1

Power Control for D2D Underlaid CellularNetworks: Modeling, Algorithms and Analysis

Namyoon Lee, Xingqin Lin, Jeffrey G. Andrews, and Robert W. Heath Jr.

Wireless Networking and Communication GroupDepartment of Electrical and Computer Engineering

The University of Texas at Austin, Austin, TX 78712 USAE-mail : {namyoon.lee, xlin, rheath}@utexas.edu, jandrews@ece.utexas.edu

Abstract—This paper proposes a random network model fora D2D underlaid cellular system using stochastic geometry anddevelops centralized and distributed power control algorithms.The goal of the centralized power control is two-fold: ensure thecellular users have sufficient coverage probability by limiting theinterference created by underlaid D2D users, while schedulingas many D2D links as possible. For the distributed powercontrol method, the optimal on-off power control strategy isproposed, which maximizes the sum rate of D2D links. Analyticalexpressions are derived for the coverage probabilities of cellular,D2D links, and the sum rate of the D2D links in terms of thedensity of D2D links and the path-loss exponent. The analysisreveals the impact of key system parameters on the networkperformance. For example, the bottleneck of D2D underlaidcellular networks is the cross-tier interference between D2D linksand the cellular user, not the D2D intra-tier interference whenthe density of D2D links is sparse. Simulation results verify theexactness of the derived coverage probabilities and the sum rateof D2D links.

I. INTRODUCTION

Device-to-device (D2D) communication underlaid with cel-lular networks allows direct communication between mobileusers [1]–[3]. D2D is an attractive approach for dealingwith local traffic in cellular networks. The initial motivationfor incorporating D2D communication in cellular networksis to support proximity-based services, e.g. social network-ing applications or media sharing [3]. Assuming there areproximate communication opportunities, D2D communicationmay also increase area spectral efficiency, improve cellularcoverage, lower end-to-end latency, or reduce handset powerconsumption [4], [2]. In spite of these potential gains, thecoexistence of D2D and cellular communication in the samespectrum is challenging due to the difficulty of interferencemanagement [2]. Specifically, the underlaid D2D signal be-comes a new source of interference. As a result, cellular linksexperience cross-tier interference from the D2D transmissionswhereas the D2D links need to combat not only the inter-D2D interference but also the cross-tier interference from thecellular transmissions. Therefore, interference management isessential to ensure successful coexistence of cellular and D2Dlinks.

Power control is an effective approach to mitigate interfer-ence in wireless networks; it is broadly used in current wirelesssystems. In this paper, we propose power control methodsfor interference coordination and analyze their performance in

D2D underlaid cellular networks. In particular, we consider ahybrid random network model using stochastic geometry anddevelop two different power control algorithms for the pro-posed network model. With a carefully designed (centralized)power control technique, we show that multiple D2D linksmay communicate successfully while guaranteeing reliablecommunication for the existing cellular link. This shows that,with an appropriate power control technique, underlaid D2Dlinks help to increase the network sum-throughput withoutcausing unacceptable performance degradation to existing cel-lular links.

A. Related WorkThere has been considerable interest in power control

techniques for D2D underlaid cellular networks. A simplepower control scheme was proposed in [5] for a single-cellscenario and deterministic network model, which regulatesD2D transmit power to protect the existing cellular links. Tomaximize the sum rate of the network, a D2D transmit powerallocation method was proposed in [6] for the deterministicnetwork model. A dynamic power control mechanism for asingle D2D link communication was proposed in [7], whichtargets improving the cellular system performance by mit-igating the interference generated by D2D communication.The main idea was to adjust the D2D transmit power viabase station (BS) to protect cellular users. In [8], a powerminimization solution with joint subcarrier allocation, adaptivemodulation, and mode selection was proposed to guaranteethe quality-of-service demand of D2D and cellular users. Inprior work [1], [2], [5], [7]–[17], D2D power control strategiesare developed and evaluated in a deterministic D2D linkdeployment scenario. For a random network model, spectrumsharing between ad hoc and cellular networks was studied in[18]–[20] but power control – an essential component of D2Dunderlaid cellular networks – has not been addressed. Powercontrol has been studied in other random ad hoc networkswithout considering cellular networks (see e.g. [21]–[23]). Inour paper, we propose power control algorithms and analyzetheir performance in a D2D underlaid cellular network.

B. ContributionsIn this paper, we consider a D2D underlaid cellular network

in which an uplink cellular user intends to communicate with

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the BS while multiple D2D links coexist in the commonspectrum. In such a network, we model the D2D user’s(transmitter’s) locations using a spatial Poisson point process(PPP). The rationale is that stochastic geometry is an usefultool to model irregular spatial structure of D2D locationsand analytically quantify the interference in D2D underlaidcellular networks. In this D2D underlaid cellular system, wepropose a centralized and a distributed power control algo-rithm. The main idea of the centralized algorithm is to designthe transmit power of mobile users so as to maximize thesignal-to-interference-plus-noise ratio (SINR) of the cellularlink while satisfying the individual target SINR constraintsfor D2D links. Using the fact that the centralized powerallocation problem is convex, we solve it with a feasibility setincrement technique. A main observation is that the centralizedpower control approach is possible to significantly improve theoverall cellular network throughput due to the newly underlaidD2D links while guaranteeing the coverage probability of pre-existing cellular links.

We also propose a simple distributed on-off power controlalgorithm. Note that the centralized algorithm requires globalchannel state information (CSI) possibly at a centralizedcontroller, which may incur high CSI feedback overhead. Toresolve this issue, the proposed on-off power control methodrequires CSI knowledge about the direct link between thetransmitter and its corresponding receiver only. In particular,for the distributed power control method, we derive analyticexpressions including the coverage probabilities of both cel-lular and D2D links and the sum rate of D2D links. Oneimportant insight obtained from the analysis is that on-offpower control strategy for the uplink user is actually optimal interms of the coverage probability of the cellular link, agreeingwith the finding in ad hoc networks [23]. Further, we derivethe optimal D2D transmission probability which maximizesthe sum rate of D2D links when the distributed on-off powercontrol algorithm is used. In contrast to the centralized powercontrol method, the distributed power control algorithm is notsufficient to guarantee reliable cellular communication, thoughit does improve the cellular network throughput by additionalD2D communication. We verify the results by simulating twodifferent D2D link deployment scenarios.

The remainder of this paper is organized as follows. InSection II, the proposed model for D2D underlaid cellularnetworks is described. The proposed power control algorithmsare presented in Section III. In Section IV, the analyticalexpressions for the coverage probabilities of the cellular andtypical D2D link are derived and validated through comparisonwith the simulation results. For the distributed power control,the sum rate of D2D links is derived in Section V. Simulationresults are provided in Section VI to compare the performanceof the proposed algorithms, which are followed by our con-clusions in Section VII.

