IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. …home.eps.hw.ac.uk/~cw46/2013_GeX_TW_13_03_961.pdf · power consumption model for a typical PVT cell, con-sidering path loss,

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 3, MARCH 2013 961

Energy Efficiency Evaluation of Cellular NetworksBased on Spatial Distributions of

Traffic Load and Power ConsumptionLin Xiang, Xiaohu Ge, Senior Member, IEEE, Cheng-Xiang Wang, Senior Member, IEEE,

Frank Y. Li, Senior Member, IEEE, and Frank Reichert

Abstract—Energy efficiency has gained its significance whenservice providers’ operational costs burden with the rapidlygrowing data traffic demand in cellular networks. In this paper,we propose an energy efficiency model for Poisson-Voronoi tessel-lation (PVT) cellular networks considering spatial distributionsof traffic load and power consumption. The spatial distributionsof traffic load and power consumption are derived for a typicalPVT cell, and can be directly extended to the whole PVT cellularnetwork based on the Palm theory. Furthermore, the energyefficiency of PVT cellular networks is evaluated by taking intoaccount traffic load characteristics, wireless channel effects andinterference. Both numerical and Monte Carlo simulations areconducted to evaluate the performance of the energy efficiencymodel in PVT cellular networks. These simulation results demon-strate that there exist maximal limits for energy efficiency in PVTcellular networks for given wireless channel conditions and userintensity in a cell.

Index Terms—Poisson-Voronoi tessellation cellular networks,traffic load, power consumption, energy efficiency, interferencemodel.

Manuscript received December 4, 2011; revised May 22 and October 7,2012; accepted November 2, 2012. The associate editor coordinating thereview of this paper and approving it for publication was L. Deneire.

L. Xiang and X. Ge (corresponding author) are with the Depart-ment of Electronics and Information Engineering, Huazhong Universityof Science and Technology, Wuhan 430074, Hubei, China (e-mail: [email protected], [email protected]).

C.-X. Wang is with the School of Information Science and Engineering,Shandong University, Jinan 250100, Shandong, China, and the Joint ResearchInstitute for Signal and Image Processing, School of Engineering & PhysicalSciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK (e-mail: [email protected]).

F. Y. Li and F. Reichert are with the Department of Information andCommunication Technology, University of Agder (UiA), Grimstad, Norway(e-mail: {frank.li, frank.reichert}@uia.no).

L. Xiang, X. Ge, and C.-X. Wang would like to acknowledge the supportfrom the RCUK for the UK-China Science Bridges Project: R&D on (B)4GWireless Mobile Communications. L. Xiang, X. Ge, F. Y. Li and F. Reichertacknowledge the support from the EU FP7-PEOPLE-IRSES program, projectacronym S2EuNet (Grant no.: 247083). L. Xiang and X. Ge also acknowledgethe support from the National Natural Science Foundation of China (NSFC)(Grant No.: 61210002 and 61271224), the National 863 High TechnologyProgram of China (Grant No.: 2009AA01Z239), and the Ministry of Scienceand Technology (MOST), China, the International Science and TechnologyCollaboration Program (Grant No.: 0903), and the Hubei Provincial Scienceand Technology Department (Grant No.: 2011BFA004). C.-X. Wang acknowl-edges the support from the Scottish Funding Council for the Joint ResearchInstitute in Signal and Image Processing with the University of Edinburgh,which is a part of the Edinburgh Research Partnership in Engineering andMathematics (ERPem), and the Opening Project of the Key Laboratory ofCognitive Radio and Information Processing (Guilin University of ElectronicTechnology), Ministry of Education (No.: 2011KF01).

Digital Object Identifier 10.1109/TWC.2013.011713.112157

I. INTRODUCTION

DATA traffic in cellular networks is expected to have alasting exponential increase at least in the next five years

[1]. Currently, one of the biggest challenges is the continu-ous growth in energy consumption by cellular infrastructureequipment, base stations (BSs) especially, which make upto about 80% in the total energy consumption of cellularinfrastructure [2]. To satisfy such traffic demand while stillkeeping energy consumption comparatively low, the need forenergy efficiency improvement, i.e., the reduction of energyconsumption per traffic bit has gained paramount importancefor both economical and environmental advances [3].

Traditional traffic engineering in cellular networks wasfocused on traffic measurement, characterization and control,aiming to carry the largest amount of traffic load while satis-fying the required quality of service (QoS) using limited radioresources, e.g., wireless channels [4]. The new dimension ofenergy consumption can be similarly added into this problem,since energy is also an important type of radio resource,except that the objective now becomes to minimize energyconsumption per traffic bit, for energy efficiency maximizationunder given QoS constraints [5]. Typically, cellular networkoperators allocate their radio resources based on the worstcase principle, which is to allocate available resources basedon usage requirements at peak traffic load [6]. This leads tosignificant waste of resources, including energy consumption,during periods with low traffic load [7]. To save energy, a shut-down or switch-off under-utilized BSs scheme was proposedat low traffic load conditions [8]. A method to accommodatepeak traffic load with secondary (micro-, pico- or femto-) BSsthat have smaller coverages was presented in [9]. Furthermore,combining these two methods was explored by, for example,employing the shut-down strategy to smaller secondary BSsin [10]. An adaptive traffic coalescing (ATC) scheme wasproposed to monitor and adaptively coordinate traffic packetsin a mobile platform with sleep mechanisms [11]. In ourearlier work [12], we explored the tradeoff between theoperating power and the embodied power contained in themanufacturing process of infrastructure equipments from alife-cycle perspective. From an operational perspective, wefurther investigated the tradeoff between transmission powerand circuit power in a cellular network by optimizing thenumber and the location of BSs [13]. However, most of theabove traffic studies in wireless networks neglect realistic

1536-1276/13$31.00 c© 2013 IEEE

962 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 3, MARCH 2013

physical transmission processes, especially under wirelesschannel effects and interference which are significant formulti-cell wireless networks or large-scale ad hoc networks.

Improving spectral efficiency has been the primary forcethat drives wireless communications research over the pastyears. To increase spectral efficiency, adaptive modulationtechniques were proposed by adapting rate [14], transmitpower [15], constellation size [16], modulation modes [17]etc., according to variations of fading channels. Besides,transmitter power control has shown significant capabilityto increase system capacity by suppressing co-channel in-terference [18]. The impact of power control on terminalbattery duration is further explored in [19]. However, untillquite recently, energy efficiency in wireless communicationshas begun to attract lots of attention in the literature [20]–[23]. Energy efficiency in interfering channels was analyzedin a multi-cell wireless network [20]. Considering frequencyselective channels in orthogonal frequency division multipleaccess (OFDMA) systems, a new scheme which adapts theoverall transmission power and its allocation according tochannel states and circuit power consumption was proposed in[21], in order to maximize energy efficiency. In [22], the linkadaptation problem was formulated as a convex optimizationproblem and a set of link parameters were derived to maximizeenergy efficiency under given QoS constraints. Furthermore,a mechanism was proposed to switch between multiple inputmultiple output (MIMO) and single input multiple output(SIMO) modes to conserve mobile terminals’ energy in theuplinks of cellular networks [23]. However, most existingresearch in energy efficiency evaluation was limited at thelink level of cellular networks. To analyze energy efficiencyat the network level of cellular networks, a few structurecharacteristics of cellular networks were investigated in [24]–[28]. Due to random motions, the locations of users cannot bemodeled deterministically. Thus, statistical models, e.g., thePoisson point process [24], are preferable in system modelingand evaluation. The impact of spatial node distributions onthe performance of wireless networks was analyzed in [25], inwhich the power of spatial models was illustrated for networklevel design. On the other hand, the connectivity issues havebeen studied in ad hoc networks with nodes distributed accord-ing to the Poisson point process [26]. Furthermore, at a largescale, for instance, a regional or national cellular network, aprominent characteristic of the network topology is its irregu-larity in shape [27]. Traditional cellular network studies underthe simple hypothesis of hexagon topology are, however, farfrom enough to capture this macroscopic deployment feature.Therefore, considering that BSs and mobile stations (MSs)are generally assumed to be randomly located in an infiniteplane [28], the network level energy efficiency needs to befurther investigated in stochastic geometry cellular networkswith complex wireless channel conditions.

In this paper, we propose an energy efficiency model con-sidering traffic load and wireless channel effects in a stochasticgeometry cellular network. The main contributions of thiswork are summarized as follows.

