IRS-aided Wireless Powered MEC Systems: TDMA or NOMA for Computation Offloading?

08/13/2021 ∙ by Guangji Chen, et al. ∙ University of Southampton UNSW Shanghai Jiao Tong University 0

An intelligent reflecting surface (IRS)-aided wireless powered mobile edge computing (WP-MEC) system is conceived, where each device's computational task can be divided into two parts for local computing and offloading to mobile edge computing (MEC) servers, respectively. Both time division multiple access (TDMA) and non-orthogonal multiple access (NOMA) schemes are considered for uplink (UL) offloading. Given the capability of IRSs in intelligently reconfiguring wireless channels over time, it is fundamentally unknown which multiple access scheme is superior for MEC UL offloading. To answer this question, we first investigate the impact of three different dynamic IRS beamforming (DIBF) schemes on the computation rate of both offloading schemes, based on the flexibility for the IRS in adjusting its beamforming (BF) vector in each transmission frame. Under the DIBF framework, computation rate maximization problems are formulated for both the NOMA and TDMA schemes, respectively, by jointly optimizing the IRS passive BF and the resource allocation. We rigorously prove that offloading adopting TDMA can achieve the same computation rate as that of NOMA, when all the devices share the same IRS BF vector during the UL offloading. By contrast, offloading exploiting TDMA outperforms NOMA, when the IRS BF vector can be flexibly adapted for UL offloading. Despite the non-convexity of the computation rate maximization problems for each DIBF scheme associated with highly coupled optimization variables, we conceive computationally efficient algorithms by invoking alternating optimization. Our numerical results demonstrate the significant performance gains achieved by the proposed designs over various benchmark schemes.

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I Introduction

With the rapid development of popular Internet-of-Everything (IoE) technologies, the unprecedented proliferation of mobile sensors, electronic tablets, and wearable devices is set to continue in support of smart transportation, smart homes, and smart cities [8879484]. For realizing the IoE, next generation wireless networks are expected to support massive number of connections and accommodate huge data traffic. As such, superior multiple access (MA) schemes are required to attain high spectral efficiency (SE) for a massive number of IoE devices in next generation wireless networks[8114722]. Recently, it has been shown that non-orthogonal multiple access (NOMA) is capable of improving the SE by allowing multiple users to simultaneously access the same spectrum. Therefore, NOMA has been recognized as one of the key technologies in next generation wireless networks[8114722].

On the other hand, an IoE device is often equipped with a low-performance processor and limited battery capacity, given their practical size and cost constraints. In particular, the emerging applications, such as unmanned driving and automatic navigation, generally rely on the execution of low-latency and computation-intensive tasks, thus imposing new challenges on IoE devices [8016573]. As a remedy, by incorporating radio frequency (RF)-based wireless power transmission (WPT) and mobile edge computing (MEC), wireless powered MEC (WP-MEC) becomes a promising solution for granting self-sustainability and high computational capabilities to IoE systems [7442079, 8304010, 8334188, 8434285, 9140412, 8986845, 9312671, 8537962, 9179779, 8234686]. Specifically, RF-based WPT enables energy harvesting (EH) from RF signals and it is capable of prolonging the battery recharge-period of devices [8234686, 7843670, 8421584]. To improve the computational capabilities for IoE systems, the MEC technology enables IoE devices to offload their tasks to nearby MEC servers in real time, which can compute their tasks remotely [8016573].

To enhance the computational efficiency of traditional WP-MEC systems, sophisticated resource allocation relying on optimization objectives, such as computation rate maximization [8304010, 8334188, 8434285], energy consumption [8986845, 9312671, 8537962], and latency minimization [9179779], etc, has been proposed. For instance, in [7442079]

, the WP-MEC framework was proposed for a single-user setup, where the probability of successfully processing a given amount of data was maximized subject to both end-to-end latency and EH constraints. In general, MEC supports a pair of basic operational modes, namely binary and partial offloading modes