II. SYSTEM MODEL

In this section, we present the system model and describenetwork metrics that will be used in this paper.

We consider a D2D underlaid cellular network, as shownin Fig. 1. In this model, let the circular disk C with radius R

Uplink user

Base station

D2D receiver

D2D transmitter

Macro user D2D link 6

D2D link 7

D2D link 8

D2D link 2

D2D link 3

D2D link 1

D2D link 4

D2D link 5

D2D link 9

D2D link 10

Fig. 1. A single-cell D2D underlaid cellular system: one macro userestablishes a cellular link with the BS while five active D2D links areestablished in a circular disk centered at the BS and with radius R. In thismodel, the active D2D links outside the circular disk are considered as out-of-cell D2D interference; however, out-of-cell interference from the marcrousers belonging to the other cells is ignored.

denotes the coverage region of a BS centered at the origin. Weassume that one cellular uplink user is uniformly located inthis region. Further, we assume that the locations of the D2Dtransmitters are distributed in the whole R2 plane according toa homogeneous PPP Φ with density λ. The associated receiverwith a D2D transmitter is located at a fixed distance away withisotropic direction. We assume all nodes have one antenna.

Under the given assumptions, the number of D2D trans-mitters in C is a Poisson random variable with mean E[K] =λπR2. Given a particular realization of the PPP Φ, the receivedsignals at D2D receiver k and the BS are written as

yk =hk,kd−α2k,k sk+hk,0d

−α2k,0 s0+

K∑`=1, 6=k

hk,`dα2

k,`s`+nk, (1)

y0 =h0,0d−α/20,0 s0 +

K∑k=1

h0,kd−α/20,k sk + n0, (2)

where subscript 0 is used for the uplink signal to the BS andsubscript k, k 6= 0, are used for D2D links; sk and s0 denotethe signal sent by D2D transmitter k and the uplink user; ykand y0 represent the received signal at D2D receiver k and theBS; nk and n0 denote the additive noise at D2D receiver kand the BS distributed as CN (0, σ2); hk,` and h0,k representthe distance-independent fading from D2D transmitter ` toreceiver k and the channel from D2D transmitter k to theBS, and are independently distributed as CN (0, 1). Here, weassume the distance dependent path-loss model, i.e., d−αk,` forall k, ` where dk,j denotes the distance from transmitter jto receiver k and α is the path-loss exponent. The transmitpower satisfies the peak power constraints, i.e., |s0|2 ≤ Pmax,c,|sk|2 ≤ Pmax,d for k ∈ {1, 2, . . . ,K}.

Then the SINR at D2D receiver k and the BS are given by

SINRk(K,p) =|hk,k|2d−αk,kpk

|hk,0|2d−αk,0p0 +∑K6=k |hk,`|2d

−αk,` p` + σ2

,(3)

SINR0(K,p) =|h0,0|2d−α0,0 p0∑K

k=1 |h0,k|2d−α0,kpk + σ2, (4)

where p = [p0, p1, . . . , pK ]T denotes transmit power profilevector with pi being the transmit power of transmitter i.

Note that our system model ignores out-of-cell interferencefrom macro users in the other cells. Nevertheless, the proposed

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model is able to capture the effect of dominant interference,which mainly determines the network performance. For thecase of the uplink user transmission, the dominant interferer isthe nearest D2D transmitter in the cell because the interferencepower of the nearest D2D transmission is strong than thepower of the out-of-cell interference with high probability.For the case of the typical D2D transmission, the dominantinterferer to any D2D link is either the uplink transmissionor the nearest D2D transmission in the cell. Therefore, oursystem model captures the dominant interference effect forthe both uplink and D2D links; it is possible to offer a tightupper performance of the D2D underlaid cellular system.

We are interested in the coverage probability of the cellularlink and D2D links. The cellular coverage probability isdefined as

P (C)cov (β0) = E[P (C)

cov (p, β0)] = E[P(SINR0(K,p) ≥ β0)], (5)

where β0 represents the minimum SINR value for reliableuplink connection. Similarly, the D2D coverage probability isdefined as

P (D)cov (βk) = E[P (D)

cov (p, βk)] = E[P(SINRk(K,p) ≥ βk)], (6)

where βk represents the minimum SINR value for reliableD2D link connections. Further, we define the ergodic sum rateof D2D links as

R(D) = E

[K∑k=1

log2 (1+SINRk(K,p))

]. (7)

III. POWER CONTROL ALGORITHMS

When the global CSI is available at the central controller,a centralized power control algorithm is proposed, whichmaximizes the SINR of the cellular link while satisfying theSINR constraints for both the cellular link and D2D links.Further, when the transmitter has CSI of the direct link of thecorresponding receiver only, a distributed on-off power controlalgorithm is proposed.

A. Centralized Power Control

A main difference between ad hoc networks and underlaidD2D cellular networks is that centralized power control ispossible when the D2D links are managed by the BS. Forother management strategies, centralized power control is ableto provide an upper bound on what can be achieved with moredecentralized algorithms.

Suppose that the BS has global channel state information(CSI). Under this assumption, the centralized power controlproblem is formulated as

max{p0,p1,...,pK}

G0,0p0∑Kk=1G0,kpk + σ2

subject toG0,0p0∑K

k=1G0,kpk + σ2≥ β0

Gk,kpk

Gk,0p0 +∑K` 6=kGk,`p` + σ2

≥ βk,

0 ≤ p0 ≤ Pmax, c,

0 ≤ pk ≤ Pmax, d, (8)

where Gk,` = |hk,`|2d−αk,` and ` ∈ {1, 2, . . . , } and k ∈{1, 2, . . . , }. This optimization problem is compactly writtenin a vector form as

maxp

gT0 p

gcT0 p + σ2

subject to (I− F)p ≥ b

0 ≤ p ≤ pmax, (9)

where gT0 = [G0,0, 0, . . . , 0], gc0T = [0, G0,1, G0,2, . . . , G0,K ],

pmax = [Pmax, c, Pmax, d, . . . , Pmax, d]T , and the normalizedchannel gain matrix F and target SINR vector b are definedas

Fk,` =

{0, k = `,βkGk,`Gk,k

, k 6= `.

b =

[β0σ

2

G0,0,β1σ

2

G1,1,β2σ

2

G2,2, . . . ,

βKσ2

GK,K

]T.

