# An Access Selection Algorithm for Heterogeneous Wireless Networks Based on Optimal Resource Allocation

Authors: Gen Liang; Guoxi Sun; Jingcheng Fang; Xiaoxue Guo; Hewei Yu
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

Wireless networks form heterogeneous wireless networks (HWNs) with overlapping signal coverage, and access selection is one of the key technologies of HWNs. Today, most access selection algorithms select appropriate networks mainly from the perspective of users while failing to consider resource allocation and optimize the overall transmission performance of HWNs. This paper proposes an access selection algorithm for HWNs based on optimal resource allocation by analyzing the wireless link transmission rate model, focusing on maximizing the HWN transmission rate, using the dynamic programming theory to derive the optimal value of bandwidth allocated to users. Experimental results show that the algorithm proposed in this paper can effectively improve network throughput and resource utilization and can connect users to the appropriate network according to QoS rate requirements.


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## Body

## 1. Introduction

In recent years, various wireless communication technologies, such as cellular networks, wireless local area networks (WLAN), and wireless metropolitan area networks (WMAN), have achieved rapid development. Within the signal coverage of cellular networks, a variety of other wireless access networks are deployed, forming heterogeneous wireless networks (HWNs) in which multiple networks coexist with overlapping signal coverage [1, 2].Access selection is one of the key technologies of HWNs. Their main function is to control users’ access requests and select a network that provides connection services [3, 4]. For traditional access selection algorithms, the decision parameters are mainly related to the quality of the wireless link (e.g., received signal strength (RSS) or signal to interference plus noise ratio (SINR)). The basic idea of such algorithms is that user terminals are allowed to access the network with the highest RSS and all users in the same network share the network resources relatively equally. While such algorithms are simple, too many users may end up accessing the same wireless network at once, creating insufficiency in the utilization of network resources and low network capacity [5].Additionally, some studies comprehensively consider multiple decision parameters (e.g., RSS, bandwidth, network load, delay, delay jitter, packet loss ratio, movement speed, service price, and energy consumption) and use mathematical models to study the access selection of HWNs, such as multiple attribute decision-making [6, 7], utility theory [8, 9], game theory [10, 11], Markov decision process [12, 13], and fuzzy logic [14, 15]. These algorithms are mainly based on the principle of “always best connected” [16] and comprehensively calculate a plurality of decision parameters to obtain the ranking of comprehensive scores of candidate networks. As a result, a network with the highest score is selected from a group of candidate networks.While the above algorithms solve the network access problem and can select a suitable network for users, they fail to optimize the performance of HWNs, which is not conducive to the effective utilization of the overall network resources. Moreover, they do not calculate the specific resource allocation. In addition, with the increase of the number of decision parameters and users, these algorithms become increasingly complex [17]. Therefore, how to optimize the utilization of wireless resources and improve the overall performance of networks is the research motivation of this paper, along with selecting the appropriate network for users.In this paper, an optimal resource allocation access selection algorithm (ORAAS) is proposed. Firstly, the transmission rate model of a wireless link is analyzed, and the problem is modeled as maximizing the network transmission rate. Then, we use the dynamic programming theory to calculate the optimal value of the bandwidth resources allocated to each user under the maximum network transmission rate. Finally, we connect users to the appropriate network according to the bandwidth resources allocated by the network and the QoS rate requirements of users. The main contributions of this paper are as follows:(1)
Divide the solution of the target problem into several interrelated phases according to the number of users and break down the problem of maximizing the system transmission rate for all users into subproblems of maximizing the system transmission rate for different users(2)
Utilize the bandwidth resource allocation state of the previous user when solving each subproblem in order to calculate the optimal bandwidth resource allocation to the next user, thereby gradually obtaining the optimal bandwidth allocation value for each user at the maximum system transmission rate(3)
Solve the maximum value of the network transmission rate and the optimal value of the user bandwidth resource allocation simultaneously(4)
Improve the network transmission performance under the HWN access selection simulation and optimize the resource utilization efficiency through the proposed algorithmThe rest of this paper is organized as follows: Section2 reviews the research work related to this article, Section 3 describes the system model and problem definition in detail, Section 4 solves the problem and designs the algorithm, Section 5 introduces in detail the experimental environment used for the performance analysis and discusses the experimental results, and Section 6 summarizes the paper and introduces further research work.


## 2. Related Work

According to different optimization objectives, resource allocation in HWNs is currently divided into user-oriented utility optimization and system-oriented utility optimization [18, 19]. User-oriented utility optimization mainly selects parameters related to user QoS, such as RSS, bandwidth, blocking rate, and price, to evaluate the network performance, and allocates resources on the premise of optimizing user utility [20]. System-oriented utility optimization is mainly to maximize the utility of HWNs through resource allocation, traffic flow scheduling, load allocation, and other means in access selection, including maximizing the number of accessing users, maximizing network capacity, and maximizing energy conservation [21].Jiang et al. [22] proposed different delay constraints for different services and designed a new and effective network selection strategy in a HWN communication environment, which incorporates delay in the calculation of transmission rate. Users can access the appropriate network after considering the delay requirements of different services. The algorithm greatly improves the throughput of HWNs and guarantees the QoS.Niyato and Hossain [23, 24] proposed a game theory framework for bandwidth allocation and access control in HWNs. The optimal bandwidth allocation is obtained by calculating the Nash equilibrium of noncooperative games, which maximizes the utility of all connections in the network. Then, based on the obtained bandwidth allocation, the capacity reservation threshold is calculated by a bargaining game to meet the QoS requirements of different types of connections.BenMimoune et al. [25] proposed an access selection and power allocation scheme in HWNs based on the Voronoi diagram to increase the number of access users and system throughput. As a result, this reduces the call blocking rate and improves energy efficiency.Choi et al. [26] studied the interface and frequency band selection of mobile users and the power allocation of selected links in an HWN environment. In addition, the authors analyzed the parallel transmission of multiradio access, proposed a distributed joint allocation algorithm to maximize the total system capacity, and solved the bandwidth and power allocation problems of access links.Wang et al. [27] studied the joint optimization of user connection and resource allocation in HWNs. First, the problem of user association and subchannel allocation for fixed power allocation is derived based on the graph theory, then the power allocation problem in the case of fixed user association and subchannel allocation is derived by using the convex function approximation approach, and finally, the authors proved that the proposed algorithm improves the overall network throughput through the simulation.This paper proposes an access selection algorithm based on optimal resource allocation. The algorithm calculates the optimal value of bandwidth resources allocated to each user at the maximum network transmission rate and connects users to the appropriate network according to the user’s QoS rate requirements. The algorithm improves the network performance and resource utilization.


