RETRACTED: Resource Allocation Algorithm Based on Power Control and Dynamic Transmission Protocol Configuration for HAPS-IMT Integrated System

Abstract

The high altitude platform station (HAPS) system is an essential component of the air-based network. It can shorten transmission delay and make a better user experience compared with satellite networks, and it can also be easily deployed and cover a larger area compared with international mobile telecommunications (IMT). In order to meet the needs of users with asymmetric and random data flow, the spectrum sharing and dynamic time division duplexing (TDD) mode are used in HAPS-IMT heterogeneous network. However, the cross-link interference brought by TDD mode will lead to the degradation of system performance. In this paper, a resource allocation algorithm based on power control and dynamic transmission protocol configuration is proposed. Firstly, a specific timeslot, “low power almost-bank subframe (LP-ABS)”, is introduced into the frame structure of the HAPS physical layer. The transmission protocol designing could mitigate inter-layer interference efficiently by power control in “LP-ABS”. Secondly, the utilization function is adopted for assessing the system performance, which gives attention to both diversified requirements on the quality of services (QoS) and the throughput of the HAPS-IMT system. Simulation results show that power control and resource allocation technologies proposed in this paper can effectively improve system performance and user satisfaction.

1. Introduction

1.1. Background

The rapid development of human information society has seen the demands for ubiquitous network access and rapid growth of various services. As a result, increasingly more attention has been paid to integrated systems and future-oriented traffic types (as follows).

• Layers of Integrated System

For the terrestrial part, 3GPP Rel. 15 was completed and frozen in the third quarter of 2019, which is confirmed by ITU as the 5th generation (5G) international mobile broadband standard [1]. Advanced research of 5G evolution and 6G has thus been conducted. For the airborne part, HAPS has received much attention [2,3]. TR 38.811 Rel. 15 of 3GPP is a detailed standard on the HAPS air interface technology. The HAPS has advantages of being of relatively lower altitude, smaller transmission delay, and higher networking flexibility. The same as with 5G systems, HAPS adopts the OFDMA multiple address access mechanism. It can lower the manufacturing cost and can take full advantage of the scale effect of the mobile telecommunication industry effectively. The HAPS-IMT integrated system studied in this paper is convenient for 5G users to access.

B.

Traffic Types and QoS guarantee

ITU launched the research on future-oriented service requirements more than 10 years ago. Three important traffic types for different service requirements have been defined, which are ultra-high reliability and ultra-low latency service (URLLC), enhanced mobile broadband service (eMBB) and mass machine communication services (mMTC). Given the characteristics of HAPS-IMT integrated systems, it is difficult to guarantee the QoS requirement of URLLC service. For this reason, this paper mainly focuses on the eMBB and mMTC services. eMBB is a kind of differentiated service (DiffServ), also called soft QoS services, such as AR/VR and streaming media. eMBB requires a certain QoS guarantee to make the transmission smoothly. There is no difference when the data rate is beyond the higher threshold. In the meantime, the data rate cannot be below the lower threshold to avoid a poor user experience. mMTC should be considered as a best-effort service, which does not require a QoS guarantee.

C.

Spectrum Utilization

For the HAPS system, it will substantially benefit future development for sharing IMT frequency bands. On the one hand, the scale effect of the worldwide mobile broadband industry can be acquired easily. On the other hand, the spectrum resource is an essential prerequisite for the development of the HAPS industry. The 2023 World Radiocommunication Conference’s (WRC-23) agenda item 1.4 is on frequency identification for HIBS (high altitude platform stations as IMT base stations). Agenda item 1.4 focuses on the probable frequency bands below 2.7 GHz for HIBS, which have already been identified for IMT. The conclusion that will be reached at this conference will impact the development of HAPS for the next 5 to 10 years.

1.2. Related Work

In order to obtain better performance, many types of research has been conducted on the resources management of the HAPS network, such as bearing layer selection, transmission resource allocation, and power control [4]. A centralized resource allocated strategy called ORA-LF for HAPS is proposed in [5], aiming to maximize system throughput. However, the strategy is proposed for the WCDMA system in [5], which requires further research on the OFDM system. In the meantime, the fairness of the same service among users has also been reasonably considered. A centralized call admission control (CAC) algorithm under power limitation was proposed by [6,7,8]. Centralized scheduling could optimize system performance through better resource allocation. However, these three papers only optimized the HAPS systems internally, and the problem modelling under system co-existence scenarios needs further study. A joint learning (FL) algorithm, which is based on a support vector machine (SVM) scheme, is proposed in [9]. The FL algorithm can actively figure out user association, service sequence, and a task allocation plan. Thus, mobile edge computing (MEC) is reasonably deployed on each HAPS node to reduce the delay caused by task transmission. The aim of [9] is to minimize the energy and time consumption for task computing and transmission by adjusting the user association, service sequence, and task allocation scheme. Further study of system throughput performance is required, as highlighted by [9]. The research on wireless resource allocation of OFDMA-based HAPS network is conducted in [10], which formulates and solves specific optimization problems to identify the best resource allocation plan for transmission power, sub-channels allocation, and time slots allocation. Consequently, the resource allocation scheme for a single-layer network is difficult to be applied in integrated networks. Therefore, the optimization of resource allocation for the integrated network is an important research field.

Research on resource allocation mechanisms of the multi-layer network, including HAPS layer, satellite layer and terrestrial layer, are mainly carried out in [11,12,13,14,15,16]. The research in [11] is about the method of improving QoS of different services under the same frequency band and coverage of multiple HAPS networks. However, the impact of terrestrial networks is not considered in [11]. A new framework for airborne heterogeneous wireless networks has been developed in [12], which can effectively support real-time statistics in a heterogeneous network. The optimal system transmission power plan can be established with the guarantee of system QoS requirements. However, the mutual interference suppression method in the TDD mode needs further study. QoS statistics frameworks for 5G mobile heterogeneous wireless networks and a heterogeneous network QoS statistics framework for mobile wireless networks with the two-way full-duplex transmission have been proposed also [13,14]. Therefore, statistically delayed QoS statistics can provide an effective alternative to guarantee delay-limited QoS services for real-time traffic of various types of mobile wireless networks, including AMWNs. The research in [15,16] is mainly aimed at optimization of resource allocation in downlink transmission scenarios.

In order to obtain more valuable references in resource management algorithms for future actual integrated networks, this paper models a two-layer HAPS-IMT integrated system. Both the IMT layer and HAPS layer adopt OFDMA as their multiple address access mechanisms. Considering the research purpose of agenda item 1.4 of WRC-23, both IMT and HAPS layers employ TDD mode and operate in the same frequency bands.

1.3. Contribution

Although the importance of radio resource management (RRM) is apparent, RRM research for integrated networks are relatively rare. This paper is mainly focused on the RRM of the HAPS-IMT integration system. The main work and contributions of this paper are as follows:

• Decompose the optimization of resources allocation for integrated networks into two steps: long-cycle and sub-cycle optimization. The goal of the long-cycle optimization step is to determine a transmission protocol of the physical layer of the HAPS-IMT integrated system to maximize system throughput. The purpose of the sub-period optimization step is to maximize the utility of the integration system by allocating resource blocks (RB) and assigning transmission power reasonably and effectively.
• The concept of “utility function” has been introduced in this paper, which quantifies users’ QoS requirements. The utility function can facilitate the performance optimization of multi-layer integrated systems when allocating resources to multiple user traffic flows.
• Based on the recent fairness research of the air-ground integrated network [16], this paper optimizes the fairness evaluation criteria of HAPS-IMT networks further to promote the development of a fairness evaluation scheme.

The rest of this article is organized as follows. Section 2 describes the system model and establishes the HAPS-IMT system network model. In Section 3, for integrated cellular networks in TDD mode, configuring transmission protocol is designed to alleviate the interference from downlink to uplink. At the same time, the utility function is used to describe the characteristics of various services, and a resource allocation scheme for cross-link interference mitigation is designed. Section 4 evaluates the research result. Section 5 reviews the work in its entirety and concludes.