Since the objective function (linear-fractional function) isquasi-convex and the constraint set is convex (a polytope inparticular) with respect to power profile vector p, the optimalsolution is able to be obtained by using standard convexprogramming tools, provided that the feasible set is nonempty.Note that the matrix F = [f0, f1, . . . , fK ] is comprised ofnonnegative elements and is irreducible because all the activeD2D links interfere each other. By the Perron-Frobenioustheorem, the following well-known lemma proved in [24] givesa necessary and sufficient condition on the feasibility of theoptimization problem (9).

Lemma 1: [24] The constraint set in the optimizationproblem (9) is nonempty if and only if the maximum moduluseigenvalue of F is less than one, i.e., ρ(F) < 1, where ρ(·)denotes the spectral radius of a matrix.

We next describe our proposed centralized algorithm tosolve the optimization problem (9). First we assume that D2Dreceivers can feedback all the perfect normalized channel gainsGi,k and target SINR information βi to the BS. Using thisassumption, the BS then computes the transmit power usedfor both D2D transmitters and the uplink user. Note thatthe feasible set should be nonempty to obtain the optimalsolution, i.e., ρ(F) < 1. Since the normalized channel gainsGi,k, however, are random variables (the locations of all thetransmit nodes are random variables), there exists a non-zero probability that the power control solution is infeasible,i.e., P({ρ(F) ≥ 1}) 6= 0, especially when the number ofD2D links K is large. When the solution is infeasible, anadmission control method is needed in conjunction with thepower control algorithm to provide a feasible solution tothe power control problem by selecting a subset of D2Dlinks. This D2D link selection problem may be solved bybrute-force search, which requires

∑Kr=1

(Kr

)computations.

The computational complexity grows exponentially with K.Instead of brute-force search, we propose an efficient D2D linkselection algorithm with low computational complexity for thisproblem. The key idea is to drop D2D communication linkssuccessively that causes the maximum sum of the interferencepower in the network until the feasibility condition is satisfied.For K given D2D links, we first test feasibility condition

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TABLE IPROPOSED CENTRALIZED POWER CONTROL ALGORITHM

Step AlgorithmInitialization Set initial F` for ` = 0, assuming K D2D links are all active.

Step 1 Test feasibility condition ρ(F`) < 1. If this condition is satisfied, go to Step 5. Otherwise, go to Step 2.Step 2 Pick the column of F` such that k = argmaxk∈K/{0} ‖f `k‖2.Step 3 Generate a reduced matrix F` by removing the k-th column and row vectors in F`.Step 4 Update F`+1 = F` by increasing ` = `+ 1. Go to Step 1.Step 5 Solve the optimization problem in (9).

of the optimization problem in (9). If ρ(F) > 1, i.e., thefeasibility set is empty, we select the k-th D2D transmittersuch that it creates the maximum sum of interference power toall other receivers, i.e., k = arg maxk∈K/{0} ‖fk‖2, and thenaccordingly we remove the k-th row and column to reducethe size of matrix F. We keep reducing the size of matrix Funtil the feasibility condition is satisfied. Table I summarizesthe proposed D2D link selection method in conjunction withpower control.

B. Distributed On-Off Power Control Algorithm

In this subsection, we provide a distributed power con-trol algorithm. The distributed power control is an effectiveinterference mitigation method that requires no coordinationbetween transmitters; the signaling overheads for sharing CSITis not needed. In the absence of coordination, each D2Dtransmitter chooses its transmit power to maximize its ownrate towards its intended receiver, disregarding the interferencecaused to the others. The proposed on-off method is to selectthe D2D transmit power from the decision set {0, Pmax, d}solely based on knowledge of the direct link informationand a nonnegative threshold Gmin that is fixed and knownby all users. Specifically, the power used by D2D pair kis Pmax, d when the link quality is good in the sense that|hk,k|2d−αk,k > Gmin, and 0 otherwise. Mathematically,

pk =

{Pmax, d with Ps0 with 1− Ps.

(10)

where Ps denotes the transmit probability given by

Ps = P[|hk,k|2d−αk,k > Gmin] = exp(−Gmind

αk,k

). (11)

Note that the proposed power control method is distributedas each D2D transmitter decides its transmit power by theown channel gain |hk,k|2 and threshold Gmin. For a givendistribution of the channel gain, selecting a proper thresholdGmin (the transmission probability Ps) plays an important rolein determining the sum rate performance of the D2D links.On the one hand, choosing a large Gmin (a small Ps), reducesthe inter-D2D interference. On the other hand, larger Gmin

(smaller Ps) leads to smaller number of active D2D linkswithin the disk. Therefore, a good choice of Gmin balancingthese two competing factors leads to achieve a high D2D sumrate performance. This motives us to optimize the Gmin (Ps)for maximizing the D2D sum rate performance. That problemis tackled in Section V.

Remark: The proposed power control algorithm may beuseful in non-random networks because it can be applicablein any realization of the proposed random network. Therefore,

the randomness in the network modeling is not a key part ofthe algorithm, rather it is a component of the analysis to showthat it works.

IV. COVERAGE PROBABILITY ANALYSIS FORDISTRIBUTED POWER CONTROL

In this section, we derive the cellular link coverage probabil-ity, propose an optimal power control strategy for the cellularlink under the average transmit power constraint, and derivethe D2D link coverage probability. To analyze the coverageprobabilities using the tool of stochastic geometry, we assumethat the transmit power of each D2D transmitter is i.i.d. withdistribution function Fpk(·) and that the transmit power ofthe uplink user is independent and has distribution functionFp0

(·). Note that the coverage probability analysis we providein this section is valid for any distributed power controlalgorithms that select its own transmit power independentlyof the transmit power used at the other D2D transmitters.

A. Cellular Link Coverage Probability

Assume that the BS is located at the origin. The SINR ofthe typical uplink is given by

SINR0 =p0|h0,0|2d−α0,0∑

k∈Φ pk|h0,k|2d−α0,k + σ2. (12)

Further, since the cellular user’s location is distributed uni-formly in the circle with radius R, the distribution function ofthe distance d0,0 of the cellular link is given by

Fd0,0(r) =

0 if r < 0;r2

R2 if 0 ≤ r ≤ R;1 if r ≥ R.

The following theorem provides an analytical formula for theuplink coverage probability.

Theorem 1: The cellular link coverage probability is

P (C)cov (β0) = EX

[e−a1X−a2X

2α

], (13)

where a1 = σ2β0, a2 =πλβ

2α0

sinc( 2α )

E[p

2α

k

], X = p−1

0 dα0,0 with

cdf FX(x) =∫Fd0,0(x

1α p

1α )dFp0(p).

Proof: See Appendix A.

