Performance Analysis of Uplink Opportunistic Scheduling for Multi-UAV-Assisted Internet of Things

Highlights

What are the main findings?

• This paper proposes a low-overhead uplink opportunistic scheduling framework leveraging channel reciprocity. To address the prohibitive uplink training overhead resulting from conventional downlink scheduling methods, we innovatively utilize minimum downlink interference (MDI) and the maximum downlink signal-to-interference-plus-noise ratio (MD-SINR) as criteria for scheduling uplink users, effectively reducing system overhead while fully exploiting multiuser diversity gains.
• Rigorous closed-form sum rate performance are provided for both dual-unmanned aerial vehicle (UAV) and three-UAV deployment scenarios under the MDI scheduling criterion. We further derive closed-form expressions for the asymptotic sum rate and prove that the dual-UAV system can achieve a total of $2\alpha$ degrees of freedom (DoF) when the number of users scales as $K={\rho }^{\alpha }$, with $\rho$ denoting the transmitted signal-to-noise ratio (SNR).

What are the implications of the main findings?

• The proposed scheme offers an efficient scheduling solution for low-power, large-scale Internet of Things (IoT) data collection. By significantly reducing uplink channel training overhead, it enables UAVs to efficiently serve massive low-power ground sensors, making it particularly suitable for infrastructure-less scenarios such as environmental monitoring and disaster relief in remote areas.
• The findings reveal an adaptive relationship between the number of deployed UAVs and the transmission strategy. Simulation results demonstrate that a dual-UAV deployment achieves superior performance over single- or three-UAV deployments at medium transmitted SNR levels, providing key theoretical guidance for dynamically selecting the optimal deployment strategy based on transmit power in practical systems.

Abstract

Due to the high mobility, flexibility, and low cost, unmanned aerial vehicles (UAVs) can provide an efficient way for provisioning data communication and computing offloading services for massive Internet of Things (IoT) devices, especially in remote areas with limited infrastructure. However, current transmission schemes for unmanned aerial vehicle-assisted Internet of Things (UAV-IoT) predominantly employ polling scheduling, thus not fully exploiting the potential multiuser diversity gains offered by a vast number of IoT nodes. Furthermore, conventional opportunistic scheduling (OS) or opportunistic beamforming techniques are predominantly designed for downlink transmission scenarios. When applied directly to uplink IoT data transmission, these methods can incur excessive uplink training overhead. To address these issues, this paper first proposes a low-overhead multi-UAV uplink OS framework based on channel reciprocity. To avoid explicit massive uplink channel estimation, two scheduling criteria are designed: minimum downlink interference (MDI) and the maximum downlink signal-to-interference-plus-noise ratio (MD-SINR). Second, for a dual-UAV deployment scenario over Rayleigh block fading channels, we derive closed-form expressions for both the average sum rate and the asymptotic sum rate based on the MDI criterion. A degrees-of-freedom (DoF) analysis demonstrates that when the number of sensors, K, scales as ${\rho }^{\alpha }$, the system can achieve a total of $2\alpha$ DoF, where $\alpha \in \left[0,1\right]$ is the user-scaling factor and $\rho$ is the transmitted signal-to-noise ratio (SNR). Third, for a three-UAV deployment scenario, the Gamma distribution is employed to approximate the uplink interference, thereby yielding a tractable expression for the average sum rate. Simulations confirm the accuracy of the performance analysis for both dual- and three-UAV deployments. The normalized error between theoretical and simulation results falls below 1% for K > 30. Furthermore, the impact of fading severity on the system’s sum rate and DoF performance is systematically evaluated via simulations under Nakagami-m fading channels. The results indicate that more severe fading (a smaller m) yields greater multiuser diversity gain. Both the theoretical and simulation results consistently show that within the medium-to-high SNR regime, the dual-UAV deployment outperforms both the single-UAV and three-UAV schemes in both Rayleigh and Nakagami-m channels. This study provides a theoretical foundation for the adaptive deployment and scheduling design of UAV-assisted IoT uplink systems under various fading environments.

1. Introduction

Since the advent of the 4G and 5G eras, the application domains and market scale of the Internet of Things (IoT) have witnessed rapid and exponential growth. In the increasingly diverse IoT application scenarios, a massive number of IoT terminal devices such as wearable electronics, various sensors, and smart home appliances rely on a wide range of wireless and wired networks to support data transmission and network services [1,2].

As a promising IoT technology, cellular IoT can leverage existing cellular network infrastructures to provide ubiquitous coverage to numerous IoT connections, without investing and deploying dedicated IoT networks [3]. Some popular cellular IoT technologies developed in the 3rd-Generation Partnership Project (3GPP), such as Narrow-Band Internet of Things (NB-IoT), have been extensively deployed and have contributed about 80 percent of the increased IoT connections [4]. It is also forecasted that the cellular IoT connections will reach 5.5 billion by 2027 [5]. With the support of emerging technologies such as edge intelligence, Reconfigurable Intelligent Surfaces (RISs), space–air–ground–underwater communications, Terahertz communications, new modulation/demodulation technologies, and massive ultra-reliable and low-latency communications, the sixth-generation (6G) wireless communication networks will continue to promote the proliferation of IoT applications in new domains, including healthcare IoT, vehicular IoT and autonomous driving, unmanned aerial vehicles, satellite IoT, and industrial IoT [6,7,8,9].

A fundamental constraint of cellular IoT, however, is its reliance on existing infrastructure, which limits its applicability to support IoT devices in remote or inaccessible regions. Consequently, the Internet of Remote Things (IoRT) has emerged to deploy IoT solutions in remote and geographically isolated areas without terrestrial nor cellular networks. The typical prominent applications include environmental monitoring, wildlife tracking, disaster relief management, and internet of Underground/Underwater things [5]. Additionally, traditional cellular networks exhibit several limitations such as coverage gaps and low communication energy efficiency [10]. To overcome these challenges, the emerging unmanned aerial vehicle-assisted Internet of Things (UAV-IoT) has garnered substantial attention from both academia and industry in recent years. In this framework, highly mobile and low-cost UAVs can extend radio coverage to inaccessible and infrastructure-less areas or fill coverage gaps in terrestrial networks. Flexible three-dimensional trajectory planning can shorten the communication distance between UAVs and terminal devices and avoid obstructive paths that induce high signal attenuation. The resultant improvement in communication energy efficiency significantly extends the battery life of ground IoT devices [10,11].

In typical UAV-IoT scenarios, flight trajectories are pre-planned based on the positions of ground IoT devices. After takeoff, the UAVs disseminate data to or collect raw data from ground IoT devices when flying by. In data collection applications, the gathered data are sent forward to a cloud data center for further analysis. Due to the constraints of the low cost, the long deployment duration, and the need for unmanned operation maintenance, ground devices in UAV-IoT systems are typically limited in battery capacity, computing capability, transmit power, and data rate. In more stringent zero-power communication scenarios, battery-less IoT devices can harvest energy from ambient radio waves. These devices are characterized by ultra-low power consumption, minimal size, and extremely low cost [12].

In IoT systems, UAVs are capable of performing multiple functions. They can serve as aerial relays to expand communication range and ensure seamless multi-hop connectivity, act as Mobile Edge Computing (MEC) servers for processing offloaded tasks locally, and function as energy transmitters in Wireless Power Transfer (WPT) networks [13]. Therefore, compared with traditional wireless sensor networks and satellite networks, UAV-IoT offers advantages in performing information dissemination, data collection, and task execution with low cost and high energy efficiency [14,15,16]. In addition to communication services, UAV-mounted Edge Computing (UMEC) can further provide computation offloading for IoT devices with limited processing capabilities, where UAVs are equipped with computing servers and can process tasks onboard [17]. UMEC can significantly reduce end-to-end communication delays for many emerging applications, as well as perform preliminary processing of raw data to decrease the volume of data transmitted back to terrestrial data centers [18,19].

In UAV-IoT scenarios, a large number of ground terminal devices establish wireless links with multiple UAVs, forming a typical Interfering Broadcast Channel (IBC) or Interfering Multiple-Access Channel (IMAC). In such settings, different users experience independent time-varying channel fadings, and methods such as opportunistic scheduling (OS) or opportunistic beamforming (OBF) offer significant advantages [20,21].

The OBF method, which is also called random beamforming (RBF), was initially proposed to enhance multiuser diversity in slow-fading downlink multiuser Multiple-Input Single-Output (MISO) broadcast channels (BCs). By employing random beamforming vectors, it artificially induces larger and faster fluctuations in the effective channel gain. Then, the user whose instantaneous effective channel gain or SNR is near the peak is scheduled to establish communication connections [22]. In an independent Rayleigh fast-fading environment, when the base station transmits only a single random beamforming vector, the distribution of the users’ effective channel gain becomes identical to that of the single-antenna case. In this situation, OBF degenerates into OS [23]. In addition to multiuser diversity gain, OBF offers another advantage in multiuser Multiple-Input Multiple-Output (MIMO) systems in that it requires only partial Channel State Information (CSI), thereby achieving high spectral efficiency with lower implementation complexity and reduced feedback overhead [24,25,26]. In downlink MIMO BC or IBC scenarios, conventional beamforming requires the estimation and feedback of global Channel State Information (CSI). The transmitter then computes the transmit precoding and decoding matrices based on this global CSI, leading to high computational complexity. In contrast, OBF randomly selects the precoding matrix, and each user only feeds back channel side information instead of the full channel matrix, thereby reducing both the feedback overhead and the computational complexity for precoding and decoding matrices [27,28]. Although OS or OBF transmission strategies are well-suited for UAV-IoT scenarios, most current research still predominantly relies on simple polling-based scheduling and resource allocation schemes like Time Division Multiple Address (TDMA) or Frequency Division Multiple Access (FDMA). This approach fails to fully exploit the multiuser diversity gain offered by the massive number of IoT devices [29,30,31].

