I Introduction
With the everincreasing demands on mobile data streams and the number of connected multimedia devices, some industry and academic experts predict the data rate is expected to increase 1000fold by year 2020 [1]. Many efforts have been made to tackle this issue. Heterogeneous cellular networks (HCNs) with small cells densely deployed underlaying the macrocells have shown great potential to increase frequency reuse and system capacities [2]. Besides, due to the scarcity of practical radio frequency resources, many mobile network operators have dedicated to exploit new spectrum bands. Among them, millimeter wave (mmwave) has emerged as a promising candidate for the fifthgeneration (5G) mobile network and attracted tremendous attention for huge bandwidth [3, 4]. It can achieve throughput in the range of gigabits per second. Moreover, there are already several standards defined for indoor wireless personal area networks (WPANs) or wireless local area networks (WLANs) in the mmwave band, such as ECMA387 [5], IEEE 802.15.3c [6], and IEEE 802.11ad. In view of the large bandwidth, it is necessary to divide the mmwave into multiple bands, as well as for cellular communications. Therefore, in order to meet the fast growth of mobile Internet traffic demands, one promising way is to exploit HCNs in multiple microwave bands and multiple mmwave bands.
To efficiently use the available spectrum resources and maintain a desired quality of service at local users, HCNs with small cells densely deployed have been brought into many studies [7, 8, 9]. Essentially, implementing HCNs decreases the distance between terminals, which results in lower path losses, reduction in battery consumption, increased energy efficiency and spectrum efficiency. Apart from this, in this paper, the combination of cellular network and mmwave network makes the advantages of the two networks maximized, and the disadvantages are complementary. It turns out that cellular network has more stable and reliable propagation conditions, and is responsible for network control [10]. However, its transmission rate is limited to fail to meet the continuous growing data traffic. Mmwave network offers huge bandwidth and provides multiple gigabit data rate, while to some extent, much larger distance associated propagation loss is suffered [11, 12]. For example, the free space path loss at 60 GHz band is 28 dB more than that at 2.4 GHz [13]
. In addition, a dense deployment of the small cells can increase lineofsight (LOS) probability and compensate for the blockage of mmwave networks. Considering the different characteristics of both networks, the benefits of the HCNs with many small cells deployed will be obvious. At last, dividing the cellular and mmwave communications into multiple bands makes our research more practical.
Devicetodevice (D2D) communication has emerged as a promising component to further improve spectral efficiency. In the conventional cellular network, cellular users communicate with each other via the central coordinator, such as base stations (BSs). Different from the infrastructure based cellular network, D2D communications allow two closely located users to communicate directly without involving central controllers [14, 15, 16]. Due to the proximity of D2D users, the benefits such as reduced endtoend latency and lower energy consumption are reaped. Then, D2D pairs need to share uplink spectrum resources with cellular users or use the resources in mmwave bands in multicell HCNs. Thus, various interference due to the spectrum sharing needs to be considered, and effective interference management is meaningful.
In this paper, we consider a scenario of multicell D2Denabled heterogeneous cellular network aggregating multiple microwave bands and multiple mmwave bands. The challenges of concerned multiband resource allocation issues have followed, such as the uniqueness of HCNs, differences in propagation conditions and power gains of cellular and mmwave networks, complicated intra and intercell interferences caused by coband cellular links and D2D links, and algorithm design with priority. Considering these together, the optimization problem of D2D communication spectrum resource allocation among microwave bands and mmwave bands is formulated to maximize the metric of system transmission rate [17], which is defined as the sum rate of all cellular users and D2D pairs in all involved cells, ( and D denote the set of all cellular users and D2D pairs respectively, denotes the transmission rate). To address this problem, we propose a heuristic algorithm, which takes advantage of the mmwave communications preferentially after taking into account the characteristics of two networks. As a result, the caused interference is effectively managed and the system performance in terms of the total transmission rate is enhanced. The main contributions of the paper can be summarized as follows.

In HCNs with small cells densely deployed underlying the conventional macrocells, we outline the system model consisting of multiple microwave bands and mmwave bands for multiple cellular users and D2D pairs.

