Power Control and Channel Allocation for D2D Underlaid Cellular Networks

Transcription

1 1 Power Control and Channel Allocation for DD Underlaid Cellular Networs Asmaa Abdallah, Student Member, IEEE, Mohammad M. Mansour, Senior Member, IEEE, and Ali Chehab, Senior Member, IEEE arxiv: v1 [cs.it Mar 18 Abstract Device-to-Device (DD) communications underlaying cellular networs is a viable networ technology that can potentially increase spectral utilization and improve power efficiency for proximitybased wireless applications and services. However, a major challenge in such deployment scenarios is the interference caused by DD lins when sharing the same resources with cellular users. In this wor, we propose a channel allocation (CA) scheme together with a set of three power control (PC) schemes to mitigate interference in a DD underlaid cellular system modeled as a random networ using the mathematical tool of stochastic geometry. The novel aspect of the proposed CA scheme is that it enables DD lins to share resources with multiple cellular users as opposed to one as previously considered in the literature. Moreover, the accompanying distributed PC schemes further manage interference during lin establishment and maintenance. The first two PC schemes compensate for large-scale path-loss effects and maximize the DD sum rate by employing distance-dependent pathloss parameters of the DD lin and the base station, including an error estimation margin. The third scheme is an adaptive PC scheme based on a variable target signal-to-interference-plus-noise ratio, which limits the interference caused by DD users and provides sufficient coverage probability for cellular users. Closed-form expressions for the coverage probability of cellular lins, DD lins, and sum rate of DD lins are derived in terms of the allocated power, density of DD lins, and path-loss exponent. The impact of these ey system parameters on networ performance is analyzed and compared with previous wor. Simulation results demonstrate an enhancement in cellular and DD coverage probabilities, and an increase in spectral and power efficiency. Index Terms Device-to-device communications, Poisson point process, power control, resource allocation, stochastic geometry. The authors are with the Department of Electrical and Computer Engineering, American University of Beirut, Lebanon. {awa18,mmansour,chehab}@aub.edu.lb.

2 I. INTRODUCTION The main motivation behind using Device-to-Device (DD) communication underlaying cellular systems is to enable communication between devices in close vicinity with low latency and low energy consumption, and potentially to offload a telecommunication networ from handling local traffic [1 [5. DD is a promising approach to support proximity-based services such as social networing and file sharing [4. When the devices are in close vicinity, DD communication improves the spectral and energy efficiency of cellular networs [5. Despite the benefits of DD communications in underlay mode, interference management and energy efficiency have become fundamental requirements [6 in eeping the interference caused by the DD users under control, while simultaneously extending the battery lifetime of the User Equipment (UE). For instance, cellular lins experience cross-tier interference from DD transmissions, whereas DD lins not only deal with the inter-dd interference, but also with cross-tier interference from cellular transmissions. Therefore, power control (PC) and channel allocation (CA) have become necessary for managing interference levels, protecting the cellular UEs (CUEs), and providing energy-efficient communications. Power control and channel allocation schemes have been presented in the literature as strategies to mitigate interference in wireless networs [7 [3. In [7, open loop and closed loop PC schemes (OLPC, CLPC), used in LTE [4, are compared with an optimization based approach aimed at increasing spectrum usage efficiency and reducing total power consumption. However, such schemes require a large number of iterations to converge. In [8 [11, a power allocation scheme is presented based on a soft dropping PC algorithm, in which the transmit power meets a variable target signal-to-interference-plus-noise ratio (SINR). However, the system considered is not random, and the DD users in [9 [11 are confined within a hotspot in a cellular region. In [1, a DD mode is selected in a device based on its proximity to other devices and to its distance to the enb. However, the inaccuracy of distance derivation is a ey aspect that is not addressed in [1. In [16, a two-phase auction-based algorithm is used to share uplin spectrum. The authors assume that all the channel information is calculated at the enb and broadcasted to users in a timely manner, which will cause an excessive signaling overhead. In [17, a heuristic delay-tolerant resource allocation is presented for DD underlying cellular networs; however, power control is ignored since DD users always transmit at maximum power.

3 3 In the above schemes, power control and channel allocation methods [7 [11, [16, [17, [, [3 are developed and evaluated assuming deterministic DD lin deployment scenarios. On the other hand, PC in [13 is presented for unicast DD communications by modeling a random networ for a DD underlaid cellular system, using stochastic geometry. DD users are distributed using a (-dimensional) spatial Poison point process (PPP) with density λ. Stochastic geometry is a useful mathematical tool to model irregular spatial structures of DD locations, and to quantify analytically the interference in DD underlaid cellular networs using the Laplace transform [5 [7. Two PC schemes are developed in [13; a centralized PC and a simple distributed on-off PC scheme. The former requires global channel state information (CSI) possibly at a centralized controller, which may incur high CSI feedbac overhead, whereas the latter is based on a decision set and requires only direct lin information. However, the authors assume a fixed distance between the DD pairs, and that the DD devices for the distributed case operate at maximum power leading to severe co-channel interference. Moreover, the distributed PC scheme of [13 does not guarantee reliable cellular lins, especially at high SINR targets. In [14, similar PC algorithms to [13 are presented but with channel uncertainty considered; the results in [13 are regarded as ideal best-case scenarios with perfect channel nowledge. In [18, [19, a framewor based on stochastic geometry to analyze the coverage probability and average rate with different channel allocations in a DD overlaid cellular systems is presented. In [, PC and resource allocation schemes are considered; however, the interference between DD pairs is ignored. In [1, a transmission cost minimization problem using hypergraph model is investigated based on a content encoding strategy to download a new content item or repair a lost content item in DD-based distributed storage systems. Moreover, [1 considers the one-to-one matching case, in which only one DD lin shares resources with only one uplin cellular user. In [, [3, resource allocation is considered where one DD lin shares resources with only one cellular user in the underlay case. Obviously, these schemes in [ [3 are not spectrally efficient because DD pairs are restricted to use different resource allocations. In [8, [9, power control is studied in random ad hoc networs without taing into consideration the underlaid cellular networ. In this paper, we propose power control methods along with channel allocation and analyze their performance assuming a random DD underlaid cellular networ model. A main shortcoming in most papers in the literature is that unrealistic assumptions are considered. For instance, in [13, [14 the authors rely on deterministic values such as fixed distance between

