Load-Aware Coordinated Multi-point Joint Transmission in Dense Heterogeneous Networks: Downlink Coverage and Throughput Limits

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1 Load-Aware Coordinated ulti-point Joint Transission in Dense Heterogeneous Networs: Downlin Coverage and Throughput Liits Chun-Hung Liu and Po-Chia Chen Departent of Electrical and Coputer Engineering National Chiao Tung University, Hsinchu, Taiwan e-ail: Abstract In this paper, we propose a cell) load-aware approach to perforing the downlin coordinated ulti-point joint transission JT-CoP) schee in a dense heterogeneous cellular networ HetNet) consisting tiers of different inds of base stations BSs). Inspiring fro the fact that there could exist a considerable nuber of the void BSs in a dense HetNet that are not associated by any user this load-aware JT-CoP schee is able to coordinate the void BSs so as to enhance the downlin transission of the non-void BSs. Using such a CoP schee, the fundaental upper and lower liits on the coverage and ergodic lin throughput of a user with downlin ISO channel are accurately found in a low-coplexity approxiated for and they straightforwardly show that the load-aware JT-CoP schee has the capability of significantly increasing the coverage and ergodic lin throughput of a user as long as the user intensity is not very uch higher than the total intensity of the BSs. Nuerical results validate the accuracy and correctness of the analytical results and deonstrate the huge gains of the coverage and the ergodic lin throughput exploited by the proposed loadaware JT-CoP schee. I. INTRODUCTION In a heterogeneous cellular networ HetNet), any different inds of sall cell base stations BSs) could coexist with the acrocell BS such as icrocell picocell fetocells and distributed antenna etc. Densely deploying these sall cell BSs is able to boost the strength of transitted signals between users and BSs owing to an enorous reduction in the pathloss of transitted signals. However, such a dense HetNet could suffer a serious interference proble that significantly offsets the advantage of the dense HetNet in the pathloss reduction so that the signal-to-interference power ratio SIR) of users ay not be effectively iproved. Coordinated ultipoint CoP) transission in a HetNet has the rearable capability of alleviating the interference proble through the spatially cooperation aong the heterogeneous BSs. Especially in a cloud-ran architecture, the transission perforance of CoP can be exploited uch ore effectively due to the low latency in ipleenting the centralized resource anageent and optiization,. Since the resources of all BSs can be anaged and optiized in a centralized fashion under the cloud-ran architecture, we would not be surprised to foresee that the CoP schee under the could-ran architecture is able to bring a huge cooperative transission gain. Thu here arises an interesting fundaental proble, that i how to evaluate the transission perforance liits of the CoP schee in a large-scale dense HetNet with a cloud-ran architecture. To investigate this proble, first we need to use an appropriate and tractable networ odel in oder to facilitate atheatical analysis and obtain soe accurate results since the traditional grid odel or the Wyner odel have been shown to be an inaccurate odel for HetNets 3, 4 owing to the randoly located nature of heterogeneous sall cell BSs. In this wor, we use stochastic geoetry to odel the locations of all the BSs in an -tier HetNet as ultiple planar Poisson Point Processes PPPs) so that the large-scale and randoly deployed nature of a HetNet can be characterized. In addition, the PPPbased HetNet odel usually brings us a very tractable approach to analyzing the perforance etric such as coverage and throughput 4 7. We ai to analytically characterize the fundaental liits on the downlin coverage probability and ergodic lin throughput of a user under the scenario that all void/idle BSs in the HetNet can be coordinated to do joint transission. In the early wors on the JT-CoP schee in HetNet coordinated BSs need to allocate part of their resources to help other BSs enhance their throughput and coverage if they still need to serve their own user siultaneously. For exaple, reference 8 studied a JT-CoP schee of scheduling