Joint Transmission and Resource Optimization in NOMA-Assisted IoVT With Mobile Edge Computing

Internet of Video Things (IoVT) brings much higher requirements on the transmission and computing capabilities of wireless networks than the traditional Internet of Things (IoT). Non-orthogonal multiple access (NOMA) and mobile edge computing (MEC) have been considered as two promising technologies to satisfy these requirements. However, successive interference cancellation (SIC) and grouping operations in NOMA as well as delay-sensitive IoVT video tasks with different priorities make it challenging to achieve optimal performance in NOMA-assisted IoVT with MEC. To address this issue, we formulate a joint optimization problem where both NOMA operations and MEC offloading are involved, to minimize the weighted average total delay. To tackle such an intractable problem, we proposed a graph theory-based optimization framework, then decompose and transform the problem into finding negative loops in the weighted directed graph. Specifically, we design a priority-based SIC decoding mechanism and propose convex optimization-based power allocation and computing resource allocation algorithms to calculate the adjacency matrix. Then, two negative loop searching algorithms are adopted to obtain the device association and grouping strategies. Simulation results demonstrate that compared with existing algorithms, the proposed algorithm reduces the weighted average total delay by up to $92.43\%$ as well as improves the transmission rate of IoVT devices by up to $79.1\%$.


Joint Transmission and Resource Optimization in NOMA-Assisted IoVT With Mobile Edge Computing
Langtian Qin , Hancheng Lu , Senior Member, IEEE, Yuang Chen , Graduate Student Member, IEEE, Baolin Chong , and Fengqian Guo Abstract-Internet of Video Things (IoVT) brings much higher requirements on the transmission and computing capabilities of wireless networks than the traditional Internet of Things (IoT).Non-orthogonal multiple access (NOMA) and mobile edge computing (MEC) have been considered as two promising technologies to satisfy these requirements.However, successive interference cancellation (SIC) and grouping operations in NOMA as well as delay-sensitive IoVT video tasks with different priorities make it challenging to achieve optimal performance in NOMA-assisted IoVT with MEC.To address this issue, we formulate a joint optimization problem where both NOMA operations and MEC offloading are involved, to minimize the weighted average total delay.To tackle such an intractable problem, we proposed a graph theorybased optimization framework, then decompose and transform the problem into finding negative loops in the weighted directed graph.Specifically, we design a priority-based SIC decoding mechanism and propose convex optimization-based power allocation and computing resource allocation algorithms to calculate the adjacency matrix.Then, two negative loop searching algorithms are adopted to obtain the device association and grouping strategies.Simulation results demonstrate that compared with existing algorithms, the proposed algorithm reduces the weighted average total delay by up to 92.43% as well as improves the transmission rate of IoVT devices by up to 79.1%.Index Terms-Internet of video things (IoVT), non-orthogonal multiple access (NOMA), mobile edge computing (MEC), graph theory, resource allocation.

I. INTRODUCTION
W ITH the proliferation of smart visual sensors (including closed circuit television, 3-D cameras, etc.), as well as the explosive growth of video data in the Internet of Things (IoT), a new sub-field of IoT, called the Internet of Video Things (IoVT), has had a significant impact on various applications, such as smart traffic monitoring and vehicle sensing [1].Unlike most conventional sensor data, which are typically highly structured, the audio or visual sensor data in IoVT is often unstructured and requires additional processing to extract the "semantics" needed for information collection and decision making [2].Additionally, to ensure real-time video applications in IoVT, such as virtual reality (VR) and augmented reality (AR), wireless transmission with high throughput and capacity is essential.As a result, compared to traditional IoT, IoVT presents higher demands for the transmission and computing capabilities of wireless networks.
Numerous research efforts have been made to enhance service quality in IoVT from the perspective of transmission and computing resource optimization.To improve the transmission performance, an energy-efficient IoVT transmission scheme is considered in [3], where the encoding rate, the modulation and coding scheme, and the transmit power are jointly optimized.The authors in [4] propose a collaboration mechanism based on the social attributes of IoVT devices and their owners.They have also designed a collaborative video streaming strategy to mitigate the impact of network instability on video services.However, in these studies, the use of orthogonal multiple access (OMA)-based transmission methods cannot guarantee the high transmission requirements of IoVT applications.To address this issue, some researchers have introduced Non-orthogonal multiple access (NOMA) to IoVT [5], [6].NOMA-assisted wireless transmission has the potential to serve a larger number of users than traditional OMA because of power domain multiplexing, resulting in improved diversity gain, spectral efficiency, and total throughput of IoVT [7], [8].To satisfy IoVT's requirements on the computing capability, the authors in [9] propose a cloudbased solution of real-time 3D visualization of outdoor scenes in IoVT, where the optimization algorithms are executed in the clouds to calculate the location of moving objects and detect abnormal events.By deploying servers in wireless access networks to users, mobile edge computing (MEC) can offer IoVT devices robust computing, storage, networking, and communication capabilities [10].A few studies have been done on resource allocation optimization in IoVT with MEC [2], [11], [12], [13], where IoVT devices offload visual processing tasks to edge servers to obtain better quality of service (QoS).The research topics, proposed solutions, distinctive aspects, and shortcomings of all highly relevant works are summarized in Table I.
Existing studies have demonstrated the effectiveness of NOMA and MEC in breaking the transmission and computation capacity limitations of IoVT.However, it is still challenging to integrate both NOMA and MEC into IoVT to achieve optimal performance.The authors in [14] first propose a NOMA-assisted IoVT with MEC and analyze the challenges in the system.They also demonstrate that better performance in terms of total delay can be obtained by resource optimization.Nevertheless, NOMA operations like successive interference cancellation (SIC) decoding, device grouping, and the IoVT video task priority are not fully considered with resource optimization in existing studies.Neglecting these factors hinders the NOMA-assisted IoVT with MEC from achieving optimal performance for several reasons.Firstly, the SIC decoding order in NOMA plays a crucial role in determining the power allocation strategy and will affect the transmission rate of IoVT devices.It means that the SIC decoding order needs to be carefully set to ensure that devices with high delay sensitivity have smaller interference during decoding.Secondly, device association and grouping strategies are coupled with resource allocation.The device association strategy will directly decide the MEC server that allocates computing resources.Also, the interference received by an IoVT device will be determined by other devices in the same group [15], [16], [17], [18], [19], [20], [21].Therefore, transmit power should be allocated based on the association and grouping strategies to ensure a high transmission rate of IoVT devices.And devices with computing-intensive tasks should be associated with the MEC servers with high computing capabilities.The existing user grouping algorithms in NOMA mostly consider a single cell [17], [18], or assume that users associate with the nearest BS [14], [22], which ignores the impact of the computing capability of different MEC-BSs on delay performance.Other joint optimization algorithms for association and grouping mostly assume the number of users in each subchannel has to be consistent [23], [24], which is far from optimal.Thirdly, existing studies on NOMA-assisted IoVT with MEC fail to consider that IoVT visual processing tasks have different priorities [25].Taking priorities into account creates a correlation between IoVT device association, grouping, and power allocation, as devices with high-priority tasks require more transmission and computing resources.Based on the above analysis, NOMA transmission and resource allocation, along with video task priority, should be jointly considered to ensure the reliability and efficiency of NOMA-assisted IoVT with MEC.In this paper, we propose a joint NOMA transmission and resource allocation optimization framework in NOMA-assisted IoVT with MEC.Considering the priorities of different video tasks, the weighted average total delay of the system is minimized by jointly optimizing the transmit power allocation, IoVT device association, grouping, and computing resource allocation strategies.Then, a graph theory-based optimization framework is proposed, which transforms the problem into finding negative loops in the weighted directed graph.We also derive efficient convex optimizationbased algorithms and negative loop searching algorithms to solve the NOMA transmission and resource allocation problem, respectively.The main contributions of this paper are described as follows: r Considering the different priorities of visual processing tasks, we formulate a weighted average total delay minimization problem, where power allocation, IoVT device association, grouping, and computing resource allocation are jointly performed in NOMA-assisted IoVT with MEC.To tackle such an intractable problem, we decompose the original problem into three subproblems, i,e., power allocation problem, resource allocation problem, and device association and grouping problem.
r By applying graph theory, we transform the three subprob- lems into finding negative loops in a weighted directed graph, using the concepts of transfer set, swap set, and steady-state strategy.In the proposed optimization framework, the resource allocation is performed to calculate the adjacency matrix, and the device association and grouping strategies can be obtained according to negative loops.
r We propose the priority-based SIC decoding mechanism and transform the power allocation problem based on majorization minimization (MM).Then, the power allocation and computing resource allocation problem are solved by convex optimization methods.
r To find all the negative loops in the graph effectively, we adopt two negative loop searching algorithms.One is the extended Bellman-Ford searching algorithm (EBFSA) which is extended from the traditional Bellman-Ford algorithm.The other is the fast greedy searching algorithm (FGSA) based on the ideology of greed, which has lower computational complexity than EBFSA.Extensive simulations are carried out to demonstrate the effectiveness of the proposed algorithms.The results show that the proposed joint optimization scheme can effectively improve the transmission rate of users by up to 79.1% and reduce the weighted average total delay of the system by up to 92.43% compared with the relevant algorithms.
The rest of the paper is organized as follows.Section II gives the system model and the problem formulation.A graph theory-based joint optimization framework is proposed in Section III, including the problem decomposition and transformation, the priority-based SIC decoding mechanism and the power allocation algorithm, the IoVT device association and grouping algorithm, as well as the computing resource allocation algorithm.Section IV presents the evaluation results and analysis, followed by the conclusion and future work in Section V.
Notations: In this paper, vectors and sets are denoted by bold and calligraphic letters, respectively.∪ and \ represent set union and set difference operators, respectively.• denotes the ceiling function.Given a set A, A denotes the number of elements in A. The key notations used in this paper are listed in Table II.