4 4 the DD transmitter and receiver, fixed transmission power, and fixed SINR targets and they only consider one cellular user sharing the resources with the DD lins. These deterministic assumptions simplify the derivation of the analytical models, but are in many cases unrealistic. In our wor, we study a general scenario by randomly modeling the distance between the DD pairs, assigning different transmission power to DD lins, varying the SINR targets, and consider multiple cellular users sharing the resources with the DD lins. Therefore, the presented analytical expressions in this paper give more insight into the performance of a DD underlaid cellular system in a rather more realistic approach. Contributions: The main contributions of this wor are the following: 1) A new channel allocation scheme is proposed based on how far the DD users are from the cellular users. It enables DD lins to share resources with multiple cellular users as opposed to one as previously considered in the literature. It also decreases the density of active DD users sharing the same resources, thus the interference generated by the DD users is decreased, which in turn enhances the cellular as well as the DD coverage probabilities. ) Analytical expressions for the coverage probability for cellular and DD lins are derived taing into account varying distances between the pairs of devices, in contrast to [13, [14. Therefore, the random variables that model distances and allocated power will significantly add to the complexity of the equations derived in [13, [14; however, the randomness of the DD underlaid system is efficiently captured, and accurate insights of the performance aspects of the DD system are provided. 3) Two distributed power control algorithms are proposed for lin establishment. One scheme maximizes the sum rate of the DD lins, while the other minimizes the interference level at the enb. Both schemes depend on a distance-based path-loss parameter between the DD transmitter, DD receiver and the enb. In addition, the inaccuracy of distance estimation is handled by incorporating an estimation error margin. A closed-form expression of various moments of the power allocated to the DD lins is derived. Moreover, an analytical expression of the sum-rate of DD lins is derived to determine the optimal DD transmission probability that maximizes this sum rate. 4) A distributed adaptive power control scheme (soft dropping distance-based PC) is proposed for lin maintenance. This PC scheme adapts to channel changes in a more realistic manner. Furthermore, this dynamic approach maintains the lin quality over time by softly dropping the target SINR as the distance between the DD pairs changes, and thus the power transmitted

5 5 DD user sharing resources with c1 Cellular UE c1 DD user sharing resources with c DD lin E-Node B Cellular UE c Fig. 1. A single-cell DD underlaid cellular networ. Two cellular users c 1 and c establish a lin with the enb while several active DD lins are established in a dis centered at the enb with radius R C. For the case m =, a subset of active DD lins share resources with cellular UE c 1 ( ), while other DD lins share resources with c ( ). E-Node B h,c1 d,c1 Cellular user c1 RX-a TX- TX-a Uplin cellular user TX-b RX-b RX- DD transmitter DD receiver Uplin Direct lin DD Direct lin Interference lin Fig.. The system model shows the channel model for one of the cellular users and a subset of active DD lins that share resources with c 1. The active DD lins outside the cell are considered as out-of-cell DD interference, whereas out-of-cell interference from cellular users belonging to cross-tier cells is ignored. is adjusted to meet this variable SINR. Hence, this scheme limits the interference caused by the DD users while varying the target SINR for the DD lins. The rest of the paper is organized as follows. The system model for a DD underlaid cellular networ is described in Section II. In Section III, the proposed channel allocation is introduced. In Section IV, analytical expressions for the coverage probabilities are derived. In Section V, the proposed PC schemes are presented. Case studies with numerical results are simulated and analyzed based on the proposed schemes in Section VI. Section VII concludes the paper. II. SYSTEM MODEL In this section, the system model and the corresponding networ parameters are presented. As shown in Fig. 1, we study a DD underlaid cellular networ in which a pool of K active DD users is divided into M groups such that each group shares distinct resources with one of M cellular users, as opposed to the assumption taen in [13, [14 where all the K DD users share the same resource with one cellular user. The enb coverage region is modeled as a circular dis

6 6 C with radius R C and centered at the enb. We assume that two cellular users are uniformly distributed in this dis, while the DD transmitters are distributed in the whole R plane by the homogeneous PPP Φ with density λ, where P[Φ = n = exp ( λ) λn. The PPP assumption n! corresponds to having the expected number of nodes per unit area equal to λ, and the nodes being uniformly distributed in the area of interest. Hence, the number of DD transmitters in C is a Poisson random variable K with mean E[K = λπrc. In addition, the associated DD receiver is uniformly distributed in a dis centered at its transmitter with radius R D. We consider a particular realization of the PPP Φ and a transmission time interval (TTI) t to describe the system model. In the following, we use subscript to refer to the uplin signal received by the enb, c m to refer to the mth transmitting cellular user, and to refer to the th DD user. Denote by s (t),c m the signal transmitted by the mth cellular user in the uplin, and by s (t), the signal transmitted by the th DD transmitter to its th DD receiver, during the TTI t. We assume distance-independent Rayleigh fading channel models between the enb and the UEs, between the enb and the DD users, and between the DD users themselves. Let h (t),c m denote the uplin channel gain between the mth cellular user and enb, h (t), the direct lin channel gain between the th DD transmitter (TX) and corresponding th DD receiver (RX), h (t), the channel gain of the interfering lin from the th DD TX to the enb, h(t),c m channel gain of the interfering lin from the mth cellular UE to the th DD RX, and h (t),l the lateral channel gain of the interfering lin from the lth DD TX to the th DD RX. Random variables n and n denote additive noise at the enb and the th DD RX, and are distributed as CN (, σ ), where σ is the noise variance. We also assume a distance-dependent path-loss model, i.e., a factor of the form d α,l the that modulates the channel gains, where d,l represents the distance between the lth TX and the th RX, with α being the path-loss exponent. Moreover, we assume that each cellular user and a subset K < K of the DD transmitters share the same uplin physical resource bloc (PRB) during the same TTI (t) as depicted in Fig.. Furthermore, we assume that the channel coherence bandwidth is larger than the bandwidth of a PRB, leading to a flat fading channel over each PRB. Therefore, the received signals y (t), at the th DD receiver, and y (t),c m at the enb can be expressed as y (t), = α/ h(t), d(t) (t), s, + h(t),c m d (t) α/ (t),c m s,c m + K l=1,l h (t) α/,l d(t) (t),l s,l + n(t), (1)