[7442079, 8304010, 8334188, 8434285, 9140412]. Specifically, for the partial offloading mode, the computational task can be divided into two parts for partial local computing and offloading, respectively, while for the binary offloading mode, the computational task cannot be partitioned, hence it is either executed at the local device or offloaded to MEC servers [9140412]. Based on the concept of binary and partial offloading modes, the corresponding computation rate maximization problem was investigated in [8304010, 8334188, 8434285] for a multi-user setup, where time division multiple access (TDMA) was adopted for uplink (UL) offloading. As a further advance, the superiority of employing NOMA over TDMA in WP-MEC systems was quantified in terms of its energy efficiency improvement [8986845, 9312671, 8537962] and latency reduction [9179779]. Therefore, NOMA is regarded as an attractive scheme for UL offloading in traditional WP-MEC systems.

However, the efficiency of both the downlink (DL) WPT and UL offloading may become severely degraded by the wireless channel attenuation between transceivers, which thus fundamentally limits the performance of WP-MEC systems. With the goal of tacking this issue, the authors of [8234686] exploited the multiple-input multiple-output (MIMO) technique for improving the WPT efficiency and studied the corresponding energy consumption minimization problem. Although the massive MIMO technology considerably improves the efficiency of both WPT and offloading by exploiting the huge beamforming (BF) gain [8271992], [9027954], the associated high hardware cost and energy consumption are still grave obstacles in the way of its practical implementation. Recently, intelligent reflecting surfaces (IRSs) have been proposed as a cost-effective technology for improving the spectral efficiency and energy efficiency of next generation wireless networks [8910627, 9326394, 9424177]. Specifically, an IRS is a planar array comprised of a large number of low-cost passive reflecting elements, which can reflect incident signals and intelligently adapt their phase shifts according to the real-time channel conditions[8910627]. As such, IRSs are capable of reconfiguring the wireless propagation environment for achieving e.g., signal enhancement and/or interference suppression. In particular, the fundamental squared-power gain of IRSs was originally unveiled in [8811733], which then inspired intensive research interests in investigating various IRS-aided wireless systems.

The new research paradigms of IRS-aided wireless information transmission (WIT), WPT, and MEC have been extensively studied in the literature. For IRS-aided WIT systems, joint passive BF at IRSs and active BF at base stations (BSs) was designed either for minimizing the transmit power of BSs or for maximizing the system capacity, e.g., [8811733, 9139273, 9110869, 9279253, 9039554, 9427474]. As a further practical development, the analysis and optimization of IRS-aided wireless communications were studied by considering both discrete phase shifts [8930608, 9295369] and amplitude-dependent phase shifts [9115725]. In addition to exploiting IRSs for improving the WIT performance, the IRS technology is also appealing for WPT in IoE applications, thanks to its beneficial passive BF gain. Specifically, a promising line of research focused on passive BF design for simultaneous wireless information and power transfer (SWIPT) systems [8941080, 9257429, 9133435]. Another line of research investigated IRS-aided wireless powered communication networks (WPCNs), where the devices first harvest energy in the DL and then transmit information in the UL [9214497, 9298890, 9400380]. However, in traditional MEC systems, the task offloading efficiency may not be satisfactory due to the harsh propagation conditions of the wireless links. To address this issue, the authors of [9270605, 9133107] exploited the IRS technology for improving the offloading efficiency of MEC systems by studying the associated computation rate maximization and offloading latency minimization problems, respectively.