A. Network Model
As shown in Fig. 1, we consider a multi-cell uplink NOMAassisted IoVT with MEC.We assume the MEC servers are integrated into BSs and referred to as MEC-BSs.MEC-BSs have varying computing capabilities and occupy several subchannels.There are M MEC-BSs and N IoVT devices randomly distributed in the system.We denote the set of MEC-BSs and the set of all IoVT devices as M = {1, . . ., M} and N = {1, . . ., N}, respectively.MEC-BSs have different computing capabilities, and we assume the computing capability of MEC-BS m is C b m .When running the video application, each IoVT device will generate a visual processing task to be offloaded.To characterize the specificity of the visual processing task of each IoVT device, we assume that different video processing tasks have varied priorities, data sizes, and computing requests.For the visual processing task of IoVT device n, we assume the task workload, the task priority, and the task data size of IoVT device n are W n , ω n , and D n , respectively, and a larger ω n represents IoVT device n has a higher priority.When the IoVT device offloads its task to the MEC-BS, let the binary association variable a mn = 1 denotes IoVT device n ∈ N is associated with MEC-BS m ∈ M, otherwise not.

B. Transmission Model
To improve the uplink transmission performance during the task offloading, the NOMA technique is adopted in the network for uplink data transmission.Assume there are G channels in each MEC-BS, and the subchannels occupied by adjacent MEC-BSs are orthogonal.Let the binary grouping variable Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.a scalar coefficient modeling the large-scale fading, and P L(d mn ) (expressed in dB), d mn (expressed in km) is the path-loss and the distance between IoVT device n and MEC-BS m, respectively.γ mg n is a random variable modeling the small-scale fading between IoVT device n and MEC-BS m on subchannel g.
In an uplink NOMA-assisted IoVT system, MEC-BSs could perform channel estimation by receiving pilot sequences from IoVT devices.Channel state information (CSI) can be obtained through emerging deep learning-based channel estimation methods [26], [27].Similar to [13], [16], [21], [28], we assume all the IoVT devices and MEC-BSs can obtain the perfect channel state information (CSI) in advance. 1 Let x n , p n and y mg denote the signal transmitted from IoVT device n, transmit power of IoVT device n and the signal received by MEC-BS m on subchannel g, respectively.Thus the signal received by MEC-BS m on subchannel g can be expressed as where n ∼ N (0, σ 2 ) is additive white Gaussian noise (AWGN).
If IoVT device n associate with MEC-BS m on subchannel g, the signal-to-interference plus noise ratio (SINR) of IoVT device n is where S mg n is the SIC decoding order of IoVT device n, and the IoVT device with greater S mg n should be decoded earlier.p n is the transmit power of IoVT device n and let p = {p n , ∀n ∈ N } represent the power allocation vector.Assume the bandwidth of each subchannel is B, the uplink transmission rate of IoVT device n can be obtained by Shannon theorem:

C. Computing Model
Let T pro n , T tra n , and T tot n denote the processing delay, the wireless transmission delay, and the total delay of task type v of IoVT device n.When IoVT device n offloads the video stream data to MEC-BSs, the transmission delay is The MEC-BSs will allocate certain amount of computing resources to process the tasks of IoVT devices.The task processing delay of IoVT device n is expressed as where c mn is the amount of computing resources allocated to IoVT device n by MEC-BS m.Since the results returned by MEC-BSs are relatively small, we ignore the downlink delay in this paper.Thus the total delay of IoVT device n can be expressed as the sum of the transmission delay and the task processing delay, i.e.,

D. Problem Formulation
Since the tasks of IoVT devices have different priorities, we aim to minimize the weighted average total delay of all IoVT devices in the system.the weighted average total delay is defined as 1 ω n is the average priority weight that used to balance the weighted average total delay value.The joint power allocation, computation resource allocation, IoVT association, and grouping optimization problem can be formulated as follows: where A = {a mn }m∈M n∈N denote the IoVT device association matrix, B = {b gn }g∈G n∈N is the IoVT device grouping matrix and C = {c mn }m∈M n∈N represent the computing resource allocation matrix.Constraint (7b) represents that the transmit power of IoVT devices cannot exceed the maximum transmit power; (7c) means that the computing resources allocated by MEC-BS cannot exceed its computing capability; (7d) specifies the domain of computing resource allocation variables, association variables and the grouping variables.

