 Research
 Open Access
Low energy indoor network: deployment optimisation
 Siyi Wang^{1}Email author,
 Weisi Guo^{1} and
 Tim O'Farrell^{1}
https://doi.org/10.1186/168714992012193
© Wang et al; licensee Springer. 2012
 Received: 1 October 2011
 Accepted: 11 June 2012
 Published: 11 June 2012
Abstract
This article considers what the minimum energy indoor access point deployment is in order to achieve a certain downlink qualityofservice. The article investigates two conventional multipleaccess technologies, namely: LTEfemtocells and 802.11n WiFi. This is done in a dynamic multiuser and multicell interference network. Our baseline results are reinforced by novel theoretical expressions. Furthermore, the work underlines the importance of considering optimisation when accounting for the capacity saturation of realistic modulation and coding schemes. The results in this article show that optimising the location of access points both within a building and within the individual rooms is critical to minimise the energy consumption.
Keywords
 Distribute Coordination Function
 Single Room
 Fractional Frequency Reuse
 Contention Window Size
 Femtocell Network
1 Introduction
Recent research shows that more than 50% of voice calls and more than 70% of data traffic are generated indoors [1]. Two main wireless technologies have been used for serving users indoors. The first one is the traditional outdoor cellular system which deals with real time voice, short messages and mobile broadband (MBB) applications. The other is wireless local area networks (WLANs) which focus on providing nonreal time data applications [2]. Due to the increasing demands for indoor higher datarate wireless applications, existing cellular systems will be insufficient to meet the expected quality of service (QoS) requirement for indoor users from both service coverage and capacity perspectives. The femtocell access points (FAPs) have been proposed to address the above challenge, which uses lowpower, shortrange and lowcost base stations. Femtocells are compatible with the existing outdoor macrocell cellular base stations which can easily support seamless handoff but provide better indoor signal strength. With the introduction of femtocell technology, serving base stations are becoming similar to the closest competing WiFi technology in reality. Instead of the conventional cellular network infrastructure, femtocells use the IP Network as a backhaul architecture which has a lot in common with the existing 802.11 technology. Despite the huge potentials of femtocells, they still face many technical and business challenges. Whilst existing research has focused on dynamic power control and radio resource management to mitigate interference between indoor and the outdoor network, the issue of the physical deployment of the indoor network (the number and location of indoor access points (APs)) is unresolved.
1.1 Review of existing work
Several existing research [3–5] has been focusing on the improvement of femtocell network throughput. AlRubaye et al. [3] outlined the cognitive radio technologies for the future MBB era by proposing a cognitive femtocell solution for indoor communications in order to increase the network capacity in serving indoor users and to solve the spectrumscarcity problems. Ko and Wei [4] proposed a desirable resource allocation mechanism, taking into account mobile users' selfish characteristics and private traffic information to improve the femtocell network performance. The aggregate throughput of twotier femtocell networks has been improved by a beamforming codebook restriction strategy and an opportunistic channel selection strategy in [5]. However, the above study did not take into account the location and the number of FAPs deployed in the indoor environment.
There also have been various approaches to investigate the optimal base station (BS) placement to achieve the operator's desired QoS or coverage targets. Much of the previous work [6–8] has focused on minimising the transmitting power of BS. Fagen et al. [9] proposed a new automated method of simultaneously maximising coverage while minimising interference for a desired level of coverage overlap. However, such an approach is not always practical as network optimisation is constrained by a number of restrictions on BS placements, interference and power emissions. Ashraf et al. [10] described an approach of adjusting the transmitting power of fixed positions of FAPs in the enterprise offices to achieve coverage optimisation and load balance, but did not consider the evaluation of the effect on the positions of FAPs. In [11], a joint power and channel allocation method was proposed to improve the uplink throughput, but the calculation of the throughput, which was the key parameter of the algorithm, was relatively simple and might be inaccuracy by just using Shannon capacity equation. Ki Wo et al. [12] derived the downlink SINR formula for the residential femtocell but the formula did not take the throughput into account while researchers in [13] provided system simulation to evaluate the femtocell network performance, however the simulation was relatively simple without an extensive theoretical model verification. A theoretical framework was proposed in [14] to analyse the interference characteristics of different femtocell subbands for Orthogonal FrequencyDivision Multiple Access (OFDMA) systems employing the Fractional Frequency Reuse scheme which could be extended to optimise power and frequency allocation, but the pathloss model employed in this framework is far too simple to reflect the real characteristics of the indoor scenario.
Bianchi proposes a classic twodimensional Markov chain to determine the saturation throughput of a Wireless Local Area Network (WLAN) using the Distributed Coordination Function (DCF) [15]. Tay and Chua [16] proposed a model based on average value analysis and studied the effects of contention window sizes on the throughput performance. Both of the above models assumed an ideal wireless channel with no physical layer (PHY) channel errors. In fact, wireless channels are usually errorprone and the packet errors have an impact on the system performance. Several articles has extended the above system models to study the throughput performance under different channel error conditions.
1.2 Contribution
Given the large number of propagation variables in indoor buildings and its relation to the outdoor cellular network, the article provides a best practice in optimising AP deployment with very little signal to interferenceplusnoise ratio (SINR) degradation for microcell users. The novel contribution is the simulation results and the theoretical framework that reinforces the key deployment solutions. Our baseline results are reinforced by novel theoretical expressions. Moreover, for a given building size, the tradeoff between increased QoS and power consumption, as well as the capacity saturation points are demonstrated. An approach has been introduced to study the saturated throughput, user QoS and energy consumption performances of 802.11n networks under errorprone channels by extending Bianchi's model.
2 Body of investigation