Given the aforementioned benefits of the IRS technique, its employment in WP-MEC systems is attractive for realizing IoE, since both the efficiency of DL WPT and UL offloading can potentially be improved. Additionally, next generation wireless networks require superior MA schemes for supporting a massive number of IoE devices. Therefore, integrating IRSs with efficient MA schemes in WP-MEC systems is essential for granting satisfactory experience of IoE applications in next generation wireless networks. For traditional MEC and WP-MEC systems operating without IRSs, the authors of [8986845, 9312671, 8537962] demonstrated the superiority of NOMA over TDMA for UL offloading under the assumption of given wireless channels within a channel coherence time duration. However, these results may not be applicable to the new family of IRS-aided WP-MEC systems, since the IRS is capable of proactively establishing favorable time-varying wireless channels, which could introduce different impacts on MA schemes. As such, it still remains unknown which MA scheme is more efficient for UL offloading in IRS-aided WP-MEC systems. This knowledge-gap motivates us to investigate the achievable computation rate in such scenarios by considering the interplay between IRSs and MA schemes. To characterize the achievable computation rate of IRS-aided WP-MEC systems, the main challenges we identify are as follows: 1) the performance comparison between NOMA and TDMA has to be carried out for IRS-aided UL offloading by considering the favorable time-varying wireless channels controlled by IRSs; 2) the specific IRS configuration required for reaping the potential benefits of WP-MEC systems has to be identified; 3) the design of IRS BF and resource allocation for WP-MEC systems is generally intractable.

To address the above issues, this paper investigates the achievable computation rate maximization problems of IRS-aided WP-MEC systems by considering two types of offloading schemes, i.e., TDMA and NOMA. Specifically, we focus our attention on a typical setup, where a hybrid access point (HAP) is exploited both as the energy transmitter and the MEC server. Moreover, an IRS is deployed for enhancing the efficiency of both DL WPT and UL offloading. Our main contributions are summarized as follows:

  • We propose an offloading framework for investigating the performance of IRS-aided WP-MEC systems, where three different levels of dynamic IRS beamforming (DIBF) schemes are considered: Case 1: both DL WPT and UL offloading share the same IRS BF vector; Case 2: two different IRS BF vectors are exploited for DL WPT and UL offloading, respectively; Case 3: the IRS BF vectors can be further adapted for UL offloading with respect to each individual device. Under this framework, we formulate the corresponding computation rate maximization problems by jointly optimizing the resource allocation and the IRS BF for the aforementioned three cases.

  • We analytically show that appropriately adjusting the IRS BF vectors for UL offloading is capable of improving the computation rate of TDMA, while it is not beneficial for that of NOMA. By analyzing the relationship between the computation rate maximization problems of TDMA and NOMA, we prove that the computation rate achieved by TDMA is the same as that by NOMA for both Case 1 and Case 2. By contrast, since TDMA-based IRS-aided WP-MEC systems are capable of benefiting from varying the IRS BF vectors in the UL offloading stage, the computation rate of TDMA exceeds that of NOMA for Case 3.

  • To gain insights into the beneficial effect of IRSs on WP-MEC systems, we first consider a single-user setup, where we derive a threshold-based UL offloading activation condition. Specifically, we demonstrate that UL offloading is activated iff the transmit power of the HAP is above a certain threshold and increasing the number of IRS elements is capable of reducing the threshold. For the more general multi-user setup, we develop an efficient alternating optimization (AO) algorithm for solving the resultant problems, where the resource allocation and the IRS BF design subproblems are solved alternatingly.

  • Our numerical results show that the proposed IRS-aided WP-MEC designs are capable of substantially improving the computation rate compared to the benchmark schemes. It is also found that exploiting IRSs not only increases the total energy harvested via DL WPT, but also leaves more time available for UL offloading, which unveils a further benefit of IRSs for WP-MEC systems. Moreover, the computation rate of Case 3 significantly exceeds that of Case 1 and Case 2, while the performance loss of Case 1 compared to Case 2 is negligible. The results imply that the associated signaling overhead can be reduced by opting for Case 1 instead of Case 2 at the cost of a modest performance erosion.

The rest of this paper is organized as follows. Section II presents our system model and problem formulations. Section III provides the theoretical performance comparison of NOMA and TDMA-based UL offloading. Section IV focuses on studying the impact of IRSs on the UL offloading activation condition. Section V proposes computationally efficient algorithms for solving the formulated problems for the different scenarios. Section VI provides numerical results for evaluating the proposed designs. Finally, Section VII concludes the paper.

Notations: stands for the set of complex matrix. represents the set of positive number. For a complex-valued vector , the -th entry is denoted by , and denote Hermitian transpose and transpose, respectively, denotes a diagonal matrix with each diagonal entry being the corresponding entry in . The real part and the phase of a complex number are denoted by and , respectively. is the big-O computational complexity notation. denotes the partial derivation of a function with respect to a variable .

