1 Introduction
1.1 Motivation and related work
1.2 Contributions and main insights
2 System model
c
centered at an MBS. To show the impact of macro–femto distance on femtocell access control, we analyze a target femtocell, its location denoted as X0, which is located at a distance D away from the MBS. Since FAPs are installed by the end customer, they are distributed with randomness rather than regular pattern. FAPs are thus distributed according to a homogeneous PPP Φ = {X
j
}j=1,2,… with intensity λ, where each X
j
is the location of the j th FAP. The mean number of femtocells per cell cite is given as. Home users are uniformly deployed indoors (home), a disc of radius R
i
centered at their FAP. Cellular users are uniformly distributed outdoors. A summary of notation is given in Table1.Symbol | Description | Sim. value |
---|---|---|
Indoor area covered by the FAP at D > D
th
(a disc with the radius R
i
) | N/A | |
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
) | ||
Indoor area covered by the FAP at D ≤ D
th
(a disc with the radius R
f
) | N/A | |
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 M-ary modulation (M-QAM) | 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 multi-level modulation
c
and P
f
, respectively. We assume that orthogonal multiple access is used (TDMA or OFDMA on a per subband basis), thus no intracell interference is considered. Interference from neighboring macrocell BSs is ignored for analytical tractability.b We consider multi-level M-ary modulation single carrier transmission that is adapted to the received SIR γ, thus each user is assumed to estimate its SIR and provide perfect SIR feedback to their MBS (or FAP). Define N SIR regions as, where Γ1 is the minimum SIR providing the lowest discrete rate and ΓN + 1= ∞. Then, the instantaneous transmission rate (in bps/Hz) is2.2 Femtocell coverage and cell association
Lemma 1
f
isProof
th
as a nominal threshold distance at which the femtocell coverage area is exactly equal to the indoor (home) area, i.e., R
f
= R
i
. We accordingly partition the macrocell into two regions, inner region (D ≤ D
th
) and outer region (D > D
th
). Note that D
th
is the same for all FAPs, since we assume all indoor regions with same area.
-
: indoor area where home users are served by the FAP—a disc with radius R
f
. -
: indoor area where home users are served by the MBS—a circular annulus with inner radius R
f
and outer radius Ri
.
-
: indoor area where home users are served by the FAP—a disc with radius R i .
-
: 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 Rf
.
3 Per-zone average SIR and throughput
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,,, and. We analyze the throughput for the each zone, and then, based on it, derive per-tier throughput in next section. We assume small-sized femtocell R ≪ D resulting D
c
≈ D.3.1 Neighboring cellular user in zone
Lemma 2
f
and inner radius R
i
, is given as follows Re
{z} and Im
{z} represent the real and imaginary parts of z, respectively, is the Exponential integral function.Proof
3.2 Home user SIR In zone,, or
Lemma 3
i
, is given asProof
th
(in the inner region). The SIR distribution of the home users in is given in the following corollary.Corollary 1
f
. Thus, the CDF of spatially averaged SIR over the zone is given by (10) with R
f
replacing R
i
, i.e.,th
(in the inner region). The user SIR isLemma 4
i
and inner radius R
f
, is given as3.3 Numerical results
f
degrades the averaged throughput of all the zones.
4 Per-tier throughput: closed access versus open access
a
, U
b
, U
i
, and U
o
are the number of users in the zone,,, and, respectively. Let U
c
and U
h
denote the number of outdoor cellular users and the number of home users per femtocell, respectively, and λ
c
and λ
h
are the corresponding user density. Average number of user in a macrocell cite is then given byf1
= 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
Theorem 1
a
(D), T
b
(D), T
i
(D), and is given from Equation (17). and ρ
o
are the fraction of time-slot dedicated to the home users in and the cellular users in, respectively, among all users supported by the MBS, which is given asProof
Remark 1.
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. 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.
i
, T
a
, and T
b
are not affected by λ
c
. From Equation (21), increasing λ
c
reduces. 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, 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 is independent on U
c
, higher cellular user density increases the average sum throughput of neighboring cellular user.4.2 Open access
Theorem 2.
i
is the fraction of time-slot dedicated to the home users in among home and cellular users supported from the FAP. They are given asProof.