Concept demonstration: Single Room with and without an outdoor interference source (Simulation and theory);

Generic building with multiple rooms on multiple floors with an outdoor interference source (Simulation).
The combined results of the two scenarios will lead to a general low energy indoor deployment rule. Moreover, the single room femtocell results are reinforced with a novel theoretical framework that can optimise the location of an AP with respect to the interference and propagation parameters. It is shown that the key results hold for a multiple room building. There is only 1 outdoor interference source considered, because given that a building is inside the coverage of a cell, the interference of that cell will be far greater than neighbouring cells that are further away.

LTEfemtocell Simulation: Cochannel FAPs employing SISO transmission and Round Robin (RR) scheduling. The link level capacity is derived from adaptive modulation and coding schemes. This is simulated for a single room and a building with multiple rooms.

802.11n WiFi Simulation: WiFi APs employing frequency reuse 1 and 3, and SISO transmission with a theoretical contention model. The link level capacity is derived from adaptive modulation and coding schemes. This is simulated for a single room.

Single Room Theory: Single LTEfemtocell deployed in a single room in the presence of a fully loaded outdoor microcell BS.
2.2 LTEfemtocells simulator model
The user distribution for each room or building is random even distribution. The position of each user and the traffic conditions are updated within each loop and the simulation results are run enough times to reach convergence. RR scheduler is employed, which evenly partitions the resource blocks between users. In this simulator resource allocation is performed at intervals of 1 ms in the time domain. This interval is called a transmission time interval (TTI) [17].
2.2 802.11n simulator
The 802.11n simulator is based on an existing throughput analytical model [15]. The model is concerned with infrastructure mode WLANs that use the DCF medium access control (MAC) protocol. The model assumes that there are a number of 802.11n APs operating on three different frequency channels in conventional scenario and one frequency channel in deployed indoors and a fixed number of client stations in the WLAN. Each user is associated with exactly one AP which provides the highest SINR to that user and each AP with its associated stations defines a cell. Therefore, DCF is used for singlehop only communication within the cells and users access data through their serving APs. Each user is assumed to have saturated traffic. The wireless channel bit error rate (BER) is P_{ b } . The minimum contention window size is W and the maximum backoff stage is m. In 802.11 WLANs, control frames are transmitted at the basic rate which is more robust in combating errors. They have a much lower frame error rate as the size of these control frames are much smaller than an aggregated data frame. Therefore, the frame error probabilities for control frames and preambles are assumed to be zero.
where the probability of an idle slot p_{ σ }is (1 − τ)^{ n }, the probability of a noncollided transmission p_{s _nc}is $\frac{n{\left(1\tau \right)}^{n1}}{{p}_{tr}}$, the probability for a transmission in a time slot p_{ tr }is 1 − p_{ σ } = 1 − (1 − τ) ^{ n } , the probability of a successful transmission (without collisions and transmission errors) is p_{ tr }p_{s _nc}(1 − p_{ e } ) and τ is computed by (4). T_{ σ } is equal to the system's empty slot time of 9μ s. T_{ σ } , T_{ c }T_{ s } and T_{ e } are the idle, collision, successful and error virtual time slot's length and are defined as follows: T_{ c } = EIFS, T_{ s } = DATA + BACK + 3SIFS + DIFS, T_{ e } = DATA + EIFS + 2SIFS, where BACK = 5.63μ s and DATA are the transmission time for backoff stage and the transmission time for aggregated data frame, SIFS = 16μ s, DIFS = SIFS +T_{ σ } , EIFS = SIFS + DIFS + BACK, respectively. 48 of the 52 OFDM subcarriers are for data and the remaining 4 are for pilot subcarriers.
3 Power and energy metrics
