Cooperative Non-Orthogonal Multiple Access with Simultaneous Wireless Information and Power Transfer

Transcription

1 1 Cooperative Non-Orthogonal Multiple Access with Simultaneous Wireless Information and Power Transfer Yuanwei Liu, Zhiguo Ding, Maged Elkashlan, and H Vincent Poor Abstract In this paper, the application of simultaneous wireless information and power transfer SWIPT to non-orthogonal multiple access NOMA networks in which users are spatially randomly located is investigated A new cooperative SWIPT NOMA protocol is proposed, in which near NOMA users that are close to the source act as energy harvesting relays to help far NOMA users Since the locations of users have a significant impact on the performance, three user selection schemes based on the user distances from the base station are proposed To characterize the performance of the proposed selection schemes, closed-form expressions for the outage probability and system throughput are derived These analytical results demonstrate that the use of SWIPT will not jeopardize the diversity gain compared to the conventional NOMA The proposed results confirm that the opportunistic use of node locations for user selection can achieve low outage probability and deliver superior throughput in comparison to the random selection scheme Index Terms Non-orthogonal multiple access, simultaneous wireless information and power transfer, stochastic geometry, user selection I INTRODUCTION Non-orthogonal muliple access NOMA is an effective solution to improve spectral efficiency and has recently received significant attention for its promising application in fifth generation 5G networks [1] The key idea of NOMA is to realize multiple access MA in the power domain which is fundamentally different from conventional orthogonal MA technologies eg, time/frequency/code division MA The motivation behind this approach lies in the fact that NOMA can use spectrum more efficiently by opportunistically exploring users channel conditions [] In [3], the authors investigated the performance of a downlink NOMA scheme with randomly deployed users An uplink NOMA transmission scheme was proposed in [4], and its performance was evaluated systematically In [], the impact of user pairing was characterized by analyzing the sum rates in two NOMA systems, namely, fixed power allocation NOMA and cognitive radio inspired NOMA In [5], a new cooperative NOMA scheme Y Liu and M Elkashlan are with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, UK {yuanweiliu, magedelkashlan}@qmulacuk Z Ding and H V Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NJ 8544, USA poor@princetonedu Z Ding is also with the School of Computing and Communications, Lancaster University, LA1 4WA, UK zding@lancasteracuk This research was supported in part by the UK EPSRC under grant number EP/L57/1 and the US National Science Foundation under Grant EECS was proposed and analyzed in terms of outage probability and diversity gain In addition to improving spectral efficiency which is the motivation of NOMA, another key objective of future 5G networks is to maximize energy efficiency Simultaneous wireless information and power transfer SWIPT, which was initially proposed in [6], has rekindled the interest of researchers to explore more energy efficient networks In [6], it was assumed that both information and energy could be extracted from the same radio frequency signals at the same time, which does not hold in practice Motivated by this issue, two practical receiver architectures, namely time switching TS receiver and power splitting PS receiver, were proposed in a multi-input and multi-output MIMO system in [7] Since point-to-point communication systems with SWIPT are well established in the existing literature, recent research on SWIPT has focused on two common cooperative relaying systems: amplify-andforward AF and decode-and-forward DF On the one hand, for the AF relaying, a TS-based relaying protocol and a PSbased relaying protocol were proposed in [8] On the other hand, for the DF relaying, a new antenna switching SWIPT was proposed in [9] to lower the implementation complexity In [1], the application of SWIPT to DF cooperative networks with randomly deployed relays was investigated using stochastic geometry in a cooperative scenario with multiple source nodes and a single destination A scenario in which multiple source-destination pairs are randomly deployed and communicate with each other via a single energy harvesting relay was considered in [11] A Motivation and Contributions One important advantage of the NOMA concept is that it can squeeze a user with better channel conditions into the bandwidth channel which has been occupied by a user with poor channel conditions [] For example, consider a practical scenario, in which there are two groups of users: 1 near users, which are close to the base station S and have better channel conditions; and far users, which are close to the edge of the cell controlled by the S and therefore have poor channel conditions While the spectral efficiency of NOMA is superior compared to orthogonal MA, the fact that the near users co-exist with the far users brings performance degradation at the far users In order to improve the reliability of the far users, an efficient method was proposed in [5] by applying cooperative transmission to NOMA The key idea