Recently, several studies have established performance analysis methods for OBF-based scheduling in downlink BC and IBC scenarios. In a multi-cell MISO BC setting, Sung-Hyun Moon et al. derived a closed-form expression for the asymptotic sum rate under homogeneous user deployment. They concluded that the system’s average achievable sum rate scales as $M_s{log}_2{log}_{2}K$, where $M_s$ represents the number of concurrently scheduled users and K denotes the total number of users [24]. Suo et al. further extended the analysis framework to multi-cell MIMO broadcast channels, developed a more comprehensive analytical model, and derived a closed-form expression for the average sum rate, effectively accounting for heterogeneous user deployment and heterogeneous scheduling probabilities [25]. Aljumaily et al. investigated the impact of user mobility on the performance of OBF in millimeter-wave communication systems, concluding that user movement leads to beam misalignment, which in turn causes a significant throughput degradation of up to 27% [26]. Bao et al. conducted a study on the physical layer security of OBF. They compared the impact of eavesdropper collusion strategies and the establishment of a secrecy guard zone on the system’s secrecy performance and derived closed-form expressions for the secrecy outage probability under various scenarios [27]. Su et al. addressed the problem that conventional opportunistic beamforming suffers a severe performance degradation when the number of users is small. They proposed a scheme utilizing selection combining, equal-gain combining, or maximal-ratio combining in a multi-receive-antenna configuration. Closed-form expressions for the system achievable sum rate are derived over Nakagami-m fading channels. The results show that the proposed method can effectively exploit the combining gain from multiple antennas to compensate for the insufficient multiuser diversity gain [28].

In addition, researchers have integrated OS or RBF techniques with RISs, where the random phases introduced by the IRS are used to induce channel fluctuations. This approach can significantly enhance spectral efficiency while avoiding the increased CSI training requirements and overhead associated with large IRS arrays [32,33,34,35,36]. Psomas et al. analyzed the outage probability and energy-efficiency performance of an RBF transmission scheme in RIS-assisted single-user single-input single-output (SISO) systems. The effectiveness of the theoretical analysis was validated through numerical simulations, demonstrating that, in this setting, the RBF approach offers significant advantages in terms of low complexity, reduced CSI requirements, and high energy efficiency [32]. In [33], Nadeem et al. derived the closed-form expressions for the average achievable sum rate in a RIS-assisted SISO BC scenario that considers only the RIS-reflected link. Their analysis covers independent Rayleigh fading, independent Rician fading, and correlated Rayleigh fading channels. Subsequently, for RIS-assisted SISO/MISO BCs incorporating both the direct link and the RIS-reflected links, Nadeem et al. further derived the expressions for the average and asymptotic sum rate under slow fading and correlated Rayleigh fading conditions. Leveraging these expressions, they then optimized system configuration parameters such as the number of base station antennas, the number of RIS elements, and the transmit power with the objective of maximizing system energy efficiency [34,35]. Dimitropoulou et al. proposed an RBF scheduling method based on multiple beam-groups for IRS-assisted MISO BCs. During the channel training stage, multiple groups of random beams are transmitted, and the optimal group is then selected based on the signal-to-interference-plus-noise-ratio (SINR) feedback from users. The authors revealed a tradeoff between system performance and channel training overhead and derive an integral-form expression for the average sum rate along its upper bound by using extreme value theory tools. However, this upper bound shows a noticeable deviation from the simulation results [36].

Most existing performance analyses of OS and RBF systems focus on downlink BC and IBC scenarios. However, in UAV-IoT applications, uplink data collection is much more prevalent. Jung et al. analyzed the system performance of uplink IMACs employing opportunistic interference nulling (OIN) and opportunistic interference alignment (OIA) in terms of degrees of freedom (DoF), where DoF is defined as the ratio of the achievable sum rate to the logarithm of the transmitted signal-to-noise ratio (SNR), as the SNR approaches infinity [37].

Moreover, in uplink UAV-IoT scenarios, the number of ground devices is typically much larger than the number of UAVs. If traditional OS or OBF scheduling methods are applied directly, each ground device must perform uplink channel training and estimation, resulting in prohibitively high training overhead. Although employing traditional designed beamforming can reduce the channel training cost, it fails to fully exploit multiuser diversity gains. Therefore, it is necessary to design OS schemes specifically tailored for uplink UAV-IoT scenarios.

The workflow of the proposed uplink scheduling schemes and performance analysis results is shown in Figure 1, and the main contributions of this paper are summarized as follows:

• We first propose a low-complexity and low-overhead OS strategy for uplink UAV-IoT scenarios. By leveraging channel reciprocity, we utilize downlink interference and downlink SINR as scheduling metrics for uplink users and design user scheduling procedures based on minimum downlink interference (MDI) and maximum downlink SINR (MD-SINR).
• For the dual-UAV deployment case over Rayleigh block fading channels with the MDI scheduling criterion, we derive closed-form expressions for the sum rate and the asymptotic sum rate as the transmitted SNR is finite and approaches infinity, respectively, and analyze the DoF performance. The DoF analysis results show that when the number of sensors K scales as ${\rho }^{\alpha }$, the system can sustain a total DoF of $2\alpha$ where $\rho$ is the transmitted SNR.
• For the three-UAV deployment case over Rayleigh block fading channels, by approximating the uplink user interference with a Gamma distribution, we obtain the average sum rate expression under the MDI scheduling criterion.
• We perform Monte Carlo simulations to validate the correctness of the theoretical analysis showing a normalized error of less than 1%. The simulation results under Nakagami-m fading channels reveal that the system sum rate and DoF decrease with the parameter m. Moreover, in the medium-to-high transmitted SNR regime, deploying two UAVs yields a higher system sum rate compared with deploying more UAVs or only one UAV.

Figure 1. Flowchart of proposed methods.

Figure 1. Flowchart of proposed methods.

The remainder of this paper is organized as follows. Section 2 introduces the system model. Section 3 presents the principles and procedures of the uplink OS scheme based on channel reciprocity. Section 4 and Section 5 are dedicated to the performance analysis for the dual-UAV and three-UAV scenarios, respectively. The simulation results and analyses are shown in Section 6, and Section 7 concludes the paper.

2. System Model

As illustrated in Figure 2, we consider a multi-UAV uplink data collection scenario for IoT applications. A massive number of IoT sensors belonging to various IoT applications, such as contamination detectors, environmental parameter collectors, fire and smoke sensors, and wildlife trackers, are scattered across remote areas lacking cellular infrastructure. Those sensors possess limited battery capacity and computing capability, and their collected data cannot be processed by themselves. Multiple UAVs are deployed to collect data from these ground sensors efficiently. The UAVs take off from a base, hover above the target area for a period, and subsequently relay the data to cellular base station. The offloaded data can be further processed at an edge data center or a remote cloud data center. If the UAVs are equipped with onboard servers, they can also provide computation services for ground sensors. We assume the number of sensors, denoted by K, is significantly larger than the number of UAVs, denoted by N, i.e., $K\gg N$. Both UAVs and sensors are assumed to be equipped with a single antenna. The symbols used are listed in

Table 1. List of symbols.

Symbol	Description
N	Number of deployed UAVs
K	Number of ground sensors
$U_i$	The i-th UAV, $i=1,2,\dots ,N$
$s_k$	The k-th ground sensor, $k=1,2,\dots ,K$
$P_d$	Downlink transmit power of each UAV
$P_u$	Uplink transmit power of each sensor
$P_n$, ${\sigma }^2$	Noise power
$\rho$	Uplink transmitted SNR, $\rho =P_u/{\sigma }^2$
$h_{k,i}$	Downlink channel coefficient from UAV $U_i$ to sensor $s_k$
$g_{i,k}$	Uplink channel coefficient from sensor $s_k$ to UAV $U_i$
$I_{k,i}^d$	Downlink interference power at sensor $s_k$ when $U_i$ is the desired UAV
${\mathrm{SINR}}_{k,i}^d$	Downlink SINR at sensor $s_k$ corresponding to UAV $U_i$
${\mathrm{SINR}}_{i,k}^u$	Uplink SINR at UAV $U_i$ corresponding to sensor $s_k$
$i^{\ast }$	Index of the UAV providing the minimum interference or maximum SINR
$Q^{\ast }$, $T^{\ast }$	Indices of the scheduled sensors under MDI or MD-SINR criteria
${\Delta }_{I^{\ast }}$	The normalized MDI for sensor $S_{I^{\ast }}$ in the three-UAV scenario
A	The normalized uplink received interference power for $U_2$ in the three-UAV scenario
$E_1\left(x\right)$	Exponential integral function, $E_1\left(x\right)={\int }_1^{\infty }{\displaystyle \frac{e^{-tx}}{t}}dt$
$\Gamma \left(n\right)$	Gamma function, $\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}\mathrm{d}t,x>0$
$\Gamma (n,x)$	Upper incomplete Gamma function, $\Gamma (n,x)={\int }_x^{\infty }{t}^{n-1}{e}^{-t}dt$

.

The UAVs are denoted as $U_1,U_2,\dots ,U_N$ and the sensors are denoted as $s_1,s_2,\dots ,{\mathrm{s}}_K$. It is assumed that all UAVs transmit with the same downlink power, denoted by $P_d$, and all sensors transmit with the same uplink power, denoted by $P_u$. The downlink and uplink complex channels between UAV $U_i$ and sensor $s_k$ are denoted by $h_{k,i}$ and $g_{i,k}$, respectively.