## 3. System Model and Problem Definition

### 3.1. System Model

Assume that a HWN includesn radio access networks (RAN) with overlapping signal coverage and that there is no interference in wireless signals. Each radio access network has its own architecture, and the networks communicate with each other through core network interconnection [28]. m mobile user terminals are randomly distributed in the signal ranges of HWNs, and the user terminals are multimode terminals capable of processing all radio access technologies. In addition, assume that a user terminal can access one or more wireless networks in parallel at the same time and that the access selection is controlled by the user terminal. The HWN selection scenario in this paper is shown (Figure 1).Figure 1
Heterogeneous wireless network selection scenario.For the convenience of explanation, the main mathematical symbols used in this paper and their descriptions are shown (Table1).Table 1
Main symbols and their descriptions.
SymbolDescriptionnTotal number of RANsmTotal number of mobile usersbi,jBandwidth allocated by Networkj to User iBjTotal bandwidth resources of Networkjri,jTransmission rate of Useri in Network jRi,mSum of transmission rates from Useri to User mpi,jTransmission power of Useri in Network jri,jreq_minMinimum transmission rate requirement of Useri in Network jsiBandwidth resource state assignable to UseribisiWhen resource state is insi, bandwidth allocation decision for User ibi∗siWhen resource state is insi, optimal bandwidth allocation decision for User iDisiA set of bandwidth allocation decisions in Useri under state sipi,msiBandwidth allocation policy from Useri to User mpi,m∗siOptimal bandwidth allocation policy from Useri to User mRi,m∗siMaximum value of sum of transmission rates from Useri to User m


### 3.2. Problem Definition

Assume that Useri is connected to wireless Network j and that the bandwidth resource allocated by User i in Network j is bi,j. According to the Shannon capacity formula [22, 26], the maximum data transmission rate that can be reached by User i in Network j can be defined as
(1)ri,j=βjbi,jlog21+pi,jNobi,j.In Formula (1), βj is the network effectiveness coefficient of Network j, which can be set according to the characteristics of different networks. For example, for Network j with better coding, the value of its βj is higher than that of other networks using traditional coding. In this paper, βj represents the bandwidth efficiency of wireless networks, that is, the bandwidth effectiveness of each network, and 0≤βj≤1. In addition, bi,j indicates the bandwidth resources obtained by User i in Network j, pi,j indicates the transmission power of User i in Network j, and No is the thermal noise power spectral density.In this paper, the user terminal supports multilink parallel access ton networks, and according to Formula (1), the maximum data transmission rate that User i can achieve is
(2)ri=∑j=1nβjbi,jlog21+pi,jNobi,j.Since bandwidth allocation may affect the transmission rate of users, from the perspective of the whole HWNs, it is always expected that all users receive a reasonable bandwidth in each wireless network, enabling HWNs to reach the maximum transmission rate. Therefore, the problem is defined as follows:(3)maxRHWNsb=max∑i=1mri,=max∑j=1n∑i=1mβjbi,jlog21+pi,jNobi,j,s.t.∑i=1mbi,j≤Bj,bi,j≥0,i=1,2,⋯,m,j=1,2,⋯,n.The first constraint condition∑i=1mbi,j≤Bj in Formula (3) represents that the total bandwidth allocated to all users in Network j should be less than or equal to the total bandwidth of Network j. In addition, the second constraint condition bi,j≥0 represents that the bandwidth allocated to users should be greater than or equal to 0.


## 3.1. System Model

Assume that a HWN includesn radio access networks (RAN) with overlapping signal coverage and that there is no interference in wireless signals. Each radio access network has its own architecture, and the networks communicate with each other through core network interconnection [28]. m mobile user terminals are randomly distributed in the signal ranges of HWNs, and the user terminals are multimode terminals capable of processing all radio access technologies. In addition, assume that a user terminal can access one or more wireless networks in parallel at the same time and that the access selection is controlled by the user terminal. The HWN selection scenario in this paper is shown (Figure 1).Figure 1
Heterogeneous wireless network selection scenario.For the convenience of explanation, the main mathematical symbols used in this paper and their descriptions are shown (Table1).Table 1
Main symbols and their descriptions.
SymbolDescriptionnTotal number of RANsmTotal number of mobile usersbi,jBandwidth allocated by Networkj to User iBjTotal bandwidth resources of Networkjri,jTransmission rate of Useri in Network jRi,mSum of transmission rates from Useri to User mpi,jTransmission power of Useri in Network jri,jreq_minMinimum transmission rate requirement of Useri in Network jsiBandwidth resource state assignable to UseribisiWhen resource state is insi, bandwidth allocation decision for User ibi∗siWhen resource state is insi, optimal bandwidth allocation decision for User iDisiA set of bandwidth allocation decisions in Useri under state sipi,msiBandwidth allocation policy from Useri to User mpi,m∗siOptimal bandwidth allocation policy from Useri to User mRi,m∗siMaximum value of sum of transmission rates from Useri to User m


## 3.2. Problem Definition

Assume that Useri is connected to wireless Network j and that the bandwidth resource allocated by User i in Network j is bi,j. According to the Shannon capacity formula [22, 26], the maximum data transmission rate that can be reached by User i in Network j can be defined as
(1)ri,j=βjbi,jlog21+pi,jNobi,j.In Formula (1), βj is the network effectiveness coefficient of Network j, which can be set according to the characteristics of different networks. For example, for Network j with better coding, the value of its βj is higher than that of other networks using traditional coding. In this paper, βj represents the bandwidth efficiency of wireless networks, that is, the bandwidth effectiveness of each network, and 0≤βj≤1. In addition, bi,j indicates the bandwidth resources obtained by User i in Network j, pi,j indicates the transmission power of User i in Network j, and No is the thermal noise power spectral density.In this paper, the user terminal supports multilink parallel access ton networks, and according to Formula (1), the maximum data transmission rate that User i can achieve is
(2)ri=∑j=1nβjbi,jlog21+pi,jNobi,j.Since bandwidth allocation may affect the transmission rate of users, from the perspective of the whole HWNs, it is always expected that all users receive a reasonable bandwidth in each wireless network, enabling HWNs to reach the maximum transmission rate. Therefore, the problem is defined as follows:(3)maxRHWNsb=max∑i=1mri,=max∑j=1n∑i=1mβjbi,jlog21+pi,jNobi,j,s.t.∑i=1mbi,j≤Bj,bi,j≥0,i=1,2,⋯,m,j=1,2,⋯,n.The first constraint condition∑i=1mbi,j≤Bj in Formula (3) represents that the total bandwidth allocated to all users in Network j should be less than or equal to the total bandwidth of Network j. In addition, the second constraint condition bi,j≥0 represents that the bandwidth allocated to users should be greater than or equal to 0.


## 4. Problem Solving and Algorithm Design

In this section, the problem in this paper is first analyzed according to the principle of dynamic programming. Next, the calculation process of the optimal bandwidth allocation, which maximizes the network rate, is detailed. Finally, the pseudocode of the access selection algorithm is provided based on optimal bandwidth resource allocation.