2 of this cooperative NOMA scheme is that the near users that are close to the S are used as relays to help the far users with poor channel conditions The advantage of implementing cooperative transmission in NOMA systems is that successive interference cancelation is used at the near users and hence the information of the far users is known by these near users In this case, it is natural to consider the use of the near users as decode-and-forward DF relays to transmit information to the far users In this paper, we consider that the near users are energy constrained and hence harvest energy from their received RF signals To improve the reliability of the far NOMA users without draining the near users batteries, we consider the application of SWIPT to NOMA, where SWIPT is performed at the near NOMA users Therefore, the aforementioned two communication concepts, cooperative NOMA and SWIPT, can be naturally linked together, and a new spectrally and energy efficient wireless multiple access protocol, namely, the cooperative SWIPT NOMA protocol, is proposed in this paper In order to investigate the impact of the locations of randomly deployed users on the performance of the proposed protocol, stochastic geometry tools are used Particularly, users are spatially randomly deployed in two groups via homogeneous Poisson point process PPP Here, the near users are grouped together and randomly deployed in an area close to the S The far users are in the other group and deployed close to the edge of the cell controlled by the S Since NOMA is co-channel interference limited, it is important to combine NOMA with conventional orthogonal MA technologies and realize a new hybrid MA network For example, we can first group users in pair to perform NOMA, and then use conventional time/frequency/code division MA to serve the different user pairs Note that this hybrid MA scheme can effectively reduce the system complexity since less users are grouped together for the implementation of NOMA ased on the proposed protocol and the considered stochastic geometric model, a natural question arises: which near NOMA user is to help which far NOMA user? To investigate the performance of one pair of selected NOMA users, three opportunistic user selection schemes are proposed, based on locations of users to perform NOMA as 1 random near user and random far user RNRF selection, where both the near and far users are randomly selected from the two groups; nearest near user and nearest far user NNNF selection, where a near user and a far user closest to the S are selected from the two groups; and 3 nearest near user and farthest far user NNFF selection, where a near user which is closest to the S is selected and a far user which is farthest from the S is selected The insights obtained from these opportunistic user selection schemes provide the precise guidance for the design of dynamic user clustering algorithms, which is beyond the scope of the paper The primary contributions of our paper are summarized as follows We propose a new SWIPT NOMA protocol to improve the reliability of the far users with the help of the near users without consuming extra energy With this in mind, three user selection schemes are proposed by opportunistically taking into account the users locations We derive closed-form expressions for the outage probability at the near and far users, when considering the three proposed user selection schemes In addition, we analyze the delay-sensitive throughput based on the outage probability at the near and far users We derive the diversity gain of the three proposed selection schemes for the near and far users We conclude that all three schemes have the same diversity For the far users, it is worth noting that the diversity gain of the proposed cooperative SWIPT NOMA is the same as that of a conventional cooperative network without radio frequency energy harvesting Comparing RNRF, NNNF, and NNFF, we confirm that NNNF achieves the lowest outage probability and the highest throughput for both the near and far users Organization The rest of the paper is organized as follows In Section II, the network model considering cooperative SWIPT NOMA is presented In Section III, new analytical expressions are derived for the outage probability, diversity gain, and throughput when the proposed selection schemes, RNRF, NNNF, and NNFF, are used Numerical results are presented in Section IV, which is followed by the conclusion in Section V II NETWORK MODEL We consider a network with a single source S ie, the base station S and two groups of randomly deployed users {A i } and { i } We assume that the users in group { i } are deployed within disc D with radius R D The far users {A i } are deployed within ring D A with radius R DC and R DA assuming R DC R D, as shown in Fig 1 Note that the S is located at the origin of both the disc D and the ring D A The locations of the near and far users are modeled as homogeneous PPPs Φ κ κ {A, } with density λ Φκ Here the near users are uniformly distributed within the disc and the far users are uniformly distributed within the ring The number of users in R Dκ, denoted by N κ, follows a Poisson distribution as Pr N κ = k = µ k κ/k!e µ κ, where µ κ is the mean measure, ie, µ A = π RD A RD λφa C and µ = πrd λ Φ All channels are assumed to be quasi-static Rayleigh fading, where the channel coefficients are constant for each transmission block but vary independently between different blocks In the proposed network, we consider that the users in { i } are energy harvesting relays that harvest energy from the S and forward the information to {A i } using the harvested energy as their transmit power The DF strategy is applied at { i } and the cooperative NOMA system consists of two phases, detailed in the following In this work, without loss of generality, it is assumed that the two phases have the same transmission periods, the same as in [8, 1, 11] It is worth pointing out that dynamic time allocation for the two phases can further improve the performance of the proposed cooperative NOMA scheme, which is beyond the scope of the paper

3 3 A 3 A 6 A 5 R D C 6 R DA h Ai S R 5 4 DC h i R R D 1 3 g i A i i D A 1 A 4 Direct Transmission Phase with SWIPT Cooperative Tansmission Phase Fig 1 An illustration of SWIPT NOMA considering a S blue circle The spatial distributions of the near users yellow circles and the far users green circles follow homogeneous PPPs A Phase 1: Direct Transmission Prior to transmission, the two users denoted by A i and i, are selected to perform NOMA, where the selection criterion will be discussed in the next section During the first phase, the S sends two messages p i1 x i1 + p i x i to two selected users A i and i based on NOMA [3], where p i1 and p i are the power allocation coefficients and x i1 and x i are the messages of A i and i, respectively The observation at A i is given by y Ai,1 = h Ai P S p ik x ik + n 1 + d Ai,1, 1 Ai k {1,} where P S is the transmit power at the S, h Ai models the small-scale Rayleigh fading from the S to A i with h Ai CN, 1, n Ai,1 is the additive Gaussian white noise AWGN at A i with variance σa i, d Ai is the distance between S and A i, and is the path loss exponent Without loss of generality, we assume that p i1 > p i with p i1 + p i = 1 The received signal to interference and noise ratio SINR at A i to detect x i1 is given by γ x ρ h i1 Ai p i1 S,A i = ρ p i h Ai, d A i where ρ = P S σ is the transmit signal to noise radio SNR assuming σa i = σ i = σ We consider that the near users have the rechargeable storage ability [8] and power splitting [7] is applied to perform SWIPT From the implementation point of view, this rechargeable storage unit can be a supercapacitor or a short-term highefficiency battery [9] The power splitting approach is applied as explained in the following: the observation at i is divided into two parts One part is used for information decoding by A directing the observation flow to the detection circuit and the remaining part is used for energy harvesting to power i for helping A i Thus, y i,1 = P S k {1,} 1 βi h i p ik x ik + n 1 + d i,1, 3 i where β i is the power splitting coefficient which is detailed in 7, h i models the small-scale Rayleigh fading from S to i with h i CN, 1, n i is AWGN at n i,1 with variance σ i, and d i is the distance between S and i We use the bounded path loss model to ensure that the path loss is always larger than one even for small distances [1] Applying NOMA, the successive interference cancellation SIC [1] is carried out at i Particularly, i first decodes the message of A i, then subtracts this component from the received signal to detect its own information Therefore, the received SINR at i to detect x i1 of A i is given by γ x ρ h i1 i p i1 1 β i S, i = ρ h i p i 4 1 β i d i The received SNR at i to detect x i of i is given by γ xi S, i = ρ h i p i 1 β i 1 + d i 5 The power splitting coefficient β i is used to determine the amount of harvested energy ased on 4, the data rate supported by the channel from S to i for decoding x i1 is given by R xi1 = 1 1 log ρ h i p i1 1 β i + ρ h i p i 6 1 β i d i We assume that the energy required to receive/process information is negligible compared to the energy required for information transmission [8] In this work, we apply the dynamic power splitting protocol which means that the power splitting coefficient β i is a variable and opportunistically tuned to support the relay transmission Our aim is to first guarantee the detection of the message of the far NOMA user, A i, at the near NOMA user i, then i can harvest the remaining energy In this case, based on 6, in order to ensure that i can successfully decode the information of, A i, at a given target rate, ie, R 1 = R xi1 Therefore, the power splitting coefficient is set as β i = max, 1 τ d i ρ p i1 τ 1 p i h i, 7 where τ 1 = R1 1 Here β i = means that all the energy is used for information decoding and no energy is remaining for energy harvesting ased on 3, the energy harvested at i is given by E i = T ηp Sβ i h i 1 + d i, 8 where T is the time period for the whole transmission including the direct transmission phase and the cooperative transmission phase, and η is the energy harvesting coefficient