Consider the homogeneous UAV and sensor deployment scenario with block fading, where all large-scale fading coefficients are equal to one, and the small-scale fading coefficients $h_{ki}$ for $1\le k\le K$ and $1\le i\le N$ are independently and identically distributed (i.i.d.). The channel coefficient $h_{ki}$ remains constant within each time block or frame but varies independently across different blocks. In this paper two typical channel models are considered, i.e., the Rayleigh channel and Nakagami-m channel. The Rayleigh channel model is suitable for scattering environments. For example, the field measurements in [38] verify that the air–to–ground (AG) propagation fading channel follows a Rayleigh distribution in mixed-urban scenarios with large elevation angles. In a Rayleigh channel, the complex channel coefficient $h_{ki}$ is distributed according to $\mathcal{C}\mathcal{N}\left(0,1\right)$, and the channel fading $\left|h_{ki}\right|$ follows a Rayleigh distribution. The Nakagami-m channel model is appropriate for characterizing the UAV fading channels in high-altitude applications [39]. The shape factor m describes the severity of channel fading. When $m=1$, the Nakagami-m channel model reduces to the Rayleigh fading channel model, and when $m>1$, it can be approximated by the Rician channel model. Although Rayleigh or Nakagami-m fading is assumed, the uplink and downlink channels are considered reciprocal under time-division duplexing (TDD) operation, i.e., $g_{ik}=h_{ki}^H$ holds within one channel coherence interval. This assumption does not contradict the Rayleigh or Nakagami-m fading model, as reciprocity applies to instantaneous channel realizations rather than their statistical distributions. In the following sections, the Rayleigh channel model is adopted in the closed-form sum rate analysis, while both channel models are used in the simulation part to evaluate the performance of proposed uplink scheduling schemes.

3. Uplink Opportunistic Scheduling Based on Channel Reciprocity

3.1. Principle of Uplink Opportunistic Scheduling

In downlink BC and IBC scenarios, the number of base stations is far smaller than the number of users. Consequently, when OS or OBF is employed, the overhead associated with downlink channel training is relatively small. Moreover, each user only needs to feed back channel-quality indicators such as the SNR, SINR, or interference chord distance, rather than the full channel matrix, which keeps the feedback overhead low. Furthermore, the feedback cost can be reduced even more through limited feedback or threshold-based feedback mechanisms [40,41]. However, in uplink MAC or IMAC scenarios, directly applying OS or OBF would require all users to sequentially transmit channel training symbols, which leads to excessively large uplink training overhead. In addition, because ground sensors typically transmit with much lower power than UAVs, the error in uplink channel estimation is generally larger than that in downlink channel estimation. Therefore, when designing low-overhead uplink scheduling strategies, one can exploit channel reciprocity and let UAVs or the base station perform downlink channel training in a time-division manner to avoid substantial uplink training costs. The uplink link channel quality used for scheduling can then be inferred indirectly from the downlink CSI channel information measured by the ground sensors.

Guided by this principle, this section designs two uplink opportunistic scheduling methods based on channel reciprocity: the MDI and the MD-SINR scheduling schemes. The MDI scheduling approach exploits a key property: if a ground sensor experiences weak interference from non-target UAVs during downlink transmission, it will also generate weak interference toward those non-target UAVs in the uplink. This property stems from channel reciprocity. In the MDI scheduling approach, each ground sensor first measures the received downlink signal powers from all UAVs based on downlink channel estimation. Then, treating the signal from a particular UAV as the desired signal, the sensor computes the total interference power contributed by the remaining UAVs. Let $I_{k,i}^d$ denote the total downlink interference power received by sensor $s_k$, when the signal of UAV $U_i$ is considered the desired signal, expressed as

$$I_{k,i}^d={\sum _{1\le j\le N,j\ne i}{P}_d\left|h_{k,j}\right|}^2,\forall k.$$

(1)

To reduce feedback overhead, each sensor $s_k$ compares the downlink interference power from all UAVs and identifies the UAV whose signal experiences the minimum level of downlink interference at $s_k$, denoted by $U_{i\ast }$. $U_{i\ast }$ is determined by $s_k$ according to

$$i\ast =\underset{1\le i\le N}{argmin}{I}_{k,i}^d.$$

(2)

Sensor $s_k$ then feeds back the corresponding minimum interference power value, denoted by

$$I_{k,i\ast }^d=\underset{1\le i\le N}{min}{I}_{k,i}^d,$$

(3)

to UAV $U_{i\ast }$ via the uplink feedback channel. Based on the received feedback, each UAV determines its target sensor according to the MDI criterion: UAV $U_{i\ast }$ selects the sensor that reported the smallest downlink interference value, denoted by $S_{Q^{\ast }}$, as its target sensor for uplink transmission, i.e.,

$$Q^{\ast }=arg\underset{1\le k\le K}{min}{I}_{k,i\ast }^d.$$

(4)

During the uplink data transmission phase, UAV $U_{i\ast }$ treats the signal from sensor $S_{Q^{\ast }}$ as the desired signal while treating signals from other sensors as interference. Due to channel reciprocity, since sensor $S_{Q^{\ast }}$ experiences weak interference from its non-target UAVs besides $U_{i\ast }$ in the downlink, it will also generate relatively weak interference to those UAVs during the uplink transmission phase.

When all UAVs adopt the MDI scheduling method, the interference received by the scheduled sensor from its interfering UAVs in the downlink decreases as the number of sensors K increases. Due to channel reciprocity, the interference that this sensor generates to its non-target UAVs in the uplink similarly decreases. As K becomes sufficiently large, the system performance approaches that of an interference-free parallel transmission system. Furthermore, to reduce feedback overhead, a threshold can be introduced, allowing a sensor to report only if its minimum downlink interference power falls below a certain value. This avoids scheduling sensors that may cause significant interference. When the number of ground sensors K is sufficiently large, this threshold-based feedback method approaches the ideal performance of the ideal non-threshold scheme. By setting an appropriate threshold, the system can achieve a good tradeoff between performance, complexity, and feedback overhead [40].

The MD-SINR scheduling method leverages the principle that when a sensor achieves the maximum downlink SINR, it is also likely to achieve good SINR performance during uplink transmission. To validate this result, Monte Carlo simulations were conducted for a dual-UAV deployment scenario with Rayleigh fading channels. In each snapshot, the channel coefficients were randomly generated. During the downlink, the SINR values of each sensor were calculated, and the uplink sum rate was computed for all possible $C_K^2=K(K-1)/2$ schedulable sensor combinations. Each combination consisted of two distinct sensors, and these combinations were ranked in descending order of their sum rates. Finally, the probability that the sensor with the MD-SINR value would be included in one of the top-ten uplink scheduling sensor groups was analyzed. As shown in Figure 3, when $K=10$, the sensor with the MD-SINR still belongs to the optimal uplink scheduling group with a probability greater than 95%, and this probability increases to over 98% when $K=30$.

In the MD-SINR scheduling method, each ground sensor measures the downlink signal power received from all UAVs. It then computes a downlink SINR value for each UAV by treating that UAV’s signal as the desired signal and the signals from the remaining UAVs as interference. When sensor $s_k$ considers the signal from UAV $U_i$ as the useful signal, the corresponding downlink SINR is given by

$$SIN{R}_{k,i}^d={\displaystyle \frac{P_d|{\mathrm{h}}_{k,i}{|}^2}{{\displaystyle \sum _{1\le j\le N,j\ne i}}{P}_d|h_{k,j}{|}^2+P_n}},\forall k,$$

(5)

where $P_n$ is the noise power.

Then, sensor $s_k$ selects its target UAV $U_{i\ast }$ that provides its MD-SINR value according to

$$SIN{R}_{k,i\ast }^d=\underset{1\le i\le N}{max}SIN{R}_{k,i}^d,\forall k,$$

(6)

and feeds the corresponding MD-SINR value $SIN{R}_{k,i\ast }^d$ back to $U_{i\ast }$. Subsequently, UAV $U_{i\ast }$ selects the largest MD-SINR from multiple MD-SINR values it receives and chooses the corresponding sensor, denoted by $S_{T^{\ast }}$, as its target sensor for uplink transmission, i.e.,

$$T^{\ast }=arg\underset{1\le k\le K}{max}SIN{R}_{k,i\ast }^d.$$

(7)

3.2. Procedures of MDI and MD-SINR Scheduling

The complete four-step implementation procedure for the uplink MDI and MD-SINR-based OS schemes is illustrated in Figure 4. The detailed steps are described below.

(1) Downlink Channel Training All UAVs sequentially transmit the training signal $\sqrt{P_d}{\widehat{x}}_i$, where ${\widehat{x}}_i$ is the unit-power training symbol, satisfying $E\left({\left|{\widehat{x}}_i\right|}^2\right)=1$, and $P_d$ is the downlink transmit power. Thus, the received training signal at sensor $s_k$ from UAV $U_i$ can be expressed as

$$y_{k,i}=\sqrt{P_d}{h}_{k,i}{\widehat{x}}_i+n_k,$$

(8)

where $n_k\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }^2\right)$ denotes the additive white Gaussian noise (AWGN) at sensor $s_k$. Based on the received signal, sensor $s_k$ estimates the channel coefficient $h_{ki}$. It is assumed that, in this step, the UAVs employ a sufficiently high transmit power $P_d$ for channel training to ensure that sensors can obtain accurate channel estimates.

(2) Channel Quality Feedback Based on the channel information measured from different UAVs in the previous step, sensor $s_k$ computes the downlink interference or the downlink SINR. Under the MDI scheduling criterion, sensor $s_k$ calculates the total downlink interference $I_{k,i}^d,1\le i\le N$ for each UAV according to Equation (1) and feeds back the minimum value $I_{k,i\ast }^d={\displaystyle \underset{1\le i\le N}{min}}{I}_{k,i}^d$ to the corresponding UAV $U_{i\ast }$. Under the MD-SINR scheduling criterion, sensor $s_k$ computes the downlink SINR value $SIN{R}_{k,i}^d,1\le i\le N$ corresponding to each UAV according to Equation (5) and feeds back the maximum value $SIN{R}_{k,i\ast }^d={\displaystyle \underset{1\le i\le N}{max}}SIN{R}_{k,i}^d$ to the corresponding UAV $U_{i\ast }$.

