I Introduction
The main motivation behind using DevicetoDevice (D2D) 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 network from handling local traffic [1, 2, 3, 4, 5]. D2D is a promising approach to support proximitybased services such as social networking and file sharing [4]. When the devices are in close vicinity, D2D communication improves the spectral and energy efficiency of cellular networks [5].
Despite the benefits of D2D communications in underlay mode, interference management and energy efficiency have become fundamental requirements [6] in keeping the interference caused by the D2D users under control, while simultaneously extending the battery lifetime of the User Equipment (UE). For instance, cellular links experience crosstier interference from D2D transmissions, whereas D2D links not only deal with the interD2D interference, but also with crosstier 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 energyefficient communications.
Power control and channel allocation schemes have been presented in the literature as strategies to mitigate interference in wireless networks [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In [7], open loop and closed loop PC schemes (OLPC, CLPC), used in LTE [24], 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, 9, 10, 11], a power allocation scheme is presented based on a “soft dropping” PC algorithm, in which the transmit power meets a variable target signaltointerferenceplusnoise ratio (SINR). However, the system considered is not random, and the D2D users in [9, 10, 11] are confined within a hotspot in a cellular region.
In [12], a D2D “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 key aspect that is not addressed in [12]. In [16], a twophase auctionbased algorithm is used to share uplink 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 delaytolerant resource allocation is presented for D2D underlying cellular networks; however, power control is ignored since D2D users always transmit at maximum power.
In the above schemes, power control and channel allocation methods [7, 8, 9, 10, 11, 16, 17, 22, 23] are developed and evaluated assuming deterministic D2D link deployment scenarios. On the other hand, PC in [13] is presented for unicast D2D communications by modeling a random network for a D2D underlaid cellular system, using stochastic geometry. D2D users are distributed using a (2dimensional) spatial Poison point process (PPP) with density . Stochastic geometry is a useful mathematical tool to model irregular spatial structures of D2D locations, and to quantify analytically the interference in D2D underlaid cellular networks using the Laplace transform [25, 26, 27]. Two PC schemes are developed in [13]; a centralized PC and a simple distributed onoff PC scheme. The former requires global channel state information (CSI) possibly at a centralized controller, which may incur high CSI feedback overhead, whereas the latter is based on a decision set and requires only direct link information. However, the authors assume a fixed distance between the D2D pairs, and that the D2D devices for the distributed case operate at maximum power leading to severe cochannel interference. Moreover, the distributed PC scheme of [13] does not guarantee reliable cellular links, 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 bestcase scenarios with perfect channel knowledge. In [18, 19], a framework based on stochastic geometry to analyze the coverage probability and average rate with different channel allocations in a D2D overlaid cellular systems is presented.
In [20], PC and resource allocation schemes are considered; however, the interference between D2D pairs is ignored. In [21], 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 D2Dbased distributed storage systems. Moreover, [21] considers the onetoone matching case, in which only one D2D link shares resources with only one uplink cellular user. In [22, 23], resource allocation is considered where one D2D link shares resources with only one cellular user in the underlay case. Obviously, these schemes in [20, 21, 22, 23] are not spectrally efficient because D2D pairs are restricted to use different resource allocations. In [28, 29], power control is studied in random ad hoc networks without taking into consideration the underlaid cellular network.
In this paper, we propose power control methods along with channel allocation and analyze their performance assuming a random D2D underlaid cellular network 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 the D2D transmitter and receiver, fixed transmission power, and fixed SINR targets and they only consider one cellular user sharing the resources with the D2D links. These deterministic assumptions simplify the derivation of the analytical models, but are in many cases unrealistic. In our work, we study a general scenario by randomly modeling the distance between the D2D pairs, assigning different transmission power to D2D links, varying the SINR targets, and consider multiple cellular users sharing the resources with the D2D links. Therefore, the presented analytical expressions in this paper give more insight into the performance of a D2D underlaid cellular system in a rather more realistic approach.
Contributions: The main contributions of this work are the following:

A new channel allocation scheme is proposed based on how far the D2D users are from the cellular users. It enables D2D links to share resources with multiple cellular users as opposed to one as previously considered in the literature. It also decreases the density of active D2D users sharing the same resources, thus the interference generated by the D2D users is decreased, which in turn enhances the cellular as well as the D2D coverage probabilities.