resource in order to axiize the throughput for the cell-edge users. In 9, it studied how to do JT-CoP so that transit powers of the BSs can be effectively used. The resource allocation proble of JT-CoP and cell-range expansion was investigated in and three different solutions are proposed and copared in ters of load balancing. In, the coverage proble of the JT-CoP schee in a PPPbased HetNet was forulated and how to iprove the coverage by resource sharing was addressed. The JT-CoP schees in these aforeentioned wors essentially cannot achieve the throughput optiality because they cannot fully and optially utilize all resources of the entire HetNet without a cloud-ran architecture. In a dense HeNet, there could exist a considerable nuber of the void BSs that are not associated with any users

2 3. This void cell phenoenon inspires us to propose a load-aware JT-CoP schee in this wor that is able to coordinate all void BSs under a cloud-ran architecture to do joint transission with the non-void BSs. By coparing with the previously proposed joint transission schee, this load-aware JT-CoP schee possesses a salient feature, that i void BSs are able to fully use their power as well as bandwidth resource to do joint transission. To quantitatively understand how the load-aware JT-CoP schee enhances the downlin transission perforance, we analyze the coverage probability and the downlin) ergodic lin throughput of a typical user when all void BSs are coordinated to do joint transission together with the serving BS of the user. Our first ain contribution in this wor is to successfully derive the explicit expression of the coverage probability with a downlin ISO channel for the proposed load-aware JT-CoP schee and the fundaental upper and lower bounds on the coverage probability are characterized. These analytical results help us understand what are the coverage liits achieved by the load-aware JT-CoP schee. Our second contribution is to derive a very neat approxiated expression of the ergodic lin throughput of a downlin ISO channel for the proposed load-aware JT-CoP schee and clearly shows how the load-aware JT-CoP schee iproves the lin throughput. Nuerical results deonstrate that our analytical results are very accurate and the coverage probability and ergodic lin throughput both can be effectively enhanced. II. NETWORK ODEL AND VOID CELL PHENOENON In this paper, we consider a large-scale interference-liited planar heterogeneous cellular networ HetNet) under a cloud- RAN architecture in which there are different tiers of base stations BSs). The BSs in each particular tier are of the sae type and perforance and they for an independent hoogeneous Poisson point process PPP) of a certain intensity. Specifically, the set of the BSs in the th tier, denoted by Φ, is a ared hoogeneous PPP of intensity λ given by Φ X,i, P, N, V,i ) : X,i R, P, V,i, }, N, i N + }, ) where,..., }, X,i denotes the ith BS in the th tier and its location, P is the transit power of a tier- BS, N denotes the nuber of antennas equipped with a tier- BS, V,i is a Bernoulli rando variable RV) that is one if BS X,i is not void i.e., it is associated with at least one user) and zero otherwise. All users in the HetNet also for an independent PPP of intensity µ. All users are equipped with a single antenna. Without loss of generality we assue there is a typical user located at the origin and the following location-dependent expressions and analysis are based on the location of this typical user since the statistical properties observed at this typical user are sae as those observed at any other locations in the networ according to Slivnya s theore 4. All users use the following generalized user association schee based on the location of the typical user) to select their serving BS X arg sup X,i Φ W,i X,i, ) where X is the associated BS, Φ Φ denotes the entire set of the independent PPPs of the BS W i, denotes the rando) association weight of BS X,i, > is the pathloss exponent and X Y denotes the Euclidean distance between nodes X and Y. The schee in ) can cover several power-law-pathloss-based user association schees 7, 3. For exaple, if W,i for all and i N +, users associate with their nearest BS and the schee in this case is called unbiased) nearest BS association NBA). If