(3) Sensor Scheduling Based on the received channel quality, each UAV selects its corresponding target sensor and broadcasts the scheduling result via the downlink channel. Under the MDI scheduling criterion, UAV $U_{i\ast }$ selects the sensor $S_{Q^{\ast }}$ that reported the MDI value according to (4). Under the MD-SINR scheduling criterion, UAV $U_{i\ast }$ selects the sensor $S_{T^{\ast }}$ that reported the MD-SINR value according to (7). (4) Uplink Data Transmission The scheduled sensors transmit uplink data to their respective target UAVs simultaneously. They are allowed to share the same resource block while still maintaining favorable sum rate performance.

Figure 4. Scheduling procedures of MDI- and MD-SINR-OS.

Figure 4. Scheduling procedures of MDI- and MD-SINR-OS.

3.3. Complexity Analysis

The computational complexity is measured in terms of the number of floating-point operations (flops). Specifically, an addition, subtraction, multiplication, or division of real numbers is counted as one flop, while a complex addition and a complex multiplication are counted as two flops and six flops, respectively.

In the MDI scheme, each sensor first needs to calculate the interference experienced by the signal from each UAV, that is, sensor $S_{\mathrm{k}}$ computes $I_{k,i}^d$ for $1\le i\le N$ according to Equation (1) and then finds its MDI term according to Equation (3). For sensor $S_{\mathrm{k}}$, the received power $P_d{\left|h_{k,i}\right|}^2$ from UAV $U_i$ has already been measured during the channel training phase. Therefore, obtaining $P_d{\left|h_{k,i}\right|}^2$ does not incur additional computational complexity. To compute $I_{k,i}^d$ for $1\le i\le N$, the total interference term $I_k^d={\displaystyle \sum _j}{P}_d{\left|h_{k,j}\right|}^2$ can be first calculated, which requires $N-1$ real addition operations, and then $I_{k,i}^d=I_k^d-P_d{\left|h_{k,i}\right|}^2$ can be determined for $1\le i\le N$, which requires N real subtraction operations. Determining the MDI term $I_{k,i^{\ast }}^d$ requires $N-1$ comparisons in $S_{\mathrm{k}}$. To reduce the uplink feedback overhead, by using the threshold-based feedback method, each sensor further needs to compare the obtained MDI value with the feedback threshold. Assume that each UAV can receive MDI values fed back from $n^u$ sensors on average, where the value of $n^u$ is affected by the feedback threshold. Then, each UAV needs to perform $n^u-1$ comparisons. Therefore, in the MDI scheme, each sensor involves $3N-1$ arithmetic and comparison operations, and each UAV involves $n^u-1$ comparison operations. The overall computational complexity of the MDI-OS is $\mathcal{O}\left(KN+n^{u}N\right)$, where $\mathcal{O}\left(·\right)$ is the big-O notation.

In the MD-SINR scheme, sensor $S_{\mathrm{k}}$ first computes $SIN{R}_{k,i}^d$ for $1\le i\le N$ according to Equation (5) and then finds its MD-SINR term according to Equation (6). Note that the interference term in Equation (5) is exactly $I_{k,i}^d$ for $1\le i\le N$, which requires $N-1$ real addition operations and N real subtraction operations. In $S_{\mathrm{k}}$, the computation of $SIN{R}_{k,i}^d={\displaystyle \frac{P_d|{\mathrm{h}}_{k,i}{|}^2}{I_{k,i}^d+{\sigma }^2}}$ for $1\le i\le N$ requires N real addition operations and N real division operations, and determining the MD-SINR value $SIN{R}_{k,i^{\ast }}^d$ requires $N-1$ comparisons. When the same threshold-based method is adopted, in the MD-SINR scheme, each sensor involves $5N-1$ arithmetic and comparison operations, and each UAV involves $n^u-1$ comparison operations. Therefore, the overall computational complexity of the MD-SINR-OS is also $\mathcal{O}\left(KN+n^{u}N\right)$.

4. Performance Analysis of MDI-Based OS for the Dual-UAV Scenario

After the scheduling method is specified, a natural question arises: given the number of ground sensors K and the uplink and downlink transmit powers, how many UAVs should be deployed or, equivalently, how many sensors should be scheduled concurrently and participate in simultaneous uplink transmission to achieve better system performance? To answer this question, it is necessary to analyze the system’s average achievable rate performance under different scenarios. This section begins with the dual-UAV scenario, in which two ground sensors are scheduled to perform concurrent uplink communication in each transmission round.

4.1. Average Achievable Sum Rate Analysis

In the uplink data transmission stage, when the target sensor corresponding to UAV $U_i$ is $s_k$, the received signal at $U_i$ can be written as

$$y_i^u=g_{i,k}\sqrt{P_u}{x}_k+\sum _{j\ne k}{g}_{i,j}\sqrt{P_u}{x}_j+n_i,$$

(9)

where $P_u$ is the uplink transmit power, and $n_i\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }^2\right)$ is the AWGN at $U_i$. By exploiting channel reciprocity $g_{i,k}=h_{k,i}^H$, the corresponding uplink SINR can be expressed as

$$SIN{R}_{i,k}^u={\displaystyle \frac{P_u|g_{i,k}{|}^2}{{{\displaystyle \sum _{j\ne k}}{P}_u\left|g_{i,j}\right|}^2+{\sigma }^2}}={\displaystyle \frac{|h_{k,i}{|}^2}{{{\displaystyle \sum _{j\ne k}}\left|h_{j,i}\right|}^2+\frac{{\sigma }^2}{P_u}}}.$$

(10)

In the SINR expression (10), the relationship between the uplink SINR and the downlink channel is established. When the number of UAVs is $N=2$, closed-form expressions for the average sum rate of the dual-UAV system can be derived.

Theorem 1.

In a dual-UAV and K-sensor scenario with Rayleigh block fading channels, when MDI-based OS is performed, the average achievable sum rate is given by

$$R_{\mathit{sum}}={\displaystyle \frac{2K}{ln2\left(K-1\right)}}\left[e^{{\displaystyle \frac{1}{\rho }}}{E}_1\left({\displaystyle \frac{1}{\rho }}\right)-e^{{\displaystyle \frac{K}{\rho }}}{E}_1\left({\displaystyle \frac{K}{\rho }}\right)\right],$$

(11)

where $E_1\left(x\right)={\int }_1^{\infty }{\displaystyle \frac{e^{-tx}}{t}}dt$ is the exponential integral function, and $\rho ={\displaystyle \frac{P_u}{{\sigma }^2}}$ denotes the transmitted SNR.

Proof. 

Let the two UAVs be denoted by $U_1$ and $U_2$. Based on the MDI-based OS method, suppose that in a given scheduling round, $U_1$ and $U_2$ select sensors $S_{I^{\ast }}$ and $S_{J^{\ast }}$ as their respective target sensors. The uplink SINR corresponding to $U_1$ can then be simplified as

$$SIN{R}_{1,I^{\ast }}^u={\displaystyle \frac{P_u|g_{1,I^{\ast }}{|}^2}{P_u|g_{1,J^{\ast }}{|}^2+{\sigma }^2}}={\displaystyle \frac{{\left|h_{I^{\ast },1}\right|}^2}{{\left|h_{J^{\ast },1}\right|}^2+\frac{{\sigma }^2}{P_u}}}={\displaystyle \frac{Z}{Y+1/\rho }},$$

(12)

where Z and Y separately represent the normalized desired signal power and normalized interference power.

According to the MDI criterion, sensor $S_{J^{\ast }}$ experiences the MDI from $U_1$. Consequently, we have

$$J^{\ast }=arg\underset{1\le k\le K}{min}{\left|h_{k,1}\right|}^2$$

(13)

and

$$Y={\left|h_{J^{\ast },1}\right|}^2=\underset{1\le k\le K}{min}{\left|h_{k,1}\right|}^2.$$

(14)

In Rayleigh fading channels, $Z={\left|h_{I^{\ast },1}\right|}^2$ follows a chi-square distribution with two DoFs, which is equivalent to an exponential distribution. Its probability density function (PDF) and cumulative density function (CDF) are $f_Z\left(x\right)=e^{-x},x\ge 0$ and $F_Z\left(x\right)=1-e^{-x}, x\ge 0$, respectively.

Let $X_{k1\le k\le K}={\left|h_{k,1}\right|}^2$ denote K random variables with $h_{k,1}\sim \mathcal{CN}(0,1)$ and then $X_{k1\le k\le K}$ are i.i.d. exponential random variables with the same CDF and PDF as Z, i.e., $F_k\left(x\right)=1-e^{-x},x\ge 0$ and $f_k\left(x\right)=e^{-x},x\ge 0$. Let $F_Y\left(x\right)$ denote the CDF of Y. With $Y={min}_{1\le k\le K}{\left|h_{k,1}\right|}^2={min}_{1\le k\le K}{X}_k$, the CDF of Y can be derived as

$$\begin{array}{cc}\hfill F_Y\left(x\right)& =\mathbb{P}(Y\le x)=1-\mathbb{P}(Y>x)\hfill \\ \hfill & =1-\mathbb{P}\left(\underset{1\le k\le K}{min}{X}_k>x\right)\hfill \\ \hfill & =1-\mathbb{P}(X_1>x,\dots ,X_K>x)\hfill \\ \hfill & =1-\prod _{k=1}^K\mathbb{P}(X_k>x)=1-\prod _{k=1}^K\left[1-F_k\left(x\right)\right]\hfill \\ \hfill & =1-{\left[1-F_k\left(x\right)\right]}^K=1-e^{-Kx},x\ge 0.\hfill \end{array}$$

(15)

Accordingly, the associated PDF is $f_Y\left(x\right)=K{e}^{-Kx}$.

