Resource Partitioning and User Association With Sleep-Mode Base ...

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Resource Partitioning and User Association With Sleep-Mode Base Stations in Heterogeneous Cellular Networks Chenlong Jia, Student Member, IEEE, and Teng Joon Lim, Senior Member, IEEE

Abstract—In this paper, we minimize the total energy used in a two-tier heterogeneous cellular network (HCN), through the optimization of resource partitioning and user association, assuming that both macro and small-cell base stations may be put into sleep mode. With resource partitioning, one tier of base stations can be put into sleep mode on a fraction of available time/frequency resources to reduce network power consumption. A user association scheme is adopted to alleviate the SINR degradation due to severe inter-tier interference for users close to tier boundaries. By deriving tractable throughput characterizations, we formulate the network-wide energy usage minimization problem and determine the optimal user association and resource partitioning strategies. The maximum achievable network coverage probabilities using the optimal strategies are also investigated. Numerical results show that the proposed resource allocation and user association scheme reduces network energy consumption and improves coverage probability in co-channel heterogeneous networks. Index Terms—Two tier network, stochastic geometry, user association, resource allocation.

I. I NTRODUCTION

C

ELLULAR networks of the future must deliver vastly improved quality of experience and quality of service to users, at a much higher energy efficiency (in Joules per bit) [1]. A heterogeneous or multi-tier network architecture is widely seen as an essential part of future cellular networks [2], [3]. In a heterogeneous cellular network (HCN), low-powered small-cell base stations (BSs) are deployed within existing macro-cells, providing high-rate transmissions to nearby users while macrocell BSs are used to ensure network coverage. Due to their short transmission distances, small cells consume less energy than macro-cells do. At the same time, high spectral efficiency can be achieved in an HCN because of increased cell density [4], which leads to lower user populations on average per cell and hence more resources for each user. This motivates the focus on HCNs for improving energy efficiency in the present work.

Manuscript received March 18, 2014; revised July 29, 2014, November 30, 2014, and February 21, 2015; accepted February 25, 2015. Date of publication March 10, 2015; date of current version July 8, 2015. This work was supported in part by Ministry of Education grant R-263-000-A48-112. The associate editor coordinating the review of this paper and approving it for publication was M. Rossi. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2015.2411737

The improvement of energy efficiency in HCNs could be achieved through cell load adaptation and resource allocation. 1) Cell Load Adaptation: If cell coverage can be adaptively changed and/or BSs can be switched to sleep mode, performance gains in terms of energy saving may be obtained through cell load adaptation. Cell zooming was proposed in [5] where BSs in a green cellular network adjusted their coverage areas in response to the traffic load and hence reduce energy consumption. Besides changing cell coverage area, cell load adaptation can also be achieved by putting lightly loaded BSs into sleep mode. A comparative study of these two cell load adaptation schemes was conducted in [6], which showed that the optimal energy saving strategies in the high and low load regimes are cell range adaptation and BS sleep mode, respectively. Nevertheless, the results were obtained based on a single cell assumption where the effects of interference were ignored. Adopting cell load adaptation to reduce power consumption in HCNs was investigated in [7]. In that work, a repulsive cell activation scheme was proposed where small-cell base stations located too close to each other were turned off to alleviate co-tier interference and hence reduce network power consumption. However, the repulsive cell activation strategy in that work does not consider the problem of cross-tier cell load adaptation. A tractable model to analyze inter-tier offloading in HCNs was given in [8], but the optimal load adaptation strategy that maximizes heterogeneous network energy efficiency still remains unknown. 2) Resource Allocation: Smart resource allocation provides another possibility to reduce inter-tier interference and improve network energy efficiency in HCNs. In [9], the problem of network throughput maximization in a two-tier femtocell network was investigated with fixed BS transmit power, where the two tiers of base stations were active either on all the frequencies or on orthogonal sub-channels. In [10], each small cell randomly accessed a fraction of the macro-cells’ radio resources, and the resource fraction was optimized to minimize inter-tier interference and maximize energy efficiency. However, it is still possible for BSs located close to each other to use the same time/frequency resources to transmit and cause severe interference to each other. Resource partitioning was studied in [11], with a whole tier of BSs being muted on some radio resources to mitigate inter-tier interference in heterogeneous networks. An analytical resource partitioning framework was proposed in [12], but only a homogeneous network was discussed. Results in [12] showed that resource partitioning increased network

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JIA AND LIM: RESOURCE PARTITIONING AND USER ASSOCIATION WITH SLEEP-MODE BASE STATIONS IN HCNs

sum-rate and cell edge user coverage. The throughput gains achieved by resource partitioning in heterogeneous networks was discussed in [13]. However, the case where macro users were offloaded to small-cell BSs was not considered. While load adaptation and resource allocation have been separately discussed in the above papers, a system in which load adaptation and resource allocation are jointly optimized remains to be further investigated. Efficient algorithms for admission control and resource allocation that maximize network utility of OFDMA system were given in [14], but only a single cell scenario was investigated. With stochastic geometry modeling employed, signal to interference plus noise ratio (SINR) and rate coverage probabilities were derived in [15], where resource partitioning was jointly discussed with offloading. The theoretical expressions in [15] provided insights into improving SINR and rate coverage by jointly applying resource partitioning and offloading strategy in HCNs. However, the method for minimizing network power consumption was not investigated. To address the problems mentioned above, in this paper, we consider a resource partitioning and user association scheme in two-tier HCNs, where the locations of all BSs and of the users are modeled as independent Poisson Point Processes (PPP). Users that experience high inter-tier interference are allocated resources on which only one tier of base stations are active. The inactive base stations are put into sleep mode to reduce energy consumption. The proposed user association scheme adjusts cell load on different resources to improve network performance. Based on this framework, we formulate energy minimization and coverage probability maximization problems with throughput constraints, which lead to the jointly optimal resource partitioning and user association strategy. Specifically, the contributions of this paper are as follows: • By determining the fraction of time/frequency resources on which BSs are put into sleep mode and adjusting the average cell load on the partitioned resources, we derive the optimal strategies for resource partitioning and user association that minimize energy usage and maximize coverage probability. Note that although [15] addressed offloading and resource partitioning in a HCN and derived analytical expressions for coverage probabilities, it did not provide the optimal strategy to minimize energy consumption of the HCN. • The performance of muting macro BSs and muting smallcell BSs is investigated. We show numerically that muting macro BSs on a fraction of resources with users offloaded to small-cells typically saves more energy than muting small-cell BSs does. However, muting macro BSs and offloading users to small cells may lead to a higher outage probability than muting small-cell BSs and activating macro BSs do. These discoveries provide guidelines for base station muting in heterogeneous networks. The paper is structured as follows. The system model is described in Section II. The user association and resource allocation scheme is proposed and BS power consumption model are characterised. Based on the proposed model, we give expressions for association probability and coverage probability in Section III and then derive the average throughput

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constraints for each user set. In Section IV, we formulate the optimization problems to minimize the BS power consumption and maximize coverage probability over the entire network and provide a method to find the optimum solutions. Section V gives numerical results that verify our analytical discoveries. Finally, Section VI concludes the paper. Notations: For consistency, in this paper, we use calligraphic fonts to denote sets and blackboard bold fonts to denote probabilities. II. S YSTEM M ODEL In this paper, we consider the downlink of a two-tier network, where each tier consists of BSs of the same type. Without loss of generality, let macro-cell BSs constitute tier 1 and the smallcell BSs be tier 2 base stations. Base stations in the kth tier are assumed to form a homogeneous Poisson Point Process (HPPP) Φk with intensity λk . The spatial distribution of user equipments (UEs) is another HPPP Φu with constant intensity λu . Moreover, Φ1 , Φ2 , and Φu are independent. The total spectrum resource has a bandwidth W . All tier k BSs are assumed to have constant transmit power spectral density Sk over W . We assume the BS density is large enough that the interference power dominates the additive noise. In the rest of the paper we will therefore ignore the additive noise. A typical UE i at distance di,j away from a BS j in the kth tier has the received signal power Pk (di,j ) = hi,j gk (di,j )Sk Δi , where hi,j is an exponential random variable with unit mean modelling the power attenuation due to Rayleigh fading, gk (di,j ) is the path loss from the kth tier BS j to UE i, and Δi is the bandwidth allocated to UE i. As in [7]–[9], we characterize the path loss model as k gk (di,j ) = L0 d−α i,j ,

where L0 is a constant and αk is the kth tier path loss exponent factor. By assuming identical path loss environments in both tiers, i.e., α1 = α2 = α, we can benefit from more compact and useful expressions of user association and coverage probabilities, which will be given in the next section. Furthermore, the value of path loss exponent for urban area cellular radio environment typically lies within a small range between 2.7 and 3.5 [24, Table 4.2]. Therefore, the approximation of α1 = α2 = α does not cause much loss in accuracy and it has already been adopted in many previous papers [7], [10], [21]. For simplicity, in the rest of the paper, we use the path loss model notation g(di,j ) = L0 d−α i,j without the subscript k. A. User Association and Resource Partitioning Our model applies a user association and resource partitioning scheme similar to the one proposed in [15]. 1) Resource Partitioning: For clarity, the term “resources” used in this paper refers to a set of time/frequency 3GPP resource elements. Throughout this paper, a resource element is said to be “shared” by the two tiers of BSs if both the macro and small-cell BSs are active on it to transmit to the target users. Similarly, we denote the resource elements that are deliberately allocated to one of the two BS tiers as “unshared” resources in the rest of the paper.