A. Optimization Framework Overview
Problem ( 7) is a mixed-integer non-linear programming (MINLP) problem, which can be proved as non-convex and NP-hard [30].The coupling between optimization variables and the non-convexity of objective function makes it intractable to obtain the solution in a reasonable time when the number of IoVT devices and the subchannels grows [31].In this section, we first decompose the original problem into three subproblems, i.e., the SIC decoding and power allocation subproblem, the computing resource allocation subproblem, and the device association and grouping subproblem.Then, we introduce the proposed graph theory-based joint transmission and resource allocation optimization framework, which transforms the subproblems into finding negative loops in the weighted directed graph.Compared to traditional matching-based association and grouping optimization algorithms, the proposed graph theory-based optimization algorithms can demonstrate the impact of one user on the total delay of other users, as well as the impact on system performance after any grouping and association transformation [32], [33].
The overview of the framework is shown in Fig. 2.After initializing the device association and grouping strategies (step 1), resource allocation can be performed to calculate each element in the adjacency matrix (step 2).After obtaining the adjacency matrix (step 3), a weighted directed graph can be generated (step 4), where the weight of each node can be calculated according to the transmission and resource allocation strategies (the detail is introduced in Section III-B).By searching the negative loop in the graph (step 5), the device association and grouping strategies can be updated (step 6) to ensure the weighted average total delay can decrease.Then, the resource allocation will be performed again and will be used to calculate the adjacency matrix together with the updated association and grouping strategies.The blue block diagram (steps 2-6) in Fig. 2 will keep looping until the delay gradually converges.Finally, the transmission and resource allocation strategies will be output (step 7).

B. Problem Decomposition and Transformation
We decompose problem (7) into three subproblems.with given SIC decoding order, device association and grouping strategies and computing resource allocation strategy, the transmit power allocation subproblem can be expressed by With given power allocation, device association and grouping strategies, the computing resource allocation subproblem is And the device association and grouping subproblem under given power allocation, computing resource allocation strategies can be written as ) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Then we introduce graph theory to convert three subproblems into finding negative loops in a weighted directed graph.For convenience, the g-th subchannel of MEC-BS m ∈ M is indexed by subchannel mg.We define a subchannel assigning vector π = [π n ] 1≤n≤N , where π n is the subchannel index of IoVT device n.Let π −n denote the vector of subchannel indexes of other IoVT devices except for IoVT device n, i.e., π −n = (π 1 , . . ., π n−1 , π n+1 , . . ., π N ).Let n → n denote that IoVT device n is moved into the group containing IoVT device n .First, we introduce the concept of the transfer set and the swap set.Definition 1: For any i IoVT devices indexed by {n 1 , . . ., n i } in different subchannels, if the IoVT association matrix A and IoVT grouping matrix }, which will make the weighted average total delay decrease, these i IoVT devices compose an i-swap set.
The distinction between the transfer set and the swap set lies in the impact of changing association and grouping strategies.Specifically, in the transfer set, the number of IoVT devices in the subchannel of the first and last IoVT devices is modified when implementing such changes, whereas it remains the same in the swap set.Based on the definitions of these sets, we define the steady-state strategy as follows.
Definition 3: An IoVT device association and grouping strategy {A, B} is a steady-state strategy if, for ∀i ∈ N , there exists no i-transfer set or i-swap set.
The concept of the steady-state strategy is derived from the stable matching solution [34].The stable matching solution denotes that the total welfare of all agents is maximized when no two agents can benefit from switching their assigned partners.In this paper, we adopt the term steady-state strategy to describe a strategy of grouping and allocation that satisfies the condition that the weighted average total delay of any given IoVT devices cannot be reduced by transferring or swapping subchannels with any other devices.
It is noteworthy that the actions that lead to a decrease in the weighted average total delay can be identified by searching for i-transfer set or i-swap set.Moreover, to achieve the steadystate strategy, it is imperative to identify all the i-transfer set or i-swap set and adjust the association and grouping strategies accordingly.Let (n ñ) denote the operation that transfers IoVT device n to the subchannel of IoVT device ñ, i,e, g, and moves IoVT device ñ out of its original subchannel g.Assuming that IoVT device ñ is originally assigned to subchannel g of MEC-BS m.After the operation, the set of IoVT devices in subchannel mg, i.e., U mg becomes Ũmg .The difference in the weighted average total delay before and after (n ñ) can be expressed as follows: A weighted directed graph G(V, E; π) can be constructed based on (11), where V = N is the set of nodes, i.e., all the IoVT devices, and the weight of each node represents the weighted total delay of this IoVT device.E denotes the set of edges.It should be noted that these edges only exist between two IoVT devices in the different subchannels.For ∀n i , n j ∈ V, the edge between these two IoVT devices φ ij in adjacency matrix Φ is defined as follows: where φ ij = ∞ means that node n i is not connected with node n j .As can be seen from ( 12), each element in the adjacency matrix Φ can be calculated according to device association and grouping, power allocation, and computing resource allocation strategies.Next, we will introduce the concept of negative loop.Definition 4: In a weighted directed graph, if a loop whose sum of the weights of the edges is negative, this loop is referred to as a negative loop.
Based on the concept of the negative loop, we can derive the following lemma Lemma 1: For any i IoVT devices in different subchannels, the following statements are equivalent: 1 These i IoVT devices compose an i-swap set.
2 These i IoVT devices compose a negative loop in graph G m (V, E; π).
Proof: Consider there are i IoVT devices n 1 , .., n i grouped in the different subchannels.Those subchannels may belong to the different MEC-BS.For a given subchannel assigning vector π, assume that π n 1 = g 1 , . .., π n i = g i .If we change the association and grouping strategies by taking the action T {(n 1 → π g 2 ), . . ., (n i−1 → g i ), (n i → π g 1 )}, according to (18), textcircled1 is equivalent that the difference of weighted average total delay is negative, i.e., And by the same token, 2 is equivalent to the sum of edge weights of loop According to ( 13) and ( 14), the proof of Lemma 1 is concluded [28], [32].
Based on Lemma 1, appropriate modifications to IoVT device association and grouping strategies in accordance with the swap set can lead to a reduction in the weighted average total delay.However, it is worth noting that this approach does not guarantee the optimality of the association and grouping strategies, as the number of IoVT devices in each subchannel remains unchanged.Therefore, searching for the transfer set presents a better alternative for reducing delay.To make the searching process more convenient, we introduce a virtual IoVT device for each subchannel of each MEC-BS, resulting in a total of GM virtual IoVT devices.Then graph G(V, E; π) can be extended to G(V e , E e ; π e ), where V e = N ∪ N v , and the size of the adjacency matrix becomes N + GM × (N + GM ).Assume Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.π e is the subchannel assigning vector of all IoVT devices in V e .Since the visual processing requirement of these virtual IoVT devices is 0, the transmit power and computing resources allocated to the real IoVT devices will not be affected by the virtual IoVT devices.In other words, any transfer sets can be converted into the swap set with a virtual IoVT device.
Lemma 2: We can convert the transfer set into the swap set with a virtual IoVT device.
Proof: We assume that for the subchannel assigning vector π e , IoVT devices n 1 , . . ., n i can compose an i-transfer set, i.e., we can reduce the weighted average total delay by operations (n 1 → π n 2 ), . . ., (n i−1 → π n i ).Assume that the virtual IoVT device added to the group containing IoVT device n i is n v .If we change the subchannel assigning vector π e by (n 1 → π n 2 ), . . ., (n i−1 → π n v ), (n v → π n 1 ), since the virtual IoVT devices will not affect the transmit power and computing resources allocated to the real IoVT devices, the weighted average total delay will be reduced.In other words, IoVT devices n 1 , . . ., n i−1 can compose an i-swap set with IoVT device n v .
To describe the concept of the i-transfer set, the i-swap set, and the negative loop in the weighted directed graph more clearly, we show a simple case in Fig. 3. Assume there are five IoVT devices (solid borders) and three virtual IoVT devices (dashed borders) are assigned into three subchannels g 1 of MEC-BS m 1 , subchannel g 1 of MEC-BS m 2 and subchannel g 2 of MEC-BS m 2 , respectively.The edges in this graph only exist between the IoVT devices in different subchannels.As shown in Fig. 3(a), Combining Lemmas 1 and 2 with Definitions 1-4, we can obtain Theorems 1 and 2.
Theorem 1: In weighted directed graph G m (V e , E e ; π e ), if there is no negative loop can be found, the IoVT device association and grouping strategy is a steady-state strategy.
Proof: For any i IoVT devices that can compose an i-swap set based on subchannel assigning vector π, according to Lemma 1, these IoVT devices can compose a negative loop in weighted directed graph G m (V e , E e ; π e ).Similarly, For any i IoVT devices that can compose an i-shift league based on subchannel assigning vector π, based on Lemmas 1 and 2, these IoVT devices can compose a negative loop in a weighted directed graph G m (V e , E e ; π e ).Therefore, if no negative loop can be found in a graph G m (V e , E e ; π e ), there is no i-shift league or i-swap set can be found in the IoVT device grouping strategy based on subchannel assigning vector π e .So the proof of Theorem 1 is concluded.
Theorem 2: In weighted directed graph G m (V e , E e ; π e ), if IoVT devices n 1 , . . ., n i can compose a negative loop the weighted average total delay can be decreased by operations 13), ( 14), we have i k=1 Therefore, the weighted average total delay of all IoVT devices can be reduced by (n 1 → π n 2 ), . . ., (n i−1 → π n i ), (n i → π n 1 ).As mentioned in (11), the weighted total delay of virtual devices is 0, hence the weighted average total delay of actual devices can also be reduced.So the proof of Theorem 2 is concluded.
Based on the analysis above, the weighted average total delay can be reduced by identifying the negative loops in the weighted directed graph G(V e , E e ; π e ).The association and grouping strategies are then updated based on these loops until there is no negative loop remains in the graph.It should be noted that, before searching for negative loops, it is necessary to calculate the adjacency matrix Φ, and the calculation of adjacency matrix elements φ ij relies on power allocation and computing resource allocation decisions.Next, we will first introduce how to obtain the power allocation and computing resource allocation decisions, followed by the negative loop searching method.

