Analysis of Multi-tier Uplink Cellular Networks with Energy Harvesting and Flexible Cell Association

Analysis of Multi-tier Uplin Cellular Networs with Energy Harvesting and Flexible Cell Association Ahmed Hamdi Sar and Eram Hossain Abstract We model and analyze a K-tier uplin cellular networ with flexible cell association where all transmissions are powered by energy harvesting from ambient interference. Each cellular user transmits data to the corresponding base station (BS) only when the amount of energy harvested is sufficient to perform channel inversion towards the serving BS. Furthermore, the data transmitted can be successfully decoded only when the signal-to-interference-plus-noise ratio (SINR) at the receiver is above a predefined threshold. With flexible cell association, users are not necessarily associated with their nearest BS where a different bias factor is added to each networ tier. We use tools from stochastic geometry to evaluate the performance of the proposed system model in terms of the coverage probability of a generic user associated with the -th tier. We show that energy harvesting can be a reliable source to power cellular users with short-range communication, e.g., small cell users. In addition, we show that energy harvesting can achieve high coverage performance by optimizing different networ parameters such as the BS receiver sensitivity as well as the bias factors. Keywords: Energy harvesting, K-tier cellular networs, uplin transmission, flexible association, power control, coverage probability, stochastic geometry. I. INTRODUCTION Radio frequency (RF) energy harvesting in cellular networs has recently attracted significant attention to power wireless devices motivated by the issue of global greenhouse gas emissions increase ]. On the other hand, overlaying macrocells by different classes of smaller and lower-power base stations (BSs) such as femtocells and picocells is considered as one solution to improve the spectral efficiency of cellular networs, hence, it is called a multi-tier networ. In the context of ambient RF energy harvesting in wireless networs, the authors in ] use power beacons to power uplin transmissions where no power control is assumed under an outage constraint. In the cognitive radio networ in 3], the authors use the RF energy transmitted by primary users to power underlaying secondary users where all users transmit with the same power. The authors in 4] consider a deviceto-device networ that is powered by harvesting energy from the concurrent transmissions of a downlin cellular networ where all DD transmitters have a fixed power level. On the other hand, in the context of modeling uplin cellular A. H. Sar and E. Hossain are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Canada (emails: Ahmed.Sar@umanitoba.ca, Eram.Hossain@umanitoba.ca). This wor was supported by a Strategic Project Grant (STPGP 4385) from the Natural Sciences and Engineering Research Council of Canada (NSERC). networs, the authors in 5] provide a framewor to model a single-tier networ in which all users perform fractional power control where users are assumed to constitute a Poisson Point Process (PPP) and each user has one BS in her vicinity. In 6], the authors present a general framewor for modeling uplin transmission in multi-tier networs where users use truncated channel inversion power control to satisfy a certain received power threshold. In this wor, we consider uplin transmission when all users are powered only by the harvested RF energy from the ambient interference that results from the concurrent downlin transmissions by all networ tiers. Channel inversion power control is used by all users to ensure that the power received at the corresponding BS is higher than the receiver s sensitivity. We also consider the case when users do not necessarily associate with the nearest BS for any reason such as the load per BS. That is, each networ tier has a specific bias factor that is added to the decision criterion of the association. Note that the idea of flexible cell association has been used for downlin networs, e.g., in 7]. After harvesting energy and associating with a certain BS, each user transmits only when power harvested is enough to perform channel inversion. Note that a user may suffer outage due to either insufficient harvested energy or low SINR received at the serving BS. We use statistical modeling based on stochastic geometry to capture the randomness of the networ topology, e.g., BSs and users locations 8], 9]. In particular, for analytical tractability, we use independent PPPs to model the locations of BSs and users. We evaluate the performance of our system model in terms of SINR coverage probability, transmission probability, and overall coverage probability. We show the effect of varying the different parameters of the networ on (such as receiver sensitivity and bias factors) on the performance metrics. The contributions of the paper can be summarized as follows: Using stochastic geometry, we provide a tractable analytical framewor to model and analyze the performance of energy harvesting in multi-tier uplin cellular networs with flexible cell association. Furthermore, we consider a practical system model in which users perform power control in order to mitigate the near-far problem. We show that energy harvesting can provide an acceptable performance for uplin transmissions in cellular networs especially for users with short-range communication lins. Furthermore, we show that adjusting networ