Since the MDI scheduling procedure does not involve the desired signal, Z and Y are mutually independent. By applying the law of total probability, the PDF of $SIN{R}_{1,I^{\ast }}^{\mathrm{u}}$ can be written as

$$\begin{array}{cc}\hfill f_{SIN{R}_{1,I^{\ast }}^{\mathrm{u}}}\left(x\right)& ={\int }_0^{\infty }{f}_{Z|Y}\left(x|y\right)f_Y\left(y\right)dy={\int }_0^{\infty }{e}^{-\left({\displaystyle \frac{1}{\rho }}+y\right)x}·\left({\displaystyle \frac{1}{\rho }}+y\right)·K{e}^{-Ky}dy\hfill \\ \hfill & ={\displaystyle \frac{K{e}^{-\frac{x}{\rho }}·(1+\frac{x+K}{\rho })}{{\left(x+K\right)}^2}}.\hfill \end{array}$$

(16)

Accordingly, the corresponding CDF of $SIN{R}_{1,I^{\ast }}^u$ is

$$F_{{\mathrm{SINR}}_{1,I^{\ast }}^u}\left(x\right)={\int }_0^x{f}_{SIN{R}_{1,I^{\ast }}^u}\left(y\right)dy=1-{\displaystyle \frac{K{e}^{-\frac{x}{\rho }}}{x+K}}.$$

(17)

Across scheduling processes in different time blocks, although the sensor selected by $U_1$ may vary, the uplink SINRs of $U_1$ in each scheduling instance are i.i.d. in a homogeneous sensor deployment scenario with negligible large-scale fading. Consequently, the average achievable rate of $U_1$ can be written as

$$R_1=\mathbb{E}\left[{log}_2(1+SIN{R}_{1,I^{\ast }}^u)\right]={\displaystyle \frac{1}{ln2}}\mathbb{E}[ln(1+SIN{R}_{1,I^{\ast }}^u)].$$

(18)

For any nonnegative random variable X, it holds that

$$\mathbb{E}[ln(1+X)]={\int }_0^{\infty }{\displaystyle \frac{\mathbb{P}(X>t)}{1+t}}dt={\int }_0^{\infty }{\displaystyle \frac{1-F_X\left(t\right)}{1+t}}dt.$$

(19)

Therefore, according to Equation (19), $R_1$ can be further derived as

$$\begin{array}{cc}\hfill R_1& ={\displaystyle \frac{1}{ln2}}\mathbb{E}[ln(1+SIN{R}_{1,I^{\ast }}^u)]={\displaystyle \frac{1}{ln2}}{\int }_0^{\infty }{\displaystyle \frac{1-F_{SIN{R}_{1,I^{\ast }}^u}\left(x\right)}{1+x}}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{ln2}}{\int }_0^{\infty }{\displaystyle \frac{K{e}^{-\frac{x}{\rho }}}{\left(1+x\right)\left(K+x\right)}}dx\hfill \\ \hfill & ={\displaystyle \frac{K}{ln2(K-1)}}\left[{\int }_0^{\infty }{\displaystyle \frac{e^{-\frac{x}{\rho }}}{1+x}}dx-{\int }_0^{\infty }{\displaystyle \frac{e^{-\frac{x}{\rho }}}{K+x}}dx\right].\hfill \end{array}$$

(20)

Let $x_1=\rho t-1$ and $x_2=\rho t-K$, and we have

$$\begin{array}{cc}\hfill R_1& ={\displaystyle \frac{K}{ln2(K-1)}}\left[{\int }_{1/\rho }^{\infty }{\displaystyle \frac{e^{-(\rho t-1)/\rho }}{1+(\rho t-1)}}\rho dt-{\int }_{K/\rho }^{\infty }{\displaystyle \frac{e^{-(\rho t-K)/\rho }}{K+(\rho t-K)}}\rho dt\right]\hfill \\ \hfill & ={\displaystyle \frac{K}{ln2(K-1)}}\left[{\int }_{1/\rho }^{\infty }{\displaystyle \frac{e^{-t+1/\rho }}{\rho t}}\rho dt-{\int }_{K/\rho }^{\infty }{\displaystyle \frac{e^{-t+K/\rho }}{\rho t}}\rho dt\right]\hfill \\ \hfill & ={\displaystyle \frac{K}{ln2(K-1)}}\left[e^{1/\rho }{\int }_{1/\rho }^{\infty }{\displaystyle \frac{e^{-t}}{t}}dt-e^{K/\rho }{\int }_{K/\rho }^{\infty }{\displaystyle \frac{e^{-t}}{t}}dt\right]\hfill \\ \hfill & ={\displaystyle \frac{K}{ln2(K-1)}}\left[e^{1/\rho }{E}_1\left({\displaystyle \frac{1}{\rho }}\right)-e^{K/\rho }{E}_1\left({\displaystyle \frac{K}{\rho }}\right)\right].\hfill \end{array}$$

(21)

Since the dual-UAV case is a symmetric scenario, the average achievable rate of $U_1$ and $U_2$ are almost the same. Therefore, the average achievable sum rate of the dual-UAV system can be approximated as twice the achievable rate of a single UAV, expressed as

$$R_{\mathrm{sum}}=2R_1={\displaystyle \frac{2K}{ln2\left(K-1\right)}}\left[{\mathrm{e}}^{{\displaystyle \frac{1}{\rho }}}{E}_1\left({\displaystyle \frac{1}{\rho }}\right)-{\mathrm{e}}^{{\displaystyle \frac{K}{\rho }}}{E}_1\left({\displaystyle \frac{K}{\rho }}\right)\right].$$

(22)

□

4.2. Asymptotic Sum Rate Analysis

In conventional downlink OS or RBF schemes, the asymptotic sum rate is a commonly used performance metric, referring to the system performance as the transmitted SNR $\rho$ tends to infinity for a given number of sensors K. This case is also known as the interference-limited or high-SNR regime [42]. As $\rho \to \infty$, the PDF of the asymptotic SINR is given by

$$f_a\left(x\right)=\underset{\rho \to \infty }{lim}{f}_{SIN{R}_{1,I^{\ast }}^u}\left(x\right)={\displaystyle \frac{K}{{\left(x+K\right)}^2}},$$

(23)

and the corresponding CDF is $F_a\left(x\right)=1-{\displaystyle \frac{K}{x+K}}$. Accordingly, the asymptotic average sum rate of the system can be expressed as

$$\begin{array}{cc}\hfill R_{sum}^0& =2\mathbb{E}\left[{log}_2(1+\underset{\rho \to \infty }{lim}SIN{R}_{1,I^{\ast }}^u)\right]\hfill \\ \hfill & =2{\int }_0^{\infty }{log}_2(1+x)·{\displaystyle \frac{K}{{(x+K)}^2}}dx={\displaystyle \frac{2K}{K-1}}{log}_{2}K.\hfill \end{array}$$

(24)

4.3. DoF Analysis

DOF is also a commonly used performance metric in wireless networks. By definition, the DoF can be interpreted as the number of interference-free data streams that can be simultaneously supported in the network [37]. In the dual-UAV uplink transmission scenario, the signals sent by the two scheduled sensors inevitably interfere with each other; however, this interference diminishes as the number of ground sensors increases. Therefore, based on the closed-form expressions of the sum rate derived in the previous subsection, we can further obtain the DoF performance of the dual-UAV IMAC under the MDI criterion.

Theorem 2.

In a dual-UAV K-sensor scenario using the MDI-based OS over Rayleigh block fading channels, $2\alpha$ DoFs can be achieved if K scales as $K={\rho }^{\alpha }$, where $\alpha \in \left[0,1\right]$ and ρ denotes the transmitted SNR.

Proof. 

Let the number of sensors in the system scale with $K=\beta {\rho }^{\alpha }$, where $\alpha >0$ and $\beta >0$ separately represent the exponential scaling factor and linear coefficient. When $\rho \to \infty$, the achievable DoF of the system can be expressed as

$$DoF=\underset{\rho \to \infty }{lim}{\displaystyle \frac{R_{sum}}{{log}_2\rho }}=\underset{\rho \to \infty }{lim}{\displaystyle \frac{2K}{K-1}}{\displaystyle \frac{{log}_{2}K}{{log}_2\rho }}.$$

(25)

Substituting $K=\beta {\rho }^{\alpha }$ into the expression, we obtain that when $\rho \to \infty$,

$$DoF=\underset{\rho \to \infty }{lim}{\displaystyle \frac{2\beta {\rho }^{\alpha }}{\beta {\rho }^{\alpha }-1}}{\displaystyle \frac{{log}_2\left(\beta {\rho }^{\alpha }\right)}{{log}_2\rho }}=2\underset{\rho \to \infty }{lim}{\displaystyle \frac{\alpha {log}_2\rho +{log}_2\beta }{{log}_2\rho }}=2\alpha .$$

(26)

□

It can be observed that the linear coefficient $\beta$ has no effect on the DoF performance. In the dual-UAV scenario, at most two sensors can be scheduled for uplink transmission in each time slot, which forms a typical two-user SISO interference channel (IC) model [43]. When treating interference as noise, the maximum DoF for the SISO IC is two. However, achieving this bound requires the inter-user interference to be asymptotically zero. Consequently, the maximum achievable DoF in the dual-UAV system is also upper-bounded by two, thus resulting in $\alpha \le 1$.

According to Theorem 2, maintaining the system’s DoF performance under the MDI scheduling criterion requires a sufficiently large number of schedulable sensors to ensure that the sum rate continues to scale with ${log}_2\rho$. Moreover, the fact that the total DoF of the dual-UAV opportunistic scheduling system exceeds one indicates that, when the number of ground users is large, deploying two UAVs within the same area is more effective than deploying a single UAV.

Based on the results reported in [44,45], in a two-cell MISO IBC with single-antenna users and single-stream transmission at each base station, the OBF scheduling yields a per-cell DoF of $\alpha$ if the number of users per cell scales as ${\rho }^{\alpha }$ with $0\le \alpha \le 1$. The dual-UAV scenario considered in this work can be viewed as a special case of a two-cell deployment where the two base stations are co-located and jointly serve all users. Consequently, it can be seen that OS achieves the same multiuser diversity gain in both the uplink and downlink of the dual-UAV system.