Theorem 1 provides an intuition that how important networkparameters affect the cellular link coverage probability. Forexample, we observe that P (C)

cov (β0) depends on two D2D-related network parameters: λ and E

[p

2α

k

]. In particular,

5

P(C)cov (β0) decreases as the density λ of D2D transmitters

increases, which is intuitive as higher D2D link density causesmore interference to the cellular link. Further, the randomD2D power control pk affects P (C)

cov (β0) only through its 2α -th

moment. This implies that the system can control the impactof D2D links on the cellular link by constraining E

[p

2α

k

]and

then find the optimal distribution of p0 to maximize cellularlink coverage probability.

Example 1 (A Closed Form Expression): Let us considerthe case where the uplink user uses a constant transmit powerp0 = Pmax,c and ignore the noise σ2 = 0. With path-lossexponent α = 4, a standard value in terrestrial outdoor wirelesssystems, the expression of P (C)

cov (β0) simplifies substantially as

P (C)cov (β0) =

∫ R

0

exp

(a2√Pmax,c

r2

)2r

R2dr

=

1− exp

(− πλ

√β0

sinc(1/2)√Pmax,c

E[√pk]R2

)πλ√β0

sinc(1/2)√Pmax,c

E[√pk]R2

. (14)

Further, if the D2D transmitters send the signal using powerPmax,d with probability Ps = 0.5, the coverage probabilitybecomes

P (C)cov (β0) =

1− exp(−π(λ/2)R2

sinc(1/2)

√Pmax,d

Pmax,c

√β0

)π(λ/2)R2

sinc(1/2)

√Pmax,d

Pmax,c

√β0

. (15)

This expression explicitly shows that the coverage perfor-mance of the cellular link is jointly determined by threefactors: 1) the average number of active D2D transmittersE[K] = π(λ/2)R2, 2) the power ratio between the cellularand the D2D user

√Pmax,d

Pmax,c, and 3) the target threshold β0.

To validate our analysis, we compare the coverage probabilityexpression in (15) with the simulation result. As illustratedin Fig 2, the coverage probability performance of the uplinkuser is well matched with the corresponding Monte Carlosimulation over the entire range of β0 and different λ ∈{0.00002, 0.00005}.

We next provide a simple lower bound for P (C)cov (β0), which

is useful for an arbitrary path-loss exponent value and noise-limited case. The lower bound simply depends on the certainmoments of p0 and d0,0 (rather than the distributions). Thislower bound is formalized in the following corollary.

Corollary 1: Cellular link coverage probability P(C)cov can

be lower bounded as

P (C)cov (β0) ≥ P (C)

cov,lb(β0)

= e−a12

2+αRα·E[p−1

0 ]−a2( 22+α )

2αR2·(E[p−1

0 ])2α. (16)

Proof: See Appendix B.

B. Optimal Cellular Link Power Control Strategy

In this subsection, we provide an optimal power controlstrategy for the cellular link when the cellular user has locationinformation of distance d0,0. As shown in Theorem 1, the

−5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR (dB)

P [S

IR>

β0]

λ=0.00002 (Analytic)

λ=0.00005 (Simulation)

λ=0.00005 (Analytic)

λ=0.00002 (Simulation)

Fig. 2. Coverage probability performance of the uplink user with a set ofparameters dk,k = 50 m, R = 500 m, Pmax,c = 100 mW, Pmax,d = 0.2mW, P[Gk,k > Gmin = 0.5], and λ ∈ {0.00002, 0.00005}.

coverage probability of the cellular link is a function ofthe transmit power p0 and the distance d0,0 of the uplinkuser. Conditioning on the location for the uplink user, i.e.d0,0 = d, the cellular link coverage probability is reduced

as φ(p0) = e−a1dαp−1

0 −a2d2p− 2α

0 . Under the average and peakpower constraints of the uplink transmission power p0, theoptimal distribution function of p0, i.e., Fp0

, is obtained bysolving the following optimization problem:

maximize∫φ(p0)dFp0

(p0)

subject to∫p0dFp0

(p0) = Pavg, c∫dFp0

(p0) = 1

p0 ≤ Pmax,c. (17)

Note that (17) is an infinite-dimensional optimization problembecause the distribution function Fp0 may have an infinitenumber of degrees of freedom. Although this class of opti-mization problem is not solvable in general, we are able to findthe optimal solution of the distribution function Fp0

thanks toa remarkably simple structure for the optimization problem in(17), which is stated in the following theorem.

Theorem 2: There exists a non-degenerate p?(d) ∈ (0,∞)

that maximizes φ(p0)p0

. Further, conditional on d0,0 = d theoptimal power allocation strategy maximizing the cellular linkcoverage probability is on-off power control.

Proof: See Appendix C.

From Theorem 2, the on-off power control strategy providesthe cellular user with the optimal coverage probability perfor-mance and the optimal transmission power p?0(d) is a maxi-mizer of the function φ(p0)

p0. Although the exact expression of

p?0(d) is difficult to obtain, we are able to find a closed formexpression in the interference limited regime, i.e., σ2 = 0.

6

Corollary 2: For the interference limited regime (σ2 = 0),the optimal transmit power of the cellular user placed at thedistance d0 = d with respect to the BS is

p?0(d) = max {min {p0, Pmax,c} , Pavg, c} . (18)

where p0(d) =

(2E[p

2αk ]

αsinc( 2α )

)α2

E[K]α/2β0

(dR

)α.

Proof: Let x = 1p0

. Then, for the interference limitedregime, the objective function for the uplink power optimiza-tion problem becomes φ(x)x = x exp

(−a2d

2x2/α). From the

first order optimality condition, i.e., ∂φ(x)x∂x = 0, we obtain

the maximizer x∗ = ( α2a2d2 )

α2 . Putting a2 =

πλβ2α0

sinc( 2α )

E[p2α

k ] =

E[K]β2α0

R2sinc( 2α )

E[p2α

k ] and using p∗0 = 1x∗ , we obtain the p0(d) in

(18). Since the transmit power should satisfy the maximumtransmit power constraint, we take the minimun value betweenp0(d) and Pmax,c. Note that since the cellular user uses binarypower control, the cellular user’s transmit probability becomesPavg, c

p?0(d) ≤ 1 .

Using the optimal cellular user transmit power obtained inCorollary 2, we have a closed form expression on the cellularuser coverage probability for the interference-limited regime.

Corollary 3: For the interference limited regime (σ2 = 0)and a given uplink user distance d, the cellular user coverageprobability becomes as in (19).

Proof: Recall that under the assumptions of interferencelimited regime and the fixed distance of uplink user, thecellular user coverage probability becomes P

(C)cov (d;β0) =

Ep0

[e−a2d

2p− 2α

0

]. Since the optimal power control strategy

of the uplink user is the binary power control, i.e., p0 = p?0(d)with probability Pavg, c

p?0(d) and p0 = 0 with probability 1− Pavg, c

p?0(d) ,the coverage probability expression is reduced as

P (C)cov (d;β0) =

Pavg, c

p?0(d)exp

(−a2d

2p?0(d)−2/α). (20)

Using the solution of p?0(d) in (18) and a2 =E[K]β

2α0

R2sinc( 2α )

E[p2α

k ],we obtain the desired coverage probability expression.

