Open, closed, and shared access femtocells in the downlink
 HanShin Jo^{1}Email author,
 Ping Xia^{2} and
 Jeffrey G Andrews^{2}
https://doi.org/10.1186/168714992012363
© Jo et al.; licensee Springer. 2012
Received: 21 November 2011
Accepted: 14 November 2012
Published: 15 December 2012
Abstract
A fundamental choice in femtocell deployments is the set of users which are allowed to access each femtocell. Closed access restricts the set to specifically registered users, while open access allows any mobile subscriber to use any femtocell. Which one is preferable depends strongly on the distance between the macrocell base station (MBS) and femtocell. The main results of the this article are lemmas which provide expressions for the signaltointerferenceplustonoise ratio (SINR) distribution for various zones within a cell as a function of this MBSfemto distance. The average sum throughput (or any other SINRbased metric) of home users and cellular users under open and closed access can readily be determined from these expressions. We show that unlike in the uplink, the interests of home and cellular users are in conflict, with home users preferring closed access and cellular users preferring open access. The conflict is most pronounced for femtocells near the cell edge, when there are many cellular users and fewer femtocells. To mitigate this conflict, we propose a middle way which we term shared access in which femtocells allocate an adjustable number of timeslots between home and cellular users such that a specified minimum rate for each can be achieved. The optimal such sharing fraction is derived. Analysis shows that shared access achieves at least the overall throughput of open access while also satisfying rate requirements, while closed access fails for cellular users and open access fails for the home user.
Keywords
1 Introduction
Femtocells are small formfactor base stations that can be installed within an existing cellular network. They can be installed either by an enduser or by the service provider and are distinguished from pico or microcells by their low cost and power and use of basic IP backhaul, and from WiFi by their use of cellular standards and licensed spectrum[1]. Femtocells are a very promising and scalable method for meeting the everincreasing demands for capacity and highrate coverage. Since femtocells share spectrum with macrocell networks, managing crosstier interference between femto and macrocells is essential[2–4]. Furthermore, a basic question, particularly for enduserinstalled femtocells, is which users in the network should be allowed to use a given femtocell.
1.1 Motivation and related work
The uplink performance of femtocell access schemes has been investigated in[16, 17]. The interrelationship between the traffic type, access policy, and performance of highspeed packet access (HSPA) was examined in a simulationbased study[16]. In[17], an analytical framework was presented from open versus closed access. Both studies suggested a hybrid access approach with an upper limit to the number of unregistered users to access the femtocell. We term this approach shared access since the femtocell is shared with cellular users, but within limits and hence not fully open. With respect to the uplink throughput of registered home users, open (or shared) access reduces interference by handing over the loud neighbor at the expense of FAP resource sharing. The tradeoff is such that open access is generally preferred for both home users and cellular users, since the interference reduction is so important[17]. Does the same tradeoff hold in the downlink?
It would seem that the tradeoff is different in the downlink since here the FAP is the loud neighbor (see Figure1). Therefore, serving unregistered users with the FAP benefits them at the cost of FAP resources. Meanwhile, there is at best a very small decrease in downlink interference to the home user. The downlink capacity of open versus closed access has been studied using simulations for HSPA femtocells[16, 18] and OFDMA femtocells[19–21]. These studies propose and analyze shared access methods with limits on the number of unregistered users[18] and frequency subchannels for them[20, 21]. Indeed, these studies find that cellular user performance is improved with open (or shared) access at the cost of reduced home user performance.
Recently, a theoretical analysis of signaltointerferenceplustonoise ratio (SINR) in general Ktier downlink heterogeneous cellular networks with open or closed access was proposed in[22–24]. For K = 2, macrocells plus femtocells can be considered as a special case. Analytical results on downlink outage probability for open and closed access have also been been presented in[25]. However, these models do not classify mobile users into femtocellregistered (home users) and femtocellunregistered (cellular users), and does not define the distribution area of each type of users. It is thus hard to quantify the throughput of the registered home users and unregistered cellular users separately. Furthermore, since all BSs are distributed as a Poisson Point Process (PPP), the feasible distance between macrocell base station (MBS) and femtocell is arbitrarily given within a range from zero to infinity (i.e., not fixed). Such a random BS model appears useful for obtaining outage/throughput averaged over the entire network, but may not quantify the throughput of open, closed, and shared access for a particular femtocell at a given distance from the MBS.
