- Research
- Open Access
Multichannel analysis of soft frequency reuse and user association in two-tier heterogeneous cellular networks
- Lili Guo^{1},
- Shanya Cong^{1}Email author and
- Zhiguo Sun^{1}
https://doi.org/10.1186/s13638-017-0954-8
© The Author(s) 2017
- Received: 27 October 2016
- Accepted: 27 September 2017
- Published: 17 October 2017
Abstract
The cell range expansion (CRE) is encouraged to be applied in the heterogeneous cellular networks (HCNs), to enhance the capacity by offloading macro users to small cells. However, the enhanced inter-cell interference coordination (eICIC) techniques are supposed to be used for mitigating the strong cross-tier interference suffered by the offloaded users and small cell edge users. To address this, a novel soft frequency reuse (SFR) scheme is adopted in this paper. We analyze multichannel downlink scenarios for the SFR scheme using the tools of stochastic geometry. In consideration of the random resource allocation and practical cell load model, the analytical results of coverage probability and average user rate are derived and validated through Monte Carlo methods. Furthermore, our results can reduce to simple closed-form under reasonable special case for modern urban cellular networks. The main evaluation of the performance in terms of average user rate is presented, and the optimal combination of association bias and parameters of the SFR scheme is also investigated. Numerical results show that the SFR scheme outperforms the frequency resource partitioning (FRP) scheme in any load condition. Moreover, the CRE with SFR scheme can improve the average user rate significantly.
Keywords
- Heterogeneous cellular networks
- Cell range expansion
- Cell load
- eICIC
- SFR
- Stochastic geometry
1 Introduction
Heterogeneous cellular networks (HCNs) are expected to be one of primary technologies for the emerging fifth generation (5G) mobile networks [1, 2], which can be considered as an efficient solution for dealing with the exploding data traffic demands of users. In a HCN, it is proved that the co-channel deployment of low-power base stations (BSs) (also known as small cells such as micro, pico, and femto BSs) with conventional macro BSs can yield the largest sum rate [3].
For a typical network, its user association rule that connects a user to a specific serving BS could substantially affect the network performance. In the existing conventional homogeneous networks, the maximum received signal strength (max-RSS) is widely adopted, which can achieve the anticipated performance. However, the max-RSS user association rule is not suitable for HCNs, since the large difference in transmit powers between small cells (e.g., pico BSs ≈30 dBm) and macro BSs(≈46 dBm) [4], and thereby most of users will be associated with the macro BSs. The unbalanced user load leads to overloaded macro BSs and inefficient resource utilization in small cells. To cope with this problem, the biased user association also known as cell range expansion (CRE) has been proposed [5], wherein the macro users are proactively offloaded to small cells. Nevertheless, the drawback of biased user association is that the offloaded macro users referred to as range-expanded small cell users are liable to experience severe interference from the nearby macro BSs. Naturally, the small cell edge users also are vulnerable users. In this context, the enhanced inter-cell interference coordination (eICIC) techniques [6] are expected to mitigate such strong interference.
1.1 Motivation and related work
Previous works on the CRE in conjunction with suitable eICIC techniques can be divided into two general groups, namely, time-domain strategies [7–13] and frequency-domain strategies [14–18]. One of time-domain strategies is the so-called almost blanking subframes (ABS) introduced in 3GPP Release 10, where macro BSs are periodically muted in order to mitigate interference to offloaded users [7–10]. The ABS can be considered as a resource partitioning in the time domain. It can be seen in [8] that the user throughput can be enhanced through simulations. Using the tools of stochastic geometry, tractable expressions which can give clear insight into the performance gain are derived in [7, 9]. Furthermore, the work [10] analyzes the required number of ABS based on user throughput requirement, but the CRE is not captured. However, a larger association bias is required in a heavily loaded scenario, which causes macro BSs should be shut off about half the time. This is counter-intuitive and unrealistic. Hence, the reduced power subframes (RPS) are encouraged to be applied to address this issue [11–13]. Instead of ABS, the RPS are allocated to macro interior users with lower transmit power. The capacity loss caused by ABS is thereby reduced because the sever interference to the small cell edge users is mitigated. The results in [13] show that the RPS provide better rate coverage than the ABS, with a given association bias.
We focus on the frequency-domain strategies in this paper. In [14, 15], the authors propose a frequency resource partitioning (FRP) scheme, wherein a certain fraction of frequency resources with full transmit power is preserved for offloaded users. Analytical expressions for coverage probability and rate distribution are presented in [14] using the tools of stochastic geometry. The optimal system performance can be achieved through jointly tune association bias and resource partitioning fraction. However, the full-load model that all the BSs are always active is not reasonable, especially for the network with CRE. This is because the assumption of full-load model cannot adequately reflect the enhancement in performance through load balancing. In detail, the interference from a BS is mainly dependent on its load. That is, the probability of a BS becomes the source of interference is directly proportional to its users. Taking such effects into consideration, the authors of [15, 19] propose a more practical cell load model that a BS is active on a given sub-channel only if the sub-channel is allocated for at least one user. In [15], both the coverage probability and average user rate of FRP scheme are dependent on user density and the resource partitioning fraction, apart from association bias. The results in [15] are more practical for design guidelines.