5. Sum Rate Analysis of MDI-Based OS for the Three-UAV Scenario

In this subsection, we continue to analyze the sum rate performance of the three-UAV scenario under the MDI scheduling criterion over Rayleigh block fading channels.

5.1. Distribution of Normalized Downlink Interference

In each scheduling round, three ground sensors can be scheduled for simultaneous uplink transmission. Denote the three UAVs by $U_1$, $U_2$, and $U_3$, respectively. According to the MDI criterion, suppose that in a given scheduling instance the three UAVs select ground sensors $S_{I^{\ast }}$, $S_{J^{\ast }}$, and $S_{Q^{\ast }}$, respectively, as their target sensors.

According to the MDI scheduling rules, sensor $S_{I^{\ast }}$ experiences the smallest total downlink interference from UAVs $U_2$ and $U_3$. Accordingly, the SINR of the signal received at sensor $S_{I^{\ast }}$ can be expressed as

$$\begin{array}{cc}\hfill SIN{R}_{I^{\ast },1}^d& ={\displaystyle \frac{P_d|h_{I^{\ast },1}{|}^2}{P_d|h_{I^{\ast },2}{|}^2+P_d|h_{I^{\ast },3}{|}^2+{\sigma }^2}}={\displaystyle \frac{|h_{I^{\ast },1}{|}^2}{{\displaystyle \underset{1\le k\le K}{min}}\left\{{\left|h_{k,2}\right|}^2+{\left|h_{k,3}\right|}^2\right\}+\frac{{\sigma }^2}{P_d}}}\hfill \\ \hfill & ={\displaystyle \frac{|h_{I^{\ast },1}{|}^2}{{\Delta }_{I^{\ast }}+\frac{{\sigma }^2}{P_d}}},\hfill \end{array}$$

(27)

where ${\Delta }_{I^{\ast }}={\displaystyle \underset{1\le k\le K}{min}}\left\{{\left|h_{k,2}\right|}^2+{\left|h_{k,3}\right|}^2\right\}$ denotes the normalized MDI. Define K random variables ${\Omega }_k={\left|h_{k,2}\right|}^2+{\left|h_{k,3}\right|}^2,1\le k\le K$. Under a Rayleigh block fading channel, ${\Omega }_k$ follows a chi-square distribution with four DoFs. The PDF and CDF of ${\Omega }_k$ are given, respectively, by $f_{\Omega }\left(x\right)=x{e}^{-x},x\ge 0$ and $F_{\Omega }\left(s\right)={\int }_0^{x}t{e}^{-t}dt=1-(x+1)e^{-x},x\ge 0$.

Since ${\Omega }_k,1\le k\le K$ consists of K i.i.d. random variables and ${\Delta }_{I^{\ast }}={\displaystyle \underset{1\le k\le K}{min}}{\Omega }_k$, the CDF of ${\Delta }_{I^{\ast }}$ can be derived as

$$F_{\Delta }\left(s\right)=1-{[1-F_{\Omega }\left(x\right)]}^K=1-{\left[(x+1)e^{-x}\right]}^K,x\ge 0,$$

(28)

and the corresponding PDF can thus be obtained as

$$f_{\Delta }\left(x\right)=Kx{(x+1)}^{K-1}{e}^{-Kx},x\ge 0.$$

(29)

Furthermore, let ${\Delta }_{I^{\ast }}={\Delta }_{I^{\ast },2}+{\Delta }_{I^{\ast },3}=|h_{I^{\ast },2}{|}^2+|h_{I^{\ast },3}{|}^2$, where ${\Delta }_{I^{\ast }2}$ and ${\Delta }_{I^{\ast },3}$ denote the normalized downlink interference powers sent from $U_2$ and $U_3$, respectively, when the total interference experienced at sensor $S_{I^{\ast }}$ from $U_2$ and $U_3$ is minimized. Before scheduling, ${\Delta }_{I^{\ast }2}$ and ${\Delta }_{I^{\ast },3}$ are i.i.d. exponential random variables, whereas after sensor $S_{I^{\ast }}$ is scheduled, their distributions are no longer independent. For two i.i.d. exponential random variables X and Y, conditioned on their sum $X+Y=S$, the conditional distribution of either X or Y is uniform over $[0,S]$. Therefore, the PDF of ${\Delta }_{I^{\ast },2}$ can be formulated as

$$\begin{array}{cc}\hfill f_{{\Delta }_{I^{\ast },2}}\left(x\right)& ={\int }_0^{\infty }{f}_{{\Delta }_{I^{\ast }2}|\Delta }\left(x\right|s)f_{\Delta }\left(s\right)ds={\int }_x^{\infty }{\displaystyle \frac{1}{s}}·Ks{(s+1)}^{K-1}{e}^{-Ks}ds\hfill \\ \hfill & =K{\int }_x^{\infty }{(s+1)}^{K-1}{e}^{-Ks}ds=K{e}^K{\int }_{x+1}^{\infty }{t}^{K-1}{e}^{-Kt}dt.\hfill \end{array}$$

(30)

By using the definition of the upper incomplete Gamma function

$$\Gamma (n,x)={\int }_x^{\infty }{t}^{n-1}{e}^{-t}dt,$$

(31)

and applying the change in variables $t={\displaystyle \frac{u}{p}}$, we have

$$\begin{array}{cc}\hfill & {\int }_x^{\infty }{t}^{n-1}{e}^{-pt}dt={\int }_{px}^{\infty }{\left({\displaystyle \frac{u}{p}}\right)}^{n-1}{e}^{-u}{\displaystyle \frac{du}{p}}\hfill \\ \hfill & ={\displaystyle \frac{1}{p^n}}{\int }_{px}^{\infty }{u}^{n-1}{e}^{-u}du=p^{-n}\Gamma (n,px).\hfill \end{array}$$

(32)

Substituting the above result into Equation (30), the expression of $f_{{\Delta }_{I^{\ast },2}}\left(x\right)$ can be further written in a closed form as

$$\begin{array}{cc}\hfill f_{{\Delta }_{I^{\ast },2}}\left(x\right)& =K{e}^K{\int }_{x+1}^{\infty }{t}^{K-1}{e}^{-Kt}dt=K{e}^K{K}^{-K}\Gamma \left(K,K(x+1)\right)\hfill \\ \hfill & =K^{1-K}{e}^K\Gamma \left(K,K(x+1)\right).\hfill \end{array}$$

(33)

Under a homogeneous sensor deployment where large-scale fading is not considered, the normalized downlink interference from the same UAV to different sensors, such as ${\Delta }_{J^{\ast },1}$ and ${\Delta }_{Q^{\ast },1}$, are i.i.d. random variables. Therefore, the subscripts in $f_{{\Delta }_{I^{\ast },2}}\left(x\right)$ can be simplified, and the PDF of these interference random variables can all be denoted by $f_{\Delta \ast }\left(x\right)$.

5.2. Approximation of Normalized Uplink Interference A

During the uplink transmission, the received uplink SINR at $U_2$ can be represented as

$$\begin{array}{cc}\hfill SIN{R}_{2,J^{\ast }}^u& ={\displaystyle \frac{P_u|g_{2,J^{\ast }}{|}^2}{P_u{\left|g_{2,I^{\ast }}\right|}^2+P_u{\left|g_{2,Q^{\ast }}\right|}^2+{\sigma }^2}}={\displaystyle \frac{|h_{J^{\ast },2}{|}^2}{|h_{I^{\ast },2}{|}^2+|h_{Q^{\ast },2}{|}^2+\frac{{\sigma }^2}{P_u}}}\hfill \\ \hfill & ={\displaystyle \frac{Z}{{\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}+1/\rho }}.\hfill \end{array}$$

(34)

In this expression, $Z={\left|h_{I^{\ast },1}\right|}^2$ follows an exponential distribution with PDF $f_Z\left(x\right)=e^{-x}$. To derive the PDF of $SIN{R}_{2,J^{\ast }}^u$, the key lies in characterizing the distribution of the interference term $A={\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}$. It is worth noting that the term A represents the ratio of the uplink received interference power to the transmit power $P_u$, i.e., the normalized uplink received interference power rather than the absolute interference power. In addition, ${\Delta }_{I^{\ast },2}$ and ${\Delta }_{Q^{\ast },2}$ are i.i.d. with the same pdf as Equation (33). However, Equation (33) possesses a complex structure, which makes deriving the accurate pdf of A intractable. Therefore, in this paper we approximate the distribution of $A={\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}$ by a Gamma distribution,

$$f_A\left(x\right)\approx f_{A^{\ast }}(x,\alpha ,\beta )=x^{\alpha -1}{\displaystyle \frac{e^{-\frac{x}{\beta }}}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}},x\ge 0,$$

(35)

where $\alpha >0$ is the shape parameter, $\beta >0$ is the scale parameter, and $\Gamma \left(\alpha \right)$ denotes the Gamma function with $\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}\mathrm{d}t,x>0$.

Lemma 1.

Let Z and A be independent random variables with PDFs $f_Z\left(x\right)=e^{-x}$ and $f_A(x,\alpha ,\beta )=x^{\alpha -1}{\displaystyle \frac{e^{-\frac{x}{\beta }}}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}},x\ge 0$, respectively. Then, for the random variable $S={\displaystyle \frac{Z}{A+1/\rho }}$, the CDF and PDF are given as follows.

$$F_S\left(x\right)=1-e^{-x/\rho }{(1+\beta x)}^{-\alpha }$$

(36)

$$f_S\left(x\right)=\alpha \beta e^{-x/\rho }{(1+\beta x)}^{-\alpha -1}+{\displaystyle \frac{e^{-x/\rho }}{\rho }}{(1+\beta x)}^{-\alpha }$$

(37)

Proof. 