1) Compared with regular hexagonal cellular scenarios,a stochastic geometry cellular scenario with PVT cellcoverages, is proposed to analyze spatial distributions

of traffic load and power consumption in cellular net-works. A PVT cellular scenario is able to capture therandomness of cell coverage in a realistic propagationenvironment in cellular networks compared with tradi-tional regular hexagonal cellular scenarios.

2) An aggregate traffic load model used for a typical PVTcell is derived through a characteristic function, whichcan be extended to a spatial distribution of traffic loadin PVT cellular networks.

3) Based on the signal-to-interference-ratio (SIR)-basedpower control process, we propose an analytical BSpower consumption model for a typical PVT cell, con-sidering path loss, log-normal shadowing and Rayleighfading effects in wireless channels. The spatial distri-bution of power consumption in a PVT cellular net-work can be extended from this BS power consumptionmodel.

4) Based on the spatial distributions of traffic load andpower consumption in PVT cellular networks, an energyefficiency model is proposed for performance analysis.

The rest of the paper is organized as follows. Section IIdescribes the system model. In Section III, an aggregate trafficload model in a typical PVT cell is derived. In Section IV,a BS power consumption model in a typical PVT cell withcomplex wireless channel effects is derived. Furthermore,based on the spatial distributions of aggregate traffic loadand power consumption, an energy efficiency model for PVTcellular networks is presented and evaluated in Section V.Finally, Section VI concludes this paper.

II. SYSTEM MODEL

A. Poisson-Voronoi Tessellation Cellular Networks

Assume that both MSs1 and BSs are located randomly inthe infinite plane R

2. Besides, users’ motions are isotropicand relatively slow, such that during an observation pe-riod, e.g., a time slot [29], the relative positions of MSsand BSs are assumed to be stationary. Then, following thework in [27], [28], the locations of MSs and BSs can bemodeled as two independent Poisson point processes [30],which are denoted as ΠM = {xMi : i = 0, 1, 2, · · ·} andΠB = {yBk : k = 0, 1, 2, · · ·}, where xMi and yBk are two-dimensional Cartesian coordinates, denoting the locations ofthe ith MS MSi and the kth BS BSk, respectively. Thecorresponding intensities of the two Poisson point processesare λM and λB .

For a traditional cellular network, assume that an MSassociates with the closest BS, which would suffer the leastpath loss during wireless transmission. Moreover, every cellis assumed to include only one BS and a few MSs. Thenthe cell boundary, which can be obtained through the De-launay Triangulation method by connecting the perpendicularbisector lines between each pair of BSs [31], splits the planeR

2 into irregular polygons that correspond to different cellcoverage areas. Such stochastic and irregular topology formsa so-called Poisson-Voronoi tessellation [30]. An illustration

1Throughout this work, MSs refer to active mobile users equipped withonly one antenna.

XIANG et al.: ENERGY EFFICIENCY EVALUATION OF CELLULAR NETWORKS BASED ON SPATIAL DISTRIBUTIONS OF TRAFFIC LOAD AND POWER . . . 963

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

km

km

MS

BS1

BS2

BS3

BS4

BS5

BS6

BS0

0

Fig. 1. Illustration of PVT cellular structure: both BSs and MSsare randomly located; the solid lines depict cell boundaries insidewhich a polygon corresponds a cell coverage; and the dashed lines,which are perpendicularly bisected by corresponding cell boundaries,demonstrate how to build tessellations through the Delaunay Trian-gulation method.

of PVT cellular network is depicted in Fig. 1, where each cellis denoted as Ck (k = 0, 1, 2, · · ·).

Despite of its complexity, an outstanding property of PVTcellular networks is that the geometric characteristics of anycell Ck with the kth BS centered at location yBk (yBk ∈ ΠB)coincide with that of a typical PVT cell2 C0 where BS0 locatesat a fixed position, e.g., origin, according to the Palm theory[28], [30]. This feature implies that the analytical results for atypical PVT cell C0 can be extended to the whole PVT cellularnetwork.

The propagation effects of path loss, shadowing andRayleigh fading can be found in [33], [34]. The channel gain,defined as the ratio between the received power Prx at anMS and the corresponding transmission power Ptx from itsassociated BS, is [35], [36]

L(r, ξ, ζ) =Prx

Ptx= Kr−β · ecσξ · ζ2, (1)

where K is a constant depending on antenna gains and theterm Kr−β models the path loss effect between the transmit-ting BS and the receiving MS at distance r away with path lossexponent β; the term ecσξ accounts for log-normal shadowingwith deviation σ, where ξ ∼ Gaussian(0, 1) is a standardnormal random variable (r.v.) and constant c = ln 10/10; theterm ζ2 is exponentially distributed with mean value 1 in theRayleigh fading environment. Here, the path loss term Kr−β

depends on the stochastic geometry of PVT cellular networksdefined above, instead of being deterministic. For the sakeof simplicity, we assume that channel parameters β, σ areconsistent across the whole PVT cellular network. Hereafter

2In stochastic geometry, the typical cell with certain geometric chatacteristicΞ is strictly defined, in ergodic means, as the empirical distribution of Ξover a square area with infinite length, which would converge to the Palmdistribution of Ξ and further coincide with the Palm distribution over thecell containing the origin, according to Slivnyak’s theorem [28], [30]. Moreformal discussions can be found in Chapter 5 of [32].

abbreviation Lki will be alternatively used to denote the chan-nel gain of a link between BSk and MSi (k, i = 0, 1, 2, · · · ).

B. Problem Formulation

Under the assumption of PVT cellular networks, a basicframework is used to evaluate spatial distributions of trafficload and power consumption in a typical PVT cell C0. Anadditive functional to characterize the geometrical structureof PVT cells was proposed in [27], [28], as

Ωwdef=

∑xi∈ΠM

w (xi)1 {xi ∈ C0}, (2)

where w(x) : R2 → R+ is a non-negative weight functiondefined on each point xi ∈ ΠM ; 1 {...} is an indicatorfunction, which equals to 1 when the condition inside thebracket is satisfied and 0 otherwise; and C0 is a typical PVTcell under investigation. Based on (2), we step further andstudy the aggregate traffic load and total BS transmissionpower considering spatial distributions in a typical PVT cell:

(1) Aggregate traffic load TC0 in a typical PVT cell C0The traffic rate, ρ, of the ith MS MSi at any location xMi

is denoted as ρ(xMi), referred as the spatial traffic intensityat MSi. Moreover, assume that the traffic load at each MS isgenerated independently. Then the aggregate traffic load TC0,which is the sum of spatial traffic intensity of all associatedMSs in a typical PVT cell C0, can be defined by a revisedadditive functional as follows

TC0def=

∑xMi∈ΠM

ρ(xMi)1 {xMi ∈ C0}. (3)

(2) Total BS transmission power PC0 in a typical PVT cellC0

By summing up the transmission power over all activedownlinks in a typical PVT cell C0, the total BS transmissionpower can be defined as follows

PC0def=

∑xMi∈ΠM

ε(xMi)1 {xMi ∈ C0}, (4)

where ε(xMi) is the per-link transmission power from the BSto its associated MS located at xMi. According to the SIR-based power control process [37], the per-link transmissionpower ε(xMi) depends on user’s traffic rate, interference andwireless channel conditions. The detailed modeling of per-link transmission power and total BS transmission power isextended in Section IV.

III. SPATIAL DISTRIBUTION OF TRAFFIC LOAD IN A PVTCELLULAR NETWORK

A. Spatial Distribution of Traffic Load in a Typical PVT Cell

Prior to characterizing the traffic load distribution, the areadistribution of PVT cells needs to be investigated as a basis forsubsequent derivations. Due to the complexity of the definedPVT structure, no analytical result or even approximationabout the distribution of a PVT cell area has been achievedcurrently. However, through large number of simulations, theempirical area distribution of PVT cells has been demonstratedto fit a Gamma distribution [31].


Given that the intensity of BS Poisson point process is λB ,a typical PVT cell area, denoted as AC0, follows a Gammadistribution with the probability density function (PDF) ex-pressed as,

fAC0(x) =(bλB)

a

Γ(a)xa−1 exp(−bλBx), (5)

where the Gamma function Γ (x) =∫∞0tx−1e−tdt; a is the

shape parameter and bλB is the inverse scale parameter for aGamma distribution, respectively.