We propose a heuristic algorithm to make full use of the advantages of cellular network and mmwave network, while, minimizing interference and maximizing the system transmission rate. Then, we show that the algorithm always yields the nearoptimal solution with fairly low computational complexity.

Through extensive simulations under various system parameters, we evaluate the system performance of the proposed heuristic algorithm compared with other practical schemes. Besides, the optimality and complexity are also analyzed. On average, the proposed algorithm enhances the system performance in terms of total rate than full mmwave transmission strategy by about .
The rest of the paper is organized as follows. In Section II, we summarize the related work. Section III outlines the system model and formulates a resource optimization problem. We propose an effective and efficient heuristic algorithm in Section IV. Section V gives the performance evaluation of the proposed scheme in terms of optimality, complexity and comparison with other three schemes under various system parameters. Finally, we conclude this paper in Section VI.
Ii Related Work
In this section, we partition the related works of resource allocation and interference management into four categories. 1) cellular networks, 2) heterogeneous cellular networks, 3) heterogeneous networks in the mmwave bands, 4) heterogeneous networks consisting of mmwave and microwave bands.
1) There have been several related works studying power control under a restricted cellular network [18], resource allocation for D2D communications under a realistic cellular network [19, 20, 21]. For example, RamezaniKebrya et al. [18] proposed an efficient approximate power control algorithm to maximize the sum rate of a cellular user and a D2D pair with the consideration of the worstcase intercell interference (ICI) limit in multiple neighboring cells. To improve spectrum efficiency and enhance system capacity, Li et al. [19] proposed a coalition formation game to deal with the interference problem of multiple cellular users and D2D pairs underlaying cellular networks by reasonable resource allocation. Xu et al. [20] proposed an innovative reverse iterative combinatorial auction mechanism to allocate resources to D2D communications underlaying downlink cellular networks. Dai et al. [21] proposed a framework in which users uploaded data to BSs at most two hops for D2D overlaying multichannel uplink cellular networks.
2) Different from traditional cellular network, heterogeneous network, which is a multicell topology, significantly boosts the overall network capacity. Inspired by it, existing literatures on resource allocation and interference management for heterogeneous cellular networks include [22] and [23]. Wang et al. [22] proposed a biased cell association scheme with coordinated subchannel allocation and channel inversion power control for mitigating the cotier and crosstier interferences caused by spectrum resource sharing and densification of the small cells in HCNs. Tan et al. [23] proposed a joint access selection and resource allocation scheme to maximize the network capacity in the cacheenabled HCNs with D2D communications.
3) Considering the advantages of mmwave, such as huge bandwidth, the latest emerging works [24, 2, 25, 26] open the direction for studying heterogeneous networks in the mmwave bands. Su et al. [24] studied the user association and wireless backhaul allocation in a twotier heterogeneous network operating in the mmwave band. Niu et al. [2] developed an energyefficient mmwave backhauling scheme to deal with the joint optimization problem of concurrent transmission scheduling and power control of small cells densely deployed in HCNs. Niu et al. [25] jointly designed the scheduling problem of radio access and backhaul for small cells in the mmwave band. Niu et al. [26] proposed a coalition formation game based algorithm for optimal subchannel allocation of access and D2D links in densely deployed multiple mmwave small cells.
4) Deng et al. [10] considered a hierarchical network control framework to address the problems of resource allocation and interference coordination in mmwave/sub6 GHz multiconnectivity with relaying scenarios. Chen et al. [27] investigated the D2D communications resource allocation considering singlecell multimicroband singlemmwaveband in HCNs. In this paper, we investigate the scenario of D2D underlaying multicell HCNs consisting of multiple microwave bands and mmwave bands, in which resources have to be allocated across frequencies with disparate propagation conditions. The lower microwave band is responsible for network control and relatively reliable communications, while mmwave communications provide highthroughput enhancement. Cellular uplink spectrum resources or mmwave radio resources are shared by D2D pairs and as a result, intra and intercell interferences are involved and becoming a challenge to the followup research. Thus, effective resource allocation and interference handling are the keys to improve the system performance.