4 4 We assume that the two phases have the same transmission period, therefore, the transmit power at i is expressed as Phase : Cooperative Transmission P t = ηp Sβ i h i 1 + d i 9 During this phase, i forwards x i1 to A i by using the harvested energy during the direct transmission phase In this case, A i observes Pt x i1 g i y Ai, = + n 1 + d Ai,, 1 Ci where g i models the small-scale Rayleigh fading from i to A i with g i CN, 1, n Ai, is AWGN at A i with variance σa i, d Ci = d A i + d i d Ai d i cos θ i is the distance between i and A i, and θ i denotes the angle A i S i ased on 9 and 1, the received SNR for A i to detect x i1 forwarded from i is given by γ x P i1 t g i A i, i = 1 + d = β i h i g i 11 Ci σ 1 + d Ci 1 + d i At the end of this phase, A i combines the signals from S and i using maximal-ratio combining MRC Combining the SNR of the direct transmission phase and the SINR of the cooperative transmission phase 11, we obtain the received SINR at A i as γ x i1 A = ρ h Ai p i1 i,mrc ρ h Ai p i + β i h i g i d A i 1 + d i 1 + d Ci III NON-ORTHOGONAL MULTIPLE ACCESS WITH USER SELECTION 1 In this section, the performance of three user selection schemes are characterized in the following A RNRF Selection Scheme In this scheme, the S randomly selects a near user i and a far user A i This selection scheme provides a fair opportunity for each user to access the source with the NOMA protocol The advantage of this user selection scheme is that it does not require the knowledge of instantaneous channel state information CSI To make meaningful conclusions, in the rest of the paper, we only focus on β i > and the number of the near users and the far users both satisfy N 1, N A 1 1 Outage Probability of the Near Users of RNRF: In the NOMA protocol, an outage of i can occur due to two reasons The first is when i cannot detect x i1 The second is when i can detect x i1 but cannot detect x i To guarantee that the NOMA protocol can be implemented, the condition p i1 p i τ 1 > should be satisfied [3] ased on this, the outage probability of i can be expressed as ρ h i p i1 P i = Pr ρ h i p i < τ d 1 i ρ h i p i1 + Pr ρ h i p i > τ d 1, γ x i S, i < τ, 13 i where τ = R 1 with R being the target rate at which i can detect x i The following theorem provides the outage probability of the near users in RNRF for an arbitrary choice of Theorem 1: Conditioned on PPP, the outage probability of the near users i can be approximated in closed-form as P i 1 N ω N 1 ϕ n 1 e cnεa i ϕn + 1, 14 if ε Ai ε i, otherwise P i = 1, where ε Ai = τ 1 τ and ε ρ p i1 p i τ 1 i =, N is a parameter to ensure a complexity-accuracy tradeoff, c n = 1 + ρ p i RD, ϕ n + 1 ωn = π N, and ϕ n = cos n 1 N π Proof: Define X i = i ha 1+d, Y i = i h A 1+d, and Z i = g i i 1+d i C i Substituting 4 and 5 into 13, the outage probability of the near users is given by P i = Pr Y i < ε Ai + Pr Y i > ε Ai, ε Ai < ε i 15 If ε Ai < ε i, the outage probability at the near users is always one For the case ε Ai ε i, note that the users are deployed in D and D A according to homogeneous PPPs Therefore, the NOMA users are modeled as independently and identically distributed iid points in D and D A, denoted by W κi κ {A, }, which contain the location information about A i and i, respectively The probability density function PDF of W Ai and W i is given by and f Wi ω i = λ Φ µ RD = 1 πr D, 16 f WAi ω Ai = λ Φ A 1 = µ RDA π, RD A RD 17 C respectively Therefore, for the case ε Ai ε i, the cumulative distribution function CDF of Y i is given by F Yi ε = 1 e 1+d i ε f Wi ω i dω i D = R D RD 1 e 1+r ε rdr 18 For many communication scenarios >, it is challenging to obtain exact closed-from expressions In this case,

5 5 we provide Gaussian-Chebyshev quadrature [13] to find the approximation of 18 as F Yi ε 1 N ω N 1 ϕ n 1 e c nε ϕ n Applying ε Ai ε into 19, 14 is obtained, and the proof of the theorem is completed Corollary 1: For the special case =, the outage probability of i can be obtained as exact expression as e εa i P i = = 1 RD ε Ai + e 1+R D ε Ai RD, ε Ai if ε Ai ε i, otherwise P i = = 1 Proof: ased on 18, when = and after some manipulations, we can easily obtain F Yi ε = = 1 e ε RD ε + e 1+R RD ε D ε 1 Applying ε Ai ε into 1, can be obtained The proof is completed Outage Probability of the Far Users of RNRF: With the proposed cooperative SWIPT NOMA protocol, the outage experienced by A i can occur due to two reasons The first is when i can detect x i1 but the overall received SNR at A i cannot support the targeted rate The second is that neither A i nor i can detect x i1 ased on this, the outage probability can be expressed as P Ai = Pr γ x i1 A < τ i,mrc 1, γ x i1 βi S, > τ i 1 = + Pr γ x i1 S,A i < τ 1, γ x i1 βi S, < τ i 1 = The following theorem provides the outage probability of the far users in RNRF for an arbitrary choice of Theorem : Conditioned on PPP, assume R DC R D, the outage probability of A i can be approximated in closed-form as N P Ai ζ 1 ϕ n K ϕ n c n 1 ψk s k1 + s k M + a 1 N m e 1+s k t m χ tm ln χ t m 1 + s k c n + c 1 ϕ n c n ϕ n + 1 K 1 ψ k 1 + s k s k, 3 where M and K are parameters to ensure a complexityaccuracy tradeoff, ζ 1 = ε A i R Di ω N ω K ω M 8 R DAi +R DCi, χ t m = τ 1 ρt m p i1, t ρt m p i +1 m = ε A i φ m + 1, ω M = π M, φ m = cos m 1 M π, s k = R D A R DC ψ k R DC, ω K = π K, ψ k = cos k 1 K π, c = φ1 φ, and a 1 = ω K ω N ε A 1 R DA +R DC Proof: See Appendix A Corollary : For the special case =, the outage probability of A i can be simplified in closed-form as K P Ai = ζ 1 ψk s k 1 + s M k m χ tm e 1+s kt m ln χ t m 1 + s k c n + b + 1 e 1+R D C ε Ai ε Ai R DA RD + e 1+R D A ε Ai C ε Ai R DA RD C 1 e ε A i RD + e 1+R D ε Ai ε Ai RD, 4 ε Ai where ζ = ω Kω M ε Ai R D + and b 8R DA +R = DC 1+R D ln1+r D + R RD D + c 1 4 Proof: See Appendix 3 Diversity Analysis of RNRF: To obtain further insights for the derived outage probability, we provide the diversity analysis of both the near and far uses of RNRF Near users: For the near users, based on the analytical results, we carry out high SNR approximations as follows When ε, a high SNR approximation of 19 with 1 e x x is given by F Yi ε 1 N ω N 1 ϕ n c n ε Ai ϕ n The diversity gain is defined as log P ρ d = lim ρ log ρ 6 Substituting 5 into 6, we obtain that the diversity gain for the near users is one, which means that using NOMA with energy harvesting will not decrease the diversity gain Far users: For the far users, substituting 3 into 6, we obtain log 1 ρ log 1 d = lim ρ ρ log ρ = lim ρ log log ρ log ρ = 7 log ρ As we can see from 7, the diversity gain of RNRF is two, which is the same as that of the conventional cooperative network [14] This result indicates that using NOMA with an energy harvesting relay will not affect the diversity gain In addition, we see that in high SNRs, the dominant factor for the outage probability is 1 ρ ln ρ Therefore we conclude that the outage probability of using NOMA with SWIPT decays ln SNR at a rate of However, for a conventional cooperative SNR system without energy harvesting, a faster decreasing rate of can be achieved 1 SNR