Many empirical measurement results have demonstratedthat the traffic load in both wired and wireless networks,including cellular networks, is self-similar and bursty [38].Different from traditional traffic sources, self-similar trafficexhibits characteristics of slow-decaying tails and burstinessat different time-scales. Self-similar traffic models, e.g., Paretodistributions with infinite variance, have gained significantattention in modeling wireless networks [39]. To evaluate theimpact of self-similar traffic load on energy efficiency, thespatial traffic intensity ρ (xMi) at MSi is assumed to begoverned by a Pareto distribution with infinite variance inour study. Moreover, the spatial traffic intensity of all MSs isassumed to be independently and identically distributed (i.i.d.).Then, a PDF of spatial traffic intensity is given by

fρ(x) =θρθmin

xθ+1, x ≥ ρmin > 0, (6)

where θ ∈ (1, 2] reflects the heaviness of the distribution tail.When the value of heaviness index θ is closer to one, thedistribution tail of spatial traffic intensity becomes heavier.This result implies slower decaying in the tail of PDF curveand more burstiness in the defined traffic load. Parameter ρmin

represents the minimum traffic rate preserved for MS’s QoSguarantee. Furthermore, the average spatial traffic intensity atMSs is E (ρ) = θρmin

θ−1 , where E(·) denotes an expectationoperation.

The spatial distribution of aggregate traffic load dependsstochastically on both MS locations in space and traffic inten-sities that each MS point carries. To capture the randomness ofboth MS location and traffic intensity processes, the aggregatetraffic load TC0 in a typical cell C0 can be viewed as a markedPoisson point process, which is defined on the MS Poissonpoint process ΠM [24], [30] with the MS points assignedi.i.d. traffic intensity marks. The intensity measure is a randomcounting measure in characterizing the distribution of a spatialpoint process that is either homogenous or heterogeneous. Forthe marked traffic load point process, the intensity measure isgiven by

Λ (dx, dϕ) = λMAC0dx · fρ (ϕ) dϕ. (7)

Then, the characteristic function of the aggregate traffic loadTC0 can be derived by conditioning on the BS Poisson pointprocess ΠB [28]

φTC0 (ω) = EAC0 {exp [−λMAC0 (1− φρ(ω))]}

=

[1 +

λMbλB

− λMbλB

φρ (ω)

]−a

,(8)

where φρ(ω) is the characteristic function of spatial trafficintensity. Furthermore, an average aggregate traffic load ex-pression can be derived as follows

E (TC0) = λMEρ

{exp

[∫ϕ

1 {xMi ∈ C0}ϕfρ (ϕ) dϕ]}

=λMλB

E (ρ) .

(9)

Considering that the spatial traffic intensity is governed by aPareto distribution, the characteristic function of TC0 is furtherderived as follows

φTC0 (jω) =

[1 +

λMbλB

− λMbλB

θ(−jρminω)θ

×Γ (−θ,−jρminω)

]−a

,

(10a)

with

Γ(−θ,−jρminω) =

∫ +∞

−jρminω

t−θ−1e−tdt. (10b)

Then the average aggregate traffic load expression E(TC0) issimplified as

E (TC0) =λMθρmin

λB(θ − 1). (11)

From the characteristic function in (10), the distribution func-tion of the aggregate traffic load can be uniquely determinedby inverse Fourier transforms [40]. Furthermore, applyingthe Palm theory [28], [30], the aggregate traffic load modelproposed in (8) in a typical PVT cell can be safely extendedto the whole PVT cellular network.

B. Discussions on the Traffic Load Model

Based on the above proposed spatial distribution of trafficload, numerical results are illustrated in this subsection. Thedefault values for the following parameters are configured as:a = 3.61 and b = 3.57, which are based on the best-fit results[31]; θ =1.8 and ρmin = 10.75 kbps for typical wirelesscommunication applications [41]; and the intensity ratio ofMSs to BSs is configured as λM/λB = 15 [42].

Fig. 2 shows the cumulative density function (CDF) ofaggregate traffic load with different intensity ratios of MSs toBSs based on the Fourier transform of (10a). Fig. 2 indicatesthat the probability mass shifts to the right with the increasingintensity ratio of MSs to BSs. This is due to the fact that theaverage aggregate traffic load increases with the increasingvalue of λM/λB , i.e., the average aggregate traffic load willincrease when there are more MSs in a typical PVT cell.

Fig. 3 illustrates the impact of the heaviness index θ andthe minimum traffic rate ρmin on the aggregate traffic load ina typical PVT cell. When the heaviness index θ is fixed, theprobability mass would shift to the right with the increasingminimum traffic rate ρmin, resulting in slight changes in thetail probability. When the minimum traffic rate ρmin is fixed,the tail of PDF curves would become heavier, i.e., slowerdecay, with the decreasing heaviness index θ, which implieshigher probability of large traffic load. The above results also


0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aggregate traffic load [kbps]

CDF

of a

ggre

gate

traf

fic lo

ad

M/ B=15

M/ B=30

M/ B=45

Fig. 2. The CDF of aggregate traffic load in a typical PVT cell withdifferent intensity ratios of MSs to BSs.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5

3

3.5x 10-3

Aggregate traffic load [kbps]

PDF

of a

ggre

gate

traf

fic lo

ad

=1.8, min=10

=1.8, min=15

=1.2, min=10

=1.2, min=15

Fig. 3. Impact of heaviness index and minimum traffic rate on theaggregate traffic load in a typical PVT cell.

demonstrate that the minimum traffic rate and heaviness indexhave inverse effects on the average amount of aggregate trafficload in a typical PVT cell.

IV. SPATIAL DISTRIBUTION OF POWER CONSUMPTION IN

A PVT CELLULAR NETWORK

A. Transmission Power in Wireless Downlinks

The transmission processes of the point-to-point downlinksare power controlled by the associated BSs, in order to keepthe MSs’ received SIR above a given threshold. In Fig. 4,without loss of generality, a wireless downlink transmissionis considered between an arbitrary MS, e.g., MS0, and itsassociated BS, e.g., BS0, inside a typical PVT cell C0. ForMS0, no more than one co-channel interfering MS is assumedto exist in each adjacent PVT cell. The downlinks of co-channel MSs in the adjacent PVT cells contribute interferenceto MS0, which are transmitted from its associated BSs. Fordistinction, a co-channel interfering MS for MS0 is denoted asIMSk(k = 1, 2, 3, · · · ) and it is associated to a corresponding

Fig. 4. Wireless downlinks of a PVT cellular network. An exampleof interfering BS is illustrated at IBSk with detailed channelparameters.

interfering BS, denoted as IBSk(k = 1, 2, 3, · · · ), in PVT cellCk(k = 1, 2, 3, · · · ). The receiving power of IMSk is denotedas Sk(k = 1, 2, 3, · · · ) and the receiving power of MS0

is S0. Moreover, all random variables Sk(k = 0, 1, 2, · · · )are assumed as i.i.d.. Then, the interference power that MS0

receives from IBSk can be expressed as Sk · Lk0/Lkk [37],where Lkk is the channel gain of the link from IBSk to IMSk

and Lk0 is the channel gain of the link from IBSk to MS0.Neglecting the noise at MSs, the instantaneous SIR of MS0,denoted as γ , is given by

γ =S0

Iagg=

S0∑k∈ΦBS

Sk ·L(rk0, ξk0, ζk0)

L(rkk, ξkk, ζkk)· 1D(Lk0/Lkk)

,

(12a)with

1D(Lk0/Lkk) = 1D{rkk/rk0 ≤ 1}, (12b)

where Iagg is the aggregate interference received at MS0, ΦBS

is the index set of interfering BSs, and the indicator function1D(Lk0/Lkk) is a constraint on MS distance distributionsunder the closest association rule3 in PVT cellular networks.

Without considering the additional transmission power foroutage loss in downlinks, the relationship of the requiredreceiving SIR4 γ and the spatial traffic intensity ρ (xM0) atMS0 is given by [43]

BW · log2(1 + γ/Δ) = ρ (xM0) , (13)

where BW is the bandwidth allocated for MS0; Δ defines theSIR gap between the channel capacity and a practical codingand modulation scheme achieving a target bit error rate atMS0. Based on the PDF of spatial traffic intensity in (6), thePDF of the receiving SIR γ can be further derived as follows

fγ(z) =θρθminB

−θW

ln 2 · (Δ + z)(log2 (1 + z/Δ))

−θ−1,

z > z0 = Δ ·(2ρmin/BW − 1

),

(14)

3An MS cannot change its associated BS unless it moves into another PVTcell.

4The SIR here approximates the signal-to-interference-noise-ratio (SINR)in an interference-limited cellular network.


where z0 is the minimum required SIR to achieve the mini-mum traffic rate ρmin for an MS.