### 4.1. Concepts to the Dynamic Programming Theory

According to the definition of the problem given in the previous section and to make HWNs obtain the maximum transmission rate, namely, making every radio access network in HWNs reach the maximum transmission rate, the maximum data transmission rate of networkj can be expressed as follows:
(4)maxRjb=max∑i=1mβjbi,jlog21+pi,jNobi,j,s.t.∑i=1mbi,j≤Bj,bi,j≥0,i=1,2,⋯,m,j=1,2,⋯,n.To make the network reach the maximum transmission rate, it is necessary to obtain the optimal solution of the bandwidth allocation to each user in the network. In this paper, the dynamic programming theory is used to solve this problem, with the key terms defined as follows:(i)
Stage: to solve the optimal bandwidth allocation to m users and maximize the network transmission rate, the problem is divided into several interrelated “stages” in order to solve it in a certain order. In this paper, for the same network, the bandwidth allocation to m users can be seen as a sequential process (i.e., the bandwidth is first allocated to User 1 and then User 2, up to User m). Therefore, m users are divided into m stages. Each stage is to allocate bandwidth to the user in that stage. The purpose is to breakdown the problem solving process into a multistage decision-making process.(ii)
State: from the dynamic programming model, “state” describes the natural state at the beginning of each stage throughout the whole research process of the problem. Usually, a stage contains several states, and the set they constitute is called the reachable state set. Here, si is used to represent state variables of Stage i, and si is the bandwidth resources assignable by User i in the network. In addition, Si represents the set of the reachable states at Stage i. For example, when the state variable is si=10, the set of the reachable states is Si=si∣si∈0,10. By definition, Si is both the starting point of the bandwidth resource state of User i (i.e., Stage i) and the end point of the resource state of User i−1 for completing bandwidth allocation (i.e., Stage i−1). Moreover, after the state of User i is determined, the bandwidth resource allocation to User i and all following users are not affected by the users before User i.(iii)
Decision: “decision” refers to the action performed in a multistage decision-making process at the specific state of a certain stage. In this paper, the decision variable bi represents the bandwidth value allocated to User i when the bandwidth resource state is si, and for Network j, it is bi,j. By definition, bi is the function of state variable si, which can be expressed as bisi. The value of decision variable bi is generally not unique but allowed within a certain range. This paper uses Disi to represent the set of decisions allowed by User i in the state si, so as to obtain bisi∈Disi. For example, when the state variable si=10, bisi∈0,10.(iv)
Policy: “policy” refers to the set established by the above-mentioned decisions in order. In this paper, the bandwidth allocation policies adopted from User i to User m are the set of bandwidth allocation decisions from User i to User m in sequence, namely,(5)pi,msi=b1s1,b2s2,⋯,bmsm.Wheni≠1 in Formula (5), it represents a subpolicy composed of decisions at each stage from User i to User m, which is recorded as pi,msi. When i=1, this decision sequence is the policy of the whole process, which is recorded as p1,m, that is, a set of decisions for bandwidth allocation to all users in the network in sequence, namely,
(6)p1,ms1=b1s1,b2s2,⋯,bmsm.In addition, since there are many different policies for bandwidth allocation to all users, the optimal policy is the one that maximizes the network speed among all the policies, which is recorded asp1,m∗, namely,
(7)p1,m∗s1=b1∗s1,b2∗s2,⋯,bm∗sm.(v)
State transition equation: the “state transition equation” represents the transition relationship between two adjacent states from Stage i to Stage i+1, that is, the value of the state variable si+1 of Stage i+1 is determined jointly by the state variable si of Stage i and the decision variable bi of Stage i. The transition relationship between the above two states can be recorded as follows:(8)si+1=Tisi,bi.In this paper, if the state variable of Useri is Si and the bandwidth allocation decision for User i is bi, then the state variable of User i+1 can be obtained as follows:
(9)si+1=si−bi.(vi)
Index function and optimal index function value: in this paper, Ri,m represents the sum of transmission rates from User i to User m, and according to Formula (4) and the above definition, Ri,m can be expressed as a function of si, bi, and Ri+1,m, and Ri,m satisfies the recurrence relation:(10)Ri,msi,bi,si+1,⋯,sm+1=∑j=1mrisi,bi=risi,bi+Ri+1,msi+1,bi+1,si+2,⋯,sm+1i=1,2,⋯,m.In Formula (10), risi,bi indicates the stage index value of User i, that is, the rate that User i can reach after obtaining bandwidth. In addition, the optimal index function value achieved by adopting the optimal bandwidth allocation policy from User i to User m is recorded as Ri∗si, namely,
(11)Ri∗si=maxbi,⋯,bmRi,msi,bi,si+1,⋯,sm+1.According to the above definition, the maximum transmission rate for a certain wireless network is determined by breaking down this problem into a multistage decision-making process (Figure2).Figure 2
Multistage process for determining the maximizing system transmission rate.According to the problem defined in Formula (4) and the principle of optimality of dynamic programming, the user bandwidth allocation policy is determined to be a sufficient and necessary condition for the optimal policy.Theorem 1.
Assume that there arem users in the signal range of a network, with i=1,2,⋯,m as the number of users. The sufficient and necessary condition for bandwidth allocation policy p1,m∗=b1∗,b2∗,⋯,bm∗ enables the network to reach the maximum transmission rate, with the following equation being established for any User i:
(12)R1,ms1,p1,m∗=maxp1,i−1∈p1,i−1s1R1,i−1s1,p1,i−1+maxpi,m∈pi,msiRi,msi,pi,m.
In Formula (12), p1,m=p1,i−1,pi,m, and si=Ti−1si−1,bi−1 is the state of User i determined by the given initial state s1 of bandwidth resources and the bandwidth allocation subpolicy p1,i−1 from User 1 to User i−1.
See AppendixA for the proof of the theorem.Inference 2.
If the bandwidth allocation policyp1,m∗ is the optimal policy to maximize the network transmission rate, then for any User i, its subpolicy pi,m∗ must be the optimal policy for the subprocess with the starting point as si∗=Ti−1si−1∗,bi−1∗ from i to m. si∗ is determined by s1 and p1,i−1∗.See AppendixB for the proof of the inference.


### 4.2. Calculation of Optimal Bandwidth Allocation Value

In the above section, the stages, states, decisions, and policies are defined and explained, and the problem of determining the maximum network rate is divided into several interrelated stages according to the number of users based on the dynamic programming model. This means that a huge problem is broken down into a series of subproblems of the same type, and they are solved one by one. This section determines the user’s optimal bandwidth allocation value and the maximum network rate that can be reached according to the state transition equation, the recurrence relation of adjacent stages, and the boundary conditions.There are two methods to solve problems based on the dynamic programming model: the reverse order method and the sequential method. This paper uses the reverse order method. First, according to the boundary conditionRm+1sm+1=0 and starting from i=m, the optimal bandwidth resource value allocated to users in each stage and the corresponding maximum network rate value are obtained step by step, from backwards to forwards. In the solution of subproblems in each stage, the optimal solutions of subproblems in the previous stage are used. For the last user and when determining R1,m∗s1, the maximum value of the network rate for all users is obtained. According to Formula (4), the index function is the sum of rates for all users in the network, and according to the reverse order solution, its basic equation is
(13)Ri,m∗=maxbi∈Disirisi,bi+Ri+1,msi+1,i=m,m−1,⋯,1,Rm+1sm+1=0.To solve the bandwidth allocation at the maximum network transmission rate, according to the reverse order solution, and using the theorem and inference derived in the previous section, from the last Userm, the following is obtained:
(14)Rm∗=maxbm∈Dmsmrmsm,bm,in which, Dmsm is a set of decisions to allocate bandwidth size for User m determined by bandwidth resource state sm. By solving this problem, the optimal bandwidth resource allocation value bm∗sm of User m and the maximum value Rm∗sm of the network transmission rate when User m is included are obtained.For the sum of transmission rates from Userm−1 to User m, the following can be obtained:
(15)Rm−1,m∗sm−1=maxbm−1∈Dm−1sm−1rm−1sm−1,bm−1+Rmsm,in which, according to Formula (9), sm=sm−1−bm−1 can be obtained. In the previous step for calculating User m, the function Rmsm is obtained. The optimal bandwidth resource allocation value bm−1∗sm−1 of User m−1 and the maximum network transmission rate Rm−1,m∗sm−1 when User m−1 and User m are included are obtained by solving the User m−1.Similarly, for the sum of transmission rates from Useri to User m, the following is concluded:
(16)Ri,m∗si=maxbi∈Disirisi,bi+Ri+1,msi+1,in which si+1=si−bi, and the bandwidth resource allocation value bi∗si of User i and the maximum network transmission rate Ri,m∗si under the circumstance from User i to User m are obtained from the results of the previous User i+1.Similarly, up to User 1, for the sum of transmission rates from User 1 to Userm, the following is obtained:
(17)R1,m∗s1=maxb1∈D1s1r1s1,b1+R2,ms2,in which s2=s1−b1, and by solving the maximum value of the above formula, the bandwidth resource allocation value b1∗s1of User 1 and the maximum transmission rate R1,m∗s1 of the entire network can be obtained.Since the total bandwidth resources of the network are known (i.e., the initial states1 is known), b1∗s1 and R1,m∗s1 can be determined, and the value of the state s2 can also be obtained from s2=s1−b1, so b2∗s2 and R2,m∗s1 can also be determined. The bandwidth allocation decision and transmission rate for each user can be determined step by step according to the reverse order of the above recursive process, which reveals the maximum network transmission rate.In addition, since both the bandwidth resource statesi and the bandwidth allocation value bi are continuous real variables, the set of the reachable states Si and the set of the resource allocation decisions Disi are both real number sets. In addition, the real closed interval of the set of the reachable states Si in each stage is si¯,si¯. The above-mentioned problems in the simulation are solved by first discretizing si and selecting the appropriate increment Δs before calculating. In Stage i, the calculation will be carried out on the point column si¯,si¯+Δs,⋯,si¯+tiΔs, where ti is a positive integer and satisfies
(18)si¯+tiΔs≤si¯≤si¯+ti+1Δs.