The power consumption model employed by this article considers the FAP and 802.11n APs to have the same model. This assumption is reinforced by existing literature [20]. The model considers the power consumption of an AP to have two distinct parts: a radiohead (RH) and an overhead (OH). Together the RH and OH constitute the operational (OP) power consumption of the AP. During transmission, the RH is active, and irrespective of transmission, the fixed OH is always active.
where ${P}_{i}^{\text{RF}}$, ${P}_{i}^{\text{RH}}$ and ${P}_{i}^{\text{OH}}$ are RF power, RH power and OH power, respectively. The RH power is defined as ${P}_{i}^{\text{RF}}/{\mu}_{\Sigma}={P}_{i}^{\text{RH}}$, where μ_{Σ} is the RH efficiency [20]. The throughput of the system is defined as ${R}_{\text{AP},i}=M/{T}_{\text{AP}}^{\text{RH},i}$, which is greater or equal to the offered load: ${R}_{\text{tra c}}=M/{T}_{\text{AP}}^{\text{OH}}$. The term $\frac{{P}_{i}^{\text{RH}}}{{R}_{\text{AP},i}}$ in (8) is an indication of the average radio transmission efficiency, which does not consider the OH energy. This is commonly used to measure energy consumption in literature, and is known as the energyconsumptionratio (ECR) [21]. n refers to the total number of APs.
For a given offered load demanded by users, a more spectral efficient deployment is able to transmit the same data for a short transmission time. Over time, this amounts to a reduction of the RH energy consumption. The energy saving caused by spectral efficiency alone is upperbounded (ERG threshold) by the ratio of OH to OP. This upperbound $\left({\text{ERG}}_{\text{RANupper}\text{bound}}^{\text{OP}}=\frac{{P}_{\text{test}}^{\text{OH}}}{{P}_{\text{ref}\text{.}}^{\text{RH}}+{P}_{\text{ref}\text{.}}^{\text{OH}}}=\frac{{P}_{\text{test}}^{\text{OH}}}{{P}_{\text{ref}\text{.}}^{\text{OP}}}\right)$ can be obtained when R_{AP,test} in Equation (8) approaches infinity on the condition that the same number of APs deployed for both reference and test systems is considered. In order to significantly reduce energy consumption further, a reduction in the number of APs is required to meet the QoS needed. This can only be accomplished by significantly improving the overall throughput of the AP deployment. This article proposes novel location optimisation simulation and theoretical results to achieve this.
4 Simulation and theoretical results
4.1 Single room AP number
4.1.1 Conventional scenario
As the number of APs is increased, the 802.11n deployment is always more spectrally and energy efficient due to the increased operating bandwidth of 60 MHz with frequency reuse pattern 3, compared to the LTE bandwidth of 20 MHz with frequency reuse pattern 1. As the number of 802.11n APs increases to beyond 3, the interference that arises between the APs will cause a degradation of overall downlink user QoS and average user data rate performance. For the number of APs greater than 2, 802.11 APs provides 2.61 to 21.80% ERG over the FAP deployment. The results are shown in Figure 4b,d.
Therefore, for a single AP deployment, an LTE FAP is more spectrally and energy efficient than an 802.11n AP. This is true both with and without a fully loaded microcell interference source. In order to achieve a higher user QoS performance, deploying more 802.11n APs is the more spectrally and energy efficient. No more than 3 802.11n APs should be deployed in the same room; any more causes mutual interference and degrades the aggregate QoS received by the users.
4.1.2 Alternative scenario
The alternative scenario is defined in the body of investigation, in which both FAPs and 802.11n APs have a total bandwidth of 20 MHz with different frequency reuse pattern 1 and 3, respectively. The result of 1 FAP and 1 802.11n AP for both conventional and alternative scenarios are identical as shown in Figure 4a. Figure 4c,d, average user data rate and ERG performance for 1 FAP and 3 FAPs in both conventional and alternative scenario with a baseline 802.11n network. In the alternative scenario, FAP outperforms 802.11 AP when the number of APs is 3. The average user data rate for three FAPs is 1.12 Mbit/s while this value for 3 802.11APs is 0.22 Mbit/s. This is because 802.11 AP suffers server interference from other APs in the alternative scenario and its PHY adopts convolution codes which is less efficient than turbo codes used in LTEfemtocell. Figure 4d indicates three FAPs provides an ERG of 20.08% in alternative scenario while 3 802.11n APs offers an ERG of 21.80% in conventional scenario. Hence FAP investigation is particular interest in the following sections.