Nevertheless, the drawback of FRP scheme is the sacrifice of the spectral efficiency of macro users. Hence, the work [16] presents an evolved FRP scheme, wherein a certain fraction of frequency resources is allocated to not only offloaded macro users with full transmit power but also macro users with lower transmit power. By means of the transmit power reduction in macro tier, the spectral efficiency of macro users is guaranteed. However, the transmit power reduction is randomly applied to macro users, which probably leads to that the performance of macro users in poor situation gets worse. In addition, the small cell edge users also are vulnerable to interfered by the nearby macro BSs. Similar to the RPS, a novel soft frequency reuse (SFR) scheme proposed by [17, 18, 20] can mitigate the strong cross-tier interference by means of joint resource partitioning and transmit power reduction for macro interior users. The strong cross-tier interference suffered by offloaded macro users and pico edge users is commonly mitigated. More recently, the work [17] presents the spectral efficiency analysis for the proposed SFR scheme based on stochastic geometry. As a result, the system spectral efficiency can be significantly improved. Although the proportional fair resource allocation is adopted in [17], the analytical results cannot adequately capture the variation in performance because of the assumption of full-load model. To the best of our knowledge, none of the earlier works considered the impact of resource allocation and appropriate load model on the SFR based on stochastic geometry, which is fulfilled in this paper.
1.2 Approach and contributions
Motivated by the works in [15, 19], we propose a general and tractable framework to analyze joint SFR, appropriate load model and CRE in a two-tier HCNs. Based on stochastic geometry, the locations of the BSs in each tier are modeled as a two-dimensional homogeneous Poisson Point Process (HPPP) [21–23], which have been proved to be as accurate as the hexagonal grid model. Furthermore, the Fractional Frequency Reuse (FFR) was analyzed using PPPs in order to tackle the strong interference from nearby macro BSs [24, 25]. In these works of [21–25], all analytical results have been verified via Monte Carlo methods. Note that all these works just adopt a simple resource allocation scheme, namely, the entire bandwidth of each BS is time-shared for its users. We consider a multichannel downlink based on orthogonal frequency division multiple access (OFDMA) technique, where each user is only served by one sub-channel. In this paper, we use the metric of average user rate, which can reflect the high spectral efficiency obtained from the proposed SFR scheme.
- 1)
Based on the cell load model, we first derive the coverage probability of the network without SFR scheme, which can be used for the derivation of coverage probability and average rate of the network with SFR scheme.
- 2)
Next, we derive the evaluation for network performance of the proposed SFR scheme is performed in terms of average user rate incorporated with the random resource allocation and cell load. For special case, the expressions of coverage probability are simple closed-form.
- 3)
We compare the proposed SFR scheme with FRP scheme. Moreover, we comprehensively analyze the average user rate under different parameters by varying the association bias, resource partitioning factor, power control factor, and SINR thresholds. Then, we show the impact of the aforementioned parameters on the average user rate and investigate the optimal combination finally.
- 4)
We show that the proposed SFR scheme is promising for improving the average user rate while considering the cell load.
The remainder of the paper is organized as follows: the system model, user association scheme, cell load analysis, and SFR scheme are presented in Section 2. Mathematical coverage probability and average user rate expressions are derived in Section 3. In Section 4, numerical results on the performance evaluation for the proposed SFR scheme are analyzed. Furthermore, the comparison with FRP scheme is presented. Finally, the paper is concluded in Section 5.
2 System model
2.1 Two-tier cellular network model
We consider a two-tier downlink cellular network based on OFDMA technique, i.e., the intra-cell interference is not considered. Without any loss of generality, let the first tier be macro (higher-power BS) tier, while let the second tier be pico (lower-power BS) tier. The locations of the BSs in jth tier are modeled as a two-dimensional HPPP Φ _{ j } with density λ _{ j }. Furthermore, the users are located according to another HPPP Φ _{ u } with density λ _{ u }, which is independent of {Φ _{ j }}_{ j=1,2}. For the co-channel deployment of the network, both the network tiers share the same set of available sub-channels (denoted by C). Then, let |C| denote the total number of available sub-channels. Moreover, every BS in jth tier transmits with same power \(P_{j}^{T}\) and thus, the power per sub-channel of a BS in jth tier is kept constant at \({P_{j}} = {{P_{j}^{T}} \left / {\left | \mathbf{C} \right |}\right.}\). For tractability, the standard power loss propagation model is applied in both tiers with the same path loss exponent α>2. As far as random channel fluctuations, Rayleigh fading with mean 1 (denoted as H _{ x }∼ exp(1)) is applied at each channel. The noise is assumed to be additive with power σ ^{2}. All macro and pico BSs are assumed to be open access in this paper. That is, the number of users served by each BS is unlimited.