### 4.3. Design of Access Selection Algorithm Based on Optimal Bandwidth Resource Allocation

After solving the bandwidth value allocated to users by each network, the transmission rateri,j that can be reached by User i in Network j is obtained according to Formula (1). Assuming that the minimum rate requirement of User i in Network j is ri,jreq_min, the user is connected to the corresponding network according to the user rate requirement. If Network j cannot meet the minimum rate requirement of User i in the network (i.e., ri,j<ri,jreq_min), User i will not access Network j. On the contrary, if Network j can meet the minimum rate requirement of User i in the network, then that network is accessed. In particular, if multiple networks can satisfy the rate requirements after allocating bandwidth to users, users can access these networks simultaneously and in parallel. Therefore, the pseudocode of the access selection algorithm based on optimal bandwidth resource allocation is as follows:Algorithm 1: Optimal resource allocation access selection algorithm.
1 Initializationm, n, Rm+1sm+1, Δs, Bj, ri,jreq_min2 forj=1:n do3  fori=m:1 do4si=si¯5   whilesi≤si¯ do6Ri,m∗=maxbi∈Disirisi,bi+Ri+1,msi−bi=risi,bi∗+Ri+1,m∗si+17    storageRi,m∗si and bi∗si8si=si+Δs9   end while10Ri+1,m∗si+1⟵Ri,m∗si11  end for12s1∗=Bj13  outputR1,m∗=R1,ms1∗14  Fori=1:m do15   read outbi∗si16   According to Formula (1), calculate and output ri,jbi∗17   ifri,jbi∗≥ri,jreq_min then18    useri access the network j19   else20    useri does not access the network j21   end if22si+1=si∗−bi∗23  end for24 end for


## 4.1. Concepts to the Dynamic Programming Theory

According to the definition of the problem given in the previous section and to make HWNs obtain the maximum transmission rate, namely, making every radio access network in HWNs reach the maximum transmission rate, the maximum data transmission rate of networkj can be expressed as follows:
(4)maxRjb=max∑i=1mβjbi,jlog21+pi,jNobi,j,s.t.∑i=1mbi,j≤Bj,bi,j≥0,i=1,2,⋯,m,j=1,2,⋯,n.To make the network reach the maximum transmission rate, it is necessary to obtain the optimal solution of the bandwidth allocation to each user in the network. In this paper, the dynamic programming theory is used to solve this problem, with the key terms defined as follows:(i)
Stage: to solve the optimal bandwidth allocation to m users and maximize the network transmission rate, the problem is divided into several interrelated “stages” in order to solve it in a certain order. In this paper, for the same network, the bandwidth allocation to m users can be seen as a sequential process (i.e., the bandwidth is first allocated to User 1 and then User 2, up to User m). Therefore, m users are divided into m stages. Each stage is to allocate bandwidth to the user in that stage. The purpose is to breakdown the problem solving process into a multistage decision-making process.(ii)
State: from the dynamic programming model, “state” describes the natural state at the beginning of each stage throughout the whole research process of the problem. Usually, a stage contains several states, and the set they constitute is called the reachable state set. Here, si is used to represent state variables of Stage i, and si is the bandwidth resources assignable by User i in the network. In addition, Si represents the set of the reachable states at Stage i. For example, when the state variable is si=10, the set of the reachable states is Si=si∣si∈0,10. By definition, Si is both the starting point of the bandwidth resource state of User i (i.e., Stage i) and the end point of the resource state of User i−1 for completing bandwidth allocation (i.e., Stage i−1). Moreover, after the state of User i is determined, the bandwidth resource allocation to User i and all following users are not affected by the users before User i.(iii)
Decision: “decision” refers to the action performed in a multistage decision-making process at the specific state of a certain stage. In this paper, the decision variable bi represents the bandwidth value allocated to User i when the bandwidth resource state is si, and for Network j, it is bi,j. By definition, bi is the function of state variable si, which can be expressed as bisi. The value of decision variable bi is generally not unique but allowed within a certain range. This paper uses Disi to represent the set of decisions allowed by User i in the state si, so as to obtain bisi∈Disi. For example, when the state variable si=10, bisi∈0,10.(iv)
Policy: “policy” refers to the set established by the above-mentioned decisions in order. In this paper, the bandwidth allocation policies adopted from User i to User m are the set of bandwidth allocation decisions from User i to User m in sequence, namely,(5)pi,msi=b1s1,b2s2,⋯,bmsm.Wheni≠1 in Formula (5), it represents a subpolicy composed of decisions at each stage from User i to User m, which is recorded as pi,msi. When i=1, this decision sequence is the policy of the whole process, which is recorded as p1,m, that is, a set of decisions for bandwidth allocation to all users in the network in sequence, namely,
(6)p1,ms1=b1s1,b2s2,⋯,bmsm.In addition, since there are many different policies for bandwidth allocation to all users, the optimal policy is the one that maximizes the network speed among all the policies, which is recorded asp1,m∗, namely,
(7)p1,m∗s1=b1∗s1,b2∗s2,⋯,bm∗sm.(v)
State transition equation: the “state transition equation” represents the transition relationship between two adjacent states from Stage i to Stage i+1, that is, the value of the state variable si+1 of Stage i+1 is determined jointly by the state variable si of Stage i and the decision variable bi of Stage i. The transition relationship between the above two states can be recorded as follows:(8)si+1=Tisi,bi.In this paper, if the state variable of Useri is Si and the bandwidth allocation decision for User i is bi, then the state variable of User i+1 can be obtained as follows:
(9)si+1=si−bi.(vi)
Index function and optimal index function value: in this paper, Ri,m represents the sum of transmission rates from User i to User m, and according to Formula (4) and the above definition, Ri,m can be expressed as a function of si, bi, and Ri+1,m, and Ri,m satisfies the recurrence relation:(10)Ri,msi,bi,si+1,⋯,sm+1=∑j=1mrisi,bi=risi,bi+Ri+1,msi+1,bi+1,si+2,⋯,sm+1i=1,2,⋯,m.In Formula (10), risi,bi indicates the stage index value of User i, that is, the rate that User i can reach after obtaining bandwidth. In addition, the optimal index function value achieved by adopting the optimal bandwidth allocation policy from User i to User m is recorded as Ri∗si, namely,
(11)Ri∗si=maxbi,⋯,bmRi,msi,bi,si+1,⋯,sm+1.According to the above definition, the maximum transmission rate for a certain wireless network is determined by breaking down this problem into a multistage decision-making process (Figure2).Figure 2
Multistage process for determining the maximizing system transmission rate.According to the problem defined in Formula (4) and the principle of optimality of dynamic programming, the user bandwidth allocation policy is determined to be a sufficient and necessary condition for the optimal policy.Theorem 1.
Assume that there arem users in the signal range of a network, with i=1,2,⋯,m as the number of users. The sufficient and necessary condition for bandwidth allocation policy p1,m∗=b1∗,b2∗,⋯,bm∗ enables the network to reach the maximum transmission rate, with the following equation being established for any User i:
(12)R1,ms1,p1,m∗=maxp1,i−1∈p1,i−1s1R1,i−1s1,p1,i−1+maxpi,m∈pi,msiRi,msi,pi,m.
In Formula (12), p1,m=p1,i−1,pi,m, and si=Ti−1si−1,bi−1 is the state of User i determined by the given initial state s1 of bandwidth resources and the bandwidth allocation subpolicy p1,i−1 from User 1 to User i−1.
See AppendixA for the proof of the theorem.Inference 2.
If the bandwidth allocation policyp1,m∗ is the optimal policy to maximize the network transmission rate, then for any User i, its subpolicy pi,m∗ must be the optimal policy for the subprocess with the starting point as si∗=Ti−1si−1∗,bi−1∗ from i to m. si∗ is determined by s1 and p1,i−1∗.See AppendixB for the proof of the inference.