W,i P G,i and G,i > denotes the shadowing power gain in the association signals of BS X,i, users associate with a BS that provides the the ean axiu received power P). Such an association schee, called PA, can be used in the context that non-stationary users who are oving very fast are only able to estiate the ean received powers without sall-scale fading fro BSs. Furtherore, we assue that all W,i /EW,i } are i.i.d. RVs for all and i N +. The fractional oent of W,i always exits for all and i N +, i.e., EW,i a EW a < for all a, ). According to our previous wor in 7, the probability that there are n users associating with a tier- BS is given by p n, Γn + ζ ) ) µλ n ζ λ ) n+ζ n!γζ ) ζ ζ λ, 3) + µ where λ λ E W /E W and ζ 7 E W E W. Thu we now that the void probability that a tier- BS does not have any users is p, + µ ) ζ ζ λ. 4) In general, in a densely-deployed HetNet, p, is not so sall that the void BSs would be a considerable portion of all the BSs in the HetNet. Fig. shows an illustrative exaple of the void cell BS) phenoenon in a two-tier HetNet where users adopt the PA schee to associate with their BS and we indeed see that soe void picocells do not have any users. Since the void BSs do not serve any user they potentially can be re-utilized to iprove the networ perforance, such as switching the to a dorant ode in order to save energy, offloading traffic to the in order to balancing the load of each load, and coordinating the to do ulti-point joint transission, etc. 7,, 3, 5. In the following section, we will investigate the fundaental liits on the downlin coverage and lin throughput by coordinating the void BSs to do joint transission. III. COVERAGE AND LINK THROUGHPUT WITH LOAD-AWARE JT-COP According to the non-void probability of the tier- BSs in 4), we propose a cell) load-aware JT-CoP schee in which

3 Void Cell Void Cell acrocell Picocell User to 5), the coverage probability of the typical user is defined as S p c θ) Pγ θ P θ. 7) I When the load-aware JT-CoP is used, the received signal power S of the typical user is given by S P H G X + Sc, 8) Void Cell Fig.. An illustrative exaple of the void cell BS) phenoenon in a twotier HetNet. Users adopt the PA schee to associate with their BSs and weighted Voronoi tessellation is used to generate the cells of the BSs. The cell boundary is represented by the solid blue line and the void cells are represented by the shaded ovals. The networ paraeters used to generate this figure are: µ 5 users/, P 5 W, P W, and intensity ratio λ : λ : µ is : 3 :. a considerable nuber of the void BSs in a dense HetNet under a cloud-ran architecture can be copletely coordinated to enhance the downlin transission perforance of the nonvoid BSs. In this section, our interest is in the analysis of how uch the downlin coverage and ergodic lin throughput of users can be iproved by this load-aware JT-CoP approach. Before proceeding the analysis of downlin coverage and ergodic lin throughput, we need to define the signal-tointerference SIR) based on the location of the typical user as follows γ S, 5) I where S is the total desired signal power received by the typical user associating with BS X and I is the interference given by I,i:X,i Φ\X H,iG,iV,iP X,i 6) in which H,i Exp) and G,i ln N, σ ) represent the Rayleigh fading channel gain that is an exponential RV with unit ean and variance and the shadowing channel gain that is a log-noral RV whose log-scale ean is zero and logscale variance is σ for BS X,i, respectively. All H,i s are i.i.d. for all and i N +, and all G,i s are independent for all and i N +. We will use γ to define the coverage probability and ergodic lin throughput of users and analyze the for the load-aware JT-CoP in the following two subsections. A. Coverage Probability with Load-Aware JT-CoP Consider the SIR threshold for success decoding at users is θ and the user association schee in ) is adopted. According where P P,..., P } is the transit power of BS X, H and G are the ISO fading channel gain and log-noral channel gain of X, respectively, and S c X ) is the signal power fro all the coordinated void BSs given by S c,i:x,i Φ\X P H c,i G,i V,i ) X,i, 9) where H,i c χ N is the ISO fading channel gain that is a chi-square RV with N degrees of freedo 5. The coverage probability p c