C. Priority-Based SIC Decoding and Power Allocation
In this subsection, we propose the priority-based SIC decoding and power allocation strategy to calculate the adjacency matrix.By implementing NOMA, IoVT devices in the same subchannel will adopt a non-orthogonal transmission method, which will inevitably introduce interference.Therefore, the SIC technique is applied at the MEC-BSs to ensure the correct demodulation of the superimposed signals and realize the power domain multiplexing.In the traditional NOMA systems, the signals of different IoVT devices are distinguished at the MEC-BS by SIC according to their different power levels.The signal with the highest power level is preferentially decoded, and the signals of other mobile devices are regarded as noise at the same time.So the signal with the highest power level will suffer the most interference.
Therefore, the SIC decoding order will directly determine the transmit power of IoVT devices, thus further affecting the transmission rate [35].In traditional uplink NOMA systems, the receiver generally decodes signals in descending order of channel conditions.However, in the NOMA-assisted IoVT, the visual processing tasks of IoVT devices have different delay sensitivity, i.e., different IoVT devices have different priorities.
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If an IoVT device with high priority is decoded first, the IoVT device will be interfered with by the signals from all other IoVT devices associated with the same subchannel.Since the uplink transmit power of an IoVT device is limited, the achievable throughput of the IoVT device will be relatively low, which may lead to task time out and greatly affect the QoS of the IoVT device.Therefore, we propose a priority-based SIC decoding mechanism.The SIC decoding order is set as the ascending order of the priority of IoVT devices.
By expanding (8a) and eliminating irrelevant constants, the optimization problem (8) becomes As can be seen, the formulated problem ( 15) is non-convex and intractable because the transmit power variable p n , ∀n ∈ N not only exists on the molecule of ζ n , but also exists on the denominator of ζ n in the form of addition, making the problem intractable.We handle this issue by transforming the weighted average total delay minimization problem (15) into the weighted average total transmission rate maximization problem to seek a sub-optimal solution where R n = B log 2 (ζ n + 1) can be further represented as a difference of convex (DC) function as Therefore, we can solve the problem of optimizing p through DC programming.To obtain the local optimal solution of p, we propose a MM-based power allocation algorithm.which is described in Algorithm 1.In Algorithm 1, and T max are the maximum tolerance error and the maximum number of iterations, respectively.We define Rn as and p (t) i is the transmit power of IoVT device i in the t-th iteration.By adopting the majorization-minimization (MM) method [36], (18) can be iteratively approximated to a convex term.Problem ( 19) is a convex problem, which can be solved using solvers such as CVX [37].
In practical NOMA systems, SIC is always imperfect due to residual self-interference (RSI).Similar to [38], [39], assume 0 ≤ ξ ≤ 1 is the level of RSI.After imperfect SIC decoding, the SINR of IoVT device n in (2) will become It should be noted that the power allocation algorithm mentioned above will still be applicable to imperfect SIC.In this case, an additional RSI item ξ S mg k >S mg n ,i∈Umg |h mg k | 2 p k needs to be added.Similarly, (17) can still be converted to a convex term like (18) through MM under imperfect SIC.Obviously, imperfect SIC will reduce the SINR of users, thereby reducing the uplink transmission rate and increasing the transmission delay and total delay.Therefore, imperfect SIC will reduce the system performance and QoS.Since the focus of the paper is resource allocation in NOMA-assisted IoVT with MEC, we consider perfect SIC for the convention of analysis [40].