## 4.2. Calculation of Optimal Bandwidth Allocation Value

In the above section, the stages, states, decisions, and policies are defined and explained, and the problem of determining the maximum network rate is divided into several interrelated stages according to the number of users based on the dynamic programming model. This means that a huge problem is broken down into a series of subproblems of the same type, and they are solved one by one. This section determines the user’s optimal bandwidth allocation value and the maximum network rate that can be reached according to the state transition equation, the recurrence relation of adjacent stages, and the boundary conditions.There are two methods to solve problems based on the dynamic programming model: the reverse order method and the sequential method. This paper uses the reverse order method. First, according to the boundary conditionRm+1sm+1=0 and starting from i=m, the optimal bandwidth resource value allocated to users in each stage and the corresponding maximum network rate value are obtained step by step, from backwards to forwards. In the solution of subproblems in each stage, the optimal solutions of subproblems in the previous stage are used. For the last user and when determining R1,m∗s1, the maximum value of the network rate for all users is obtained. According to Formula (4), the index function is the sum of rates for all users in the network, and according to the reverse order solution, its basic equation is
(13)Ri,m∗=maxbi∈Disirisi,bi+Ri+1,msi+1,i=m,m−1,⋯,1,Rm+1sm+1=0.To solve the bandwidth allocation at the maximum network transmission rate, according to the reverse order solution, and using the theorem and inference derived in the previous section, from the last Userm, the following is obtained:
(14)Rm∗=maxbm∈Dmsmrmsm,bm,in which, Dmsm is a set of decisions to allocate bandwidth size for User m determined by bandwidth resource state sm. By solving this problem, the optimal bandwidth resource allocation value bm∗sm of User m and the maximum value Rm∗sm of the network transmission rate when User m is included are obtained.For the sum of transmission rates from Userm−1 to User m, the following can be obtained:
(15)Rm−1,m∗sm−1=maxbm−1∈Dm−1sm−1rm−1sm−1,bm−1+Rmsm,in which, according to Formula (9), sm=sm−1−bm−1 can be obtained. In the previous step for calculating User m, the function Rmsm is obtained. The optimal bandwidth resource allocation value bm−1∗sm−1 of User m−1 and the maximum network transmission rate Rm−1,m∗sm−1 when User m−1 and User m are included are obtained by solving the User m−1.Similarly, for the sum of transmission rates from Useri to User m, the following is concluded:
(16)Ri,m∗si=maxbi∈Disirisi,bi+Ri+1,msi+1,in which si+1=si−bi, and the bandwidth resource allocation value bi∗si of User i and the maximum network transmission rate Ri,m∗si under the circumstance from User i to User m are obtained from the results of the previous User i+1.Similarly, up to User 1, for the sum of transmission rates from User 1 to Userm, the following is obtained:
(17)R1,m∗s1=maxb1∈D1s1r1s1,b1+R2,ms2,in which s2=s1−b1, and by solving the maximum value of the above formula, the bandwidth resource allocation value b1∗s1of User 1 and the maximum transmission rate R1,m∗s1 of the entire network can be obtained.Since the total bandwidth resources of the network are known (i.e., the initial states1 is known), b1∗s1 and R1,m∗s1 can be determined, and the value of the state s2 can also be obtained from s2=s1−b1, so b2∗s2 and R2,m∗s1 can also be determined. The bandwidth allocation decision and transmission rate for each user can be determined step by step according to the reverse order of the above recursive process, which reveals the maximum network transmission rate.In addition, since both the bandwidth resource statesi and the bandwidth allocation value bi are continuous real variables, the set of the reachable states Si and the set of the resource allocation decisions Disi are both real number sets. In addition, the real closed interval of the set of the reachable states Si in each stage is si¯,si¯. The above-mentioned problems in the simulation are solved by first discretizing si and selecting the appropriate increment Δs before calculating. In Stage i, the calculation will be carried out on the point column si¯,si¯+Δs,⋯,si¯+tiΔs, where ti is a positive integer and satisfies
(18)si¯+tiΔs≤si¯≤si¯+ti+1Δs.


## 4.3. Design of Access Selection Algorithm Based on Optimal Bandwidth Resource Allocation

After solving the bandwidth value allocated to users by each network, the transmission rateri,j that can be reached by User i in Network j is obtained according to Formula (1). Assuming that the minimum rate requirement of User i in Network j is ri,jreq_min, the user is connected to the corresponding network according to the user rate requirement. If Network j cannot meet the minimum rate requirement of User i in the network (i.e., ri,j<ri,jreq_min), User i will not access Network j. On the contrary, if Network j can meet the minimum rate requirement of User i in the network, then that network is accessed. In particular, if multiple networks can satisfy the rate requirements after allocating bandwidth to users, users can access these networks simultaneously and in parallel. Therefore, the pseudocode of the access selection algorithm based on optimal bandwidth resource allocation is as follows:Algorithm 1: Optimal resource allocation access selection algorithm.
1 Initializationm, n, Rm+1sm+1, Δs, Bj, ri,jreq_min2 forj=1:n do3  fori=m:1 do4si=si¯5   whilesi≤si¯ do6Ri,m∗=maxbi∈Disirisi,bi+Ri+1,msi−bi=risi,bi∗+Ri+1,m∗si+17    storageRi,m∗si and bi∗si8si=si+Δs9   end while10Ri+1,m∗si+1⟵Ri,m∗si11  end for12s1∗=Bj13  outputR1,m∗=R1,ms1∗14  Fori=1:m do15   read outbi∗si16   According to Formula (1), calculate and output ri,jbi∗17   ifri,jbi∗≥ri,jreq_min then18    useri access the network j19   else20    useri does not access the network j21   end if22si+1=si∗−bi∗23  end for24 end for


## 5. Experiment and Performance Analysis

This section analyzes the performance of the algorithm proposed in this paper. First, the experimental environment is described, including the simulations and experimental parameter settings. Then, the optimal resource allocation access selection algorithm (ORAAS) is analyzed. Finally, the ORAAS algorithm is compared with the RSS algorithm.