4.1.3 Remarks
The results in Figure 4a covers the results of all four possible combinations of comparison between one FAP and one 802.11n AP in either conventional or alternative scenario while the results in Figure 4c contains the same number of combinations results for the case of 3 FAPs and 3 802.11n AP. These four possible combinations are conventional FAP versus alternative AP and conventional AP versus alternative FAP besides the other two which have already been covered in the above sections. It is worth mentioning that 3 FAPs in conventional scenario is more energy efficient than 3 APs in alternative scenario. This is due to the mutual impact from the the different scheduler mechanism and coding scheme applied in both systems.
4.2 Single room AP placement
Previously, the optimal number of APs to deploy in a single room has been considered. The conclusion was that for a low QoS target, 1 FAP is the most energy efficient deployment. For higher user QoS targets, 23 802.11n APs should be deployed. Next, simulation is used to determine where to place the 1 FAP given that there is an outdoor interference source from a microcell, and where to place 26 cofrequency FAPs that interfere with each other. Furthermore, result of 1 FAP with a theoretical background has been reinforced, which can be found in the Appendix of this article.
4.2.1 1 FAP
The optimal placement of 1 FAP is judged according to the strength of the outdoor microcell source. Another baseline FAP deployment has been considered for comparison. In this baseline scenario, FAP is placed at the corner of the room where the power socket is typically located.
where ${P}_{1}=\frac{\left(b+{d}_{bp2}\right){C}_{\text{s}}}{Y}$, ${Q}_{1}=\frac{\left(Yb{d}_{bp2}\right){\text{log}}_{2}{K}_{\gamma}}{Y}$, ${R}_{1}=\frac{\beta {\text{log}}_{2}10}{2Y}\left[{Y}^{2}{\left(b+{d}_{bp2}\right)}^{2}\right]$, α=1.87 and Y is the length of the room. T_{1} = Y log_{2}(Y −b)−(b+d_{bp 2}) log_{2}d_{bp 2}, ${U}_{1}=\left(\frac{Yb{d}_{bp2}}{\text{ln}2}+b{\text{log}}_{2}\frac{Yb}{{d}_{bp2}}\right)$. C_{s} in P_{1} equals ${\text{log}}_{2}\left(1+1{0}^{\frac{\gamma s}{10}}\right)$ bit/s/Hz where γ_{ s }is the saturation SNR threshold. ${P}_{2}=\frac{\left({d}_{bp1}+{d}_{bp2}\right){C}_{\text{s}}}{Y}$, ${Q}_{2}=\frac{\left(Y{d}_{bp1}{d}_{bp2}\right){\text{log}}_{2}{K}_{\gamma}}{Y}$, ${R}_{2}=\frac{\beta {\text{log}}_{2}10}{2Y}{\left(b{d}_{bp1}\right)}^{2}$, T_{2}=(bd_{bp 1})log_{2}d_{bp 1}and ${U}_{2}=\frac{b{d}_{bp1}}{\text{ln}2}+b{\text{log}}_{2}\frac{{d}_{bp1}}{b}$. K_{ γ }in Q_{2} is $\frac{{P}_{\text{FAP}}{D}^{3.67}\times {{10}^{\left(22.7+\text{P}{\text{L}}_{\text{wall}}\right)}}^{/10}\times \phantom{\rule{0.3em}{0ex}}{\left(\frac{f}{5}\right)}^{2.6}}{{P}_{\text{micro}}{\u1e20}_{\text{micro}}\times {10}^{46.8/10}\times {\left(\frac{f}{5}\right)}^{2}}$, where P_{FAP} and P_{micro} are the transmitting power of FAP and microcell station, respectively. ${\u1e20}_{\text{micro}}$ is the expected value of the antenna gain from the microcell BS. f is the operation frequency. d_{bp 1}= B exp [−W (F) and d_{bp 2}= B exp [−W (−F)] where $B={10}^{{\scriptscriptstyle \frac{b}{20\alpha}}}{\left({\scriptscriptstyle \frac{K\gamma}{{10}^{\frac{{}_{\gamma s}}{10}}}}\right)}^{{\scriptscriptstyle \frac{1}{\alpha}}}$, $F=\frac{\text{ln}10}{20\alpha}B$, W is Lambert W function. Finally, ${P}_{3}=\frac{\left(Yb+{d}_{bp1}\right){C}_{\text{s}}}{Y}$, ${Q}_{3}=\frac{\left(b{d}_{bp1}\right){\text{log}}_{2}{K}_{\gamma}}{Y}$, ${b}_{1}=\sqrt[\alpha ]{\frac{{K}_{\gamma}}{{10}^{\frac{{\gamma}_{s}}{10}}}}$ and ${b}_{2}=Y\sqrt[\alpha ]{\frac{{K}_{\gamma}{10}^{\beta Y}}{{10}^{\frac{{\gamma}_{s}}{10}}}}$.