Ii System Models and Problem Formulations

Ii-a System Model

As shown in Fig. 2, an IRS-aided WP-MEC system is considered, which consists of a HAP, an IRS, and wireless-powered devices. In particular, a MEC server and an RF energy transmitter are integrated at the HAP so that it can broadcast wireless energy to devices and execute computational tasks, while each device has a rechargeable battery and an EH circuit component which can store the harvested energy to power its operation. The HAP and all the devices are equipped with a single-antenna111To unveil the potential benefits of IRSs in WP-MEC systems for improving the computation rate performance, we assume that the HAP is equipped with a single-antenna for tractability. It is commonly adopted in previous works [8334188, 8434285, 9140412]. The case of multiple antennas is left for our future work. and the IRS has reflecting elements. To ease their practical implementation, all the devices and the HAP operate over the same frequency band. The DL WPT and UL offloading are assumed to operate in time-division multiplexing manner by following the typical “harvest-and-then offload” protocol of [8334188, 8434285], as shown in Fig. 2. Without loss of generality, we assume that each channel coherence block consists of multiple frames and the operation time of each frame is denoted by . To characterize the maximum achievable computation rate, the channel state information (CSI) of all channels is assumed to be perfectly acquired by the HAP, based on the channel acquisition methods discussed in [8910627]. The channels from the HAP to device , from the HAP to the IRS, and from the IRS to device are denoted by , , and , , respectively.

In this paper, we assume that the partial computation offloading mode is used. Specifically, the computational tasks of each device can be partitioned into two parts: one for local computing and the other for offloading to the HAP. Similar to [8334188, 8434285, 8986845], we assume that the local computing at each device adopts a different component from that used for EH and task offloading. Thus, local computing can be executed throughout the entire frame of duration . The number of central processor unit (CPU) cycles required for computing one bit of raw data at each device is denoted by and its value is determined by the properties of the specific application [8334188]. Let denote the CPU’s chosen frequency (cycles per second) at device . Therefore, the bits computed locally by device and the corresponding dissipated energy by local computing are and , respectively [8334188]. Note that represents the computational energy efficiency of specific CPU chip, which depends on the architecture of the chip [7442079].

Fig. 1: An IRS-aided wireless powered MEC system.
Fig. 2: The structure of a transmission frame.

As shown in Fig. 2, the transmission frame is comprised of four segments. First, the HAP broadcasts wireless energy to all devices with the aid of the IRS. Then, all the devices can decide to offload their tasks to the HAP by using TDMA or NOMA. In the third stage, the MEC server at the HAP executes the computational tasks offloaded from all devices. Finally, the computational results are downloaded from the HAP to each device. Since the MEC server has much higher computational capability than those of the devices and the amount of data representing the computational results is negligible [8234686], the time duration of the third and fourth stages can be neglected as in [8334188, 8434285]. The details of the first and second stages are described as follows.

For DL WPT, an energy signal is broadcasted by the HAP at a constant transmit power for a time duration of . The reflection phase-shift matrix of the IRS for DL WPT is denoted by , where . Since the noise power is much lower than the power received from the HAP[6678102], we assume that the energy harvested from noise is negligible. Based on the linear EH model222Although the non-linear EH model can capture the relationship between the harvested RF power and the converted direct current power more precisely [7843670], the key results regarding to theoretical performance comparison between NOMA and TDMA for UL offloading are directly applicable to a more general non-linear EH model. It will be discussed later in Remark 3. The linear EH model is adopted here to facilitate us to explicitly demonstrate the impact of IRS on UL offloading activation condition. of [8304010, 8334188, 8434285], the energy harvested at device is

(1)

where represents the energy conversion efficiency of each device, and denotes the IRS BF vector of the DL WPT.