θ) with load-aware JT-CoP can be explicitly found as shown in the following theore. Theore. Suppose all users adopt the PA schee i.e., using W,i P G,i in )) to select their BSs and the loadaware JT-CoP is used in the HetNet. If void probability p, exists for all, the coverage probability in 7) can be shown as ϑ d N p c θ) N )! dh N + ϑ p, ) θ l H h, ) p, l H c h ), + ) } h N, ) λ P e σ σ λ P e h where ϑ,x) + axx, }, H Exp) H c χ N, and l Z x, y) is defined as l Z x, y) x y Γ y)ez y + L Z xt y ) dt. ) Proof: See Appendix II. According to the result in ), we can see that the loadaware JT-CoP schee induces the ter p, l H c, ) that weaens the ter p, )l H, ) induced by the interference. In other word load-aware JT-CoP is able to significantly iprove the coverage probability by largely increasing p, l H c, ). Naely, the ore void probability and transit antenna the better coverage. To ae this point uch clear, consider a special case of N and θ so that p c θ) in ) reduces to p c θ) + l H, ) ϑ p, ) +, ) which iplies that we can achieve alost % coverage at the typical user if p, for all. Without using

4 the load-aware JT-CoP schee, the coverage probability in this case is only achieved by p c θ) + l H, ) ϑ p, ), 3) i.e., it is ipossible to achieve % coverage by increasing p, since < p, always holds. The coverage probability in ) is found based on the PA schee that brings the axiu SIR at the user. As a result, it represents the fundaental upper liit on the coverage probability achieved by the load-aware JT-CoP schee. any prior wors on the coverage analysis overloo the void cell issue existing in the networ and their derived coverage probabilities are intensity-free. For instance, if the void BSs are not odeled in the interference odel, the coverage probabilities in ) and 3) reduces to p c θ) that does not depend on the intensities of BSs and +l H, ) users. This indicates that the coverage cannot be iproved even ore and ore BSs are deployed, which is certainly not true. Note that as the user intensity goes to infinity p c θ) in ) reduces to li p cθ) µ ϑ N )! d N dh N h N + l H θ h, ) h, 4) and this is the fundaental lower liit on the coverage probability under the PA schee since no void BSs can be coordinated to do joint transission. B. Ergodic Lin Throughput with Load-Aware JT-CoP According to the SIR defined in 5), the ergodic lin throughput achievable rate) of a user can be defined as c Elog + γ ) bps/hz). 5) If the load-aware JT-CoP schee is adopted, it can be further written as c E log + P H G + S X c ) I X. 6) The explicit result of c in 6) is shown in the following theore. Theore. Suppose all users adopt the PA schee to select their serving BSs and the load-aware JT-CoP is used in the HetNet. If void probability p, exists for all, then the ergodic lin throughput in 6) can be explicitly found as shown in 7) in which function l Z x, y) and ϑ are already defined in Theore. Proof: See Appendix III. Fro the ergodic lin throughput in 7), we also can see that the load-aware JT-CoP schee indeed iproves the ergodic lin throughput since the ter p, l H c ) induced by the load-aware JT-CoP schee helps to reduce the second ter in the integral. To ae this point uch clear, consider TABLE I NETWORK PARAETERS FOR SIULATION Paraeter \ BS Type Tier ) acrocell ) Picocell ) Power P W) 5 Intensity λ BPs/ ).5 5 5λ Nuber of Antennas N 4 SIR Threshold θ G ln N, σ ) ln N, 5 db) Pathloss Exponent 4 the special case of N for all. In this signal antenna case, c in 7) reduces to c ln) s + + l H p,)ϑ ) + s) + l H ) ds 8) > ds ln) + + s) + l H ), p,)ϑ where the lower bound is the ergodic lin throughput without using load-aware JT-CoP. Hence, the load-aware JT-CoP schee always attains a higher lin throughput in this siple case. Since all void BSs in the HetNet are used to perfor the load-aware JT-CoP schee, c in 7) is the upper liit on the ergodic lin throughput. Whereas the lower liit on the ergodic lin throughput is given by ) ϑ + s) N ds c ln)s + l H ) p,)ϑ, 9) i.e., the ergodic lin throughput without using the load-aware JT-CoP schee in the HetNet. The upper and lower liits in 7) and 9) both further reduce and converge to li c µ ϑ + s) N ) ln)s + l H ) ds ) as µ goes to infinity and this is also the ergodic lin throughput if the interference odel overloos