Mmwave bands from 28 GHz to 300 GHz are considered to be a promising candidate for new spectrum in the 5G networks [28, 29, 30, 31, 32]. Meanwhile, channel measurements have confirmed some unique characteristics of mmwave signals, which are different from traditional cellular signals [33]. On the one hand, based on large and continuous bandwidth, mmwave spectrum can offer much higher throughput. On the other hand, mmwave signal suffers much larger propagation loss due to the high carrier frequency, and it’s more vulnerable to blocking and sensitive to obstacles. Consequently, network congestion may happen in mmwave networks [34]. Another distinguishing characteristic is the directional transmission. The highly directional narrow beams are utilized and the power gains are closely related to the angles of departure/arrival (AoD/AoA). Yu et al. [35] presented a general framework for the analysis of the coverage probabilities in mmwave networks, and then conducted a thorough investigation on the impact of directional antenna arrays. Ai et al. [11] performed some measurements and simulations on indoor mmwave massive multipleinput multipleoutput (MIMO) channel at 26 GHz. In this paper, integrating mmwave into HCNs with densely deployed small cells can increase the mmwave coverage, tackle the problems of path loss and blockage, and further achieve both high capacity and consistent user experience.
Iii System Model and Problem Formulation
Iiia System Description
Differing from a simple singlecell scenario, we consider a multicell heterogeneous cellular network with multiple potential D2D pairs in this paper. The small cells can be separated or overlapped. In the investigated communication pattern, both the cumulated interference from neighbor cells and the intracell interference are considered. Since the HCN is a combination of the cellular network and mmwave network, we assume two operating modes for each potential D2D pair, the cellular mode and the mmwave mode. If the D2D pair chooses to operate in the cellular mode, that means it shares the uplink spectrum resource of one cellular user in the same located cell. Otherwise, it uses the radio resource in one of the mmwave bands. Next, we will discuss system interference in detail.
For the cellular D2D system, the BS is equipped with omnidirectional antennas for cellular communications. Both cellular users and D2D pairs have only one antenna element and its pattern is still omnidirectional, i.e., the gain over all directions is the same. In order to make the complicated interference problem tractable and achieve the maximum spectral efficiency, one cellular user’s uplink spectrum resource can be shared simultaneously with multiple D2D pairs. At the same time, the D2D pair is allowed to share no more than one cellular user’s spectrum to keep the computational costs low. Besides, the interferences between different cellular subchannels in the same cell are supposed to be nonexistent for the independence of subchannels. When cellular users of different small cells transmit on the same channel, there exists interference between each other. In general, four kinds of interferences need to be taken into consideration in cellular D2D system, such as cellular link to cellular link interference, cellular link to D2D link interference, D2D link to cellular link interference, and D2D to D2D interference. For the mmwave D2D system, it is fundamentally different. Highly directional antennas are leveraged for D2D pairs in order to achieve the directional transmission and reception between D2D users in the mmwave band. Furthermore, significant beamforming gains are provided. Since there is no forwarding architectures involved, such as BSs, we only need to consider one kind of interference, D2D link to D2D link interference.
In such a system, our goal is to maximize the total transmission rate. To achieve it, a critical challenge should be addressed for efficient D2D communications in HCNs. We need to concentrate on assigning appropriate spectrum resources for all D2D pairs, while satisfying the constraints and mitigating inter and intracell interferences as much as possible. As shown in Fig. 1, we introduce the resource sharing relationship of D2D communications in the HCN underlaying the macrocell, and elaborate on inter and intracell interferences. To make the figure clearer, we simplify the multicells into two small cells. The cellular users in each cell are controlled by the corresponding BS, which is connected to the macrocell BS via a direct and highrate wired connection, called gateway. It should be noted in advance that, 1) the letter represents a cellular user in cell assigning to subchannel , 2) the letter represents the mmwave subchannel in cell , 3) the letter represents the D2D in cell .