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Let η be the fraction of resources shared by the macro and small-cell BSs. The remaining 1 − η fraction of the resources are unshared resources, which may be allocated to macroor small-cell BSs, unlike in [15]. η is called the resource partitioning factor. 2) User Association: We assume that user association discussed in this paper is based on the pilot/reference signal power. The average pilot signal power received by user i from the kth tier BS j is E[Pk (di,j )] = g(di,j )Sk Δi . For the purpose of load adaptation, the average received pilot signal power is multiplied by a bias factor, which is called the biased received power [8]. Without loss of generality, the bias factor for macro (tier 1) BSs is 1 and a bias factor B2 is used for small-cell (tier 2) BSs. Two kinds of rate requirements for users are determined based on the biased received power from BSs. If the biased received power at a user from the macro tier is higher than that from the small-cell tier, we denote the user’s target rate as R1 . Otherwise, the target rate is R2 . One important goal of deploying small cells is to provide high-data-rate coverage for users. Thus, it is reasonable to assume that users with larger biased received power from small-cells than from macro cells have a higher rate requirement, which means R1 ≤ R2 . Specifically, the target rate Rj for a typical user can be determined using the biased received power association:  R1 , if S1 g(D1 ) ≥ B2 S2 g(D2 ) Rj = (1) R2 , if B2 S2 g(D2 ) > S1 g(D1 ) where Dk denotes the distance between the typical user and its nearest BS in tier k, and B2 is the association bias factor for tier 2. B2 determines how many UEs can be served with rate requirements R1 and R2 . As B2 increases, more UEs will be served by small-cell BSs with target rate R2 . Note that a user measures the received pilot signal power from the femto and macro BSs over the same bandwidth, and therefore the comparison of signal powers is equivalent to the above comparison of spectral densities. We let U2 denote the set of users with target rate R2 . The set of users with target rate R1 is split into two disjoint sets U1 and UD , where U1 consists of users closer to the macrocell centres and UD represents the set of users located closer to the cell boundaries between the two tiers. A user in UD suffers from inter-tier interference if both tiers of BSs transmit on its allocated resources. To determine whether a user belongs to U1 or UD , another association bias factor B1 is introduced, where B1 ≥ B2 . The mapping of user i with rate requirement Rj to sets U1 , U2 and UD is summarized as follows: ⎧ U1 , ⎪ ⎨ UD , i∈ ⎪ ⎩ U2 ,

if Rj = R1 and g(D1 )S1 ≥ B1 g(D2 )S2 if Rj = R1 and B1 g(D2 )S2 > g(D1 )S1 ≥ B2 g(D2 )S2 if Rj = R2 and B2 g(D2 )S2 > g(D1 )S1 , (2)

By changing the value of B1 , we can control the number of users in U1 and UD . Set UD will be empty only if B1 = B2 . To resolve the inter-tier interference problem, we combine the user association rule in (2) with resource partitioning, where

Fig. 1. Resource allocation scheme: if macro BSs are used to transmit on the unshared resources and small-cell BSs are muted, then UD = UD1 ; if smallcell BSs are used to transmit on the unshared resources and macro BSs are muted, then UD = UD2 .

U1 and U2 users are served by macro and small-cell BSs on the shared resources, respectively. The unshared resources are allocated to users in UD .1 Since either small-cell BSs or macro BSs can be muted on the unshared resources, we will consider the two cases separately. For simplicity, in the subsequent analysis we use Case 1 to denote the scenario where UEs in UD associate with macro BSs and use UD1 to represent the set UD . If small-cell BSs are assigned to serve users in UD on the unshared resources, it is similar to the offloading scheme in [15]. This scenario is denoted as Case 2 and UD2 is used to represent UD . For clarity, the considered system is illustrated in Fig. 1. Some additional remarks are listed as follows: 1) In this paper, we consider the scenario where only users with target rate R1 are offloaded to the unshared resources. We would also like to point out that the method proposed in this paper can be easily generalised to analyse the case where both users with required rate R1 and R2 can be allocated to the unshared resources. The details are not provided here in order not to detract from the main contributions of this paper. 2) The traditional fully shared resource allocation scheme is a special case of our proposed model with B1 = B2 , which means UD = ∅ in (2). Thus, all resources are shared among users in U1 and users in U2 . 3) By setting B1 = ∞, U1 = ∅. If macro BSs are selected to serve users in UD (UD = UD1 ), we have the macro and small-cell BSs occupy orthogonal resource elements. This is in fact the traditional unshared resource allocation scheme. On the other hand, if small-cell BSs are used to transmit to UD (UD = UD2 ) users and B1 = ∞, macro BSs can be completely muted on all resources and all users are served by only small-cell BSs. In that case, the two tier heterogeneous network becomes a homogeneous network consisting of only small-cell BSs.

1 By increasing B /decreasing B , the inter-tier interference level at users in 1 2 U1 /U2 can be reduced. Due to the difference in coverage areas, the number of U1 users in each macro cell is typically larger than the number of U2 users in each small cell. Therefore, reducing inter-tier interference for U1 users is more favourable. Thus, it is typically desirable to let B1 ≥ 1 (0 dB), but B2 is not restricted to be smaller than 1.

JIA AND LIM: RESOURCE PARTITIONING AND USER ASSOCIATION WITH SLEEP-MODE BASE STATIONS IN HCNs

4) For simplicity, in this paper we treat B2 as a given constant. For any association bias B2 , we present the method of finding B1 to optimize network performance in terms of power consumption and UE coverage. Joint optimization of B1 and B2 will be discussed in the future. 5) We assume that all BSs are active—if in fact a k-th tier BS is on with probability pak , then, without loss of generality, we replace the Poisson density λk with pak λk .

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TABLE I N OTATIONS U SED IN A NALYSIS

B. Base Station Power Consumption Model A simple power consumption model based on measurements done on real hardware was given in [17]. Using that model, the power consumption of macro- and small-cell BSs during downlink transmission is given respectively by Pma = ama PT 1 + bma ,

Psm = asm PT 2 + bsm .

(3)

In the above models, PT 1 and PT 2 are the transmit powers of macro- and small-cell BSs, respectively. The coefficients ama and asm account for the power consumption that scales with the transmit power. The terms bma and bsm represent the transmission-independent power consumption due to signal processing, battery backup, site cooling, etc. The model in (3) reflects the fact that the average power consumption of a base station comprises both transmit power and non-transmit power. In [17], the authors gave the numerical values for the parameters (ama , bma ) and (asm , bsm ) as ama = 22.6, bma = 412.4 W, and asm = 5.5, bsm = 32.0 W. The average BS power consumption per unit area is therefore given by EP = λ1 Pma + λ2 Psm .

(4)

Whether time or frequency resource partitioning is used impacts the formulation of energy saving problem, as explained next. 1) Frequency Domain Resource Partitioning: For frequency domain resource partitioning, muting BSs on unshared bands will reduce transmit power but the non-transmit power consumption remains unchanged. If UD = UD1 (i.e. Case 1), the macro BSs will transmit over the entire band of W Hz while the small-cell BSs will transmit over a band of ηW Hz, thus PT 1 = S1 W and PT 2 = S2 W η. We use Ef1 P to denote the power consumption EP . On the other hand, when UD = UD2 (Case 2), we have PT 1 = S1 W η, PT 2 = S2 W and EP = Ef2 P . The average BS power consumption per unit area is therefore, from (4), given respectively by Ef1 P = λ1 [ama S1 W + bma ] + λ2 [asm S2 W η + bsm ],

if UD = UD1

Et1 P = λ1 [ama S1 W + bma ] + ηλ2 [asm S2 W + bsm ],

if UD = UD1

(7)

Et2 P = ηλ1 [ama S1 W + bma ] + λ2 [asm S2 W + bsm ],

if UD = UD2

(8)

Although the EP expressions for resource partitioning in frequency and time domains are different, minimizing EP in both cases can be done similarly. To be more concrete and concise, we will only elaborate on time domain resource partitioning in t2 the rest of this paper. Using Et1 P and EP expressions (7) and (8), we will give the optimal scheme that minimizes network-wide average power consumption under certain rate constraints. For clarity, we summarize the notations used in this paper in Table I.