D. Computing Resource Allocation
For a given association and grouping strategies, the SIC decoding order and transmission power of each IoVT device in the subchannel can be determined.After removing unrelated variables and expressions, the original optimization problem (9) Since problem ( 21) is a simple convex problem, it can be also solved by CVX [37].

E. Device Association and Grouping
With the given power allocation and computing resource allocation strategies, as well as the device association and grouping strategies in the last iteration, the adjacency matrix can be determined.As previously mentioned, the weighted average total delay can be reduced by identifying the negative loops in the weighted directed graph G(V e , E e ; π e ).The association and grouping strategies are then updated based on these loops until there is no negative loop remains within the graph.To find all the negative loop efficiently, we adopt the extended Bellman-Ford searching algorithm (EBFSA) [32].EBFSA is derived from the Bellman-Ford algorithm (BFA), which is widely used for searching the single-source shortest path (SSSP) in weighted directed graphs.The details of EBFSA are described in Algorithm 2. A super node is added to the graph, which is connected to all other nodes.The main procedure of EBFSA is searching for the shortest path from the super node to all other nodes and relaxing the outgoing edges from the super node until no outgoing edges can be relaxed.When relaxing these outgoing edges, IoVT devices in the same subchannel are avoided from appearing in the same path, which is the main difference from the BFA.If the length of the shortest path from the super node to a node is greater than GM , it means there exists a negative loop in this graph.In EBFSA, we relax the shortest path from the super node to node n, i.e., Z n , (∀n ∈ V) in step 6 and step 11 repeatedly until Z n (∀n ∈ V) does not change anymore.Step 9 is a recursive procedure by calling EBFSA repeatedly.The computational complexity of EBFSA increases exponentially with the dimensions of G, M , and N .
We also adopt another greedy-based loop searching algorithm called fast Greedy searching algorithm (FGSA) [28] to reduce the computational complexity of negative loop searching process.The details of FGSA are shown in algorithm 3. FGSA first finds the smallest edge in the graph (for example, i → j), and then finds the node pointing to the smallest edge starting from node j as the next hop.The searching process will be repeated until reaching the maximum iteration steps or there is no negative loop can be found in the graph.It should be noted that steps 4-15 in FGSA will repeat α|U mg | + GM times, where α is an adjustable factor that satisfied α ∈ [ 1 |U mg |+G , |U mg |].By adjusting α, a large number of unnecessary calculations can be reduced at the cost of a small amount of delay.
To more intuitively show the process of IoVT device association and grouping strategies updating according to the negative loops, we give a simple example in Fig. 4. As shown in Fig. 4(a), suppose there are 4 MEC-BSs in a NOMA-assisted IoVT.Each MEC-BS is allocated with 2 subchannels, and there are 10 IoVT devices that request the task offloading service.According to the initial association and grouping strategies, a weighted directed graph can be constructed.Assume a negative loop 2− > 6− > 3− > 2 is found by running EBFSA or FGSA, as shown in Fig. 4(b).Then the IoVT device association and grouping strategies will change according to the direction of the negative loop.As shown in Fig. 4(c), IoVT device 2 will move to the subchannel of IoVT device 6, IoVT device 6 will move to IoVT device 3's subchannel, and IoVT device 3 will move to the subchannel of IoVT device 2. It should be noted that the corresponding associated MEC-BS will also change.
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F. Optimization Framework Summarize
The proposed joint transmission and resource optimization framework in NOMA-assisted IoVT with MEC is shown in Algorithm 4. In the initialization phase, the IoVT will first associate with the nearest MEC-BS and randomly be allocated to a subchannel.Then the transmit power of each IoVT device will be determined by Algorithm 1 and each MEC-BS will allocate computing resources to the associated IoVT devices.Since there are MG virtual IoVT devices are added, a weighted directed graph with the size of MG + N will be constructed, where each node represents an IoVT device and the weight of each node represents the weighted total delay of this IoVT device.In addition, the adjacency matrix Φ with the size of MG + N will be initialized, and the value of φ ij represents the difference of weighted average total delay of all the IoVT devices in subchannel π n j after moving n i to the π n j and remove n j from π n j .the negative loop in the graph can be found by running the searching algorithm EBFSA or FGSA.After that, the association and grouping strategies can be updated according to the negative loop, and the graph weight and the adjacency matrix will also be re-initialized.This process will repeat until there is no negative loop in the graph or achieve the maximum number of iterations.
Computational Complexity Analysis: In Algorithm 1, the optimization variable is an N -dimensional vector, thus the Algorithm 3: Fast Greedy Searching Algorithm (FGSA).

Algorithm 4: Joint Transmission and Resource Optimization Algorithm.
worse-case complexity of Algorithm 1 is O(N 3 ).When solving problem (16), the optimization variable is a matrix of M rows and N columns, so the worse-case computational complexity is O(M 3 N 3 ).In addition, the computational complexity of EBFSA increases exponentially with the dimensions of GM Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
and the number of IoVT devices in each subchannel.The computational complexity of FGSA is O(GM (GM + N ) 2 ) [28].Assume that Algorithm FGSA is repeated τ times in Algorithm 2, thus the computational complexity of Algorithm 2 by adopting FGSA is O(τ (GM 2 (GM + N ) + N 3 + M 3 N 3 )).According to the simulation results in Section IV, τ is much less than (GM + N ).