For a separate cell, we give detailed intracell interference. In the small cell numbered 1, the utilized microwave band is divided into two subchannels. There are two cellular users and occupying them respectively, and 3 D2D pairs with and shared the spectrum resource of , shared with . Similarly, the available mmwave band is divided into two subchannels. and use the mmwave band in , and in band. In addition, a similar network is configured in small cell numbered 2. In each small cell, there are four kinds of interference, 1) cellular user to D2D interference, 2) D2D to cellular user interference, 3) cellular D2D to D2D interference, and 4) mmwave D2D to D2D interference. Then, the intercell interference is also briefly described. There are four subchannels consisting microwave and mmwave bands in this case, and each subchannel is represented by a colored rectangle. The links sharing the same subchannel in the same cell or different cell are marked with the same colored rectangle, that is to say, there must be intra or intercell interferences between any two of them. For example, the received signals at the BS from are interfered by the transmitters of , , and sharing the same spectrum resource. The signal at the D2D receiver is interfered by the transmitters of , , and . In contrast, in the mmwave band, the D2D pairs are mutually interfered as they use the same radio resource. The receiver of is interfered by the transmitters of , and . Regarding to the resource allocation problem to be studied, when all D2D pairs select the cellular mode, since the transmission rate of the cellular link is particularly low, the total system rate will be relatively small. While, when all of them select the mmwave mode, the interference generated may be serious, and it is not beneficial to the overall performance improvement. Thus, we need to rationally allocate spectrum resources to increase bandwidth utilization and maximize system utility.
IiiB System Model
IiiB1 System Model Overview and Assumptions
In the system, we assume that there are
small cells, in which multiple cellular users and D2D pairs are uniformly distributed with the BS in the center. The BS associated with cell
is denoted by . Taking a cell as an example, since the spectrum resources of the HCN are consisted by multiple microwave bands and multiple mmwave bands, for the part of the cellular D2D network, we assume that cellular bands are defined. Due to the frequency division multiplexing (FDM), we also suppose there are cellular users occupying different subchannels without interference between each other. For cell , we denote the set of cellular users as . Thus, the set of all cellular users in the system can be expressed as . In order to be closer to the actual scenario, the number of D2D pairs in each cell is randomly generated. Suppose there are D2D pairs, the set of D2D pairs in cell can be denoted by . Thus, the set of all D2D pairs in the system can be expressed as . For the part of mmwave D2D network, we assume that there are mmwave bands denoted as in cell . Therefore, the set of mmwave bands in the system is expressed as .Note that there exists two modes for D2D pair to choose. One is to share the uplink resources of cellular users , and the other is to use mmwave spectrum
. To better reflect the spectrum resource usage relationship, we define a binary variable
to represent whether the cellular mode or the mmwave mode is selected by D2D pair in cell . If the cellular mode is selected, . Another case means that the mmwave spectrum is chosen. For more convenient representation of the relationship of sharing frequency band, we define two other binary variables and . The former represents the th subchannel of the cell is the same as the th of the cell when it equals to , otherwise, it is not the same. Similarly, the latter represents D2D pair in cell shares the spectrum with cellular user when it equals to . For mmwave D2D network, we also define two binary variables. One is , representing mmwave band of cell and mmwave band of cell are the same if , otherwise, . Another is . The variable indicates D2D pair in cell occupies the resource of mmwave band . For all binary variables defined above, we place some reasonable constraints. One is . In other words, each D2D pair in each cell must share the spectrum of the cellular users or use the radio resource of mmwave. It cannot transmit directly through the BS. Another one is .Notation  Description 

number of small cells  
BS of cell  
number of cellular bands  
the set of cellular users of cell  
the set of all cellular users  
number of D2D pairs of cell  
the set of D2D pairs of cell  
the set of all D2D pairs  
number of mmwave bands  
the set of mmwave bands of cell  
the set of all mmwave bands  
D2D pair of cell  
cellular user of cell  
mmwave spectrum of cell  
mode selection of  
whether and share the same band  
whether and share the same band  
whether and are the same  
whether shares the band of  
power or secondorder statistic of the channel  
pathloss exponent  
distance between and  
received power at from in cellular system  
received SINR at from in cellular system  
transmit antenna gain in cellular system  
receive antenna gain in cellular system  
interference power received by in cellular system  
received power at from in mmwave system  
received SINR at from in mmwave system  
transmit antenna gain of in mmwave system  
receive antenna gain of in mmwave system  
interference power received by in mmwave system 
IiiB2 System Channel Model and SINR Representation
To maximize the network performance in terms of the system transmission rate, we should pay more emphasis on the key part of signal to interference plus noise ratio (SINR). Considering the cellular D2D network first, we adopt the Rayleigh channel model for smallscale fading due to shadowing and attenuation, while the distancebased path loss is also considered. For communication link , we denote its sender and receiver by and , respectively. The corresponding channel coefficient of the link is denoted by , which can be written as under the free space propagation pathloss model, where is the power or secondorder statistic of the channel, is the distance between sender and receiver , and is the pathloss exponent [36, 37]. represents the uplink channel state. On the contrary, is considered as a constant within the BS’s coverage area, and
is a complex Gaussian random variable with zero mean and unit variance. Based on the above channel model and pathloss model, for communication link
, we derive the expression of the received power at from as , where is the cellular transmission power, is the transmit antenna gain and is the receive antenna gain. All of them are fixed value for the sake of tractability. The received SINR at from , denoted by , can be expressed as(1) 
where is the interference signal power received by user . The channel is assumed to experience additive white Gaussian noise. is the noise power spectral density of the cellular networks, and is the cellular subcarrier bandwidth.