6 6 4 System Throughput in Delay-Sensitive Transmission Mode of RNRF: In this paper, we will focus on the delaysensitive throughput In this mode, the transmitter sends information at a fixed rate and the throughput is determined by evaluating the outage probability ased on the analytical results for the outage probability of the near and far users, the system throughput of RNRF in the delay-sensitive transmission mode is given by R τrnrf = 1 P Ai R P i R, 8 where P Ai and P i are obtained from 3 and 14, respectively NNNF Selection Scheme In this subsection, we characterize the performance of NNNF, which exploits the users CSI opportunistically We first select a user within the disc D which has the shortest distance to the S as the near NOMA user denoted by i This is because the near users also act as energy harvesting relays to help the far users The NNNF scheme can enable that the selected near user to harvest more energy Then we select a user within the ring D A which has the shortest distance to the S as the far NOMA user denoted by A i The advantage of NNNF scheme is that it can minimize the outage probability of both the near and far users 1 Outage Probability of the Near Users of NNNF: Using the same definition of the outage probability as the near users of NOMA, we can characterize the outage probability of the near users of NNNF The following theorem provides the outage probability of the near users of NNNF for an arbitrary choice of Theorem 3: Conditioned on PPP, the outage probability of i can be approximated in closed-form as P i b 1 N 1 ϕ n 1 e 1+cn ε Ai c n e πλ Φ c n, 9 if ε Ai ε i, otherwise P i = 1, where c n = R D ϕ n + 1, b 1 = ξω N R D πλ, and ξ = Φ 1 e πλ Φ R D Proof: Similar to 15, the outage probability of i can be expressed as P i = Pr Y i < ε Ai N 1 = F Yi ε Ai, 3 where Y i = h i 1+d and d i is the distance from the nearest i i to the S The CDF of Y i can be calculated as F Yi ε = RD 1 e 1+r ε f di r dr, 31 where f di is the PDF of the shortest distance from i to the S The probability Pr {d i > r N 1} conditioned on N 1 is with the event that there is no point located in the disc Therefore we can express this probability as Pr {d i > r N 1} = Pr {d i > r} Pr {d i > r, N = } Pr {N 1} = e πλφ r e πλ Φ RD 3 1 e πλ Φ RD Then the corresponding PDF of i is given by f di r = ξ r e πλ Φ r 33 Substituting 33 into 31, we obtain F Yi ε = ξ RD 1 e 1+r ε r e πλφ r dr 34 Applying the Gaussian-Chebyshev quadrature approximation to 19, we obtain F Yi ε ξ ω N R D N 1 ϕ n 1 e ε 1+c n c n e πλ Φ c n 35 Applying ε Ai ε, we obtain the outage probability of i in 9 ased on 34 and after some manipulations, the following corollary can be obtained Corollary 3: For the special case =, the outage probability of i can be expressed in exact closed-form as ξ e R D πλ Φ +ε Ai ε Ai e ε Ai P i = = πλ Φ + ε Ai ξ e πλφ R D 1, 36 πλ Φ if ε Ai ε i, otherwise P i = = 1 Outage Probability of the Far Users of NNNF: Using the same definition of the outage probability of the far users of NOMA, and similar to, we can characterize the outage probability of the far users of NNNF The following theorem provides the outage probability of the far users of NNNF for an arbitrary choice of Theorem 4: Conditioned on PPP, assume R DC R D, the outage probability of A i can be approximated in closed-form