2. System Model

Since both HAPS and 5G networks adopt OFDMA as their multiple address access mechanism, the HAPS-IMT network can be easily and quickly deployed when needed, especially in disaster relief scenarios. An integrated system is structured as below for further study. HAPS-IMT network supports dynamic frame structure configuration, as shown in Figure 1. This paper mainly focuses on systematic research on the resource management of the HAPS-IMT network to solve the scheduling issue of the eMBB and mMTC services.

2.1. Network Topology

The HAPS-IMT network topology includes two layers. The first is the HAPS layer, in which there is only one HAPS node in the network topology. The HAPS node is the central node of the network, which can obtain all system information. The other is the 5G layer, containing several 5G ground base stations (GBS). Since the two layers adopt the OFDMA mechanism, every user can access one selected HAPS node or GBS of either layer during a period according to a scheduling algorithm. In this model, a ground station (GS) is set for communication with the outside network. The backhaul link is used for returning data traffic back to the GS. The two layers share the same spectrum resources, and dynamic TDD mode is adopted to minimize the inter-layer interference in this model. Resource allocation and transmission power control are two effective technologies to optimize the performance of the HAPS-IMT system in this paper. Users can access to HAPS node station or any one of the GBS as scheduled. We suppose two research cycles: a long cycle and a sub-cycle. The long cycle is 10 radio frames long. And one radio frame is 10 ms. The transmission protocol configuration is stable during the long cycle. One subframe, which is 1 ms long, is considered a sub-cycle. The system allocates time slots in each subframe to different users. The HAPS layer and 5G layer within the integrated system cover the same geographical area. The HAPS node could acquire all status information within the system in real-time since it has a connection with other GBSs and GS. Therefore, the HAPS node is also an RRM network element of the integrated network, which manages the system resources, the situation of channel link, resources remaining in the integrated network and allocated those resources (including resource block and power).

Symbols and their meanings of essential parameters used in this paper are offered in

Table 1. Parameter symbols and physical meaning.

Symbol	Meaning
$U^S$$(U^B$)	Utility of soft QoS service. (Utility of best effort service).
${\mathsf{\Gamma }}^d$$({\mathsf{\Gamma }}^u$)	Downlink user set in integrated networks (Uplink user set in integrated networks).
$P_i^u$	Uplink transmission power of user i.
$P_o^l$	Downlink transmission power of HAPS node in the LP-ABS.
$P_o^u$$(P_o^d$)	Uplink transmission power of HAPS node in the normal subframe. (Downlink transmission power of HAPS node in the normal subframe).
$P_k^d$	Transmission power of GBS
${\gamma }_i^d$$({\gamma }_i^u$)	Downlink SNR of user (Uplink SNR of user).
$h_{i,j}$	Channel gain from user i to BS j.
$I_{k,j}$	Average interference from Cell k to users in cell j.
$c_i^{l,d}$$(c_i^{l,u}$)	Spectrum efficiency of uplink scheduling of user i in normal/LP-ABS
$c_i^{n,d}$$(c_i^{n,u}$)	Spectrum efficiency of downlink scheduling of user i in normal/LP-ABS
${\mathsf{\Phi }}^{n,d}$$({\mathsf{\Phi }}^{n,u}$$, {\mathsf{\Phi }}^l$)	Proportion of normal downlink frames, normal uplink frames, and LP-ABS in each transmission period
$B=\{0,1,2,\dots ,M-1\}$	$B$ is the set of all BSs in the system model. 0 is represented to HAPS node. 1 to M-1 represented to M-1 5G Ground Base Stations.
$B_i$	Base station for User $i$.

. Superscript “$d$” represents downlink data, “$u$” represents uplink data. The superscript “$B$” indicates Best Effort service, and the superscript “$S$” indicates the soft quality of service (soft QoS). Symbol $i$ is used for user index, and Symbol $j$ is used for base station index. Superscript “$n$” and “$l$” denote normal subframe and low power almost-bank subframe (LP-ABS), respectively.

2.2. Transmission Protocol

The system adopts a dynamic TDD transmission mechanism to avoid link interference in the HAPS-IMT network effectively. All cells (including a HAPS cell and GBS cells) are configured synchronously. Each radio frame (10 ms duration) consists of 10 subframes (1 subframe is 1 ms duration).

The asynchronous transmission protocol configuration of different layers will introduce cross-link interference. Cross-link interference includes downlink to uplink interference caused by BS in neighbor cells and uplink to downlink interference caused by users in neighbor cells. Because the transmission power of the HAPS node is much greater than that of GBS, the interference from the HAPS node to neighbor cells is the major reason for the decline of system performance. To effectively support dynamic TDD in an integrated system and mitigate cross-link interference, this paper designs the transmission protocol as shown in Figure 2. The transmission protocol combines power control and time slot resource configuration. HAPS divides each subframe into two parts, one is called a normal subframe, and the other one is called LP-ABS. To mitigate HAPS-to-GBS interference, the HAPS node and GBS configure synchronous uplink and downlink slots on the normal subframe. In addition, to provide better flexibility to support the asymmetry of data flow, each GBS dynamically configures the proportion of uplink and downlink slots on the dynamic subframe. In this case, the asynchronous transmission configuration between GBS may lead to serious interference from the HAPS node to GBS. HAPS configure downlink transmission with low transmission power P on LP-ABS in order to reduce unnecessary interference and utilize spectrum resources efficiently.

In each transmission cycle, the proportions of downlink normal frames, uplink normal subframe and LP-ABS is denoted as ${\mathsf{\Phi }}^{n,d}$, ${\mathsf{\Phi }}^{n,u}$, and ${\mathsf{\Phi }}^l$. Each configuration $\left({\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^l\right)$ corresponds to a transmission protocol configuration.

2.3. Spectrum Efficiency Analysis

Reasonable configuration of RB allocation and transmission power for the integrated system within a sub-cycle will be provided under the condition of ensuring the QoS requirements of different traffic flows. The goal of the sub-cycle optimization is to maximize the overall network utility, which is the sum of the utility of all users’ traffic flows in the integrated system. The users’ utility is directly related to their data rate.

The downlink and uplink SINR of the system can be expressed as:

$${\gamma }_i^{n,d}=\frac{P_{B_i}\cdot h_{i{B}_i}}{{\displaystyle {\sum }_{k\in B\backslash \left\{B_i\right\}}{P}_k^d{h}_{k,B_k}+}{\sigma }^2};$$

(1)

$${\gamma }_i^{l,d}=\frac{P_i^{l,d}\cdot h_{i, B_i}}{{\displaystyle {\sum }_{k\in B\backslash \left\{B_i\right\}}{P}_k^d{h}_{k,B_k}}+{\sigma }^2};$$

(2)

$${\gamma }_i^{n,u}=\frac{P_i^u\cdot h_{i, B_i}}{{\displaystyle {\sum }_{k\in B\backslash \left\{B_i\right\}}{I}_{k,B_k}+{\sigma }^2}};$$

(3)

$${\gamma }_i^{l,u}=\frac{P_i^{l,u}\cdot h_{i, B_i}}{{\displaystyle {\sum }_{k\in B\backslash \left\{B_i\right\}}{I}_{k,B_k}+}{\sigma }^2};$$

(4)

where ${\sigma }^2$ represents noise power, $h_{i,B_i}$ is channel fading from user $i$ to BS $B_i$:

$$B_i=\underset{j\in B}{\arg \max }\{P_j{h}_{i,j}+\beta \};$$

(5)

Due to the limitation of user transmission power, the uplink transmission power is

$$P_i^u=\min \{\gamma {\sigma }^2/h_{i,B_i}, P_{i,\max }^u\};$$

(6)

Since the system adopts a dynamic TDD mode, the frequency bands for both the HAPS layer and IMT layer are the same. The interference of uplink transmission depends on the users scheduled together. Therefore, the uplink interferences from the system are different in each time slot. In this paper, average interference is adopted instead of real-time interference in order to avoid the complicated calculation caused by scheduling. The average interference is calculated as follows.

$$I_{k,j}=\left({\displaystyle {\sum }_{i\in N_k^u}{P}_i^u\cdot h_{i, j}}\right)/|{\mathsf{\Gamma }}_k^u|$$

(7)

where $j$ is the number of the interfered cell and $k$ is the number of the interfering cell. $\left|{\mathsf{\Gamma }}_k^u\right|$ is the number of the uplink only users in the interfering cell. The data rate of the user is:

$$c_i^d={\log }_2(1+{\gamma }_i^d)$$

(8)

$$c_i^u={\log }_2(1+{\gamma }_i^u)$$

(9)

where $c_i^d$ represents the data rate of downlink only User $i$, $c_i^u$ represents the data rate of the uplink only user $i$.