It can be shown that the function is convex and that according to the first rule of finding the maximum value of a function, stationary points can be determined by differentiating Equation (9) $\left(\text{Note}:\frac{\partial {d}_{bp1}}{\partial b}=\frac{B\text{exp}\left[W\left(F\right)\right]\text{ln}10}{20\alpha \left[1+W\left(F\right)\right]}\text{and}\frac{\partial {d}_{bp2}}{\partial b}=\frac{B\text{exp}\left[W\left(F\right)\right]\text{ln}10}{20\alpha \left[1+W\left(F\right)\right]}\right)$and then solving the differentiated function for zeros. The resulting expression is a closed form expression, but is unfortunately too long for the scope of this article. All the stationary points are tested in order to verify the type of the stationary points (max) by checking if the corresponding value in the secondorder differential function of Equation (9) is negative. Finally, the mean capacity value(s) corresponding to all the stationary points are compared with all endpoints of the interval of each subfunction in Equation (9) and the global maximum value is selected as the maximum of mean capacity. The solution b_{opt} is the optimal coordinate for FAP placement. Detailed derivations of Equation (9) can be found in the Appendix.
Figure 5b shows the OP ERG performance. The result in line is calculated based on the throughput from the theory while the result in symbols is obtained based on the throughput from the simulation. They were both obtained from the Equation (8). The results in Figure 5 show that 1 FAP should be located between the middle of the room to the wall closest to the outdoor interference source. As the strength of the microcell interference decreases due to it being either further away, stronger wall loss, or lower transmitting power, the FAP should be moved closer to the wall. This is because most of the room is in capacity saturation, and the FAP should be moved to compensate for regions which are not. It is important to consider capacity saturation, without which the FAP's optimal location is always likely to be in centre of the room.
4.2.2 26 FAPs
4.3 Multiroom multifloor FAP placement
 (1)
In the presence of no strong outdoor interference, deploy a single FAP at centre of building. In the presence of outdoor microcell interference, deploy the FAP near the wall that is closest to the outdoor interference source. The floor level should be one that is closest to the height of the microcell.
 (2)
Any single additional FAP should be deployed also near the aforementioned wall on the same floor, but not in the same room as the first FAP.
 (3)
Any multiple additional FAPs should be deployed not on the same floor, and at the opposite side of the building in corner rooms. These FAPs should not be on the same floor as FAPs placed in Steps 1 and 2 and with each other in Step 3.
 (4)
Any additional FAPs that do not satisfy rule 3. is likely to cause energy inefficiency.
Generally speaking, this rule can cover the optimisation of FAP placement for up to 6 FAPs, which can provide a sufficiently high QoS. The RAN QoS increases as the number of FAPs increases. This optimal deployment offers an average 12% ERG compared to the baseline even distributional deployment. This is shown in Figure 8b. Figure 8c illustrates that how much energy can be reduced while deploying the optimal FAPs in this building when comparing to the baseline scenario for a certain RAN QoS. As the number of FAPs needed for different targeted RAN QoS is not always same for optimal and baseline deployment, ERG threshold is waived is this comparison.
It can be noted that the solution of optimising the FAP location does not significantly degrade the outdoor network performance. By moving the FAP from centre to a point that is closer to the outdoor interference source, the interference from the FAP to the outside network is increased by up to 2.5 dB. Given that the outdoor interference is more dominated by outdoor interference from other microcells the total interference is not significantly increased. This is to say that the interference generated by the FAP to outdoor users is not significantly increased.
5 Discussions