Remark 3.
a
and T
b
in Figure3. Whereas, in the outer region, as D increases, 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.Remark 4.
i
≤ 1 from Equation (26). Second, we observe. Since, comparing and Equation (53) yields 1 − ρ
i
> ρ
o
. Moreover, is obviously larger than. In conclusion, home user and neighboring cellular user prefer opposite access schemes.4.3 Numerical results
f
and cellular users U
c
. Since R
f
= R
i
at D = D
th
, we obtain the distance D
th
of 130 m by substituting R
i
= 20 into Equation (3). In closed access, home user throughput decreases with D(D ≤ D
th
), while it increases with D(D > D
th
) per Remark 1. For D > D
th
, the home user throughput in open access first increases then decreases with increasing D. Additionally, Figure5 shows that the turning point moves into the cell interior with increasing U
c
. This is because increasing D and U
c
increases the number of neighboring cellular users, and thus, the time resource allocated to home user in femtocell downlink is reduced. In Figure4, the throughput for both open and closed accesses is degraded, since the aggregated interference from other femtocells increases with N
f
. We observe that unlike the case of D > D
th
, open access outperforms closed access for D ≤ D
th
. However, the throughput loss of open access at D > D
th
dominates the throughput gain at D ≤ D
th
. Thus, closed access is better for home users
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 |
th
, open access is inferior to closed access with respect to the network throughput. The reason is from the inequality given by. Intuitively, since from Figure3, the decrement of home user throughput T
i
due to time resource sharing with cellular users in open access prevails against the increment of cellular user throughput by substituting for. In a different point of view, this implies that a slight increase in the time fraction ρ
i
provides a high increase at the cost of a slight drop in, i.e., an increase in network throughput T
OA
. This, as well as the extremely low throughput in closed access motivates the shard access femtocells using time slot allocation, which will be discussed in the next section.
5 Shared access: time-slot allocation
i
is dependent on the number of home users and cellular users, the time-slot allocation in the shared access optimizes η to maximize the network throughput T
SA
while satisfying QoS requirement. The network throughput is given ash
(home user) and Ω
c
= ε Ω
h
with ε ∈ (0,1] (cellular user), respectively. Satisfying the QoS, the time-slot allocation problem to maximize the network throughput T
SA
is formulated asTheorem 3.
Proof
Remark 5.
h
= 0.01 (corresponding to 500 kbps for 5-MHz bandwidth) and Ω
c
= 0.01ε. Since R
f
= R
i
at D = D
th
, we obtain the distance D
th
of 130 m by substituting R
i
= 20 into Equation (3). For D > D
th
, the throughput of shared access increases with decreasing ε. Considering lower ε results in higher η, this indicates that increasing home user throughput η T
i
counteracts the effects of decreasing cellular user throughput. Moreover, this implies that the shared access with higher ε (more time-slot allocation to cellular users) provides lower throughput than open access as shown in the result for ε = 0.1. For D > D
th
, closed access always provides higher throughput than shared access because shared access with η = 1, which does not satisfy the QoS requirement, is the same as closed access. For D ≤ D
th
, shared access obtains the same throughput as open access regardless of ε. The reason is that like open access, the shared access with time-slot allocation allows access from all neighboring cellular users located in the zone. Note that the shared access with appropriate value of ε achieves higher (at D > D
th
) or equal (at D ≤ D
th
) network throughput than open access. We summarize these observations in Table2.
6 Conclusion
Appendix
Proof of Lemma 2
f
and inner radius R
i
. Then, probability density function (PDF) of the distance R is, R ∈ [R
i
,R
f
]. The spatially averaged SIR distribution over is given asProof of Lemma 3
i
]. Thus, the spatially averaged SIR distribution of the home users is given asProof of Lemma 4
i
and inner radius R
f
. Then, PDF of R is, R∈[R
f
,R
i
]. The spatially averaged SIR distribution over is given asProof of Theorem 1
th
, the home users in connect to the FAP, while the remaining home users in communicate to the MBS. Thus, the average sum throughput of the home users is given as, where T1 and T2 are the average sum throughput of the home users in and, respectively. Since the FAP supports the home users in only, we obtain T1 = T
a
, On the other hand, since the MBS transmits data to cellular users as well as the remaining U
b
home users in, T2 is given as, and thus we getb
home users among all users supported by the MBS with TDMA and RR scheduling, which is given asb
is given asth
and (b) follows from D
th
= κ1/αR
i
. Moreover, U
c
is given asth
supports home users in the zone only, the average sum throughput,, of the home users is equal to T
i
, which proves Equation (19). For a reference FAP at D > D
th
, its neighboring cellular users in the zone connect to the central MBS. Since the MBS transmits a data to some home users in inner region as well as U
c
cellular users including the neighboring cellular users, the average sum throughput of the neighboring cellular users is given aso
is the fraction of time-slot dedicated to the U
o
neighboring cellular users in among all users supported by the MBS with TDMA and RR scheduling, which is given aso
is given asProof of Theorem 2
th
, the femtocell/macrocell access scenario of the home users in the zone and is the same as that in closed access. Thus, from Equation (47) the average sum throughput of the home users is thus givenb
home users among all users supported by the MBS, which is given asth
) with open access. The average number of users in,, is given byth
. Plugging Equations (49), (50), (51), (57) into (56) gives the desired result in Equation (25).th
, since the MBS transmits a data to the neighboring cellular users in as well as the home users in, the average sum throughput of the home users is given asi
is the fraction of time slot dedicated to the home users in among home and cellular users supported from the FAP, which is given as