(5) III. T HROUGHPUT C HARACTERIZATION

Ef2 P = λ1 [ama S1 W η + bma ] + λ2 [asm S2 W + bsm ],

the unshared time slots to reduce both transmit and nontransmit power consumption, which in fact is a generalization of “eICIC” in LTE discussed in [22]. The transmit power for tier 1 and 2 BSs are respectively PT 1 = S1 W and PT 2 = S2 W . For t2 clarity, we use EP = Et1 P for Case 1 and EP = EP for Case 2. Therefore,

if UD = UD2

(6)

2) Time Domain Resource Partitioning: For time domain resource partitioning, BSs can be put into sleep mode on

In this section, user throughput constraints are characterized and then used in the next section to find the optimal resource allocation scheme. To derive throughput constraints, user association probability and coverage probability in sets U1 , U2 , UD1 , and UD2 are required.

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Let Aj (j ∈ {1, 2, D1, D2}) denote the user association probability for set Uj . The expressions of Aj are determined in the following lemma. Lemma 1: The probability that a randomly selected UE belongs to set Uj , j ∈ {1, 2, D1, D2} can be obtained as A1 =

 λ1 +

λ1 B1 S2 S1

 α2

AD1 = AD2 =

, λ2

 λ1 +

A2 =

λ1 B2 S2 S1

 λ2 +

 α2

λ2 S1 B2 S2

− λ2

 λ1 +

 α2

λ1 B1 S2 S1

, λ1

 α2

. λ2 (9)

(Note that it does not matter whether the unshared resources are used by the macro or small-cell BSs.) Proof: The derivation of Aj expressions in (9) follows that of Lemma 1 of [16].  The coverage probability of a typical user i in set Uj is defined as Pj = Pr(SIR ≥ γ|i ∈ Uj ), where γ is the target SIR. Recall that we ignored additive noise in the system model. This assumption will simplify the analysis and closed-form expressions can then be obtained. Lemma 2: For a typical user i ∈ Uj , the coverage probabilities Pj are given in (10)–(13) for j ∈ {1, 2, D1, D2} (see equations at bottom of the page) where ρ(γ, α) = 2 ∞ γ α γ − α2 1 α2 du. 1+u Proof: The results follow from Lemma 2 of [15] and closed-form expressions were obtained by letting α1 = α2 = α and setting the noise power to zero and then completing the integrals in that lemma.  The overall coverage probability is used as the metric to characterize the network coverage performance. Based on Lemmas 1 and 2, the overall coverage probability can be calculated as ⎧

Pj Aj for Case 1 ⎪ ⎨ j∈{1,2,D1}

(14) Pc = Pj Aj for Case 2. ⎪ ⎩ j∈{1,2,D2}

 P1 =

λ1 +

B1 S2 S1

We define the coverage spectrum efficiency for users in set Uj as rj = log2 (1 + γ)Pj .

rj is the average achieved throughput if users in outage are assumed to not transmit at all, while users that are not in outage transmit at log2 (1 + γ) bps/Hz. The cell load for a BS is defined as follows. On the shared resources, we use N1 to denote the number of U1 users within a tagged macro-cell, and let N2 be the number of U2 users in a small-cell. In Case 1, macro BSs are allowed to transmit on the unshared resources. The load of a macro BS on the unshared resources, consisting of UD1 users, is denoted as ND1 . In Case 2, the number of UD2 users served by a small-cell BS on the unshared resources is ND2 . The mean cell load averaged ¯ j (j ∈ {1, 2, D1, D2}). over the entire network is denoted by N According to [8] and [15],  0 if Aj = 0 ¯j = (16) N 1 + 1.28λu A otherwise, λm(j)

j



1 if j ∈ {1, D1} . 2 if j ∈ {2, D2} In this paper, we assume that all Uj users associated with a particular BS are allocated equal resources, which can be achieved by round-robin scheduling in the time-sharing resource allocation scheme. Then, we characterize the average user rate in a similar manner to what was done in [16], [18], [19]. For a typical user in set Uj (j ∈ {1, 2, D1, D2}) and given SIR target γ, the user’s average throughput is defined as where we used the mapping: m(j) =

Wj rj Wj log2 (1 + γ)Pj T¯j = ¯ = , ¯j Nj N

(17)

where W1 = W2 = ηW and WD1 = WD2 = (1 − η)W . Using this average user throughput expression, we do not need to integrate the Laplace functional of a Poisson point process, which leads to the tractability of the optimization problem in this

 α2

λ2  α2 

λ1 [1 + ρ(γ, α)] + λ2 BS1 S1 2 1 + ρ( Bγ1 , α)  α2  λ2 + BS2 S1 2 λ1 P2 =   α2 λ2 [1 + ρ(γ, α)] + λ1 BS2 S1 2 [1 + ρ(γB2 , α)]

2  2    B1 S2 α B2 S2 α λ2 λ2 λ1 + S 1 λ1 + S1 

PD1 =   α2   α2  , λ1 [1 + ρ(γ, α)] + λ2 BS1 S1 2 λ1 [1 + ρ(γ, α)] + λ2 BS2 S1 2

2  2    B1 S2 α B2 S2 α λ2 λ2 λ1 + S 1 λ1 + S1 

PD2 = 2  α   α2  , λ1 + [1 + ρ(γ, α)] λ2 BS1 S1 2 λ1 + [1 + ρ(γ, α)] λ2 BS2 S1 2 

(15)

(10)

(11)

if UD = UD1

(12)

if UD = UD2

(13)

JIA AND LIM: RESOURCE PARTITIONING AND USER ASSOCIATION WITH SLEEP-MODE BASE STATIONS IN HCNs

paper. As discussed in Section II-A, we assumed  that the target rate is R1 for U1 UD1 in Case 1 and U1 UD2 in Case 2, and R2 for U2 . A user’s average throughput in set Uj should be no smaller than the corresponding target rate. Therefore, the following throughput constraints can be applied for set Uj .  ¯j if j ∈ {1, 2} ηW Pj log2 (1 + γ) ≥ Rj N ¯ j , if j ∈ {D1, D2} (1 − η)W Pj log2 (1 + γ) ≥ R1 N (18) ¯ j are given in Lemma 2 and where the expressions of Pj and N (16). Additionally, we can observe from (9) and (10)–(13) that ¯ j is related Aj and Pj all depend on B1 . Also, (16) shows that N with Aj . Thus, both the left and right hand sides of (18) depend on B1 . We acknowledge that there are other metrics to measure throughput. For example, the fifth and the median percentile throughput were used in [15]. However, these performance metrics result in tremendous mathematical complexity. Hence, the resource partitioning and user association strategy that achieves the fifth or the median percentile throughput requirements can only be found numerically. In that case, it is hard to model and analyze the problem. On the contrary, based on the user average rate characterization (17), which has been adopted in [16], [18], and [19], the average throughput constraints (18) used in this paper have the merit of mathematical tractability.

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¯

ND1 is dethe target rate. Finally, ηD1 (B1 ) = 1 − W PD1R1log 2 (1+γ) rived from UD1 user rate requirements. The feasible region F1 is non-empty if and only if the upper bound ηD1 (B1 ) is bigger than the lower bounds η1 (B1 ) and η2 . Since η1 (B1 ) and ηD1 (B1 ) are functions of B1 , a feasible set for B1 can be derived. The feasible values of B1 must satisfy both

ηD1 (B1 ) ≥ η1 (B1 )

(20)

and ηD1 (B1 ) ≥ η2 .

(21)

Let M1 denote the set of B1 values that satisfies ηD1 (B1 ) ≥ η1 (B1 ). By finding the first order derivative of η1 (B1 ), we can 1 (B1 ) ≤ 0 and hence, η1 (B1 ) is a non-increasing function get dηdB 1 of B1 . However, ηD1 (B1 ) is not necessarily monotonic, since dη 1) the sign of D1(B depends on the value of other parameters dB1 (node density, transmit power, SIR threshold, etc.). Thus, a method is proposed as follows to determine the set M1 . Corollary 1: The range of B1 values that satisfies ηD1 (B1 ) ≥ η1 (B1 ) is the set M1 = {B1 |B1 ≥ b1 }, where b1 is given by 1) b1 = ∞, if lim ηD1 (x) < lim η1 (x). x→∞

x→∞

2) b1 = B2 , if lim ηD1 (x) ≥ lim η1 (x). x→B2

x→B2

3) b1 is the single root of ηD1 (x)−η1 (x)=0, if lim ηD1 (x) < x→B2

lim η1 (x) and lim ηD1 (x) ≥ lim η1 (x). x→∞

Proof: Please see Appendix A.  Constraint ηD1 (B1 ) ≥ η2 defines another range of B1 , which

In this section, we derive resource allocation and user association schemes that are used to minimize power consumption and maximize coverage probability. It is possible to shut down (put to sleep) macro BSs or small-cell BSs on the unshared resources. In this section, we will first formulate and solve the optimization problems for the two cases separately. Then we give a discussion of how to determine which of the two cases should be used.