parameters increases the achievable gain by balancing the trade-off between the probability of harvesting sufficient power and the probability of transmitting with acceptable power level to satisfy a certain SINR threshold at the receiver. II. SYSTEM MODEL AND ASSUMPTIONS A. K-tier Cellular Networ Model We consider a K-tier cellular networ where the locations of BSs belonging to tier are modeled by an independent PPP Φ = {x i : i =,,... } with intensity λ where x i R denotes the location of the i-th BS. BSs belonging to the same tier have the same receiver sensitivity ρ and transmit power P. The complete set of users is also modeled by an independent PPP Φ u = {y i : i =,,... } with intensity λ u. In the uplin, each user performs channel inversion to adjust her transmit power to ensure that the average received power at the serving BS is equal to its receiver sensitivity. Orthogonal channel access is assumed to avoid intra-cell interference. B. Channel Model and User Association The networ is assumed to operate in the TDD mode such that the downlin and uplin transmissions are separated from each other in time. That is, there is no interference between the downlin networ and the uplin networ even though both networs use the same set of channels. The power of the signal transmitted on any channel decays at a rate of r where is the path-loss exponent and r is the propagation distance. In addition, the power envelope of each channel is modeled by an independent exponential random variable h with unit mean, i.e., Rayleigh fading assumption. In the uplin, flexible cell association is used such that association with BSs belonging to the same tier is biased with a positive bias factor β. That is, each user is assumed to be served in the uplin by the BS that offers the best biased average channel gain (i.e., average channel gain plus bias factor). For example, the bias factors can be chosen according to the receiver sensitivity of each tier in order to mae users biased to associate with the tier that requires lowest transmit power. Another scenario is when β = P in which each users is served by the same BS in both downlin and uplin. For a user located at y R, let x and x o denote the BS with the best biased channel gain from the -th tier and the serving BS, respectively. Hence, the flexible cell association criterion is described as x = arg max x Φ {β x y } () x o = arg max {β x y } () x {x :=,,...,K} where is the Euclidean distance. Without loss of generality, Fig. shows a realization of a downlin-uplin cellular networ with three different tiers, e.g., a macrocell networ tier, a picocell networ tier, and a femtocell networ tier. Fig.. A 3-tier cellular networ in an area of 3m 3m. A macro-tier (red squares) with intensity.5(5 π) BS/m is overlaid with lower-power and 3 times denser picocells (green circles) and 5 times denser femtocells (cyan diamonds). Solid lines show the coverage area of macro (tier ), pico (tier ), and femto (tier 3) BSs for biased uplin association criterion defined in () where β = 5, β =, and β 3 =. Dashed lines show the coverage area for uplin when association is unbiased, i.e., β =, {,, 3}. C. Energy Harvesting Model Each user is equipped with an energy harvesting unit that converts the ambient RF power received from the interference caused by the downlin cellular transmissions into useful DC power. Hence, the total power received at the harvesting unit of a user located at y is given by P H (y) = a K = x Φ P h x y where a is the RF-to-DC power conversion efficiency. Based on the TDD operation, users are assumed to adopt a time-slotted harvest-then-transmit strategy in which a user transmits only when the power harvested in one time slot (i.e., the downlin time slot) is sufficient to perform channel inversion power control (i.e., in the uplin time slot). There is no energy storage assumed such that no user can save the extra harvested energy for the next time slot. III. UPLINK TRANSMISSION PROBABILITY As mentioned earlier, for a user to be able to mae an uplin transmission, the energy harvested by this user should be sufficient to perform channel inversion towards the serving BS. Hence, we define η as the probability of transmission by a user in the -th tier after harvesting sufficient energy. In this section, we derive the probability density function (PDF) of the users transmit power as well as the harvested power, and then we obtain the transmission probability. A. Uplin Transmit Power Analysis Since each user performs channel inversion in order to satisfy a power level requirement ρ at her serving BS, we define the transmit power of a user when it associates with a BS from the -th tier at a distance R as, (3) p = ρ R. (4)