Considering the mmwave D2D network, the received power at from can be written as
(2) 
where is a constant coefficient and proportional to ( denotes the wavelength) [38]. We specify the gain which is different from the setting in cellular D2D networks. The antenna gain of pointing at direction of is denoted by . The antenna gain of pointing at direction of is denoted by . Both of them are related with the angles of AoDs/AoAs. is the mmwave transmission power. There inevitably exists interference between two mutually independent mmwave communication links and . Thus, the received interference at from can be calculated as
(3) 
where denotes the multiuser interference (MUI) factor related to the cross correlation of signals from different links.
Combining useful received power, interference, and noise power, we can obtain the received SINR at as follows.
(4) 
where is the interference signal power received by user , the noise onesided power spectral density in the mmwave D2D network is symbolized by , and is the bandwidth of mmwave communication.
IiiB3 System Transmission Rate
In the case of cellular D2D communication, we denote the transmit and receive antenna gain of user equipments and BS as and , respectively. For simplicity, we take them to fixed and reasonable value. Then, we analyze the interference experienced by cellular users and D2D pairs in each cell, and further obtain the uplink transmission rate of each individual contributed to the system. The cumulative interference of the cellular link receiver, which is the BS, partly comes from D2D pairs occupying the same spectrum resource with the cellular user in the same cell, and the remaining comes from cellular users and D2D pairs sharing the same frequency band in other cells. In summary, the interference power at BS for cellular user in the small cell can be expressed as
(5)  
According to the Shannon theory, the achievable transmission rate in bit/s of the cellular user , denoted by , can be expressed as
(6) 
Taking D2D pair in cell as an example, which is denoted as . The received cumulative interference signal of the receiver is from the cellular user and the other D2D pairs sharing the same spectrum resource of in cell , and cellular users and D2D pairs occupying the same subchannel in other cells. Therefore, the calculation formula of the interference power for the receiver of D2D , denoted by , is
(7)  
With the interference power, we can get the SINR for the receiver of D2D pair , denoted by , as follows.
(8) 
Thus, the transmission rate of the D2D pair , denoted by , is expressed as
(9) 
In the case of mmwave D2D communication, interference is more complicated. Different from the singlecell scenario, each D2D pair in the multicell scenario suffers interference from all D2D pairs sharing with the same spectrum, not only in the located cell, but also other cells. Thus, we can get the interference power for the receiver of D2D , denoted by , as follows.
(10)  
Similarly, the expression of the SINR of the D2D receiver , denoted by , is shown as follows.
(11) 
Thus, the achievable data rate for the D2D pair in mmwave band, denoted by , is given by
(12) 
Combining in the cellular D2D network and in the mmwave D2D network, we can obtain the transmission rate of D2D pair in the heterogeneous cellular network system, denoted by , as
(13) 
where denotes the probability of blockage in the line of sight (LOS) path between the sender and the receiver of D2D pair in mmwave band [39]. The probability of blockage is mainly added to better reflect the characteristics of the mmwave link, such as mmwave links are easily blocked by various obstacles. It can be expressed as , where is the distance between user equipments and , and is the parameter used to reflect the density and size of obstacles, which result in an interruption caused by blockage.
