Resource Allocation for D2D Communication in Cellular Networks Based on Stochastic Geometry and Graph-coloring Theory

Abstract

In a device-to-device (D2D) underlaid cellular network, there exist two types of co-channel interference. One type is inter-layer interference caused by spectrum reuse between D2D transmitters and cellular users (CUEs). Another type is intra-layer interference caused by spectrum sharing among D2D pairs. To mitigate the inter-layer interference, we first derive the interference limited area (ILA) to protect the coverage probability of cellular users by modeling D2D users' location as a Poisson point process, where a D2D transmitter is allowed to reuse the spectrum of the CUE only if the D2D transmitter is outside the ILA of the CUE. To coordinate the intra-layer interference, the spectrum sharing criterion of D2D pairs is derived based on the (signal-to-interference ratio) SIR requirement of D2D communication. Based on this criterion, D2D pairs are allowed to share the spectrum when one D2D pair is far from another sufficiently. Furthermore, to maximize the energy efficiency of the system, a resource allocation scheme is proposed according to weighted graph coloring theory and the proposed ILA restriction. Simulation results show that our proposed scheme provides significant performance gains over the conventional scheme and the random allocation scheme.

1. Introduction

Growing demands for providing high data rate and low energy cost to support the rapidly increased mobile data traffic have been a great impetus to the development of novel technologies of improving system capacity [1-6]. Device-to-device (D2D) communication has drawn much attention due to its short mutual distance and high spectrum reuse gain by allowing direct communications between two or more user equipment in proximity [2]. As a result, D2D has been considered in the fifth generation (5G) as an underlay to cellular networks [1].

On the other hand, the energy efficiency (EE) of the system is becoming mainstream pursuit with the growth of energy demand and the rise of energy prices [4]. A D2D network underlaying a cellular network can improve the spectrum efficiency (SE) and energy efficiency (EE) of the system [5-6], and reduce the traffic load of the macro BS [3]. However, resource sharing among cellular users and D2D users will cause serious co-channel interference.

To solve this problem, there has been extensive research on design and analysis of resource allocation schemes for D2D underlaid cellular networks [7-16]. Authors in [7] derived the average coverage probability of cellular users with corresponding density of potential D2D users based on stochastic geometry theory. In [8], a distance-based power control scheme was devised to enhance SE and coverage probability of users. The authors in [9] proposed a resource allocation and power control algorithm by exploiting the properties of fractional programming and penalty function to maximize the EE of DUEs, while the number of DUEs is less than that of CUEs. A two-loop iterative algorithm was designed in [10] to solve the energy efficiency fairness problem with the constraint of the maximum transmit power and the minimum data rate requirement. Yang et al. [11] investigated the energy-efficient power control problem for DUEs to maximize the total EE and the individual EE. The authors in [12] proposed a resource allocation algorithm which aimed to maximize the minimum weighted energy efficiency of DUEs as well as satisfy the minimum data rates of CUEs. In [13], a reverse iterative combinatorial auction (ICA) game was formulated to improve the system EE, while the QoS requirements of users were ignored. In [14], the authors formulated two resource allocation problems to optimize system EE and total individual EE considering the QoS of CUEs. An energy-efficient scheduling scheme among DUEs was proposed in [15], while only the QoS of DUEs was considered. A user grouping and power control scheme were proposed in [16] to maximize the system EE, while it only considered the impact of interference on DUEs when grouping, which may cause serious interference to CUEs.

Apart from these high complexity optimizing methods, the interference limited method was considered in [17-18] to reduce the computational complexity. A δ𝐷-interference limited area (ILA) scheme was designed to reduce the interference from CUEs to D2D users in [17]. The authors in [18] proposed a minimum distance restriction to alleviate the interference from cellular users to D2D links reusing the same resources. However, these are generally designed to improve SE. Different from the exiting work, we consider the EE programming problem based on the QoS constraints of both CUEs and DUEs based on stochastic geometry modeling and weighted graph coloring to mitigate co-channel interference in D2D underlaid cellular networks.