7 7 y (t),c m = h (t),c m d (t),c m α/ s (t),c m + K =1 h (t) α/, d(t) (t), s, + n(t). () The transmit powers p and p are conditioned to meet certain pea power constraints, i.e. p s,cm P max,c and p s, P max,d for all lins. The channel gains are estimated at each DD receiver using the reference signal received power (RSRP), and are fed bac to the corresponding DD transmitter. In addition, it is worth noting that E[K represents the average number of DD lins (or transmitters) before channel allocation, whereas E[K represents the number of DD lins (or transmitters) sharing resources with c m. The SINR of any typical lin is defined as SINR W, where W represents the power of I + N the intended transmitted signal, I represents the power of the interfering signals, and N denotes the noise power. Therefore, the SINR at the enb and DD receiver can be written as SINR (K, p) = SINR (K, p) = p h,cm d α,c m K =1 p, (3) h, d α, + σ K i, p i h,i d α,i p h, d α, + p h,cm d α,c m + σ, > (4) where p = [p, p 1,, p T represents the transmit power profile vector, with p i being the transmit power of the ith UE transmitter, and K is the number of DD transmitters. The supersubscript (t) is suppressed for simplicity. The proposed system model ignores the out-of-cell interference transmission from other uplin users from cross-tier cells. However, the density of the DD lins is a networ parameter that captures the expected interference on cellular and DD lins. Moreover, when the density of the DD lins is high, the proposed system is able to capture the effect of the dominant interferer for both cellular (uplin) and DD lins, since there is a high probability that the nearest DD interferer would become the dominant interference of a DD lin and that of the cellular lin. Furthermore, when this networ parameter is high, it can provide an upper bound on the performance of a DD underlaid cellular networ with out-of-cell interference. In addition, one can note that the radius of the dis R C is large enough to encompass all the DD pairs, since the dominant interference is generated from the nearest DD interferers. Based on the above defined SINRs, we use the coverage probability and achievable sum rate as metrics to evaluate system performance. Precisely, the proposed CA and PC algorithms aim to maximize those quantities while maintaining a minimum level of Quality-of-Service (SINR threshold β). The coverage probabilities of both the cellular lin and DD lins are derived in

8 8 TABLE I SYSTEM PARAMETERS Cell radius R C PPP of all DD users in the cell Φ PPP of all DD users in the cell after channel allocation Φ Density of DD lins (DD/m ) λ Channel gain from the cellular UE c m to enb h,cm Channel gain from DD TX to DD RX h, Channel gain from DD TX to enb h, Channel gain from the cellular UE c m to DD RX h,cm Channel gain from DD TX l to DD RX h,l Distribution of channel fading (h x,y ) Rayleigh fading h x,y exp (1) Distance between DD lins (d, ) Uniformly distributed Distance between uplin user and enb (d,cm ) Uniformly distributed Distance between DD TX and enb (d, ) Uniformly distributed Expectation of an event E[ Probability of an event P[ Laplace transform of a variable X L X Coverage probability of lin L P cov,l Transmit probability P tx Ergodic sum rate of DD lins R (D) s Maximum transmit power for cellular user P max,c Maximum transmit power for DD user P max,d Receiver sensitivity (dbm) ρ rx Cumulative distribution function (cdf) of variable X F X ( ) Probability density function (pdf) of variable X f X ( ) this wor. The cellular coverage probability P cov,c (β ) is defined as P cov,c (β ) = E[P cov,c (p, β ) = E[P(SINR (K, p) β ), (5) where β denotes the minimum SINR value for reliable uplin connection. Similarly, the DD coverage probability P cov,d (β ) is defined as P cov,d (β ) = E[P cov,d (p,β ) = E[P(SINR (K, p) β ), (6) where β denotes the minimum SINR value for a reliable DD lin connection. In addition, the ergodic sum rate of DD lins is defined as [ K = E log (1 + SINR (K, p)). (7) R (D) s =1 The main system parameters are summarized in Table I.

9 9 III. PROPOSED CHANNEL ALLOCATION SCHEME In this section, we propose a channel allocation scheme that enables active DD users to share the same resource blocs used by M >1 (two or more) CUEs. Its main objective is to decrease the density of DD users sharing the same resource with a particular cellular user by dividing all active DD pairs into M groups, such that each group shares resources with one of M distinct CUEs. By this, we further extend the system model in [13, [14, where only the case M = 1 is considered and all the active DD pairs are assumed to share resources with only one CUE. However, we include in our study a new resource allocation scheme for M > 1. Initially, mode selection determines whether a DD pair can transmit in DD mode or in cellular mode, and time and/or frequency resources are allocated accordingly. For simplicity, we study the case of two CUEs (M = ); the same approach is also generalized for any M. Using the independent thinning property [3, we independently assign random binary mars {1, } to the subset of active DD users that can share resources with cellular users c 1 and c, respectively. The assignment is based on the following criterion: when the distance between the cellular UE c 1 and the th DD RX is greater than the distance between cellular c and the th DD RX (d,c1 > d,c ), the th DD TX at instant (t) will be assigned the value {1}; otherwise the DD TX will be assigned {}. Consequently, all DD users assigned with value {1} will share the same resources as c 1, while the rest will share the same resources as c. Therefore, sharing resources with the farthest cellular user reduces the interference at the enb by decreasing the density of the DD TXs sharing the same resources, and reduces the interference generated from the cellular user at the DD RXs. Remar 1. Independent thinning of a PPP alters the density of the point process. If we independently assign random binary mars {1, } with P[Q =1=q and P[Q ==1 q to each point in a PPP and collect all the points which are mared as 1, the new point process will be a PPP Φ but with density qλ, while the remaining points mared as will have a PPP Φ with density (1 q)λ. In our case, the arrival of DD users such that d,c1 >d,c is independent of the arrival of another pair of DD users such that d,c >d,c1. Hence the thinning property applies. Lemma 1. Using the above remar, half of the active DD users will share the same resources with one of the cellular users and the other half will share them with the other cellular user. Proof. The proof relies on the pdf of the distance between two uniformly distributed points,