2.2 User association
where B≥0 dB is the association bias for tier 2 (pico tier). For simplicity, it is assumed that the association bias for tier 1 (macro tier) is unity in this paper. As the association bias B increases, more macro users will be offloaded to the corresponding pico BSs. Here, we let \({\mathcal {U}_{k}}\) denote the set of users in the kth tier, which satisfies \({\mathcal {U}_{1}} \cup {\mathcal {U}_{2}} = \mathcal {U}\).
where \(\Gamma \left (z \right) = \int _{0}^{\infty } {{t^{z - 1}}{e^{- t}}dt} \) is the standard gamma function. The statistical property of the number of users is crucial for calculating a cell load.
2.3 The main results of network without SFR
Before discussing the proposed SFR scheme, we firstly focus on the network’s SINR distribution without SFR, which is extremely essential for the classification of users. Following the analysis [22, 23], the main results for the biased user association are briefly presented here for the purpose of analytical evaluating the proposed SFR scheme.
2.3.1 Resource allocation and load statistics
For the sake of keeping simplicity and tractability of the PPP model, we adopt a simple random resource allocation scheme for both macro and pico users. That is, each BS, independent of the other BSs, randomly and uniformly selects one sub-channel for each of its users. If the number of users in a serving BS is greater than the number of available sub-channels, the resources can be equally allocated in time-division way. Hence, together with the use of OFDMA, the intra-cell interference can be ignored. With regard to other sophisticated resource allocation schemes like opportunistic and fair resource allocation, they are not analytically tractable and will be taken in consideration for our future work.
Different from the assumptions in [14–17] that all BSs of each tier are always active, we utilize a more practical load model that a BS is active when it has at least one user to serve. When a BS has no user to serve, its corresponding sub-channel set C will be left idle. Furthermore, it is also assumed that each BS has full buffer traffic downlink transmission for its each user. Hence, a BS is the source of interference when the BS must be active and simultaneously use the same sub-channel.
To obtain the SINR statistics on a given sub-channel, [15, 19] have derived the probability that a typical BS of each tier accesses a given sub-channel, which depends on the PMF of the number of users associated with that BS.
Lemma 1
Note that, we can term ρ _{ k } as the load of a typical BS in the kth tier. It also can be viewed as the probability that a typical BS becomes the source of interference because the BSs use the same sub-channel. Therefore, the macro BSs and pico BSs using the same sub-channel will form two independent homogeneous PPPs Ψ _{1} of density λ _{1} ρ _{1} and Ψ _{2} of density λ _{2} ρ _{2}, respectively. In other words, the interfering sets Ψ _{1} and Ψ _{2} are independent thinning of the original PPPs Φ _{1} and Φ _{2}, respectively, with retention probabilities ρ _{1} and ρ _{2}, respectively [ 15 , 19 ].
2.3.2 SINR distribution
where \({I_{x,j}} = {P_{j}}{\sum \nolimits }_{y \in {\Psi _{j}}\backslash {b_{k}}} {{H_{y}}{{\left \| y \right \|}^{- \alpha }}} \) is the cumulative interference from all the loaded BSs in the jth tier (except the user’s serving BS in the kth tier), and H _{ x } is the channel fading gain from the serving BS b _{ k } at a distance x.
Theorem 1
where the conditional coverage probabilities without SFR are given by (9) and (10), \(Q\left ({a,b,c,d} \right) = {c^{{2 / b}}} + {a^{{2 / b}}} d\int _{{{\left ({\frac {c}{a}} \right)}^{{2 / b}}}}^{\infty } {\frac {{du}}{{1 + {u^{{b / 2}}}}}} \), and \({\text {SNR}_{k}}\left (x \right) = \frac {{{P_{k}}{x^{- \alpha }}}}{{{\sigma ^{2}}}}\).
Proof
The proof is given in Appendix A. □
Unlike the results in [22], the coverage probability is dependent of ρ _{ k }, i.e., the load of each BS. Additionally, the conditional coverage probabilities without SFR are of special importance with respect to the proposed SFR scheme.
2.4 SFR scheme
2.4.1 Resource partitioning and power control
Similarly, both the network tiers are co-channel deployment. However for each cell, the entire available sub-channel set C is divided into two different subsets C _{1} and C _{2} with size |C _{1}| and |C _{2}|, respectively. The two different sunsets C _{1} and C _{2} have no intersection, i.e., C _{1}∩C _{2}=∅ and C=C _{1}∪C _{2}. Therefore, a resource partitioning factor η is defined as η=|C _{1}|/|C|(0≤η≤1).