Denote the interfering downlink set ΠInf as the collec-tion of co-channel active downlinks, which contains thelink pairs from the interfering BS IBSk at location yIkto the interfering MS IMSk at location xIk , i.e., ΠInf ={(xIk, yIk) : k = 1, 2, 3, · · ·}. The interfering downlink setΠInf can be viewed as an independent thinning process on theBS Poission point process ΠB . Thus, ΠInf is still a Poissonpoint process with intensity λInf , which generally satisfies0 ≤ λInf ≤ λB

5.Based on the stochastic geometry theory, the co-channel

interference Iagg aggregated at MS0 can be further expressedas follows

Iagg =∑

(xIk,yIk)∈ΠInf

Sk · ‖yIk − xIk‖β

‖yIk − xM0‖β·Qk

· 1D1 {xM0 ∈ C0} · 1D2 {xIk ∈ Ck} ,(15a)

with

Qk = ecσ(ξk0−ξkk) · ζ2k0

ζ2kk, (15b)

where ‖yIk − xIk‖ and ‖yIk − xM0‖ denote the Euclid dis-tances from IBSk to IMSk and from IBSk to MS0, theindicator functions 1D1 and 1D2 specify the range of distancesfrom MSs to their associated BSs in residing PVT cells. Notethat constraint (12b) is met immediately.

Under the assumption that channel parameters β and σare identical across the whole cellular network, Qk(k =1, 2, 3, · · · ) are i.i.d. random variables with the following PDF

fQk(x) =

1

2√πcσ

∫ ∞

0

exp(− t2

4c2σ2 + t)

(et + x)2dt, x ≥ 0. (16)

Furthermore, based on the result from the Appendix, the char-acteristic function of co-channel interference Iagg aggregatedat MS0 is derived as follows

φIagg (ω) = exp

{−δ|ω| 2β

[1− jsign(ω) tan

(π

β

)]},

(17a)with

δ =λInf4λB

Γ(1− 2

β) cos(

π

β)E(S

2/βk )E(Q

2/βk ), (17b)

E(Q2/βk ) =

2π

β sin(2π/β)exp

(4c2σ2

β2

), (17c)

where sign(ω) is a sign function [44]. From (17a), theaggregate interference is an α-stable process with infiniteexpectation [36], [45].

Based on the results of the required SIR and interferencederivations, the required receiving power can then be derivedbased on (12a). Given that the moment of the required

5Actually, for cellular networks with full frequency reuse, 0 ≤ λInf ≤ λB ;while for traditional cellular networks with frequency reuse Nf (Nf is aninteger and Nf > 1), 0 ≤ λInf ≤ λB

Nf. The results in this study are also

applicable for the latter case.

receiving power is E(S2/βk ), then, the characteristic function

of Sk(k = 0, 1, 2, · · · ) is derived [40]

φSk(ω) =

∫x

φIagg (ωx)fγ(x)dx

=

∫ ∞

0

exp

{−δ|ω|

2β x

2β ×[

1− jsign(ω) tan πβ

] }fγ(x)dx.

(18)

Furthermore, based on (18), the corresponding requiredtransmission power for the downlink from BS0 to MS0 isderived as follows

ε (xM0) =S0

L (‖xM0 − yB0‖ , ξ0, ζ0)

=‖xM0 − yB0‖β · e−cσξ0

Kζ20S0.

(19)

B. Total BS Transmission Power

It is assumed that all wireless downlinks are assignedappropriate transmission power to transmit data with requiredtraffic rate. Denote r.v.s U = S0 · V and V = e−cσξ0

/ζ20 . By

substituting (19) into (4), the required total BS transmissionpower in a typical PVT cell C0 is given by

PC0 req =∑

xMj∈ΠM

‖xMj − yB0‖β

KU · 1 {xMj ∈ C0}, (20)

where we similarly have E (1 {xMj ∈ C0}) = e−πλB‖xMj‖2

and the PDF of V, fV (x), as follows

fV (x) =1√

2πcσx2

∫ ∞

−∞exp

(− t2

2c2σ2+ t− et

x

)dt. (21)

Furthermore, the required total BS transmission power canbe also formulated by a marked Poisson point process based onMS Poisson point process with marks capturing the requiredtransmission power of downlinks to each MS point. Differentfrom the spatial traffic load process, here the probabilisticmarks of transmission power depend on the distances betweenMS and BS points instead of being i.i.d. random variables.Moreover, the spatial geometry of PVT cellular networkshas greater impact on the spatial power consumption processdue to its effects on downlink distance distributions, co-channel interference, and per-downlink transmission power, asillustrated in (19). Following the derivations in the previoussection, the intensity measure of the marked transmissionpower process can be similarly given as

Λ(dx, du) = λMdx · fU (u)du, (22)

where fU (u) is the PDF of U . Applying the Campbell theoremfor marked Poisson point processes [30], the characteristicfunction of the required total BS transmission power is derivedas (23a), where φU (·) is the characteristic function of U , andwe have φU (ω) =

∫xφS0(ωx)fV (x)dx [40]. By substituting

φSk(ω) into (18), we can further express φPC0 req(ω) as (23b)

with (23c).However, numerical results show that the expectation value

of the required total BS transmission power approximatesinfinity in the case of perfect power control. It is caused


φPC0 req(ω) = E

{exp

[2πλM

∫x

∫u

(e

jωxβuK − 1

)fU (u)du · 1 {x ∈ C0}xdx

]}= exp

[−2πλM

∫ ∞

0

(1− φU

(ωxβ

K

))e−πλBx2

xdx

],

(23a)

φPC0 req(ω) = exp

⎧⎨⎩−λM

∫∫∫r,x,y

[1− exp

(−G( ω

K )r2x2β y

2β

)]2πre−πλBr2fV (x)fγ(y)drdxdy

⎫⎬⎭

= exp

{−λMλB

[1−Eγ,V

(πλB

G( ωK )V

2β γ

2β + πλB

)]},

(23b)

G(ω) = δ|ω|2β

[1− jsign(ω) tan

π

β

]. (23c)

by a result that an infinite demand for transmission power isrequired to maintain a given SIR level when the expectationof aggregate interference approaches infinity.

Considering that perfect power control is an ideal case, amaximal BS transmission power threshold Pmax is configuredin the following. In this case, a portion of wireless downlinksmay be interrupted when the required total BS transmissionpower exceeds Pmax. Therefore, by truncating PC0 req in theinterval (0, Pmax], the PDF of realistic total BS transmissionpower in a typical PVT cell C0 can be derived as

fPC0 real(x) =

⎧⎨⎩

PC0 req(x)

FPC0 req(Pmax), x ≤ Pmax;

0, x > Pmax;(24)

where FPC0 req(·) is the CDF of PC0 req.

C. BS Power Consumption Model

The BS power consumption can be decomposed into a fixedpower consumption part and a dynamic power consumptionpart respectively [46]. The fixed power consumption, e.g., thecircuit power consumption, is the baseline power consumedduring processes such as signal processing, site cooling, powersupply and battery backup. The circuit power consumptionusually depends on the hardware and software configurationsof BSs and is independent of the spatial distribution of trafficload. The dynamic power consumption, depending on thespatial distribution of traffic load however, accounts for thetransmission power consumed in the radio frequency (RF)transmission circuits including power dissipation such as heat.

Based on the decomposition of BS power consumption, theprobability distribution of total BS power consumption canbe derived from the transmission power distributions if theefficiency of RF transmission circuits is properly modeled.However, only the average RF efficiency is normally availablein practical applications. Thus, a linear average BS powerconsumption model is simply built as follows

E(PBS) =E (PC0 real)

ηRF+ PCircuit

=

∫ Pmax

0 xfPC0 req(x)dx

ηRF ·∫ Pmax

0 fPC0 req(x)dx+ PCircuit,

(25)

where ηRF is the average efficiency of RF transmission circuitsand the circuit power PCircuit is fixed as a constant. Applyingthe Palm theory [28], [30], (25) can be directly extended toa spatial distribution of power consumption in PVT cellularnetworks.

