Optimal Base Station Density for Energy-Efficient Heterogeneous ...
Optimal Base Station Density for Energy-Efficient Heterogeneous Cellular Networks Dongxu CAO, Sheng ZHOU, Zhisheng NIU Tsinghua National Laboratory for Information Science and Technology Dept. of Electronic Engineering, Tsinghua Univ., Beijing, 100084, P.R. China [email protected], {sheng.zhou, niuzhs}@tsinghua.edu.cn Abstract—In this paper, we adopt stochastic geometry theory to analyze the optimal macro/micro BS (base station) density for energy-efficient heterogeneous cellular networks with QoS constraints. We first derive the upper and lower bounds of the optimal BS density for homogeneous scenarios and, based on these, we analyze the optimal BS density for heterogeneous networks. The optimal macro/micro BS density can be calculated numerically through our analysis, and the closed-form approximation is also derived. Our results reveal the best type of BSs to be deployed for capacity extension, or to be switched off for energy saving. Specifically, if the ratio between the micro BS cost and the macro BS cost is lower than a threshold, which is a function of path loss and their transmit power, the micro BSs are preferred, i.e., deploy more micro BSs for capacity extension or switch off certain macro BSs for energy saving. Otherwise, the optimal choice is the opposite. Our work provides guidance for energy efficient cellular network planning and dynamic operation control.1
I. I NTRODUCTION Future cellular networks are expected to provide more data service with high QoS (quality of service) requirement. Heterogeneous cellular networks composed of macro, outdoor pico and indoor femto BSs (base station) are promising to enhance the network capacity and reduce the network cost [1], [2]. However, the deployment of the pico/femto BSs will be mostly ad hoc [2]. Then one must consider how to plan heterogeneous networks, especially for capacity extension through deploying more BSs based on existing networks. On the other hand, today’s cellular networks are consuming lots of energy, and will call for more in the future. As a result, energy efficient approaches are urgently required by network vendors. For instance, China Mobile promises to reduce power consumption 20% per traffic unit in 2012 compared with 2008 [3]. Recent researches demonstrate that a system-wide approach by adapting the BS density through turning on/off BSs according to the actual traffic load is effective [4]. Thus energy saving through BS sleeping will be another important issue for heterogeneous cellular networks. Current researches about cellular network planning mainly focus on the practical deployment algorithm design, e.g., [5] and references therein. Among the very limited work on BS density, Ref. [6] is particularly related, since it attempts to find the maximal inter-BS distance for CDMA cellular networks 1 This work was supported in part by the National Basic Research Program of China (973 Program: No. 2012CB316001), the Nature Science Foundation of China (No. 61021001, No.60925002), and Hitachi R&D Headquarter.
with the hexagonal cellular network model. However, this work cannot be used for capacity extension or dynamic BS sleeping, and is difficult to be extended to heterogeneous cellular networks. There are also many studies on energy efficiency of heterogeneous cellular networks ([7] and references therein). Most of them focus on improving energy efficiency through micro BS deployment, showing the benefits of low power micro BS sites with system-level simulations. However, few focus on BS density, or jointly consider dynamic BS sleeping in heterogeneous cellular networks. In this paper, we focus on network planning and energy efficient operation for heterogeneous cellular networks. We tackle both problems through analyzing the optimal macro/micro BS density, by modeling heterogeneous cellular networks with stochastic geometry, in which both macro and micro BSs are located independently according to PPPs (Poisson Point Process) [8], [9], [10]. This model has several advantages, e.g., suitable to analyze capacity extension through deploying more BSs and energy saving through BS sleeping, since the superposition of two or more independent PPPs, and the independent thinning of a PPP are still PPPs [8]. Based on this model, we solve the optimal macro/micro BS density problem: given the predefined QoS requirement and traffic load, what is the optimal BS density to minimize the network cost. For network planning, it can provide information about the type and the number of BSs required. For energy saving, it can determine which type of and how many BSs can be switched off when the traffic load is low. II. C ELLULAR S YSTEM M ODEL Both homogeneous and heterogeneous cellular networks are considered in this paper. In homogeneous cellular networks, only one type of BS is deployed in the systems, while for heterogeneous cellular networks, there consist of two types of BSs: macro BSs with high transmit power and high cost, and micro BSs with low transmit power and low cost. In both homogeneous and heterogeneous networks, the transmit power of BSs is fixed, and no dynamic power control is assumed. We also assume that both the BSs and mobile users are located in the Euclidean plan according to homogeneous PPPs. For convenience, the mathematical symbols used in this paper are summarized in Table I. Each user is always associated with the BS from which the mean received signal strength is the largest. Specifically, in homogeneous cellular networks, each
TABLE I S UMMARY OF THE N OTATIONS
Y coordinate
70 65
ρ
BS density
60
ρM
macro BS density
55
ρm
micro BS density
50
CM
deployment and/or operation cost of macro BS
45
Cm
deployment and/or operation cost of micro BS
40
PM
transmit power of macro BS
35
Pm
transmit power of micro BS
30 25 20 20
25
30
35
40
45
50
55
60
65
70
X coordinate
Fig. 1.
An example of the general homogeneous cellular network model.
100
λ
user density
u
downlink rate threshold
η
service outage probability threshold
W
system wireless bandwidth
α
path loss exponent, valued in (2,4]
c
M α c = (P ) P
Δ
τ
τ =
e
e=
m ρm ρM
Δ Cm CM
K
90
2
Δ
constant value 3.575
80
where, N is a random integer variable, indicating the total number of users in a certain cell (N ≥ 1).