1.2 Contributions and main insights
Clearly, it is desirable to quantify the throughput of open, closed, and shared access as a function of the FAP–MBS distance. It should provide the best access or optimal resource allocation (in shared access) for each femtocell based on its location, and even the density of femtocells and cellular users. It would be more realistic if the femtocell and cellular user positions were not fixed, but rather were modeled as a spatial random process (see[26, 27] and references therein). Unlike the closed access model[9, 28] with a fixed radio range (coverage) for the femtocell, we develop a unified model for both open and closed accesses, where the femtocell coverage area depends on its distance from the MBS. In addition, the femtocell coverage is divided into zones, each of which has unique SINR model due to path loss and wall loss (indoor/outdoor), type of serving BS (MBS/FAP), and access control (open/closed). Ideally, a general statistical distribution of the SINR for the various zones could be found. Since metrics such as outage probability, error probability, and throughput follow directly from SINR, once the SINR distribution is known these metrics can be computed quite quickly and easily[27]. Deriving such an SIR distribution (we neglect both thermal noise and interference from other cells) is the main contribution of this article, and is used to draw a few conclusions about access strategies in the downlink.
First, we see that unlike the uplink[17], the preferred access schemes for home and cellular users are incompatible, with home user preferring closed access. For femtocells with coverage area extending outside the home, i.e., far from the MBS, closed access provides higher sum throughput for home user and lower sum throughput for neighboring cellular users, when compared to open access. For example, in a cell edge femtocell, open access causes at least 20% throughput loss to home user compared to closed access, while the neighboring cellular user experiences outage for typical data service (less than 15 kbps for 5 MHz bandwidth) in closed access. Furthermore, we observe that the open access femtocells far from MBS reduce the macrocell load, thereby open access rather than closed access offers higher throughput for a few home users (in its femtocell coverage area smaller than home area) located near and connected to the MBS. Nevertheless, most home users in cell site accessing FAPs are still reluctant to use their femtocells in open access.
Since neither open nor closed access can completely satisfy the need of both user groups, we next propose a shared access approach where the femtocell has a timeslot ratio η between the home and cellular users it serves, where η = 1 is closed access. An optimal value of η is found to maximize network throughput subject to QoS requirements on the minimum throughput per home and cellular user. For example, given a cell edge femtocell with minimum throughput of 50 kbps/cellular user and 500 kbps/home user, this shared access approach achieves about 80% higher network throughput than open access. When the QoS requirements increase in favor of significantly higher throughput of cellular user, shared access provides the lower network throughput than open access.
2 System model
Notation and simulation values
Symbol  Description  Sim. value 

${\mathcal{F}}_{\mathtt{\text{i}}}$  Indoor area covered by the FAP at D > D _{ th } (a disc with the radius R_{ i })  N/A 
${\mathcal{F}}_{\mathtt{\text{o}}}$  Outdoor area covered by the FAP at D > D _{ th } in open access or  
covered by the MBS in closed access  N/A  
(a circular annulus with inner radius R _{ i } and outer radius R _{ f })  
${\mathcal{F}}_{\mathtt{\text{a}}}$  Indoor area covered by the FAP at D ≤ D _{ th } (a disc with the radius R _{ f })  N/A 
${\mathcal{F}}_{\mathtt{\text{b}}}$  Indoor area covered by the MBS (a circular annulus with inner radius R _{ f }  N/A 
and outer radius R _{ i } with respect to the FAP at D ≤ D _{ th })  
D  Distance between FAP and central MBS  Not fixed 
D _{ th }  Threshold distance (Radius of inner region)  Not fixed 