D. Discussions on BS Power Consumption

Based on the proposed model, we illustrate BS power con-sumption numerically in this subsection. The parameters for aPVT cellular network shown in Fig. 4 are configured as fol-lows: the moment of receiving power is E(S

2/βk ) = 10−10 W

(or -70 dBm), which corresponds to a receiving power levelof MSs in the order of 10−15 W (or -120 dBm) [41]; theBS intensity is configured as λB = 1

/(π ∗ 8002) m-2; the

intensity ratio of MSs to BSs is configured as λM/λB =30 and the intensity ratio of interfering links to BSs isconfigured as λInf/λB = 0.9; the path-loss exponent β =3.5, shadowing deviation σ = 6 and K = −31.54 dB inan urban micro-cell environment [47]; the heaviness indexθ = 1.8 and the SIR gap Δ = 8.6 dB are configured fora quadrature amplitude modulation (QAM) system [43]; thechannel bandwidth BW is normalized, then the normalizedminimum rate is ρmin = 2 bits/s/Hz, which corresponds to anequivalent average spectral efficiency 4.5 bits/s/Hz for typicalcellular networks [42].

Based on the inverse Fourier transform of (23b), the CDF ofrequired total BS transmission power PC0 req under differentintensity ratios of MSs to BSs is illustrated in Fig. 5. For amicro BS, the values of the required transmission power varyfrom several Watts to tens of Watts in this numerical result[46]. With the increase of the intensity ratio of MSs to BSs,the average number of MSs in a PVT cell increases, leadingto the increased total BS transmission power. Different fromthe CDF of aggregate traffic load, these curves exhibit a heavytail feature, indicating a high probability of large required totalBS transmission power.

Fig. 6 shows the CDF of required total BS transmissionpower with different path loss exponents. The increased pathloss exponent would shift the probability mass to the left,indicating that lower transmission power is needed with largerpath-loss exponent on average due to greater attenuation ofinterference power. The heavy tail is also easily observed.


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Required total BS transmission power [W]

CDF

of r

equi

red

tota

l BS

tran

smis

sion

pow

er

M/ B=45

M/ B=30

M/ B=15

Fig. 5. The CDF of required total BS transmission power withdifferent intensity ratios of MSs to BSs.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


CDF

of r

equi

red

tota

l BS

tran

smis

sion

pow

er

=3.6=3.8=4

Fig. 6. The CDF of required total BS transmission power withdifferent path loss exponents.

From Fig. 5 and Fig. 6, the required total BS transmissionpower exhibits a significant degree of bustiness, indicatinghigher demand for large transmission power to support self-similar traffic load.

In Fig. 7, the PDF of required total BS transmission powerPC0 req is depicted with different heaviness indices θ. Withthe decrease of heaviness index, indicating that the traffic atMSs is more bursty, the probability mass of the required totalBS transmission power remains rather stable except that theincreasingly heavier tail decays slower.

Finally, Fig. 8 evaluates the required total BS transmissionpower PC0 req with different interfering link intensities λInf .With the increase of the interfering link intensity λInf , thePDF of the required total BS transmission power would shiftits probability mass to the right. This result indicates that, toguarantee the traffic rate for a specified MS, more transmissionpower is required to compensate for the corresponding SIRdegradation with increased interference from adjacent PVTcells.

0 5 10 15 20 25 30 35 40 45 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05


PDF

of r

equi

red

tota

l BS

trans

mis

sion

pow

er

=1.9=1.5=1.1

Fig. 7. The PDF of required total BS transmission power withdifferent heaviness indices.

0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


PDF

of r

equi

red

tota

l BS

tran

smis

sion

pow

erInf=3.0 10-7

Inf=4.0 10-7

Inf=5.0 10-7

Fig. 8. The PDF of required total BS transmission power withdifferent interfering link intensities.

V. ENERGY EFFICIENCY OF PVT CELLULAR NETWORKS

A. Energy Efficiency Model

Based on the obtained spatial distributions of traffic loadand power consumption in PVT cellular networks, we furtherinvestigate the energy efficiency of PVT cellular networks,which is represented by the BS energy efficiency in a typicalPVT cell according to the Palm theory [28], [30]. The utilityfunction of energy efficiency ηEE is defined as the ratio ofthe effective average traffic load6 over the average total powerE(PBS) consumed at a BS in a typical PVT cell

ηEE =E(TC0) · (1− pout)

E(PBS), (26)

6Here “effective” means the realistic traffic load eliminating the traffic loadin outage, by multiplying the average aggregate traffic load E(TC0) with theprobability 1− pout.


ηEE =λMθρmin

λB(θ − 1)·

(∫ Pmax

0fPC0 req(x)dx

)2

1ηRF

∫ Pmax

0xfPC0 req(x)dx + PCircuit ·

∫ Pmax

0fPC0 req(x)dx

, (28)

where the outage probability7, defined as the probability thatthe required transmission power fails to be satisfied under themaximal transmission power constraint due to variations ofwireless channel status and effects of interference, is givenby,

pout = Pr {PC0 req > Pmax}= 1− FPC0 req (Pmax) .

(27)

Based on (11) and (25), the energy efficiency model can befurther derived as shown in (28), where the PDF of the re-quired BS transmission power fPC0 req(x) can be numericallycalculated by its characteristic function in (23b).

B. Simulation Results and Discussions

Based on the proposed energy efficiency model, the effectof traffic load and wireless channel parameters on the energyefficiency of PVT cellular networks is investigated in details.Furthermore, the analysis results are validated through Monte-Carlo (MC) simulations in a PVT cellular network withradius R = 10 km. Different from the statistical treatmentof transmission power in analytical models (e.g., in (24)),the maximal available transmission power Pmax = 40 Wis defined in the MC simulations and MSs are randomlyselected in outage when the total transmission power reachesthe maximum. In the following, some default parametersare specified: the moment of receiving power is configuredas E(S

2/βk ) = 10−10 W (or -70 dBm); the BS intensity is

given by λB = 1/(π × 8002) m-2, while the intensity ratioof interfering links to BSs is configured as λInf/λB = 0.8;the path-loss exponent β = 3.8, shadowing deviation σ = 6and K = −31.54 dB corresponding to an urban micro-cellenvironment; the heaviness index is θ = 1.8 and the SIRgap is Δ = 8.6 dB for a QAM system; the normalizedminimum rate is ρmin = 2 bits/s/Hz, which corresponds to anapproximate average spectral efficiency level of 4.5 bits/s/Hzfor typical cellular networks; the maximal transmission poweris Pmax = 40 W [46]; the average efficiency of RF circuit fora micro BS is configured as ηRF = 0.047 and the fixed BSpower consumption is set as Pconst = 354.4 W [51].

Fig. 9 illustrates the energy efficiency of PVT cellularnetworks with respect to the heaviness index θ and theminimum traffic rate ρmin at MSs, in which “Num” labelsthe numerical results and “MC” represents the MC simulationresults. First, we fix the values of minimum traffic rate ρmin at2 or 3 bits/s/Hz, and analyze the impact of the heaviness indexθ on the energy efficiency of PVT cellular networks. Bothnumerical and MC simulation results consistently demonstratethat the energy efficiency of PVT cellular networks is in-creased when the heaviness index θ is decreased from 1.8to 1.2. Secondly, we fix the values of the heaviness index θ

7Outage probability can be also similarly defined as the probability that thebit (symbol) error probability exceeds a maximal threshold, to measure thebit (symbol) error outage in wireless fading channels [48]–[50].

at 1.2 or 1.8, and analyze the impact of the minimum trafficrate ρmin on the energy efficiency of PVT cellular networks.The MC simulation results show that the energy efficiencyof PVT cellular networks monotonically increases when theminimum traffic rate increases from 2 to 3 bits/s/Hz. However,the numerical results show that when the heaviness index θis fixed at either 1.2 or 1.8, there exist turning points forthe intensity ratios of MSs to BSs (the turning points are 25for θ = 1.2 and 68 for θ = 1.8). Below the turning point,the energy efficiency of PVT cellular networks increases andabove the turning point the energy efficiency decreases, as theminimum traffic rate ρmin is increased from 2 to 3 bits/s/Hz.

The numerical results in Fig. 9 illustrate that there existsa maximal value for energy efficiency under each parameterconfiguration. The maximal energy efficiency values are 0.55,0.45, 0.29 and 0.26, corresponding to the intensity ratiosof MSs to BSs of 110, 80, 130 and 90, respectively. Thisresult can be explained from (28). When the intensity ratioof MSs to BSs is low, indicating a few MSs in a typicalPVT cell, the increase of the intensity ratio of MSs to BSsleads to a moderate increase in total BS power consumptionincluding mainly fixed BS power consumption and a smallportion of dynamic BS power consumption. In this case,the energy efficiency of PVT cellular networks is increased.However, when the intensity ratio of MSs to BSs in a PVTtypical cell exceeds a given threshold, a high aggregate trafficload resulted from a large number of MSs will significantlyincrease the total BS power consumption including mainlydynamic BS power consumption and a small portion of fixedBS power consumption. In this case, the energy efficiency ofPVT cellular networks is decreased. Such an energy efficiencypattern with respect to traffic load was also observed inwireless LANs [52], where random medium access protocolsare deployed. However, the MC simulation results in Fig. 9demonstrate that the energy efficiency turns into saturationafter having reached the maximal threshold.