Given $A=a$, $S={\displaystyle \frac{Z}{a+\frac{1}{\rho }}}$ is a linear function of Z. Thus, the conditional PDF of S with $A=a$ can be expressed as

$$f_{S|A}\left(x\right|a)=\left(a+{\displaystyle \frac{1}{\rho }}\right)exp\left[-\left(a+{\displaystyle \frac{1}{\rho }}\right)x\right].$$

(38)

By the law of total probability, the PDF of S is obtained by

$$\begin{array}{cc}\hfill f_S\left(x\right)& ={\int }_0^{\infty }{f}_{S|A}\left(x\right|y)f_A\left(y\right)dy\hfill \\ \hfill & ={\int }_0^{\infty }\left(y+{\displaystyle \frac{1}{\rho }}\right)e^{-(y+{\displaystyle \frac{1}{\rho }})x}{\displaystyle \frac{y^{\alpha -1}{e}^{-y/\beta }}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}}dy\hfill \\ \hfill & ={\displaystyle \frac{e^{-x/\rho }}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}}{\int }_0^{\infty }\left(y+{\displaystyle \frac{1}{\rho }}\right)y^{\alpha -1}{e}^{-y\left(x+{\displaystyle \frac{1}{\beta }}\right)}dy.\hfill \end{array}$$

Let $\lambda =x+{\displaystyle \frac{1}{\beta }}$. Using the Gamma integral identities

$${\int }_0^{\infty }{y}^{\alpha }{e}^{-\lambda y}dy={\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^{\alpha +1}}}={\displaystyle \frac{\alpha \Gamma \left(\alpha \right)}{{\lambda }^{\alpha +1}}},$$

(39)

and

$${\int }_0^{\infty }{y}^{\alpha -1}{e}^{-\lambda y}dy={\displaystyle \frac{\Gamma \left(\alpha \right)}{{\lambda }^{\alpha }}},$$

(40)

we obtain

$$\begin{array}{cc}\hfill f_S\left(x\right)& ={\displaystyle \frac{e^{-x/\rho }}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}}\left[{\int }_0^{\infty }{y}^{\alpha }{e}^{-\lambda y}dy+{\displaystyle \frac{1}{\rho }}{\int }_0^{\infty }{y}^{\alpha -1}{e}^{-\lambda y}dy\right]\hfill \\ \hfill & =\alpha \beta e^{-x/\rho }{(1+\beta x)}^{-\alpha -1}+{\displaystyle \frac{e^{-x/\rho }}{\rho }}{(1+\beta x)}^{-\alpha },x\ge 0.\hfill \end{array}$$

(41)

Define function

$$g\left(x\right)={\displaystyle \frac{e^{-x/\rho }}{{(1+\beta x)}^{\alpha }}}.$$

(42)

Then, its derivative is

$$g^{\prime }\left(x\right)=-\alpha \beta {\displaystyle \frac{e^{-x/\rho }}{{(1+\beta x)}^{\alpha +1}}}-{\displaystyle \frac{e^{-x/\rho }}{\rho {(1+\beta x)}^{\alpha }}}=-f_S\left(x\right).$$

(43)

By integration by parts, the CDF of S is given by

$$F_S\left(x\right)={\int }_0^x{f}_S\left(t\right)dt=g\left(0\right)-g\left(x\right)=1-{\displaystyle \frac{e^{-x/\rho }}{{(1+\beta x)}^{\alpha }}},x\ge 0.$$

(44)

□

To determine the parameters $\alpha$ and $\beta$ in the Gamma distribution, the mean and variance of the approximating Gamma variable $A^{\ast }$ must match those of the original distribution $A={\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}$. Based on this requirement, Theorem 3 is obtained as follows.

Theorem 3.

For the Gamma-distributed random variable $A^{\ast }$ used to approximate the normalized uplink interference term $A={\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}$ the parameters α and β that ensure the moment-matching conditions $\mathbb{E}\left[A\right]=\mathbb{E}\left[A^{\ast }\right]$ and $Var\left(A\right)=Var\left(A^{\ast }\right)$ are given as follows.

$$\alpha ={\displaystyle \frac{6{\left[\frac{1}{K}+\frac{(K-1)!}{K^K}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}\right]}^2}{-3{\left[\frac{(K-1)!}{K^K}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}\right]}^2+2\frac{(K-1)!}{K^{K+1}}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}+\frac{8}{K}+\frac{5}{K^2}}}$$

(45)

$$\beta ={\displaystyle \frac{-3{\left[\frac{(K-1)!}{K^K}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}\right]}^2+\frac{2(K-1)!}{K^{K+1}}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}+\frac{8}{K}+\frac{5}{K^2}}{6\left[\frac{1}{K}+\frac{(K-1)!}{K^K}{\displaystyle \sum _{j=0}^{K-1}}\frac{K^j}{j!}\right]}}$$

(46)

The proof is provided in the Appendix A.

To demonstrate the effectiveness of approximating the uplink interference $A={\Delta }_{I^{\ast },2}+{\Delta }_{Q^{\ast },2}$ by the Gamma-distributed random variable $A^{\ast }$, Figure 5 presents a comparison of their PDFs under different numbers of sensors. It can be observed that the two PDF curves exhibit the same trend and match closely.

Based on Lemma 1 and Theorem 3, the CDF and PDF expressions of the uplink SINR for each UAV in the three-UAV scenario under the MDI criterion can be derived as Equations (36) and (37) over Rayleigh block fading channels. Owing to the symmetry of the three-UAV setting, the system’s average sum rate equals three times the achievable rate of a single UAV. The resulting sum rate expression is given as follows in an integral form, which can be efficiently evaluated via numerical methods to obtain accurate results.

$$\begin{array}{cc}\hfill R_{sum}& =3\mathbb{E}\left[{log}_2(1+SIN{R}_{2,J^{\ast }}^u)\right]={\displaystyle \frac{3}{ln2}}{\int }_0^{\infty }{\displaystyle \frac{1-F_S\left(x\right)}{1+x}}dx\hfill \\ \hfill & ={\displaystyle \frac{3}{ln2}}{\int }_0^{\infty }{\displaystyle \frac{e^{-x/\rho }{(1+\beta x)}^{-\alpha }}{1+x}}dx\hfill \end{array}$$

(47)

5.3. Kolmogorov–Smirnov Test

To quantitatively validate the accuracy of the proposed Gamma approximation, we further conduct the Kolmogorov–Smirnov (KS) goodness-of-fit test between the empirical interference distribution of A obtained via Monte Carlo simulations and the theoretical Gamma distribution of $A^{\ast }$ with analytically derived parameters. It is well known that the KS test becomes overly sensitive with extremely large sample sizes and may reject the null hypothesis even for very small deviations. Therefore, the KS test is performed on a randomly selected subset of Monte Carlo samples with size $N=500$, which is a common practice for distributional validation.

The detailed KS test results are summarized in

Table 2. Kolmogorov–Smirnov test results for Gamma approximation.

K	KS Statistic D	p-Value	Decision ($\mathit{\alpha }=0.05$)
50	0.0595	0.0556	Not rejected
200	0.0500	0.1580	Not rejected

. For $K=50$ and $K=200$, the p-values are greater than $0.05$, indicating that the null hypothesis cannot be rejected at the $5\%$ significance level. This confirms the statistical validity of the proposed Gamma approximation. The corresponding KS statistics further show that the maximum discrepancy between the empirical and theoretical CDFs of A and $A^{\ast }$ remains small.

6. Simulation Results

In this section, Monte Carlo simulations are conducted to validate the correctness of the theoretical analysis derived in the previous two sections for the dual-UAV and three-UAV scenarios and to compare the performance of different transmission strategies and channel models. Unless otherwise specified, the simulation parameters used throughout this section are summarized in

Table 3. Simulation Parameters.

Parameter	Value/Description
Number of UAVs (N)	$N=2$ or $N=3$
Number of sensors (K)	$K=10,30,50,100,300$
Channel fading model	Rayleigh fading channel and Nakagami-m channel
Nakagami-m parameter	$\Omega$ = 1, m = 0.75 or 2
Uplink transmitted SNR ($\rho =P_u/{\sigma }^2$)	0–50 dB
Scheduling schemes	MDI-OS, MD-SINR-OS, TDMA
Number of snapshots	${10}^5$ independent channel realizations

.

6.1. Comparison of Theoretical and Simulation Results Under Rayleigh Channel

For the dual-UAV case with Rayleigh block fading channels, comparisons among the theoretical average sum rate, the asymptotic sum rate, and the simulation results under the MDI criterion scheduling scheme for different numbers of sensors are shown in Figure 6. The horizontal axis represents the transmitted SNR $\rho$. The theoretical average sum rate results are obtained from Equation (11), whereas the asymptotic sum rate results are based on Equation (24). It can be observed that the theoretical average sum rate agrees well with the simulated values, which verifies the accuracy of the theoretical analysis, and the discrepancy decreases as K increases. The asymptotic sum rate appears as a straight line independent of the transmit power, representing an upper bound on the achievable sum rate performance. The average sum rate of MDI-based OS first increases with and then converges to the asymptotic sum rate curve at high transmit-power levels. Moreover, the upper bound of the achievable sum rate increases with K, indicating that a larger number of scheduled sensors yields greater exploitable multiuser diversity gain.

For the three-UAV scenario with Rayleigh block fading channels, comparisons among the theoretical average sum rate and the simulation results under the MDI criterion with different numbers of sensors are shown in Figure 7. The theoretical average sum rate results are computed according to Equation (47). It can be seen that the theoretical average sum rate also matches the simulated values closely, thereby confirming the validity of approximating the uplink interference distribution with a Gamma distribution to reduce computational complexity. The discrepancy between theory and simulation decreases as K increases. Similar to the dual-UAV case, as the transmit power grows, the average sum rate curve first increases and then converges to a horizontal line, and the corresponding convergence level rises with increasing K.