The coverage probability of the cellular user P (C)cov (d;β0)

behaves in three different ways according to the location ofthe user d. When p0(d) < Pavg, c (i.e., the user is locatedat around the cell center), the cellular user uses the constanttransmit power Pavg, c; this results in the coverage proba-bility decreases exponentially with respective to β

2α0 . When

Pavg, c < p0(d) < Pmax, c (i.e., the user is located at midrange of the cell edge), the on-off power control strategy isactivated. In this regime, the cellular user increases its transmitpower proportionally to dα, implying that the cellular usershould increase the transmit power according to the inverse ofpath-loss, agreeing with intuition. Further, the uplink user isrequired to increase the transmit power linearly according toE[K]

α2 where K is the random number of D2D links in the

coverage of the BS. From this on-off power control strategy,the coverage probability decreases linearly with respect to the

target SIR β0. In the regime of p0(d) ≥ Pmax, c (i.e., the user islocated at around the cell edge), the cellular user sends a signalwith its maximum transmit power with probability Pavg,c

Pmax,cdue

to the peak power constraint; thus, the coverage probabilitydecreases linearly with respective to β

2α0 again.

We provide an example to help the understanding of threedifferent behaviors on the coverage probability performance.

Example 2 (Three Different Behaviors of the CellularLink Coverage Probability): In this example, let us considera set of typical parameters: the path-loss exponent α = 4, thecell radius R = 500m, the target SINR β0 = 6 dB, the averagenumber of D2D links E[K] = λπR2 = 39, and the averageand maximum transmit power constraints of the cellular userPavg, c = 0.1 W and Pmax, c = 0.2 W. Further, we assume thatthe D2D links use a constant transmit power Pmax,d = 0.0001W, which gives us E[

√pk] =

√Pmax, d =

√0.0001. In this

set of parameters, the transmit power of the cellular link isexpressed in terms of the distance d as

p0(d) =

(E[p

12

k ]

2sinc( 12 )

)2

E[K]2β0

(d

R

)2

' 0.375×(d

R

)2

W. (21)

If the cellular user is located in the half of cell ra-dius d = R

2 , the optimal uplink transmission powerp?0(R/2) = max {min {p0, Pmax, c} , Pavg, c} = 0.1 W becauseof p0(R/2) ' 0.093 < 0.1 W, implying that the averagetransmit power is used in this regime. Thus, the cellular usercoverage probability becomes P (C)

cov (R/2, β0) ' 0.743. Alter-natively, if we consider that the cellular user is located at midrange of the cell edge with d = 0.7R, then the transmissionpower of the cellular user becomes p?0(0.7R) = 0.183 W,which means that the uplink user opportunistically sends itsuplink signal using transmit power 0.183 W with probabilityof 100

183 . Therefore, it gives a coverage probability performanceP

(C)cov (0.7R, β0) ' 0.3298. For the regime of d = R, the

optimal transmit power of the cellular link p?0(R) = 0.2 W(the maximum transmit power); thus, the coverage probabilityperformance in this regime is P (C)

cov (R, β0) ' 0.216.

C. D2D Link Coverage Probability

We derive an expression for the coverage probability forthe typical D2D link. Consider an arbitrary communicationD2D pair k and assume that the D2D receiver is located atthe origin. Then,

SINRk =pk|hk,k|2d−αk,k∑

x∈Φ\{k} pi|hk,i|2||xi||−α + p0|hk,0|2d−αk,0 + σ2,

(22)

where ||xi|| = dk,i. Using the same approach we used toprove Theorem 1, we need to compute two Laplace transformsE[e−sp0|hk,0|2d−αk,0

]and E

[e−s

∑x∈Φ\{k} pi|hk,i|

2||xi||−α]

toderive the distribution of SINRk.

First let us focus on E[e−sp0|hk,0|2d−αk,0

]. As we assume the

uplink user and the D2D receiver are randomly positioned in

7

P (C)cov (d;β0) =

Pavg, c

Pmax, cexp

{−E[K]β

2α0

sinc( 2α )

E[p2α

k ](dR

)2P−2/αmax,c

}, for p0(d) ≥ Pmax, c,

Pavg, c exp(− 2α ) 2E[p

2αk

]

αsinc( 2α

)

α/2E[K]α/2β0( dR )α

, for Pavg, c < p0(d) < Pmax, c,

exp

{−E[K]β

2α0

sinc( 2α )

E[p2α

k ](dR

)2P−2/αavg, c

}for p0(d) ≤ Pavg, c.

(19)

the disk with radius R, the pdf fdk,0(r) is given by [25]

fdk,0(r)=2r

R2

(2

πcos−1

( r

2R

)− r

πR

√1− r2

4R2

), 0≤ r≤ 2R.

(23)

Besides, |hk,0|2 is a random variable with the exponentialdistribution, i.e. |hk,0|2 Exp(1) and p0 has cdf Fp0

(p). Notingfurther that p0, |hk,0|2, and dk,0 are independent, we have

E[e−sp0|hk,0|2d−αk,0

]=

∫∫ ∞0

∫ 2R

0

e−sphr−α−hfdk,0(r)drdhdFp0

(p).

(24)

With a similar approach as in the previous subsection, it ispossible to derive the complementary cumulative distributionfunction (ccdf) of SINRk, and the coverage probability for thetypical D2D link is given in the following theorem.

Theorem 3: The coverage probability of the typical D2Dlink is given by

P (D)cov (β) = EZ

[e−b1Z−b2Z

2α LY (βZ)]

], (25)

where b1 = σ2β, b2 = πλβ2α

sinc( 2α )

E[p2α

k ], Z = p−1k dαk,k with

cdf FZ(z) =∫Fdk,k(x

1α p

1α )dFpk(p), Y = p0|hk,0|2d−αk,0 and

LY (s) = E[e−sp0|hk,0|2d−αk,0

].

Proof: See Appendix D.

To shed further light on the significance of the expressionderived in Theorem 3, it is instructive to consider a specialcase where all D2D transmitters communicate with theircorresponding receivers with a fixed distance dk,k using a fixedtransmit power pk. When the BS uses also a constant transmitpower p0, we are able to derive a closed from expressionof the coverage probability for the typical D2D link in theinterference limited regime σ2 = 0 as

P (D)cov (β)=exp

(− πλβ

2α

sinc( 2α )p− 2α

k p2α

k d2k,k

)E[e−β p0

pk

(dk,kdk,0

)α|hk,0|2

]

= exp

(− πλβ

2α

sinc( 2α )d2k,k

)E

1

1 + β p0

pk

(dk,kdk,0

)α ,(26)

where the second equality comes from the fact that |hk,0|2 ∼exp(1) and the expectation in (26) is over dk,0.