In the proposed SFR scheme, a typical BS classifies users with average SINR as two types of users: interior users and edge users. Instead of a geographic classification criterion, the SINR threshold can more adequately capture the randomness: the locations of the BSs and users [24,25]. Let T _{FR,k } denote the SFR threshold of a typical BS in the kth tier. With SFR, each user calculates its SINR to the serving BS in the kth tier, and if it is less than the threshold T _{FR,k }, then the user is an edge user, and otherwise the user is an interior user.
In this paper, we consider the following random resource allocation scheme. Let η be the fraction of resources (namely, C _{1}) allocated to the macro interior users and pico edge users. The remaining 1−η fraction of resources (namely, C _{2}) are allocated to the macro edge users and pico interior users. Nevertheless, the macro edge users and pico edge users, especially the offloaded macro users, usually suffer severe interference from the neighboring macro BSs. To accomplish this, a power control factor β(0<β<1) is introduced to the sub-channel set C _{1} used by the macro BSs, i.e., \(P_{1}^{\mathrm {i}} = \beta {P_{1}}\) and \(P_{1}^{\mathrm {e}} = {P_{1}}\), where \(P_{1}^{\mathrm {i}}\) is the transmit power of macro BSs for the interior users and \(P_{1}^{\mathrm {e}}\) is the transmit power of macro BSs for the edge users. For all the pico users, the transmit power keeps full power transmission, the same as the macro edge users. Therefore, both high spectral efficiency and good user experience of the pico edge users can be achieved.
2.4.2 User association and load statistics
where \(\mathcal {U}_{1}^{\mathrm {i}} \) is the set of interior users associated with the kth tier BSs and \(\mathcal {U}_{1}^{\mathrm {e}} \) is the set of edge users associated with the kth tier BSs. Clearly, \({\mathcal {U}_{1}} \buildrel \Delta \over = \mathcal {U}_{1}^{\mathrm {i}} \cup \mathcal {U}_{1}^{\mathrm {e}}\) is the set of macro users, \({\mathcal {U}_{2}} \buildrel \Delta \over = \mathcal {U}_{2}^{\mathrm {i}} \cup \mathcal {U}_{2}^{\mathrm {e}}\) is the set of pico users, and \(\mathcal {U}_{1}^{\mathrm {i}} \cup \mathcal {U}_{1}^{\mathrm {e}} \cup \mathcal {U}_{2}^{\mathrm {i}} \cup \mathcal {U}_{2}^{\mathrm {e}} = \mathcal {U}\).
For a randomly chosen user in Φ _{ u }, it will exactly belong to specific one of above four sets, according to the user association strategy in (11). Then, the probabilities that a randomly chosen user belongs to the sets \(\mathcal {U}_{k}^{\mathrm {i}} \) and \(\mathcal {U}_{k}^{\mathrm {e}} \) are \({{\mathcal A}_{k}}{p_{c,k}}\left ({{T_{\text {FR},k}}} \right)\) and \({{\mathcal A}_{k}}\left ({1 - {p_{c,k}}\left ({{T_{\text {FR},k}}} \right)} \right)\), which also can be interpreted as the average fraction of users belonging to the sets \(\mathcal {U}_{k}^{\mathrm {i}} \) and \(\mathcal {U}_{k}^{\mathrm {e}} \). Irrespective of the exact distribution of user locations, the numbers of interior and edge users in a typical BS are significant for characterizing the cell load. To make the proposed framework analytically tractable, the sets \(\mathcal {U}_{k}^{\mathrm {i}} \) and \(\mathcal {U}_{k}^{\mathrm {e}} \) can be equivalently modeled as independent homogeneous PPPs with densities \({{\mathcal A}_{k}}{p_{c,k}}\left ({{T_{\text {FR},k}}} \right)\) and \({{\mathcal A}_{k}}\left ({1 - {p_{c,k}}\left ({{T_{\text {FR},k}}} \right)} \right)\), respectively [15]. Based on the same lines as the derivation in (4), their PMFs are given in the following lemma.
Lemma 2
Let \(N_{k}^{\mathrm {i}}\) and \(N_{k}^{\mathrm {e}}\) denote the numbers of interior and edge users associated with a BS in the kth tier, respectively. Their PMFs are given by (12) and (13).