In this paper, a resource allocation scheme in D2D underlaid cellular networks was proposed, where D2D transmitters reuse the uplink spectrum of CUEs. Resource reusing can improve the energy efficiency of the system but will cause serious interference to CUEs. To guarantee the QoS of CUEs, an interference limited area (ILA), which ensure that the SINR of the CUE be greater than a specified threshold with a large probability, is derived for CUEs based on the stochastic analysis. That is to say, the DUE who is out of ILA of the CUE is allowed to reuse the CUE's resource. Otherwise, resource sharing between the DUE and CUE is forbidden. Then, based on the above restrictions, we formulated a user grouping scheme that meet the DUE’s distance restriction to avoid the DUEs suffer from serious interference. Finally, we propose an optimal resource selection scheme to maximize system EE based on the weighted graph-coloring theory. The main contributions in this paper are summarized as follows.

1) We propose a resource reuse scheme, which allows multiple D2D users reuse cellular user’s spectrum resource, without increasing the overhead of conventional cellular system based on the stochastic geometry theory. And the energy efficiency of system can be greatly enhanced.

2) To depress the D2D-to-cellular interference, we propose an interference limited area for CUEs by modeling the D2D locations as a Poisson point process. Compare to the scheme in [16], the proposed scheme can efficiently protect the CUE’s performance from serious interference.

3) In order to reduce the intra-layer interference among DUEs, we formulate a user grouping scheme based on the limitation of user distance. Based on the result of grouping scheme, we select an optimal DUE according to the interference graph to reuse the resource.

4) We demonstrate the effectiveness of our proposed resource reuse scheme by simulations.

The rest of the paper is organized as follows. In Section 2, we describe the system model and problem formulation to improve EE for D2D underlaid cellular networks. Section 3 describes the proposed resource allocation scheme based on ILA and the weighted graph coloring theory. The simulation results are given in Section 4. And we draw the conclusions in Section 5. Proof of the Theorem is given in the Appendix.

2. System Model and Problem Formulation

2.1 System Model

We consider an uplink scenario in a single-cell cellular system, in which there exist two kinds of communications, as illustrated in Fig. 1. One is called cellular communication. Another is D2D communication which is an underlay to the cellular communication. In the system, M cellular user equipment (CUE) and K D2D user equipment (DUE) (M≤K) are randomly distributed in the coverage area of the base station (BS). DUEs are distributed according to the Poisson point process (PPP) Φ of intensity 𝜆_𝑑. Each D2D pair consists of a D2D transmitter (DT) and a D2D receiver (DR) with a fixed distance 𝑅_𝐷. There are M orthogonal spectrum resources allocated to M CUEs respectively. Without loss of generality, we assume that the ith resource block is allocated to the i-th CUE. DUEs share the same resource set with CUEs. Therefore, the co-channel interference consists of three parts: 1) interference received by BS from DT; 2) interference received by DR from CUE; 3) interference received by DR from other DT sharing the same spectrum resource with that D2D pair.

Fig. 1. System model and the illustration of resource allocation

In the following, we use 𝐶_𝑖 (i=1, 2,...,M) and 𝐷_𝑗 (j=1,2,...,K) to denote the i-th CUE and j-th D2D pairs respectively; 𝐷𝑇_𝑗, 𝑎𝑛𝑑 𝐷𝑅𝑗 represent the transmitter of 𝐷_𝑗 and the receiver of 𝐷_𝑗 respectively. The channel gain can be expressed as

\(g=h \cdot \beta\)       (1)

where h represents the small scale fading, h∼exp(𝜇), and 𝛽 = 𝑑^−𝛼 is the large scale fading, in which 𝑑 is the distance between transmitter and receiver; 𝛼 is the path loss exponent. The useful symbol information is summarized as Table 1.