Y coordinate
70 60 50 40 30 20 10 0
0
10
20
30
40
50
60
70
80
90
100
X coordinate
Fig. 2.
An example of the general heterogeneous cellular network model.
user is served by the nearest BS. Figs. 1-2 are the network topology examples of the homogeneous and heterogeneous cellular network model, respectively. Wireless channel gain is modeled as the typical path loss fading multiplied by Rayleigh fading. We also assume that the noise is ignored, since a typical cellular network is interference-limited, especially for downlink data transmission. Universal frequency reuse is assumed in this paper. For simplicity and to make it tractable, we assume that the resource allocation scheme of each BS is to allocate the resource (e.g., time slots or wireless spectrum) equally to the associated users. The user QoS requirement is that the service outage probability of a random user should be less than η. Note that outage occurs when the instantaneous downlink data rate is lower than a threshold (denoted by u). Thus, the QoS requirement can be expressed as E{P r(
W log2 (1 + SIN R) < u)} < η, N
(1)
III. H OMOGENEOUS C ELLULAR N ETWORKS In homogeneous cellular networks, since the transmit power of all the BSs is the same, users are always associated with their nearest BS. Thus, these cells are polygonal, and form the well-known PV (Poisson-Voronoi) tessellation. PV tessellation is the final stage of the growth process in which seeds generated according to the Poisson point distribution grow simultaneously at the same isotropic rate until they grow into contact. The cell size probability density function (PDF) of a PV tessellation is known to be accurately predicted by a gamma distribution [11] with K = 3.575: f (A) = ρK
K K K−1 exp(−KρA) A Γ(K)
(2)
where, A denotes the cell size, ρ is the BS density, and ∞ Γ(K) = 0 xK−1 exp(−x)dx is the gamma function. Since the users are located according to PPP, the number of users in a cell with cell size A follows the Poisson distribution: PA (n) =
(λA)n exp(−λA). n!
(3)
On the other hand, a striking property of PPPs is Slivnyak’s theorem [8], which means that if we condition on having a user in a cell with cell size A, the number of coexisting users still follows the Poisson distribution. With the above knowledge, we get the service outage probability, W E{P r( log2 (1 + SIN R) < u)} N ∞ ∞ (4) W = P r( log2 (1 + SIN R) < u)PA (n)f (A)dA. n+1 0 n=0
As proved in Ref. [9], in the Rayleigh fading, no noise and PPP BS deployment scenario, the coverage probability of a random user is expressed as 1+
T 2/α
1 ∞
0.4
. 1 dx T −2/α 1+xα/2
(5)
To summarize, the optimal BS density problem for homogeneous cellular networks is formulated as the following: min ρ ∞ ∞ s.t. 0 n=0 n
1 + (2
(n+1)u W
− 1)
2 α
1 ∞
(n+1)u (2 W
Upper Bound Lower Bound Optimal Density
0.45
2 −1)− α
1 dx 1+xα/2
(λA) K K K−1 exp(−KρA)dA exp(−λA)ρK A n! Γ(K) ≥ 1 − η. (6) The above problem has a unique solution, since the left side of the constraint is a strictly monotone increasing function. Thus, the optimal result can be achieved numerically through the binary search algorithm. However, the calculation of service outage probability consists of two integrals and one summation to the infinity, one embedded into another, which is of high computational complexity. To get some theoretical analysis, we derive both the upper and lower bounds of the optimal BS density, and for the special case α = 4, the upper bound has a closed form. In the following, detailed proof of the bounds is omitted due to limited space. 1) Upper Bound:
network density ρ
P r(SIN R ≥ T ) =
0.5
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
Fig. 3.
0.15
0.2
0.25
0.3
0.35
user outage probability η
0.4
0.45
0.5
BS density versus outage probability threshold.
both the upper bound and lower bound are close to the optimal. u u Furthermore, since K{(2 W (1−η))−1/K −1} ≈ − log(2 W (1− η)) (for more details, refer to the approximation (16) in Section 1 IV), we have that the upper bound ρˆ ∼ − log (1−η)+ u . Due 2 W to the same trend as shown in this figure, it is reasonable to 1 1 u ∗ have ρ∗ ∼ − log (1−η)+ ∼ − log(1−η) when W is u , and ρ 2 W approaching to 0. IV. H ETEROGENEOUS C ELLULAR N ETWORKS