7 7 as P Ai ς N 1 ϕ n 1 + c n c n e πλ Φ c n K 1 ψ k 1 + sk s k e πλ Φ A s k R DC M m e 1+s k t m χ tm ln χ t m 1 + s k 1 + c n + c K + b b 3 1 ψ k 1 + s k s k e πλ Φ A s k N 1 ϕ n 1 + c n c n e πλ Φ c n, 37 where ς = ξ ξ A ω N ω K ω M ε Ai R D R DA R DC 8, b = ξ A e πλ Φ A R D C ωk ε Ai R DA +R DC, and b 3 = ξ ω N R D ε Ai Proof: See Appendix C Corollary 4: For the special case =, the outage probability of A i can be simplified in closed-form as 38 on the top of this page Proof: For the special case =, after some manipulations, we can express C11 in closed-form as ξ A e πλ Φ A R D R C D A 1 F Xi ε = = πλ ΦA + ξ Ae πλφ A R D C e ε e R D A πλ ΦA +ε e R D C πλ ΦA +ε πλ ΦA + ε 39 ased on C1, combining 39 and 36, and applying = into C9, we can obtain 38 The proof is completed 3 Diversity Analysis of NNNF: Similarly, we provide diversity analysis of both the near and far users of NNNF Near users: For the near users, based on the analytical results, we carry out the high SNR approximation as follows When ε, a high SNR approximation of 9 with 1 e x x is given by N P i b 1 ε Ai 1 ϕ n 1 + c n c n e πλφ c n 4 Substituting 4 into 6, we obtain that the diversity gain for the near users of NNNF is one, which indicates that using NNNF will not affect the diversity gain Far users: For the far users, substituting 37 into 6, we obtain that the diversity gain is still two It indicates that NNNF will not affect the diversity gain 4 System Throughput in Delay-Sensitive Transmission Mode of NNNF: ased on the analytical results of the outage probability of the near and far users, the system throughput of NNNF in the delay-sensitive transmission mode is given by R τnnnf = 1 P Ai R P i R, 41 where P Ai and P i are obtained from 37 and 9, respectively C NNFF Selection Scheme In this scheme, we first select a user within disc D which has the shortest distance to the S as a near NOMA user Then we select a user within ring D A which has the farthest distance to the S as a far NOMA user denoted by A i The use of this selection scheme is inspired by an interesting observation described in [3] that NOMA can offer a larger performance gain over conventional MA when user channel conditions are more different 1 Outage Probability of the Near Users of NNFF: Since the same criterion for the near users is used, the outage probabilities of near nears for an arbitrary and the special case = are the same as those expressed in 9 and 36, respectively Outage Probability of the Far Users of NNFF: Using the same definition of the outage probability of the far users, and similar to, we can characterize the outage probability of the far users of NNFF The following theorem provides the outage probability of the far user of NNFF for an arbitrary choice of Theorem 5: Conditioned on PPP, assume R DC R D, the outage probability of A i can be approximated as N P Ai ς 1 ϕ n 1 + c n c n e πλ Φ c n K 1 ψ k 1 + sk s k e πλφ AR DA s k M m e 1+s k t m χ tm ln χ t m 1 + s k 1 + c n + c K + b 3 b 4 1 ψ k 1 + s k s k e πλ Φ A s k N 1 ϕ n 1 + c n c n e πλ Φ c n, 4 ΦA where b 4 = ξae πλ R D A ωk ε Ai R DA +R DC Proof: See Appendix D Corollary 5: For the special case =, after some manipulations, the outage probability of A i can be simplified in closed-form as 43 on the top of the next page 3 Diversity Analysis: Similarly, we provide diversity analysis of both the near and far uses of NNFF Near users: Since the same criterion for selecting a near user is used, the diversity gain is one which is the same as NNNF Far users: Substituting 4 into 6, we find that the diversity gain is still two Therefore, we conclude that using opportunistic user selection schemes NNNF and NNFF based on distances will not affect the diversity gain 4 System Throughput in Delay-Sensitive Transmission Mode of NNFF: ased on the analytical results of the outage probability of the near and far users, the system throughput of the NNFF in the delay-sensitive transmission mode is given by R τnnff = 1 P Ai R1 + 1 P i R, 44

8 8 P Ai = ς N 1 ϕ n 1 + c n cn e πλ Φ c n K 1 ψk 1 + s sk k e πλ Φ A s k R D C M m e 1+s kt m χ tm ln χ t m 1 + s k 1 + c n + c + ξ Ae πλ Φ A RD C e εa i e R D A πλ ΦA +ε Ai e R D C πλ ΦA +ε Ai e πλ Φ A R DA e πλ Φ A R DC πλ ΦA + ε Ai πλ ΦA ξ e R D πλ Φ +ε Ai ε Ai e ε Ai R e πλφ D 1 38 πλ Φ + ε Ai πλ Φ P Ai = ς N 1 ϕ n 1 + c n cn e πλφ c n M m e 1+s kt m χ tm + ξ Ae πλ Φ A RD A e πλ Φ A R D A e πλ Φ A R D C πλ ΦA e R D πλ Φ +ε Ai ε Ai e ε Ai ξ πλ Φ + ε Ai K ln χ t m 1 + s k 1 + c n 1 ψk 1 + s sk k e πλφ AR D s A k + c e ε A i e R D A πλ ΦA ε Ai e R D C πλ ΦA ε Ai πλ ΦA ε Ai e πλ Φ R D 1 πλ Φ 43 where P Ai and P i are obtained from 4 and 9, respectively IV NUMERICAL RESULTS In this section, numerical results are presented to facilitate the performance evaluations including the outage probability of the near and the far users and the delay sensitive throughput of the proposed cooperative SWIPT NOMA protocol In the considered network, we assume that the energy conversion efficiency of SWIPT is η = 7 and the power allocation coefficients of NOMA is p i1 = 8, p i1 = In the following figures, we use red, blue and black color lines to represent the RNRF, NNNF and NNFF user selection schemes, respectively A Outage Probability of the Near Users In this subsection, the outage probability achieved by the near users with different choice of density and path loss coefficients for the three user selection schemes is demonstrated Note that the same user selection criteria is applied for the near users of NNNF and NNFF, we use NNNFF to represent these two selection schemes in Fig, Fig 3, and Fig 4 Fig plots the outage probability of the near users versus SNR with different path loss coefficients for both RNRF and NNNFF The solid red and blue curves are for the special case = of RNRF and NNNFF, corresponding to the analytical results derived in and 36, respectively The dashed red and blue curves are for an arbitrary choice of, corresponding to the analytical results derived in 14 and 9, respectively Monte Carlo simulation are marked as to verify our derivation The figure shows precise agreement between the simulation and analytical curves One can observe that by performing NNNF and NNFF which we refer to as NNNFF in the figure, lower outage probability is achieved than RNRF since shorter distance makes less path loss and leads to better performance The figure also demonstrates that as increases, outage will occur more because of higher path loss For NNNF and NNFF, the performance is very close for different values of This is because we use the bounded path loss model ie 1 + d i > 1 to ensure that the path loss is always larger than one When selecting the nearest near user, d i will approach to zero and the path loss will approach to one, which makes the performance difference of the three selection schemes insignificant It is worth noting that all curves have the same slopes, which indicates that the diversity gains of the schemes are the same This phenomenon validates the insights we obtained from the analytical results derived in 6 Fig also shows that if the choices of rates for users are incorrect ie, R 1 = 5 and R = 1 in this figure, the outage probability of the near users will be always one, which verifies the analytical results in 14 and 9 Fig 3 plots the outage probability of the near users versus the density with different R D RNRF is also shown in the figure as a benchmark for comparison Several observations are drawn as 1 The outage probabilities of RNRF and NNNFF decrease by decreasing R D because path loss is reduced; The outage probability of NNNFF decreases as the density of the near users increases This is due to the