Table 1. The Implication of Symbols

The received Signal to Interference plus Noise Ratio (SINR) at the BS from 𝐶_𝑖 and the SINR of 𝐷_𝑗 using the i-th resource block can be represented as

\(\operatorname{SINR}_{i}^{C}=\frac{P_{c} g_{i}^{C}}{\sum_{j=1}^{K} \pi_{i j} P_{d} g_{j}^{D}+N_{0}}\)       (2)

\(\operatorname{SINR}_{j}^{D(i)}=\frac{P_{d} g_{j j}^{d d}}{P_{c} g_{i j}^{c d}+\sum_{k \neq j}^{K} \pi_{i k} P_{d} g_{k j}^{d d}+N_{0}}\)       (3)

where 𝑃_𝑐, and 𝑃_𝑑 is the transmit power of CUE and DT respectively; 𝑁₀ is the power of the additive white Gaussian noise; 𝜋_𝑖𝑗 is a resource reuse metrics. 𝜋_𝑖𝑗 = 1 indicates that 𝐷_𝑗 shares the resources of 𝐶_𝑖, while 𝜋_𝑖𝑗 = 0 indicates that 𝐶_𝑖 and 𝐷_𝑗 are not allowed to share the same resources. According to SINR in (2) and (3), we can write the energy efficiency (EE) of 𝐶_𝑖 as

\(E E_{i}^{C}=\frac{\log _{2}\left(1+\operatorname{SINR}_{i}^{C}\right)}{P_{c}+P_{c i r}}\)       (4)

where 𝑃_𝑐𝑖𝑟 represent the circuit power consumption of user. Similarly, the energy efficiency (EE) of 𝐷_𝑗 can be expressed as

\(E E_{j}^{D}=\frac{\log _{2}\left(1+\operatorname{SINR}_{j}^{D}\right)}{P_{d}+2 P_{c i r}}\)       (5)

2.2 Problem Formulation

The EE of the system can be expressed as

\(\begin{aligned} E E_{s u m} &=\sum_{i=1}^{M} E E_{i}^{C}+\sum_{j=1}^{K} E E_{j}^{D} \\ &=\sum_{i=1}^{M} \frac{\log _{2}\left(1+\operatorname{SIN} R_{i}^{C}\right)}{P_{c}+P_{c i r}}+\sum_{i=1}^{M} \sum_{j=1}^{K} \pi_{i j} \frac{\log _{2}\left(1+\operatorname{SINR}_{j}^{D(i)}\right)}{P_{d}+2 P_{c i r}} \end{aligned}\)       (6)

We aim to maximize the system EE with the SINR constrain of CUEs and DUEs, i.e.,

\(\max . E E_{\text {sum }}\)       (7)

\(\text { s.t. SINR }_{i} \geq S I N R_{c} \quad \forall C_{i} \in \mathrm{C}\)       (7a)

\(S I N R_{j}^{D(l)} \geq S I N R_{d} \quad \forall D_{j} \in \mathrm{D}\)       (7b)

\(\sum_{i=1}^{M} \pi_{i j} \leq 1 \quad \forall j \in\{1, \ldots, K\}\)       (7c)

where constraints (7a) and (7b) guarantee the QoS of CUE and DUE respectively, and constraint (7c) indicates that 𝐷_𝑗 is only allowed to reuse at most one cellular user’s resource at each time. However, in the optimization problem (7), the number of spectrum allocation schemes for K D2D pairs is as huge as 𝑀^𝐾. Hence, we try to obtain a low complexity scheme based on the stochastic geometry and graph coloring.

3. The Proposed Resource Allocation Scheme

In this section, we proposed a two-step resource allocation scheme to mitigate the inter-layer interference between CUEs and DUEs and intra-layer interference among users. Firstly, we derive an ILA based on stochastic geometry theory, where only DUEs outside the ILA of the CUE are allowed to reuse the spectrum with the CUE. Then, based on the ILA, we propose a multi-user resource sharing solution according to the interference graph.

3.1 ILA of Cellular Users

Since the cellular users are the primary users of cellular systems, we formulate an ILA to guarantee the QoS of the CUEs where DUEs outside the ILA are allowed to reuse the spectrum of the CUE.