2

is denoted by M2 . To find M2 , we first let x = B1α . Then, ηD1 (B1 ) ≥ η2 can be rewritten in terms of x using Lemma 1, Lemma 2 and (16). As the derivation is tedious, we only give the final result as follows: ux2 + vx + w ≤ 0,

(22)

where the coefficients u, v and w are  2  p2 λ22 (1 − η2 ) λ1 + λ2 pB2α  u = p2 λ22 + 2 − 2  λ1 + λ2 pB2α C1 λ1 (1 + ρ(γ, α)) + λ2 pB2α     1.28λu 1.28λu v = pλ1 λ2 (2 + ρ(γ, α)) 1 + − 2 λ1 λ1 + λ2 pB2α  2  2pλ1 λ2 (1 − η2 ) λ1 + λ2 pB2α  − 2  C1 λ1 (1 + ρ(γ, α)) + λ2 pB2α    1.28λu 2 w = (1 + ρ(γ, α)) 1+ λ1 − 1.28λu λ1 2 λ1 + λ2 pB2α  2  λ21 (1 − η2 ) λ1 + λ2 pB2α  − 2 , C1 λ1 (1 + ρ(γ, α)) + λ2 pB2α   α2 where C1 = W logR1(1+γ) and p = SS21 . 1.28p2 λ22 λu

A. Case 1: Unshared Resources Allocated to Macro BSs By optimizing over the resource sharing fraction η and user association threshold B1 , we can reduce the network power consumption Et1 P and improve user coverage Pc . The values of η and B1 should satisfy the minimum throughput constraints expressed in (18). Hence, a feasible (η, B1 ) set F1 can be described by ¯ 1, F1 = {(η, B1 ) s.t. ηW P1 log2 (1 + γ) ≥ R1 N ¯ 2, ηW P2 log2 (1 + γ) ≥ R2 N ¯ D1 , (1 − η)W PD1 log2 (1 + γ) ≥ R1 N B1 > B 2 } .

x→∞

x→B2

IV. P OWER M INIMIZATION AND C OVERAGE I MPROVEMENT

(19)

We can rewrite F1 as {(η, B1 )| max{η1 (B1 ), η2 } ≤ η ≤ ¯ 1 N1 ηD1 (B1 ), B1 > B2 }, where η1 (B1 ) = W P1 Rlog is the 2 (1+γ) minimum resource percentage required to serve U1 users in ¯ 2 N2 each cell. Similarly, η2 = W P2 Rlog is the minimum frac2 (1+γ) tion of resources that should be allocated to U2 users to achieve

2

Let the solution set of x for (22) be X . The following results can be used to determine X .

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Corollary 2: The set X is determined according to the sign of u as follows. 2 Otherwise, X = 1) u  > 0: If√ v − 4uw < 0, X√ = ∅.  2 −v− v −4uw −v+ v 2 −4uw . x: ≤x≤ 2u 2u 2) u < 0: If v 2 − 4uw < 0, X =R (R is the set of√ real numbers). Otherwise, X =   √ v 2 −4uw v 2 −4uw . x : x ≤ −v− 2u , x ≥ −v+ 2u   w 3) u  = 0: If vw> 0, X = x : x ≤ − v . If v < 0, X = x : x ≥ −v . Since the expressions of u, v and w are known, X for specific parameter settings can be derived using Corollary 2. 2

Then using the relation x = B1α , we obtain the feasible set M2 for constraint (21) from X . Thefeasible set of B1 satisfying both (20) and (21) is finally M1 M2 . With the feasible sets of η and B1 determined, the energy reduction problem can be stated as (P1a) :

Et1 P

minimize (η,B1 )∈F1

Since Et1 P = λ1 [ama S1 W +bma ]+ηλ2 [asm S2 W + bsm ] increases with η, problem (P1a) is solved at the minimum η in F1 , i.e., η ∗ = max{η1 (B∗1 ), η2 }, where (η ∗ , B∗1 ) is the globally optimal solution of (P1a). Using the expressions of Aj and Pj (j ∈ {1, 2, D1}), the following results are derived: Lemma 3: When UEs in UD are served by the macro BSs, η ∗ and B∗1 are given by the following expressions. For simplicity, we denote max  (x) by b2 . M2

x∈M1



¯

2 N2 1) If M1 M2 = ∅ and b2 < ∞, η ∗ = η2 = W P2 Rlog 2 (1+γ)  and B∗1 can take any value in the set M1 M2 that satisfies η1 (B∗1 ) ≤ η2 . In other words, Et1 P is a constant for all values of B1 in this range. The optimal power consumption per unit area is Et1∗ P = λ1 [ama S1 W + bma ] + η2 λ2 [a sm S2 W + bsm ]. 2) If M1 M2 = ∅ and b2 = ∞, macro-and small-cell BSs occupy unshared resources, i.e., a fully unshared scheme ¯2 R2 N is used. The optimal B∗1 = ∞ and η ∗ = W P log 2 (1+γ)  B S  α2 2 2 λ1 + S λ2 1 with P =  B S  2 . The minimum power

λ1 +[1+ρ(γ,α)]λ2

2 2 S1

(P1b) :

maximize (η,B1 )∈F1

Pc .

From (14), the overall coverage probability Pc only depends on B1 and is a non-decreasing function of B1 . As a result, to solve (P1b), we simply select the maximum value of B1 from its feasible range. Based on the discussion in the proof of Lemma 3, the optimum η for (P1a) can also be achieved when B1 takes  the biggest value in its feasible set M1 M2 . Thus, we can simultaneously minimize the network power consumption and maximize UE coverage probability for Case 1 (using macro BSs to serve UEs in UD ). When resource partitioning is fea∗ max sible, the optimal B∗1 is B∗1 =  (x) and η is obtained x∈M1

M2

using Lemma 3.

α

consumption per unit area in this case is Et1∗ P = (1 − η ∗ )λ1 [ama S1 W + bma ] + η ∗ λ2 [asm S2 W + bsm ]. 3) If M1 M2 = ∅, resource partitioning is infeasible. In this case, if throughput constraints in sets U1 and U2 can be satisfied with η = 1 and B1 = B2 , then the fully shared scheme is feasible and Et1∗ P = λ1 [ama S1 W + bma ] + λ2 [asm S2 W + bsm ]. Otherwise, the network cannot support the given rate requirement.  Proof: Firstly, we consider the case M1 M2 = ∅. According to the discussion above Corollary 1, η1 (B1 ) decreases as B1 increases. Thus, η ∗ is the larger of η1 (b2 ) and η2 , where max b2 =  (B1 ). From Corollary 1, M1 = {B1 |B1 ≥ B1 ∈M1

η1 (b2 ). Thus, it can be shown that η1 (b2 ) ≤ η2 for all b2 < ∞ and the minimum η therefore equals η2 . Secondly, if b2 = ∞ is feasible, all the UEs with required rate R1 can be served by macro BSs on the unshared resources (i.e., B1 = ∞). The two tiers of BSs will not cause inter-tier interference to each other. For small-cell BSs, the coverage probability P is derived in [18] and hence the minimum required resource fraction η ∗ equals ¯2 R2 N W P log2 (1+γ) . In that case, a fully unshared allocation scheme is used to achieve a minimum power consumption Et1∗ P = (1 − η ∗ )λ1 [ama S1 W + bma ] + η ∗ λ2 [asm S2 W + bsm ]. Thirdly,  it is possible that no feasible region exists for B1 , i.e., M1 M2 = ∅. In that scenario, resource partitioning cannot be applied on the network with the given parameter settings. Instead, macroand small-cell BSs should adopt the fully shared scheme to allocate the resources.  According to our user rate definition, the average user P throughput T¯j is approximated by N¯jj Wj log2 (1 + γ), where Pj is the average probability of set Uj users’ SIR exceeding the target value γ, i.e. the event that these users are served. Additionally, as can be seen from Lemma 3, when the optimal η ∗ = η2 , B∗1 takes any value in the set M1 M2 that satisfies η1 (B∗1 ) ≤ η2 . Thus, there exist more than one B∗1 values that solve Problem (P1a). By further taking coverage maximization into account we can now determine a unique optimal value of B1 . The coverage maximization problem is