D _{ c }  Distance between central MBS and homeuser (or neighboring cellular user)  Not fixed 
R  Distance between FAP and homeuser (or neighboring cellular user)  Not fixed 
R _{ f }  Femtocell radius  Not fixed 
R _{ c }  Macrocell radius  500 m 
R _{ i }  Indoor (home) area radius  20 m 
P _{ c }  Transmit power at macrocell  43 dBm[29] 
P _{ f }  Transmit power at femtocell  13 dBm[29] 
α  Outdoor path loss exponent  4 
β  Indoor path loss exponent  4 
L  Wall penetration loss  0.5 (−3 dB) 
G _{ n }  Shannon gap  3 dB 
N  Number of discrete levels for Mary modulation (MQAM)  8 
Ω_{ c }  Required minimum throughput of cellular user for hybrid access  0.01 bps/Hz 
Ω_{ h }  Required minimum throughput of home user for hybrid access  0.1 bps/Hz 
2.1 Channel model and multilevel modulation
2.2 Femtocell coverage and cell association
We assume a maxSINR cell association where each mobile users is served by the station (MBS or FAP) providing the strongest average power to them. This is desirable in that it maximizes the user SINRs in downlink. We therefore define a femtocell coverage$\mathcal{F}\subset {\mathbb{R}}^{2}$ as the area inside which the average received power from the FAP is stronger than that from the central MBS. The coverage is mathematically modeled by the following lemma.^{c}
Lemma 1
where$\kappa =\frac{{P}_{\mathtt{\text{c}}}}{{P}_{\mathtt{\text{f}}}L}\gg 1$.
Proof
Consider a central MBS located at (0,0) and an FAP at distance D from the MBS. Without loss of generality, the FAP is assumed to be located at (D,0). The received power at the position (x,y) with distances${D}_{\mathtt{\text{c}}}=\sqrt{{x}^{2}+{y}^{2}}$ and$R=\sqrt{{(Dx)}^{2}+{y}^{2}}$ from MBS and FAP are, respectively, given as${P}_{\mathtt{\text{r}}}^{\left(c\right)}={P}_{\mathtt{\text{c}}}{P}_{0}{\left(\frac{{D}_{\mathtt{\text{c}}}}{{d}_{0}}\right)}^{\alpha}$ and${P}_{\mathtt{\text{r}}}^{\left(f\right)}={P}_{\mathtt{\text{f}}}L{P}_{0}{\left(\frac{R}{{d}_{0}}\right)}^{\alpha}$, where P_{0} is the path loss at a reference distance d_{0}. The contour with${P}_{\mathtt{\text{r}}}^{\left(c\right)}={P}_{\mathtt{\text{r}}}^{\left(f\right)}$ (zero dB SIR) satisfies$\frac{{x}^{2}+{y}^{2}}{{(Dx)}^{2}+{y}^{2}}={\kappa}^{\alpha /2}$, where$\kappa =\frac{{P}_{\mathtt{\text{c}}}}{{P}_{\mathtt{\text{f}}}L}\gg 1$. The equation is rewritten as${\left(x\frac{{\kappa}^{2/\alpha}D}{{\kappa}^{2/\alpha}1}\right)}^{2}+{y}^{2}=\frac{{\kappa}^{2/\alpha}{D}^{2}}{{({\kappa}^{2/\alpha}1)}^{2}}$. Note that$\frac{{\kappa}^{2/\alpha}}{{\kappa}^{2/\alpha}1}$ is very close to 1 because κ ≫ 1. For example,$\frac{{\kappa}^{2/\alpha}}{{\kappa}^{2/\alpha}1}=1.02$ for the values in Table1. Thus, the equation is simplified to (x−D)^{2} + y^{2} = κ^{−2/α}D^{2}, which is the equation of circle and completes the proof. □
For all types of femtocell access control, the home users can freely choose between the MBS and their own femtocell according to the association policy. Thus, they will be served by their own femtocell if in its coverage, and by the MBS otherwise. On the other hand, whether a cellular user will be served by a femtocell is more complicated and dependent on the type of access control: (1) in closed access, a femtocell will not serve any cellular users, even if they are in its coverage; (2) in the open access, a femtocell will serve any cellular users as long as they are in its coverage, they are named neighboring cellular users. In the inner region where the femtocell coverage area is smaller than the indoor area, some home users outside the femtocell coverage actually communicate with the MBS, while in the outer region where the femtocell coverage area is larger than the indoor area, the neighboring cellular users would like to connect to the open access FAP. In conclusion, the SIR of each type of users needs a slightly different SIR model for the femtocell location (inner region or outer region) and femtocell access control (open or closed). We thus classify femtocell coverage into four geographic zones, whereby users in the same zone have the same SIR model (See Figure2).