10 1 which is given by [31: ( f d,cm (r)= r ( ) ) r RC π cos 1 r 1 r, r R R C πr C 4RC C. (8) Using (8), we have P[Q = 1 = q = 1. See detailed proof in Appendix VII-A. A similar ap- proach can be applied for a more general case of M CUEs. A DD UE shares resources with the CUE that is furthest away from it. For instance, if resources are shared with M CUEs, then after M 1 comparisons, P[d,cm max n m {d,cn,, d,cm } = P[Q = value assigned to c m = q = 1 M, where M i q i = 1. We show in Section IV that the coverage probabilities for cellular and DD lins depend on the density of the DD users sharing the same resource. With the proposed CA scheme, the density of the DD users is decreased by a factor of q < 1 to be qλ. Therefore, the interference at the enb is further reduced compared to the scenario considered in [13, because here a smaller number of DD users (E[K = E[K M = 1 M λπr C ) share the same resources with the each CUE. It should be noted that sharing resources with more than one CUE increases the coverage probability, which is intuitive as the interference caused by the DD lins is reduced. However, upon increasing M (implying decreasing K ), the spectral efficiency of the system will decrease according to (7), and hence we would lose one of the main advantages of DD communications that is increasing the spectral efficiency of the cellular system. Therefore, a trade-off exists between enhancing the lin coverage probability and increasing the system throughput. In addition, the complexity of the proposed channel allocation is O(MK) where K is the total number of the DD lins that will share resources with (M > 1) uplin cellular users. This is due to the fact that the base station will compute, for each DD lin, the distance (d,cm ) from the th DD receiver to the all M cellular users (where < K and 1 < m M). Therefore, the base station computes a total of MK distance parameters (d,cm ) to perform the comparisons (d,cm max n m {d,cn,, d,cm }) as discussed above. IV. ANALYSIS OF COVERAGE PROBABILITY In this section, the cellular and DD coverage probabilities are derived using the tool of stochastic geometry. In order to analyze the coverage probabilities, the transmit powers p of the DD transmitters are assumed to be i.i.d. with cdf F p ( ), = 1,, K, and the transmit power p of the uplin cellular user is independent having distribution F p ( ).

11 11 A. Cellular Lin Coverage Probability Based on the system model and assuming that the enb is located at the origin, the SINR of the uplin is given by (3). We are interested in the cellular coverage probability P cov,c (β ), which is the probability that the SINR of cellular lin is greater than a minimum SINR β for a reliable uplin connection as defined (5). Using Lemma 1, we derive an analytical expression for the coverage probability of a cellular lin. Proposition 1. The cellular coverage probability is given by P cov,c (β ) = E X [e a 1X θ X /α, (9) where a 1 = β σ, θ = πp[q =1λβ /α E[p /α sinc(/α), and X = d α,c m p 1 is a random variable with cdf F X (x)= F d,cm (x 1/α p 1/α )df p (p). Proof. See Appendix VII-B. One can note that the SINR of the uplin signal given in (3) is independent of d,cm ; however, it depends on K, which is the number of DD users sharing the resource bloc with a particular uplin cellular user c m. Therefore, the base station depends on how far the DD users are from it and not how far the DD users are from the cellular users; therefore, the joint probability distribution with respect to the random location of c m s is not needed when deriving the cdf of the SINR at the enb. The coverage probability depends on three DD-related networ parameters: P[Q = 1 = q, λ, and E[p /α. As the density qλ of DD transmitters decreases, Pcov,C (β ) increases because a lower DD lin density causes less interference to the cellular lin. Moreover, the random DD PC parameter p, affects P cov,c (β ) only through its (/α)th moment. Since the cellular user is uniformly distributed in a circle with center enb and radius R = R C, the cdf of the distance d = d,cm of the uplin is given by if r < ; F d (r) = r if r R; R 1 if r R. Using (1), we consider the case when the uplin user employs a constant transmit power p = P max,c, and assume a noise variance of σ = (so SINR is reduced to SIR (signal- (1)