2.4. Transmission Protocol

This paper assumes that each user will generate one downlink flow but will generate uplink flow on a probabilistic basis. Services generated by different users are independent of each other. ${\mathsf{\Gamma }}^d$ and ${\mathsf{\Gamma }}^u$ represent the downlink user (data flow) set and single uplink user (data flow) set in integrated networks, respectively. The data flow is temporarily stored in the buffer and scheduled by the system according to the resource status.

The eMBB and mMTC services studied in this paper have different QoS requirements. The eMBB service requires the guarantee of soft QoS, which means the eMBB service can tolerate a specific QoS reduction. However, the data flow cannot be available once the data rate decreases beyond a necessary threshold. mMTC service isn’t sensitive to transmission delay and requires limited system resources for its transmission. In order to make the relationship between the QoS and data rate, we adopt the method of constructing utility function [1,2,3,4]. This paper adopts the definition of the utility function in [5].

$$U^S(R)=\left\{\begin{array}{l}(1-p_1)e^{q_1(R-R_{th})},R<R_{th}\\ 1-p_1{e}^{-q_1(R-R_{th})},R\ge R_{th}\end{array}\right.$$

(10)

$$U^B(R)=P_2(1-e^{-q_{2}R}),R\ge 0$$

(11)

Parameters $p_1$, $q_1$, $p_2$, $q_2$ are used to characterize the slope of the utility function. Symbol $R$ indicates the data rate for soft QoS service. The symbol $R_{th}$ is a threshold of data rate. QoS requirements can be met effectively for the best-effort service, as long as some RBs are allocated for the flow.

3. Transmission Protocol Configuration and Resource Allocation Scheme

Firstly, this paper determines the configuration of the transmission protocol by maximizing the expected throughput of the system. Secondly, for diversified services, this paper introduces utility functions to characterize the satisfaction of services. Finally, the traffic flow resource allocation problem in the integrated system is modelled as a utility maximization problem, which can be decomposed into an independent cell utility maximization problem. By optimizing the transmission power on LP-ABS and the resource allocation of each traffic flow, the network utility is maximized under resource constraints. Figure 3 shows the flow and relationships between the different problems and subproblems of the algorithm design.

3.1. Transmission Protocol Configuration

To realize the practical and high-quality transmission of the diversified services carried by a system, according to the characteristics of integrated networks, the overall performance optimization problem of the system is divided into two stages: long-period and sub-period optimization stages. Considering the whole system performance, the long-period optimization goal is to maximize the expected throughput of the system. Then the frame structure of the TDD transmission mechanism can be figured out, that is, the configuration ratio of normal subframe and LP-ABS. The normal subframe is divided into uplink time slots and downlink time slots configurations. Then, from the perspective of cell performance optimization, the goal is to maximize the cell utility function based on the QoS requirements of different traffic flows and the limitations of system resources. The objective of sub-period optimization is to maximize the utility of the system.

Due to many users and various service types in an integrated network, the proportion of uplink and downlink data flow of the system is relatively stable when many traffic flows are generated. Therefore, the strategy for setting the frame structure of integrated networks in this paper is that: (1) the proportion of frames consumed by the uplink is equal to the system resource requirement of the uplink traffic flow; (2) the proportion of frames consumed by the downlink is equal to the system resource requirement of the downlink data flow. On the one hand, the resource demand of traffic flow depends on the number of services. The more the traffic flows, the more system resource is needed. On the other hand, it depends on the data rate.

Therefore, the transmission subframe configured by the system can be expressed as:

$$\frac{{\mathsf{\Phi }}^{n,u}}{{\mathsf{\Phi }}^{n,d}+\frac{{\displaystyle \sum _{i\in {\mathsf{\Gamma }}^d}{c}_i^{l,d}}}{{\displaystyle \sum _{i\in {\mathsf{\Gamma }}^d}{c}_i^{n,d}}}{\mathsf{\Phi }}^l}=\frac{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^u}\frac{1}{c_i^{n,u}}}}{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^d}\frac{1}{c_i^{n,d}}}}$$

(12)

$${\mathsf{\Phi }}^{n,d}+{\mathsf{\Phi }}^{n,u}+{\mathsf{\Phi }}^l=1$$

(13)

Let $X={\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^d}\frac{c_i^{l,d}}{c_i^{n,d}\left|M_0^d\right|}}$, $Y=\frac{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^d}\frac{1}{c_i^{n,u}}}}{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^u}\frac{1}{c_i^{n,d}}}}$, from Equations (12) and (13):

$${\mathsf{\Phi }}^{n,d}=\frac{1+(X-1){\mathsf{\Phi }}^l}{1+Y}$$

(14)

$${\mathsf{\Phi }}^{n,u}=\frac{Y-(X+Y){\mathsf{\Phi }}^l}{1+Y}$$

(15)

The expected throughput of an integrated network is:

$$C=\frac{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^d}{c}_i^{n,d}}}{\left|{\mathsf{\Gamma }}^d\right|}{\mathsf{\Phi }}^{n,d}+\frac{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^u}{c}_i^{n,u}}}{\left|{\mathsf{\Gamma }}^u\right|}{\mathsf{\Phi }}^{n,u}+\frac{{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^d}{c}_i^{l,d}+{\displaystyle {\sum }_{i\in {\mathsf{\Gamma }}^u}{c}_i^{l,u}}}}{\left|{\mathsf{\Gamma }}^d\right|+\left|{\mathsf{\Gamma }}^u\right|}{\mathsf{\Phi }}^l$$

(16)

By maximizing the expected throughput of the system, ${\mathsf{\Phi }}^l$ can be determined as

$${\mathsf{\Phi }}^l=\underset{{\mathsf{\Phi }}^l\in \left\{0,0.1,\dots ,1\right\}}{\arg \max }{\displaystyle \sum _{j\in {\rm B}}{C}_j\left({\mathsf{\Phi }}^l\right)}$$

(17)

Then, by bringing Equations (14) and (15) into (17), the expected throughput of the system can be obtained.

The transmission power $P_0^l$ of the HAPS on the LP-ABS is a one-dimensional discrete variable. Therefore, we consider fixed power $P_0^l$ to eliminate the coupling relationship firstly. When $P_0^l$ is given, the specific frame allocation strategy $\left({\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^l\right)$ is determined by Equation (16). Then, the network utility maximization problem is an optimization problem about resource allocation variable $s_i$, which can be further decomposed into $N+1$ independent sub-problems. Each subproblem is the maximization of the utility of the cell, where the utility of the cell is defined as the sum of the utility of all the business flows in the cell. Each base station distributes resources to maximize cell utility independently so as to realize network utility maximization.

The transmission power $P_0^l$ of HAPS on LP-ABS can be determined by the following two methods. The first approach is that the expected accessible throughput of HAPS and GBS is related to $P_0^l$ and ${\mathsf{\Phi }}^l$. $P_0^l$ and ${\mathsf{\Phi }}^l$ can be determined by maximizing expected throughput, i.e.,

$$\left\{{\mathsf{\Phi }}^l,P_0^l\right\}=\arg \max {\displaystyle \sum _{j\in {\rm B}}{C}_j\left({\mathsf{\Phi }}^l,P_0^l\right)}$$

(18)

Another method is to solve the corresponding cell utility maximization problem under the condition of a given power $P_0^l$. By continuously updating the utility value of the cell between HAPS and GBS, the power $P_0^l$ is updated to determine the optimal solution to maximize the network utility. The determination of power $P_0^l$ in this resource allocation scheme considers specific traffic flow allocation, so the network utility may be higher than that determined in the first approach. However, the second method will bring some additional signaling overhead. Therefore, this paper adopts the first method to determine $P_0^l$.