When an FAP is in inner region, its coverage is divided into${\mathcal{F}}_{\mathtt{\text{a}}}$ and${\mathcal{F}}_{\mathtt{\text{b}}}$ as follows.

${\mathcal{F}}_{\mathtt{\text{a}}}$: indoor area where home users are served by the FAP—a disc with radius R_{ f }.

${\mathcal{F}}_{\mathtt{\text{b}}}$: indoor area where home users are served by the MBS—a circular annulus with inner radius R_{ f }and outer radius R_{ i }.
When an FAP is in outer region, its coverage is divided into${\mathcal{F}}_{\mathtt{\text{i}}}$ and${\mathcal{F}}_{\mathtt{\text{o}}}$ as follows.

${\mathcal{F}}_{\mathtt{\text{i}}}$: indoor area where home users are served by the FAP—a disc with radius R_{ i }.

${\mathcal{F}}_{\mathtt{\text{o}}}$: nearby outdoor area where neighboring cellular users are served by the FAP (in open access) or the MBS (in closed access)—a circular annulus with inner radius R _{ i } and outer radius R _{ f }.
Although the cell coverage model based on multiple geographic zones, i.e., with multiple SIR model, is more intricate than conventional model[9, 12] with single SIR model for closed access only, it is essential for a conclusive comparative study of open, closed, and shared access. We do not compute the throughput of cellular users outside the four zones, since the throughput is not hardly affected by the femtocell access control, so it does not change the cellular user’s preference on femtocell access.
3 Perzone average SIR and throughput
Consider a reference FAP at distance D from a central MBS, and its home users (or neighboring cellular users) at distance R and D _{ c } from the FAP and the MBS, respectively. As shown in Section 2, according to SIR model, all the users divided into four groups located in the zone${\mathcal{F}}_{\mathtt{\text{a}}}$,${\mathcal{F}}_{\mathtt{\text{b}}}$,${\mathcal{F}}_{\mathtt{\text{i}}}$, and${\mathcal{F}}_{\mathtt{\text{o}}}$. We analyze the throughput for the each zone, and then, based on it, derive pertier throughput in next section. We assume smallsized femtocell R ≪ D resulting D _{ c }≈ D.
3.1 Neighboring cellular user in zone${\mathcal{F}}_{\mathtt{\text{o}}}$
where g_{0} is the exponentially distributed channel power (with unit mean) from the MBS. h_{ j }is the exponentially distributed channel power (with unit mean) from the FAP. X_{ j } denotes the distance between the user and the FAP. The following lemma quantifies the user SIR for the zone${\mathcal{F}}_{\mathtt{\text{o}}}$.
Lemma 2
 (1)Closed access:$\begin{array}{l}{S}_{\mathtt{\text{o}}}^{\mathtt{\text{CA}}}\left(\mathrm{\Gamma}\right)=\phantom{\rule{0.3em}{0ex}}1\phantom{\rule{0.3em}{0ex}}\frac{{e}^{\lambda C{\left(K\mathrm{\Gamma}\right)}^{2/\alpha}}}{{R}_{\mathtt{\text{f}}}^{2}{R}_{\mathtt{\text{i}}}^{2}}\phantom{\rule{0.3em}{0ex}}\left(A\left({R}_{\text{f}},\alpha \right)A\left({R}_{\text{i}},\alpha \right)\right)\phantom{\rule{2em}{0ex}}\end{array}$(5)$\begin{array}{l}A(x,y)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{x}^{2}\left(1\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{}_{2}{F}_{1}\left[2/y,1;1+2/y;{x}^{y}/\left(K\mathrm{\Gamma}\right)\right]\right)\end{array}$
where$C=\frac{2{\Pi}^{2}}{\alpha}csc\left(\frac{2\Pi}{\alpha}\right)$,$K=\frac{{P}_{\mathtt{\text{f}}}L{D}^{\alpha}}{{P}_{\mathtt{\text{c}}}}$, and _{2}F_{1}[·] is the Gauss hypergeometric function.