12 1 to-interference ratio)). For a given path-loss exponent value, the coverage probability in the interference-limited regime becomes P cov,c (β ) = RC where E[K = P[Q =1λπR C. ( exp θ r p /α ) r dr = RC 1 exp ( E[K β /α sinc(/α)p /α [ E[K β /α sinc(/α)p /α E [ E p /α p /α ), (11) Expression (11) explicitly shows that the coverage probability of the cellular lin depends on: 1) the average number of active DD transmitters E[K, ) certain moments of the power transmitted from the DD transmitters, 3) the power transmitted by the cellular user p, 4) path-loss exponent α, and 5) the target SINR threshold β. B. DD Lin Coverage Probability Using the same approach in the previous subsection, the SINR of the th DD lin, based on the system model, is given in (4). Then: Proposition. The DD coverage probability is given by P cov,d (β ) = E Z [e a Z θ Z /α L Y (β Z), (1) where β is the minimum SINR required for reliable transmission, a = β σ, θ = πp[q =1λβ /α E[p /α sinc(/α), Z = d α, p 1 is a random variable with cdf F Z (z) = F d, (x 1/α p 1/α )df p (p), Y = h,cm d α,c m p, and L Y (β Z) = E Y [e (β Z)Y. Proof. See Appendix VII-C. Using the fact that h,cm exp (1), which implies P( h,cm x) = e x, and the expectation is over d,cm in L Y (β Z), we derive a closed form expression for the DD coverage probability (1) in an interference-limited regime (where noise variance σ =, and SINR reduces to SIR ) as P cov,d (β ) = E d α, p 1 = E d α, p 1 [ ( ( exp θ d α, p 1 [ ( ( exp θ d α, p 1 ) /α ) E d,cm [e β (d α, p 1 [ ) ) /α E d,cm 1 ) h,cm d α,cm p 1 + β p p d α, d α,c m We next simplify (13) by deriving expressions for the various expectations involved.. (13)

13 13 Corollary 1. Using Lemma 1 and considering the proposed channel allocation scheme for the case of CUEs, then the first moment of the distance between two uniformly distributed points can be approximated as E [d,cm 51R C /(45π ). Proof. See Appendix VII-D. [ 1 We next employ the following approximation E d,cm 1+κd 1 as in [13. α,cm 1+(κ) /α E[d,cm Using this approximation together with the result from corollary 1, equation (13) reduces to C. Discussion P cov,d (β ) E d α, p 1 1+ ( exp θ (d α, p 1 ) /α) ( ) /α p β p d α (51R C /(45π )),. (14) The coverage probability depends on the following DD-related networ parameters: density of the DD lins (λ), thinning probability q, target SINR (β), the moments of the power transmitted from the DD transmitters, and the power transmitted by the cellular user. This modeling approach allows us to analyze the coverage probability and ergodic rate for a DD underlaid cellular networ with high accuracy. It also enables networ designers/operators to optimize networ performance by efficiently determining the optimal networ parameters mentioned above. The system can control the impact of DD lins on the cellular lin through 1) the proposed channel allocation scheme, which constrains the density of the DD lins that uses the same resources with a particular cellular user, and ) through the proposed power control schemes, which control the transmit power of the DD users. V. PROPOSED DD DISTRIBUTED POWER CONTROL SCHEMES In order to minimize the interference caused by the DD users, we propose distributed power control schemes that only require the CSI of the direct lin. For lin establishment, two static distributed PC are proposed, and both rely on the distance-dependent path-loss parameters [3, [33. On the other hand, for lin maintenance, a more adaptive distributed PC is proposed that compensates the measured SINR at the receiver with a variable target SINR. A. Proposed Distance-based Path-loss Power Control (DPPC) In this PC scheme, each DD transmitter selects its transmit power based on the channel conditions, namely the distance-based path-loss d α,, so as to maximize its own DD lin rate.

14 14 In order to realize our proposed scheme, we define DD proximity as the area of a dis centered ( ) Pmax,D 1/α, at the transmitting UE, with radius R D = where Pmax,D is the maximum transmit ρ rx power of the DD UE, and ρ rx is the minimum power for the DD RX to recover a signal (sometimes referred to as receiver sensitivity). The th DD TX can only use transmit power p with transmit probability P tx, if the channel quality of the th DD lin is favorable, in the sense that it exceeds a nown non-negative threshold Γ min : P tx P [ h, d α, Γ min exp ( Γmin E [ d α,). (15) Furthermore, an estimation error margin ε is introduced to compensate for the error in estimating the distance between the DD pairs. Hence, the proposed power allocation, based on the channel inversion for the DD lin, is given by ρ rx d α, (1 + ε) p = with P tx, with 1 P tx, where d, is the distance between the th DD pair, α is the path-loss exponent, and ε is the estimation error margin of d α,, such that ε 1. Each DD transmitter decides its transmit power based on its own channel gain and a nown non-negative threshold Γ min. For a given distribution of the channel gain, selecting a proper threshold Γ min plays an important role in determining the sum rate performance of the DD lins. For instance, if a large Γ min is chosen (implying a small P tx ), the inter-dd interference is reduced. However, a larger Γ min (implying a smaller P tx ) means a smaller number of active DD lins within the cell. Thus, Γ min needs to be carefully chosen to balance these two conflicting factors, while providing a high DD sum rate. We optimize the choice Γ min so as to maximize the DD sum rate as discussed in Section V-A3. Moreover, the DD transmitter checs if the lin quality degrades (i.e., h, d α, <Γ min), then the DD communication is dropped. Also, the DD receiver checs if the estimated distancebased path-loss increases, and reports it to the DD transmitter, conditioned on the fact that the DD communication lin remains active if this distance remains within R D. Note here that channel inversion only compensates for the large-scale path-loss effects and not for small-scale fading effects. For instance, instantaneous CSI is not required at the transmitter, since the loss due to distance is compensated. Moreover, the proposed scheme captures the randomness of the distance between the DD pairs, and if the DD pairs are close to each other, (16)