Deploying one FAP is always more spectral and energy efficient than one 802.11n WiFi AP;

Deploying up to 3 multiple 802.11n APs is always more energy efficient than deploying multiple FAPs within the same building in conventional scenario. For buildings with more than one room, no APs should be deployed in the same room of a building;

In the presence of a strong outdoor interference source (i.e., from a microcell), the location of some of the cofrequency FAPs, should be placed near the wall facing the interference source to counteract the high level of indoor interference.

A single FAP versus A single 802.11n AP saves 4.44% OP ERG;

Three 802.11n APs versus three FAPs saves 21.80% OP ERG in conventional scenario;

Three FAPs versus three 802.11n APs saves 20.08% OP ERG in alternative scenario;

Optimising placement of a single FAP in a single room saves 5% OP ERG;

Optimising placement of multiple FAPs in a multiroom building saves average 12% OP ERG.
In general, improving the location of the FAPs, whilst keeping the number the same can improve the RAN throughput by 33%. Whilst, improving the location of the FAPs and reducing the number of FAPs can save 40% energy.
6 Conclusions
This body of investigation has proposed a novel strategy for indoor AP deployment for both LTEFemtocells and 802.11n WiFi multiple access technologies. This is done for a single room and multiple room scenarios. Our comparisons show that between 5 and 40% ERG can be saved by optimising the AP number and location with respect to the propagation parameters and interference scenario. Our baseline simulation results are reinforced with a novel theoretical framework, which can be used to develop and optimise indoor deployment while causing very little SINR degradation for microcell users.
7 Methods
The analysis in this paper is conducted by using both MonteCarlo computation in a MATLAB software environment and theoretical expressions.
Appendix
Theoretical framework of 1 FAP optimisation
Introduction

The throughput experienced by indoor users are always in the high SINR regime, and that at least some region of the room experiences saturated throughput. This has been verified to be accurate after extensive simulation runs.

The variation of throughput is mainly along the axis of the FAPMicrocell, and not orthogonal to it.