 (2)Open access:${S}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}\left(\mathrm{\Gamma}\right)=1\frac{1}{{R}_{\mathtt{\text{f}}}^{2}{R}_{\mathtt{\text{i}}}^{2}}{\int}_{{R}_{\mathtt{\text{i}}}^{2}}^{{R}_{\mathtt{\text{f}}}^{2}}\frac{{e}^{r\lambda C{\mathrm{\Gamma}}^{2/\alpha}}}{{K}^{1}\mathrm{\Gamma}{r}^{\alpha /2}+1}\mathtt{\text{d}}r.$(6)If α = 4, the closedform expression is given as${S}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}\left(\mathrm{\Gamma}\right)=1\frac{B\left({R}_{\mathtt{\text{f}}}^{2}\right)B\left({R}_{\mathtt{\text{i}}}^{2}\right)}{({R}_{\mathtt{\text{f}}}^{2}{R}_{\mathtt{\text{i}}}^{2})},$(7)$\begin{array}{ll}\phantom{\rule{6pt}{0ex}}B\left(x\right)& =\frac{1}{\sqrt{z}}\left[\left(\mathtt{\text{Re}}\left\{E\right(\mathit{iw}\left)\right\}+\mathtt{\text{Re}}\left\{E\right(\mathit{xy}+\mathit{iw}\left)\right\}\right)sin\left(w\right)\right.\\ \phantom{\rule{1em}{0ex}}+\left.\left(\mathtt{\text{Im}}\left\{E\right(\mathit{iw}\left)\right\}\mathtt{\text{Im}}\left\{E\right(\mathit{xy}+\mathit{iw}\left)\right\}\right)cos\left(w\right)\right],\end{array}$$\phantom{\rule{2.50em}{0ex}}y=\lambda C\sqrt{\mathrm{\Gamma}},\phantom{\rule{.3em}{0ex}}z=\mathrm{\Gamma}{K}^{1},\phantom{\rule{.3em}{0ex}}w=y/\sqrt{z}=\lambda C\sqrt{K},$(8)
where Re{z} and Im{z} represent the real and imaginary parts of z, respectively,$E\left(z\right)={\int}_{\infty}^{z}\frac{{e}^{t}}{t}\mathit{dt}$ is the Exponential integral function.
Proof
See Proof of Lemma 2 in Appendix. □
3.2 Home user SIR In zone${\mathcal{F}}_{\mathtt{\text{i}}}$,${\mathcal{F}}_{\mathtt{\text{a}}}$, or${\mathcal{F}}_{\mathtt{\text{b}}}$
This is the same as the cellular user SIR for open access given in (4) except for the distinction that this home user is indoors and so the propagation terms are adjusted accordingly. The SIR distribution of the home user in${\mathcal{F}}_{\mathtt{\text{i}}}$ is given in the following lemma.
Lemma 3
where w, y, and z for calculating$B\left({R}_{\mathtt{\text{i}}}^{2}\right)$ are given in (12).
Proof
See Proof of Lemma 3 in Appendix. □
Note that the user SIR in the zone${\mathcal{F}}_{\mathtt{\text{a}}}$ is given in (9), since the zone is the indoor area covered by an FAP at D ≤ D _{ th } (in the inner region). The SIR distribution of the home users in${\mathcal{F}}_{\mathtt{\text{a}}}$ is given in the following corollary.
Corollary 1
The SIR is the same as the cellular user SIR for closed access given in (4) except for the distinction that this home user is indoors and so the propagation terms are adjusted accordingly. Thus, the SIR distribution for${\mathcal{F}}_{\mathtt{\text{b}}}$ is given in similar form to (5) as shown in the following lemma.
Lemma 4
where$K=\frac{{P}_{\mathtt{\text{f}}}{D}^{\alpha}}{{P}_{\mathtt{\text{c}}}L}$.