In this section, we turn our attention to the optimal macro/micro BS density of heterogeneous cellular networks which consists of macro and micro BSs with different cost and different coverage ability. The QoS constraint is that ∞ 4 − α −u m K ρˆ α − 2 −u }K = 1−η. the service outage probability in both macro and micro cells ( 2W 2W ) { −u 2 2 (1 − 2(m+1) W )λ + K ρˆ should be less than η. The heterogeneous cellular network is m=0 (7) not a PV tessellation any more, because a user may choose a For the special case α = 4, the upper bound has a closed-form macro BS far away as his home BS, rather than the nearest expression: micro BS, due to the difference in transmit power. Actually u it is a weighted PV tessellation, an extension of PV. A 1 − 2− W ρˆ = λ. (8) u weighted-PV is the final stage of the growth process in which K{(2 W (1 − η))−1/K − 1} seeds are generated according to a PPP, and each seed grows 2) Lower Bound: simultaneously at its own isotropic rate until they grow into 1/α 1/α Theorem 2. The optimal BS density in the general contact. The growing speed of macro BSs is PM , while Pm interference-limited homogeneous cellular network (6) has a for micro BSs. lower bound ρˇ, which satisfies the following expression: However, to our best knowledge, there is no formula about ∞ the domain size distribution of the general weighted-PV tessel 3u u K ρˇ 1 }K = 1 − η. 2−( 4W +1)m− 4W { lation. Due to that PV is the extreme case of the weighted-PV 3m+1 u 2 m=0 (1 − 2− 4 W )λ + K ρˇ when ρ M or ρm is 0, we do numerical fitting of the macro (9) and micro cell size distributions using the gamma function. Now that the upper and lower bounds are derived, it is We take a scenario as a case study, in which Pm is 10dB important to see how tight the upper and lower bounds are. lower than PM , and α is 4. Due to limited space, only fitting In Fig. 3, we set α = 4 and λ = 1. The optimal density is results are provided. It turns out that the fitting functions of found through the binary search algorithm. Fig. 3 shows how the macro and micro cell size distribution for this case are: the optimal network density ρ∗ varies with the service outage cρM +ρm KM x) KM −1 exp(− c threshold η. In this figure, user rate threshold u = 1Mbps and f (x) = x , (10) M u ( cρMc+ρm K1M )KM Γ(KM ) network bandwidth W = 20MHz ( W = 0.05). We can see that Theorem 1. The optimal BS density in the general interference-limited homogeneous cellular network (6) has an upper bound ρˆ, which satisfies the following expression:
0.025
upper bound, as
0.045 0.04
m
0.035
PDF of A
PDF of AM
0.02
0.015
0.01
W log2 (1 + SIN R) ≥ u)} N cρM +ρm u KM c ≥2− W { }KM . u (1 − 2− W )λ + cρMc+ρm KM Emacro {P r(
0.03 0.025 0.02 0.015 0.01
0.005
0.005 0
0
5
10
15
20
25
0
0
Macro cell size
5
10
15
20
25
By letting the upper bound (13) equal to 1 − η, we have,
Micro cell size
(a)
(13)
(b)
u
1 − 2− W
ρM =
Fig. 4. Verification chart of the cell size distribution with parameters setting ρM = 0.1 and ρm = 0.27: (a) macro cells; (b) micro cells.
c+τ c
− K1
u
KM {[2 W (1 − η)]
M
− 1}
λ,
(14)
which leads to the following reinforced problem: u
+0.4106ρm where, KM = 3.575 ρρM ; and M +0.1673ρm
fm (x) = xKm −1
exp(−(cρM + ρm )Km x) , ( cρM1+ρm K1m )Km Γ(Km )
min (11)
+2.5327ρM where, Km = 3.575 ρρm . m +5.1952ρM Figs. 4(a) and 4(b) are verification charts of the macro and micro cell size distribution respectively, with the parameter setting ρM = 0.1 and ρm = 0.27. The red lines are from our fitting functions. These figures verify that the macro and micro cell size distributions are well matched by our fitting functions. The coverage probability of a random user in our heterogeneous cellular network model still follows the distribution (5), the same as that in homogeneous scenario. The reason is that, if only focusing on the signal strength level, each macro m 1/α BS is the same as a micro BS with ( PPM ) times of shorter distance for each user. Thus, the heterogenous cellular network is equivalent to the homogeneous cellular network deployed with ( PPM )2/α ρM + ρm density of micro BSs. m Therefore, the optimal BS density problem for heterogeneous cellular networks to minimize the network cost CM ρM + Cm ρm or ρM + eρm equivalently is formulated as:
min ρM + eρm s.t.
W Emacro {P r( log2 (1 + SIN R) ≥ u)} ≥ 1 − η, N W Emicro {P r( log2 (1 + SIN R) ≥ u)} ≥ 1 − η, N ρ0 ≤ ρM ≤ ρ2 , (12) ρ1 ≤ ρm ≤ ρ3 .
This formulation (12) can be applied to capacity extension and energy saving issues. For capacity extension, (ρ0 , ρ1 ) may be the initial state, while (ρ2 , ρ3 ) for energy saving. The service outage probability expressions of macro and micro users are similar to Eq. (4), just by changing f (A) into fM (A) in Eq. (10) and fm (A) in Eq. (11). Generally, the macro cell size is much larger than the micro’s. The micro cell users can get more bandwidth, and their QoS requirement is easier to be satisfied. Thus, it is reasonable to relax the micro cell QoS constraint in (12). Furthermore, based on the previous results on homogeneous scenarios, we reinforce the macro cell QoS constraint by the
s.t.
(1 + eτ ) ρ0 ≤
1 − 2− W − K1
u c+τ W c KM {[2
(1 − η)] u 1 − 2− W
− K1
u c+τ W c KM {[2
ρ1 ≤ τ
(1 − η)] u 1 − 2− W
u c+τ W c KM {[2
M
M
− K1
(1 − η)]
− 1}
− 1}
M
λ
λ ≤ ρ2 ,
− 1}
λ ≤ ρ3 .
Actually, x1 (ax − 1) ≈ log(a) as x ≈ 0. Since 1 ] is near to 0, we have that (0, 3.575 − K1
u
KM {[2 W (1 − η)]
M
− 1} ≈ log(
(15)
1 KM
1 ). u 2 W (1 − η)
∈
(16)
Therefore, the reinforced problem can be further approximated as u
min ρM + eρm ≈
1 + eτ 1 − 2− W λ, 1 + 1c τ log( Wu 1 ) 2
(1−η)
max{τ1 , τ2 } ≤ τ ≤ min{τ0 , τ3 },
s.t.