We further consider a simple but approximated expressionof the coverage probability for the typical D2D link. With

−5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR (dB)

P[S

INR

>β]

λ=0.00002 (Simulation)

λ=0.00002 (Analytic)

λ=0.00002 (Approximated)

λ=0.00005 (Simulation)

λ=0.00005 (Analytic)

λ=0.00005 (Approximated)

λ=0.00005

λ=0.00002

Fig. 3. Coverage probability performance of the typical D2D link with a setof parameters dk,k = 50 m, p0 = 100 mW, pk = 0.1 mW, R = 500 m,and λ ∈ {0.00002, 0.00005}.

an approximation of E[

11+ κ

dαk,0

]' 1

1+ κ2/α

E[dk,0]2

, which is ob-

tained from numerical observations, we have an approximatedexpression on the coverage probability as

P (D)cov (β) ' exp

(− πλβ

2α

sinc( 2α )d2k,k

)1

1+(β p0

pk

)2/α d2k,k

E[dk,0]2

,

= exp

(− πλβ

2α

sinc( 2α )d2k,k

)1

1 +(β p0

pk

)2/α d2k,k

(128R/(45π))2

,

(27)

where the equality follows from the first moment of dk,0,E[dk,0] = 128R

45π given in [25].To validate our analysis, we compare the analytic expres-

sions (exact and approximated) in (26) and (27) with simula-tion results. Fig. 3 depicts the analytical expressions alongsidethe result of the corresponding Monte Carlo simulation forthe entire range of β and different λ. The agreement isaccurate. Further, the approximated expression on the coverageprobability in (27) provides a very precise approximatedperformance, especially when α = 4 case.

V. SUM RATE ANALYSIS OF D2D LINKS

In this section, we analyze the sum rate of D2D links whenthe proposed on-off power control is applied and characterize

8

the optimal threshold of the on-off power control, whichmaximizes the sum-rate of D2D links.

A. Sum Rate of D2D Links

Let us denote the normalized inter-D2D link interferencepower at the k-th D2D receiver as Ik =

∑` 6=k |hk,`|2d

−αk,` for

k, ` ∈ {1, 2, . . . , |S|} where |S| denotes the number of activelinks selected by the proposed on-off power control algorithm,i.e., |S| = λP[|hk,k|2d−αk,k ≥ Gmin]πR2 = λπR2. Further,let p = Pmax, c

Pmax, ddenote the transmission power ratio between

the D2D transmitter and the uplink user. Assuming Gaussiansignal transmission from all the active links, the distribution ofthe interference becomes Gaussian. Then, the achievable sumrate of D2D links is written as

R(D) = E

[K∑k=1

log2

(1+

|hk,k|2d−αk,kIk + |hk,0|2d−αk,0 p

)]= |S|E [log2(1 + SIRk)] ,

= λπR2 × Rd2d. (28)

Using the SIR distribution of the typical D2D link given in(26), the ergodic rate of the typical D2D link can be rewrittenas

Rd2d =

∫ ∞0

log2(1+x)P[SIRk ≥ x]dx

=

∫ ∞0

P(D)cov (x)

1 + xdx

=

∫ ∞0

1

1+xexp

(−πλx

2α d2

k,k

sinc( 2α )

)E

1

1+xp(dk,kdk,0

)αdx

(29)

where the expectation is taken over dk,0. Using an approxi-

mation of E[

11+ κ

dαk,0

]' 1

1+ κ2/α

E[dk,0]2

, we have an approximated

expression of the ergodic rate of the typical D2D link in (29)as

Rd2d '∫ ∞

0

1

1+xexp

(−πλx

2α d2

k,k

sinc( 2α )

)1

1+(xp)2α

(dk,k

E[dk,0]

)2 dx

(30)

=

∫ ∞0

1

1+xexp

(−πλx

2α d2

k,k

sinc( 2α )

)1

1+ κx2α

dx, (31)

where κ=(Pmax,c

Pmax,d

) 2α(

dk,k128R/(45π)

)2. Interestingly, the approxi-

mated expression of the ergodic rate of the typical D2D link in(31) is determined by two factors: (1) the Laplace transform ofthe total interference power created by all active links on the

entire network, i.e., exp

(−πλx

2α d2

k,k

sinc( 2α )

)and (2) the approximated

effect of the uplink interference 1

1+κx2α

.

B. Optimizing D2D ON-Off Threshold

With the characterized ergodic sum rate of D2D links, weoptimize the D2D on-off threshold Gmin by maximizing the

approximated transmission capacity of D2D links given as

R(D)(β) = λπR2 log2(1 + β)P[SIRk ≥ β]

' λπR2 exp

(−πλβ

2α d2

k,k

sinc( 2α )

)log2(1 + β)

1+ κβ2α

(32)

= λPsπR2 exp

(−πλPsβ

2α d2

k,k

sinc( 2α )

)log2(1 + β)

1+ κβ2α

. (33)

To this end, we first compute the optimal transmission prob-ability Ps by solving the optimization problem:

max R(D)(β)

subject to 0 < Ps ≤ 1 (34)

Although the objective function is not concave, the optimalsolution of Ps can be obtained by using the first orderoptimality condition since the objective function has an uniqueoptimum point. The first order optimality condition yields

1−πλβ

2α d2

k,k

sinc( 2α )

Ps = 0, (35)

from which we have P ?s = min

{sinc( 2

α )

πλβ2α d2

k,k

, 1

}. Finally, since

Ps = P[|hk,k|2d−αk,k > Gmin], the optimal on-off threshold canbe obtained as

G?min =− ln(P ?s )

dαk,k. (36)

Using the solution of P ?s , the approximated transmissioncapacity in (33) can be expressed as

R(D)(β) '

λπR2 exp

(−πλβ

2α d2

k,k

sinc( 2α )

)log2(1+β)

1+κβ2α, for β < β,

sinc( 2α )

exp(1)

(Rdk,k

)2β−

2α

log2(1+β)

1+κβ2α

for β > β.

