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Entropy 2018, 20, 88 5 of 22
However, too many stations will increase the collision probability and decrease the capacity [
23
].
Conversely, too few stations will not make full use of the channel spectrum resource.
Thus, the number
of stations in one AP should be carefully determined to maximize the capacity of the WiFi system.
To best utilize the spectrum resources and maximize the system capacity, a coalition game among
WiFi APs in the first layer of our CAGT optimization scheme is designed, and our goal is to
optimize the station number
n
∗
in one AP coalition to maximize the throughput of the WiFi system.
Accordingly, we reshape (2) and the optimization problem can be formulated as
max
n∈N
+
R =
anτ(1 − τ)
n−1
E[P]
(T
σ
− T
c
)(1 − τ)
n
+ (T
S
− T
c
)nτ(1 − τ)
n−1
+ T
c
, (5)
where
a =
T
o f f
T
on
+T
o f f
. Define
f (n) = anτ(
1
− τ)
n−1
E[P]
and
g(n) = (T
σ
− T
c
)(
1
− τ)
n
+ (T
S
− T
c
)nτ(
1
−
τ)
n−1
+ T
c
. Thus, Equation (5) can be expressed as
R =
f (n)
g(n)
=
− f (n)
−g(n)
, where
− f ( n)
is concave and
−g(n)
is convex, which are judged by the Hessian Matrix. Then, according to [
26
], the function
R
is
quasi-concave. Thus, the optimization problem can be expressed as
η(n)
n∈N
+
= max
n∈N
+
R =
f (n)
g(n)
. (6)
As mentioned in [
27
], any optimization problem in fractional form can be transformed into an
equivalent optimization problem in a subtractive form. Hence, the non-linear fractional optimization
problem in Equation (6) can be transformed into a parameterized function and the optimal solution
can be determined by finding the root to the
U(n)
as shown in Equation (7) using various root finding
methods [28,29],
U(n) = max
n∈max
{b
R
+
c
,
d
R
+
e}
{
− f ( n) − η(−g(n))
}
= max
n∈
{b
R
+
c
,
d
R
+
e}
{
− f ( n) + ηg(n)
}
. (7)
The solution to Equation (7) could be formulated as an iterative two-layer solution combining
Dinkelbach type method and Lagrangian dual decomposition approach. In addition, this problem can
be solved by mathematical tools such as CVX or other build-in functions in MATLAB [29].
Due to concavity of the function analyzed above, balancing the stations and making the station
number as close as the optimal value
n
∗
could improve the performance of the whole WiFi system.
Therefore, we form a coalition game by encouraging individuals in the WiFi system to form groups in
the first layer of CAGT mechanism, and then make a traffic transfer between LAA and WiFi in the
second layer of CAGT.
3. Modified CSAT Mechanism
Before discussing the CAGT scheme, the basic and modified CSAT will be devised firstly.
The primary principle of CSAT is to let LTE in unlicensed bands cycle between state “on” and “off”
within a given single cycle while the LTE in licensed band remains connected all the time. During the
activated state “on”, LTE transmits data on the unlicensed band and blocks the transmission of
WiFi systems. However, within the deactivated state “off”, LAA shuts down its transmission to
yield the channel sources to WiFi [
11
], just as shown in Figure 2. Adjusting “on” and “off” time
dynamically by identifying the number of WiFi terminals can be an affordable way for proper “on”
and “off” time assignment and meet better needs between system aggressiveness and fairness [
12
].
However, WiFi nodes
may not be aware of the LAA activity in some actual scenarios. For example,
WiFi causes interference to LTE in the “on” state period since WiFi cannot identify the structure
of signals from non-IEEE systems. When the Energy Detection for Clear Channel Assessment
(CCA-ED) measurement is below the predefined inter Radio Access Technology (inter-RAT) threshold,
the ongoing LAA messages cannot be identified and will be regarded as noises. Therefore, a LBT