Hence, we firstly need to derive the PDF of the distance R in order to derive the PDF of the transmit power. Conditioned on the event of a generic user to be served by a BS belonging to the -th tier, the following lemma presents the PDF of the distance between this user and her serving BS. Lemma. The PDF of the distance between a generic user associated with tier and her serving BS is f R (r) = πλ r exp πλ r ] (5) where Λ = β K j= λ jβ j. Proof: See Appendix A. Using Lemma and (4) we derive the PDF of a generic user s transmit power when it associates with the -th tier as stated in the following lemma whose proof is left to the reader. Lemma. The PDF of the transmit power a generic user associated with tier to achieve a received power of ρ at her serving BS is f p (t) = πλ ρ ( t ρ and its -moment is given by where Λ = β ) ( t exp πλ ρ ] E p K j= λ jβ j. ) ] (6) = ρ (7) πλ Note that, for the unbiased association (i.e., β = ) and when all BSs have the same receiver sensitivity, the transmit power of any user is independent of which tier the user is associated with. That is because the association in this case is made with the nearest BS regardless of its tier. B. Analysis of Harvested Power Based on Slivnya s theorem, Theorem 8.], we use (3) to derive the Laplace transform of the harvested power at a typical user located at the origin from which the PDF is obtained as in the following lemma. Lemma 3. The PDF of the power received at the energy harvesting module of a generic user is given by ( π ) a ( π ) ] 4 aξ f PH (t) = ξ πt 3 exp (8) t where ξ = K = λ P. Note that the expression in Lemma 3 is derived for the special case when = 4. However, does not need to be 4 for the rest of the analysis, for example when < 4, this expression provides a pessimistic bound on the amount of harvested power and vice versa. Due to space limitations, we only present the detailed proof of the main result of this wor (i.e.,theorem ) along with the outline of other proofs where the complete proofs are left to an extended version of this paper. C. Transmission Probability As described above, the transmission probability of a user associated with the -th tier can be defined as η = PP H > p ] (9) and the following theorem provides a closed-form expression for this probability. Theorem. For a user served by a BS belonging to the -th tier, the probability that this user harvests sufficient energy to perform channel inversion towards her serving BS is given by η = ( ( π ) 6 aξ Λ π G3,,3, ρ, ) () where ξ = K j= λ jp j, Λ = β K j= λ jβ j, and G 3,,3 (x a, a, a 3 ) denotes the Meijer G-function. Proof: See Appendix B. Note that the function G 3,,3 (x,, ) in () is a decreasing function of x, furthermore, it can be calculated numerically by many computer algebra systems, e.g., MuPAD, MATLAB, Maple, and Mathematica. Also note that, for the unbiased association (i.e., β = ), the transmission probability is independent of which tier the user is associated with when all BSs have the same receiver sensitivity. IV. UPLINK COVERAGE PROBABILITY In this section we use Slivnya s theorem to characterize the SINR for a typical BS located at the origin, then we derive closed-form expressions for the uplin coverage probability. A. SINR Definition Based on the association criterion defined in () for the uplin, the SINR received at a typical BS belonging to the -th tier and located at (, ) R can be written as SINR = ρ h K j= u i Ψ j\{u o} p jg u i + σ () where h and g are the small-scale fading coefficients between the typical BS and the served and interfering users, respectively. σ is the variance of the additive noise at the BS where no specific distribution is assumed. u o is the user served by the typical BS at a specific time slot. In order to define the set of interfering users at a specific time slot, we define a point process Ψ with intensity η λ that represents the set of users who harvested sufficient energy and ready for transmission at this time slot. Although this thinning is not independent and Ψ is not a PPP, for analytical tractability, we assume that the interfering users at a certain time slot constitute a PPP. The channel coherence time is assumed to be greater than the frame duration. 6]. This assumption has been used and validated in the literature, e.g., 5],