9 Outage probability of the near users1 NNNFF RNRF Simulation Incorrect choice of rate RNRF analytical = RNRF analytical-appro = 3 RNRF analytical-appro = 4 NNNFF analytical = NNNFF analytical-appro = 3 NNNFF analytical-appro = 4 R1 = 5, R = 1 PCU R1 = R = 1 PCU SNR d Fig Outage probability of the near users versus SNR with different, where R D = m, and λ Φ = 1 Outage probability of the near users R1 PCU NNNFF RNRF 1 5 R PCU Fig 4 Outage probability of the near users versus R 1 and R, where =, R D = m, and SNR = 3 d Outage probability of the near users 1 RNRF NNNFF 1 1 RD = 3 m 1 RD RD = 4 m = m RD = 1 m Density of the near users Fig 3 Outage probability of the near users versus density with different R D, where λ Φ = 1, and SNR = 3 d multiuser diversity gain, since there is an increasing number of the near users; 3 The outage probability of RNRF is a constant, ie, independent of the density of near users, and is the outage ceiling of the NNNFF This is due to the fact that no opportunistic user selection is carried out for RNRF; and 4 An outage floor exits even if the density of the near users goes to infinity This is due to the used bounded path loss model When the number of the near users exceeds a threshold, the selected near user will be very close to the source, which makes the path loss approach to one Fig 4 plots the outage probability of the near users versus the rate of the near users and far users for both RNRF and NNNFF One can observe that the outage of the near users occurs more frequently as the rate of the far user, R 1, increases This is because in our proposed protocol, the near user i needs to first decode x i1 which is intended to the far user A i, and then decode of its own Therefore increasing R 1 makes it harder to decode x i1, which will lead to more outage An important observation is that incorrect choices of R 1 and R will make the outage probability always one Particularly, for the choice of R 1, it should satisfy the condition p i1 p i τ 1 > in order to ensure that successive interference cancelation can be implemented For the choice of R, it should satisfy the condition that the split energy for detecting x i1 is also sufficient to detect x i ε Ai ε i Outage Probability of the Far Users In this subsection, we demonstrate the outage probability of the far users with different choices of the density, path loss coefficients, and user zone of the three user selection schemes Fig 5 plots the outage probability of the far users versus SNR with different path loss coefficients of RNRF, NNNF, and NNFF The dashed red, blue, and black curves circled together and pointed by =, are the analytical approximation for the special case of RNRF, NNNF, and NNFF, which are obtained from 4, 38 and 43, respectively The dashed red, blue, and black curves circled together and pointed by = 3, are the analytical approximation for an arbitrary choice of of RNRF, NNNF, and NNFF, which are obtained from 3, 37 and 4, respectively We use the solid marked lines to represent the Monte Carlo simulation results for each case As can be observed from the figure, the simulation and the analytical approximation are very close, particularly in the high SNR region Several observations are drawn as 1 NNNF achieves the lowest outage probability among the three selection schemes since both the near and far users have the smallest path loss; NNFF achieves lower outage than RNRF, which indicates that the distance of the near users has more impact than that of the far users; 3 it is clear that all of the curves in Fig 5 have the same slopes, which indicates that the diversity gain of the far users for the three schemes are the same In the diversity analysis part, we derive the diversity gain of the three selection schemes is two The simulation validates the analytical results and indicates that the achievable diversity gain is the same for different user selection schemes

10 1 Outage probability of the far users RNRF simulation NNNF simulation NNFF simulation RNRF analytical-appro NNNF analytical-appro NNFF analytical-appro SNR d = = Fig 5 Outage probability of the far users with different, R 1 = 3 PCU, R DA = 1 m, R D = m, R DC = 8 m, λ ΦA = 1, and λ Φ = 1 Outage probability of the far users RNRF Cooperative NOMA NNNF Cooperative NOMA NNFF Cooperative NOMA RNRF Non-cooperative NOMA NNNF Non-cooperative NOMA NNFF Non-cooperative NOMA SNR d Fig 7 Comparison of outage probability with non-cooperative NOMA, = 3, R 1 = 3 PCU, R DA = 1 m, R D = m, R DC = 8 m, λ ΦA = 1, and λ Φ = 1 Outage probability of the far users RD C = 1 m, RD A = 14 m RD C = 8 m,rd A = 1 m RNRF NNNF NNFF R1 PCU Fig 6 Outage probability of the far users versus R 1, where =, R D = m, and SNR = 3 d Fig 6 plots the outage probability of the far users versus R 1 with different R DC and R D One can observe that the outage probabilities of the three schemes increase as R 1 increases This is because increasing R 1 will make the threshold of decoding higher, which in turn leads to more outage It can also be observed that increasing the radius of the user zone for the far users will deteriorate the outage performance The reason is that the path loss of the far users becomes larger Fig 7 plots the outage probability of the far users versus SNR for both cooperative NOMA and non-cooperative NO- MA 1 Several observations are drawn as 1 by using an energy constrained relay to perform cooperative NOMA 1 It is common to use outage probability as a criterion to compare the performance of cooperative transmission and non-cooperative transmission schemes [14] In the context of cooperative NOMA, the use of outage probability is particularly useful since cooperative NOMA is to improve the reception reliability of the far users transmission, the outage probability of the far users has a larger slope than that of non-cooperative NOMA, for all user selection schemes This is due to the fact that cooperative NOMA can achieve a larger diversity gain and guarantees more reliable reception for the far users in high SINR region; NNNF achieves the lowest outage probability among these three selection schemes both for cooperative NOMA and noncooperative NOMA because of its smallest path loss; 3 it is worth noting that NNFF has higher outage probability than RNRF in non-cooperative NOMA, however, it achieves lower outage probability than RNRF in cooperative NOMA This phenomenon indicates that it is very helpful and necessary to apply cooperative NOMA in NNFF due to the largest performance gain over non-cooperative NOMA C Throughput in Delay-Sensitive Transmission Mode Fig 8 plots the system throughput versus SNR with different targeted rates One can observe that NNNF achieves the highest throughput since it has the lowest outage probability among three selection schemes The figure also demonstrates the existence of the throughput ceilings in the high SNR region This is due to the fact that the outage probability is approaching to zero and the throughput is determined only by the targeted data rate It is worth noting that increasing R from R = 5 PCU to R = 1 PCU can improve the throughput, however, for the case R = PCU, the throughput is lowered This is because, in the latter case, the energy remaining for information decoding is not sufficient for message detection of the near user, and hence an outage occurs, which in turn affects the throughput Therefore, it is important to select appropriate transmission rates when designing practical NOMA downlink transmission systems V CONCLUSIONS In this paper, the application of SWIPT to NOMA has been considered A novel cooperative SWIPT NOMA pro-