A modified Shannon expression is used for throughput, which accounts for the mutual information saturation of modulation and coding schemes.
Where $\frac{{P}_{\text{FAP}}{D}^{3.67}\times {10}^{\left(22.7+\text{P}{\text{L}}_{\text{wall}}\right)/10}\times {\left(\frac{f}{5}\right)}^{2.6}}{{P}_{\text{micro}}{\u1e20}_{\text{micro}}\times {10}^{46.8/10}\times {\left(\frac{f}{5}\right)}^{2}}$ is denoted as K_{ γ } , α = 1.87 and β = 0.05. P_{FAP} and P_{micro} are the transmitting power of FAP and microcell station, respectively. ${\u1e20}_{\text{micro}}$ is the expected value of the antenna gain from the microcell BS. d_{x,y,FAP}is the i th FAPtogrid distance and d_{x,y,micro}is the i th microcelltogrid distance. d_{x,y,in}is the distance between the wall and i^{ th } grid. As D ≫ d_{x,y,in}, d_{x,y,micro}can be accurately estimated as D. γ_{x,y}can then be rewritten as ${K}_{\gamma}\frac{{10}^{\beta {d}_{x,y,\text{in}}}}{{d}_{i,\text{FAP}}^{\alpha}}$.
where ${C}_{\text{s}}={\text{log}}_{2}\left(1+{10}^{\frac{{\gamma}_{{}_{S}}}{10}}\right)\text{bit/s/HZ}$. Y_{s} and Y_{ns} are the length of saturated and nonsaturated regions, respectively. Equation (12) can be split into three subfunctions with respect to FAP position b.
 (1)
Scenario 1: the FAP is close to the window (0 < b ≤ b _{1}) so that the only nonsaturated capacity region is away towards the wall.
 (2)
Scenario 2: the FAP is in the middle region of the room (b _{1} < b ≤ b _{2}) so that the nonsaturated capacity regions exist both towards the window and the wall.
 (3)
Scenario 3: the FAP is close to the far wall (b _{2} < b ≤ Y ) so that the only nonsaturated capacity region is towards the window.
We now consider these scenarios in turn and combine their theoretical formulation.
Formulation Scenario 1
Formulation Scenario 2
${P}_{2}=\frac{\left({d}_{bp1}+{d}_{bp2}\right){C}_{\text{s}}}{Y}$, ${Q}_{2}=\frac{\left(Y{d}_{bp1}{d}_{bp2}\right){\text{log}}_{2}{K}_{\gamma}}{Y}$, ${R}_{2}=\frac{\beta {\text{log}}_{2}10}{2Y}{\left(b{d}_{bp1}\right)}^{2}$, ${T}_{2}=\left(b{d}_{bp1}\right){\text{log}}_{2}{d}_{bp1}$, and ${U}_{2}=\frac{b{d}_{bp1}}{\text{ln}2}+b{\text{log}}_{2}\frac{{d}_{bp1}}{b}$.
Formulation Scenario 3
where ${P}_{3}=\frac{\left(Yb+{d}_{bp1}\right){C}_{\text{s}}}{Y}$, and ${Q}_{3}=\frac{\left(b{d}_{bp1}\right){\text{log}}_{2}{K}_{\gamma}}{Y}$. Equations (14), (19) and (21) complete the expression of the mean capacity of the network with respect to the position of the FAP b as shown in Equation (9).
Declarations
Acknowledgements
The work reported in this paper has formed part of the Green Radio Core 5 Research Programme of the Virtual Centre of Excellence in Mobile and Personal Communications, Mobile VCE. Fully detailed technical reports on this research are available to Industrial Members of the Mobile VCE. http://www.mobilevce.com
Authors’ Affiliations
References
 Chandrasekhar V, Andrews J, Gatherer A: Femtocell networks: a survey. Communications Magazine, IEEE 2008, 46(9):5967.View ArticleGoogle Scholar
 Hasan SF, Siddique NH, Chakraborty S: Femtocell versus WiFiA survey and comparison of architecture and performance. Wireless Communication, Vehicular Technology, Information Theory and Aerospace & Electronic Systems Technology, 2009. Wireless VITAE 2009. 1st International Conference on, Aalborg, Denmark, IEEE 2009, 916920.Google Scholar
 AlRubaye S, AlDulaimi A, Cosmas J: Cognitive Femtocell. Vehicular Technology Magazine, IEEE 2011, 6: 4451.View ArticleGoogle Scholar
 Ko CH, Wei HY: OnDemand Resource Sharing Mechanism Design in TwoTier OFDMA Femtocell Networks. Vehicular Technology, IEEE Transactions on 2011, (99):1.Google Scholar