(17)
where, u
τ0 ≈
1 − 2− W ρ0 log( Wu 1 2
(1−η)
)
cλ − c, τ1 ≈
2 W (1−η)
u
1 − 2− W τ2 ≈ ρ2 log( Wu 1 2
(1−η)
1 u
1−2− W ρ1 log( u 1
)
cλ − c, τ3 ≈
)
λ−
1 c
)
λ−
1 c
1 u
1−2− W ρ3 log( u 1
2 W (1−η)
,
.
(18) According to (17), it is obvious that when e is larger than the target is an increasing function, which suggests that τ should be as small as possible; otherwise, τ needs to be as large as possible. 1 c,
∗
τ =
⎧ ⎨ min{τ0 , τ3 },
0 ≤ e < 1c ;
⎩ max{τ , τ }, 1 2
1 c
< e ≤ 1.
(19)
Specifically, for capacity extension, if e > 1c , it is better to deploy more macro BSs, while if e < 1c , the optimal strategy is to deploy micro BSs. For energy saving, if e > 1c , it is better to switch off certain micro BSs, while if e < 1c , the optimal strategy is to switch off certain macro BSs. The approximate closed-form solution can be obtained through the formulas (14), (18) and (19).
1.2
0.18 e=0.6 e=0.4 e=0.32 e=0.3 e=0.1
m
total network cost ρ +eρ
0.12
1
M
M
ρ +eρ
0.14
total network cost
m
0.16
0.1
0.08
0.6
0.4 near−optimal network cost by the approximated solution optimal network cost for the reinforced problem optimal network cost for the orignal problem
0.2
0.06
0.04
0.8
0
1
2
3
4
5
6
7
8
9
10
0 0.1
ratio of micro−BS density over macro−BS density (τ)
Fig. 5.
Total network cost ρM + eρm versus BS density ratio τ
Fig. 5 verifies the rule (19). The parameter setting is that λ = 1, u = 1Mbps, W = 20MHz, Pm = 0.1PM , and α = 4. This figure shows that the network cost is a decreasing function of τ when e < 0.3, and an increasing function when e > 0.32. Noting that 1/c = 0.3162, from this figure we can see that rule (19) holds for most cases. On the other hand, the lines of e = 0.3 and e = 0.32 are almost constant and close to each other, which means, even through our rule may not be the optimal in the critical region, the gap between them is quite small. In the critical region, a 2 more precise approximation as x1 (ax − 1) ≈ log(a) + log2(a) x can be used if a more accurate result is required. Fig. 6 takes the capacity extension as an example to show the gaps among the approximate solution (18), the optimal of the reinforced problem (15), and the optimal of the original problem (12). The optimal results of the reinforced and original problems are obtained through the exhaustive search. Assume there has been a homogenous cellular unetwork with 1−2− W macro BSs with the density ρ0 = λ0 = u W K{(2 (1−η))−1/K −1} 0.4770, where λ0 =1. However, capacity extension is needed, since λ = 2λ0 . From this figure, we can see that the approximate solution (18) is accurate, and our rule on BS type selection is verified. When e ≤ 0.3, the optimal BS type is the micro, therefore, the optimal network cost is linear with e; when e ≥ 0.35, the optimal network cost is constant for the optimal BS type, macro. V. C ONCLUSION In this paper, we analyze the optimal BS density for both homogeneous and heterogeneous cellular networks with service outage probability constraint. For homogeneous cellular networks, we derive the upper and lower bounds of the optimal BS density, and the upper bound has a closed-form formula when the path loss exponent α = 4. For heterogeneous cellular networks, we do numerical fitting of the macro and micro cell size distribution with the gamma distribution. Based on these
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ratio of micro−BS cost over macro−BS cost C /C m
Fig. 6.
M
0.55
0.6
(e)
Total network cost ρM + eρm versus BS cost ratio e
results, we solve two important issues: capacity extension and energy saving, and provide a rule to determine which type of BSs should be deployed or slept with higher priority. The rule reveals that if e, the ratio between the micro cost and the macro cost, is less than a threshold 1/c, which is a function of path loss and their transmit power, deploying or switching on more micro BSs is more beneficial, otherwise the optimal choice is the opposite. With this rule, the optimal macro/micro BS density can be obtained numerically through the onedimensional binary search, or alternatively be approximated with our conservative closed-form solution. R EFERENCES [1] A. Damnjanovic, et. al., “A Survey on 3GPP Heterogeneous Networks”, IEEE Wireless Communications Magazine, vol. 18, no. 3, pp. 10-21, June 2011. [2] Qualcomm Inc., “LTE advanced: Heterogeneous Networks, White Paper, Jan. 2011. Available at: http://www.qualcomm.com/documents/lteadvanced-heterogeneous-networks-0. [3] http://it.people.com.cn/GB/1068/42899/10360952.html [4] Z. Niu, Y. Wu, J. Gong, and Z. Yang, “Cell zooming for cost-efficient green cellular networks”, IEEE Communication magazine, vol. 48, no. 11, pp. 74-79, Nov. 2010 [5] E. Amaldi, A. Capone, and F. Malucelli, “Radio Planning and Coverage Optimization of 3G Cellular Networks”, Wireless Networks, vol. 14, no. 4, pp. 435-447, August 2008. [6] S. Hanly, and R. Mathar, “On the Optimal Base-Station Density for CDMA Cellular Networks”, IEEE. Trans. Commun., vol. 50, no. 8, pp. 1274-1281, Aug. 2002. [7] W. Wang, and G. Shen, “Energy Efficiency of Heterogeneous Cellular Network”, Proc. VTC 2010-Fall, Ottawa, Canada 2010. [8] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks (invited paper)”, JSAC, vol. 27, no. 7, pp.1029-1046, Sep. 2009. [9] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A Tractable Approach to Coverage and Rate in Cellular Networks”, IEEE Trans. on Commun., vol. 59, no. 11, pp. 3122-3134, Nov. 2011. [10] H. S. Dhillon, R. K. Ganti, F. Baccelli and J. G. Andrews, “Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks,” to appear, IEEE Journal on Sel. Areas in Comm.,2012. [11] E. Pineda, and V. Garrodo, “Domain Size Distribution in a Poisson Voronoi Nucleation and Growth Transformation”, Physical Review E., vol. 75, no. 4 , 2007. Available: http://pre.aps.org/abstract/PRE/v75/i4/e040107.