M2

b1 }. Hence, the value of b2 is determined by constraint (21). We can conclude that ηD1 (b2 ) = η2 , when b2 < ∞. In addition, quantity b2 also satisfies constraint (20) where ηD1 (b2 ) ≥

B. Case 2: Unshared Resources Allocated to Small-Cell BSs Similar to the analysis for Case 1, a feasible (η, B1 ) set F2 can be determined using the minimum throughput constraints for UEs in U1 , U2 and UD2 . Since small-cell BSs are activated on the unshared resources, we have the throughput constraint ¯ D1 in (19) replaced by (1 − (1 − η)W PD1 log2 (1 + γ) ≥ R1 N ¯ η)W PD2 log2 (1 + γ) ≥ R1 ND2 . The throughput constraints for UEs in U1 and U2 remain the same as those in (19). Thus, the feasible set F2 is derived as ¯ 1, F2 = {(η, B1 ) s.t. ηW P1 log2 (1 + γ) ≥ R1 N ¯ 2, ηW P2 log2 (1 + γ) ≥ R2 N ¯ D2 , (1 − η)W PD2 log2 (1 + γ) ≥ R1 N B1 > B 2 } ,

(23)

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The set F2 is re-expressed as {(η, B1 )| max{η1 (B1 ), η2 } ≤ ¯ 1 N1 η ≤ ηD2 (B1 ), B1 > B2 } with η1 (B1 ) = W P1 Rlog (1+γ) , η2 = 2

¯2 R2 N W P2 log2 (1+γ)

¯ D2 R1 N W PD2 log2 (1+γ) .

The feasi-

ηD2 (B1 ) ≥ η1 (B1 ), and ηD2 (B1 ) ≥ η2 .

(24)

and ηD2 (B1 ) = 1 − ble set of B1 is determined by

The procedure for finding the feasible set is similar to that in Case 1. From constraints in (24), two sets M1 and M2 can be respectively found. The η upper bound ηD2 is a monotonically decreasing function of B1 , unlike in Case 1 where the monotonicity of ηD1 depends on specific parameter settings. Hence, finding M1 and M2 is much easier for Case 2. Specifically, M1 = {b1 ≤ B1 } and M2 = {B1 ≤ b2 }, where b1 can be obtained by applying Corollary 1 with minor modifications and b2 is given as 1) b2 = 0, if lim ηD2 (x) < η2 . x→B2

2) b2 = ∞, if lim ηD2 (x) > η2 . x→∞

3) Otherwise, the value of b2 is obtained by solving ηD2 (x) − η2 = 0. The feasible range of B1 is then M1 ∩ M2 = {b1 ≤ B1 ≤ b2 }. Similarly, the energy reduction and coverage improvement problems for Case 2 can be stated as (P2a) :

minimize (η,B1 )∈F2

Et2 P,

(P2b) :

maximize (η,B1 )∈F2

Pc .

The optimal (η ∗ , B∗1 ) for (P2a) and (P2b) are denoted as (ηa∗ , B∗1a ) and (ηb∗ , B∗1b ), respectively. As discussed for (P1a), solving (P2a) is equivalent to minimizing the term max  (η1 (B1 ), η2 ). The optimal solution B1 ∈M1

M2

for (P2a) is given in the following lemma. Due to its similarity to Lemmas 3, we omit the proof of it. Lemma 4: When UEs in UD are served by the small-cell BSs, the optimal ηa∗ and B∗1a for (P2a) are found from the intersection of feasible sets M1 and M2 .  1) If M1 M2 = ∅ and b2 < ∞, the optimal ηa∗ for ∗ (P2a) equals any value in the   η2 , and B1a can take  set M1 M2 that satisfies η1 (B∗1a ) ≤ η2 . The optimal power consumption per unit area is calculated as Et2∗ P = η2 λ1 [a ma S1 W + bma ] + λ2 [asm S2 W + bsm ]. 2) If M1 M2 = ∅ and b2 = ∞, all the UEs can be served by small-cell BSs, i.e. turning down macro BSs on all resources, Et2∗ P = λ2 [asm S2 W + bsm ].  and   3) If M1 M2 = ∅, resource partitioning is infeasible. In that case, if the fully shared scheme is feasible, η = 1 and Et2∗ P = λ1 [ama S1 W + bma ] + λ2 [asm S2 W + bsm ]. Otherwise, the network cannot support the given rate requirement. Optimizing coverage probability is not as straightforward in Case 2 as in Case 1. Two results are given below to solve (P2b). Firstly, when all the users are served by small-cell BSs, the macro tier can be completely muted on all the resources. Then the overall coverage probability is denoted as P∞ . Referring to the derivation of coverage probability for homogeneous 1 . The second networks in [20], we can obtain P∞ = 1+ρ(γ,α) result is summarized in the following lemma.

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Lemma 5: With resource partitioning adopted, the overall

coverage probability is calculated using (14) as Pc = Pj Aj . If there exists a Bc that satisfies j∈{1,2,D2} 1  2 2 λ1 + (1 + ρ(γ, α)) λ2 pBcα Bc /(γ + Bc ) =  2 , (25) 2  λ1 (1 + ρ(γ, α)) + λ2 pBcα 1 + ρ Bγc , α the coverage probability Pc is a unimodal function with its maximum value achieved at B1 = Bc . Otherwise, Pc is monotonic in B1 . Proof: Please refer to Appendix B.  Based

on Lemma 5, the optimal B1 = b0 that maximizes Pc = Pj Aj can be found within the feasible range j∈{1,2,D2}

b1 ≤ B1 ≤ b2 . The value of b0 is determined as 1) If Bc ≤ b1 , b0 = b1 . 2) If b1 < Bc ≤ b2 , b0 = Bc . 3) If Bc > b2 , b0 = b2 . 4) If Bc does not exist, b0 = arg max Pc . b2

{b1 ,b2 }

Note that when = ∞, the network becomes homogeneous and there is a sudden change from Pc to P∞ . Therefore, when b2 = ∞ the optimal B∗1b for (P2b) is selected from b0 and ∞ that maximizes Pc . Finding the optimal B∗1b and ηb∗ for (P2b) is summarized in the following lemma. Lemma6: When UEs in UD are served by the small-cell BSs and M1 M2 = ∅, the optimal ηb∗ and B∗1b for (P2b) are 1) B∗1b = b0 and ηb∗ = max{η1 (b0 ), η2 }, when b2 < ∞. 2) B∗1b = b0 and ηb∗ = max{η1 (b0 ), η2 }, when b2 = ∞ and Pc |B1 =b0 ≥ P∞ . 3) B∗1b = ∞ and the network is homogeneous consisting of only small-cell BSs, when b2 = ∞ and Pc |B1 =b0 < P∞ . C. Additional Comments In the above two subsections, we showed how to find the optimal user association for load adaptation and resource allocation schemes in Case 1 and Case 2. According to the discussion for Case 1, the minimum power consumption per unit area and maximum overall network coverage probability can be achieved simultaneously, where B∗1 = max  (x) x∈M1

M2

and η ∗ is obtained using Lemma 3. In Case 2, however, the optimum power consumption and coverage probability may be achieved at different (η, B1 ) pairs. In other words, problems (P2a) and (P2b) do not always have a common solution. The optimal (ηa∗ , B∗1a ) for (P2a) and (ηb∗ , B∗1b ) for (P1b) are given in Lemmas 4 and 6, respectively. Whether to assign the unshared resources to the macro BSs (Case 1) or to the small-cell BSs (Case 2) thus depends on the relative importance of power consumption and network coverage. To find the optimal (η, B1 ) that minimizes network power consumption, we can first solve (P1a) and (P2a) separately and then choose the smaller optimal power. Similarly, for maximizing network coverage, we can find the optimal coverage probabilities derived from (P1b) and (P2b) and then choose the better one.

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Fig. 2. Effects of B1 on η and overall coverage probability Pc with SIR threshold γ = 0.5. Top: Case 1, bottom: Case 2. The thick horizontal lines in the figures indicate the feasible ranges of B1 . (a) Case 1: η bounds v.s. B1 . (b) Case 1: Pc v.s. B1 . (c) Case 2: η v.s. B1 . (d) Case 2: Pc v.s. B1 .