Lemma 3
The Interfering sets of a typical user
User type | Interfering sets |
---|---|
Macro interior user (\(u\ \epsilon \ \mathcal {U}^{\text {i}}_{1}\)) | Ξ1i,Ξ2e |
Macro edge user (\(u\ \epsilon \ \mathcal {U}^{\text {e}}_{1}\)) | Ξ1e,Ξ2i |
Pico interior user (\(u\ \epsilon \ \mathcal {U}^{\text {i}}_{2}\)) | Ξ1e,Ξ2i |
Pico edge user (\(u\ \epsilon \ \mathcal {U}^{\text {e}}_{2}\)) | Ξ1i,Ξ2e |
where 1(A) is the indicator function that takes the value 1 if the event A is true, H _{ x } is the channel fading gain from the serving BS b _{ k } at a distance x, and \(I_{x,j}^{\mathrm {i}} = {P_{j}}{\sum \nolimits }_{y \in \Xi _{j}^{\mathrm {i}}\backslash {b_{k}}} {{H_{y}}{{\left \| y \right \|}^{- \alpha }}} \) and \(I_{x,j}^{\mathrm {e}} = {P_{j}}{\sum \nolimits }_{y \in \Xi _{j}^{\mathrm {e}}\backslash {b_{k}}} {{H_{y}}{{\left \| y \right \|}^{- \alpha }}} \) are the cumulative interference from the interior and edge loaded BSs in the jth tier, respectively.
3 Average user rate
This section is our main technical part. We first derive the general coverage probability for the proposed SFR scheme. Then, the methods of derivation are subsequently used for the average user rate. Moreover, we present a special case where α=4 and σ ^{2}=0, representing for the modern cellular networks. For this case, the expressions of coverage probability reduce to simple closed-form, which can provide clear insight into the performance analysis of each user.
3.1 Coverage probability
As discussed in Section 2, we observe that the conditional coverage probability of a typical user depends on two SINR thresholds [24,25], first the SINR threshold (i.e., T _{FR,k }) on the allocated a sub-channel from the entire available sub-channel set C to determine its status (interior or not), and second the actual SINR threshold (i.e., T) on the newly allocated a sub-channel from the subset C _{1} or C _{2} to determine whether it is covered or not. Actually, these two SINR thresholds are correlated because the interference may be generated by the same set of BSs, which makes our analysis challenging. The following theorems give the conditional coverage probabilities for a typical user under different status.
Theorem 2
(Macro tier, interior user): the coverage probability of the macro interior user is given by (22), where \(\text {SNR}_{k}\left ({x,a} \right) = \frac {{a{P_{k}}{x^{- \alpha }}}}{{{\sigma ^{2}}}}\), p _{ c,1}(T _{FR,1}) is given by (9), and \(\xi \left ({a,b,c,d,e} \right) = \int _{1}^{\infty } \left [ 1 - \left (1 - a\left ({1 - \frac {1}{{1 + c{v^{- e}}}}} \right) \right) \left (1 - b\left ({1 - \frac {1}{{1 + d{v^{- e}}}}} \right) \right) \right ]vdv \).
Proof
The proof is given in Appendix B. □
As we can see, ξ(a,b,c,d,e) is similar to ρ _{ j } Z(a,b,c) given by previous results in Theorem 1, they are different because of the dependence of two SINR thresholds. Moreover, ξ(a,b,c,d,e) can efficiently capture the disparity of the intra-tier and inter-tier interference before and after the proposed SFR scheme is applied.
Now, we turn our attention to the special case where α=4 and σ ^{2}=0, which is significant in practice that is widely applied in lots of literatures [21–25]. It is noted that the typical HCNs are interference-limited which the noise can be ignored compared to the interference. Furthermore, the special case can be considered as the scenario corresponding to an interference-limited urban cellular network [24,25,27], where FFR has been generally applied. In the special case, the expression (22) will be further simplified to a simple closed-form.
Corollary 1
Proof
Combining with (22) gives the desired result.
Theorem 3
(Macro tier, edge user): the coverage probability of the macro edge user is given by (24).
Proof
The proof is given in Appendix C. □
Corollary 2
Theorem 4
(Pico tier, interior user): the coverage probability of the pico interior user is given by (26).
Proof
Following the method of Theorem 2 gives the desired result. □
Corollary 3
Theorem 5
(Pico tier, edge user): the coverage probability of the pico edge user is given by (28).
Proof
Following the methods of Theorems 1 and 3 give the desired result. □
Corollary 4
3.2 Average user rate
where \({\mathcal R}_{\text {FFR},k}^{\mathrm {i}}\) and \({\mathcal R}_{\text {FFR},k}^{\mathrm {e}}\) are the average rate of a typical user when it is an interior user and edge user associated with the kth tier BS, respectively.
where the expectation is taken with respect to the distance x, \(t_{k}^{\mathrm {i}}\), and \(t_{k}^{\mathrm {e}}\) are the faction of time the users \(u \in \mathcal {U}_{k}^{\mathrm {i}} \) and \(u \in \mathcal {U}_{k}^{\mathrm {e}} \) are served on a sub-channel, respectively. \(t_{k}^{\mathrm {i}}\) and \(t_{k}^{\mathrm {e}}\) can be considered as the reciprocal of average number of users sharing a given sub-channel, i.e., the load at a sub-channel.