G. Optimization Framework Implementation
In this section, we will introduce two ways, i.e., centralized and decentralized approaches to implement the proposed graph theory-based optimization frameworks, which can be flexibly selected and deployed to practical NOMA-assisted IoVT with MEC systems according to the actual network situation.
Centralized implementation: When IoVT devices in the network generate video processing tasks that require offloading, they first send pilot signals to MEC-BSs for channel estimation.Subsequently, IoVT devices transmit task summary information (such as data size, processing density, and priority) to a central processing unit (CPU) with sufficient computing capacity through the nearest MEC-BS via OFDMA.After decoding the user signal, MEC-BSs will forward the signals to the CPU through the wired link.The CPU then executes the proposed optimization framework based on IoVT device network conditions, priority, task information, and MEC-BS computing capacity information (steps 1-6 in Fig. 4), obtaining power allocation decisions, device association and grouping decisions, and computing resource allocation decisions.Subsequently, the CPU sends the decision information to all IoVT devices through MEC-BSs (step 7 in Fig. 4), and all IoVT devices formally offload task data to the corresponding MEC-BS subchannels via NOMA, following the policy information provided by the CPU.After decoding the user signal by priority-based SIC decoding, MEC-BSs allocate computing resources accordingly to process the tasks.Once the processing is complete, the processing results will be returned to the IoVT device.
Decentralized implementation: Centralized optimization architecture may face the risk of a single point of failure, and the running time will significantly increase as the network scale expands.In such cases, all MEC-BSs can collaboratively run the proposed optimization framework.A specific MEC-BS can be chosen as the leader, responsible for aggregating and processing intermediate optimization variables.Each MEC-BS will transmit the received task summary information and its own computing capacity information to all other MEC-BS through the X2 interface [41].As the adjacency matrix is an N + MG square matrix and requires global information to obtain power allocation and computing resource allocation decisions, the adjacency matrix calculation is the most time-consuming step.Therefore, each MEC-BS is responsible for calculating the adjacency matrix elements of its associated users and its own subchannels (steps 1-3 in Fig. 4).Afterward, all MEC-BSs will transmit the adjacency matrix calculation results back to the leader MEC-BS.The leader MEC-BS will then run EBFSA or FGSA to find negative loops and update the association and grouping strategies (steps 4-6 in Fig. 4), followed by transmitting the updated results to all other MEC-BSs.This process will continue until no negative loop can be found.Finally, the leader MEC-BS will send the policy to IoVT devices and other MEC-BSs (step 7 in Fig. 4), after which IoVT devices and MEC-BSs will execute task offloading and resource allocation accordingly.
It should be noted that the proposed optimization framework can also be implemented when considering the time-varying wireless channel and device task requirement.In this case, we can assume that the network operations are in a time-slotted fashion.At each time slot, each IoVT device will generate a video processing task, and the channel state remains unchanged [42], [43].At this point, the proposed optimization framework can be deployed according to the above implementation methods at each time slot to obtain instantaneous transmission and resource allocation strategies.

IV. NUMERICAL SIMULATION AND DISCUSSION
In this section, the simulation results and discussion are presented to evaluate the performance of our joint transmission and resource allocation framework.We conduct our simulation on MATLAB R2021b, with 2.9 GHz Intel Core CPU i7 and 32 GB RAM.The simulation code is available on https: //github.com/qlt315/NOMA-MEC-IoVT.

A. Simulation Setup
Assume all the IoVT devices are randomly distributed in a 600 m ×600 m rectangular area.There are four MEC-BSs placed in (200 m, 200 m), (400 m, 200 m), (200 m, 400 m), (400 m, 400 m).The large-scale channel fading is −128.1 − 37.6 log 10 (d n [km]) dB.Additionally, the Rayleigh fading coefficient of each IoVT device is assumed to follow an i.i.d.Gaussian distribution γ mg n ∼ CN (0, 1).The subchannel bandwidth is fixed at B = 2 MHz by default and the power spectrum density of the Gaussian white noise is −174 dBm/Hz.The computation workload and data size requirements of each IoVT device are randomly distributed in the ranges of [15,30] Mbits and [100,200] Kbits, respectively.Furthermore, the computation capability of each MEC-BS is selected uniformly from the range of 5 to 15 Gbits/s.

B. Comparison Algorithms
To verify the feasibility and effectiveness of the proposed graph-based joint optimization schemes, we compare the proposed EBFSA and FGSA with the following four benchmarks: r Gale-Shapley-based Association and Grouping (GSAG) [23], [44]: IoVT devices are associated and grouped by Gale-Shapley algorithm, which is a classic deferred-acceptance method for yielding the stable matching between IoVT devices and subchannels.The number of devices in each subchannel cannot exceed r Hungarian-based Association and Grouping (HAG) [45], [46]: IoVT devices are associated and grouped by Hungarian algorithm, which is a common method to solve the maximum matching problem of borderless weight bipartite graph.
r Channel Difference-based Association and Grouping (CDAG) [47]: A low complexity method that associates and groups the IoVT devices according to the difference of channel gain between IoVT devices and MEC-BSs, in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.which the devices with a larger difference will be assigned in the same subchannel.
r Device Preference-based Association and Grouping (DPAG) [33], [48]: The preference lists of all IoVT devices are developed based on the channel condition.Then each IoVT device will be assigned one by one according to their preference lists.In the simulations, the number of IoVT devices is set to an integer multiple of the number of subchannels.Because in all the baselines, the number of IoVT devices in each subchannel should be equal.Moreover, the preference lists in original GSAG, HAG, CDAG, and DPAG are established only based on channel gain, which does not consider the different computing capacities of MEC-BSs.For the fairness of comparison, we normalize the MEC-BS computing capacity and channel gain and add them together to be the basis for the preference list.

C. Convergence Analysis
We first evaluate the convergence performance of our proposed approach.Assume the number of subchannels of each MEC-BS G = 5.The convergence process of the EBFSA is depicted in Fig. 5(a) for different numbers of IoVT devices.Our findings reveal that as the number of IoVT devices increases, more iteration steps are required for EBFSA to achieve convergence.This can be attributed to the fact that an increase in the number of IoVT devices or subchannels results in a larger directed graph, which in turn leads to more negative loops.Consequently, more policy updates are required to find all negative loops.Furthermore, since network resources are limited, the increased number of IoVT devices accessing the network leads to an increase in interference and insufficient resources allocated to IoVT devices, thereby resulting in increased transmission delay and edge processing delay.Therefore, the total weighted average total delay also increases.Conversely, we observe a decrease in the weighted average total delay when each MEC-BS has more subchannels.
Next, we analyze the convergence process of FGSA under different values of α.In this simulation, we set the number of IoVT devices to 20.As shown in Fig. 5(b), increasing the value of α from 1 to 5 results in an increase in the required iteration steps for convergence.This is due to the fact that increasing α improves the searching accuracy of the FGSA, which enables the algorithm to find more negative loops, leading to better delay performance.However, a higher value of α also increases the running time of the algorithm.Moreover, when α exceeds 5, the delay performance does not improve significantly.
Fig. 6 depicts the positioning, association, and grouping results of four algorithms for 40 IoVT devices and 4 MEC-BSs.The same shape of IoVT devices indicates that they will be connected to the same MEC-BS, and different colors represent different subchannels.It can be seen that IoVT devices prefer the MEC-BSs with better channel conditions and larger computing capabilities.Moreover, due to the random distribution of IoVT devices, if all devices are associated with the nearest MEC-BS, the computing resource allocation of all MEC-BSs may become unbalanced.Thus, the traditional distance-based association strategy is no longer effective.The proposed graphbased optimization algorithms take into account both the channel conditions and computing capabilities.Therefore, when the computing resources of the nearest MEC-BS are insufficient, the IoVT device can associate with another MEC-BS and be grouped optimally on the available subchannel.