The different trends between numerical and MC simulationsafter having reached the maximal threshold are caused bydifferent outage control methods. In the numerical analysis, ifthe required transmission power exceeds the maximal trans-mission power Pmax with a large number of users, MSs’ trafficload requests are interrupted with the probability calculated by(27). This will cause the energy efficiency to decrease afterhaving reached the maximal threshold. In the MC simulation,for the same scenario, an MS’ traffic load request would beinterrupted instantly when the transmission power is unavail-able. This will achieve better utilization, on the average, ofthe BS transmission power and turns the energy efficiencyinto saturation after having reached the maximal threshold.

In Fig. 10, the effect of interfering link intensity on theenergy efficiency of PVT cellular networks is investigated.When the BS intensity λB is fixed, both numerical and MCsimulation results show that the energy efficiency of PVT cel-


0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Intensity ratio of MSs to BSs

Ener

gy e

ffici

ency

of P

VT c

ellu

lar n

etw

orks

[bits

/Hz/

Joul

e]=1.2, min=2, Num

=1.2, min=2, MC

=1.2, min=3, MC

=1.2, min=3, MC

=1.8, min=2, Num

=1.8, min=2, MC

=1.8, min=3, Num

=1.8, min=3, MC

Fig. 9. Energy efficiency of PVT cellular networks with respect tothe intensity ratio of MSs to BSs considering the heaviness indexand the minimum traffic rate.

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Ener

gy e

ffici

ency

of P

VT c

ellu

lar n

etw

orks

[bits

/Hz/

Joul

e]

=4, Num=4, MC=3.8, Num=3.8, MC=3.6, Num=3.6, MC

Fig. 10. Energy efficiency of PVT cellular networks with respectto the intensity ratio of MSs to BSs considering the interfering linkintensity.

lular networks is decreased when the interfering link intensityλInf is increased from 3.0 × 10−7 m−2 to 5.0 × 10−7 m−2.Moreover, the maximal values of the energy efficiency inthree different cases are 0.39, 0.29, and 0.23 bits/Hz/Joule,which correspond to the intensity ratios of MSs to BSs as170, 130, and 100, respectively. The MC simulation curvesexhibit a good match with the numerical curves before theenergy efficiency reaches the maximum, with deviations afterthe maximum value similar to the results shown in Fig. 9.These results imply that interference has obvious impact onthe energy efficiency of PVT cellular networks.

Finally, the impact of path loss exponent β on the energyefficiency of PVT cellular networks is evaluated in Fig. 11.Both numerical and MC simulation results illustrate that theenergy efficiency of PVT cellular networks is increased when

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Ener

gy e

ffici

ency

of P

VT c

ellu

lar n

etw

orks

[bits

/Hz/

Joul

e]

inf=3.0*10-7, Num

inf=3.0*10-7, MC

inf=4.0*10-7, Num

inf=4.0*10-7, MC

inf=5.0*10-7, Num

inf=5.0*10-7, MC

Fig. 11. Energy efficiency of PVT cellular networks with respect tothe intensity ratio of MSs to BSs considering the path loss exponent.

the path loss exponent is increased from 3.6 to 4. Moreover,the maximal values of the energy efficiency in three differentcases are 0.17, 0.29, and 0.46 bits/Hz/Joule, which correspondto the intensity ratios of MSs to BSs as 80, 130, and 190,respectively.

VI. CONCLUSIONS

In this paper, we have proposed a novel model to evaluatethe energy efficiency of PVT cellular networks consideringspatial distributions of traffic load and power consumption. Toderive this model, an aggregate traffic load model in a typicalPVT cell is derived through its characteristic function basedon stochastic cellular geometry. Moreover, taking into accountpath loss, log-normal shadowing and Rayleigh fading effectsin wireless channels, an analytical total BS power consumptionmodel in a typical PVT cell is derived. Based on the proposedenergy efficiency model of PVT cellular networks, numericalresults have shown that there is a maximal limit of energyefficiency in PVT cellular networks considering the tradeoffbetween the traffic load and BS power consumption. More-over, wireless channel conditions have great impact on theenergy efficiency of PVT cellular networks. Our analysis in-dicates that interference reduction or interference coordinationcan effectively improve the energy efficiency of PVT cellularnetworks, especially in scenarios with high intensity ratio ofMSs to BSs. Furthermore, our results provide guidelines forthe design of energy-efficient traffic control techniques andinsights into the energy efficiency optimization in PVT cellularnetworks.

APPENDIX

DERIVATIONS OF (17A), (17B) AND (17C)

Based on (15a), we set R1 = ‖yIk − xIk‖ in the followingderivations. By conditioning on the BS Poisson point processΠB [28], the event of {R1 > r|ΠB} is equivalent to the eventthat no BSs (ΠB points) locate inside the circle centered at


xIk with radius r, Then the conditional PDF of R1 is givenby

fR1|ΠB(r) = −dPr{R1 > r}

dr= 2πλBxe

−πλBr2 . (29)

Here the BS intensity λB is used instead of the interferinglink intensity λInf due to the closest association rule underthe stochastic geometry assumed previously.

The random variable R2 = ‖yIk − xM0‖ in (15a) dis-tributes in the same way as ‖yIk − yB0‖ (k = 1, 2, 3, · · · ) al-most surely due to the stationarity of Poisson point processes,which further coincides with the distribution of ‖yIk‖ (k =1, 2, 3, · · · ) based on Slivnyak’s theorem [30], behaving as ifBS0 locates at origin. Moreover, we set Ψ = Sk · Qk forsimplifying notation, where Sk and Qk are independent in(15a). Therefore, (15a) is reduced as follows

Iagg =∑

k∈ΦBS

Rβ1

‖yIk‖β·Ψ · 1D1 · 1D2, (30)

which is a marked Poisson point process defined on the BSPoisson point process ΠB . The intensity measure of (30) isgiven by

Λ(dx, dr, dψ) = λInfdx · fR1|ΠB(r)dr · fΨ(ψ)dψ, (31)

where λInf is the interfering link intensity.Based on (30), (31) and the Campbell theorem for marked

Poisson point processes [30], a log-characteristic function ofIagg is derived as follows

lnφIagg (ω) = −(2π)2λInfλB

×∫ ∞

0

∫ ∞

0

[1− φΨ

(ωxβ

yβ

)]ydy︸︷︷︸

=H(ω,x)

·xe−2πλBx2

dx.

(32)

(32) holds with E (1D2 {xIk ∈ Ck}) = e−πλB‖yIk−xIk‖2

dueto the fact that a point at xIk belongs to a cell Ck if and onlyif there is no BS point locating inside the circle centered atxIk with radius ‖yIk − xIk‖.

Let t = |ω|xβy−β . Then H(ω, x) in (32) can be calculatedby [36]

H(ω, x) =Γ(2− 2

β ) cos(πβ )

2− 4β

|ω| 2β x2E(Ψ2β )

×[1− jsign(ω) tan

(π

β

)].

(33)

Due to the independence between Sk and Qk, we haveE(Ψ2/β) = E(S

2/βk )E(Q

2/βk ). Then by applying the integral∫∞

0 x3e−2πλBx2

dy = 1/8(πλB)

2 [53], the characteristicfunction of Iagg is derived as follows

φIagg(ω) = exp{−δ|ω| 2β

[1− jsign(ω) tan

(πβ

)]}, (34a)

with

δ =λInf

4λBΓ(1− 2

β ) cos(πβ )E(S

2/βk )E(Q

2/βk ), (34b)

E(Q2/βk ) =

2π

β sin(2π/β)exp

(4c2σ2

β2

). (34c)

This completes the derivations.

REFERENCES

[1] Cisco Visual Networking Index: Global Mobile Data Traffic ForecastUpdate, 2010-2015. White Paper, Feb. 2011.

[2] A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok, “The globalfootprint of mobile communications: the ecological and economic per-spective,” IEEE Commun. Mag., vol. 49, no.8, pp. 55–62, Aug. 2011.