IiiC Problem Formulation
From the rate formula given in (14), the system transmission rate is only relevant to the binary variables we defined, such as , and . In view of the relation of equality between and , we only take the first four variables into consideration in the analysis of the system transmission rate. For simplicity, we design a matrix to represent the spectrum resource sharing relationship. Thus, (14) can be simplified as a function, denoted by . In general, based on the above analysis, the optimization problem of D2D communication resource allocation in multicell multiband heterogeneous cellular networks can be expressed as follows. The goal of the optimization problem is to maximize the system transmission rate and significantly improve system performance.
(15) 
Obviously, the formulated optimization problem is considered to be a nonlinear integer programming problem. The characteristic of nonlinear can be easily seen from the function. In addition, all involved variables are taken values 0 or 1, and both of them are integer. This problem is NPcomplete and it is more difficult to solve compared with the 01 Knapsack problem [40]. From the defined function in (15), we determine the optimization problem and the goal we intend to achieve. Table I summarizes the notations adopted in this section.
Iv Heuristic Algorithm
Iva Motivation and Main Idea
To maximize the overall transmission rate, it is important to come up with an effective and efficient resource allocation scheme so that the spectrum resources are fully utilized. At the same time, the system performance is significantly enhanced.
Without loss of generality, the mmwave channel transmission rate is about four to five orders of magnitude higher than that of cellular channels. Based on this situation, we first allocate only mmwave bands to all D2D pairs and leave the cellular subchannels free. The purpose of this operation is to make the channel with a higher transmission rate can be used preferentially and achieve the optimization goal. Assigning more D2D pairs to share the cellular spectrum resources is of little significance. In order to make the complicated problem tractable, we put all the D2D pairs in the system sharing the same spectrum resource into one set. For example, carrier frequencies of mmwave are selected in this paper, and thus all included D2D pairs at the beginning form sets. The main idea is to perform an appropriate number of switch operations between the mmwave sets to preliminarily maximize the system transmission rate. For each set , we define the rate as , which is equivalent to the sum of all D2D rate in the set. It is noted that if the set is sharing the uplink spectrum of cellular users, adding the rate of all cellular users is necessary. After a reasonable allocation of the mmwave bands, we turn our target to the cellular bands. If the D2D pair in the mmwave set is exchanged to the cellular set that can make the system rate increased, such switch operation is feasible. By manipulating all the D2D pairs in the mmwave set, we can move a part of the D2D into the cellular set, eventually making the interferences caused in the mmwave set and the system transmission rate reach a tradeoff.
IvB Heuristic Algorithm
In this subsection, we describe the proposed algorithm in details. Its pseudo code is shown in Algorithm 1. Initially, we randomly generate a certain number of D2D pairs for each cell. To make it reasonable, we set an upper bound for the number of D2D in each cell. Next, we give the initial resource allocation for cell , where
. Then, we integrate all D2D pairs and classify them sharing the same frequency band into a set. From step 1 to step 10, we complete the optimization between
mmwave bands. Indeed, it is feasible to select users randomly in step 2. However, in order to allow each D2D pair to participate in this iterative process, we label them and iterate in the order of labels. Knowing the D2D’s current located set , we uniformly randomly search for another one . From steps 4 to 9, if we make the system transmission rate increased by exchanging the D2D from to , the switch operation is performed. What needs to be emphasized is that our decision condition in step 4 only involves two sets, because the rest of the sets have not changed. Thus, the increase in the rate of these two sets means that the system rate increases. If the relation is satisfied, D2D pair leaves its set , and joins the new set . At the same time, we update the set . In addition, we set the corresponding iteration termination condition by defining the parameter , which is the number of consecutive recordings without switch operations. Thus, when we perform the switch operation, we zero the parameter; when not executed, . From steps 11 to 20, considering the cellular spectrum resources, we make the second improvement in system performance. Similarly, in step 13, we randomly search for another possible cellular set . If switching users to the cellular band is beneficial to system performance, we also perform this operation. Steps 11 to 20 are similar to steps 1 to 10. Parameter is another record parameter established to distinguish two iterations. Finally, we return a union of each set of frequency bands.In Algorithm 1, we set the number of iterations for the two loops from steps 1 to 10 and from steps 11 to 20 to be and , respectively. In each iteration, based on the judgment criterion that whether the system utility will increase, the D2D pair decides to perform a switch operation or not. Thus, there will be at most 1 switch operation in each iteration, and then, the computational complexity lies in the number of total iterations and can be expressed as .