D. Transmission Performance Analysis
The uplink transmission rate is a crucial performance metric for wireless communication systems, as it directly impacts the transmission delay of IoVT devices.Therefore, in this subsection, we aim to evaluate the weighted average total transmission rate obtained by different algorithms under varying parameters.The weighted average total transmission rate can be calculated by 1   N ω N n=1 ω n R n .Fig. 7(a) illustrates the impact of the number of IoVT devices on the weighted average total transmission rate.As the number of IoVT devices increases, the average transmission rate decreases.This is due to the increased interference among IoVT devices associated with the MEC-BSs, resulting in a lower SINR.Among the evaluated algorithms, FGSA and EBFSA exhibit the best average transmission rate performance, followed by the GSAG, DPAG and CDAG schemes.On the other hand, HAG has the worst performance.This can be attributed to the effectiveness of the proposed algorithms in searching for better association and grouping strategies.Notably, the proposed algorithms can still achieve an uplink transmission rate of 17.3 Mbps when N = 100, which is 79.1% higher than that of HAG.
The average transmission rate under the different number of subchannels is shown in Fig. 7(b).It is observed that when the number of subchannels per MEC-BS increases, the average transmission rate also increases.This can be explained by the fact that having more subchannels leads to a decrease in the average number of IoVT devices in each subchannel, resulting in reduced inter-user interference in the same subchannel and hence increased transmission rate.Similar to the findings in Fig. 7(a), the proposed FGSA and EBFSA can obtain the highest transmission rate compared to the other algorithms by effectively exploring the association and grouping strategies in the weighted directed graph.The proposed algorithms achieve an average transmission rate of 60.3 Mbps when G = 20, which is 49.7% more than that of HAG.

E. Weighted Average Total Delay Performance Analysis
Next, we turn our attention to the weighted averaged total delay performance under different parameters.Fig. 8(a) shows Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.[23], [44].(c) HAG [45], [46].(d) CDAG [47].(e) DPAG [33], [48].the weighted average total delay under the different numbers of IoVT devices.When the number of IoVT devices increases, the weighted average total delay in the system also increases.This is because an increase in the number of IoVT devices associated with MEC-BSs leads to an increase in the number of devices in each subchannel, thereby increasing inter-user interference and reducing the uplink transmission rate, leading to higher transmission delays.Additionally, the limited computing capabilities of MEC-BSs can result in reduced computing resources allocated to some of the IoVT devices, increasing the processing delay at the edge.The proposed algorithms can effectively mitigate these issues by optimizing the association and grouping strategies through a graph theory-based approach.Specifically, the weighted average total delay achieved by the proposed algorithms is 0.1228 s when N = 100, which is 67.18% lower than that of HAG.
The performance of the weighted average total delay is further investigated under different numbers of subchannels.As illustrated in Fig. 8(b), with N = 40.The increase in the number of subchannels per MEC-BS leads to a decrease in the weighted average total delay.This is because more subchannels will reduce the number of IoVT devices in each subchannel, which in turn reduces the inter-user interference and increases the uplink transmission rate, resulting in a decrease in transmission delay.The proposed algorithms achieve the lowest weighted average total delay among all algorithms, with a value of 0.0281 s when G = 20, which is 92.43% less than that of HAG.This result highlights the effectiveness of the proposed algorithms in reducing the total transmission delay.
Fig. 8(c) shows the weighted average total delay under different computing capabilities of MEC-BSs.The results demonstrate that the weighted average total delay decreases as the computing capability of each MEC-BS increases.This is because a higher computing capability of MEC-BSs allows for more computing resources to be allocated to each IoVT device, which reduces the processing delay and ultimately results in a lower total delay.Specifically, the proposed algorithms achieve a weighted average total delay of 0.0254 s when C b m = 10 Gbit/s, which is 85.7% less than that of HAG.
In Fig. 8(d), the impact of increased computing requirements for video tasks on the weighted average total delay of IoVT devices in the system is shown.As can be observed, as the computing requirements increase, the weighted average total delay also increases.This is due to the fact that with fixed computing capabilities of MEC-BSs, processing tasks of IoVT devices takes longer time, resulting in increased total delay.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.For instance, when W n = 80 Mbit/s, the proposed algorithms achieve a weighted average total delay of 0.0731 s, which is 74.5% less than that of HAG.
Fig. 8(e) shows the weighted average total delay of MEC-BS under different maximum transmit power.It can be observed that increasing the transmission power of IoVT devices will only slightly improve the SINR and result in a slight decrease in the weighted average total delay of all IoVT devices in the system.However, the increase in transmission power will also increase the interference to other IoVT devices, offsetting most of the gains from the improved SINR.Therefore, blindly increasing the transmit power is not recommended due to the power consumption overhead of IoVT devices.From Fig. 8(e), it can be seen that when P max = 0.7 W, the proposed algorithms achieve a weighted average total delay of 0.0244 s, which is 66.67% less than that of HAG.
Finally, we evaluate the impact of the bandwidth of each subchannel on the weighted average total delay.As shown in Fig. 8(f), when the bandwidth of each subchannel increases, the uplink transmission rate of IoVT devices within the subchannel also increases.Consequently, the transmission delay decreases, leading to an overall decrease in the total delay.Specifically, the proposed algorithms achieve a weighted average total delay of 0.0405 s when the bandwidth is set to 0.5 MHz, which represents a reduction of 72.52% compared to the performance of HAG.

F. Impact of SIC Decoding Orders
SIC decoding is an important part of NOMA transmission, which will directly affect the transmission rate of NOMA.In order to verify the effectiveness of priority-based SIC decoding proposed in this article, we will investigate the impact of different SIC decoding sequences on system performance in this section.We chose the following two SIC decoding methods for comparison.Channel-based SIC [20], [21]: IoVT devices are decoded based on decreasing order of their channel gains.Random SIC: The SIC decoding order is randomly generated.Specifically, we validate the system performance using EBFSA, GSAG, and HAG under different SIC decoding orders.
Fig. 9(a) shows the weighted average total uplink transmission rate versus the number of IoVT device slots with different SIC decoding orders.Similar to Fig. 9, as the number of users increases, the uplink transmission rate will decrease because more users bring greater interference.It is worth noting that the proposed priority-based SIC decoding can achieve the highest throughput performance, since the priority-based SIC decoding can ensure that users with higher priorities suffer less interference, while the other two SIC decoding methods ignore the priority of devices in IoVT.For EBFSA, when the number of IoVT devices is 100, the weighted average total uplink transmission rate of priority-based SIC decoding is 15.01% and 45.91% higher than channel-based SIC decoding and random SIC decoding, respectively.
Similar conclusion can also be validated in Fig. 9(b), which demonstrates that priority-based SIC decoding can achieve optimal delay performance under different IoVT device numbers.Since the SIC decoding order may affect the resource allocation, device association and grouping optimization, therefore the uplink transmission rate of high-priority IoVT devices will increase, resulting in a decrease in uplink transmission delay.Specifically, for EBFSA, when the number of IoVT devices is 100, the weighted average total delay of priority-based SIC decoding is reduced by 7.86% and 28.27% compared to channelbased SIC decoding and random SIC decoding, respectively.