### 5.1. Setting of Experimental Parameters

The simulation network environment in this paper includes one LTE eNB node, one WLAN AP node, and one WiMAX BS node. Here, the total bandwidth of WLAN network access points isB1=20MHz, with coordinates of (0, 0); the total bandwidth of LTE network access points is B2=25MHz, with coordinates of (200, 0); and the total bandwidth of WiMAX network access points is B3=10MHz, with coordinates of (0, 200), and the network effectiveness coefficients β for WLAN, LTE, and WiMAX are set to 0.95, 0.9, and 0.85, respectively. In addition, in this experiment, the user emission power is set to 20 mW, the thermal noise power spectral density to N0=−174dBm/Hz, and the increment Δs of discretization processing of si is set to 0.02.To calculate the RSS between the access point and the user, the experiment uses an improved model based on the COST-231 Hata model to calculate the path lossPLdB. According to literature [29], the COST-231 Hata model is defined as follows:
(19)PLdB=46.3+33.9lgf−13.82lghb+44.9−6.55lghblgd−FhM+C,in which
(20)FhM=1.1lgf−0.7×hM−1.56×lgf−0.8smallcities,3.2×lg1.75×hM2largecities,C=0dBsmallcities,3dBlargecities.Thef, hb, hM, and d in the above formula represent frequency (MHz), height of access point from the ground (m), height of user from the ground (m), and distance between the access point and the user (km), respectively. This experiment sets f=2000MHz, hb=15m, and hM=1.5m. Additionally, the value of FhM and C are small cities, while the simplified path loss model is obtained as follows:
(21)PLdB=144+38lgd.


### 5.2. Performance Analysis of Algorithms

Five users are first randomly generated within a circular area with a radius of 100 m, taking (100, 100) as the center of the circle. The coordinates for the five users are User 1 (101.97, 183.39), User 2 (104.37, 32.04), User 3 (97.53, 104.91), User 4 (54.97, 117.17), and User 5 (152.78,124.74) (Figure3).Figure 3
Setup of Simulation I.The distanced from each user to each access point is calculated based on the coordinates of users and network access points. Thus, the corresponding path loss PLdB is calculated according to Formula (21), with the specific data being shown in Table 2.Table 2
Distance and path loss between users and network access points.
WLANLTEWiMAXdPLdBdPLdBdPLdBUser 1209.8118.23207.9118.08103.3106.54User 2109.2107.45100.9106.14197.7117.25User 3143.2111.93146.7112.32136.2111.10User 4129.4110.26186.4116.28994.0105.90User 5197.2117.21133.4110.75170.3114.79After obtaining the path lossPLdB, the optimal bandwidth allocation value and the maximum data transmission rate are calculated for each user. The curves (Figures 4–6) show the change of wireless link capacity between each user and network with the change of bandwidth allocation. From the figures, it can be seen that the capacity of each user link is increasing logarithmically with respect to the bandwidth. At the maximum capacity of the HWN, the method given in Section 4.2 solves the optimal bandwidth allocation value of WLAN, LTE, and WiMAX networks for each user (i.e., the area marked with “∗” in each figure). As can be seen (Table 2 and Figures 4–6), for the same network access point, the closer the user is to the access point, the better the channel quality is and the more bandwidth that can be allocated. This is due to the utility generated for the same bandwidth resource in the same network by allocating the bandwidth resource to users with good channel quality being greater than that generated by allocating it to users with poor channel quality. This also conforms to the characteristics of monotonically increasing logarithmically and decreasing the marginal benefit. Therefore, the bandwidth allocation policy proposed in this paper can make the best use of the bandwidth resources of each network.Figure 4
Optimal bandwidth allocation policy of the WLAN network to the user.Figure 5
Optimal bandwidth allocation policy of the LTE network to the user.Figure 6
Optimal bandwidth allocation policy of the WiMAX network to the user.According to Figures4–6 and Formula (1), the bandwidth allocation values bi,j (MHz) and the maximum transmission rates ri,j (Mbps) that can be achieved by each user in different networks are shown (Table 3).Table 3
Optimal bandwidth allocation value and maximum transmission rate of users in each network.
WLANLTEWiMAXbi,jri,jbi,jri,jbi,jri,jUser 10.802.570.922.693.6615.48User 29.6830.9914.3041.890.301.28User 33.4411.023.4410.081.285.41User 45.0416.171.384.054.2217.87User 51.043.314.9614.510.542.29Sum20.0064.0725.0073.2210.0042.33According to the maximum transmission rates that users can achieve in each network and the minimum rate requirements of users, the network access, the utilization rate of network resources, and the maximum total capacity that the system can achieve for each user can be obtained. As shown (Table4), “√” indicates that the user is connected to the network and “×” indicates that the user is not connected to the network. When the user’s minimum rate requirement is ri,jreq_min=5Mbps, User 1 chooses to connect to WiMAX; User 2 connects to WLAN and LTE; User 3 connects to WLAN, LTE, and WiMAX; User 4 connects to WLAN and WiMAX; and User 5 connects to LTE.Table 4
Results of user’s network access selections.
WLANLTEWiMAXUser 1××√User 2√√×User 3√√√User 4√×√User 5×√×As can be seen (Table5), the total bandwidth allocated to users (i.e., User 2, User 3, and User 4) accessing the WLAN is b2,1∗+b3,1∗+b4,1∗=18.16MHz, the unused bandwidth of the WLAN network is 1.84 MHz, the resource utilization ratio is 90.8%, and the maximum rate that the WLAN network can achieve is r2,1∗+r3,1∗+r4,1∗=58.19Mbps. Users accessing LTE network include User 2, User 3, and User 5. The allocated total bandwidth is b2,2∗+b3,2∗+b5,2∗=22.70MHz, the unused bandwidth of LTE is 2.30 MHz, the resource utilization ratio of the LTE network is 90.8%, and the capacity of the LTE network is r2,2∗+r3,2∗+r5,2∗=66.48Mbps. Users accessing the WiMAX network include User 1, User 3, and User 4. The allocated total bandwidth is b1,3∗+b3,3∗+b4,3∗=9.16MHz, the unused bandwidth of WiMAX is 0.84 MHz, the resource utilization ratio of the WiMAX network is 91.6%, and the capacity of the WiMAX network is r1,3∗+r3,3∗+r4,3∗=38.76Mbps.Table 5
Bandwidth resource utilization ratio and capacities of networks.
Allocated bandwidth (MHz)Total network bandwidth (MHz)Bandwidth utilization ratio (%)Network capacity (Mbps)WLAN18.1620.0090.858.19LTE22.7025.0090.866.48WiMAX9.1610.0091.638.76