15 15 they will allocate less power than the case if they are further apart. However in [13, a fixed distance between the pairs is assumed and maximum power P max,d is always allocated for DD transmission, which needlessly increases power consumption and generates more interference. Considering the proposed DPPC scheme along with the random locations of DD users, the transmit powers and the SINRs experienced by the receivers become random as well. Therefore in what follows, we first characterize the transmit power p via its α/th moment, and then characterize the cellular and DD coverage probabilities accordingly. Finally, we derive an expression for the DD sum rate and maximize in order to optimize the DPPC threshold Γ min. 1) Analysis of Power Moments: According to the system model, the DD receivers are considered to be uniformly distributed in a circle centered at the corresponding DD transmitter with radius R D ; therefore, the cdf of the distance d, of the DD lin is similar to that of d,cm in (1), where d = d, and R = R D. Using (1), the moments of the transmit power p for the DPPC scheme, where p = ρ rx d α, (1 + ε), can be expressed as [ RD E d, p /α =ρ /α rx r r (1+ε) /α dr =ρ /α R RD D rx (1 + ε) /α. (17) [ Cellular Coverage Probability for DPPC: By substituting (17) for E d, p /α into the derived expression (11), the cellular coverage probability for DPPC can be obtained. DD Coverage Probability for DPPC: For p = ρ rx d, α (1 + ε), and using the moments of p in (17), the DD coverage probability in (13) becomes P cov,d (β ) = e θ (ρ rx(1+ε)) /α E d,cm [ β p ρ rx(1+ε) d α,c m, (18) Following the same approach as in (14), the approximated expression for P cov,d (β ) is given by P cov,d (β ) 1 + e θ (ρ rx(1+ε)) /α ( ) /α. (19) p β (51RC ρ rx(1+ε) /(45π )) ) Sum Rate of DD Lins: We analyze the sum rate of DD lins when the proposed DPPC scheme is employed, and compute the optimal threshold Γ min of the proposed PC that maximizes the sum rate of DD lins. Let A D denote the number of active lins selected by the proposed PC and CA algorithms, i.e., A D = P[Q = 1 P[ h, d α, Γ minλπr C = λπr C, where λ = P[Q = 1 P[ h, d α, Γ min λ = qp tx λ. As in [13, we assume Gaussian signal transmissions on all lins, and hence, the distribution of the interference terms becomes Gaussian. From the SIR distribution of the DD lin given in (19) with σ =, the ergodic rate of the

16 16 typical DD lin is generally expressed as R DD = log (1 + x) x [P[SIR 1 x dx 1 + x P cov,d (x) dx ( [ π λx/α (d exp E p /α ) ) α /α sinc(/α), p 1 ( ( ) /α )dx. () (1 + x) 1 + x p E [d d α, p,cm Note that the above general expression of the ergodic rate is valid for any distributed power control scheme that allocates its own transmit power independently of the transmit power used at other DD transmitters. Using (7) and (), the new achievable sum rate of DD lins is given as [ K = E log (1 + SIR ) = A D R DD = λπrc R DD. (1) R (D) s =1 3) DD Power Control Threshold for DPPC: From the ergodic sum rate of DD lins, we optimize the DD PC threshold Γ min by maximizing the derived transmission capacity of DD lins, which is given as R (D) s (β ) where κ = 1+ ( qλp txπr C log (1+β) β p (ρ rx (1 + ε)) ( λπr C log (1+β ) 1+κβ /α ( p (ρ rx(1+ε)) exp ) /α ( 51R 45π ) ) /α( 51RC 45π π λβ /α sinc(/α) ( ) exp R D ), /α [ π λβ E sinc(/α) p /α (ρ rx (1 + ε)) /α ) () and λ = qλptx. By solving the following optimization problem, we can compute the new optimal transmission probability: maximize R (D) s (β) subject to P tx 1 The optimal solution of P tx can be obtained by the 1 st order optimality solution, since the objective function has one optimum point. The first order derivative yields: dr (D) s (β ) dp tx /α R D = 1 πqλβ P sinc(/α) tx =. (3) The second derivative of R (D) s (β ) is applied to test the concavity at P tx, which is given as d R (D) s (β ) dp tx /α R D = πqλβ < for α. (4) sinc(/α)

17 17 Thus, R (D) s (β ) is maximum at P tx = sinc(/α). However, to satisfy the conditions of P πqλβ /α RD tx { } {, 1}, we have Ptx = min sinc(/α), 1. Using (16) where P tx = exp ( Γ min E[d α, ), then πqλβ /α RD the optimal threshold Γ min can be obtained as Γ min = ln (Ptx) + α R α D (5) Knowing the solution Ptx, the approximated transmission capacity in () can be rewritten as ( ) λπrc log (1 + β ) exp πqλβ/α RD for β R (D) s (β) 1 + κβ /α sinc(/α) β, sinc(/α)rc log (1 + β (6) ) β /α RD (1 + for β > β, κβ/α ) exp (1) where β = [ sinc(/α) α/. πqλr D The transmission capacity of the DD lins depends on the relationship between the minimum SINR value β and the networ parameters: path-loss exponent α, the density of the DD lins qλ, and the maximum allowable distance between the DD pairs R D. When β < β, all DD transmitters are scheduled; therefore no admission control is applied. However, when β β, the DD lins are scheduled with transmit probability P tx, which mitigates the inter-dd interference since the transmission capacity no longer depends on the density of the nodes λ. By integrating the transmission capacity in (6) with respect to β, the sum rate of DD lins is expressed as follows R (D) s β qλπr C (1 + x)(1 + κx /α ) exp ( ) πqλx/α RD dx + sinc(/α) β sinc(/α)rc (x /α RD )(1 + x)(1 + exp ( 1)dx. κx/α ) The DPPC scheme is summarized in the first part of the pseudo-code in Algorithm 1. Algorithm 1 Static Distributed Power Control 1: if DD TX is unable to acquire d, then : Apply DPPC scheme 3: Calculate Γ min that maximizes the DD sum rate R (D) s (β) according to (5) 4: if h, d α, Γ min and d, R D then 5: DD candidates transmit in DD mode 6: p ρ rx d α, (1 + ε). 7: else p 8: else 9: Apply EDPPC scheme 1: Set Γ min = G min 11: if h, d α, Γ min and d, R D then 1: DD candidates transmit in DD mode 13: U ρ rx (1 + ε), V µρ rx (1 + ε) 14: p min{ud α,, V dα, }. 15: else p (7)