Lemma 4
Proof
The proof is given in Appendix D. □
Assuming the independence between the load and SINR [14,15,23], the following theorem gives the average rate for a typical macro interior user. Note that the average rate for other type of users can be obtained following the same procedure.
Theorem 6
Proof
Since \(\mathbb {E}\left [ \tau \right ] = \int _{0}^{\infty } {\mathbb {P}\left ({\tau > t} \right)dt} \) for τ>0, we obtain □
Plugging back into (38), the average rate (37) can be obtained.
4 Simulation and numerical results
In this section, we present numerical results on the coverage probability and average user rate for the proposed SFR scheme. Furthermore, we compare the proposed SFR scheme with FRP scheme. The simulation parameters are in accordance with 3GPP technical reports [28]. Unless otherwise stated, the transmit powers of a macro BS and a pico BS are \(P_{1}^{T} = 46 \ {\text {dBm}}\) and \(P_{2}^{T} = 30 \ {\text {dBm}}\). The densities of the two tiers are λ _{1}=1 BS/km^{2} and λ _{2}=5 BS/km^{2} with α = 4. The user density is λ _{ u } = 100 users/km^{2}. For a typical LTE system with 10 MHz bandwidth, 50 sub-channels (i.e., |C| = 50) are available to each BS, each sub-channel bandwidth then is 200 kHz.
4.1 Validation of analysis
4.2 Comparison with FRP scheme
4.3 Average user rate: trends and discussion
4.3.1 Impact of SFR thresholds
4.3.2 Impact of power control
For a given power control factor, the average user rate initially increases as the association bias increases, but decreases beyond a certain association bias. Hence, the optimal association bias exists. This is because more macro users in poor situation is offloaded to pico tier with the increase in association bias and thus, the interference from macro tier decreases due to the macro cell load decreases. This leads to the average user rate initially increases. But after a certain association bias, the pico BS is overloaded so that the average user rate eventually decreases.
4.3.3 Impact of resource partitioning
4.3.4 Optimal average user rate
As discussed above, the association bias and parameters of the SFR scheme need to be carefully chosen for optimal average user rate. The average user rate is calculated from the combination of resource partitioning factor and power control factor. However, the results are obtained from varied cell load for each pair. That is, the association bias increases from 0 to 40 dB, and the threshold T _{FR,1} decreases gradually from 20 to − 20 dB and the matched threshold T _{FR,2} increases from − 20 to 20 dB. Hence, the optimal average user rate is 2.6217 nats/sec/Hz when B = 18 dB, η = 0.72, β = 0.03, T _{FR,1} = − 20 dB, and T _{FR,2} = 12 dB. Compared to the network without CRE that just gets 1.7704 nats/sec/Hz, the gain obtained from the CRE with SFR scheme can be as high as 48.1%.
5 Conclusions
In this paper, we have presented an analytical framework to evaluate average user rate of HCNs with CRE and SFR scheme in a multichannel environment using the tools of stochastic geometry. Taking the random resource allocation and cell load into consideration, our network model can adequately capture the impact of CRE and SFR scheme on the performance. The numerical results have shown that the average user rate can be improved significantly, and further, the proposed SFR scheme outperforms the FRP scheme in any load condition. In addition, the optimal average user rate can be achieved by properly tuning the association bias and parameters of the SFR scheme. With the optimal combination, the gain can be as high as 48.1%.
6 Appendix A
Proof of Theorem 1
□
where \(Z\left ({a,b,c} \right) = {a}^{2 / b}{\int \nolimits }_{\left ({\frac {c}{a}} \right)^{{2 / b}}}^{\infty }{\frac {du}{1 + {u^{b / 2}}}}\).
Using (3) in (39) along with (40–42), the conditional coverage probabilities given in Theorem 1 are obtained.
7 Appendix B
Proof of Theorem 2
□
Conditioning on x, the distance to its serving macro BS, and focusing on the numerator of (44), we observe that the noise is independence of the interference and the expectation with respect to \(\hat I_{x,1}^{\mathrm {i}}\), I _{ x,1}, \(\hat I_{x,2}^{\mathrm {e}}\) and I _{ x,2} is the joint Laplace transform \(\mathcal {L}\left ({{{\hat s}_{1}},{s_{1}},{{\hat s}_{2}},{s_{2}}} \right)\) of \(\hat I_{x,1}^{\mathrm {i}}\), I _{ x,1}, \(\hat I_{x,2}^{\mathrm {e}}\), and I _{ x,2} evaluated at \(\left ({{x^{\alpha } }P_{1}^{- 1}T,{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}}} \right.,{x^{\alpha } }{\left ({\beta {P_{1}}} \right)^{- 1}}T,\left. {{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}}} \right)\).
where (a) follows from the independence of \(\hat I_{x,1}^{\mathrm {i}} + {I_{x,1}}\) and \(\hat I_{x,2}^{\mathrm {e}} + {I_{x,2}}\).