Analytical expressions for the coverage probability for cellular and D2D links are derived taking 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 D2D underlaid system is efficiently captured, and accurate insights of the performance aspects of the D2D system are provided. 
Two distributed power control algorithms are proposed for link establishment
. One scheme maximizes the sum rate of the D2D links, while the other minimizes the interference level at the eNB. Both schemes depend on a distancebased pathloss parameter between the D2D transmitter, D2D receiver and the eNB. In addition, the inaccuracy of distance estimation is handled by incorporating an estimation error margin. A closedform expression of various moments of the power allocated to the D2D links is derived. Moreover, an analytical expression of the sumrate of D2D links is derived to determine the optimal D2D transmission probability that maximizes this sum rate.

A distributed adaptive power control scheme (soft dropping distancebased PC) is proposed for link maintenance. This PC scheme adapts to channel changes in a more realistic manner. Furthermore, this dynamic approach maintains the link quality over time by softly dropping the target SINR as the distance between the D2D pairs changes, and thus the power transmitted is adjusted to meet this variable SINR. Hence, this scheme limits the interference caused by the D2D users while varying the target SINR for the D2D links.
The rest of the paper is organized as follows. The system model for a D2D underlaid cellular network 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 network parameters are presented. As shown in Fig. 1, we study a D2D underlaid cellular network in which a pool of active D2D users is divided into groups such that each group shares distinct resources with one of cellular users, as opposed to the assumption taken in [13, 14] where all the D2D users share the same resource with one cellular user. The eNB coverage region is modeled as a circular disk with radius
and centered at the eNB. We assume that two cellular users are uniformly distributed in this disk, while the D2D transmitters are distributed in the whole
plane by the homogeneous PPP with density , where . The PPP assumption corresponds to having the expected number of nodes per unit area equal to , and the nodes being uniformly distributed in the area of interest. Hence, the number of D2D transmitters in is a Poisson random variable with mean . In addition, the associated D2D receiver is uniformly distributed in a disk centered at its transmitter with radius .We consider a particular realization of the PPP and a transmission time interval (TTI) to describe the system model. In the following, we use subscript 0 to refer to the uplink signal received by the eNB, to refer to the transmitting cellular user, and to refer to the D2D user. Denote by the signal transmitted by the cellular user in the uplink, and by the signal transmitted by the D2D transmitter to its D2D receiver, during the TTI . We assume distanceindependent Rayleigh fading channel models between the eNB and the UEs, between the eNB and the D2D users, and between the D2D users themselves. Let denote the uplink channel gain between the cellular user and eNB, the direct link channel gain between the D2D transmitter (TX) and corresponding D2D receiver (RX), the channel gain of the interfering link from the D2D TX to the eNB, the channel gain of the interfering link from the cellular UE to the D2D RX, and the lateral channel gain of the interfering link from the D2D TX to the D2D RX. Random variables and denote additive noise at the eNB and the D2D RX, and are distributed as , where
is the noise variance. We also assume a distancedependent pathloss model, i.e., a factor of the form
that modulates the channel gains, where represents the distance between the TX and the RX, with being the pathloss exponent.Moreover, we assume that each cellular user and a subset of the D2D transmitters share the same uplink physical resource block (PRB) during the same TTI () as depicted in Fig. 2. 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 at the D2D receiver, and at the eNB can be expressed as
(1)  
(2) 
The transmit powers and are conditioned to meet certain peak power constraints, i.e. and for all links. The channel gains are estimated at each D2D receiver using the reference signal received power (RSRP), and are fed back to the corresponding D2D transmitter. In addition, it is worth noting that represents the average number of D2D links (or transmitters) before channel allocation, whereas represents the number of D2D links (or transmitters) sharing resources with .