the void cell phenoenon. In other word without odeling the void cells in the interference and adopting the load-aware JT-CoP the analysis of the ergodic lin throughput would not be very accurate. Also, note that the ergodic lin throughput has a low-coplexity for of characterizing the ISO beaforing gain which is derived based on the new ethod devised in this wor. The ergodic lin rates/throughputs in the prior wors are obtained by integrating the function in ters of the coverage probability with a single-input-single-output SISO) Rayleigh fading channel so that their expressions are fairly coplicate and lac of tractability for a non-rayleigh fading channel. IV. NUERICAL RESULTS To validate the correctness and accuracy of the analytical results in the previous section, three nuerical results regarding

5 c ) ln) s + l + H p,)ϑ + ϑ ϑ + s) N ) p, )l H + p, l H c ) ) ds. 7) Void Cell Probability, p, Analytical Void Cell Probability Picocell, p, ) Analytical Void Probability arcocell, p, ) Siulated Void Cell Probability Picocell, p, ) Siulated Void Cell Probability arcocell, p, ) Ergodic Lin Throughput c bps/hz) Siulated c No Load-Aware CoP, SISO) Siulated c Load-Aware CoP, SISO) Analytical c Load-Aware CoP, SISO) Siulated c No Load-Aware CoP, ISO) Siulated c Load-Aware CoP, ISO) Analytical c Load-Aware CoP, ISO) acrocell BSs have 4 antenna Picocell BSs have antennas All BSs have a single antenna User intensity/picocell Intensity μ/λ ) Fig.. Siulation results of the void cell probability of the BSs in each tier User intensity/picocell intensity μ/λ ) Fig. 4. Siulation results of ergodic lin throughput c with load-aware JT-CoP Coverage Probability pcθ) Siulated p cθ) No Load-Aware CoP, SISO) Siulated p cθ) Load-Aware CoP, SISO) Analytical p cθ) Load-Aware CoP, SISO) Siulated p cθ) No Load-Aware CoP, ISO) Siulated p cθ) Load-Aware CoP, ISO) Analytical p cθ) Load-Aware CoP, ISO) acrocell BSs have 4 antenna Picocell BSs have antennas All BSs have a single antenna User intensity/picocell intensity μ/λ ) Fig. 3. Siulation results of coverage probability p cθ) with load-aware JT-CoP the void probability, the coverage probability and the ergodic lin throughput are provided in this section when the loadaware JT-CoP schee is used in a two-tier HetNet. All users adopt the PA schee to select their serving BS and their intensity is µ 5 users/ ). All other networ paraeters for siulation are listed in Table I. To deonstrate the void cell phenoenon, the siulation results of the void probability of the BSs in each tier are shown in Fig.. We can see that the analytical results of p, in 4) fairly coincide with the siulated results of p,, and ost iportantly the void cell probability of the picocell BSs is actually not sall at all when µ/λ is not large, whereas the void cell probability of the acrocell BSs is very sall since µ/λ is pretty large. Hence, the void picocells can be used to perfor the load-aware JT-CoP schee very liely in practice if they are densely deployed. The siulation results of the coverage probabilities are shown in Fig. 3 and in this coverage siulation we consider the two scenarios: the BSs with a single antenna the SISO scenario) and the BSs with ultiple antennas the ISO scenario). As can be seen in the figure, the nuerical results obtained by the coverage probability in ) are pretty close to their corresponding siulated results in the ISO and SISO scenarios. Hence, the approxiated coverage probability in ) is actually very accurate. As expected, the load-aware JT-CoP schee can significantly increase the coverage probabilities in the both scenarios especially when µ/λ is not large i.e., the void probability is high). For exaple, the coverage probability with load-aware JT-CoP is about % higher than that without load-aware JT-CoP when µ/λ. Fig. 4 deonstrates the siulation results of the ergodic lin throughput with and without using the load-aware JT-CoP schee. Liewise, fro Fig. 4 we observe that the load-aware JT-CoP schee iproves c in the both scenarios of SISO and ISO, especially when µ/λ is not large. The analytical results of c in 7) are very close to their corresponding siulated results so that our analytical approach and algebraic techniques that are devised to derive the approxiated c in 7) are correct and very accurate.