11 11 System Throughput PCU R1 =1, R =1 14 PCU R1 =1, R =5 1 PCU 1 8 R1 =1, R = PCU 6 4 RNRF NNNF NNFF SNR d Fig 8 System Throughput in delay-sensitive mode versus SNR with different rate, =, R DA = 1 m, R D = m, R DC = 8 m, λ ΦA = 1, and λ Φ = 1 tocol with three different user selection criteria has been proposed We have used the stochastic geometric approach to provide a complete framework to model the locations of users and evaluate the performance of the proposed user selection schemes Closed-form results have been derived in terms of outage probability and delay-sensitive throughput to determine the system performance The diversity gain of the three user selection schemes has also been characterized and proved to be the same as that of a conventional cooperative network For the proposed protocol, the decreasing rate of the outage probability ln SNR 1 of far users is SNR while it is SNR for a conventional cooperative network Numerical results have been presented to validate our analysis We conclude that by carefully choosing the parameters of the network, eg, transmission rate or power splitting coefficient, acceptable system performance can be guaranteed even if the users do not use their own batteries to power the relay transmission One promising future direction is to design a hybrid MA network with dynamic user clustering/pairing APPENDIX A: PROOF OF THEOREM Substituting 4 and 1 into, the outage probability can be expressed as P Ai = Pr γ xi1 A i,mrc < τ ρ h i p i1 1, ρ h i p i > τ d 1 }{{ i } Θ 1 ρ h + Pr γ xi1 i p i1 S,A i < τ 1, ρ h i p i < τ d 1, }{{ i } Θ A1 We express Θ 1 as A on the top of next page where f Xi x = 1 + d A i e 1+d A i x, and f Yi y = 1 + d i e 1+d i y ased on A, using t = y ε Ai, we calculate Ξ as Ξ = 1 e 1+d C i τ ρx p i1 ρx p i +1 t 1 + d i e 1+d i t+ε Ai dt A3 Applying [15, Eq 334], we rewrite A3 as Ξ = e 1+d i ε Ai 1 χλk 1 χλ, A4 where Λ = 1+d i 1+d C i and χ = τ 1 ρx p i1 ρx p i +1 We use the series representation of essel functions to obtain the high SNR approximation which is expressed as xk 1 x 1 + x ln x + c, A5 where K 1 is the modified essel function for the seconde kind, c = φ1 φ, and φ denotes the psi function [15] To obtain the high SNR approximation of A4 and using A5, we obtain Ξ χλ ln χλ + c Substituting A6 into A, we rewrite A as Θ 1 = χ Λe 1+d A i x D i D Ai 1 + d i ln χλ + c f Wi ω i dω i A6 } {{ } Φ f WAi ω Ai dω Ai dx A7 Since d Ci = d A i + d i d Ai d i cos θ i and R DC R D, we can approximate the distance as d Ai d Ci Applying 16, we calculate Φ as Φ RD RD 1 + r ln m 1 + r + c rdr, A8 where m = χ1+d C i χ1+d A i For arbitrary choice of, the integral in A8 is mathematically intractable, we use Gaussian-Chebyshev quadrature to find the approximation Then Φ can be approximated as Φ ω N N 1 ϕ n c n ln m c n + c ϕ n + 1 A9

12 1 Θ 1 = Pr Z i < τ 1 ρx i p i1 ρx i p i +1 Y i ε Ai, X i < ε Ai, Y i > ε Ai = D D A ε Ai 1 e 1+d C i τ ρx p i1 1 ρx p i +1 y ε Ai f Y i y dy } {{ } Ξ f Xi x dxf WAi ω Ai dω Ai f Wi ω i dω i, A Substituting A9 into A7, we rewrite A7 as ω εai N χ N Θ 1 = RD A RD ϕ n ϕ n C RDA r1 + r e 1+r x c n ln χ 1 + r c n + c dr R DC }{{} dx A1 Similarly as above, we use Gaussian-Chebyshev quadrature to find an approximation of in A1 as ω K R DA R DC K 1 ψk s k1 + s k e 1+s k x c n ln χ 1 + s k c n + c A11 Substituting A11 into A1, we rewrite A1 as N Θ 1 =a ϕ n K ϕ n c n 1 ψk s k1 + s k χe 1+s k x ln χ 1 + s k c n + c dx, }{{} Ψ ω N ω K A1 where a = R DA +R DC Similarly, we use Gaussian-Chebyshev quadrature to find an approximation of Ψ in A1 as Ψ ω M ε Ai M m e 1+s k t m χ tm ln χ t m 1 + s k c n + c A13 Substituting A13 into A1, we obtain N Θ 1 ζ 1 ϕ n K ϕ n c n 1 ψk s k1 + s k M m e 1+s k t m χ tm ln χ t m 1 + s k c n + c A14 We express Θ as Θ = Pr X i < ε Ai Pr Y i < ε Ai A15 The CDF of X i for A i is given by F Xi ε = 1 e 1+d A i ε f WAi ω Ai dω Ai D = RD A RD C RDA R DC 1 e 1+r ε rdr A16 For an arbitrary choice of, similarly as 19, we provide Gaussian-Chebyshev quadrature to find the approximation for the CDF of X i We rewrite A16 as ω K F Xi ε R DA + R DC K 1 ψ k 1 e 1+s k ε s k A17 When ε, a high SNR approximation of the A17 is given by ω K ε F Xi ε R DA + R DC K 1 ψ k 1 + s k s k A18 Substituting A18 and 19 into A15, we can obtain the approximation for the general case as Θ a 1 N 1 ϕ n c n ϕ n + 1 K 1 ψ k 1 + s k s k Combining A14 and A19, we can obtain 3 The proof is completed APPENDIX : PROOF OF COROLLARY A19 For the special case =, and let λ = 1 + r, we rewrite A8 as Φ = = 1 R D 1+R D 1 λ ln m λ + c dλ R = D + ln m + b, 1 where m = χ1+d C i χ1+d A i