At the UL offloading stage, all devices can offload their tasks to the HAP by the TDMA or NOMA schemes. Adopting different IRS BF vectors during the NOMA/TDMA frame, i.e., DIBF, is in principle possible and may potentially improve the computation rate at the cost of additional signaling overhead. This is because the algorithm is usually executed by the HAP due to the limited processing capability of the IRS and thus the HAP has to feed back the IRS BF vectors to the IRS for reconfiguration. Specifically, we propose three different levels of DIBF schemes as follows: Case 1: The same IRS BF vector is adopted during the entire frame; Case 2: The IRS BF vectors of the DL WPT and UL offloading can be different, but the same IRS BF vector is adopted in the UL offloading stage for all devices; Case 3: The IRS BF vectors of the DL WPT and UL offloading of each device can be different, i.e., different IRS BF vectors can be used for UL offloading. Considering the aforementioned three cases, the details of UL offloading using TDMA and NOMA are presented as follows.

Ii-A1 Offloading Using TDMA

The time duration of offloading, namely , is further partitioned into orthogonal time slots (TSs), which are denoted by . Device offloads its data in its -th TS . Let denote the transmit power of device . For Case 1, the DL WPT stage and the UL offloading stage share the same IRS BF vector . In this case, the achievable offloading sum-rate is written as

(2)

where represents the system bandwidth and denotes the power of the additive white Gaussian noise at the HAP.

For Case 2, we adopt to denote the common IRS BF vector in the UL offloading stage. The achievable offloading sum-rate is represented as

(3)

For Case 3, the IRS BF vector used for UL offloading in -th TS is denoted by . Thus, the achievable offloading sum-rate is given by

(4)

Ii-A2 Offloading Using NOMA

When NOMA is adopted for UL offloading, all the devices simultaneously transmit their respective data to the HAP throughout the whole time duration of at the transmit power . To mitigate the inter-user interference, successive interference cancellation (SIC) is performed at the HAP. Taking device as an example, the HAP will first decode the message of device , , before decoding the message of device . Then, the offloading message of device , , will be subtracted from the composite signal. The offloading message received from device , , is treated as noise. For Case 1, the IRS BF vector is denoted by for the UL offloading stage using NOMA. Thus, the achievable offloading sum-rate for all the devices is [9400380]

(5)

Accordingly, the achievable offloading sum-rate of Case 2 and Case 3 for the NOMA-based UL offloading can be written as

(6)
(7)

respectively, where denotes the time duration, when is used.

Ii-B Problem Formulation

In this paper, we aim for maximizing the total number of computed bits of our IRS-aided WP-MEC systems, by jointly optimizing the IRS BF vectors, the time allocation of WPT and offloading, the transmit power of each device, and the local CPU frequency at each device. Both TDMA and NOMA are considered for UL offloading leading to the following formulation:

Ii-B1 TDMA-based Offloading

When the TDMA scheme is applied, Case 1, Case 2, and Case 3 are considered for evaluating the impact of DIBF on the computation rate. The computation rate maximization problem of Case 1 can be formulated as333To facilitate us to compare the fundamental limits of the achievable computation rate for TDMA and NOMA-based offloading schemes, the quality of service (QoS) constraints of each device are not considered here. Note that the proposed algorithm can be applicable to the scenarios with QoS constraints. Please refer to Remark 5 in Section V for details.

(8a)
(8b)
(8c)
(8d)
(8e)

In , (8b) represents the energy harvesting causality constraint that the total dissipated energy cannot be higher than the total harvested energy444Since we consider that each channel coherence block consists of multiple frames, the consumed energy here comprises two parts. One part is used for UL offloading in the current frame, while the other part is for local computing throughout UL offloading in the current frame and DL WPT in the next frame. Note that constraint (8b) was also adopted in [8234686, 8334188, 8434285]. [8234686, 8334188, 8434285]. Furthermore, (8c) is the constraint on the time duration of the DL WPT and UL offloading, while (8d) contains the non-negativity constraints for the optimization variables and (8e) is the unit-modulus constraint for the IRS BF vector. For Case 2 and Case 3, the corresponding computation rate maximization problems can be formulated, respectively, as:

(9a)
(9b)
(9c)
(10a)
(10b)
(10c)

Ii-B2 NOMA-based Offloading

When NOMA is applied for UL offloading, the corresponding computation rate maximization problems are formulated according to the aforementioned three cases, respectively, as follows:

(11a)
(11b)
(11c)
(11d)
(11e)
(12a)
(12b)
(13a)
(13b)

Iii TDMA or NOMA for UL Offloading?