Factoring out the first term of (45), we have the derivation of (46), where (a) follows that \({\hat H_{x}},{H_{x}} \sim \exp \left (1 \right)\) and (b) is obtained from the probability generating functional (PGFL) [29] of Φ _{1}, replacing v=∥y∥ and x is the lower bound on distance between the macro user and its own tier.
Hence, \(\mathcal {L}\left ({{x^{\alpha } }P_{1}^{- 1}T,{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}},{x^{\alpha } }{{\left ({\beta {P_{1}}} \right)}^{- 1}}T,{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}}} \right)\) is given by (47).
8 Appendix C
Proof of Theorem 3
□
The first term of the numerator represents the SINR on the newly allocated sub-channel and following the method of Theorem 1 gives (51).
Hence, \(\mathcal {L}\left ({{x^{\alpha } }P_{1}^{- 1}T,{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}},{x^{\alpha } }P_{1}^{- 1}T,{x^{\alpha } }P_{1}^{- 1}{T_{\text {FR},1}}} \right)\) is given by (52).
Deconditioning on x, the second term of the numerator can be expressed as (53), where ξ(a,b,c,d,e) is originally defined in Theorem 2.
The denominator of (50) also can be obtained from (9), we have 1−p _{ c,1}(T _{FR,1}). Plugging back into (50), the conditional coverage probability given in Theorem 3 is obtained.
9 Appendix D
Proof of Lemma 4
Let \(\tilde N_{k}^{\mathrm {i}}\) and \(\tilde N_{k}^{\mathrm {e}}\) be the numbers of other interior and edge users, conditioned on the typical user being associated with that BS in the kth tier, respectively. Their PMFs are derived in a similar way in [23], given by (54) and (55). □
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 61271263).
Authors’ contributions
LG and SC conceived the proposed scheme. SC conducted the detailed derivation to evaluate the performance of the proposed scheme and wrote the manuscript. LG and ZS reviewed the manuscript. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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References
- JG Andrews, S Buzzi, W Choi, SV Hanly, A Lozano, ACK Soong, JC Zhang, What will 5g be? IEEE J. Sel. Areas Commun.32(6), 1065–1082 (2014).View ArticleGoogle Scholar
- E Hossain, M Hasan, 5g cellular: key enabling technologies and research challenges. IEEE Instrum. Meas. Mag.18(3), 11–21 (2015).View ArticleGoogle Scholar
- X Lin, JG Andrews, A Ghosh, Modeling, analysis and design for carrier aggregation in heterogeneous cellular networks. IEEE Trans. Commun.61(9), 4002–4015 (2013).View ArticleGoogle Scholar
- A Damnjanovic, J Montojo, Y Wei, T Ji, T Luo, M Vajapeyam, T Yoo, O Song, D Malladi, A survey on 3gpp heterogeneous networks. IEEE Wirel. Commun.18(3), 10–21 (2011).View ArticleGoogle Scholar
- Kyocera, Potential performance of range expansion in macro-pico deployment (r1-104355). 3GPP TSG RAN WG1 Meeting-62, (Madrid, 2010).Google Scholar
- D Lopez-Perez, I Guvenc, Gdl Roche, M Kountouris, TQS Quek, J Zhang, Enhanced intercell interference coordination challenges in heterogeneous networks. IEEE Wirel. Commun. Mag.18(3), 22–30 (2011).View ArticleGoogle Scholar
- S Mukherjee, I Guvenc, in Proceedings of 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR): 6-9 Nov. 2011. Effects of range expansion and interference coordination on capacity and fairness in heterogeneous networks (IEEE, Pacific Grove, 2011), pp. 1855–1859.View ArticleGoogle Scholar
- K Okino, T Nakayama, C Yamazaki, H Sato, Y Kusano, in Proceedings of 2011 IEEE International Conference on Communications Workshops (ICC): 5-9 June 2011. Pico cell range expansion with interference mitigation toward lte-advanced heterogeneous networks (IEEE, Kyoto, 2011), pp. 1–5.Google Scholar
- A Merwaday, S Mukherjee, I Guvenc, in Proceedings of 2013 IEEE Global Communications Conference (GLOBECOM): 9-13 Dec. 2013. On the capacity analysis of hetnets with range expansion and eicic (IEEE, Atlanta, 2013), pp. 4257–4262.View ArticleGoogle Scholar