(37)

where β =

[sinc( 2

α )πλd2

k,k

]α2

. The transmission capacity of the D2D

links behaves differently depending on the relative relationshipbetween the target SINR value β and network parameters:path-loss exponent α and the density of D2D links λ, and thedistance of D2D link dk,k. In the case where β is smaller

than[

sinc( 2α )

πλd2k,k

]α2

, all D2D transmitters are scheduled, which

leads to achieve the same performance with that of no powercontrol. Meanwhile, when β is large enough, the D2D linksare scheduled with the transmission probability P ?s , whichresults in mitigating the inter-D2D interference. In particular,

in the case of β >

[sinc( 2

α )πλd2

k,k

]α2

, the transmission capacity of

the D2D links becomes independent of the density of nodesλ. Further, the transmission capacity of underlaid D2D linksincreases linearly with the spatial packing ratio R2

d2k,k

of D2Dtransmissions.

9

TABLE IITHE SUM RATE PERFORMANCE OF D2D LINKS

Density of D2D links (λ) 0.00001 0.00003 0.00005Sum Rate (Simulation) 13.98 27.83 33.93

Sum Rate (38) 13.63 26.29 32.54

−800 −600 −400 −200 0 200 400 600 800−800

−600

−400

−200

0

200

400

600

800

BS

D2D Rx

D2D Tx

Uplink user

Fig. 4. A snap shot of link geometry for a D2D underlaid cellularnetwork when the dense D2D link deployment scenario, i.e., λ=0.00005.In our simulation, we consider out-of-cell interference created by the D2Dtransmission for the warp-around effect.

By integration of the transmission capacity in (37) withrespect to β, we have the sum rate of D2D links as

R(D) '∫ β

0

λπR2

(1+ κx2α )(1 + x)

exp

(−πλx

2α d2

k,k

sinc( 2α )

)dx

+

∫ ∞β

x−2α

(1+ κx2α )(1 + x)

sinc(

2α

)exp(1)

(R

dk,k

)2dx. (38)

To validate our analysis, we compare the analytical result ofthe D2D sum rate with that obtained through Monte Carlosimulation. Table II shows the sum rate performance of theD2D links under different D2D line densities when α = 4,and dk,k = 50 m, R = 500 m, Pmax,c = 100 mW, andPmax,d = 0.1 mW. It can be seen that the analytcial resultswell match the simulation results.

VI. SIMULATION RESULTS

In this section, we provide numerical results for the D2Dunderlaid cellular system. From our simulation results, we firstshow the performance gain of the proposed power controlmethods compared to the no power control case in terms ofthe cellular user and the D2D user coverage probability.

1) Simulation Setup: Fig. 4 shows one snap shot of thecell geometry. As illustrated, the BS is located at the centerposition (0, 0) in R2 plane and the cellular user is uniformlydropped within the range of R = 500 m. The D2D transmittersare dropped according to PPP with the density parameterλ ∈ {0.00002, 0.00005} in a ball centered at the origin andthe radius of R + 250 m so that the average number of D2Dlinks are E[K] = πR2λ ∈ {15, 39} while removing celledge effect on the D2D link performance. Further, for a givenD2D transmitter’s location, the corresponding D2D receiver

−5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR (dB)

P [S

IR>

β]

Centralized PC

On−Off PC

No PC

Centralized PC

On−Off PC

No PC

D2D Link

Cellular Link

Ps <1

Fig. 5. Coverage probability performance of both the cellular and D2D linksaccording to different power control methods when the D2D links are sparse,i.e., λ = 0.00002.

is isotropically dropped at a fixed distance dk,k = 50m awayfrom the D2D transmitter. Since the D2D communication aresupposed to be of short range compared to the cellular link,we assume that the average transmit power of the cellularuser and D2D links are equal to Pmax, c = 100 mW andPmax, d = 0.1 mW. Since the number of D2D links K isa random variable and we evaluate the coverage probabilityand sum rate performance of the proposed algorithms byaveraging 1000 independent realizations. Further, the optimaltransmission scheduling parameter Gmin is obtained as in (36).The parameters used in the simulations are summarized inTable III.

2) Coverage Probability Comparison in Sparse D2D LinkDeployment: Suppose the sparse D2D link deployment sce-nario where the average number of D2D links in the cell equalsE[K] = πR2λ = 15.7. In this scenario, we compare the cov-erage probability of the cellular link and the D2D links underdifferent D2D power control algorithms. As shown in Fig. 5,we observe that the proposed power control methods improvethe cellular user coverage probability. The proposed powercontrol methods also provide increased D2D link coverageprobability compared to the no power control case, especiallyin the high target SINR regime. This implies that the powercontrol methods are efficient to mitigate both intra-D2D andcross-tier interference when D2D links communicate with ahigh data rate. In particular, one remarkable observation is thatthe centralized power control achieves nearly perfect cellularuser coverage probability performance, i.e., (no outage) inthe low target SINR values, while successfully supportinga large number of active D2D links (48 %) when targetSNIR β = 3 dB. Meanwhile, the on-off distributed powercontrol method yields performance gains for both cellularand D2D links compared to that of no power control casewhen the target SINR is larger than 12 dB. This is becausethe proposed on-off power control method provides the same

10

TABLE IIISIMULATION PARAMETERS

Paramers ValuesCell radius (R) 500 (m)

The D2D link range (dk,k) 50 (m)D2D link density (λ) 0.00002 and 0.00005

Average number D2D links (K) E[K] = πR2λ ∈ {15, 39}Path-loss exponent (α) 4

Target SINR threshold (β) from -6 to 21 (dB)The maximum transmit power of the cellular user Pmax, c = 100 mW

The maximum transmit power of the D2D transmitters Pmax, d = 0.1 mWNoise variance (σ2) for 1MHz bandwidth -143.97 (dBm)

The number of realizations 1000 geometry drops

−5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR (dB)

P[S

IR>

β]

Centralized PC

On−Off PC

No PC

Centralized PC

On−Off PC

No PC

Cellular Link

D2D Link

Ps<1

Fig. 6. Coverage probability performance of the cellular link according todifferent power control methods when the D2D links are dense, i.e., λ =0.00005.

performance with the no power control case until β < 12

dB, i.e., P ?s = min

{sinc( 2

α )

πλβ2α d2

k,k

, 1

}= 1 while it is activated

when β > 12 dB. For example, when the target SINR is 15dB, the on-off power control method provides 13% cellularlink and 5% D2D link coverage probability performance gainscompared to the no power control case.