When multiple devices are activated for UL offloading, it still remains unknown which MA scheme is more efficient for UL offloading, especially when considering the impact of the IRS. To answer this question, the theoretical performance comparison between NOMA and TDMA-based UL offloading is provided in this section. First, the impact of DIBF on the computation rate of both NOMA and TDMA-based WP-MEC systems is analyzed. Then, we analytically compare the computation rate achieved by NOMA and TDMA schemes for Case 1, 2 and 3.

Iii-a Impact of DIBF on NOMA and TDMA

For notational simplicity, we use and to denote the sum computation rate for Case of TDMA and NOMA at the optimal solution, respectively. To shed light on the impact of DIBF on the computation rate of the NOMA and TDMA schemes, we first introduce the following lemmas.

Lemma 1

For IRS-aided WP-MEC systems employing NOMA for offloading, it follows that .

proof 1

Assume that an optimal solution of is given by . Then, the optimal value of can be expressed as

(14)

There always exists an IRS BF vector denoted by , , which satisfies . As such, we have

(15)

where . The equality holds if . Meanwhile, by setting , problem is reduced to , which yields . Thus, we have . Similarly, problem is reduced to by setting , thus we have .

Lemma 2

For IRS-aided WP-MEC systems employing TDMA for offloading, it follows that .

proof 2

By setting , problem is reduced to , which yields . For Case 3, the equivalent channel power gain of each device can be maximized by setting to align the cascaded link with the direct link . Thus, holds for device , which yields .

Lemma 1 and Lemma 2 provide the following insights and also serve as the theoretical foundation for comparing TDMA and NOMA-based offloading, which will be discussed later.

  • For NOMA-based UL offloading, varying the IRS BF vectors in the UL offloading stage does not necessarily attain performance improvements over a static IRS BF vector. By contrast, for TDMA-based UL offloading, the computation rate can be further improved by varying IRS BF vectors for UL offloading.

  • For both TDMA and NOMA-based WP-MEC systems, having different IRS BF vectors for the DL WPT and UL offloading generally outperforms its counterpart using the same IRS BF vector throughout the entire frame.

Iii-B TDMA versus NOMA-based UL Offloading

To compare the achievable computation rate performance between offloading using TDMA and NOMA, the relationship between and is presented in the following theorem.

Theorem 1

Assuming that and are the optimal solutions of and , respectively, we have with and .

proof 3

See Appendix A.

Note that the similar results presented in Theorem 1 can be directly extended to capture the interrelation between and , i.e., . Theorem 1 explicitly shows that the solutions of problem and can be directly obtained based on those of and , respectively.

Remark 1

The results presented in Lemma 1, Lemma 2, and Theorem 1 answer the fundamental question regarding the computation rate comparison between offloading using TDMA and NOMA. Specifically, the comparison outcome depends on which DIBF scheme is applied. For Case 1 and Case 2, it is shown that the same computation rate can be achieved by using TDMA and NOMA for offloading. Since the computation rate of TDMA can be further improved by adapting IRS BF vectors over different TSs in the UL offloading stage, the computation rate of TDMA becomes higher than that of NOMA for Case 3 at the cost of extra signaling overhead. As such, we have the inequality chain as follows:

(16)
Remark 2

Considering the high regime, i.e., , which implies that the device has nearly no computing capability to deal with computationally intensive tasks, the computations completely rely on offloading the tasks to MEC servers. In this case, the computation rate maximization problem is equivalent to the throughput maximization problem of WPCNs. For Case 2, our previous work [9400380] unveiled that the same IRS BF vector can be exploited for the DL and UL in WPCNs without loss of optimality, i.e., . Based on the results provided in Theorem 1, we have the following relationship in the high regime:

(17)

In contrast to (16), (17) suggests that for Case 2, DL WPT and UL offloading can adopt the same IRS BF vector without loss of optimality at a lower signaling overhead.