6 V. CONCLUDING REARKS Due to the fact that nuerous void BSs could exist in a dense HetNet with high probability, we propose a load-aware JT-CoP schee in which the all void BSs are coordinated to iprove the downlin transission perforance of the nonvoid BSs. In this wor, our interest is how uch the upper and lower liits on the coverage probability and ergodic lin rate can be achieved when all void BSs can do joint transission. The coverage probability and the ergodic lin throughput are accurately derived in a low-coplexity approxiated for that clearly indicates how the proposed load-aware JT-CoP schee is able to iprove the. Nuerical results not only validate the accuracy of our analytical results but also show the potential considerable gains of coverage and throughput exploited by the proposed load-aware JT-CoP schee. APPENDIX I THE INTEGRAL IDENTITY OF THE SHANNON TRANSFOR Theore 3. Consider a nonnegative RV Z and the Laplace transfor of its reciprocal always exist i.e., L Z s) Ee sz < for s R +. If S Z ϱ) E ln + ϱz) exits for all ρ, Z R +, the following identity always holds. E S Z ϱ) L ϱ s) s + Proof: We can rewrite S z ϱ) as follows ϱz S Z ϱ) E dy E + yϱz L Z s)ds ) /ϱz + y If L Z s) always exist for any y, we have E e uy E e u/ϱz du /ϱz + y e uy L Z u/ϱ)du dy. e ϱsy L Z s)ϱds and then substituting this result into S Z ϱ) yields S Z ϱ) ϱ + e ϱsy L Z s)dyds e ϱs ) L s Z s)d + which is exactly the result in ). e ϱs ) f se s Z z)dzds Letting s ϱz). z APPENDIX II PROOF OF THEORE For any s >, let L Z s) Ee sz denote the Laplace transfor of RV Z and L }x) denote the inverse Laplace transfor with a real-valued paraeter x. Then p c θ) can be expressed as follows p c θ) P θi S c P H G X P, γ } L L γ s s) ). ) where γ PHG X θi S. Note that θi c S c can be expressed as θi S c P H,iG,i θv,i X,i,i:X,i Φ\X P H,i c V,i ) G,i X,i,i:X,i Φ\X d,i: X,i Φ\ X θv,i H,i V,i )H c,i X,i, where d represents the equivalence in distribution, Φ Φ is a hoogeneous PPP of intensity λ λ, Φ is a hoogeneous PPP of intensity λ λ P e σ, X is the node in Φ nearest to the typical user. Then L γ s) for given X and H has an accurate approxiation as shown in the following L γ s) E Φ e s,i: Φ\ X θv,i H,i V,i )H,i c ) H X,i / X ) a) sh exp π λ c p, P Y H + x R ) ) + } sθh + p, )P Y H + x dx R ) b) sθ exp πr λ p, )l H, ) H s p, l H c, ) + ), H where a) follows fro the probability generation functional PGFL) of independent PPPs by assuing all V,i s are independent, R X, Y Exp), and the algebraic techniques devised in the proof of Proposition in 3; b) follows fro H Exp) for all and the definition of function l Z x, y) in ). Since we now the pdf of R is f R r) π λ exp πr λ) and let ) +, l sθ H, ) sθ p, )l H H, s p, l H c H, ) we have s L γ s) approxiately given by s L γ s) s E H π λe πr λ l sθ H, )+ dr s E H ϑ l sθ H, ) + c) ϑ f H c sh)dh ϑ l θ h, ) +, where c) follows fro the fact that PH H c is ϑ and the pdf of H c is f H c x) xn e x N )! with x sh. Then Such an assuption still can result in a very accurate approxiation since the location correlations aong BSs due to user association are usually fairly wea 3 5.