B. Analysis of Uplin Coverage Probability Using (), we can get the uplin SINR coverage probability C of the overall system. Let C denote the SINR coverage probability offered by the -th tier to the users associated with this tier, hence, using the law of total probability, the overall SINR coverage probability is obtained as C = K = λ Λ C () where λ Λ represents the probability of a user to associate with the -th tier and Λ is given in Lemma. Here, uplin SINR coverage probability of the -th tier is defined as the probability that the received SINR at a BS in this tier is higher than a predefined threshold τ that is chosen to satisfy certain quality-of-service requirements. Hence, C = P SINR > τ ]. (3) In the following theorem, we obtain the coverage probability of the uplin transmission for a user served by a typical BS belonging to the -th tier and located at the origin where the statistics of the networ can be generalized to a generic user. As an outline to the proof, the coverage probability is derived using the Laplace transform of the interference resulting from tier j. In addition, due to using channel inversion power control, we now for sure that the closest interferer from that tier is at least at a distance of ( β β j p j ρ j ) Theorem. The coverage probability offered by the -th tier in an uplin cellular networ for a generic user is given by C = exp τ ρ σ where Λ j = β j K j= η j λ j Λ j K i= λ iβ i. ( ) ( ) ]] ρ j β ρ τ F, ρ τ β j ρ j and Fy, ] = y u +u du. (4) Note that the function F(y, ) can be evaluated by numerical methods, in addition, it reduces to simple closed-form expressions for some special values of, Appendix 3]. For example, when = 4, Fy, 4] = arctan(y ). Now, we introduce some special cases to highlight the consistency between our model and previous results on uplin cellular networs in the literature. Corollary. (Single-tier Uplin Networ, = 4, σ = ) In the special case of an interference-limited single-tier uplin networ, the overall coverage probability is given by C = exp η τ arctan ( τ )]. (5) Proof: Substitute K =, σ =, and = 4 in (4). This result shows that, for single-tier uplin cellular networs, the coverage probability is independent of the the BS intensity, receiver sensitivity, and transmit power when η =. The same conclusion can be found in previous wor on uplin and downlin (when η = ) 5] 8], ]. The next special case is the K-tier uplin networ with best uplin channel association (i.e., unbiased association). In this case, for the uplin, the serving BS is simply the closest BS since all tiers have the same propagation path-loss exponent. Corollary. (Unbiased uplin association, σ = ) In the special case when a user associates to the nearest BS in the uplin, the coverage probability of an interference-limited K- tier networ is given by C = exp K j= η j λ j Λ ( ) ρ j τ ρ F ( τ ρ ρ j ), ]] (6) where Λ = K j= λ j. Proof: We use the fact that the association of the uplin is unbiased, hence, we set β to and σ = in (4). In this case, we can see that the K tiers reduce to a singletier networ with intensity Λ which is equal to the sum of all tiers intensities. This result is consistent with the wor on uplin (when η j = ) presented in 6, Theorem 3] for the case when P u =. V. NUMERICAL RESULTS AND DISCUSSION In this section, we evaluate the performance of the proposed system with energy harvesting and flexible association in terms of transmission probability (η ), SINR coverage probability (C ), and overall coverage probability (i.e., η η ( C ) = η C ) for a user associated with the -th tier. We consider a 3- tier networ scenario with macro, pico, and femto BSs as tier, tier, and tier 3, respectively. For numerical evaluation, unless otherwise stated, the transmit powers of BSs are assumed to be P = 53 dbm, P = 33 dbm, and P 3 = 3 dbm while the thermal noise power σ z is 4 dbm. The intensities of BSs are λ = 5(.5 π) BS/m, λ = 5λ, and λ 3 = λ. Independent and identically exponential power envelope with unit mean is considered for all lins and the path-loss exponent is = 3.3. The power conversion efficiency 3 a is set to and the SINR threshold τ is set to.5. Fig. shows the effect of the bias factors as well as the SINR threshold on the performance of users associated to different tiers. It can be seen that the overall coverage probability decreases with τ due to the reduction in the SINR coverage probability in (4) where η is independent of τ. It can also be seen that in the unbiased case (i.e., β = db), the performance of all users is the same since each user associates with the nearest BS regardless of its tier. In the flexible association case, we can notice that when a bias factor is added to one tier, this degrades the performance of this tier while improving the overall coverage of the smaller BSs tiers. For example, in this case β > β > β 3 which means that 3 Note that decreasing the conversion efficiency is equivalent to increasing the receiver s sensitivity as can be seen in (). That is, varying a only scales the resulting figures when plotting versus ρ as if we plot versus the ratio ρ a instead.