[3] C. Han, T. Harrold, S. Armour, I. Krikidis, et al., “Green radio: radiotechniques to enable energy efficient wireless networks,” IEEE Commun.Mag., vol. 49, no. 6, pp. 46–54, June 2011.

[4] D. E. Everitt, “Traffic engineering of the radio interface for cellularmobile networks,” Proc. IEEE, vol. 82, no. 9, pp. 1371–1382, Sep. 1994.

[5] H. Bogucka and A. Conti, “Degrees of freedom for energy savings inpractical adaptive wireless systems,” IEEE Commun. Mag., vol. 49, no.6, pp. 38–45, June 2011.

[6] P. Trimintzios, G. Pavlou, P. Flegkas, P. Georgatsos, A. Asgari, and E.Mykoniati, “Service-driven traffic engineering for intradomain quality ofservice management,” IEEE Netw., vol. 13, no. 3, pp. 29–36, May 2003.

[7] E. Hwang, K. J. Kim, J. J. Son, and B. D. Choi, “The power-savingmechanism with periodic traffic indications in the IEEE 802.16e/m,” IEEETrans. Veh. Technol., vol. 59, no. 1, pp. 319–334, Jan. 2010.

[8] E. Oh, B. Krishnamachari, X. Liu, and Z. Niu, “Toward dynamic energy-efficient operation of cellular network infrastructure,” IEEE Commun.Mag., vol. 49, no. 6, pp. 56–61, June 2011.

[9] W. Wei and G. Shen, “Energy efficiency of heterogeneous cellularnetwork,” in Proc. 2010 IEEE VTC – Fall, pp. 1–5.

[10] I. Ashraf, F. Boccardi, and L. Ho, “SLEEP mode techniques for smallcell deployments,” IEEE Commun. Mag., vol. 49, no. 8, pp. 72–79, Aug.2011.

[11] R. Wang, J. Tsai, C. Maciocco, T. Y. Tai, and J. Wu, “Reducing powerconsumption for mobile platform via adaptive traffic coalescing,” IEEEJ. Sel. Areas Commun., vol. 29, no. 8, pp. 1618–1629, Sep. 2011.

[12] I. Humar, X. Ge, L. Xiang, J. Ho, and M. Chen, “Rethinking energyefficiency models of cellular networks with embodied energy,” IEEENetw., vol. 25, no. 2, pp. 40–49, Mar.–Apr. 2011.

[13] L. Xiang, F. Pantisano, R. Verdone, X. Ge, and M. Chen, “Adaptivetraffic load-balancing in green cellular networks,” in Proc. 2011 IEEEPIMRC, pp. 41–45.

[14] W. T. Webb and R. Steele, “Variable rate QAM for mobile radio,” IEEETrans. Commun., vol. 43, no. 7, pp. 2223–2230, July 1995.

[15] A. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM forfading channel,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218–1230,Oct. 1997.

[16] A. Conti, M. Z. Win, and M. Chiani, “Slow adaptive M-QAM withdiversity in fast fading and shadowing,” IEEE Trans. Commun., vol. 55,no. 5, pp. 895–905, May 2007.

[17] T. Keller and L. Hanzo, “Adaptive modulation techniques for duplexOFDM transmission,” IEEE Trans. Veh. Technol., vol. 49, no. 5, pp. 1893–1906, Sep. 2000.

[18] J. Zander, “Performance of optimum transmitter power control incellular radio systems,” IEEE Trans. Veh. Technol., vol. 41, no. 1, pp.57–62, Feb. 1992.

[19] M. Chiani, A. Conti, and R. Verdone, “Partial compensation signal-level-based up-link power control to extend terminal battery duration,”IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 1125–1131, July 2001.

[20] X. Ge, J. Hu, C.-X. Wang, C.-H. Youn, J. Zhang, and X. Yang, “Energyefficiency analysis of MISO-OFDM communication systems consideringpower and capacity constraints,” ACM Mob. Netw. and App., vol. 17, no.1, pp. 29–35, Feb. 2012.

[21] M. Guowang, N. Himayat, and Y.(G.) Li, “Energy-efficient link adap-tation in frequency-selective channels,” IEEE Trans. Commun., vol. 58,no. 2, pp. 545–554, Feb. 2010.

[22] H. S. Kim and B. Daneshrad, “Energy-constrained link adaptation forMIMO OFDM wireless communication systems,” IEEE Trans. WirelessCommun., vol. 9, no. 9, pp. 2820–2832, Sep. 2010.

[23] H. S. Kim, C. B. Chae, G. de Veciana, and R. W. Heath, “A cross-layerapproach to energy efficiency for adaptive MIMO systems exploitingspare capacity,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4264–4275, Aug. 2009.

[24] C. C. Chun and S. V. Hanly, “Calculating the outage probability in aCDMA network with spatial Poisson traffic,” IEEE Trans. Veh. Technol.,vol. 50, no. 1, pp. 183–204, Jan. 2001.

[25] J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber, “Aprimer on spatial modeling and analysis in wireless networks,” IEEECommun. Mag., vol. 48, no. 11, pp. 156–163, Nov. 2010.

[26] D. Dardari, A. Conti, C. Buratti, and R. Verdone, “Mathematicalevaluation of environmental monitoring estimation error through energy-efficient wireless sensor networks,” IEEE Trans. Mob. Comput., vol. 6,no. 7, pp. 790–802, July 2007.


[27] F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev, “Stochastic geometryand architecture of communication networks,” Telecommun. Syst., vol. 7,no. 1, pp. 209–227, June 1997.

[28] S. G. Foss and S. A. Zuyev, “On a Voronoi aggregative process relatedto a bivariate Poisson process,” Advances in Applied Probability, vol. 28,no. 4, pp. 965–981, Dec. 1996.

[29] X. Yang and A. P. Petropulu, “Co-channel interference modeling andanalysis in a Poisson field of interferers in wireless communications,”IEEE Trans. Signal Process., vol. 51, no. 1, pp. 64–76, Jan. 2003.

[30] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and ItsApplications, 2nd edition. Wiley, 1996.

[31] J. S. Ferenc and Z. Neda, “On the size distribution of Poisson Voronoicells,” Physica A, vol. 385, no. 2, pp. 518–526, Nov. 2007.

[32] W. S. Kendall and I. Molchanov (editors), New Perspectives in Stochas-tic Geometry. Oxford University Press, 2010.

[33] X. Cheng, C.-X. Wang, H. Wang, X. Gao, et al., “Cooperative MIMOchannel modeling and multi-link spatial correlation properties,” IEEE J.Sel. Areas Commun., vol. 30, no. 2, pp. 388–396, Feb. 2012.

[34] C.-X. Wang, X. Hong, H. Wu, and W. Xu, “Spatial temporal correlationproperties of the 3GPP spatial channel model and the Kronecker MIMOchannel model,” EURASIP J. Wireless Commun. and Net., vol. 2007,article ID 39871, 9 pages, 2007. doi:10.1155/2007/39871.

[35] C.-X. Wang, D. Yuan, H.-H. Chen, and W. Xu, “An improved de-terministic SoS channel simulator for efficient simulation of multipleuncorrelated Rayleigh fading channels,” IEEE Trans. Wireless Commun.,vol. 7, no. 9, pp. 3307–3311, Sep. 2008.

[36] M. Z. Win, P. C. Pinto, and L. A. Shepp, “A mathematical theory ofnetwork interference and its applications,” Proc. IEEE, vol. 97, no. 2, pp.205–230, Feb. 2009.

[37] M. M. Taheri, K. Navaie, and M. H. Bastani, “On the outage probabilityof SIR-based power-controlled DS-CDMA networks with spatial Poissontraffic,” IEEE Trans. Veh. Technol., vol. 59, no. 1, pp. 499–506, Jan. 2010.

[38] X. Ge, Y. Yang, C.-X. Wang, and Y. Liu, “Characteristics analysisand modeling of frame traffic in 802.11 wireless networks,” WirelessCommun. and Mob. Comput., vol. 10, pp. 584–592, Apr. 2010.

[39] A. P. Silva and G. R. Mateus, “Performance analysis for data servicein third generation mobile telecommunication networks,” in Proc. 2002Simulation Symposium, pp. 227–234.

[40] W. Feller, An Introduction to Probability Theory and Its Applications,vol. 2. Wiley, 1971.

[41] 3GPP, “User Equipment (UE) Radio Transmission and Reception(FDD),” TS 25.101 V10.2.0, June 2011.

[42] D. Tse and P. Viswanath, Fundamentals of Wireless Communications.Cambridge University Press, 2005.