We can extend the analysis to a more general case, where the unshared resources allocated to user set UD are further divided into two parts. The macro BSs could have exclusive access to one part of the unshared resources, while the small-cell BSs have exclusive access to the other. Macro- and small-cell BSs can jointly access the shared resources as before. The problems discussed in the above sections are special cases of this general model. According to Lemmas 3 and 4, it can be concluded that when resource partitioning is feasible the optimal resource sharing factor η equals η2 . Note that the value of η2 is the percentage of resources allocated to UEs in U2 to achieve their ¯ 2 N2 below (19) target rate. As the expression η2 = W P2 Rlog 2 (1+γ) shows, η2 only depends on B2 . Similarly, for the general case, the optimal η for feasible resource partitioning only depends on B2 . Therefore, with fixed B2 , we only need to consider the problem of resource allocation and user association within the user set UD . Since there is no inter-tier interference in UD , this problem can be solved using the method proposed in [18] with minor changes. V. N UMERICAL R ESULTS In this section, numerical tests are conducted to verify the analytical results derived in the above section. The bandwidth and BS transmit power spectral density are set as follows: W =

10 MHz, S1 = 2 × 10−3 mW/Hz, and S2 = 1 × 10−4 mW/Hz. According to this parameter setting, the transmit power of macro BS is upper bounded by 20 W and the transmit power of small-cell BS is no larger than 100 mW, which are valid assumptions according to [17]. In simulation, the assumed path loss model has parameters α = 3.8 and L0 = −30 dB. These parameters have been shown to model the practical path loss environment well [23]. According to our resource partitioning and user allocation policy, macro and small-cell BSs use the shared η fraction of resources to serve users in U1 and U2 , respectively. When Case 1 is considered, small-cell BSs are switched to sleep mode on the unshared resources and only macro BSs transmit to the UD1 users. On the other hand, in Case 2, users who have been allocated the unshared resources are served by small-cell BSs. In this scenario, macro BSs are muted on the unshared resources. The effects of association bias B1 on the size of user set Uj (j ∈ {1, 2, D1, D2}) can be found in Section II-A. In Fig. 2, user density λu and base station density λ1 , λ2 15 2 are set as λu = 15λ1 = 3λ2 = 500 2 π units/m . These settings mean that macro BSs have on average 500 m coverage radius, and an average of 4 small cells and 20 UEs are within the coverage of each macro BS. The value of B2 is fixed at 1 (0 dB). The SIR threshold γ = 0.5 and small cells need at least 30% of the resources to support the required rate for UEs in U2 (i.e.,

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Fig. 3. Effects of B1 on η and overall coverage probability Pc with SIR threshold γ = 10. Top: Case 1, bottom: Case 2. The thick horizontal lines in the figures indicate the feasible ranges of B1 . (a) Case 1: η bounds v.s. B1 . (b) Case 1: Pc v.s. B1 . (c) Case 2: η v.s. B1 . (d) Case 2: Pc v.s. B1 . ¯

2 N2 η2 = 0.3, where η2 = W P2 Rlog has been defined below 2 (1+γ) (19)). In Fig. 2(a), we show how the boundaries of η (η1 (B1 ), ηD1 (B1 ) and η2 ) change with B1 in Case 1. The upper bound ηD1 (B1 ) and the two lower bounds η1 (B1 ) and η2 are derived from the feasible (η, B1 ) set F1 in (19). According to (20) and (21), the feasible range of B1 is the one that makes F1 nonempty. By comparing the η bounds in Fig. 2(a) we can get the feasible B1 range, which is [3.97,16.93] dB, and we highlight this interval in bold on the x-axis for clarity. With the feasible B1 set determined, we are able to apply Lemma 3 to find the minimum power consumption for Case 1. From Lemma 3, the minimum η is selected to minimize Et1 P . According to Fig. 2(a), η1 (B1 ) > η2 for B1 < 12.96 dB (indicated by pentagram) and η1 (B1 ) ≤ η2 otherwise. Since η2 is independent of B1 , the minimum power consumption is achieved at η = η2 with B1 ∈ [12.96, 16.93] dB. From the discussion under Lemma 3, we know that the overall coverage probability Pc for Case 1 is a non-decreasing function of B1 , as shown in Fig. 2(b). It can be observed from Fig. 2(b) that the optimum overall coverage probability is achieved at the maximum value of B1 within its feasible range determined in Fig. 2(a). A similar analysis is adopted for Case 2 to determine the feasible range of B1 and the maximum achievable coverage probability as shown in Fig. 2(c) and (d).

From Fig. 2(c), we can know that the feasible interval for B1 is [1.38, ∞) dB. By setting B1 to infinity, we can have all macro BSs muted and all UEs served by small cells so that the network is homogeneous and consumes the least amount of energy. However, as shown in Fig. 2(d), the overall coverage probability Pc for Case 2 is not an increasing function of B1 . When B1 = ∞, the coverage probability is denoted as P∞ , which is smaller than the maximum Pc achieved at Bc (5.86 dB). Therefore, a tradeoff between reducing energy consumption and improving coverage probability exists in Case 2. Fig. 3 is obtained by changing γ from  0.5 to 10. Similarly, the feasible range of B1 is [7.73, 10.31] [17, ∞] dB for Case 1 and [2.57,16.16] dB for Case 2. From Fig. 3(a), we note that B1 can be set to ∞ for Case 1, which means fully unshared resource allocation is adopted and minimum amount of power is consumed. With B1 = ∞, Fig. 3(b) shows that the overall coverage probability achieves the maximum value at B1 = ∞. Specifically, in Fig. 3(b) we denote the value of Pc at B1 = ∞ as P∞ . Using the analysis in [18], we have  B S  α2 2 2 λ2 S1 λ1 P∞ = B S  2 +  B S  2 . In λ1 [1+ρ(γ,α)]+λ2

2 2 S1

α

λ1 +[1+ρ(γ,α)]λ2

2 2 S1

α

Fig. 3(c), the feasible B1 is upper bounded by 16.6 dB for Case 2. In that scenario, we cannot mute all the macro BSs. As discussed in Fig. 2 and Lemma 4, we can similarly draw the

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Fig. 4. Effects of B2 on (a) network power consumption and (b) coverage performance. (a) Area power comsumption v.s. B2 . (b) Overall coverage probability v.s. B2 .

Fig. 5. Effects of base station densities on (a) network power consumption and (b) coverage performance. (a) Area power comsumption v.s. base station density ratio. (b) Overall coverage probability v.s. base station density ratio.

conclusion that the minimum power consumption for Case 2 is achieved when B1 ∈ F2 and η1 (B1 ) ≤ η2 , which is reflected in Fig. 3(c) as the highlighted range B1 ∈ [8.69, 16.16] dB. Furthermore, the value of Bc (8.9 dB) that maximizes Pc in Fig. 3(d) also lies within this interval. Thus, for the parameter setting in Fig. 3, the minimum power consumption and maximum coverage probability will be achieved simultaneously for both Case 1 and Case 2. By comparing Fig. 2 and Fig. 3, we can conclude that the feasible ranges of B1 depend on the values of other parameters (in this case the parameter is γ). It also supports our claim in Section IV-C that simultaneously achieving the minimum power consumption and maximum coverage probability is always possible for Case 1 but cannot be guaranteed for Case 2. According to Lemma 1 and Lemma 2, the population of user set Uj has a great effect on our decision of selecting the optimal resource allocation and user association schemes. Since B2 and base station density ratio λλ12 are the two key factors, we show how the optimal scheme varies with B2 and λλ21 in

Figs. 4 and 5, respectively. In those figures, we set γ = 0.5, 1 2 λ1 = 500π 2 units/m and the required rates are R1 = 300 kbps and R2 = 900 kbps. The value of B2 varies from −2 dB to 4 dB and λλ21 = 5 in Fig. 4. On the other hand, B2 is fixed at 1 and λλ21 ∈ [3.5, 10] in Fig. 5. Recalling the four optimization problems (P1a), (P1b), (P2a) and (P2b) in Section IV, different optimal (η, B1 ) pairs are found accordingly. Each optimal (η, B1 ) pair individually results in its power consumption and coverage probability. From Lemma 3 and 4, there may exist more than one value of B1 that minimizes network power for (P1a) and (P2a). Thus, for each of these two problems we select the B1 from the optimal values, which achieves the highest coverage probability. The power consumptions and coverage probabilities of problem (P1a), (P1b), (P2a), and (P2b) are compared with respect to B2 in Fig. 4. According to Section II-A, B2 controls the population in user classes with required rates R1 and R2 . As B2 increases, more users will be served at rate R2 by smallcell BSs. Hence, the size of set U2 grows, which is reflected