### 5.3. Performance Comparison of Algorithms

This section will verify the capacity changes of various wireless access networks (i.e., WLAN, LTE, and WiMAX) and their HWNs when the number of users changes and compare the ORAAS algorithm proposed in this paper with the RSS algorithm. In the ORAAS algorithm, according to the bandwidth resources allocated to users by the network, users choose to access the network that can meet their rate requirements. For the RSS algorithm, users measure the RSS of all networks and choose to access the network with the highest RSS, and all users in the same network share the bandwidth resources of the network on average. In the simulation, the users are randomly generated in the area of HWNs composed of three networks. Five users are generated each time with up to 50 users, and the users randomly fall into a circular area with a radius of 100 m centered on the coordinates (100, 100) (Figure7). In addition, to speed up the calculation, the Δs is set to 0.05 in the simulation.Figure 7
Setup of Simulation II.The capacities of ORAAS and RSS algorithms are shown with different numbers of users (Table6). As can be seen, with the increasing number of users, both the ORAAS algorithm and the RSS algorithm can increase the capacity of each network. As the ORAAS algorithm can dynamically allocate the optimal bandwidth resources to users, the ORAAS algorithm has a higher system capacity than the RSS algorithm if the number of users is the same. When the number of users reaches 50, the capacities of WLAN, LTE, WiMAX, and HWNs under the ORAAS algorithm are 151.62 Mbps, 164.80 Mbps, 79.51 Mbps, and 395.92 Mbps, respectively, while the capacities of WLAN, LTE, WiMAX, and HWNs under the RSS algorithm are 137.08 Mbps, 143.56 Mbps, 68.59 Mbps, and 349.23 Mbps, respectively. It can be seen that the ORAAS algorithm proposed in this paper has a better performance. In addition, the data given (Table 6) is used to obtain a fitting graph (Figure 8).Table 6
Capacities of wireless networks.
UserORAASRSSWLANLTEWiMAXHWNSWLANLTEWiMAXHWNS00.000.000.000.000.000.000.000.00571.8747.7241.02160.6064.1117.4738.82120.4010109.0096.3964.74270.13101.8674.4651.53227.8615112.56104.3467.99284.89101.8683.3657.34242.5620134.44135.3569.66339.46126.26105.7258.92290.9025137.41137.1071.03345.54122.85105.7259.53288.1030140.81148.6971.34360.84127.07121.4259.53308.0235142.19157.8571.48371.52127.07137.1459.53323.7340142.72163.2077.00382.92127.07144.3965.03336.4945150.68164.0377.47392.18136.01144.0665.49345.5650151.62164.8079.51395.92137.08143.5668.59349.23Figure 8
Comparison of network capacity under different number of users.The change of system capacity is shown with the increase of the number of users (Figure8). As can be seen, when the number of users is small, the capacity of each network and HWN system increases rapidly, and when the users reach a certain number, the capacity growth of each network and HWN system gradually tends to be stable, showing logarithmic growth. Comparing the ORAAS algorithm with the RSS algorithm, it can be seen that the ORAAS algorithm performs better than the RSS algorithm in the same environment, regardless of each wireless network or HWNs composed of each wireless network. This is because the ORAAS algorithm considers different situations of users to allocate bandwidth resources. Furthermore, each user can make full use of bandwidth resources in the multiaccess form, increasing the total system capacity due to a more reasonable use of resources.The ratio of users connected in parallel to the total number of users under different user rate requirementsri,jreq_min of the ORAAS algorithm is shown (Figure 9). As can be seen, when the user rate requirement is 1 Mbps, 2 Mbps, 3 Mbps, 5 Mbps, and 8 Mbps, respectively, the proportion of users connected in parallel decreases gradually with the increase of the number of users. As the total amount of bandwidth resources in each network is limited, having more users means that each user receives less resources. When some networks cannot meet the user rate requirements, the users will not be connected to the network. In addition, when the user rate requirement is low (e.g., when ri,jreq_min=1Mbps), the proportion of users connected in parallel decreases slowly (Figure 9). As the rate requirement of each user is not high here and the total bandwidth resources of the network can meet the demand of more users at the same time, more users can be connected to the same network at the same time. When the user rate requirement is high (for example, when ri,jreq_min=8Mbps), it is difficult for the limited total resources of the network to meet the requirements of multiple users at the same time, causing the proportion of users connected in parallel to decrease obviously. When the number of users increases to 20, the number of users connected in parallel has decreased to 0.Figure 9
Proportion of users connected in parallel.


## 5.1. Setting of Experimental Parameters

The simulation network environment in this paper includes one LTE eNB node, one WLAN AP node, and one WiMAX BS node. Here, the total bandwidth of WLAN network access points isB1=20MHz, with coordinates of (0, 0); the total bandwidth of LTE network access points is B2=25MHz, with coordinates of (200, 0); and the total bandwidth of WiMAX network access points is B3=10MHz, with coordinates of (0, 200), and the network effectiveness coefficients β for WLAN, LTE, and WiMAX are set to 0.95, 0.9, and 0.85, respectively. In addition, in this experiment, the user emission power is set to 20 mW, the thermal noise power spectral density to N0=−174dBm/Hz, and the increment Δs of discretization processing of si is set to 0.02.To calculate the RSS between the access point and the user, the experiment uses an improved model based on the COST-231 Hata model to calculate the path lossPLdB. According to literature [29], the COST-231 Hata model is defined as follows:
(19)PLdB=46.3+33.9lgf−13.82lghb+44.9−6.55lghblgd−FhM+C,in which
(20)FhM=1.1lgf−0.7×hM−1.56×lgf−0.8smallcities,3.2×lg1.75×hM2largecities,C=0dBsmallcities,3dBlargecities.Thef, hb, hM, and d in the above formula represent frequency (MHz), height of access point from the ground (m), height of user from the ground (m), and distance between the access point and the user (km), respectively. This experiment sets f=2000MHz, hb=15m, and hM=1.5m. Additionally, the value of FhM and C are small cities, while the simplified path loss model is obtained as follows:
(21)PLdB=144+38lgd.


## 5.2. Performance Analysis of Algorithms

Five users are first randomly generated within a circular area with a radius of 100 m, taking (100, 100) as the center of the circle. The coordinates for the five users are User 1 (101.97, 183.39), User 2 (104.37, 32.04), User 3 (97.53, 104.91), User 4 (54.97, 117.17), and User 5 (152.78,124.74) (Figure3).Figure 3
Setup of Simulation I.The distanced from each user to each access point is calculated based on the coordinates of users and network access points. Thus, the corresponding path loss PLdB is calculated according to Formula (21), with the specific data being shown in Table 2.Table 2
Distance and path loss between users and network access points.
WLANLTEWiMAXdPLdBdPLdBdPLdBUser 1209.8118.23207.9118.08103.3106.54User 2109.2107.45100.9106.14197.7117.25User 3143.2111.93146.7112.32136.2111.10User 4129.4110.26186.4116.28994.0105.90User 5197.2117.21133.4110.75170.3114.79After obtaining the path lossPLdB, the optimal bandwidth allocation value and the maximum data transmission rate are calculated for each user. The curves (Figures 4–6) show the change of wireless link capacity between each user and network with the change of bandwidth allocation. From the figures, it can be seen that the capacity of each user link is increasing logarithmically with respect to the bandwidth. At the maximum capacity of the HWN, the method given in Section 4.2 solves the optimal bandwidth allocation value of WLAN, LTE, and WiMAX networks for each user (i.e., the area marked with “∗” in each figure). As can be seen (Table 2 and Figures 4–6), for the same network access point, the closer the user is to the access point, the better the channel quality is and the more bandwidth that can be allocated. This is due to the utility generated for the same bandwidth resource in the same network by allocating the bandwidth resource to users with good channel quality being greater than that generated by allocating it to users with poor channel quality. This also conforms to the characteristics of monotonically increasing logarithmically and decreasing the marginal benefit. Therefore, the bandwidth allocation policy proposed in this paper can make the best use of the bandwidth resources of each network.Figure 4
Optimal bandwidth allocation policy of the WLAN network to the user.Figure 5
Optimal bandwidth allocation policy of the LTE network to the user.Figure 6
Optimal bandwidth allocation policy of the WiMAX network to the user.According to Figures4–6 and Formula (1), the bandwidth allocation values bi,j (MHz) and the maximum transmission rates ri,j (Mbps) that can be achieved by each user in different networks are shown (Table 3).Table 3
Optimal bandwidth allocation value and maximum transmission rate of users in each network.
WLANLTEWiMAXbi,jri,jbi,jri,jbi,jri,jUser 10.802.570.922.693.6615.48User 29.6830.9914.3041.890.301.28User 33.4411.023.4410.081.285.41User 45.0416.171.384.054.2217.87User 51.043.314.9614.510.542.29Sum20.0064.0725.0073.2210.0042.33According to the maximum transmission rates that users can achieve in each network and the minimum rate requirements of users, the network access, the utilization rate of network resources, and the maximum total capacity that the system can achieve for each user can be obtained. As shown (Table4), “√” indicates that the user is connected to the network and “×” indicates that the user is not connected to the network. When the user’s minimum rate requirement is ri,jreq_min=5Mbps, User 1 chooses to connect to WiMAX; User 2 connects to WLAN and LTE; User 3 connects to WLAN, LTE, and WiMAX; User 4 connects to WLAN and WiMAX; and User 5 connects to LTE.Table 4
Results of user’s network access selections.
WLANLTEWiMAXUser 1××√User 2√√×User 3√√√User 4√×√User 5×√×As can be seen (Table5), the total bandwidth allocated to users (i.e., User 2, User 3, and User 4) accessing the WLAN is b2,1∗+b3,1∗+b4,1∗=18.16MHz, the unused bandwidth of the WLAN network is 1.84 MHz, the resource utilization ratio is 90.8%, and the maximum rate that the WLAN network can achieve is r2,1∗+r3,1∗+r4,1∗=58.19Mbps. Users accessing LTE network include User 2, User 3, and User 5. The allocated total bandwidth is b2,2∗+b3,2∗+b5,2∗=22.70MHz, the unused bandwidth of LTE is 2.30 MHz, the resource utilization ratio of the LTE network is 90.8%, and the capacity of the LTE network is r2,2∗+r3,2∗+r5,2∗=66.48Mbps. Users accessing the WiMAX network include User 1, User 3, and User 4. The allocated total bandwidth is b1,3∗+b3,3∗+b4,3∗=9.16MHz, the unused bandwidth of WiMAX is 0.84 MHz, the resource utilization ratio of the WiMAX network is 91.6%, and the capacity of the WiMAX network is r1,3∗+r3,3∗+r4,3∗=38.76Mbps.Table 5
Bandwidth resource utilization ratio and capacities of networks.
Allocated bandwidth (MHz)Total network bandwidth (MHz)Bandwidth utilization ratio (%)Network capacity (Mbps)WLAN18.1620.0090.858.19LTE22.7025.0090.866.48WiMAX9.1610.0091.638.76