3.2. Traffic Flow Resource Allocation Algorithm

3.2.1. Maximization of Network Utility

In this paper, through optimizing the transmission power $P_0^l$ of HAPS on LP-ABS and the resource allocation $s_i$ of each traffic flow, the network utility can be maximized under the condition of resource quantity constraint. Therefore, the resource allocation problem of user traffic flows can be expressed mathematically as

$$\max \{{\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^d\right|}{U}_i^d{(R}_i^d)}+{\displaystyle \sum _{k=1}^{\left|{\mathsf{\Gamma }}^u\right|}{U}_k^u(R_k^u)}\}$$

(19)

The constraints are

$$\begin{array}{l}{R}_i^d={\alpha }_i^{n,d}{c}_i^{n,d}+{\alpha }_i^{l,d}{c}_i^{l,d},\forall i\in {\mathsf{\Gamma }}^d\\ R_i^u={\alpha }_i^{n,u}{c}_i^{n,u}+{\alpha }_i^{l,u}{c}_i^{l,u},\forall i\in {\mathsf{\Gamma }}^u\\ {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^d\right|}{\alpha }_i^{n,d}}\le {\mathsf{\Phi }}^{n,d};\\ {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^u\right|}{\alpha }_i^{n,u}}\le {\mathsf{\Phi }}^{n,u};\\ {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^d\right|}{\alpha }_i^{l,d}}+{\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^u\right|}{\alpha }_i^{l,u}}\le {\mathsf{\Phi }}^l;\\ {\mathsf{\Phi }}^{n,d}+{\mathsf{\Phi }}^{n,u}+{\mathsf{\Phi }}^l=1;\\ {\mathsf{\Phi }}^{n,d}+{\mathsf{\Phi }}^{n,u}+{\mathsf{\Phi }}^l=1;\end{array}$$

(20)

where $U({\alpha }_i^d{c}_i^d)$ and $U({\alpha }_i^u{c}_i^u)$ correspond to the utility functions of user i downlink and uplink traffic flow, respectively. $U({\alpha }_i^d{c}_i^d)$ represents the utility value of user i’s downlink traffic flow when the reachability rate is ${\alpha }_i^d{c}_i^d$, and $U({\alpha }_i^u{c}_i^u)$ represents the utility value of user i’s uplink traffic flow when the reachability rate is ${\alpha }_i^u{c}_i^u$. HAPS configures downlink transmission on LP-ABS, so ${\alpha }_i^{l,u}=0$. In addition, if user i does not generate downlink traffic flow, ${\alpha }_i^{n,d}={\alpha }_i^{l,d}=0$. If the user i does not generate uplink traffic flow, then ${\alpha }_i^{n,u}={\alpha }_i^{l,u}=0$.

The specific form of the utility function is determined by the service type of the corresponding traffic flow, so the objective function of the optimization problem is the sum function of multiple S-type functions and multiple concave functions, that is, a non-concave function. Maximizing network utility is a non-concave problem that is difficult to solve directly. In addition, the resource allocation result $s_i$ of the traffic flow generated by user i is affected by the transmission protocol configuration $\left({\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^l\right)$, and $\left({\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^l\right)$ is affected by the power $P_0^l$. Therefore, there is a coupling relationship between ${\alpha }_i$ and $P_0^l$, which makes maximizing network utility more challenging to solve.

3.2.2. Maximization of Cell Utility

The problem of maximizing the utility of a cell (Problem 1) can be expressed as

$$\mathrm{Maximize} {\displaystyle \sum _{i\in M_j^d}{U}_i^d}\left({\alpha }_i^{n,d}{c}_i^{n,d}+{\alpha }_i^{l,d}{c}_i^{l,d}\right)+{\displaystyle \sum _{k\in M_j^d}{U}_k^u}\left({\alpha }_k^{n,u}{c}_k^{n,u}+{\alpha }_k^{l,u}{c}_k^{l,u}\right)$$

(21)

$$\begin{array}{ll}& {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^d\right|}{\alpha }_i^{n,d}}\le {\mathsf{\Phi }}^{n,d};\\ & {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^u\right|}{\alpha }_i^{n,u}}\le {\mathsf{\Phi }}^{n,u};\\ & {\displaystyle \sum _{i=1}^{\left|{\mathsf{\Gamma }}^d\right|}{\alpha }_i^{l,d}}+{\displaystyle \sum _{k=1}^{\left|{\mathsf{\Gamma }}^u\right|}{\alpha }_k^{l.u}}\le {\mathsf{\Phi }}^l;\\ \mathrm{subject} \mathrm{to} & 0\le {\alpha }_i^{l,d},{\alpha }_k^{l.u},{\alpha }_i^{n,d},{\alpha }_k^{n,u},\forall i\in M_j^d,\forall k\in M_k^u\end{array}$$

(22)

The optimization variable is ${\alpha }_i$. If the user does not generate downlink traffic flow, ${\alpha }_i^{n,d}={\alpha }_i^{l,d}=0$. If no uplink traffic flows are generated, ${\alpha }_i^{n,u}={\alpha }_i^{l,u}=0$. To solve the network utility maximization problem, the power $P_0^l$ should be determined by Formula 18, and the cell utility maximization problem should be solved. The optimal solutions of the N + 1 cell utility maximization problem constitute the optimal solution of the network utility maximization problem.

This paper focuses on the solution of the cell utility maximization problem. For the nonconvex optimization problem, the KKT condition is a necessary condition for its optimal solution. The optimal solution of problem P1 must be in the solution set satisfying its KKT condition. In this paper, the KKT condition is used to reduce the feasible solution of problem P1 and search for the solution of the objective function of the maximization problem. Construct the Lagrange function of problem P1 as follows

$$\begin{array}{l}L\left(S_j,\lambda ,\beta ,\gamma ,\nu \right)=-{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_j^d}{U}_i^d\left(R_i^d\right)}-{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^u}{U}_k^u\left(R_k^u\right)+\lambda }\left({\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^d}{\alpha }_i^{n,d}-{\mathsf{\Phi }}^{n,d}}\right)\\ +\beta \left({\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^u}{\alpha }_k^{n,u}-{\mathsf{\Phi }}^{n,u}}\right)+\gamma \left({\displaystyle \sum _{i\in {\mathsf{\Gamma }}_j^d}{\alpha }_i^{l,d}+{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_j^u}{\alpha }_i^{l,u}-{\mathsf{\Phi }}^l}}\right)+\\ {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_j^d}\left({\alpha }_i^{l,d}{v}_i^{l,d}+{\alpha }_i^{n,d}{v}_i^{n,d}\right)+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^u}\left({\alpha }_k^{l,u}{v}_k^{l,u}+{\alpha }_k^{n,u}{v}_k^{n,u}\right)}}\end{array}$$

(23)

where, $\lambda \ge 0$, $\beta \ge 0$, $\gamma \ge 0$ and $\nu =\left\{{\nu }_i^{x,y}\right\}$ correspond to the Lagrange multiplier of the constraint of problem P1, respectively. KKT condition of the cell utility maximization problem can be expressed as:

$$\begin{array}{l}\frac{\partial L\left(S,\lambda ,\beta ,\gamma ,\nu \right)}{\partial {\alpha }_i^{x,y}}=0,\forall i\in {\mathsf{\Gamma }}_j^y\\ {\alpha }_i^{x,y}{v}_i^{x,y}=0,\forall i\in {\mathsf{\Gamma }}_j^y\\ \lambda \left({\displaystyle \sum _{i\in {\mathsf{\Gamma }}_j^d}{\alpha }_i^{n,d}}-{\mathsf{\Phi }}^{n,d}\right)=0\\ \beta \left({\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^u}{\alpha }_k^{n,u}}-{\mathsf{\Phi }}^{n.u}\right)=0\\ \gamma \left({\displaystyle \sum _{i\in {\mathsf{\Gamma }}_i^d}{\alpha }_i^{l,d}+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_j^u}{\alpha }_k^{l,u}}-{\mathsf{\Phi }}^l}\right)=0\end{array}$$

(24)

Based on the above KKT conditions, we can obtain the following lemma:

Lemma 1.