I. I NTRODUCTION Future cellular networks are expected to provide more data service with high QoS (quality of service) requirement. Heterogeneous cellular networks composed of macro, outdoor pico and indoor femto BSs (base station) are promising to enhance the network capacity and reduce the network cost [1], [2]. However, the deployment of the pico/femto BSs will be mostly ad hoc [2]. Then one must consider how to plan heterogeneous networks, especially for capacity extension through deploying more BSs based on existing networks. On the other hand, today’s cellular networks are consuming lots of energy, and will call for more in the future. As a result, energy efficient approaches are urgently required by network vendors. For instance, China Mobile promises to reduce power consumption 20% per traffic unit in 2012 compared with 2008 [3]. Recent researches demonstrate that a system-wide approach by adapting the BS density through turning on/off BSs according to the actual traffic load is effective [4]. Thus energy saving through BS sleeping will be another important issue for heterogeneous cellular networks. Current researches about cellular network planning mainly focus on the practical deployment algorithm design, e.g., [5] and references therein. Among the very limited work on BS density, Ref. [6] is particularly related, since it attempts to find the maximal inter-BS distance for CDMA cellular networks 1 This work was supported in part by the National Basic Research Program of China (973 Program: No. 2012CB316001), the Nature Science Foundation of China (No. 61021001, No.60925002), and Hitachi R&D Headquarter.
with the hexagonal cellular network model. However, this work cannot be used for capacity extension or dynamic BS sleeping, and is difficult to be extended to heterogeneous cellular networks. There are also many studies on energy efficiency of heterogeneous cellular networks ([7] and references therein). Most of them focus on improving energy efficiency through micro BS deployment, showing the benefits of low power micro BS sites with system-level simulations. However, few focus on BS density, or jointly consider dynamic BS sleeping in heterogeneous cellular networks. In this paper, we focus on network planning and energy efficient operation for heterogeneous cellular networks. We tackle both problems through analyzing the optimal macro/micro BS density, by modeling heterogeneous cellular networks with stochastic geometry, in which both macro and micro BSs are located independently according to PPPs (Poisson Point Process) [8], [9], [10]. This model has several advantages, e.g., suitable to analyze capacity extension through deploying more BSs and energy saving through BS sleeping, since the superposition of two or more independent PPPs, and the independent thinning of a PPP are still PPPs [8]. Based on this model, we solve the optimal macro/micro BS density problem: given the predefined QoS requirement and traffic load, what is the optimal BS density to minimize the network cost. For network planning, it can provide information about the type and the number of BSs required. For energy saving, it can determine which type of and how many BSs can be switched off when the traffic load is low. II. C ELLULAR S YSTEM M ODEL Both homogeneous and heterogeneous cellular networks are considered in this paper. In homogeneous cellular networks, only one type of BS is deployed in the systems, while for heterogeneous cellular networks, there consist of two types of BSs: macro BSs with high transmit power and high cost, and micro BSs with low transmit power and low cost. In both homogeneous and heterogeneous networks, the transmit power of BSs is fixed, and no dynamic power control is assumed. We also assume that both the BSs and mobile users are located in the Euclidean plan according to homogeneous PPPs. For convenience, the mathematical symbols used in this paper are summarized in Table I. Each user is always associated with the BS from which the mean received signal strength is the largest. Specifically, in homogeneous cellular networks, each
TABLE I S UMMARY OF THE N OTATIONS
Y coordinate
70 65
ρ
BS density
60
ρM
macro BS density
55
ρm
micro BS density
50
CM
deployment and/or operation cost of macro BS
45
Cm
deployment and/or operation cost of micro BS
40
PM
transmit power of macro BS
35
Pm
transmit power of micro BS
30 25 20 20
25
30
35
40
45
50
55
60
65
70
X coordinate
Fig. 1.
An example of the general homogeneous cellular network model.
100
λ
user density
u
downlink rate threshold
η
service outage probability threshold
W
system wireless bandwidth
α
path loss exponent, valued in (2,4]
c
M α c = (P ) P
Δ
τ
τ =
e
e=
m ρm ρM
Δ Cm CM
K
90
2
Δ
constant value 3.575
80
where, N is a random integer variable, indicating the total number of users in a certain cell (N ≥ 1).
Y coordinate
70 60 50 40 30 20 10 0
0
10
20
30
40
50
60
70
80
90
100
X coordinate
Fig. 2.