JIA AND LIM: RESOURCE PARTITIONING AND USER ASSOCIATION WITH SLEEP-MODE BASE STATIONS IN HCNs

by a larger value of A2 . On the other hand, due to the limited transmit power of small-cell BSs and path loss effect, the average distance of UEs in U2 increases with B2 , resulting in a decrease in P2 . To achieve the throughput requirements, the value of η is raised to allocate more resources to UEs in U2 . Since the network power consumption grows with η, the curves in Fig. 4(a) all increase with B2 . From the previous discussion we know that power minimization and coverage maximization can be achieved at the same (η, B1 ) pair in Case 1. Therefore, the power consumption in Fig. 4(a) and coverage probability in Fig. 4(b) are the same for the crossed- and x-labeled lines. In these two curves, the fraction η of the resources shared with small-cell BSs increases with B2 . For Case 2, however, the power minimization and coverage maximization are not guaranteed to be achieved simultaneously. In Fig. 4(a), when B2 < 1.19 dB the optimal energy saving scheme is obtained by setting B1 = ∞, which means using only small-cell BSs to serve all the UEs. However, the coverage optimal strategy is to allow macro BSs awake on some shared fraction of resources (the coverage performance can be seen in Fig. 4(b)). For B2 exceeds 1.19 dB, to serve UEs at required rate R2 consumes too many resources that makes muting all macro BSs impossible. Thus, macro BSs are awakened on the shared resources and a sudden change is observed at B2 = 1.19 dB for the Case 2 minimum power curve and maximum coverage probability curve in Fig. 4(a) and (b) respectively. In Fig. 5, we show how power consumption and coverage probability change with the number of small-cell BSs. The analyses for Fig. 5(a) and (b) are similar to those for Fig. 4(a) and (b). Due to the limited coverage area of small-cell BSs, when the ratio of λλ21 is low, only a small group of UEs associate with small-cell BSs and therefore resource sharing is required. However, when the number of small cells is sufficiently high, we can either adopt fully unshared resource allocation in Case 1 ( λλ12 > 7.99) or mute all macro BSs in Case 2 ( λλ12 > 4.75). As observed from Figs. 4 and 5, choosing Case 1 will generally consume higher energy than choosing Case 2 due to the lower energy consumption of small cells. However, in terms of coverage maximization, Case 1 may outperform Case 2 at most of the points in Figs. 4 and 5. This is because the signal received by small cell edge users is low in strength and thus reduces the coverage probability. Although a stochastic geometry model that assumes purely random network deployment is adopted in this paper, the derived results still give practical guidelines for real network designs. In practical network designs, macro base stations could gather the load information of the small cells within their coverage areas and then determine the optimal association bias B∗1 and the resource partitioning factor η ∗ using the lemmas proposed in Section IV for both Case 1 and Case 2. If Case 1 gives a better performance in terms of power consumption or user coverage than Case 2 does, a 1 − η ∗ fraction of the resources is used by only macro BSs, and the small-cell BSs are notified to go to sleep mode on these resources. Otherwise, the macro BSs are muted on the unshared resources. The association biases are then transmitted to the users through the control channel so that each user can determine which tier to associate with and which set of resource elements will be used.

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VI. C ONCLUSION In this work, we provided a theoretical framework to study resource partitioning and load adaptation strategies in two-tier heterogeneous cellular networks. Using stochastic geometry, closed-form expressions of coverage probability and average UE population were obtained for each user set. The minimization of average BS power consumption and maximization of overall coverage probability with given throughput constraints were formulated separately. Optimal solutions were found by finding feasible sets of association bias factor B1 and resource partition fraction η. Numerical results verified that the proposed optimal resource allocation strategy helps in reducing network power consumption and improving user coverage. Future work will include uplink performance analysis and optimization over other objective functions jointly taking power, coverage probability and throughput into consideration.

A PPENDIX A P ROOF OF C OROLLARY 1 Using the results of Lemma 1, Lemma 2 and (16), the values of ηD1 (B1 ) and η1 (B1 ) at B1 → B2 and B1 → ∞ are ⎛

  2 ⎞2 B2 S2 α [1 + ρ(γ, α)] + λ λ 2 S1 ⎜ 1 ⎟ lim ηD1 (x) =1−C1 ⎝ ⎠ 2   x→B2 B2 S2 α λ1 + S1 λ2 ⎛

⎞

⎜ lim η1 (x) =C1 ⎝1 +

x→B2

λ1 +

1.28λu  α2  B2 S2 S1

⎟ ⎠ λ2

  α2  1 + ρ( Bγ2 , α) λ1 [1 + ρ(γ, α)] + λ2 BS2 S1 2 ×  α2  λ1 + BS2 S1 2 λ2 ⎛

⎞

⎜ lim ηD1 (x) =1 − C1 ⎝1 +

x→∞

λ1 +

1.28λu  α2  B2 S2 S1

⎟ ⎠ λ2

  α2 λ1 [1 + ρ(γ, α)] + λ2 BS2 S1 2 ×  α2  λ1 + BS2 S1 2 λ2 lim η1 (x) =C1 ,

x→∞

where C1 = W logR1(1+γ) . 2 From the above equations, it can be easily obtained that lim

x→∞

(ηD1 (x) − η1 (x)) ≥ lim (ηD1 (x) − η1 (x)). Hence, we can x→B2

conclude that lim ηD1 (x) > lim η1 (x) is always true if lim x→∞

x→∞

x→B2

ηD1 (x) > lim η1 (x) holds. On the other hand, we have lim x→B2

x→B2

ηD1 (x) < limx→B2 η1 (x) when lim ηD1 (x) < lim η1 (x) is x→∞ x→∞ satisfied.

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¯    γ  1.28pλu λ2  N1 λ (1 + ρ(γ, α)) + pλ x 1 + ρ , α =− 1 2 P1 (λ1 + pλ2 x)3 xα/2      γ 1

 pλ λ ρ(γ, α) + ρ  γ , α + 2 2 + p λ x 2ρ xα/2 ,α + 1 2 α/2 α/2 2 1.28λu x 1+γx − 1+ 1 + pλ2 x (λ1 + pλ2 x)2   2

¯  α λ + λ pB 1.28pλ λ (λ1 (1 + ρ(γ, α)) + pλ2 x) u 2 1 2 2 −d ND1   =− 2 dx PD1 (λ + pλ x)3 λ (1 + ρ(γ, α)) + λ pB α d dx

1

2

 +

1+

1

2

1.28λu λ1 + λ2 pB2

2 α

−

1.28λu λ1 + pλ2 x



 (26)

2

  2 pλ1 λ2 λ1 + λ2 pB2 α ρ(γ, α) 2

(λ1 + pλ2 x)2 λ1 (1 + ρ(γ, α)) + λ2 pB2 α

dPc dPc dx = dB1 dx dB1 ⎛ ⎜ =⎝

1 1+γxα/2

,

(27)

⎞ α/2

α/2

λ1 λ2 p λ1 λ2 px /(γ + x )   2 −  (λ1 + (1 + ρ(γ, α)) λ2 px) λ1 (1 + ρ(γ, α)) + λ2 px 1 + ρ

γ α x2

,α

⎟ dx , 2 ⎠ dB1

(28)

2

Introducing a new variable x = B1α , the fist order derivative of ηD1 and η1 with respect to B1 can be derived as

¯  dη1 (B1 ) dx d N R1 1 = dB1 W log2 (1 + γ) dx P1 dB1

¯  dx dηD1 (B1 ) −d N R1 D1 = . dB1 W log2 (1 + γ) dx PD1 dB1 The relationship between

dη1 (B1 ) (B1 ) and dηD1 dB dB1  canbe stud1  ¯1 ¯ D1 N N d d with − dx dx P1 PD1 , whose

ied by comparing the term expressions are given in (26) and (27) (see equations at top of   α2 the page) where p = SS21 . It can be easily verified from (26)     ¯ ¯ D1 N1 N d d < − for all B1 ≥ B2 . and (27) that dx P1 dx PD1 Given two functions f (x) and g(x), if the first order deriva(x) tive of f (x) is always smaller than g(x), i.e., fdx < g(x) x , then the two functions have at most one intersection point. According to this result, the function ηD1 (B1 ) = η1 (B1 ) has at most one root. Based on the above analysis, when lim ηD1 (x) < lim x→∞

x→∞

η1 (x), we have ηD1 (B1 ) < η1 (B1) for all B1 > B2 . Therefore, the set M1 is empty, which can be interpreted as b1 = ∞. Similarly, when lim ηD1 (x) ≥ lim η1 (x), the constraint x→B2

x→B2

ηD1 (B1 ) ≥ η1 (B1 ) is achieved for all B1 > B2 . Thus, b1 = B2 . The last case in Corollary 1 is lim ηD1 (x) ≤ lim η1 (x) x→B2

x→B2

and lim ηD1 (x) ≥ lim η1 (x). In this scenario, there exists an x→∞

x→∞

intersection point b1 between ηD1 (B1 ) and η1 (B1 ). Once the value of B1 exceeds b1 , the constraint ηD1 (B1 ) ≥ η1 (B1 ) can be satisfied. Hence, the feasible set M1 for this case is M1 = {B1 |B1 ≥ b1 }. Because there exists only one intersection point between ηD1 (B1 ) and η1 (B1 ) with their expressions known, bisection method can be applied to find the value of b1 .