For any solution satisfying the above KKT conditions, when $\frac{c_i^{n,d}}{c_i^{l,d}}>\frac{\lambda }{\gamma }$, ${\alpha }_i^{n,d}>0$, ${\alpha }_i^{l,d}=0$, there exist specific Lagrangian multipliers $\lambda$ and $\gamma$ for user i that obtains downlink transmission service, meeting when $\frac{c_i^{n,d}}{c_i^{l,d}}<\frac{\lambda }{\gamma }$, ${\alpha }_i^{n,d}=0$, ${\alpha }_i^{l,d}>0$.

It is proven that by introducing Equation (23) into Equation (24), the following equations can be obtained:

$$\left\{\begin{array}{c}-\frac{\partial U_i^d}{\partial R_i^d}\cdot c_i^{n,d}+\lambda -v_i^{n,d}=0\\ -\frac{\partial U_i^d}{\partial R_i^d}\cdot c_i^{l,d}+\lambda -v_i^{l,d}=0\end{array}\right.$$

(25)

Further, we can get $\frac{c_i^{n,d}}{c_i^{l,d}}=\frac{\lambda -{\nu }_i^{n,d}}{\gamma -{\nu }_i^{l,d}}$. According to Equation (24), ${\nu }_i^{n,d}=0$ is equivalent to ${\alpha }_i^{n,d}\ge 0$, and ${\nu }_i^{n,d}>0$ is equivalent to ${\alpha }_i^{n,d}=0$. Similarly, ${\nu }_i^{l,d}=0$ is equivalent to ${\alpha }_i^{l,d}>0$, and ${\nu }_i^{l,d}>0$ is equivalent to ${\alpha }_i^{l,d}=0$. For user i that obtains the downlink transmission service, its downlink throughput $R_i^d=c_i^{n,d}{\alpha }_i^{n,d}+c_i^{l,d}{\alpha }_i^{l,d}>0$. Therefore, at least one of ${\alpha }_i^{n,d}$ and ${\alpha }_i^{l,d}$ is greater than 0, which means ${\nu }_i^{l,d}$ and ${\nu }_i^{n,d}$ cannot be greater than 0 at the same time. When $\frac{c_i^{n,d}}{c_i^{l,d}}>\frac{\lambda }{\gamma }$, ${\nu }_i^{n,d}>0$, ${\nu }_i^{l,d}=0$, that is, ${\alpha }_i^{n,d}\ge 0$, ${\alpha }_i^{l,d}=0$. When $\frac{c_i^{n,d}}{c_i^{l,d}}<\frac{\lambda }{\gamma }$, ${\nu }_i^{n,d}=0$, ${\nu }_i^{l,d}>0$, that is, $s_i^{n,d}=0$, $s_i^{l,d}>0$. According to this lemma, for the solution satisfying the KKT condition, if the frequency spectrum efficiency of user i’s down scheduling on the normal frame and LP-ABS of the base station meets $c_i^{n,d}>\left(\lambda /\gamma \right)c_i^{l,d}$, the user should prioritize scheduling on the normal frame. Otherwise, the user should give priority to scheduling on the LP-ABS. Similarly, the lemma related to user uplink scheduling can be obtained as follows.

Lemma 2.

For any solution satisfying the above KKT conditions, there are specific Lagrange multipliers $\beta$ and $\gamma$, such that the user i obtains the uplink transmission service, when$\frac{c_i^{n,u}}{c_i^{l,u}}>\frac{\beta }{\gamma }$, ${\alpha }_i^{n,u}>0$, ${\alpha }_i^{l,u}=0$. when $\frac{c_i^{n,u}}{c_i^{l,u}}<\frac{\beta }{\gamma }$, ${\alpha }_i^{n,u}=0$, ${\alpha }_i^{l,u}>0$.

According to the preceding two lemmas, perform the following operations on downstream and uplink traffic flows. Downlink traffic flows are sorted in descending order by the ratio of the spectral efficiency of the normal subframe to LP-ABS. $d_a$ indicates the sorted traffic flow a. Figure 4 shows the sorted traffic flow. Given set $a\in \left\{1,2,\dots ,\left|{\mathsf{\Gamma }}_j^d\right|-1\right\}$, two business flow sets ${\mathsf{\Gamma }}_{j,1}$ and ${\mathsf{\Gamma }}_{j,2}$ are constructed. Uplink traffic flows are also arranged in descending order by spectral efficiency ratio, as shown in Figure 5.

According to Lemmas 1 and 2, any solution satisfying the KKT condition of the cell utility maximization problem has the following properties. There are specific a and b so that the business flow in sets ${\mathsf{\Gamma }}_{j,1}$ and ${\mathsf{\Gamma }}_{j,3}$ is allocated to the transmission resource from the normal frame of the base station. The business flows in sets ${\mathsf{\Gamma }}_{j,2}$ and ${\mathsf{\Gamma }}_{j,4}$ obtain transport resources from the LP-ABS. Therefore, the cell utility maximization problem can be transformed into the following three steps. Firstly, when a and b are fixed, and the cell utility maximization problem can be decomposed into three subproblems of the same structure. Secondly, according to the properties of the subproblems obtained after decomposition, the P1 problem was solved. Finally, the a and b values of the maximization problem P1 and the optimal solution of problem P1 are traversed.

First, consider step 1. Under the condition of fixing a and b, problem P1 can be decomposed into the following three independent subproblems:

$$\mathrm{Sub}\text{-}\mathrm{problem} 1: \mathrm{maximize} {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_{j,1}}{U}_i^d\left({\alpha }_i^{n,d}{c}_i^{n,d}\right)}$$

(26)

$$\begin{array}{ll}& {\displaystyle \sum _{ i\in {\mathsf{\Gamma }}_{j,1}}{\alpha }_i^{n,d}\le {\mathsf{\Phi }}^{n,d}}\\ \mathrm{subject} \mathrm{to} & 0\le {\alpha }_i^{n,d},\forall i\in {\mathsf{\Gamma }}_{j,1}\end{array}$$

$$\mathrm{Sub}\text{-}\mathrm{problem} 2: \mathrm{maximize} {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_{j,1}}{U}_i^u\left({\alpha }_i^{n,u}{c}_i^{n,u}\right)}$$

$$\begin{array}{ll}& {\displaystyle \sum _{ i\in {\mathsf{\Gamma }}_{j,1}}{\alpha }_i^{n,u}\le {\mathsf{\Phi }}^{n,u}}\\ \mathrm{subject} \mathrm{to}& 0\le {\alpha }_i^{n,u},\forall i\in {\mathsf{\Gamma }}_{j,3}\end{array}$$

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Sub}\text{-}\mathrm{problem} 3: \mathrm{maximize} \\ {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_{j,2}}{U}_i^d\left({\alpha }_i^{l,d}{c}_i^{l,d}\right)}+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_{j,4}}{U}_k^u\left({\alpha }_k^{l,u}{c}_k^{l,u}\right)}\end{array}$$

$$\begin{array}{ll}& {\displaystyle \sum _{ i\in {\mathsf{\Gamma }}_{j,2}}{\alpha }_i^{l,d}+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_{j,4}}{\alpha }_k^{l,u}\le {\mathsf{\Phi }}^l}}\\ \mathrm{subject} \mathrm{to} & 0\le {\alpha }_i^{l,d},{\alpha }_k^{l,d},\forall i\in {\mathsf{\Gamma }}_{j,2},\forall i\in {\mathsf{\Gamma }}_{j,4}\end{array}$$

The optimization variables involved in each sub-problem are ${\alpha }^{n,d}=\left\{{\alpha }_i^{n,d},i\in {\mathsf{\Gamma }}_{j,1}\right\}$, ${\alpha }^{n,u}=\left\{{\alpha }_i^{n,u},i\in {\mathsf{\Gamma }}_{j,3}\right\}$, ${\alpha }^{l,u}=\left\{{\alpha }_i^{l,u},i\in {\mathsf{\Gamma }}_{j,4}\right\}$, and ${\alpha }^{l,d}=\left\{{\alpha }_i^{l,d},i\in {\mathsf{\Gamma }}_{j,3}\right\}$. To be corresponding to the above three sub-problems. Normal downlink frames of base station j are allocated to business flows in set ${\mathsf{\Gamma }}_{j,1}$ to maximize the utility sum of business flows, normal up-link frames of base station j are allocated to business flows inset ${\mathsf{\Gamma }}_{j,3}$ to maximize the utility sum of business flows, and LP-ABS of base station j are allocated to sets ${\mathsf{\Gamma }}_{j,2}$ and ${\mathsf{\Gamma }}_{j,4}$ to maximize the utility and sum of business flows. The decomposition process above is independent of a particular type of business flow type.