Proof
See Proof of Lemma 4 in Appendix. □
3.3 Numerical results
4 Pertier throughput: closed access versus open access
where N _{ f1 }= N _{ f }(D _{ th }/R _{ c })^{2} and N _{ f2 }= N _{ f }(1−(D _{ th }/R _{ c })^{2}) are, respectively, the average number of femtocells in the inner region (D ≤ D _{ th }) and the outer region (D _{ th }< D ≤ R _{ c }). Furthermore, (a) follows from that U _{ h }= U _{ i } in the outer region and U _{ h }= U _{ a } + U _{ b } in the inner region.
4.1 Closed access
The femtocell/macrocell access scenario is different for inner and outer regions, from which the following theorem quantifies sum throughput of home user and neighboring cellular user.
Theorem 1
Proof
See Proof of Theorem 1 in Appendix. □
Remark 1.
Interestingly, the average sum throughput of home users is lower for a farther FAP within the inner region (D ≤ D _{ th }) but higher for a farther FAP within the outer region (D > D _{ th }). It follows that from Figure3, increasing D enhances T _{ i } but reduces T _{ a } and T _{ b }, and from Equation (21), increasing D reduces${\rho}_{\mathtt{\text{b}}}^{\mathtt{\text{CA}}}$. Intuitively, the signal from the MBS is interference to home users in the outer region, but it is the desired signal to some home users connecting to the MBS in the inner region.
Remark 2.
The perzone throughput T _{ i }, T _{ a }, and T _{ b } are not affected by λ _{ c }. From Equation (21), increasing λ _{ c } reduces${\rho}_{\mathtt{\text{b}}}^{\mathtt{\text{CA}}}$. Intuitively, this means that an MBS load is higher at more cellular user deployment and thereby fewer time slots are allocated home user served by the MBS. For an FAP in inner region,${T}_{\mathtt{\text{h}}}^{\mathtt{\text{CA}}}$ is thus higher for a lower cellular user density, while it is independent of cellular user density in outer region. Since from Equation (22) ρ _{ o } is higher at a larger U _{ c }, but${T}_{\mathtt{\text{o}}}^{\mathtt{\text{CA}}}$ is independent on U _{ c }, higher cellular user density increases the average sum throughput of neighboring cellular user.
4.2 Open access
Theorem 2.
Proof.
See Proof of Theorem 2 in Appendix. □
Remark 3.
In the inner region,${T}_{\mathtt{\text{h}}}^{\mathtt{\text{OA}}}$ is lower at farther FAP, since increasing D reduces${\rho}_{\mathtt{\text{b}}}^{\mathtt{\text{OA}}}$ in Equation (25) as well as T _{ a } and T _{ b } in Figure3. Whereas, in the outer region, as D increases,${T}_{\mathtt{\text{h}}}^{\mathtt{\text{OA}}}$ increase and begins to decrease at sufficiently large D. This is because T _{ i } is upper limited by the highest order modulation in spite of quite high SIR at large D, while ρ _{ i } decreases to zero. Next, from Equation (26), 1−ρ _{ i } is higher for larger λ _{ c } or D, which enhances${T}_{\mathtt{\text{c}}}^{\mathtt{\text{OA}}}$.
The throughput comparison of both the access schemes is given in the remarks below.
Remark 4.
First,${T}_{\mathtt{\text{h}}}^{\mathtt{\text{CA}}}\le {T}_{\mathtt{\text{h}}}^{\mathtt{\text{OA}}}$ for the inner region, because closed access rather than open access increases the number of users supported by the MBS and thereby${\rho}_{\mathtt{\text{b}}}^{\mathtt{\text{CA}}}\le {\rho}_{\mathtt{\text{b}}}^{\mathtt{\text{OA}}}$. Intuitively, open access femtocells in the outer region admit neighboring cellular users which reduces the macrocell load. This effectively increases the throughput of some home users (supported by the MBS) in inner region. Whereas,${T}_{\mathtt{\text{h}}}^{\mathtt{\text{CA}}}\ge {T}_{\mathtt{\text{h}}}^{\mathtt{\text{OA}}}$ for the outer region because ρ _{ i }≤ 1 from Equation (26). Second, we observe${T}_{\mathtt{\text{c}}}^{\mathtt{\text{CA}}}<{T}_{\mathtt{\text{c}}}^{\mathtt{\text{OA}}}$. Since${U}_{\mathtt{\text{i}}}+{U}_{\mathtt{\text{o}}}<{U}_{\mathtt{\text{c}}}+{N}_{\mathtt{\text{f1}}}{\overline{U}}_{\mathtt{\text{b}}}$, comparing$1{\rho}_{\mathtt{\text{i}}}=\frac{{U}_{\mathtt{\text{o}}}}{{U}_{\mathtt{\text{i}}}+{U}_{\mathtt{\text{o}}}}$ and Equation (53) yields 1 − ρ _{ i }> ρ _{ o }. Moreover,${T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}$ is obviously larger than${T}_{\mathtt{\text{o}}}^{\mathtt{\text{CA}}}$. In conclusion, home user and neighboring cellular user prefer opposite access schemes.