- M Cierny, H Wang, R Wichman, Z Ding, C Wijting, On number of almost blank subframes in heterogeneous cellular networks. IEEE Trans. Wirel. Commun.12(10), 5061–5073 (2013).View ArticleGoogle Scholar
- Panasonic:Performance study on abs with reduced macro power (r1-113806). 3GPP TSG-RAN WG1 #67, (San Francisco, 2011).Google Scholar
- A Merwaday, S Mukherjee, I Guvenc, in Proceedings of 2014 IEEE Wireless Communications and Networking Conference (WCNC): 6-9 April 2014. Hetnet capacity with reduced power subframes (IEEE, Istanbul, 2014), pp. 1380–1385.View ArticleGoogle Scholar
- H Hu, J Weng, J Zhang, Coverage performance analysis of feicic low-power subframes. IEEE Trans. Wireless Commun.15(8), 5603–5614 (2016).View ArticleGoogle Scholar
- S Singh, JG Andrews, Joint resource partitioning and offloading in heterogeneous cellular networks. IEEE Trans. Wirel. Commun.13(2), 888–901 (2014).View ArticleGoogle Scholar
- Y Dhungana, C Tellambura, Multichannel analysis of cell range expansion and resource partitioning in two-tier heterogeneous cellular networks. IEEE Trans. Wirel. Commun.15(3), 2394–2406 (2016).View ArticleGoogle Scholar
- W Tang, S Feng, Y Liu, MC Reed, in Proceedings of 2015 IEEE Global Communications Conference (GLOBECOM): 6-10 Dec. 2015. Joint low-power transmit and cell association in heterogeneous networks (IEEE, San Diego, 2015), pp. 1–6.Google Scholar
- B Xie, BZ Zhang, RQ Hu, Y Qian, in Proceedings of 2016 IEEE International Conference on Communications (ICC): 22-27 May 2016. Spectral efficiency analysis in wireless heterogeneous networks (IEEE, Kuala Lumpur, 2016), pp. 1–6.Google Scholar
- Q Li, RQ Hu, Y Xu, Y Qian, Optimal fractional frequency reuse and power control in the heterogeneous wireless networks. IEEE Trans. Wirel. Commun.12(6), 2658–2668 (2013).View ArticleGoogle Scholar
- H ElSawy, E Hossain, in Proceedings of 2013 IEEE Global Communications Conference (GLOBECOM): 9-13 Dec. 2013. Channel assignment and opportunistic spectrum access in two-tier cellular networks with cognitive small cells (IEEE, Atlanta, 2013), pp. 4477–4482.View ArticleGoogle Scholar
- RQ Hu, Y Qian, An energy efficient and spectrum efficient wireless heterogeneous network framework for 5g systems. IEEE Commun. Mag.52(5), 94–101 (2014).View ArticleGoogle Scholar
- HS Dhillon, RK Ganti, F Baccelli, JG Andrews, Modeling and analysis of k-tier downlink heterogeneous cellular networks. IEEE J. Sel. Areas Commun.30(3), 550–560 (2012).View ArticleGoogle Scholar
- H-S Jo, YJ Sang, P Xia, JG Andrews, Heterogeneous cellular networks with flexible cell association: a comprehensive downlink sinr analysis. IEEE Trans. Wirel. Commun.11(10), 3484–3495 (2012).View ArticleGoogle Scholar
- S Singh, HS Dhillon, JG Andrews, Offloading in heterogeneous networks: Modeling, analysis, and design insights. IEEE Trans. Wirel. Commun.12(5), 2484–2497 (2013).View ArticleGoogle Scholar
- T Novlan, RK Ganti, A Ghosh, JG Andrews, Analytical evaluation of fractional frequency reuse for ofdma cellular networks. IEEE Trans. Wirel. Commun.10(12), 4294–4305 (2011).View ArticleGoogle Scholar
- T Novlan, RK Ganti, A Ghosh, JG Andrews, Analytical evaluation of fractional frequency reuse for heterogeneous cellular networks. IEEE Trans. Commun.60(7), 2029–2039 (2012).View ArticleGoogle Scholar
- J-S Ferenc, Z Neda, On the size distribution of poisson-voronoi cells. Phys. A Stat. Mech. Appl.385(2), 518–526 (2007).View ArticleGoogle Scholar
- TS Rappaport, Wireless communications principles and practice, 2nd Edition (Prentice Hall, New Jersey, 2002).Google Scholar
- 3GPP TR36.814, Further advancements for e-utra physical layer aspects (v9.0.0) (2010).Google Scholar
- D Stoyan, W Kendall, J Mecke, Stochastic geometry and its applications (Wiley, West Sussex, 1996).MATHGoogle Scholar