V Performance Evaluation
In this section, we evaluate the performance of the proposed heuristic algorithm under various system parameters. At the same time, the optimality and complexity of the algorithm is also simulated. In addition, we compare our scheme with three other schemes in terms of the system transmission rate. Finally, a detailed analysis of the simulation results is presented.
Parameter  Symbol  Value 

mmwave bandwidth  1080 MHz  
Cellular carrier bandwidth  15 KHz  
mmwave noise spectral density  134 dBm/MHz  
Cellular noise spectral density  174 dBm/Hz  
mmwave transmission power  20 dBm  
Cellular transmission power  23 dBm  
Path loss exponent  2  
MUI factor  1  
Halfpower beamwidth  
Blockage parameter  0.01  
Antenna gains of device  0.5 dBi  
Antenna gains of BS  14 dBi 
Va Simulation Setup
In the simulation, we consider a heterogeneous cellular network system consisting of multiple small cells. The system is designed as a square area of and each cell is a circular area with radius . In each cell, both cellular users and D2D pairs are uniformly deployed in the inner circle area, and the BS is located in the center. Obviously, it is likely that there is overlap between different small cells. Without loss of generality, we set the pathloss exponent in free space propagation model considered in this paper as 2. On the one hand, D2D communication is a linear connection channel formed when the physical distance between two users is relatively short. It is more reasonable to set an upper bound on D2D distance. However, since the radius of the small cell is small, there is no additional upper bound setting in this paper. On the other hand, the widely used realistic directional antenna model is adopted in the mmwave D2D network, which is a main lobe of Gaussian form in linear scale and constant level of side lobes [41]. It is the reference antenna model with sidelobe for IEEE 802.15.3c. Based on this model, the gain of a directional antenna in units of decibel (dB), denoted by , can be expressed as
(16) 
where denotes an arbitrary angle within the range , denotes the angle of the halfpower beamwidth, and denotes the main lobe width in units of degrees. The relationship between and is . is the maximum antenna gain, and can be expressed as
(17) 
denotes the side lobe gain, which can be obtained by
(18) 
The other simulation parameters are shown in Table II.
In this paper, we focus on the system performance in terms of system transmission rate. To show the superior performance of the proposed heuristic algorithm, we compare it, labeled as HCNheuristic (HCNheu) with the following three algorithms:
Mmwave Communication (MMW), where each D2D pair in the system can only choose the spectrum resources from one of the mmwave bands to share. In addition, all cellular users occupy the subchannels individually, with no interference between each other, and no interference from D2D pairs.
HCN (HCN), where the system spectrum resources include multiple cellular carrier frequencies and multiple mmwave carrier frequencies. In other words, the system can randomly assign D2D pairs to share one cellular user’s uplink resources or occupy one of the mmwave subchannels. The difference between this algorithm and the proposed HCNheu algorithm is that there is no effective heuristic algorithm for resource allocation to further improve the system performance, but simply allocates randomly selected resources to the D2D pairs.
Mmwave One Band (MMW1), which allocates the unique mmwave subchannel to all D2D pairs. Because of the severe spectrum interference, the advantage of mmwave communication is greatly weakened. Thus, the link’s utility contributed to the system is small. The difference between this algorithm and the MMW is just the number of mmwave bands.
VB Comparison With Optimal Solution
In this subsection, we perform some simulations and further give insights into the gap between the proposed algorithm and the optimal solution (OS). Since the optimal solution is obtained by the exhaustive search method, the complexity is extremely high. Thus, we set both the number of small cells and the number of cellular bands to be 2, the number of D2D pairs in each cell to be 4, and vary the number of mmwave bands from 1 to 5 to obtain the simulation results shown in Fig. 2(a). Besides, both the number of cellular bands and mmwave bands are set to be 1, while varying the number of cells from 1 to 5 to obtain the simulation results shown in Fig. 2(b). From these two figures, we can see the system transmission rate achieved by HCNheu, shown by the dot and dash curve, has an excellent approximation to that achieved by OS, shown by the solid line curve. In order to quantitatively analyze the approximation of the two curves, we select the average deviation between the results obtained by HCNheu and OS as an indicator, which is expressed as follows.