4.3 Numerical results
Throughput comparison of open, closed, and shared access for FAPMBS distance D , cellular user density U _{ c } , femtocell density N _{ f }
HighD  LowD  

and/or lowN _{ f }  and/or highN _{ f }  
and/or highU _{ c }  and/or lowU _{ c }  
Home user throughput  Inner  Open = Shared > Closed  Open = Shared ≫ Closed 
region  
Outer  Closed ≫ Shared ≫ Open  Closed > Shared > Open  
region  
Cellular user throughput  Open ≫ Shared ≫ Closed  Open > Shared > Closed  
Home  Closed access  Closed access  
users  
Preferred access  Cellular  Open access  Open access 
users  
Both  Shared access  Shared access  
users 
5 Shared access: timeslot allocation
where${\overline{T}}_{\mathtt{\text{h}}}=\frac{\eta {T}_{\mathtt{\text{i}}}}{{U}_{\mathtt{\text{i}}}}$ and${\overline{T}}_{\mathtt{\text{c}}}=\frac{(1\eta ){T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}}{{U}_{\mathtt{\text{o}}}}$.
Theorem 3.
The solution η^{∗} is feasible when it is equal or larger than$\frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{i}}}}{{T}_{\mathtt{\text{i}}}}$.
Proof
Denote${\mathcal{Q}}_{1}$ and${\mathcal{Q}}_{2}$ as a set of η satisfying the QoS requirement (28) in the order of description, respectively. we then obtain intersection of the three sets as$\mathcal{Q}={\mathcal{Q}}_{1}\cap {\mathcal{Q}}_{2}=\left\{\eta \mid \frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{i}}}}{{T}_{\mathtt{\text{i}}}}\le \eta \le 1\epsilon \frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{o}}}}{{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}}\right\}$ for$\frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{i}}}}{{T}_{\mathtt{\text{i}}}}\le 1\epsilon \frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{o}}}}{{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}}$. Define a function of η as$f\left(\eta \right)=\eta {T}_{\mathtt{\text{i}}}+(1\eta ){T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}=({T}_{\mathtt{\text{i}}}{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}})\eta +{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}$. Since${T}_{\mathtt{\text{i}}}>{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}$, f(η) monotonically increases with increasing η. Thus, η^{∗} is the maximum$\eta \in \mathcal{Q}$, which yields Equation (29). Moreover, since$\mathcal{Q}=\varnothing $ for$\frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{i}}}}{{T}_{\mathtt{\text{i}}}}>1\epsilon \frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{o}}}}{{T}_{\mathtt{\text{o}}}^{\mathtt{\text{OA}}}}$, η^{∗} is feasible for${\eta}^{\ast}\ge \frac{{\mathrm{\Omega}}_{\mathtt{\text{h}}}{U}_{\mathtt{\text{i}}}}{{T}_{\mathtt{\text{i}}}}$. □
Remark 5.
Shared access with η^{∗} = 1 and${\eta}^{\ast}={\rho}_{\mathtt{\text{i}}}=\frac{{U}_{i}}{{U}_{i}+{U}_{o}}$ is closed and open accesses, respectively. The QoS parameter ε ∈ (0,1] determines the priority of home users relative to cellular users with ε = 1 ensuring identical required minimum throughput to home and cellular users. In Equation (29), increasing ε reduces η^{∗} and allocates more timeslots to cellular users. This indicates that shared access provides lower network throughput than open access when ε is set high, e.g., such that${\eta}^{\ast}<\frac{{U}_{i}}{{U}_{i}+{U}_{o}}$.