An example of the general heterogeneous cellular network model.
user is served by the nearest BS. Figs. 1-2 are the network topology examples of the homogeneous and heterogeneous cellular network model, respectively. Wireless channel gain is modeled as the typical path loss fading multiplied by Rayleigh fading. We also assume that the noise is ignored, since a typical cellular network is interference-limited, especially for downlink data transmission. Universal frequency reuse is assumed in this paper. For simplicity and to make it tractable, we assume that the resource allocation scheme of each BS is to allocate the resource (e.g., time slots or wireless spectrum) equally to the associated users. The user QoS requirement is that the service outage probability of a random user should be less than η. Note that outage occurs when the instantaneous downlink data rate is lower than a threshold (denoted by u). Thus, the QoS requirement can be expressed as E{P r(
W log2 (1 + SIN R) < u)} < η, N
(1)
III. H OMOGENEOUS C ELLULAR N ETWORKS In homogeneous cellular networks, since the transmit power of all the BSs is the same, users are always associated with their nearest BS. Thus, these cells are polygonal, and form the well-known PV (Poisson-Voronoi) tessellation. PV tessellation is the final stage of the growth process in which seeds generated according to the Poisson point distribution grow simultaneously at the same isotropic rate until they grow into contact. The cell size probability density function (PDF) of a PV tessellation is known to be accurately predicted by a gamma distribution [11] with K = 3.575: f (A) = ρK
K K K−1 exp(−KρA) A Γ(K)
(2)
where, A denotes the cell size, ρ is the BS density, and ∞ Γ(K) = 0 xK−1 exp(−x)dx is the gamma function. Since the users are located according to PPP, the number of users in a cell with cell size A follows the Poisson distribution: PA (n) =
(λA)n exp(−λA). n!
(3)
On the other hand, a striking property of PPPs is Slivnyak’s theorem [8], which means that if we condition on having a user in a cell with cell size A, the number of coexisting users still follows the Poisson distribution. With the above knowledge, we get the service outage probability, W E{P r( log2 (1 + SIN R) < u)} N ∞ ∞ (4) W = P r( log2 (1 + SIN R) < u)PA (n)f (A)dA. n+1 0 n=0
As proved in Ref. [9], in the Rayleigh fading, no noise and PPP BS deployment scenario, the coverage probability of a random user is expressed as 1+
T 2/α
1 ∞
0.4
. 1 dx T −2/α 1+xα/2
(5)
To summarize, the optimal BS density problem for homogeneous cellular networks is formulated as the following: min ρ ∞ ∞ s.t. 0 n=0 n
1 + (2
(n+1)u W
− 1)
2 α
1 ∞
(n+1)u (2 W
Upper Bound Lower Bound Optimal Density
0.45
2 −1)− α
1 dx 1+xα/2
(λA) K K K−1 exp(−KρA)dA exp(−λA)ρK A n! Γ(K) ≥ 1 − η. (6) The above problem has a unique solution, since the left side of the constraint is a strictly monotone increasing function. Thus, the optimal result can be achieved numerically through the binary search algorithm. However, the calculation of service outage probability consists of two integrals and one summation to the infinity, one embedded into another, which is of high computational complexity. To get some theoretical analysis, we derive both the upper and lower bounds of the optimal BS density, and for the special case α = 4, the upper bound has a closed form. In the following, detailed proof of the bounds is omitted due to limited space. 1) Upper Bound:
network density ρ
P r(SIN R ≥ T ) =
0.5
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
Fig. 3.
0.15
0.2
0.25
0.3
0.35
user outage probability η
0.4
0.45
0.5
BS density versus outage probability threshold.
both the upper bound and lower bound are close to the optimal. u u Furthermore, since K{(2 W (1−η))−1/K −1} ≈ − log(2 W (1− η)) (for more details, refer to the approximation (16) in Section 1 IV), we have that the upper bound ρˆ ∼ − log (1−η)+ u . Due 2 W to the same trend as shown in this figure, it is reasonable to 1 1 u ∗ have ρ∗ ∼ − log (1−η)+ ∼ − log(1−η) when W is u , and ρ 2 W approaching to 0. IV. H ETEROGENEOUS C ELLULAR N ETWORKS
In this section, we turn our attention to the optimal macro/micro BS density of heterogeneous cellular networks which consists of macro and micro BSs with different cost and different coverage ability. The QoS constraint is that ∞ 4 − α −u m K ρˆ α − 2 −u }K = 1−η. the service outage probability in both macro and micro cells ( 2W 2W ) { −u 2 2 (1 − 2(m+1) W )λ + K ρˆ should be less than η. The heterogeneous cellular network is m=0 (7) not a PV tessellation any more, because a user may choose a For the special case α = 4, the upper bound has a closed-form macro BS far away as his home BS, rather than the nearest expression: micro BS, due to the difference in transmit power. Actually u it is a weighted PV tessellation, an extension of PV. A 1 − 2− W ρˆ = λ. (8) u weighted-PV is the final stage of the growth process in which K{(2 W (1 − η))−1/K − 1} seeds are generated according to a PPP, and each seed grows 2) Lower Bound: simultaneously at its own isotropic rate until they grow into 1/α 1/α Theorem 2. The optimal BS density in the general contact. The growing speed of macro BSs is PM , while Pm interference-limited homogeneous cellular network (6) has a for micro BSs. lower bound ρˇ, which satisfies the following expression: However, to our best knowledge, there is no formula about ∞ the domain size distribution of the general weighted-PV tessel 3u u K ρˇ 1 }K = 1 − η. 2−( 4W +1)m− 4W { lation. Due to that PV is the extreme case of the weighted-PV 3m+1 u 2 m=0 (1 − 2− 4 W )λ + K ρˇ when ρ M or ρm is 0, we do numerical fitting of the macro (9) and micro cell size distributions using the gamma function. Now that the upper and lower bounds are derived, it is We take a scenario as a case study, in which Pm is 10dB important to see how tight the upper and lower bounds are. lower than PM , and α is 4. Due to limited space, only fitting In Fig. 3, we set α = 4 and λ = 1. The optimal density is results are provided. It turns out that the fitting functions of found through the binary search algorithm. Fig. 3 shows how the macro and micro cell size distribution for this case are: the optimal network density ρ∗ varies with the service outage cρM +ρm KM x) KM −1 exp(− c threshold η. In this figure, user rate threshold u = 1Mbps and f (x) = x , (10) M u ( cρMc+ρm K1M )KM Γ(KM ) network bandwidth W = 20MHz ( W = 0.05). We can see that Theorem 1. The optimal BS density in the general interference-limited homogeneous cellular network (6) has an upper bound ρˆ, which satisfies the following expression:
0.025
upper bound, as
0.045 0.04
m
0.035
PDF of A
PDF of AM
0.02
0.015
0.01
W log2 (1 + SIN R) ≥ u)} N cρM +ρm u KM c ≥2− W { }KM . u (1 − 2− W )λ + cρMc+ρm KM Emacro {P r(
0.03 0.025 0.02 0.015 0.01
0.005
0.005 0
0
5
10
15
20
25
0
0
Macro cell size
5
10
15
20
25
By letting the upper bound (13) equal to 1 − η, we have,
Micro cell size
(a)
(13)
(b)
u
1 − 2− W
ρM =
Fig. 4. Verification chart of the cell size distribution with parameters setting ρM = 0.1 and ρm = 0.27: (a) macro cells; (b) micro cells.