.9.8 Macro user, flexible association Pico user, flexible association Femto user, flexible association Generic user, unbiased association.9.8 Macro user, flexible associaition Pico user, flexible associaition Femto user, flexible associaition Generic user, unbiased association Overall coverage probability.7.6.5.4.3 Transmission probability,.7.6.5.4.3.... - -5 5 SINR threshold, (dbm) -9-8 -7-6 -5-4 BS receiver sensitivity, (dbm) Fig.. Overall coverage probability of a user associated with the -th tier vs. SINR threshold for the unbiased and flexible association. For the flexible association, the bias factors are set such that β = P, and ρ = 9 dbm for =,, 3. Transmission probability,.9.8.7.6.5.4.3.. Macro user, =,, ] db Pico and femto users, =,, ] db Macro and femto users, =,, ] db Pico user, =,, ] db Macro and pico users, =,, 3 ] db Femto user, =,, 3 ] db 4 6 8 4 6 8 Bias factor, (db) Fig. 3. Transmission probability of a user associated with the -th tier vs. the bias factors (for ρ = ρ = ρ 3 = 9 db). more users are served by the macro tier and less users are served by the femto tier compared to the unbiased case. This increases the transmission probability of the femto users as the femto BSs only serve the nearby users and the far users are offloaded to other tiers, hence, less transmit power p is required by the femto users. On the other hand, macro users suffer from performance degradation as more transmit power is required to invert the channel to the corresponding BS. To show the effect of the bias factor β on the transmission probability in (), Fig. 3 compares different cases of flexible association by varying β. Compared to the unbiased case (i.e., β = db), it can be seen that biasing the association to a certain tier degrades the transmission probability of this tier while improving that of other tiers. However, it can be seen that the degradation is less severe for denser tiers because the number of users added to this tier is divided on all BSs. For the same reason, the improvement in the transmission probability of femto and pico users is higher compared to the macro users. Fig. 4 shows the relation between the BS receiver sensitivity and the users transmission probability where all BSs are assumed to have the same ρ. It can be seen that we can achieve higher transmission probability for small-cell users (i.e., pico and femto users) for the same ρ by adding a bias Fig. 4. Transmission probability of a user associated with the -th tier vs. BS receiver sensitivity for the unbiased and flexible association. For the flexible association, the bias factors are set such that β = P. factor to the macro cell association. In addition, it can be also seen that the lower ρ, the better is the performance for both the unbiased and biased association. Fig. 5 sums up all the trade-offs presented so far. It shows the behavior of the femto users performance in response to varying femto BS receiver sensitivity ρ 3 and the bias factor β. The performance is shown in terms of the SINR coverage probability (i.e., C 3 ), transmission probability (i.e., η 3 ), as well as the overall coverage probability (i.e., η 3 η 3 ( C 3 )). From this figure, the following observations can be made: For both cases (i.e., unbiased and flexible association), there exists an optimal value of ρ that maximizes the overall coverage probability. This behavior can be explained as follows: when the femto BS s receiver is more sensitive (i.e., low ρ 3 ), the femto user can achieve high transmission probability (curves with circles), however, the SINR coverage becomes very low, e.g. when ρ 3 = dbm. However, as ρ 3 increases, the probability of sufficient power η 3 starts to fall (curves with circles) and the improvement in the SINR coverage (curves with squares) starts to dominate the overall coverage probability. Hence, the overall coverage probability starts to increase. After some value of ρ 3, the decrease of η 3 starts to have more influence on the overall coverage probability compared to the SINR coverage probability, hence, the overall coverage probability starts to decrease. With the flexible (biased) association, the overall coverage probability of small-cell users can be highly improved compared to the unbiased association when β =. This improvement is introduced by decreasing the distance between a small-cell user and her serving BS. This increases the probability of the user to harvest sufficient power to perform channel inversion towards her serving BS (i.e., η ). Note that all the results in Figs. -5 are under the assumption that all users only depend on harvesting energy from the ambient RF interference as a power source. That is, the degradation in the macro users performance is mainly caused by the degradation in the transmission probability. So, if energy harvesting is considered only for users with shortrange communication lins such as femto users, the same