Remark 3

Note that the theoretical comparison provided in Remark 2 can be directly extended to a more general non-linear EH model. For a general EH model, the output direct current power can be generally expressed as a function of the input RF power, i.e, . Replacing by in Appendix A, the results can be directly obtained through similar steps.

Iv UL Offloading activation Condition in single-user Systems

Before deriving the solutions of the aforementioned computation rate maximization problems, we consider the special case of a single-user setup, i.e., , to gain important insights into the efficiency of IRSs for UL offloading activation. In this case, the MA schemes have no impact on the results, and thus the computation rate maximization problems are simplified to:

(18a)
(18b)
(18c)
(18d)
(18e)

Problem (18) has not been investigated in previous articles to the best of our knowledge. Note that for a WP-MEC system, UL offloading may not be activated, when suffering from severe wireless channel conditions. Hence, we focus our attention on a single-user case to unveil the impact of IRSs on the UL offloading activation condition. For problem (18), the optimal IRS BF vector can be directly obtained as , which aligns the cascaded channel between a typical device and the HAP via the IRS with the end-to-end channel. By setting as the optimal form, the channel power gain between a typical device and HAP is determined. Let for notational simplicity. Then, problem (18) can be further transformed into a resource allocation optimization problem (OP) as follows

(19a)
(19b)
(19c)
(19d)

where . It may be readily shown that problem (19) is a convex OP. By analyzing the KKT conditions of problem (19), the general UL offloading activation condition admitting a threshold-based structure is obtained in the following proposition.

Proposition 1

For the single-user setup, UL offloading for the typical device will be activated if and only if the following condition is satisfied,

(20)

where is the unique solution of

(21)
proof 4

Please see Appendix B.

Proposition 1 explicitly shows that a typical device would prefer UL offloading for maximizing its computation rate, when the transmit power of the HAP is higher than a threshold , which depends on the channel power gain . Based on Proposition 1, we further discuss the monotonic relationship between and .

Proposition 2

The threshold decreases with the equivalent channel power gain .

proof 5

Since has the same monotonic relationship with as that of , we focus our attention on showing that decreases with instead. Taking the first order derivative of with respect to , we obtain

(22)

Note that is a function of , which is determined by (21). As such, we use instead of in the following. Based on the method of implicit differentiation, we obtain

(23)

Substituting (23) into (22) yields

(24)

Thus, decreases with .

Proposition 1 and Proposition 2 serve as a solid theoretical foundation for further investigating the impact of IRSs on the UL offloading activation condition. For ease of exposition, we assume that the IRS can establish pure line-of-sight (LoS) links with both the device and the AP. By setting the IRS BF vector as the optimal form, the equivalent channel power gain can be formulated as

(25)

where (), (), and () denote the length (path-loss exponent) of the HAP-device, HAP-IRS, and IRS-device links, respectively, and represents the channel power gain at a reference distance of 1 meter (m).

Remark 4

For a specific dominant LoS scenario, the UL offloading activation condition can be expressed as

(26)

It is plausible that the value of using IRSs becomes times higher than that without IRSs. By increasing the number of IRS elements , the channel power gain can be significantly increased, which substantially reduces the threshold for UL offloading. Thus, a typical device is more willing to perform task offloading upon increasing of due to the improved channel conditions. This confirms the practicality of deploying IRSs in next generation communication networks.

V Proposed Solutions for General Multi-user systems

In this section, we focus our attention on solving the computation rate maximization problems of TDMA-based UL offloading, i.e., , , and . Solving the same problems for NOMA is similar to those of TDMA according to the results of Section III.

V-a AO Algorithm Proposed for Solving and

Since problem is more complex than , we commence with Case 2, i.e., . It will be shown later in this section that an algorithm designed for solving may also be directly applicable to . For problem , the optimization variable is closely coupled with the variables and , while is coupled with . Moreover, the unit-modulus constraints in (8e) and (9b) render problem