Secondly, considering step 2, the three subproblems obtained after decomposition have the same structure, so we focus on solving subproblem 1. For the user business flows in the collection, some are soft QoS business flows, and others are best-effort service business flows. As shown in Figure 6, the business flows in ${\mathsf{\Gamma }}_{j,1}$ are divided into two sets, one is the set of soft QoS business flows, and the other is the set of best-effort service business flows. In ${\mathsf{\Gamma }}_{j,1}^Q$ and ${\mathsf{\Gamma }}_{j,1}^B$, traffic flows are sorted in descending order of spectral efficiency as shown in Figure 5.

Sub-problem 1 can be expressed equivalently in the following form P2. It should be noted that P2 is formulated as an equivalent to facilitate solving sub-problem 1. A similar approach can be applied for solving the other two sub-problems, sub-problem 2 and sub-problem 3.

$$\mathrm{Maximize}{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_{j,1}^Q}{U}_i^Q\left({\alpha }_i^{n,d}{c}_i^{n,d}\right)+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_{j,1}^B}{U}_k^B\left({\alpha }_k^{n,d}{c}_k^{n,d}\right)}}$$

(27)

$$\begin{array}{ll}& {\displaystyle \sum _{ i\in {\mathsf{\Gamma }}_{j,1}}{\alpha }_i^{n,d}\le }{\mathsf{\Phi }}^{n,d}\\ \mathrm{Subject} \mathrm{to} & 0\le {\alpha }_i^{n,d},\forall i\in {\mathsf{\Gamma }}_{j,1}\end{array}$$

where the optimization variable is ${\alpha }^{n,d}=\left\{{\alpha }_i^{n,d},i\in {\mathsf{\Gamma }}_{j,1}\right\}$. $U_i^Q$ represents the utility of soft QoS traffic flow i, and $U_k^B$ represents the utility of best-effort service traffic flow k. For best-effort traffic flow k, its marginal utility function can be expressed as

$$U_k^B\left(R\right)=p_2{q}_2{e}^{-q_{2}R},R\ge 0$$

(28)

Equation (28) is a subtractive function of the reachable rate R. For soft QoS traffic flow, marginal utility function can be expressed as

$$U_i^Q\left(R\right)=\left\{\begin{array}{c}\left(1-p_1\right)q_1{e}^{q_1\left(R-R_{th}\right)},R<R_{th}\\ p_1{q}_1{e}^{-q_1\left(R-R_{th}\right)},R\ge R_{th}\end{array}\right.$$

(29)

When $R<R_{th}$, $U_i^Q\left(R\right)$ is an increasing function of the reachable rate $R$. And when $R>R_{th}$, $U_i^Q\left(R\right)$ is a decreasing function of the reachable rate R. For problem P2, the necessary conditions for the optimal solution are given by the following lemma.

Lemma 3.

For the optimal solution $s^{n,d}$ of problem P2, there is a Lagrange multiplier $\lambda >0$, so that.

$$\begin{array}{l}{c}_i^{n,d}{u}_i^Q\left(c_i^{n,d}{\alpha }_i^{n,d}\right)=c_k^{n,d}{u}_k^B\left(c_k^{n,d}{\alpha }_k^{n,d}\right)=\lambda ,\forall {\alpha }_i^{n,d},{\alpha }_k^{n,d}>0\\ {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_{j,1}}{\alpha }_i^{n,d}=}{\mathsf{\Phi }}^{n,d}\end{array}$$

(30)

At most, there is only one traffic flow that meets $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$. This traffic flow must be the last soft QoS traffic flow in ${\mathsf{\Gamma }}_{j,1}^Q$ after sorting to obtain transmission resources. It is proven that if ${\alpha }^{n,d}$ is the optimal solution of problem P2, ${\alpha }^{n,d}$ satisfies the KKT condition of problem P2, that is, Equation (30).

For any best-effort service business flow k, $d{u}_i\left(R_i^d\right)/d{R}_i^d<0$ is always true. Therefore, only soft QoS traffic flow will meet $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$. It is assumed that at least two soft QoS traffic flows meet the above conditions, in which traffic flows i and h meet $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$ and $d{u}_h\left(R_h^d\right)/d{R}_h^d>0$, respectively. Considering another resource allocation scheme ${\stackrel{⌢}{\alpha }}^{n,d}$, the resources allocated to traffic flows i and h are ${\alpha }_i^{n,d}-\Delta \alpha$ and ${\alpha }_h^{n,d}+\Delta \alpha$ respectively, and the resource allocation results of other traffic flows remain unchanged. Since the utility function is continuous and monotonically increasing, the difference between the objective function value of problem P2 under resource allocation ${\stackrel{⌢}{\alpha }}^{n,d}$ and resource allocation ${\alpha }^{n,d}$ is

$$\begin{array}{l}\left[U_i^Q\left(c_i^{n,d}\left({\alpha }_i^{n,d}-\Delta \alpha \right)\right)+U_h^Q\left(c_h^{n,d}\left({\alpha }_h^{n,d}+\Delta \alpha \right)\right)\right]-\left[U_i^Q\left(c_i^{n,d}{\alpha }_i^{n,d}\right)+U_h^Q\left(c_h^{n,d}{\alpha }_h^{n,d}\right)\right]\\ ={\displaystyle \underset{{\alpha }_h^{n,d}}{\overset{{\alpha }_h^{n,d}+\Delta \alpha }{\int }}{c}_h^{n,d}{u}_h^Q\left(c_h^{n,d}\alpha \right)}d\alpha -{\displaystyle \underset{{\alpha }_i^{n,d}-\Delta \alpha }{\overset{{\alpha }_i^{n,d}}{\int }}{c}_i^{n,d}{u}_i^Q\left(c_i^{n,d}\alpha \right)}d\alpha \\ >c_h^{n,d}{u}_h^Q\left(c_h^{n,d}{\alpha }_h^{n,d}\right)-c_i^{n,d}{u}_i^Q\left(c_i^{n,d}{\alpha }_i^{n,d}\right)=0\end{array}$$

(31)

This contradicts that ${\alpha }^{n,d}$ is the optimal solution of problem P2. Therefore, at most, one soft QoS traffic flow satisfies $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$.

The traffic flows in the set ${\mathsf{\Gamma }}_{j,1}^Q$ have been sorted in descending order of spectral efficiency $c_i^{n,d}$. Suppose that for the sorted traffic flows h and i, $h<i\le \left|{\mathsf{\Gamma }}_{j,1}^Q\right|$, ${\alpha }_i^{n,d}={\alpha }^{\prime }>0$, ${\alpha }_h^{n,d}=0$. Consider another resource allocation scheme ${\stackrel{⌢}{\alpha }}^{n,d}$, where ${\alpha }_i^{n,d}={\alpha }^{\prime }$, ${\alpha }_i^{n,d}=0$, and the resource allocation results of other traffic flows remain unchanged. It can be found that the resource allocation scheme ${\stackrel{⌢}{\alpha }}^{n,d}$ can further improve the objective function value of problem P2 compared with ${\alpha }^{n,d}$. This is contrary to the fact that ${\alpha }^{n,d}$ is the optimal solution of problem P2. Therefore, the traffic flow satisfying $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$ must be the last soft QoS traffic flow in the sorted ${\mathsf{\Gamma }}_{j,1}^Q$ to obtain transmission resources.