6 Conclusion
The overall contribution of this article is a new analytical framework for evaluating throughput tradeoffs regarding femtocell access schemes in downlink twotier femtocell networks. The framework quantifies femtocellsitespecific “loud neighbor” effects and can be used to compare other techniques, e.g., power control, spectrum allocation, and MIMO. Our results show that unlike the uplink results in[17], the preferred access scheme for home and cellular users is incompatible. In particular, closed access provides higher throughput for home users and lower throughput for neighboring cellular users; vice versa with open access. As a compromise, we suggest shared access where femtocells choose a timeslot ratio for their home and neighboring cellular users to maximize the network throughput subject to a networkwide QoS requirement. For femtocells within the outer area, shared access achieves higher network throughput than open access while satisfying the QoS of both home and cellular users. These results motivate shared access—i.e., open access, but with limits—in femtocellenhanced cellular networks with universal frequency reuse.
Appendix
Proof of Lemma 2
Here, by using substitution R^{2} = r, we obtain the desired result in Equation (6).
where$y=\lambda C\sqrt{\mathrm{\Gamma}},\phantom{\rule{1em}{0ex}}z=\mathrm{\Gamma}{K}^{1}$. (a) follows from the integral formula in[33] and (b) is given on the mirror symmetry of Exponential integral function, i.e.,$E\left(\stackrel{\u0304}{z}\right)=\overline{E\left(z\right)}$. Combining Equations (38) and (43) gives the desired result (7).
Proof of Lemma 3
which proves Equation (10).
where$y=\lambda \mathit{\text{CL}}\sqrt{\mathrm{\Gamma}},\phantom{\rule{1em}{0ex}}z=K\mathrm{\Gamma}$. (a) follows from the integral formula in[33] and (b) is given on the mirror symmetry of Exponential integral function. Combining Equations (42) and (??) gives the desired result (11).
where B(x) is given from Equation (43). This gives the desired spatially averaged SIR distribution in Equation (13).
Proof of Lemma 4
Proof of Theorem 1
Plugging Equations (49), (50), (51) into (48) gives the desired result in Equation (21).
Plugging Equations (50), (51), (54) into (53) gives the desired result in Equation (22).
Proof of Theorem 2
where (a) is given from Equation (54), and (b) follows from$\mathbb{E}\left[{D}^{2}\right]={\int}_{{D}_{\mathtt{\text{th}}}}^{{R}_{\mathtt{\text{c}}}}{D}^{2}\left(\frac{2D}{{R}_{\mathtt{\text{c}}}^{2}{D}_{\mathtt{\text{th}}}^{2}}\right)\mathtt{\text{d}}D=({R}_{\mathtt{\text{c}}}^{2}+{D}_{\mathtt{\text{th}}}^{2})/2$ for D > D _{ th }. Plugging Equations (49), (50), (51), (57) into (56) gives the desired result in Equation (25).
Plugging Equations (54) and${U}_{\mathtt{\text{i}}}={U}_{\mathtt{\text{h}}}=\Pi {\lambda}_{\mathtt{\text{h}}}{R}_{\mathtt{\text{i}}}^{2}$ into (59) gives the desired result in Equation (26).
Endnotes
^{a}Fixed power (no transmit power control of femtocells) is assumed as in[16, 18–21], to focus on the effects of femtocell access scheme.^{b}Figure3 suggests that the assumption is not a significant omission of interference effects for dense femtocell systems. See Section 2 for more discussion.^{c}A similar approach is presented in[10], which considers different path loss exponents as well as the same path loss exponents for macrocell and femtocell, but not includes wall penetration loss.
Declarations
Acknowledgements
This study was supported by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA20121291101108) and Motorola Solutions. This article was partially presented at the International Conference on Communications, Kyoto, Japan, June 5–9, 2011[34].
Authors’ Affiliations
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