The definition of the ILA can be thought of equivalently as the coverage probability of the CUEs interfered by a transmitter outside 𝐴0 should be large enough [19]. Therefore, the ILA of CUEs can be derived by the coverage probability defined as P(SINR>T). The design of ILA must, therefore, ensure that the SINR of the CUE be greater than a specified threshold with a large probability β, such that

\(P\left(S I N R_{i}^{C}>T\right) \geq \beta\)       (8)

Unlike the existing work where interference D2D pairs are placed deterministically on a regular grid, we model their location as a PPP process with intensity 𝜆_𝑑. Hence, the SINR in (2) can be rewritten as

\(S I N R_{i}^{C}=\frac{P_{c} h_{i} d_{C_{i}, B}^{-\alpha}}{N_{0}+P_{d} h_{0} R_{0}^{-\theta}+\sum_{j \in \Phi \backslash A_{0}} P_{d} h_{j} d_{D_{j}, B}^{-\theta}}\)       (9)

where 𝑃_𝑐ℎ_𝑖𝑑^−𝛼_{𝐶𝑖 ,𝐵} 𝑎𝑛𝑑 𝑃_𝑑ℎ₀𝑅₀^−𝜃 are the received signal power and the interference power caused by the D2D users at a random distance 𝑅₀ with path loss exponent 𝜃 > 2, respectively. ∑𝑗∈𝛷\𝐴₀ 𝑃_𝑑ℎ_𝑗𝑑_{𝐷𝑗,𝐵}^−𝜃 is the cumulative interference from D2D transmitters outside 𝐴₀ ; 𝐴₀ is an interference limited area with BS as the center and 𝑅₀ as its radius.

Lemma 1: 𝜉 is a random variable with probability density function (p.d.f.) 𝑓_𝜉(x), h~exp(𝜇), 𝜏 is a positive constant. Then, we have

Proof: \(\begin{array}{c} P\left(\frac{h}{\xi}>\tau\right)=E_{\xi} e^{-\mu \tau \xi} \\ P\left(\frac{h}{\xi}>\tau\right)=E_{\xi} P(h>\tau \xi) \\ =E_{\xi} \int_{\tau \xi}^{\infty} \mu e^{-\mu x} d x \\ =E_{\xi} e^{-\mu \tau \xi} \end{array}\)

Theorem 1: The ILA of a tagged CUE is an inner region of a circle with the BS as the center and 𝑅₀ as the radius, where

\(R_{0} \geq f^{-1}(\beta)\)       (10)

with

\(f\left(R_{0}\right)=\frac{\exp \left(-\frac{\mu T d_{c}^{\alpha} N_{0}}{P_{c}}\right)}{1+\frac{T d_{c}^{\alpha} P_{d} R_{0}^{-\theta}}{P_{c}}} \exp \left(-\frac{\pi \lambda_{d}}{M}\left(\frac{T P_{d} d_{c}^{\alpha}}{P_{c}}\right)^{\frac{2}{\theta}} \int_{R_{0}^{2}\left(\frac{P_{c}}{T P_{d} d_{c}^{\alpha}}\right)^{\frac{2}{\theta}}}\left(\frac{1}{1+u^{\frac{\theta}{2}}}\right) d u\right)\)       (11)

and 𝑓 ⁻¹ is the inverse function determined by f uniquely; 𝑅_𝐶 is the cell radius. Theorem 1 indicates that DUEs out of the ILA 𝐴₀ can share spectrum with the tagged CUE.The proof of theorem 1 were given at the appendix.

3.2 Interference limited threshold of DUEs

After implementing the above step, we consider the intra-layer interference among users who share the same spectrum resource. Therefore, we introduce a mutual interference restrictions mechanism based on the distance

Similar to [17], the Signal-to-interference ratio (SIR) restriction of 𝐷_𝑗 between other DUEs and CUEs can be expressed as follows

\(S I R_{D_{j} \leftarrow D_{k}}=\frac{R_{D}^{-\theta}}{d_{D_{k}, D_{j}}^{-\theta}} \geq \mu_{d}\)       (12)