3) Coverage Probability Comparison in Dense D2D LinkDeployment: Consider a dense D2D link deployment scenariowhere the average number of D2D links in the cell equalsE[K] = πR2λ = 39. For the dense D2D link deployment, asshown in Fig. 6, we observe similar trends as in the sparseD2D link deployment case. One interesting point is that theperformance degradation of the cellular user is not significantas the number of D2D links increases when the centralizedpower allocation method is applied because the proposed ad-mission control ensures that the uplink user is protected. Thisimplies that the centralized power control method is able tosupport reliable uplink performance regardless of the densityof D2D links. Meanwhile, the D2D user coverage probabilityperformance becomes deteriorated because of the increasedintra-D2D link interference. It is notable that the proposedon-off power control method improves the performance ofthe cellular and D2D link coverage probabilities comparedto that of no power control case when the target SINR is

greater than 3 dB. Although the D2D user coverage probabilityperformance decreases in the dense scenario, the total numberof successful D2D transmissions is large than that of thesparse D2D link deployment scenario. For example, whenthe target SINR is 3 dB, the total numbers of successfulD2D transmissions in both sparse and dense scenarios areabout |S|sparse = E[KP

(D)cov (3)] = 15 × 0.5 ' 7.5 and

|S|dense = E[KP(D)cov (3)] = 39× 0.27 ' 10, respectively.

VII. CONCLUSIONS

In this paper, we proposed a new network model for a D2Dunderlaid cellular system based on stochastic geometry. Inthis system, we proposed both centralized and decentralizedpower control algorithms. One consequence of the results weobserved is that the centralized power control approach leadsto improve the cellular network throughput performance due tothe additional underlaid D2D links while supporting reliablecommunication for the uplink cellular user. Meanwhile, thedistributed power control approach is not enough to guaranteereliable cellular links; however, it also improves the cellularnetwork throughput by allowing D2D links to be underlaidin the network. Future work could investigate the effect ofmultiple antennas at the base station, other cell interference,and joint optimization across the resource allocation and powercontrol.

APPENDIX

A. Proof of Theorem 1

First notice that

P (C)cov = P(SINR0 ≥ β0)

= P

(p0|h0,0|2d−α0,0∑

k∈Φ pk|h0,k|2||xk||−α + σ2≥ β0

)

= P

(|h0,0|2≥ β0p

−10 dα0,0

(∑k∈Φ

pk|h0,k|2||xk||−α+σ2

))= E

[e−β0p

−10 dα0,0(

∑k∈Φ pk|h0,k|2||xk||−α+σ2)

]=E[e−σ

2β0p−10 pα0,0

]E[e−β0p

−10 dα0,0(

∑k∈Φ pk|h0,k|2||xk||−α)

],

(39)

where in the second last equality we use the fact that |h0,0|2 ∼Exp(1) and thus P(|h0,0|2 ≥ x) = e−x. Conditioned on thetransmit power of the typical uplink transmitter p0 = p and

11

the distance d0,0 = d from the cellular transmitter to BS, wenext compute the second term (39). To this end, we need theLaplace transform LΦ(s) = E

[e−s(

∑k∈Φ pk|h0,k|2||xk||−α)

]given as

LΦ(s) = e−

2απ2

sin( 2απ)

E[p

2αk

]λs

2α

. (40)

Using LΦ(s) yields

P(C)cov|p0=p,d0,0=d = e−σ

2β0p−1dαe

−2απ2

sin( 2απ)λβ

2α0 E[p

2αk

]d2p−

2α

.

(41)

De-conditioning with respect to p0 and d0,0 yields the uplinkcoverage probability P

(C)cov . The last step is to derive the

probability distribution of X = p−10 dα0,0:

FX(x) = P(p−10 dα0,0 ≤ x)

=

∫P(d0,0 ≤ (xp)

1α )dFp0(p)

=

∫Fd0,0

(x1α p

1α )dFp0

(p). (42)

B. Proof of Corollary 1

Let φ(x) = e−a1x−a2x2α . We compute the first and second

derivative of φ(x) as follows:

φ′(x) = −e−a1x−a2x

2α

(a1 + a2

2

αx

2α−1

), (43)

φ′′(x) = e−a1x−a2x

2α

(a1 + a2

2

αx

2α−1

)2

+ e−a1x−a2x2α a2

2

α

(1− 2

α

)x

2α−2. (44)

As α > 2, φ′′(x) ≥ 0 for x ≥ 0 and thus φ(x) is convex for

x ≥ 0. Applying Jensen’s inequality, we obtain

P (C)cov = EX

[e−a1X−a2X

2α

]≥ e−a1E[X]−a2E[X]

2α , (45)

where E[X] = E[p−1

0 dα0,0]

= E[p−10 ]E[dα0,0] due to the

independence of PC and DC . Finally, E[dα0,0] can be computedexplicitly.

E[dα0,0] =

∫rαdFd0,0(r)

=

∫ R

0

rα2r

R2dr

=2

2 + αRα. (46)

C. Proof of Corollary 2

Note that φ(p) is positive-valued and continuous when p >0. Also, limp→0+

φ(p)p → 0 and limp→∞

φ(p)p → 0. These

facts imply that there exists a non-degenerate p?(d) ∈ (0,∞)

that achieves the maximum value of φ(p)p . If we ignore the

constraint∫

dFp0,0(p) = 1 and p ≤ Pmax,c for the time being

and consider the following relaxed conditional cellular linkcoverage optimization problem:

max

∫φ(p)dFp0,0(p)

subject to∫pdFPC (p) = Pavg, c. (47)

Let dL(p) = pPavg, c

dFp0,0(p). Then the above optimizationproblem is equivalently formulated as

max Pavg, c ·∫φ(p)

pdL(p)

subject to∫

dL(p) = 1. (48)

whose optimal solution is L?(p?(d)) − L?(p?−(d)) = 1 andL?(p) = 0 for p 6= p?(d). Therefore, we conclude that thebinary power control strategy is optimal.

D. Proof of Theorem 3

To prove Theorem 3, we need to derive the ccdf of SINRk.To this end, using Slivnyak’s theorem [26], it is easy to seethat

LΦ\{k}(s) = E[e−s

∑x∈Φ\{k} pi|hk,i|

2||xi||−α |k ∈ Φ]

= LΦ(s)

= e− πλ

sinc( 2α

)E[p

2αk

]s

2α

. (49)

It follows that

P(SINRk ≥ β) (50)

= P

(pk|hk,k|2d−αk,k∑

x∈Φ\{k} pi|hk,i|2||xi||−α+p0|hk,0|2d−αk,0 + σ2≥ β

)

=P

|hk,k|2≥βp−1k dαk,k

∑x∈Φ\{k}

pi|hk,i|2||xi||−α+p0|hk,0|2d−αk,0 +σ2

(51)

= E[e−βp

−1k dαk,k(

∑x∈Φ\{k} pi|hk,i|

2||xi||−α+p0|hk,0|2d−αk,0+σ2)]

(52)

= Ep−1k dαk,k

[e−σ

2βp−1k dαk,kLΦ(βp−1

k dαk,k)LY (βp−1k dαk,k)

],

(53)

which completes the proof.

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