For a soft QoS traffic flow i, $d{u}_i\left(R_i^d\right)/d{R}_i^d>0$ is equivalent to $R_i^d<R_{th}$. Lemma 3 shows that in the optimal solution to problem P2, at most, one soft QoS traffic flow satisfies $R_i^d<R_{th}$, and must be the last soft QoS traffic flow to obtain resources in the set $M_{j,1}^Q$. Therefore, the optimal solution of problem P2 can be solved by assuming that the first m business flows in the set $M_{j,1}^Q$ can be allocated to resources, and the corresponding $\lambda$ value can be calculated. At the same time, m is traversed in the set $\left\{1,2,\dots ,\left|{\mathsf{\Gamma }}_{j,1}^Q\right|\right\}$ to find the optimal solution to the problem. Assuming that m soft QoS traffic streams can obtain services from base station j, there are the following two types of situations:

Scenario 1 indicates that there is no business flow satisfying $R_i^d<R_{th}$, that is, $R_m^d\ge R_{th}$. Based on Lemma 3, the following equation $\lambda$ can be obtained:

$${\displaystyle \sum _{i=1}^m\left[\stackrel{\wedge }{u_i}{\left(\lambda /c_i^{n,d}\right)}^{-1}+R_{th}\right]}/c_i^{n,d}+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_{j,1}^B}{u}_k^B{\left(\lambda /c_k^{n,d}\right)}^{-1}/c_k^{n,d}=}{\mathsf{\Phi }}^{n,d}$$

(32)

Let ${\lambda }_1$ be the solution of the equation, then the optimal solution of problem P2 is:

$${\alpha }_i^{n,d}=\left\{\begin{array}{c}\left[\stackrel{\wedge }{u_i}{\left({\lambda }_1/c_i^{n,d}\right)}^{-1}+R_{th}\right]/c_i^{n,d},i=1,2,\dots ,m\\ 0,i=m+1,\dots ,\left|{\mathsf{\Gamma }}_{j,1}^Q\right|\\ u_k^B{\left({\lambda }_1/c_k^{n,d}\right)}^{-1}/c_k^{n,d},i=\left|{\mathsf{\Gamma }}_{j,1}^Q\right|+1,\dots ,\left|{\mathsf{\Gamma }}_{j,1}\right|\end{array}\right.$$

(33)

Scenario 2 indicates that there is a business flow that satisfies $R_i^d<R_{th}$, that is, $R_m^d<R_{th}$. Accordingly, the equation about $\lambda$ can be obtained as

$${\displaystyle \sum _{i=1}^{m-1}\left[\stackrel{\wedge }{u_i}{\left(\lambda /c_i^{n,d}\right)}^{-1}+R_{th}\right]}/c_i^{n,d}+u_i^Q{\left(\lambda /c_m^{n,d}\right)}^{-1}/c_m^{n,d}+{\displaystyle \sum _{k\in {\mathsf{\Gamma }}_{j,1}^B}{u}_k^B{\left(\lambda /c_k^{n,d}\right)}^{-1}/c_k^{n,d}=}{\mathsf{\Phi }}^{n,d}$$

(34)

Let ${\lambda }_2$ be the solution of the equation, then the optimal solution of problem P2 is:

$${\alpha }_i^{n,d}=\left\{\begin{array}{c}\left[\stackrel{\wedge }{u_i}{\left({\lambda }_2/c_i^{n,d}\right)}^{-1}+R_{th}\right]/c_i^{n,d},i=1,2,\dots ,m-1\\ \begin{array}{c}{u}_m^Q{\left({\lambda }_2/c_m^{n,d}\right)}^{-1}/c_m^{n,d},i=m\\ 0,i=m+1,\dots ,\left|{\mathsf{\Gamma }}_{j,1}^Q\right|\end{array}\\ u_i^Q{\left({\lambda }_2/c_i^{n,d}\right)}^{-1}/c_i^{n,d},i=\left|{\mathsf{\Gamma }}_{j,1}^Q\right|+1,\dots ,\left|{\mathsf{\Gamma }}_{j,1}\right|\end{array}\right.$$

(35)

Bring Equations (33) and (35) into the objective function of the P2 problem and compare the value of the obtained objective function to obtain the solution of problem P2 under the given m value, that is, to complete the optimal solution of problem P2. Subproblems 2 and 3 decomposed by problem P2 have the same structure as subproblem 1. Therefore, only the relevant parameters involved in the solution process need to be replaced, and they can be solved according to the steps of solving problem P2. The parameters involved in the sub-problem-solving process include traffic flow set, spectral efficiency, and total available resources.

Using the results of sub-problems 1, 2 and 3, we can obtain the solution of problem P1 under fixed a and b. Finally, traverse a and b in sets $\left\{1,2,\dots ,\left|{\mathsf{\Gamma }}_j^d\right|-1\right\}$ and $\left\{1,2,\dots ,\left|{\mathsf{\Gamma }}_j^u\right|-1\right\}$ respectively. The optimal solution of $N+1$ cell utility maximization problem constitutes the optimal solution of the network utility maximization problem.

3.2.3. Resource Allocation Algorithm

Through the determination of transmission protocol configuration $\left({\alpha }_i^{n,d},{\alpha }_i^{n,u},{\alpha }^l\right)$ and $P_0^l$, the resource allocation problem of diversified traffic flow can be modeled as a network utility maximization problem, which can be further decomposed into a cell utility maximization problem handled by multiple base stations independently. The cell utility maximization problem corresponding to the base station can reduce its feasible solution through the KKT condition and traverse the possible solution to find the optimal solution.

The steps of solving problem P2 are intuitively expressed in the form of Algorithm 1. Firstly, under the condition of given m, the solution satisfies the necessary condition of lemma 2. Secondly, M is traversed to search for the optimal solution of problem P2.

Algorithm 1. Algorithm for solving problem P2.

1. Initialization: business flow set ${\mathsf{\Gamma }}_{j,1}$, resource ${\mathsf{\Phi }}^{n,d}$, initial value $U=0$ of objective function of problem P2;

2. The traffic flows in ${\mathsf{\Gamma }}_{j,1}$ are divided into soft QoS traffic flow set ${\mathsf{\Gamma }}_{j,1}^Q$ and best effort traffic flow set ${\mathsf{\Gamma }}_{j,1}^B$, which are arranged in descending order of $c_j^{n,d}$ respectively;

3. for m = 1 to $\left|{\mathsf{\Gamma }}_{j,1}^Q\right|$ do 4. Calculate the resource allocation result ${\alpha }_1^{n,d}$ according to Equation (33)

5. Substitute ${\alpha }_1^{n,d}$ into the objective function in problem P2 and compute the objective function, $U_1$

6. Calculate the resource allocation result ${\alpha }_2^{n,d}$ according to Equation (35) 7. Substitute ${\alpha }_2^{n,d}$ into the objective function in problem P2 and compute the objective function, $U_2$ 8. if $U_1>U$ and $U_1>U_2$ then $U=U_1$,${\alpha }^{n,d}={\alpha }_1^{n,d}$;

9. else if $U_2>U$ and $U_2>U_1$ then $U=U_2$,${\alpha }^{n,d}={\alpha }_2^{n,d}$; 10. end if 11. end for 12. Returns the optimal solution ${\alpha }^{n,d}$ and the objective function value U

Through Algorithm 1, solution Algorithm 2 of problem P2 can be designed—the resource allocation algorithm based on utility maximization. The downlink and uplink traffic flow are divided by fixing a and b, and three subproblems with the same structure are constructed. After solving the optimal solution of the subproblem, the a and b are experienced to obtain the optimal solution of problem P1.