The SINR of any typical link is defined as , where represents the power of the intended transmitted signal, represents the power of the interfering signals, and denotes the noise power. Therefore, the SINR at the eNB and D2D receiver can be written as
(3)  
(4) 
where
represents the transmit power profile vector, with
being the transmit power of the UE transmitter, and is the number of D2D transmitters. The supersubscript is suppressed for simplicity.The proposed system model ignores the outofcell interference transmission from other uplink users from crosstier cells. However, the density of the D2D links is a network parameter that captures the expected interference on cellular and D2D links. Moreover, when the density of the D2D links is high, the proposed system is able to capture the effect of the dominant interferer for both cellular (uplink) and D2D links, since there is a high probability that the nearest D2D interferer would become the dominant interference of a D2D link and that of the cellular link. Furthermore, when this network parameter is high, it can provide an upper bound on the performance of a D2D underlaid cellular network with outofcell interference. In addition, one can note that the radius of the disk is large enough to encompass all the D2D pairs, since the dominant interference is generated from the nearest D2D 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 QualityofService (SINR threshold ). The coverage probabilities of both the cellular link and D2D links are derived in this work. The cellular coverage probability is defined as
(5) 
where denotes the minimum SINR value for reliable uplink connection. Similarly, the D2D coverage probability is defined as
(6) 
where denotes the minimum SINR value for a reliable D2D link connection. In addition, the ergodic sum rate of D2D links is defined as
(7) 
The main system parameters are summarized in Table I.
Cell radius  
PPP of all D2D users in the cell  
PPP of all D2D users in the cell after channel allocation  
Density of D2D links (D2D)  
Channel gain from the cellular UE to eNB  
Channel gain from D2D TX to D2D RX  
Channel gain from D2D TX to eNB  
Channel gain from the cellular UE to D2D RX  
Channel gain from D2D TX to D2D RX  
Distribution of channel fading ()  Rayleigh fading 
Distance between D2D links (  Uniformly distributed 
Distance between uplink user and eNB ()  Uniformly distributed 
Distance between D2D TX and eNB ()  Uniformly distributed 
Expectation of an event  
Probability of an event  
Laplace transform of a variable  
Coverage probability of link  
Transmit probability  
Ergodic sum rate of D2D links  
Maximum transmit power for cellular user  
Maximum transmit power for D2D user  
Receiver sensitivity (dBm)  
Cumulative distribution function (cdf) of variable  
Probability density function (pdf) of variable 
Iii Proposed Channel Allocation Scheme
In this section, we propose a channel allocation scheme that enables active D2D users to share the same resource blocks used by (two or more) CUEs. Its main objective is to decrease the density of D2D users sharing the same resource with a particular cellular user by dividing all active D2D pairs into groups, such that each group shares resources with one of distinct CUEs. By this, we further extend the system model in [13, 14], where only the case is considered and all the active D2D pairs are assumed to share resources with only one CUE. However, we include in our study a new resource allocation scheme for .
Initially, mode selection determines whether a D2D pair can transmit in D2D mode or in cellular mode, and time and/or frequency resources are allocated accordingly. For simplicity, we study the case of two CUEs (); the same approach is also generalized for any . Using the independent thinning property [30], we independently assign random binary marks to the subset of active D2D users that can share resources with cellular users and , respectively. The assignment is based on the following criterion: when the distance between the cellular UE and the D2D RX is greater than the distance between cellular and the D2D RX (), the D2D TX at instant will be assigned the value ; otherwise the D2D TX will be assigned . Consequently, all D2D users assigned with value will share the same resources as , while the rest will share the same resources as . Therefore, sharing resources with the farthest cellular user reduces the interference at the eNB by decreasing the density of the D2D TXs sharing the same resources, and reduces the interference generated from the cellular user at the D2D RXs.
Remark 1.
Independent thinning of a PPP alters the density of the point process. If we independently assign random binary marks with and to each point in a PPP and collect all the points which are marked as , the new point process will be a PPP but with density , while the remaining points marked as will have a PPP with density . In our case, the arrival of D2D users such that is independent of the arrival of another pair of D2D users such that . Hence the thinning property applies.
Lemma 1.
Using the above remark, half of the active D2D 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, which is given by [31]:
(8) 
Using (8), we have . See detailed proof in Appendix VIIA. A similar approach can be applied for a more general case of CUEs. A D2D UE shares resources with the CUE that is furthest away from it. For instance, if resources are shared with CUEs, then after comparisons, , where . ∎
We show in Section IV that the coverage probabilities for cellular and D2D links depend on the density of the D2D users sharing the same resource. With the proposed CA scheme, the density of the D2D users is decreased by a factor of to be . Therefore, the interference at the eNB is further reduced compared to the scenario considered in [13], because here a smaller number of D2D users () 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 D2D links is reduced. However, upon increasing (implying decreasing ), the spectral efficiency of the system will decrease according to (7), and hence we would lose one of the main advantages of D2D communications that is increasing the spectral efficiency of the cellular system. Therefore, a tradeoff exists between enhancing the link coverage probability and increasing the system throughput.