(19) 
where and denote the system transmission rate obtained by OS and HCNheu, respectively, with the number of mmwave bands or small cells . As a result, the average deviation between the HCNheu and OS is about in Fig. 2(a), while the average deviation is about in Fig. 2(b). Thus, we complete the demonstration that the proposed heuristic algorithm can obtain a sufficiently accurate solution, which is close to the optimal solution of the optimization problem.
VC Comparison With Other Schemes
In Fig. 3, we set the number of mmwave bands and the number of cellular bands as 3. Then, we plot the system transmission rate comparison of the four schemes varying the number of small cells from 1 to 8. From the figure, we can see that for all algorithms, the total transmission rate increases with the increase in the number of small cells. This is because the number of cellular users and D2D pairs has increased, and their contributions to the system can completely offset the complex interferences caused to the system. In addition, the proposed algorithm is superior to other practical schemes. At the number of small cells of 8, the system rate achieved by our scheme is higher than MMW and HCN about and , respectively. Averagely, the HCNheu outperforms MMW and HCN about and , respectively.
In Fig. 4, we set the number of small cells and mmwave bands to be 5 and 3, respectively. Then, we plot the system transmission rate comparison of the four schemes, varying the number of cellular bands from 1 to 8, where the system rate is in the unit of bit per second. From the simulation results, we can observe that the proposed algorithm achieves the highest system transmission rate and significantly exceeds other ones. As the number of cellular bands changes, the curves of HCNheu, MMW and MMW1 do not change substantially, which is because the transmission rate of mmwave is significantly better than the cellular transmission rate. Thus, most D2D pairs in HCNheu and all D2D pairs in MMW and MMW1 choose to use the spectrum resource of mmwave, and there is no point in increasing the number of cellular bands. The scheme HCN declines due to the fact that some D2D pairs are randomly assigned to the cellular subchannels with the number of cellular bands increased. At the number of cellular bands of 8, the system rate achieved by the proposed scheme is higher than MMW and HCN about and , respectively.
In Fig. 5, we set the number of small cells and cellular bands to be 5 and 3, respectively. Then, we plot the achieved system rate of four schemes under different numbers of mmwave bands. Comparing with other schemes, the proposed heuristic algorithm once again shows a good advantage. Indeed, the mmwave band has a huge contribution to the entire system. At the same time, with the number of mmwave bands increasing, D2D pairs have more choices to access one of the mmwave frequencies in algorithms HCNheu, MMW and HCN involving mmwave. Consequently, both the interference power in the cellular frequency set and the mmwave frequency set are reduced. In summary, these three curves show an upward trend. When the number of mmwave bands is equal to 8, the system transmission rate of HCNheu is larger than that of MMW and HCN about and , respectively. Besides, the scheme MMW1 is unchanged because there is only one mmwave subchannel.
In Fig. 6, the number of cellular bands and mmwave bands are all set to be 3. Then, we plot the system transmission rate of four schemes with the mmwave transmission power varied from 0 to 30 dBm. As we can observe, our scheme achieves the highest system rate among the four schemes. With increased, the useful power of the numerator part and the interference power of the denominator part are also increasing, so the four curves rise slowly. Especially we continue to increase when it is already large, the effect is no longer obvious. For the proposed heuristic algorithm, the gap at of 30 dBm is only about of the MMW and about of the HCN, respectively.
In Fig. 7, we plot the system transmission rate comparison of four algorithms varying from 0.02 to 0.16. Not surprisingly, the proposed scheme still keeps at a high level in the total transmission rate. The parameter of blockage probability is used to represent the density and size of obstacles. Therefore, when increases, the impact of obstacles on the mmwave links will be greater, resulting in a decrease in the total system rate of the four algorithms involving mmwave D2D networks. At of 0.16, the gap between HCNheu and MMW is about of MMW, and between HCNheu and HCN is about of HCN.
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