decoherence. Decoherence refers to the gradual degradation of a quantum system's state due to interactions with a noisy environment. As a result of this phenomenon, errors can occur when measuring the quantum system at the receiving end, leading to changes in probability amplitudes.

The extent of decoherence can be quantified using fidelity, which measures the closeness between quantum states. Fidelity (F) ranges from 0 to 1. Higher fidelity indicates that losses in the initial quantum system are minimised when interacting with a lossy environment. In contrast, a low fidelity value ($$F\mathit{\approx }0$$) means the probabilistic behaviour of the quantum system has been entirely lost, causing it to behave like a classical system. This decoherence can negatively impact the proposed EA-based IIoT network by reducing the advantages $$(\eta )$$ of quantum entanglement. This problem can be solved by applying entanglement purification.

From the classical side, this research assumes a fixed time slot. Its length is equal to most extended time slot classes, which causes a leakage time when used by classes needing less than one.

In future work, we will focus on enhancing quantum communication over quantum channels by enhancing quantum fidelity. We will focus on variable length slots that vary with each class requirement to improve the system's overall delay.

Conclusion and Future Works

This paper handles the traffic of classical multiclass IIoT networks challenge and proposes an EA protocol to solve it. To this end, we suggest two distinct models based on EA, centralised and distributed. The centralised model consists of classical IIoT nodes connected by quantum and classical channels to a central server. In contrast, the distributed IIoT nodes are connected to each other via classical and quantum channels. We describe the communication steps with entanglement sharing. We found that the capacity region boundaries when entanglement assistance protocol is applied for a multiclass network are much larger than the classical one. This enhancement improves the throughput of centralised and distributed schemes with different classes and traffic loads. We confirmed that the throughput is optimal using EA for a centralised scheme and suboptimal for the distributed one. In both network topologies, the increased number of nodes has less effect on total network throughput. We prove this behaviour through numerical results for our model and compare it with reference classical slotted ALOHA. In this model, we assume the fixed time slot is equal to most extended time slot classes. In future work, we will extend the EA protocol to incorporate variable-size time slots to reduce the overall delay in the network.

Author Contributions

Doaa Subhi: conceptualisation, data curation, formal analysis, investigation, methodology, resources, software, writing – original draft, writing – review and editing. Laszlo Bacsardi: supervision, writing – review and editing.

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The authors have nothing to report.

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