In addition, the complexity of the proposed channel allocation is where is the total number of the D2D links that will share resources with uplink cellular users. This is due to the fact that the base station will compute, for each D2D link, the distance from the D2D receiver to the all cellular users (where and ). Therefore, the base station computes a total of distance parameters to perform the comparisons () as discussed above.
Iv Analysis of Coverage Probability
In this section, the cellular and D2D coverage probabilities are derived using the tool of stochastic geometry. In order to analyze the coverage probabilities, the transmit powers of the D2D transmitters are assumed to be i.i.d. with cdf , , and the transmit power of the uplink cellular user is independent having distribution .
Iva Cellular Link Coverage Probability
Based on the system model and assuming that the eNB is located at the origin, the SINR of the uplink is given by (3). We are interested in the cellular coverage probability , which is the probability that the SINR of cellular link is greater than a minimum SINR for a reliable uplink connection as defined (5). Using Lemma 1, we derive an analytical expression for the coverage probability of a cellular link.
Proposition 1.
The cellular coverage probability is given by
(9) 
where , , and is a random variable with cdf .
Proof.
See Appendix VIIB. ∎
One can note that the SINR of the uplink signal given in (3) is independent of ; however, it depends on , which is the number of D2D users sharing the resource block with a particular uplink cellular user
. Therefore, the base station depends on how far the D2D users are from it and not how far the D2D users are from the cellular users; therefore, the joint probability distribution with respect to the random location of
’s is not needed when deriving the cdf of the SINR at the eNB.The coverage probability depends on three D2Drelated network parameters: , , and . As the density of D2D transmitters decreases, increases because a lower D2D link density causes less interference to the cellular link. Moreover, the random D2D PC parameter , affects only through its th moment.
Since the cellular user is uniformly distributed in a circle with center eNB and radius , the cdf of the distance of the uplink is given by
(10) 
Using (10), we consider the case when the uplink user employs a constant transmit power , and assume a noise variance of (so is reduced to (signaltointerference ratio)). For a given pathloss exponent value, the coverage probability in the interferencelimited regime becomes
(11) 
where .
Expression (11) explicitly shows that the coverage probability of the cellular link depends on: 1) the average number of active D2D transmitters , 2) certain moments of the power transmitted from the D2D transmitters, 3) the power transmitted by the cellular user , 4) pathloss exponent , and 5) the target SINR threshold .
IvB D2D Link Coverage Probability
Using the same approach in the previous subsection, the SINR of the D2D link, based on the system model, is given in (4). Then:
Proposition 2.
The D2D coverage probability is given by
(12) 
where is the minimum SINR required for reliable transmission, ,
,
is a random variable with cdf
, , and .
Proof.
See Appendix VIIC. ∎
Using the fact that , which implies , and the expectation is over in , we derive a closed form expression for the D2D coverage probability (12) in an interferencelimited regime (where noise variance , and reduces to ) as
(13) 
We next simplify (13) by deriving expressions for the various expectations involved.
Corollary 1.
Using Lemma 1 and considering the proposed channel allocation scheme for the case of 2 CUEs, then the first moment of the distance between two uniformly distributed points can be approximated as .
Proof.
See Appendix VIID. ∎
IvC Discussion
The coverage probability depends on the following D2Drelated network parameters: density of the D2D links (), thinning probability , target SINR (), the moments of the power transmitted from the D2D transmitters, and the power transmitted by the cellular user. This modeling approach allows us to analyze the coverage probability and ergodic rate for a D2D underlaid cellular network with high accuracy. It also enables network designers/operators to optimize network performance by efficiently determining the optimal network parameters mentioned above. The system can control the impact of D2D links on the cellular link through 1) the proposed channel allocation scheme, which constrains the density of the D2D links that uses the same resources with a particular cellular user, and 2) through the proposed power control schemes, which control the transmit power of the D2D users.
V Proposed D2D Distributed Power Control Schemes
In order to minimize the interference caused by the D2D users, we propose distributed power control schemes that only require the CSI of the direct link. For link establishment, two static distributed PC are proposed, and both rely on the distancedependent pathloss parameters [32, 33]. On the other hand, for link maintenance, a more adaptive distributed PC is proposed that compensates the measured SINR at the receiver with a variable target SINR.