18 18 B. Proposed Extended Distance-based Path-loss Power Control (EDPPC) EDPPC is proposed as an extended DPPC scheme for lin establishment stage. We consider in this scheme an extra distance-based path-loss parameter d α,, where d, is the distance between the enb and the DD th TX, in order to reduce the DD interference at the enb. This scheme wors only if the DD users are able to obtain estimates of d, from the enb. We apply the same conditions as in DPPC in (15), where the th DD TX can only use the transmit power p with transmit probability P tx for favorable channel conditions. However, in this PC scheme, Γ min = G min is a static value that is chosen by the enb and broadcasted to the DD transmitters. The EDPPC scheme wors as follows: each DD TX selects its transmit power based on the distance-based path-loss parameters d α, and d α,. The role of the additional parameter d α, is to suppress interference even more at the enb. Let U = ρ rx (1 + ε) and V = µρ rx (1 + ε), where µ is a PC parameter with small value chosen so that the DD transmitter does not cause excessive interference to the enb and to other DD UEs in the same cell, and ε is an estimation error margin that offsets any inaccuracy in estimating the path-loss parameters d α, and dα,. Then, the proposed power allocation for the DD lin is based on the following: min{ud α, p =, V dα, } with P tx with 1 P tx, Due to the nature of the EDPPC scheme, along with the random locations of DD users, the transmit powers and the SINRs experienced by the receivers become also random. Therefore, we derive α/th moments of the transmit power p probabilities can be characterized accordingly. (8) so that the cellular and DD coverage 1) Analysis of Power Moments: The DD TX and the corresponding DD RX are assumed to be uniformly distributed; therefore, the distance d, of the DD interfering lin with the enb and the distance d, of the direct DD lin are uniformly distributed in circles with radii R C and R D, respectively. Theorem 1. The expected value of the minimum of two random variables A, B Ω R is E[min(A, B) = E[A + E[B E[max(A, B). Proof. See Appendix VII-E.

19 19 Lemma. The expected value of the maximum of two random variables A, B Ω R with pdfs f A (a), f B (b) and cdfs F A (a), F B (b), respectively, is E[max(A, B) = Proof. See Appendix VII-F. af A (a)f B (a)da + bf B (b)f A (b)db. (9) Corollary. Using the distribution functions of d, and d,, the moments of the transmit power p are given by E d, [ p /α Proof. See Appendix VII-G. = R C V /α R4 C V 4/α 6R D U /α if R D U /α > R C V /α R D U /α R4 D U 4/α 6R C V /α if R D U /α R C V /α. Under this power control scheme, it is noted that: 1) DD UEs closer to the serving enb (where d, < d, ) normally cause a stronger uplin interference and thus their transmit powers are reduced, ) DD UEs closer to the cell edge can transmit at a higher power since their interference to the uplin cellular UE is dropped due to path-loss, and 3) DD pairs with close proximity will be allocated less power than DD pairs that are far apart. The EDPPC scheme is summarized in the second part of Algorithm 1. [ Cellular Coverage Probability for EDPPC: By substituting E d, p /α obtained in (3) into the derived expressions (11), the cellular coverage probability for EDPPC can be obtained. DD Coverage Probability for EDPPC: Using the same methodolgy as in Theorem 1, for p = min{ud α,, V dα, }, and using the moments of p in (3) and the pdf of d, and d,, the DD coverage probability in (14) becomes P cov,d (β ) ( x µ1/α R C e θ ( (ρrx(1+ε)) /α ) p /α 1+ β (51R ρ rx(1+ε) C /(45π )) RD y exp ( θ (x α y α (ρ rx(1+ε)) 1 ) /α) ( p 1+ β x α y α (ρ rx(1+ε)) ) y dy RD ) /α (51R C /(45π )) µ /α x R C µ /α x dx + RC dx y R D dy. To validate our analysis for DPPC and EDPPC, we compare the derived analytical expressions with their corresponding simulated results for λ { 1 5, }, λ =.5λ, M =, µ =.5, ε =.5, and α = 4. In Fig. 3(a) and Fig. 3(b), we validate the correctness of the analytical expressions for the cellular coverage probability of (11) and DD coverage probability of (19) and (31), while using the derived expressions of E[p /α for DPPC and EDPPC in (17) (3) (31)

20 and (3), respectively. As shown in the plots, the curves of the proposed DPPC and EDPPC schemes match well with simulated results over the entire range of β P[SIR > β DD Simu. DPPC proposed λ=e 5 DD Ana. DPPC proposed λ=e 5 DD Simu. DPPC proposed λ=5e 5 DD Ana. DPPC proposed λ=5e 5 Cell. Simu. DPPC proposed λ=e 5 Cell. Ana. DPPC proposed λ=e 5 Cell. Simu. DPPC proposed λ=5e 5 Cell. Ana. DPPC proposed λ=5e β (db) P[SIR > β DD Simu. EDPPC proposed λ=e 5 DD Ana. EDPPC proposed λ=e 5 DD Simu. EDPPC proposed λ=5e 5 DD Ana. EDPPC proposed λ=5e 5 Cell. Simu. EDPPC proposed λ=e 5 Cell. Ana. EDPPC proposed λ=e 5 Cell. Simu. EDPPC proposed λ=5e 5 Cell. Ana. EDPPC proposed λ=5e β (db) (a) Fig. 3. Analytical vs. simulated coverage probability for cellular and DD users using (a) DPPC, and (b) EDPPC scheme. (b) C. Proposed Soft Dropping Distance-based Power Control (SDDPC) The PC schemes proposed earlier provide a static power allocation where varying channel quality during DD transmissions is not taen into consideration. An adaptive PC with variable target SINR would be an attractive approach to guard cellular and DD communications against mutual interference and maintain good lin quality. We propose a soft dropping distance-based PC (SDDPC) scheme that gradually decreases the target SINR as the required transmit power increases. This increases the probability of finding a feasible solution for the PC problem in which the target SINR values for all co-channel lins can be achieved. Hence, lins with bad quality, where the receiver is far from the transmitter and requires higher power, would target lower SINR values. On the other hand, lins with better quality, where the receiver is near the transmitter and requires lower power, would target higher SINR values. In the SDDPC scheme, each UE iteratively varies its transmit power so that a power vector p for all UEs in the system is found such that the SINR of the th UE satisfies SINR (K, p) β (d, ), where β (d, ) is the target SINR of the th UE that varies according to the distance between the DD pairs d,. The SDDPC scheme uses a target SINR that varies between a maximum value β max and a minimum β min as the distance between the DD pairs varies between R min,d and a maximum value R D, while satisfying a power constraint of P min,d p P max,d.