Femto user overall coverage probability.9.8.7.6.5.4.3.. SINR coverage, flexible Transmission probability, flexible Overall coverage, flexible SINR coverage, unbiased Transmission probability, unbiased Overall coverage, unbiased - -5 - -95-9 -85-8 -75-7 -65-6 -55-5 BS receiver sensitivity, 3 (dbm) Fig. 5. Femto user overall coverage probability vs. BS receiver sensitivity for the unbiased and flexible association. For the flexible association, the bias factors are set such that β = P (for ρ = ρ = 9). improvement of performance will be still attainable without performance degradation of the macrocell users. Furthermore, same results can be achieved by considering the case when users are powered by a fixed source such as batteries beside harvesting energy. VI. CONCLUSION We have used stochastic geometry to present a novel framewor to model, analyze, and evaluate the performance of flexible association and RF energy harvesting in multi-tier uplin cellular networ. We have shown that energy harvesting can be a reliable source to power cellular users especially those with short-range transmissions such as femto users. Furthermore, we have shown how to balance the different trade-offs of the overall performance by varying the networ design parameters such as BSs receiver sensitivity and bias factors. APPENDIX A PROOF OF LEMMA For each tier, we define Ψ = {β x : x Φ }. Using the mapping theorem, Theorem.34 and Corollary.35], we conclude that Ψ is a homogeneous PPP with intensity β λ since β is scalar. Using the superposition property of PPPs, we define the PPP Ψ = K = Ψ which is homogeneous with intensity K j= λ jβ j. Note that the association criterion defined in () is equivalent to associate with the nearest point in Ψ, hence, the distance distribution can be obtained by the null probability from a PPP with intensity Λ = β K j= λ jβ j which is Rayleigh-distributed as given in (5). APPENDIX B PROOF OF THEOREM By definition of η given in (9), we have η = F p (t)f PH (t)dt (7) where F p (t) and f PH (t) can be obtained from (6) and (8), respectively. Given that = 4 as dictated by Lemma 3, after substitution we have η = κ (e) = κ κ 3 3 exp t 3 κ t ( G, t ), κ ] ] exp κ 3 t dt G,, ) (κ 3 t 3 dt (8) where κ = π πaξ 4, κ = aξ ( ) π 4, and κ3 = πλ ρ. G is the Meijer G-function defined in ]. (e) follows because exp( u) = G,, (u ), G,, (u c) = G,, (u c), and u h G,, (u c) = G,, (u c + h). By replacing u = κ 3 t we get ( η = κ κ 3 G, u ) ( ), κ κ G,, u du 3 (f) = κ ( κ 3 G, u ) (, π κ κ G, u,, ) du 3 4 (g) = κ ( G,3 κ κ 3 3,, πκ 4, ) (9) where (f) follows because G,, (u c) = 4 π G,, ( u 4 c, c+ ) and (g) follows because G,, (hu c)g,, (gu m, n)du = h G3,,3 ( g h c, m, n). After minor mathematical manipulations, the results in () can be verified. REFERENCES ] J. A. Paradiso and T. Starner, Energy scavenging for mobile and wireless electronics, IEEE Pervasive Computing, vol. 4, no., pp. 8 7, Jan.-Mar. 5. ] K. Huang and V. K. N. Lau, Enabling wireless power transfer in cellular networs: architecture, modeling and deployment, IEEE Trans. Wireless Commun., vol. 3, no., pp. 9 9, Feb. 4. 3] S. Lee, R. Zhang and K. Huang, Opportunistic wireless energy harvesting in cognitive radio networs, IEEE Trans. Wireless Commun., vol., no. 9, pp. 4788 4799, Sep. 3. 4] A. H. Sar and E. Hossain, Cognitive and energy harvesting-based DD communication in cellular networs: stochastic geometry modeling and analysis, under submission, Available Online]: http://home.cc. umanitoba.ca/ sara/ 5] T. D. Novlan, H. S. Dhillon, and J. G. Andrews, Analytical modeling of uplin cellular networs, IEEE Trans. Wireless Commun., vol., no. 6, pp. 669 679, 3. 6] H. ElSawy and E. Hossain, On stochastic geometry modeling of cellular uplin transmission with truncated channel inversion power control, IEEE Trans. Wireless Commun. vol. 3, no. 8, pp. 4454 4469, 4. 7] H.-S. Jo, Y. J. Sang, P. Xia, and J. Andrews, Heterogeneous cellular networs with flexible cell association: A comprehensive downlin SINR analysis, IEEE Trans. Wireless Commun., vol., no., pp. 3484 3495,. 8] J. G. Andrews, F. Baccelli, R. K. Ganti, A tractable approach to coverage and rate in cellular networs, IEEE Trans. Commun., vol. 59, no., pp. 3 334, Nov.. 9] H. ElSawy, E. Hossain, and M. Haenggi, Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networs: A survey, IEEE Commun. Surveys Tuts., vol. 5, no. 3, pp. 996 9, July 3. ] M. Haenggi, Stochastic Geometry for Wireless Networs. Cambridge University Press, 3. ] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians. New Yor: MacMillan, 985. ] A. H. Sar and E. Hossain, Location-aware cross-tier coordinated multipoint transmission in two-tier cellular networs, IEEE Trans. Wireless Commun., to appear.