c+τ c
− K1
u
KM {[2 W (1 − η)]
M
− 1}
λ,
(14)
which leads to the following reinforced problem: u
+0.4106ρm where, KM = 3.575 ρρM ; and M +0.1673ρm
fm (x) = xKm −1
exp(−(cρM + ρm )Km x) , ( cρM1+ρm K1m )Km Γ(Km )
min (11)
+2.5327ρM where, Km = 3.575 ρρm . m +5.1952ρM Figs. 4(a) and 4(b) are verification charts of the macro and micro cell size distribution respectively, with the parameter setting ρM = 0.1 and ρm = 0.27. The red lines are from our fitting functions. These figures verify that the macro and micro cell size distributions are well matched by our fitting functions. The coverage probability of a random user in our heterogeneous cellular network model still follows the distribution (5), the same as that in homogeneous scenario. The reason is that, if only focusing on the signal strength level, each macro m 1/α BS is the same as a micro BS with ( PPM ) times of shorter distance for each user. Thus, the heterogenous cellular network is equivalent to the homogeneous cellular network deployed with ( PPM )2/α ρM + ρm density of micro BSs. m Therefore, the optimal BS density problem for heterogeneous cellular networks to minimize the network cost CM ρM + Cm ρm or ρM + eρm equivalently is formulated as:
min ρM + eρm s.t.
W Emacro {P r( log2 (1 + SIN R) ≥ u)} ≥ 1 − η, N W Emicro {P r( log2 (1 + SIN R) ≥ u)} ≥ 1 − η, N ρ0 ≤ ρM ≤ ρ2 , (12) ρ1 ≤ ρm ≤ ρ3 .
This formulation (12) can be applied to capacity extension and energy saving issues. For capacity extension, (ρ0 , ρ1 ) may be the initial state, while (ρ2 , ρ3 ) for energy saving. The service outage probability expressions of macro and micro users are similar to Eq. (4), just by changing f (A) into fM (A) in Eq. (10) and fm (A) in Eq. (11). Generally, the macro cell size is much larger than the micro’s. The micro cell users can get more bandwidth, and their QoS requirement is easier to be satisfied. Thus, it is reasonable to relax the micro cell QoS constraint in (12). Furthermore, based on the previous results on homogeneous scenarios, we reinforce the macro cell QoS constraint by the
s.t.
(1 + eτ ) ρ0 ≤
1 − 2− W − K1
u c+τ W c KM {[2
(1 − η)] u 1 − 2− W
− K1
u c+τ W c KM {[2
ρ1 ≤ τ
(1 − η)] u 1 − 2− W
u c+τ W c KM {[2
M
M
− K1
(1 − η)]
− 1}
− 1}
M
λ
λ ≤ ρ2 ,
− 1}
λ ≤ ρ3 .
Actually, x1 (ax − 1) ≈ log(a) as x ≈ 0. Since 1 ] is near to 0, we have that (0, 3.575 − K1
u
KM {[2 W (1 − η)]
M
− 1} ≈ log(
(15)
1 KM
1 ). u 2 W (1 − η)
∈
(16)
Therefore, the reinforced problem can be further approximated as u
min ρM + eρm ≈
1 + eτ 1 − 2− W λ, 1 + 1c τ log( Wu 1 ) 2
(1−η)
max{τ1 , τ2 } ≤ τ ≤ min{τ0 , τ3 },
s.t.
(17)
where, u
τ0 ≈
1 − 2− W ρ0 log( Wu 1 2
(1−η)
)
cλ − c, τ1 ≈
2 W (1−η)
u
1 − 2− W τ2 ≈ ρ2 log( Wu 1 2
(1−η)
1 u
1−2− W ρ1 log( u 1
)
cλ − c, τ3 ≈
)
λ−
1 c
)
λ−
1 c
1 u
1−2− W ρ3 log( u 1
2 W (1−η)
,
.
(18) According to (17), it is obvious that when e is larger than the target is an increasing function, which suggests that τ should be as small as possible; otherwise, τ needs to be as large as possible. 1 c,
∗
τ =
⎧ ⎨ min{τ0 , τ3 },
0 ≤ e < 1c ;
⎩ max{τ , τ }, 1 2
1 c
< e ≤ 1.