Figure 8 illustrates the normalized error between the theoretical sum rate and the simulation results shown in Figure 6 and Figure 7. When K is larger than 30, the normalized error remains below $1\%$ in both scenarios and across the entire transmitted SNR range, indicating that the derived theoretical expressions can accurately characterize the average rate performance of the MDI-based OS. In the dual-UAV scenario, as K increases, the normalized error decreases and shows little sensitivity to increases in the SNR. In the three-UAV scenario, the decrease in normalized error with increasing K is more pronounced in the low- and medium-SNR regimes (0–20 dB). When the transmitted SNR exceeds 30 dB, the normalized errors for different values of K become almost identical in the three-UAV scenario.

6.2. Sum Rate Performance of MDI-OS Under Rayleigh and Nakagami Channels

Figure 9 and Figure 10 present the average sum rate performance of MDI-OS under Rayleigh and Nakagami-m fading channels (with m = 2 and m = 0.75), for the dual-UAV and three-UAV scenarios, respectively. Note that the Rayleigh fading channel corresponds to a special case of the Nakagami-m model with m = 1. For each channel scenario, the number of sensors K varies from 10 to 100. Two main observations can be drawn from these figures. First, for a given number of UAVs, transmit SNR, and K, a smaller value of the Nakagami parameter m results in a higher average sum rate. Second, the performance gap between different m values becomes more pronounced in the high SNR region and when the number of sensors is large. These observations indicate that in MDI-OS, the system performance is highly sensitive to the fading severity (captured by the parameter m), and more severe fading (smaller m) paradoxically leads to a higher multiuser scheduling diversity gain and thus a higher sum rate.

In Nakagami-m fading channels, the shape factor m characterizes the severity of channel fading. The channel gain in a Nakagami-m channel follows a Gamma distribution, and a larger m corresponds to a smaller variance of the channel gain. This implies a stronger line-of-sight component and weaker scattered components in the channel. As m increases, the differences in downlink interference or downlink SINR among different sensors decrease, thereby limiting the multiuser diversity potential. Consequently, the multiuser diversity-based OSs are more suitable for Nakagami fading environments with smaller m values.

6.3. DoF Performance

To verify the relationship between the system DoF performance and the user-scaling law in the dual-UAV scenario, Figure 11 plots the system sum rate versus the transmitted SNR for different values of $\alpha$ when the number of users scales as $K={\rho }^{\alpha }$. The curves for the MDI scheme in Rayleigh channels are obtained from theoretical asymptotic sum rate analysis, whereas those for the MD-SINR scheme and the MDI scheme in Nakagami channels are generated from simulation results. Reference curves with rate $2\alpha {log}_2\left(1+\rho \right)$ are also plotted in Figure 11, representing a DoF of $2\alpha$. In the high transmitted SNR regime (35–40 dB), the slopes of the sum rate curves for both MDI-OS and MD-SINR-OS in Rayleigh channels coincide with that of the reference curve for different values of $\alpha$, thereby validating the conclusion of Theorem 2 and indicating that the two scheduling schemes share the same DoF performance and user-scaling law.

As shown in Figure 11, the DoF performance of MDI-OS in Nakagami-m fading channels is significantly influenced by the fading parameter m. For more severe fading conditions characterized by m = 0.75, the DoF can reach the maximum value of two even with a user-scaling factor of $\alpha$ = 0.8. This performance surpasses the DoF of 1.6 achieved in Rayleigh fading channels (which correspond to m = 1). For a shape parameter of m = 2, the achievable DoF falls below $2\alpha$. These results demonstrate that the user-scaling law governing the DoF performance of MDI-OS is inherently affected by the parameter m in Nakagami channels.

6.4. Sum Rate Performance of Different Transmission Strategies in Dual-UAV Scenario

To validate the effectiveness of the dual-UAV deployment and the proposed OS schemes, Figure 12 and Figure 13 compare the sum rate performance of several transmission strategies, including MDI-OS, MD-SINR-OS, and TDMA-based round-robin methods, in Rayleigh and Nakagami channels, respectively. For the TDMA-based two-sensor round-robin scheduling, two sensors are selected sequentially in each slot to transmit. However, the presence of co-channel interference results in the lowest achievable rate. Under the TDMA-based single-sensor round-robin scheme, only one UAV is required, and inter-sensor interference is absent. Consequently, its sum rate increases monotonically with the transmit power, but it can outperform the MDI and MD-SINR schemes only when the transmit power is sufficiently high. However, due to the limited transmission power of ground sensors, the actual uplink transmission SNR cannot reach the favorable SNR range for the TDMA scheme. These results demonstrate that dual-UAV deployment combined with OS yields better system performance than single-UAV deployment with round-robin scheduling in both Rayleigh and Nakagami channels.

6.5. Sum Rate Performance of Different UAV Numbers

To identify the optimal deployment number of UAVs where both the UAVs and sensors are equipped with single antennas, Figure 14 compares the sum rate performance of several deployment scenarios under a different numbers of sensors in Rayleigh fading channels, including the dual-UAV deployment with MDI-OS, the three-UAV deployment with MDI-OS, and the single-UAV deployment with TDMA. The sum rate performance of the same deployment scenarios are compared again in Nakagami channels with K = 100 in Figure 15.

Although the three-UAV deployment allows for more sensors to transmit concurrently, the additional inter-user interference it introduces causes its achievable sum rate to remain lower than that of the two-UAV deployment at medium-to-high transmitted SNR levels. As indicated by Figure 14 and Figure 15, different strategies should be adopted at different transmitted SNR regimes. For instance, in Figure 14, when $K=50$, the three-UAV deployment with MDI-OS in Rayleigh fading channels is preferred for transmitted SNR $\rho <10$ dB; the dual-UAV deployment yields better performance for $\rho \in \left[10,36\right]$ dB; and when the transmit power becomes even larger, the single-UAV TDMA-based scheme becomes more advantageous because it completely eliminates inter-user interference. Therefore, in Figure 14 with K = 50, $\rho$ = 10 dB serves as the performance intersection point for the dual-UAV deployment and the three-UAV deployment scenarios, while $\rho$ = 36 dB serves as the performance intersection point for the dual-UAV deployment and the single-UAV deployment scenarios. As shown in Figure 14 and Figure 15, under the MDI-OS transmission strategy, the performance intersection point (denoted by an SNR threshold) between the dual-UAV and three-UAV deployment scenarios increases with the number of sensors K and with a decrease in the Nakagami fading parameter m. In addition, the performance intersection SNR between the dual-UAV and single-UAV deployments is relatively high and lies beyond the commonly used practical SNR range. Therefore, it can be concluded again that in typical scenarios with a large number of sensors K, the two-UAV deployment based on MDI-OS achieves a higher sum rate than the single-UAV deployment, under both Rayleigh and Nakagami fading channels.

The optimal number of UAVs to be deployed is a function of key system parameters, including the channel characteristics, the number of sensors, and the sensors’ uplink transmit power. In Rayleigh fading channels, the average achievable sum rate of MDI-OS has been derived in closed form for both dual-UAV and three-UAV scenarios, as given in Equations (11) and (47), respectively. Although the performance intersection SNR value for the dual-UAV and three-UAV deployment scenarios cannot be solved directly due to the complex form of the equations, it can still be quickly determined using numerical methods such as the binary search method. Therefore, these analytical results enable quick and direct performance comparison, allowing for an efficient identification of the superior deployment strategy under these specific channel conditions. However, for the more general Nakagami-m fading channels, analogous closed-form expressions for the average sum rate of different UAV deployment scenarios are not yet available. Therefore, performance evaluation and scheme comparison necessarily rely on Monte Carlo simulation methods. This simulation-based approach is essential for characterizing the average rate performance and determining the optimal number of UAVs in Nakagami-m environments.

From an engineering perspective, the number of UAVs can be dynamically selected in practice using a threshold-based or lookup-table strategy. For Rayleigh fading channels, the decision threshold can be set as the performance intersection SNR for different UAV deployment counts, which can be determined numerically using the analytical results derived in this paper. For Nakagami-m channels, a lookup table can be constructed based on extensive offline simulations covering typical parameter combinations, including the transmit SNR, the Nakagami-m parameter, and the number of sensors. During online operation, the system controller can then select the appropriate number of UAVs through a simple threshold comparison or table lookup, without introducing significant additional computational complexity.

7. Conclusions

This paper investigates low-overhead OS strategies and their performance analysis for uplink data collection in multi-UAV-IoT systems. By leveraging uplink–downlink channel reciprocity, two uplink scheduling schemes based on MDI and MD-SINR are proposed. These schemes significantly reduce uplink channel training overhead while effectively preserving multiuser diversity gain. In terms of theoretical analysis, for the dual-UAV deployment scenario under Rayleigh fading channels, closed-form expressions for the system’s average sum rate, asymptotic sum rate, and DoF are derived. This reveals the quantitative relationship between user-scaling law and system DoF. For the three-UAV deployment scenario, the average sum rate expression is achieved by approximating the uplink interference as a Gamma distribution. The simulation results show strong agreement with theoretical analyses, validating the effectiveness of the established analytical framework. Under the more general Nakagami-m fading channels, this paper further illustrates the impact of channel fading severity on the performance of OS schemes. The results indicate that a smaller Nakagami parameter m enhances channel fluctuation, thereby increasing the multiuser diversity gain. This leads to superior sum rate and DoF performance compared to Rayleigh and weaker fading scenarios. The study also demonstrates that increasing the number of UAVs does not necessarily improve performance. Due to additional interference effects, the dual-UAV deployment exhibits better system performance within the practically usable medium-to-high SNR range, in both Rayleigh and Nakagami-m channels. Based on the above conclusions, the actual system can dynamically select the UAV deployment scale using low-complexity methods such as threshold decision or lookup tables, considering transmit power, sensor number, and channel fading characteristics. The research outcomes of this paper provide generalizable theoretical support and engineering guidance for the design of UAV-assisted uplink communication systems for large-scale, low-power IoT scenarios.