Algorithm 2. Utility of base station J based on resource allocation algorithm.

1. Initialization: set ${\mathsf{\Gamma }}_j^d$$, {\mathsf{\Gamma }}_j^u$, initial value $U=0$ of objective function of problem P1, resource $\left({\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^l\right)$;

2. Sort the traffic flows in set ${\mathsf{\Gamma }}_j^d$ in descending order of $c_j^{n,d}/c_j^{l,d}$;

3. Sort the traffic flows in set ${\mathsf{\Gamma }}_j^u$ in descending order of $c_j^{n,u}/c_j^{l,u}$; 4. for $b=1$ to $\left|{\mathsf{\Gamma }}_j^d\right|$ do 5. for $a=1$ to $\left|{\mathsf{\Gamma }}_j^u\right|$ do 6. Update the business flow set ${\mathsf{\Gamma }}_{j,1}$$, {\mathsf{\Gamma }}_{j,2}$$, {\mathsf{\Gamma }}_{j,3}$$, {\mathsf{\Gamma }}_{j,4}$, and construct the corresponding three sub-problems; 7. Algorithm 1 is applied to solve subproblem 1, and the returned optimal solution ${\alpha }^{n,d}$ and objective function value $U_1$ are recorded; 8. Algorithm 1 is applied to solve subproblem 2, and the returned optimal solution ${\alpha }^{n,u}$ and objective function value $U_2$ are recorded; 9. Algorithm 1 is applied to solve subproblem 3, and the returned optimal solution ${\alpha }^l=\left\{{\alpha }^{l,d},{\alpha }^{l,u}\right\}$ and objective function value $U_3$ are recorded; 10. if $U_1+U_2+U_3>U$ then $U=U_1+U_2+U_3$; 11. According to the results of ${\alpha }^{n,d}$, ${\alpha }^{n,u}$ and ${\alpha }^l=\left\{{\alpha }^{l,d},{\alpha }^{l,u}\right\}$, the solution ${\alpha }_j$ of P1 is formed; 12. end if 13. end for 14. Returns the optimal solution ${\beta }_j$ and the objective function value U

4. Simulation Results

This section will evaluate the performance of the proposed scheme through numerical simulation. Three typical interference mitigation mechanisms are used as comparison algorithms, which are eICIC [17], UM-ABS [18] and Synchronous [19]. The performance of the algorithm in this paper and the above three algorithms are compared.

Consider a co-channel TDD heterogeneous cellular network in the simulation. One HAPS is deployed in the center, and several GBS are randomly deployed around the HAPS. In the simulation, each user generates a downlink service flow with probability 1, and an uplink service flow with probability 0.5. The probability that the service flow generated by the user is soft QoS service flow and BE service flow each account for 50%. The target signal-to-noise ratio of the user’s transmit signal is 15 dB, and the user’s maximum transmit power is 23 dBm.

Table 2. Simulation parameters.

Parameters	Value
System Bandwidth	20 MHz
Cell topology of HAPS and GBS	Regular hexagonal cell
Cell radius of HAPS	20 km
Cell radius of GBS	800 m
Number of HAPS	1
Number of GBS	20
Transmit power of HAPS and GBS	46 dBm
Antenna gain	15 dBi
Number of UE per HAPS	80
Number of UE per GBS	20
Path Loss Model of GBS	3GPP 36.942 [20]

presents the detailed simulation parameters of the HAPS-IMT network.

The minimum transmit power of HAPS on LP-ABS is 20 dBm, and the step size is 3 dB. For the utility function in Section 2.4, $p_1=0.2$, $q_1=13$, $p_2=0.4$, and $q_2=13$. The rate requirement of soft QoS service is 3.5 Mbps. In this section, the proposed resource allocation algorithm based on power control and dynamic transmission protocol is named as PCDT. Based on PCDT, the transmission protocol configuration scheme $\left({\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^l\right)$ can be obtained. In the simulation comparison scheme, it is assumed that the transmission protocol configuration of eICIC’s HAPS downlink transmission, uplink transmission and APS is also $\left({\mathsf{\Phi }}^{n,d},{\mathsf{\Phi }}^{n,u},{\mathsf{\Phi }}^l\right)$. However, each GBS and HAPS maintains synchronization on downlink and uplink transmission. Each GBS and HAPS performs dynamic TDD transmission during the ABS cycle. In the UM-ABS algorithm, the ratio of HAPS downlink and uplink transmission frames is ${\mathsf{\Phi }}^{n,d}:({\mathsf{\Phi }}^{n,u}+{\mathsf{\Phi }}^l)$. Each GBS maintains synchronization on the downlink transmission of HAPS and sets dynamic TDD transmission on the uplink transmission of HAPS. In the Synchronous algorithm, HAPS and GBS maintain a synchronized configuration, and the ratio of downlink and uplink transmission frames is $({\mathsf{\Phi }}^{n,d}+{\mathsf{\Phi }}^l):{\mathsf{\Phi }}^{n,u}$.

Figure 7 and Figure 8 evaluate the relationship between the average throughput and load of the downlink and uplink service flows. As shown in Figure 7, the PCDT algorithm can achieve better throughput performance than other algorithms in the downlink. The reason is that the interference mitigation mechanism is adopted in the PCDT algorithm. That is, the downlink transmission resources of HAPS are effectively used through the power control strategy. In addition, it can be seen from Figure 7 that the throughput performance of the eICIC algorithm is higher than that of the synchronization algorithm. This indicates that the performance gain of the ABS frame through interference avoidance is higher than the loss caused by the waste of resources.

It can be seen from Figure 8 that the PCDT algorithm can achieve almost the same throughput performance as the eICIC algorithm in the uplink. This means that the PCDT algorithm proposed in this paper will not introduce additional severe uplink interference in the uplink. The reason is that the PDCT algorithm considers inter-link interference, which can avoid inter-link severe interference when allocating resources for users. In addition, due to the small number of uplink resources of GBS in the synchronization algorithm, the uplink throughput performance obtained by the synchronization algorithm is lower than that of the PCDT algorithm. The UM-ABS algorithm can obtain good throughput performance on the downlink. However, the UM-ABS algorithm does not reserve additional uplink transmission resources for GBS, and it ignores the downlink-to-uplink interference of GBS to HAPS. Therefore, the uplink throughput performance of the UM-ABS algorithm is the lowest among the four algorithms.

Figure 9 and Figure 10 show the relationship curve between the outage probability of soft QoS service flow and the number of UEs. The outage probability is defined as the proportion of service flows whose achievable rate is less than the rate requirement in soft QoS service flows. It can be seen from the figure that the PCDT algorithm can obtain the lowest average outage probability in the downlink. The reason is that the PDCT algorithm tries its best to allocate resources to users so that the number of users meeting QoS can reach the maximum. In the uplink, the PDCT algorithm can achieve performance close to the eICIC algorithm. This is because although PDCT sacrifices some uplink resources, it makes up for them through the LP-ABS strategy. In summary, the PDCT algorithm proposed in this article can effectively improve the average throughput of uplink and downlink services and effectively reduce the outage probability of soft QoS services.

5. Conclusions

This paper proposes a transmission protocol to realize the effective support of dynamic TDD in the HAPS-IMT heterogeneous network and the QoS guarantee of various services. Power control and time slot resource allocation in the proposed transmission protocol are combined through LP-ABS, which provides GBS with higher resource configuration flexibility while mitigating cross-link interference. Further, this paper determines the specific transmission protocol configuration scheme by maximizing the expected throughput of the system. In order to realize the resource allocation of various services under this transmission protocol, a utility function is introduced to describe the degree of satisfaction of the service. Each user’s time slot resource allocation scheme is obtained by solving the problem of maximizing cell utility. The simulation results show that the PCDT algorithm proposed in this paper can improve the average throughput of the system and reduce the outage probability. In future work, we will consider further introducing heterogeneous networks on the ground, such as micro base stations and indoor base stations. Therefore, resource allocation strategies in more complex scenarios need to be studied.