(19)
Specifically, for capacity extension, if e > 1c , it is better to deploy more macro BSs, while if e < 1c , the optimal strategy is to deploy micro BSs. For energy saving, if e > 1c , it is better to switch off certain micro BSs, while if e < 1c , the optimal strategy is to switch off certain macro BSs. The approximate closed-form solution can be obtained through the formulas (14), (18) and (19).
1.2
0.18 e=0.6 e=0.4 e=0.32 e=0.3 e=0.1
m
total network cost ρ +eρ
0.12
1
M
M
ρ +eρ
0.14
total network cost
m
0.16
0.1
0.08
0.6
0.4 near−optimal network cost by the approximated solution optimal network cost for the reinforced problem optimal network cost for the orignal problem
0.2
0.06
0.04
0.8
0
1
2
3
4
5
6
7
8
9
10
0 0.1
ratio of micro−BS density over macro−BS density (τ)
Fig. 5.
Total network cost ρM + eρm versus BS density ratio τ
Fig. 5 verifies the rule (19). The parameter setting is that λ = 1, u = 1Mbps, W = 20MHz, Pm = 0.1PM , and α = 4. This figure shows that the network cost is a decreasing function of τ when e < 0.3, and an increasing function when e > 0.32. Noting that 1/c = 0.3162, from this figure we can see that rule (19) holds for most cases. On the other hand, the lines of e = 0.3 and e = 0.32 are almost constant and close to each other, which means, even through our rule may not be the optimal in the critical region, the gap between them is quite small. In the critical region, a 2 more precise approximation as x1 (ax − 1) ≈ log(a) + log2(a) x can be used if a more accurate result is required. Fig. 6 takes the capacity extension as an example to show the gaps among the approximate solution (18), the optimal of the reinforced problem (15), and the optimal of the original problem (12). The optimal results of the reinforced and original problems are obtained through the exhaustive search. Assume there has been a homogenous cellular unetwork with 1−2− W macro BSs with the density ρ0 = λ0 = u W K{(2 (1−η))−1/K −1} 0.4770, where λ0 =1. However, capacity extension is needed, since λ = 2λ0 . From this figure, we can see that the approximate solution (18) is accurate, and our rule on BS type selection is verified. When e ≤ 0.3, the optimal BS type is the micro, therefore, the optimal network cost is linear with e; when e ≥ 0.35, the optimal network cost is constant for the optimal BS type, macro. V. C ONCLUSION In this paper, we analyze the optimal BS density for both homogeneous and heterogeneous cellular networks with service outage probability constraint. For homogeneous cellular networks, we derive the upper and lower bounds of the optimal BS density, and the upper bound has a closed-form formula when the path loss exponent α = 4. For heterogeneous cellular networks, we do numerical fitting of the macro and micro cell size distribution with the gamma distribution. Based on these
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ratio of micro−BS cost over macro−BS cost C /C m
Fig. 6.
M
0.55
0.6
(e)
Total network cost ρM + eρm versus BS cost ratio e
results, we solve two important issues: capacity extension and energy saving, and provide a rule to determine which type of BSs should be deployed or slept with higher priority. The rule reveals that if e, the ratio between the micro cost and the macro cost, is less than a threshold 1/c, which is a function of path loss and their transmit power, deploying or switching on more micro BSs is more beneficial, otherwise the optimal choice is the opposite. With this rule, the optimal macro/micro BS density can be obtained numerically through the onedimensional binary search, or alternatively be approximated with our conservative closed-form solution. R EFERENCES [1] A. Damnjanovic, et. al., “A Survey on 3GPP Heterogeneous Networks”, IEEE Wireless Communications Magazine, vol. 18, no. 3, pp. 10-21, June 2011. [2] Qualcomm Inc., “LTE advanced: Heterogeneous Networks, White Paper, Jan. 2011. Available at: http://www.qualcomm.com/documents/lteadvanced-heterogeneous-networks-0. [3] http://it.people.com.cn/GB/1068/42899/10360952.html [4] Z. Niu, Y. Wu, J. Gong, and Z. Yang, “Cell zooming for cost-efficient green cellular networks”, IEEE Communication magazine, vol. 48, no. 11, pp. 74-79, Nov. 2010 [5] E. Amaldi, A. Capone, and F. Malucelli, “Radio Planning and Coverage Optimization of 3G Cellular Networks”, Wireless Networks, vol. 14, no. 4, pp. 435-447, August 2008. [6] S. Hanly, and R. Mathar, “On the Optimal Base-Station Density for CDMA Cellular Networks”, IEEE. Trans. Commun., vol. 50, no. 8, pp. 1274-1281, Aug. 2002. [7] W. Wang, and G. Shen, “Energy Efficiency of Heterogeneous Cellular Network”, Proc. VTC 2010-Fall, Ottawa, Canada 2010. [8] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks (invited paper)”, JSAC, vol. 27, no. 7, pp.1029-1046, Sep. 2009. [9] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A Tractable Approach to Coverage and Rate in Cellular Networks”, IEEE Trans. on Commun., vol. 59, no. 11, pp. 3122-3134, Nov. 2011. [10] H. S. Dhillon, R. K. Ganti, F. Baccelli and J. G. Andrews, “Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks,” to appear, IEEE Journal on Sel. Areas in Comm.,2012. [11] E. Pineda, and V. Garrodo, “Domain Size Distribution in a Poisson Voronoi Nucleation and Growth Transformation”, Physical Review E., vol. 75, no. 4 , 2007. Available: http://pre.aps.org/abstract/PRE/v75/i4/e040107.