\(S I R_{D_{j} \leftarrow C_{i}}=\frac{P_{d} R_{D}^{-\theta}}{P_{c} d_{C_{i}, D_{j}}^{-\theta}} \geq \mu_{d}\)        (13)

where 𝜇_𝑑 is the minimum SIR restriction of DUEs; 𝑆𝐼𝑅_{𝐷𝑗←𝐷𝑘} represent the SIR from 𝐷_𝑘 to 𝐷_𝑗, 𝑆𝐼𝑅_{𝐷𝑗←𝐶𝑖} represent the SIR from 𝐶_𝑖 to 𝐷_𝑗. Thus the distance restriction of 𝐷_𝑗 between CUE and DUE can expressed as

\(d_{D_{k}, D_{j}} \geq R_{D} \mu_{d}{ }^{1 / \theta}\)       (14)

\(d_{C_{i}, D_{j}} \geq R_{D}\left(\frac{P_{c}}{P_{d}} \mu_{d}\right)^{1 / \theta}\)       (15)

3.3 Optimal Resource Reuse Scheme Based on Graph-Coloring

After the above two steps, we can obtain the set of DUEs who can reuse the resources of 𝐶_𝑖, and it can be expressed as Φ_𝑖 = {𝐷_𝑗 , … , (𝑗 ∊ 1,2, … 𝐾)}; and the set of CUEs that can share the resources with 𝐷_𝑗 can be expressed as Ψ_𝑗 = {𝐶_𝑖 , … , (𝑖 ∊ 1,2, … 𝑀)}.

As depicted in Fig. 1, the protection area of cellular user 𝐶₁ and 𝐶₂ can be expressed as the circular area with the base station as their center, with 𝑅_𝐶1 using blue line and 𝑅_𝐶2 using red line as their radius, respectively. Considering the case that one DUE has more than one CUEs that can be chosen to share the spectrum. For example, 𝐷₁ is out of the ILAs of 𝐶₁ and 𝐶₂. To maximize EE of the system, we formulate a metric to choose the best resource based on weighted graph-coloring [20]. The EE of 𝐶_𝑖 and 𝐷_𝑗 can be expressed as

\(E E=\frac{\log _{2}\left(1+\frac{\beta_{C_{i}, B}}{\beta_{D_{j}, B}}\right)}{P_{c}+P_{c i r}}+\frac{\log _{2}\left(1+\frac{\beta_{D} j, D_{j}}{\beta c_{i}, D_{j}}\right)}{P_{d}+2 P_{c i r}}=\log _{2}\left(\left(1+\frac{\beta_{C_{i}, B}}{\beta_{D_{j}, B}}\right)^{\frac{1}{P_{d}+2 P_{c i r}}} \cdot\left(1+\frac{\beta_{D_{j}, D_{j}}}{\beta_{C_{i}, D_{j}}}\right)^{\frac{1}{P_{c}+P_{c i r}}}\right)\)       (16)

From (14), the total EE of 𝐶_𝑖 and 𝐷_𝑗 depends on

\(\eta_{C_{i}, D_{j}}=\left(1+\frac{\beta_{C_{i}, B}}{\beta_{D_{j}, B}}\right)^{\frac{1}{P_{d}+2 P_{c i r}}} \cdot\left(1+\frac{\beta_{D_{j}, D_{j}}}{\beta_{C_{i}, D_{j}}}\right)^{\frac{1}{P_{c}+P_{c i r}}}\)       (17)

If 𝑃_𝐶 = 𝑃_𝑑 + 𝑃_𝑐𝑖𝑟, the total EE of 𝐶_𝑖 and 𝐷_𝑗 is an increasing function of 𝜂_{𝐶𝑖 ,𝐷𝑗}

\(\eta_{C_{i}, D_{j}}=\frac{\beta_{C_{i}, B}}{\beta_{D_{j}, B}}+\frac{\beta_{D_{j}, D_{j}}}{\beta_{C_{i}, D_{j}}}\)       (18)

Hence, we choose the CUE who has the maximal 𝜂_{𝐶𝑖 ,𝐷𝑗} with 𝐷_𝑗 to share the spectrum of 𝐷_𝑗 . Fig. 2 shows the interference between different type of users as follows:

Fig. 2. Interference between different type of users

The proposed algorithm can be summarized as Algorithm 1.