Va Proposed Distancebased Pathloss Power Control (DPPC)
In this PC scheme, each D2D transmitter selects its transmit power based on the channel conditions, namely the distancebased pathloss , so as to maximize its own D2D link rate. In order to realize our proposed scheme, we define D2D proximity as the area of a disk centered at the transmitting UE, with radius , where is the maximum transmit power of the D2D UE, and is the minimum power for the D2D RX to recover a signal (sometimes referred to as receiver sensitivity).
The D2D TX can only use transmit power with transmit probability , if the channel quality of the D2D link is favorable, in the sense that it exceeds a known nonnegative threshold :
(15) 
Furthermore, an estimation error margin is introduced to compensate for the error in estimating the distance between the D2D pairs. Hence, the proposed power allocation, based on the channel inversion for the D2D link, is given by
(16) 
where is the distance between the D2D pair, is the pathloss exponent, and is the estimation error margin of , such that .
Each D2D transmitter decides its transmit power based on its own channel gain and a known nonnegative threshold . For a given distribution of the channel gain, selecting a proper threshold plays an important role in determining the sum rate performance of the D2D links. For instance, if a large is chosen (implying a small ), the interD2D interference is reduced. However, a larger (implying a smaller ) means a smaller number of active D2D links within the cell. Thus, needs to be carefully chosen to balance these two conflicting factors, while providing a high D2D sum rate. We optimize the choice so as to maximize the D2D sum rate as discussed in Section VA3.
Moreover, the D2D transmitter checks if the link quality degrades (i.e., ), then the D2D communication is dropped. Also, the D2D receiver checks if the estimated distancebased pathloss increases, and reports it to the D2D transmitter, conditioned on the fact that the D2D communication link remains active if this distance remains within .
Note here that channel inversion only compensates for the largescale pathloss effects and not for smallscale 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 D2D pairs, and if the D2D pairs are close to each other, 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 is always allocated for D2D transmission, which needlessly increases power consumption and generates more interference.
Considering the proposed DPPC scheme along with the random locations of D2D 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 via its moment, and then characterize the cellular and D2D coverage probabilities accordingly. Finally, we derive an expression for the D2D sum rate and maximize in order to optimize the DPPC threshold .
VA1 Analysis of Power Moments
According to the system model, the D2D receivers are considered to be uniformly distributed in a circle centered at the corresponding D2D transmitter with radius ; therefore, the cdf of the distance of the D2D link is similar to that of in (10), where and . Using (10), the moments of the transmit power for the DPPC scheme, where , can be expressed as
(17) 
VA2 Sum Rate of D2D Links
We analyze the sum rate of D2D links when the proposed DPPC scheme is employed, and compute the optimal threshold of the proposed PC that maximizes the sum rate of D2D links.
Let denote the number of active links selected by the proposed PC and CA algorithms, i.e., , where . As in [13], we assume Gaussian signal transmissions on all links, and hence, the distribution of the interference terms becomes Gaussian.
From the SIR distribution of the D2D link given in (19) with , the ergodic rate of the typical D2D link is generally expressed as
(20) 
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 D2D transmitters.
VA3 D2D Power Control Threshold for DPPC
From the ergodic sum rate of D2D links, we optimize the D2D PC threshold by maximizing the derived transmission capacity of D2D links, which is given as
(22) 
where and . By solving the following optimization problem, we can compute the new optimal transmission probability:
maximize  
subject to 
The optimal solution of can be obtained by the order optimality solution, since the objective function has one optimum point. The first order derivative yields:
(23) 
The second derivative of is applied to test the concavity at , which is given as
(24) 
Thus, is maximum at . However, to satisfy the conditions of , we have . Using (16) where , then the optimal threshold can be obtained as
(25) 
Knowing the solution , the approximated transmission capacity in (22) can be rewritten as
(26) 
where .
The transmission capacity of the D2D links depends on the relationship between the minimum SINR value and the network parameters: pathloss exponent , the density of the D2D links , and the maximum allowable distance between the D2D pairs . When , all D2D transmitters are scheduled; therefore no admission control is applied. However, when , the D2D links are scheduled with transmit probability , which mitigates the interD2D interference since the transmission capacity no longer depends on the density of the nodes .
VB Proposed Extended Distancebased Pathloss Power Control (EDPPC)
EDPPC is proposed as an extended DPPC scheme for link establishment stage. We consider in this scheme an extra distancebased pathloss parameter , where
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