## 5.3. Performance Comparison of Algorithms

This section will verify the capacity changes of various wireless access networks (i.e., WLAN, LTE, and WiMAX) and their HWNs when the number of users changes and compare the ORAAS algorithm proposed in this paper with the RSS algorithm. In the ORAAS algorithm, according to the bandwidth resources allocated to users by the network, users choose to access the network that can meet their rate requirements. For the RSS algorithm, users measure the RSS of all networks and choose to access the network with the highest RSS, and all users in the same network share the bandwidth resources of the network on average. In the simulation, the users are randomly generated in the area of HWNs composed of three networks. Five users are generated each time with up to 50 users, and the users randomly fall into a circular area with a radius of 100 m centered on the coordinates (100, 100) (Figure7). In addition, to speed up the calculation, the Δs is set to 0.05 in the simulation.Figure 7
Setup of Simulation II.The capacities of ORAAS and RSS algorithms are shown with different numbers of users (Table6). As can be seen, with the increasing number of users, both the ORAAS algorithm and the RSS algorithm can increase the capacity of each network. As the ORAAS algorithm can dynamically allocate the optimal bandwidth resources to users, the ORAAS algorithm has a higher system capacity than the RSS algorithm if the number of users is the same. When the number of users reaches 50, the capacities of WLAN, LTE, WiMAX, and HWNs under the ORAAS algorithm are 151.62 Mbps, 164.80 Mbps, 79.51 Mbps, and 395.92 Mbps, respectively, while the capacities of WLAN, LTE, WiMAX, and HWNs under the RSS algorithm are 137.08 Mbps, 143.56 Mbps, 68.59 Mbps, and 349.23 Mbps, respectively. It can be seen that the ORAAS algorithm proposed in this paper has a better performance. In addition, the data given (Table 6) is used to obtain a fitting graph (Figure 8).Table 6
Capacities of wireless networks.
UserORAASRSSWLANLTEWiMAXHWNSWLANLTEWiMAXHWNS00.000.000.000.000.000.000.000.00571.8747.7241.02160.6064.1117.4738.82120.4010109.0096.3964.74270.13101.8674.4651.53227.8615112.56104.3467.99284.89101.8683.3657.34242.5620134.44135.3569.66339.46126.26105.7258.92290.9025137.41137.1071.03345.54122.85105.7259.53288.1030140.81148.6971.34360.84127.07121.4259.53308.0235142.19157.8571.48371.52127.07137.1459.53323.7340142.72163.2077.00382.92127.07144.3965.03336.4945150.68164.0377.47392.18136.01144.0665.49345.5650151.62164.8079.51395.92137.08143.5668.59349.23Figure 8
Comparison of network capacity under different number of users.The change of system capacity is shown with the increase of the number of users (Figure8). As can be seen, when the number of users is small, the capacity of each network and HWN system increases rapidly, and when the users reach a certain number, the capacity growth of each network and HWN system gradually tends to be stable, showing logarithmic growth. Comparing the ORAAS algorithm with the RSS algorithm, it can be seen that the ORAAS algorithm performs better than the RSS algorithm in the same environment, regardless of each wireless network or HWNs composed of each wireless network. This is because the ORAAS algorithm considers different situations of users to allocate bandwidth resources. Furthermore, each user can make full use of bandwidth resources in the multiaccess form, increasing the total system capacity due to a more reasonable use of resources.The ratio of users connected in parallel to the total number of users under different user rate requirementsri,jreq_min of the ORAAS algorithm is shown (Figure 9). As can be seen, when the user rate requirement is 1 Mbps, 2 Mbps, 3 Mbps, 5 Mbps, and 8 Mbps, respectively, the proportion of users connected in parallel decreases gradually with the increase of the number of users. As the total amount of bandwidth resources in each network is limited, having more users means that each user receives less resources. When some networks cannot meet the user rate requirements, the users will not be connected to the network. In addition, when the user rate requirement is low (e.g., when ri,jreq_min=1Mbps), the proportion of users connected in parallel decreases slowly (Figure 9). As the rate requirement of each user is not high here and the total bandwidth resources of the network can meet the demand of more users at the same time, more users can be connected to the same network at the same time. When the user rate requirement is high (for example, when ri,jreq_min=8Mbps), it is difficult for the limited total resources of the network to meet the requirements of multiple users at the same time, causing the proportion of users connected in parallel to decrease obviously. When the number of users increases to 20, the number of users connected in parallel has decreased to 0.Figure 9
Proportion of users connected in parallel.


## 6. Conclusions and Outlook

To optimize the transmission performance of HWNs in the access process, this paper proposes an access selection algorithm for HWNs based on optimal resource allocation, which models the target as the problem of maximizing the system transmission rate. The dynamic programming theory is adopted to determine the maximum value of the system transmission rate and the optimal solution of the user bandwidth resource allocation. Finally, according to the bandwidth resources allocated to users and QoS rate requirements, users are connected to appropriate networks.This algorithm improves the transmission performance of the HWN system, but the limitation of this algorithm is that it only considers the allocation of bandwidth resources. In the future research work, the simultaneous allocation of various network resources (e.g., bandwidth, power, and time slot) in access selection should be studied further to obtain better system performance improvement and QoS optimization.



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