[43] J. M. Cioffi, A Multicarrier Primer. ANSI T1E1, 1999.[44] Wolfram Mathworld-Sign function, http://mathworld.wolfram.com/Sign.html.[45] X. Ge, K. Huang, C.-X. Wang, X. Hong, and X. Yang, “Capacity

analysis of a multi-cell multi-antenna cooperative cellular network withco-channel interference,” IEEE Trans. Wireless Commun., vol. 10, no. 10,pp. 3298–3309, Oct. 2011.

[46] O. Arnold, F. Richter, G. Fettweis, and O. Blume, “Power consump-tion modeling of different base station types in heterogeneous cellularnetworks,” in Proc. 2010 Future Netw. & Mobile Summit, pp. 1–8.

[47] 3GPP, “Spatial Channel Model for Multiple Input Multiple Output(MIMO) Simulations,” TR 25.996 V8.0.0, Dec. 2008.

[48] A. Conti, M. Z. Win, M. Chiani, and J. H. Winters, “Bit error outagefor diversity reception in shadowing environment,” IEEE Commun. Lett.,vol. 7, no. 1, pp. 15–17, Jan. 2003.

[49] A. Conti, M. Z. Win, and M. Chiani, “On the inverse symbol-errorprobability for diversity reception,” IEEE Trans. Commun., vol. 51, no.5, pp. 753–756, May 2003.

[50] P. Mary, M. Dohler, J.-M. Gorce, et al., “BPSK bit error outage overNakagami-m fading channels in lognormal shadowing environments,”IEEE Commun. Lett., vol. 11, no. 7, pp. 565–567, July 2007.

[51] F. Richter, A. J. Febske, and G. P. Fettweis, “Energy efficiency aspectsof base station deployment strategies in cellular networks,” in Proc. 2009IEEE VTC – Fall, pp. 1–5.

[52] J. Xu, C. Liu, Y. Yang, et al., “An overview of energy efficiencyanalytical models in communication networks,” in Proc. 2010 WirelessCommunications and Signal Processing, pp. 1–6.

[53] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 7th edition. Academic Press, 2007.

Lin Xiang received the Bachelor and Master de-grees in communications and information systemfrom Huazhong University of Science and Tech-nology (HUST), China, in 2009 and 2012, respec-tively. From Aug. 2010 to Feb. 2011, he was anexchange student at University of Bologna, Italy,under support from Erasmus Mundus programme.He is currently working toward the Doctorate de-gree at Friedrich-Alexander University of Erlangen-Nuremberg, Germany.

Xiaohu Ge (M’09-SM’11) is currently a Professorwith the Department of Electronics and InformationEngineering at Huazhong University of Science andTechnology (HUST), China. He received his PhDdegree in Communication and Information Engi-neering from HUST in 2003. He has been workingat HUST since Nov. 2005. Prior to that, he workedas a researcher at Ajou University (Korea) andPolitecnico Di Torino (Italy) from Jan. 2004 to Oct.2005. He was a visiting researcher at Heriot-WattUniversity, Edinburgh, UK from June to August

2010. His research interests are in the area of mobile communications,traffic modeling in wireless networks, green communications, and interferencemodeling in wireless communications. He has published about 60 papers inrefereed journals and conference proceedings and has been granted about 15patents in China. He received the Best Paper Award from IEEE Globecom2010. He is leading several projects funded by NSFC, China MOST, andindustries. He is taking part in several international joint projects, such as theRCUK funded UK-China Science Bridges: R&D on (B)4G Wireless MobileCommunications and the EU FP7 funded project: Security, Services, Net-working and Performance of Next Generation IP-based Multimedia WirelessNetworks.

Dr. Ge is currently serving as an Associate Editor for International Journalof Communication Systems (John Wiley & Sons) and KSII Transactions onInternet and Information Systems. Since 2005, he has been actively involved inthe organisation of more than 10 international conferences, such as PublicityChair of IEEE Europecomm 2011 and Co-Chair of workshop of GreenCommunication of Cellular Networks at IEEE GreenCom 2010. He is a SeniorMember of the IEEE, a Senior member of the Chinese Institute of Electronics,a Senior member of the China Institute of Communications, and a memberof the NSFC and China MOST Peer Review College.

Cheng-Xiang Wang (S’01-M’05-SM’08) receivedthe BSc and MEng degrees in Communicationand Information Systems from Shandong University,China, in 1997 and 2000, respectively, and the PhDdegree in Wireless Communications from AalborgUniversity, Denmark, in 2004.

He has been with Heriot-Watt University, Edin-burgh, UK, since 2005, first as a Lecturer, then as aReader in 2009, and was promoted to a Professorin 2011. He is also an Honorary Fellow of theUniversity of Edinburgh, UK, and a Chair/Guest

Professor of Shandong University, Huazhong University of Science andTechnology, and Southeast University, China. He was a Research Fellowat the University of Agder, Grimstad, Norway, from 2001–2005, a VisitingResearcher at Siemens AG-Mobile Phones, Munich, Germany, in 2004, anda Research Assistant at Technical University of Hamburg-Harburg, Hamburg,Germany, from 2000–2001. His current research interests include wirelesschannel modeling and simulation, green communications, cognitive radionetworks, vehicular communication networks, Large MIMO, cooperativeMIMO, and B4G wireless communications. He has edited 1 book andpublished 1 book chapter and about 170 papers in refereed journals andconference proceedings.

Prof. Wang served or is currently serving as an editor for 8 interna-tional journals, including IEEE TRANSACTIONS ON VEHICULAR TECHNOL-OGY (2011–) and IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS(2007–2009). He was the leading Guest Editor for IEEE JOURNAL ONSELECTED AREAS IN COMMUNICATIONS, SPECIAL ISSUE ON VEHICULARCOMMUNICATIONS AND NETWORKS. He served or is serving as a TPCmember, TPC Chair, and General Chair for over 70 international conferences.He received the Best Paper Awards from IEEE Globecom 2010 and IEEEICCT 2011. He is a Fellow of the IET, a Fellow of the HEA, and a memberof EPSRC Peer Review College.


Frank Y. Li (S’99, M’03, SM’09) holds a Ph.D.degree from the Norwegian University of Scienceand Technology (NTNU). He worked as a SeniorResearcher at UniK - University Graduate Center,University of Oslo before joining the Department ofInformation and Communication Technology, Uni-versity of Agder (UiA) in August 2007 where he iscurrently a Professor. During the past few years, hehas been an active participant in several Norwegianand EU FP6/FP7 research projects. He is listed asa Lead Scientist by the European Commission DG

RTD Unit A.03 - Evaluation and Monitoring of Programmes in Nov. 2007. Dr.Li’s research interest includes 3G/4G and beyond mobile systems and wirelessnetworks, mesh and ad hoc networks; wireless sensor network; cooperativecommunications; cognitive radio networks; green wireless communications;QoS, resource management and traffic engineering in wired and wireless IP-based networks; analysis, simulation and performance evaluation of commu-nication protocols and networks.

Frank Reichert has been working for over 20years on technology and strategies for fixed andwireless communication systems both on nationaland international level. Since July 2005 he is withUiA (University of Agder), undertaking researchin wireless communication networks and services.Currently he is the Dean of the Faculty of Engineer-ing & Science. Cooperations include projects withEricsson on wireless residential communications,and the Norwegian Center for Wireless Innovationwith six other institutions.

From 1995 he was with Ericsson Sweden, guiding investigations on, e.g.,Future Service Layer Architectures, Wireless Internet technologies, and 3Gapplications and terminals. Assignments included creating and managingEuropean Research projects (e.g. EUREKA Subproject PRO-COM, ACTSOnTheMove). Frank established Ericsson Cyberlab Singapore in 1999, fo-cusing on user centric, ethnographic application and terminal design, aswell as rapid prototyping of new HW/SW products exhibited at fairs likeCeBIT 2001 and COMDEX (e.g. Ericsson Cordless WebScreen, Delphipad,Nanorouter). He was board member for Singapore CWS (Centre for WirelessCommunications). He has been working as an expert, evaluator and auditorfor the European Commission in industrial R&D frameworks such as RACE,ACTS, FP5, FP6, and Celtic Calls 1-3. Dr. Reichert holds a Ph.D. degreein Electrical Engineering from Aachen University of Technology, Germany.He has published about 60 papers at international conferences on mobiletechnologies.

Documents

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. …home.eps.hw.ac.uk/~cw46/2013_GeX_TW_13_03_961.pdf · power consumption model for a typical PVT cell, con-sidering path loss,