A PPENDIX B P ROOF OF L EMMA 5 We firstly calculate the first order derivative of Pc with respect to B1 . For simplicity of notification, a change of variable 2 x = (B1 ) α is applied. Then (28), shown at the top of the page, is derived. dPc dx > 0. Thus, dB = 0 if there exists B1 that lies Note that dB 1 1 within the set Mc , where ⎧ ! ! ⎪ ⎨ ! 1 ! M c = B1 !  2 2 ! ⎪ ⎩ ! λ1 + (1 + ρ(γ, α)) λ2 pB α 1 ⎫ ⎪ ⎬ B1 /(γ + B1 ) =     2 ⎪. 2 ⎭ λ1 (1 + ρ(γ, α)) + λ2 pB1α 1 + ρ Bγ1 , α !

Then we can easily verify that

d 2 Pc ! dB21 !B ∈M 1 c

< 0, which means

set Mc has at most one element. If Mc = ∅, its element is denoted as Bc . From the above analysis, Pc is unimodal in this case and achieves maximum value at Bc . If Mc is empty, Pc is a monotonic function of B1 . R EFERENCES [1] X. Wang, A. V. Vasilakos, M. Chen, Y. Liu, and T. T. Kwon, “A survey of green mobile networks: Opportunities and challenges,” Mobile Netw. Appl., vol. 17, no. 1, pp. 4–20, Feb. 2012. [2] J. Rao and A. Fapojuwo, “A survey of energy efficient resource management techniques for multicell cellular networks,” IEEE Commun. Surveys Tuts., vol. 16, no. 1, pp. 154–180, 1st Quart. 2014. [3] D. Calin, H. Claussen, and H. Uzunalioglu, “On femto deployment architectures and macrocell offloading benefits in joint macro-femto deployments,” IEEE Commun. Mag., vol. 48, no. 1, pp. 26–32, Jan. 2010. [4] M. Alouini and A. Goldsmith, “Area spectral efficiency of cellular mobile radio systems,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1047–1066, May 1999.

JIA AND LIM: RESOURCE PARTITIONING AND USER ASSOCIATION WITH SLEEP-MODE BASE STATIONS IN HCNs

[5] Z. Niu, Y. Wu, J. Gong, and Z. Yang, “Cell zooming for cost-efficient green cellular networks,” IEEE Commun. Mag., vol. 48, no. 11, pp. 74– 79, Nov. 2010. [6] S. Luo, R. Zhang, and T. J. Lim, “Optimal save-then-transmit protocol for energy harvesting wireless transmitters,” IEEE Trans. Wireless Commun., vol. 12, no. 3, pp. 1196–1207, Mar. 2013. [7] S. Cho and W. Choi, “Energy-efficient repulsive cell activation for heterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 31, no. 5, pp. 870–882, May 2013. [8] S. Singh, H. S. Dhillon, and J. G. Andrews, “Offloading in heterogeneous networks: Modeling, analysis, design insights,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 2484–2497, 5 2013. [9] W. C. Cheung, T. Q. S. Quek, and M. Kountouris, “Throughput optimization, spectrum allocation, access control in two-tier femtocell networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 3, pp. 561–574, Apr. 2012. [10] D. Cao, S. Zhou, and Z. Niu, “Improving the energy efficiency of two-tier heterogeneous cellular networks through partial spectrum reuse,” IEEE Trans. Wireless Commun., vol. 12, no. 8, pp. 4129–4141, Aug. 2013. [11] D. Lopez-Perez et al., “Enhanced intercell interference coordination challenges in heterogeneous networks,” IEEE Wireless Commun. Mag., vol. 18, no. 3, pp. 22–30, Jun. 2011. [12] T. D. Novlan, R. K. Ganti, A. Ghosh, and J. G. Andrews, “Analytical evaluation of fractional frequency reuse for OFDMA cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 4294–4305, Dec. 2011. [13] http://arxiv.org/abs/1304.2269M. Cierny, H. Wang, R. Wichman, Z. Ding, and C. Wijting, “On number of almost blank subframes in heterogeneous cellular networks,” IEEE Trans. Wireless Commun., vol. 12, no. 10, pp. 5061–5073, Oct. 2013. [Online]. Available: http://arxiv.org/abs/1304. 2269. [14] S. Bashar and Z. Ding, “Admission control and resource allcoation in a heterogeneous OFDMA wireless network,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4200–4210, Aug. 2009. [15] S. Singh and J. G. Andrews, “Joint resource partitioning and offloading in heterogeneous cellular networks,” IEEE Trans. Wireless Commun., vol. 13, no. 2, pp. 888–901, Feb. 2014. [16] H. S. Jo, Y. J. Sang, P. Xia, and J. G. Andrews, “Heterogeneous cellular networks with flexible cell association: A comprehensive downlink SINR analysis,” IEEE Trans. Wireless Commun., vol. 11, no. 10, pp. 3484–3495, Oct. 2012. [17] A. J. Fehske, F. Richter, and G. P. Fettweis, “Energy efficiency improvements through micro sites in cellular mobile radio networks,” in Proc. IEEE GLOBECOM Workshop, 2009, pp. 1–5. [18] W. Bao and B. Liang, “Structured spectrum allocation and user association in heterogeneous cellular networks,” arXiv:1309.7527. [19] Y. Lin and W. Yu, “Optimizing user association and frequency reuse for heterogeneous network under stochastic model,” in Proc. IEEE GLOBECOM Workshop, 2013, pp. 2045–2050. [20] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3122–3134, Nov. 2011. [21] H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, “Modeling and analysis of K-tier downlink heterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 3, pp. 550–560, Apr. 2012. [22] S. Mukherjee and I. Guvenc, “Effects of range expansion and interference coordination on capacity and fairness in heterogeneous networks,” in Conf. Rec. 45th ASILOMAR Conf. Signals, Syst. Comput., Nov. 6–9, 2011, pp. 1855–1859.

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[23] A. Ghosh and R. Ratasuk, Essentials of LTE and LTE-A. Cambridge, U.K.: Cambridge Univ. Press, 2011. [24] T. S. Rappaport, Wireless Communications: Principles and Practice, Second Edition. Upper Saddle River, NJ, USA: Prentice Hall, 2002.

Chenlong Jia (S’14) received the B.Eng. degree in electrical engineering from Shandong University, Shandong, China, in 2011. He is currently working towards the Ph.D. degree in the Electrical and Computer Engineering Department, National University of Singapore, Singapore. His research interests include energy-optimized communication networks, heterogeneous networks, stochastic geometry.

Teng Joon Lim (S’92–M’95–SM’02) received the B.Eng. degree in electrical engineering (with firstclass honours) from the National University of Singapore, Singapore, in 1992, and the Ph.D. degree from the University of Cambridge in 1996. From September 1995 to November 2000, he was a Researcher at the Centre for Wireless Communications, Singapore, one of the predecessors of the Institute for Infocomm Research (I2R). From December 2000 to May 2011, he was Assistant Professor, Associate Professor, then Professor with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto. Since June 2011, he has been a Professor with the Electrical & Computer Engineering Department, National University of Singapores, and currently serves as Deputy Head of the Department for Research and Graduate Programs, having previously served as the Director of the Communications and Networking area. His research interests span many topics within wireless communications, including the Internet of Things, heterogenous networks, energy-optimized communication networks, multi-carrier modulation, MIMO, cooperative diversity, cognitive radio, and random networks, and he has published widely in these areas. He is an Area Editor of the IEEE T RANSACTIONS ON W IRELESS C OM MUNICATIONS , an Associate Editor for IEEE W IRELESS C OMMUNICATIONS L ETTERS, and an Executive Editor for Wiley Transactions on Emerging Telecommunications Technologies (ETT). Previously, he was an Associate Editor for IEEE S IGNAL P ROCESSING L ETTERS and for IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY . He has served as TPC co-chair for various conferences, including the IEEE International Conference on Communications in China (ICCC) 2014, and the IEEE W IRELESS C OMMUNICATIONS and Networking Conference (WCNC) 2014 PHY Track. He has also been TPC chair for the IEEE International Conference on Communication Systems (ICCS) three times, in 2000, 2012 and 2014, and is a regular TPC member in IEEE Globecom, ICC, WCNC and other important international conferences in communications and networking.