7 substituting this result into ) yields } p c θ) ϑ L f H c sh)dh ϑ l θ h, ) + ) ϑ L which yields the result in ). sh) N N )! e sh ϑ l θ h, ) + dh ), APPENDIX III PROOF OF THEORE Using the integral identity of the Shannon transforation in Theore 3 provided in Appendix I, c in can be expressed as c LH s)l Q c s) L Q s)d 3) ln) s + where Q c Sc X P G and Q I X P G. Using the results in the proof of Theore II, the explicit result of L Q s) for given X R can be accurately approxiated as s V,i H,i X L Q s) E Φ X,i,i: X,i Φ\ X e ) ) e π p,) λ P sh Y + x R exp πr p, ) λ l H ) }, where ) follows fro the independent assuption between all V,i s and the fact that the correlations between all V,i s are wea; X,i, Φ, and Y are already defined in the proof of Theore. Siilarly, L Q c s) can be accurately approxiated as shown in the following: L Q c s) E Φ exp s H,i c X,i: X,i Φ\ X X,i } sh c exp π p, λ P Y + x dx R ) exp πr K p, λ l H c ) }. Thu we finally have the following LH s)l Q c s) L Q s) E R L H s) e πr p, λ l H c ) ) e πr p,) λ l H ) + ϑ p, )l H ) ϑ + s) N + ϑ p, )l H ) + p ).,l H c where L H s) ϑ +s) N. Then substituting this approxiation result into 3) yields the result in 7). dx REFERENCES C.-L. I, J. Huang, R. Duan, C. Cui, J. X. Jiang, and L. Li, Recent progress on C-RAN centralization and cloudification, IEEE Acces vol., pp. 3 39, Sep. 4. A. Checo, H. L. Christiansen, Y. Yan, L. Scolari, G. Kardara. S. Berger, and L. Dittann, Cloud RAN for obile networs - a technology overview, IEEE Coun. Surveys Tuts., vol. 7, no., pp , First Quarter 5. 3 J. G. Andrew F. Baccelli, and R. K. Ganti, A tractable approach to coverage and rate in cellular networ IEEE Trans. Coun., vol. 59, no., pp ,. 4 H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrew odeling and analysis of K-tier downlin heterogeneous cellular networ IEEE J. Sel. Areas Coun., vol. 3, no. 3, pp , Apr.. 5 P. Xia, C.-H. Liu, and J. G. Andrew Downlin coordinated ultipoint with overhead odeling in heterogeneous cellular networ IEEE Trans. Wireless Coun., vol., no. 8, pp , Jun H.-C. Tsai, C.-H. Liu, and L.-C. Wang, An analytical approach to coexisting evaluation in ulti-rat heterogeneous networs with opportunistic CSA/CA, in Proc. IEEE Int. Conf. on Coun., ay 6, pp C.-H. Liu and K. L. Fong, Fundaentals of downlin green coverage and energy efficiency in heterogeneous networ IEEE J. Sel. Areas Coun., vol. 34, no., pp , Dec S. Fu, B. Wu, H. Wen, P.-H. Ho, and G. Feng, Transission scheduling and gae theoretical power allocation for interference coordination in CoP, IEEE Trans. Wireless Coun., vol. 3, no., pp. 3, Jan G. L. Y. L. Z. Xu, C. Yang and S. Xu, Energy-efficient cop precoding in heterogeneous networ IEEE Trans. Signal Process., vol. 6, no. 4, pp. 5 7, Apr. 4. Y. Cao, H. Xia, and C. Feng, Evaluation of diverse cell range expansion strategies applying CoP in heterogeneous networ, in Prof. of IEEE PIRC, Sep. 3, pp G. Niga, P. inero, and. Haenggi, Coordinated ultipoint joint transission in heterogeneous networ IEEE Trans. Coun., vol. 6, no., pp , Nov. 4. C.-H. Liu and L.-C. Wang, Rando cell association and void probability in poisson-distributed cellular networ in Proc. IEEE Int. Conf. on Co., Jun. 5, pp , Optial cell load and throughput in green sall cell networs with generalized cell association, IEEE J. Sel. Areas Coun., vol. 34, no. 5, pp. 58 7, ay 6. 4 S. N. Chiu, D. Stoyan, W. S. Kendall, and J. ece, Stochastic Geoetry and Its Application 3rd ed. John Wiley and Son Ltd., 3. 5 C.-H. Liu and H.-C. Tsai, On the liits of coexisting coverage and capacity in ulti-rat heterogeneous networ IEEE Trans. Wireless Coun., to appear, 7.