 Park S, Seo W, Choi S, Hong D: A Beamforming Codebook Restriction for CrossTier Interference Coordination in Twotier Femtocell Networks. Vehicular Technology, IEEE Transactions on 2011, (99):11.Google Scholar
 Chandrasekhar V, Andrews JG, Muharemovic T, Shen Z, Gatherer A: Power control in twotier femtocell networks. Wireless Communications, IEEE Transactions on 2009, 8(8):43164328.View ArticleGoogle Scholar
 Jo HS, Mun C, Moon J, Yook JG: Interference mitigation using uplink power control for twotier femtocell networks. Wireless Communications, IEEE Transactions on 2009, 8(10):49064910.View ArticleGoogle Scholar
 Chandrasekhar V, Andrews J: Uplink capacity and interference avoidance for twotier femtocell networks. Wireless Communications, IEEE Transactions on 2009, 8(7):34983509.View ArticleGoogle Scholar
 Fagen D, Vicharelli PA, Weitzen J: Automated wireless coverage optimization with controlled overlap. Vehicular Technology, IEEE Transactions on 2008, 57(4):23952403.View ArticleGoogle Scholar
 Ashraf I, Claussen H, Ho LTW: Distributed radio coverage optimization in enterprise femtocell networks. Communications (ICC), 2010 IEEE International Conference on, Cape Town, South Africa, IEEE 16.Google Scholar
 Torregoza JPM, Enkhbat R, Hwang WJ: Joint power control, base station assignment, and channel assignment in cognitive femtocell networks. EURASIP Journal on Wireless Communications and Networking 2010, 2010: 6.View ArticleGoogle Scholar
 Ki Won S, Harald H, Stephen ML: A semianalytical PDF of downlink SINR for femtocell networks. EURASIP Journal on Wireless Communications and Networking 2010., 2010:Google Scholar
 Yavuz M, Meshkati F, Nanda S, Pokhariyal A, Johnson N, Raghothaman B, Richardson A: Interference management and performance analysis of UMTS/HSPA+ femtocells. Communications Magazine, IEEE 2009, 47(9):102109.View ArticleGoogle Scholar
 Lee J, Bae S, Kwon Y, Chung M: Interference Analysis for Femtocell Deployment in OFDMA Systems Based on Fractional Frequency Reuse. Communications Letters, IEEE (99):13.Google Scholar
 Bianchi G: Performance Analysis of the IEEE 802.11 Distributed Coordination function. IEEE Journal on selected areas in communications 2000, 18(3):535547. 10.1109/49.840210View ArticleGoogle Scholar
 Tay YC, Chua KC: A capacity analysis for the IEEE 802.11 MAC protocol. Wireless networks 2001, 7(2):159171. 10.1023/A:1016637622896View ArticleMATHGoogle Scholar
 3GPP: Physical layer aspects for Evolved Universal Terrestrial Radio Access (UTRA). 2006.Google Scholar
 Kyösti P, Meinilä J, Hentilä L, Zhao X, Jämsä T, Schneider C, Narandzic M, Milojevic M, Hong A, Ylitalo J, Others: WINNER II channel models (d1. 1.2 v1. 1). no. IST4027756 WINNER II, D 1Google Scholar
 Mehlführer C, Wrulich M, Ikuno JC, Bosanska D, Rupp M: Simulating the Long Term Evolution physical layer. Proc of the 17th European Signal Processing Conference (EUSIPCO 2009), Glasgow, Scotland 2009, 14711478.Google Scholar
 EARTH: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown. 2010.Google Scholar
 Badic B, O'Farrrell T, Loskot P, He J: Energy efficient radio access architectures for green radio: large versus small cell size deployment. Vehicular Technology Conference Fall (VTC 2009Fall), 2009 IEEE 70th, IEEE 2010, 15.Google Scholar
 Wyner AD: Shannontheoretic approach to a Gaussian cellular multipleaccess channel. Information Theory, IEEE Transactions on 1994, 40(6):17131727. 10.1109/18.340450MathSciNetView ArticleMATHGoogle Scholar
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