G. Impact of Channel Estimation and Imperfect SIC Decoding
Since it is impractical to obtain the perfect CSI in realworld communication system, we investigate the robustness of proposed optimization schemes to channel estimation error in this subsection.Specifically, let ĥ mg n represent the imperfect channel, which satisfied ĥ is the channel estimation error with zero-mean complex Gaussian distribution, i.e., h mg n ∼ CN (0, ζ|h mg n | 2 ).Here, the variance corresponds to the channel error power, and ζ is the channel estimation error parameter [49].
Fig. 10 illustrates the influence of RSI levels and the channel estimation error parameter on the delay performance of various schemes.In Fig. 10(a), it is observed that as the RSI level increases, the average total delay experiences a slight rise.According to (1), imperfect SIC decoding introduces an interference term into the SINR expression's denominator, leading to a reduction in the uplink rate and an increase in uplink transmission delay.Notably, our proposed schemes ensure optimal delay performance even in imperfect SIC decoding scenarios, resulting in a delay reduction of 62.31% -66.29% compared to HAG.In Fig. 10(b), the delay performance of different schemes is presented under varying channel estimation error parameters.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
With an increase in the channel estimation error parameter, the average total delay exhibits an upward trend, albeit not monotonically.This behavior stems from the system allocating transmit power based on incorrect channel coefficients as the difference between the estimated and actual channels widens.Consequently, the optimal uplink transmission rate for all users cannot be guaranteed.It is crucial to highlight that the proposed schemes maintain optimal delay performance when perfect CSI is unobtainable, leading to a delay reduction of 53.06% -66.26% compared to HAG.

V. CONCLUSION AND FUTURE WORK
In this paper, we consider joint transmission and resource allocation in the NOMA-assisted IoVT with MEC.Since the visual processing tasks of different IoVT devices have different priorities, we aim to minimize the weighted average total delay of all the IoVT devices by optimizing the power allocation, device association, device grouping, and computing resource allocation strategies.To solve the problem, we propose a graph theory-based joint optimization framework to decompose and transform the original problem into finding negative loops in the weighted directed graph.A priority-based SIC decoding mechanism, power allocation, and computing resource allocation schemes are proposed to calculate the adjacency matrix.In addition, the association and grouping decision strategies of IoVT devices can be updated according to the negative loops.Then, we propose two algorithms called EBFSA and FGSA to find the negative loops efficiently.Simulation results show that compared with other latest association and grouping algorithms, the proposed algorithms can effectively improve the transmission rate of IoVT devices by at most 79.1%, and remarkably reduce the weighted average total delay of the system by at most 92.43% compared with other related algorithms.
In traditional cellular-based NOMA networks, users at the edge of the cell are prone to suffer severe inter-cell interference and signal attenuation.As an emergent network paradigm, cell-free MIMO or user-centric network (UCN) can significantly improve the wireless transmission performance of NOMA [50].Inspired by this, we plan to explore resource allocation optimization in NOMA-assisted UCN with MEC.Considering the multi-BS cooperative transmission mode in UCN, when the number of users and APs increases, the worse-case complexity will increase exponentially for traditional optimization methods like branch and bound, and the time to obtain the optimal solution will be unbearable.Therefore, it is necessary to develop efficient and reliable algorithms, especially distributed algorithms, to ensure efficient resource allocation.

Fig. 2 .
Fig. 2. Overview of the graph theory-based joint transmission and resource allocation optimization framework.

Fig. 3 .
Fig. 3. Illustration of the transfer set and swap set in the weighted directed graph.(a) IoVT devices n 1 , n 3 , n 5 can compose a 3-swap set.(b) IoVT devices n 4 , n 2 , n 5 can compose a 3-transfer set.
and all these three IoVT devices n 1 , n 3 and n 5 are real IoVT devices, so these three IoVT devices can compose a 3-swap set.Therefore, by adopting operations (n 1 → π n 3 ), (n 3 → π n 5 ), (n 5 → π n 1 ), the objective function of problem P mg can be decreased.In addition, as shown in Fig.3(b), n 4 −> n 2 −> n 8 −> n 4 is a negative loop, where IoVT device n 8 is a virtual IoVT device in subchannel g 2 of MEC-BS m 2 , so IoVT devices n 4 , n 2 , n 5 compose a 3-transfer set (n 8 , n 5 ∈ U 3 ).So the objective function of problem P mg can be also decreased by operations (n 4 → π n 2 ), (n 2 → π n 8 ).

Fig. 9 .
Fig. 9. rate performance with different SIC decoding orders.(a) Weighted average total transmission rate vs. number of IoVT devices.(b) Weighted average total delay vs. number of IoVT devices.

Fig. 10 .
Fig. 10.Delay performance with imperfect CSI and SIC decoding.(a) Weighted average total delay vs. RSI level.(b) Weighted average total delay vs. channel estimation error parameter.
channel coefficient between IoVT device n and MEC-BS m on subchannel g, h mg n

Langtian
Qin received the B.S. degree in information engineering from Xidian University, Xi'an, China, in 2021.He is currently working toward the master's degree with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, China.His research interests include mobile edge computing and resource allocation in wireless heterogeneous networks.Hancheng Lu (Senior Member, IEEE) received the Ph.D. degree in communication and information systems from the University of Science and Technology of China (USTC), Hefei, China, in 2005.He is currently a Professor with the Department of Electronic Engineering and Information Science, USTC.His research interests include multimedia communication and networking and resource optimization in wireless heterogeneous networks.Yuang Chen (Graduate Student Member, IEEE) received the B.S. degree from the Hefei University of Technology, Hefei, China, in 2021.He is currently working toward the Ph.D. degree with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, China.His research focuses on 5 G/6 G wireless network technologies, such as URLLLC services.Baolin Chong received the B.S. degree from the Nanjing University of Science and Technology, Nanjing, China, in 2021.He is currently working toward the master's degree with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, China.His research interests include nonorthogonal multiple access, Internet of Video Things systems, and user association and power control.Fengqian Guo received the B.S. degree from Jiangnan University, Wuxi, China, in 2017, and the Ph.D. degree in communication and information systems with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, China, in 2022.His research interests include wireless transmission and multiple access networks.

TABLE I SYNOPSIS
OF SELECTED REFERENCES and the weighted average total delay will decrease, these i IoVT devices compose an i-transfer set.Definition 2: For any i IoVT devices indexed by {n 1 , . . ., n i } in different subchannels.If the IoVT association matrix A and IoVT grouping matrix B is changed into Ã and B by Π