21 1 The target SINR β (d, ) of the th DD UE at TTI (t) is given according to β max if d (t), R min,d ( ) (t) υ d, β (d, ) = β max if R min,d < d (t), R < R D min,d where υ = log 1 (β min/β max) log 1 (R D /R min,d ). β min if d (t), R D, Furthermore, the power of each DD transmitter is updated with every transmission as ( p (t+1) = p (t) β (d (t), ) ) η, (33) SINR (K, p (t) ) (3) where η is a control parameter given by (1 υ) 1 [8. Finally, the achieved power p (t+1) constrained as follows is p (t+1) = min{p max,d, max{p (t+1), P min,d }}. The SDDPC scheme is a distributed approach and the target SINR (β (d, )) depends on the distance between the DD pair; therefore, decision maing is done by the DD users themselves. In particular, the DD receivers can use the sidelin control channel (e.g., Physical Sidelin Control Channel (PSCCH)) as per the LTE technical specification in 3GPP TS [34 to report bac to the corresponding DD transmitter the received SINR value and the distance based path-loss d, whenever the received SINR is below the target value. The SDDPC scheme is summarized in Algorithm. Algorithm Dynamic Distributed Power Control procedure SDDPC p (t) P min,d = ρ rx Rmin,D α (1 + ε) Calculate β (d, ) according to (3) if SINR (K, p) < β (d, ) then LOOP: While SINR (K, p) < β (d, ) and p (t) P max,d do ( p (t+1) p (t) β (d (t), ) )η SINR (K, p (t) ) p (t+1) min{p max,d, max{p (t+1), P min,d }} else p (t+1) end goto LOOP p (t)

22 6 5 4 Iterations 3 1 M=1 M= M=3 M=4 M=K Channel allocation with M cellular users Fig. 4. Number of iterations for the SDDPC scheme for λ = and different M channel allocations. D. Discussion On complexity and convergence of Algorithms 1 and, we note that Algorithm 1 is a non-iterative, low complexity algorithm O(1), which requires around 4 simple computations. Convergence is not an issue since it is non-iterative. For Algorithm, the power allocated to the DD users is chosen iteratively and in a non-decreasing manner. At each iteration, p is increasing which increases SINR until SINR approaches the target β. Since the DD TX has finite available power, the SINR achieved by the proposed algorithm is also finite. For these reasons and following the same methodology as [8, [35, the proposed algorithm is guaranteed to converge to a finite SINR. The proof is similar to Theorem 3 in [8, [35 and hence is omitted for brevity. Furthermore, figure 4 shows the number of iterations needed in this algorithm that are very low. For instance, as M increases, the number of DD lins K, sharing the resources with one of the cellular users, decreases; therefore the interference level caused by the DD users will decrease and hence increasing the SINR. This will cause Algorithm (SDDPC) to converge faster (for M = 3, it requires an average of 3 iterations to converge). Moreover, Algorithms 1 and may not necessarily converge to the global optimal solutions. The development of global optimal power allocation is otherwise done in a centralized manner at the base station. However, it would require excessive signaling overhead in which the computational complexity grows exponentially with K [13, [14. This excessive overhead is avoided

23 3 in the distributed case, with graceful degradation in performance. Furthermore, we note that using the two proposed static distributed PC schemes for lin establishment, the allocated power remains constant over the resource blocs since we apply equal power allocation to all the assigned resource blocs. On the other hand, for lin maintenance, SDDPC compensates the measured SINR at the receiver with a variable target SINR. The power allocated per PRB of each DD UE is updated every transmission as per (33). In order to realize the proposed PC schemes, each DD transmitter needs to have nowledge of: 1) the distance based path-loss parameters d α, and dα, in order to allocate power, ) the target SINR β, 3) the density of the DD lins qλ, and (4) CSI of the direct lin. Knowledge of distance based path-loss d α, and β can be acquired through feedbac from the corresponding DD receiver. During DD lin establishment [4, the density of the DD lins (which is the average number of active DD lins per unit area) as well as d α, can be estimated at the enb. The DD transmitters acquire the density qλ when the enb broadcasts it using the downlin control channel, and acquire d α, through feedbac from the enb. All DD pairs can use the sidelin channels (Physical Sidelin Broadcast Channel (PSBCH) and PSCCH) [34 to transmit reference signals to enable DD receivers to perform measurements and report them bac to the enb or to the corresponding DD transmitter. Each DD receiver can reliably estimate the distance based path-loss parameters using these signals by averaging the effects of fading over multiple resource blocs. The enb can also estimate distances through the location updates defined in 3GPP TS 3.33 [36, and the path-loss exponent can be estimated as per [37 through defining pathloss exponents based on the region of the DD pairs location. The UE s location information exchanged is expressed in shapes as defined in 3GPP TS 3.3 [38 as universal geographical area description (GAD). VI. SIMULATION RESULTS In this section, we provide numerical results for the DD underlaid cellular networ. First, we show how the estimation error margin (ε) and the PC control parameter (µ) for DPPC and EDPPC affect the coverage probability for the cellular and the DD lins. Then, we show the performance gains of using the proposed CA and PC schemes (compared to the on/off PC in [13) in terms of coverage probability, spectral and energy efficiency.