13 13 Substituting 1 and applying = into A, we obtain R D + Θ 1 = = RD A RD χ C RDA r 1 + r e 1+r x ln χ 1 + r + b dr dx R DC }{{} = We notice that the integral = in is mathematically intractable We use Gaussian-Chebyshev quadrature to find an approximation Then = can be approximated as = ω K R DA R DC K 1 ψk s k 1 + s e 1+s kx k ln χ 1 + s k + b 3 Substituting 3 into, we rewrite as Θ 1 = = ω K R D + K 1 ψk 4 R DA + R DC s k 1 + s k εai χe 1+s kx ln χ 1 + s k + b dx 4 }{{} Ψ = Similarly, we use Gaussian-Chebyshev quadrature to find the approximation of Ψ = in 4 as Ψ = ω Mε Ai M m χ tm e 1+s kt m ln χ t m 1 + s k c n + b 5 Substituting 5 into 4, we obtain K Θ 1 = = ζ 1 ψk s k 1 + s M k m χ tm e 1+s kt m ln χ t m 1 + s k c n + b 6 For the special case =, the CDF of X i in A16 can be calculated as F Xi ε = = 1 e 1+R D C ε ε RD A RD + e 1+R D A ε C ε RD A RD C 7 Substituting 7 and 1 into A15, we can obtain Θ for the special case = in exact closed-form as Θ = = 1 e 1+R D C ε Ai ε Ai R DA RD + e 1+R D A ε Ai C ε Ai R DA RD C 1 e εa i RD + e 1+R D ε Ai ε Ai RD 8 ε Ai Combining 6 and 8, we can obtain 4 The proof is completed APPENDIX C: PROOF OF THEOREM 4 Conditioned on both the number of users in group {A i } and { i } satisfy V = N A 1, N 1, we express the outage probability for A i by applying X i X i, Y i Y i, and Z i Z i in A1 then obtain ρx i p i1 P Ai = Pr ρx i p i + 1 < τ ρy i p i1 1, ρ p i Y i + 1 < τ 1 V }{{} + Pr Z i < τ 1 ρxi p i1 ρx i p i +1 Y i ε Ai, X i < ε A i, Y i > ε Ai V, } {{ } Θ 1 C1 where X i = i ha 1+d, Y i = i h A 1+d, and Z i = g i i 1+d Here i C i d Ai, d i, and d Ci are distances from the S to A i, from the S to i, and from A i to i, respectively Since R DC R D, we can approximate the distance as d Ai d Ci Using the similar approximation method to obtain A, we calculate Θ 1 as Θ 1 = RDA 1 + ra χ R DC Θ e 1+r A x Φ f dai r A dr A dx, C where Φ = R D 1 + r ln χ 1+r 1+r A + c f di r dr and f dai is the PDF for the nearest A i Similar to 33 and applying stochastic geometry within the ring D A, we obtain f dai r A as where ξ A = f dai r A = ξ A r A e πλ Φ A r A R D C, 1 e πλ Φ A πλ ΦA R DA R DC C3 Substituting C3 and 33 into C, using the Gaussian- Chebyshev quadrature approximation, Φ can be expressed as Φ ξ ω N R D N 1 ϕ n 1 + c n ln m 1 + c n + c c n e πλφ c n C4 where m = χ1+r A Substituting C4 into C, we obtain χe 1+r A x Θ 1 = ξ ξ A ω N R D N 1 ϕ n 1 + c n c n e πλ Φ c n dx, where = R DA R DC ln χ 1+r A 1 + c n + c 1 + r A r A e πλ Φ A r A R D C dra C5

14 14 Applying the Gaussian-Chebyshev quadrature approximation into, we obtain ω K R DA R DC K 1 ψk 1 + s k ln χ 1 + s k 1 + c n + c s k e πλ Φ A s k R D C C6 Substituting C6 into C, we obtain N Θ 1 = b 5 1 ϕ n 1 + c n c n e πλ Φ c n K 1 ψ k 1 + s k s k e πλ Φ A s k R D C χe 1+r A x ln χ 1 + s k 1 + c n + c dx, }{{} Ψ C7 Where b 5 = ξξaω N ω K R D R DA R DC 4 Applying Gaussian-Chebyshev quadrature approximation into Ψ, we obtain M Ψ ε Ai ω M m e 1+s k t m χ tm ln χ t m 1 + s k 1 + c n + c C8 Substituting C8 into C7, we obtain N K Θ 1 = ς 1 ϕ n 1 + c n c n e πλ Φ c n 1 ψk 1 + s k s k e πλ Φ A s k R D C M m e 1+s k tm χ tm ln χ t m 1 + s k 1 + c n + c C9 Conditioned on the number of users in group {A i } and { i }, we obtain Θ as Θ = Pr X i < ε Ai N A 1 Pr Y i < ε Ai N 1 = F Xi ε Ai F Yi ε Ai C1 Similar to 34, the CDF of A i is given by F Xi ε = ξ A RDA R DC 1 e 1+r A ε r A e πλ Φ A r A R D C dra C11 Applying the Gaussian-Chebyshev quadrature approximation, we obtain K F Xi ε b 1 ψ k 1 e ε 1+s k s k e πλ Φ A s k C1 Combining C1 and 35 into C1 and with high SNR approximation, we obtain Θ b b 3 K N 1 ψ k 1 + s k s k e πλφ A s k 1 ϕ n 1 + c n c n e πλ Φ c n C13 Combining C13 and C7, we can obtain 37 The proof is completed APPENDIX D: PROOF OF THEOREM 5 We express the outage probability for A i by applying X i X i, Y i Y i, and Z i Z i in A1 and obtain ρx i p i1 P Ai = Pr ρx i p i + 1 < τ ρy i p i1 1, ρ p i Y i + 1 < τ 1 V }{{} + Pr Z i < τ 1 ρxi p i1 ρx i p i +1 Y i ε Ai, X i < ε A i, Y i > ε Ai V, } {{ } Θ 1 D1 where X i = i ha 1+d and Z i = g i A 1+d Here d Ai and d Ci are i C i distances from the S to A i and from A i to i, respectively Since R DC R D, we can approximate the distance as d Ai d Ci Using the similar approximation method to get A, we first calculate Θ 1 as Θ 1 = RD RDA χ Θ 1 + ra R DC 1 + r e 1+r A x ln χ 1 + r 1 + r A + c f di r dr f dai r A dr A dx, D where f dai r A is the PDF for the farthest A i Similar to 33 and applying stochastic geometry within the ring D A, we can obtain f dai r A as f dai r A = ξ A r A e πλφ AR D A r A Conditioned on the number of A i and i, we obtain Θ = Pr X i < ε Ai N A 1 Pr Y i < ε Ai N 1 D3 = F Xi ε Ai F Yi ε Ai D4 Following the similar procedure to obtain Θ 1 and Θ in Appendix, we can obtain Θ 1 and Θ Then combining Θ 1 and Θ, the general case 4 is obtained For the special case =, following the similar method to calculate 38, we can obtain 43 The proof is completed