4. Simulation Results and Analysis

To validate the proposed algorithm, we compare the algorithm with conventional algorithm in [16] and random algorithm by simulation. In the random algorithm, it performs the user grouping scheme the same as the proposed scheme while select reused resource randomly. The rest parameters of our simulation are summarized in Table 2.

Table 2. Simulation Parameters

Fig. 3 describes the interference limited radius for different distance between the tagged CUE and its serving BS. Simulation results in Fig. 3 show that the ILA of CUEs is influenced by the distance between CUE and BS and density of DUEs. The higher the distance is, the larger ILA is, and the size of ILA increases with the increasing of density of DUEs as well.

Fig. 3. Interference limited radius R0 v.s. λ for different distance between the tagged CUE and its serving BS with β=0.8, and T=-4dB.

Fig. 4 gives the access number of DUEs for different density of DUEs. It shows that with the increasing of the total number of DUEs, more DUEs are allowed to access the system. As observed, the conventional scheme can access a little more users when the density of DUEs was low because it did not restrict for specific users. While when the density of users increases, the proposed scheme can access more DUEs to the system compared with conventional scheme because it does not allow users to access who may cause severe interference to others.

Fig. 4. Access number of DUEs for different density of DUEs

As we can see in Fig. 5, compared with conventional scheme, the proposed scheme can avoid severe interference to the CUEs. The reason is that we formulate an ILA for CUE to prevent it suffering from serious interference. And the proposed scheme performs well compared to the random scheme because it carried out a graph coloring algorithm to coordination the interference between users.

Fig. 5. Energy efficient of CUEs vs Different density of DUEs

Fig. 6 displays the energy efficient loss ratio of CUEs with the different density of DUEs. It can be seen that the EE loss ratio of CUEs under the proposed scheme and random scheme is lower than the conventional scheme due to only CUEs with good channel conditions share spectrum to DUEs in our proposed scheme.

Fig. 6. Average EE loss ratio of CUEs vs Different density of DUEs

The energy efficiency of CUEs and DUEs are illustrated in Fig. 7. As observed, the EE of DUEs increases with the increasing of DUEs’ density because the system can access more D2D pairs. And the EE of CUEs decreases with the increasing of DUEs’ density. For the EE of DUEs, our proposed scheme performs well than the conventional scheme because it allows more DUEs to access, and it performs well than the random scheme due to it handles the interference problem using the theory of graph-coloring. For the EE of CUEs, contrast with the original system without DUEs, the proposed scheme decreases a bit more than the conventional scheme because the resource of the CUE who has better channel condition is reused in the proposed scheme, and the proposed scheme avoids the resource of CUEs with worse channel condition from being reused.

Fig. 7. Energy efficient of users vs Different density of DUEs​​​​​​​

5. Conclusion

In this paper, we consider an uplink D2D underlay scenario where the number of DUEs is larger than that of CUEs. In order to improve the access succeed rate of DUEs, we formulate a resource reuse scheme where the available spectrum resource of a CUE is allowed to be reused by multiple DUEs. To maximize the energy efficient under the restrictions, we drive the guard region for CUEs based on random point process to ensure its coverage probability. On the other hand, the performance of DUE can be improved by the ILA which avoids two DUE pairs with small distances sharing the same resource. Furthermore, we proposed an optimal reuse scheme according to the weighted graph coloring theory. Simulation results show that the proposed scheme can significantly improve the system energy efficient and avoid CUEs suffering from serious co-channel interference

Appendix

In this Appendix, we will give the proof of Theorem 1.

Proof: The coverage probability can be described as

where (a) follows from Lemma 1. (b) follows from the independence of the small-scale fading powers. (c) follows from the probability generating functional [5] of the PPP. The integration limits are from 𝑅₀ to ∞ since the closest interferer is at least at a distance 𝑅₀. (d) follows from a change of variables \(v=\left(x\left(\frac{P_{c}}{T P_{d} d_{c}^{\alpha}}\right)^{1 / \theta}\right)^{2}\) Notice that 𝑓(𝑅₀) is a monotonically increasing function. From (8) and (17), we obtain (10).