According to the SDLS procedure, an active device may find a close received device and some usable resources to use. Therefore, we define the probability pLS that an active device can finish all the SDLS procedures. In other words, it finds a close idle received device which received the SNR larger than the pre-determined threshold and shares the
0 50 100 150 200 250 300
Figure 5.2: pLS is not a concave function.
same spectral resources with the CU without causing harmful interference.
pLS papdps(θu) = e−q1PDδ and q3 are positive numbers.
Therefore, we can model this problem as an optimization problem in upper bound scenario without limiting the transmit power PD of active devices.
maxPD≥0 pLS = e−q1PDδ According to Definition 4, the objective function pLS is not a concave function by providing an example a counterexample which is shown in Fig. 5.2. We can not use optimization tool [25] to solve this problem. We try another method to find the optimal solution.
The partial differential of pLS on PD is given by
∂pLS(PD)
The partial differential of pLS is the slope of pLS. The term that we care is its sign, not
Theorem 1 There exists only one solution to make f (z) = 0 for z ≥ 1. If the solution of f (z) = 0 is z, then
is the optimal solution of the optimization problem (5.28).
Proof: The 1st derivative and 2nd derivatives of f (z) are given respectively by
∂f (z)
and a decreasing function for PD ≥
ln z
Corollary 2 The function −pLS(PD) is a quasiconvex function.
0.5 1 1.5
−0.15
−0.1
−0.05 0 0.05 0.1
z
values of A(z), B(z), and f(z)
q1 = 0.1405 , q
2 = 0.0811 , q
3 = 0.2775 A(z) B(z) A(z)−B(z)
Figure 5.3: The intersection between z axis and f(z) (one solution).
Proof: The domain of −pLS(PD) is convex set (∵ PD ≥ 0). According to Theorem 1:
−∂pLS∂P(PDD) < 0, if PD∗ ≥ PD ≥ 0
−∂pLS∂P(PDD) > 0, if PD ≥ PD∗
(5.33) We assume x > y. We can divide into three cases for setting x and y.
1. If {PD∗ ≥ x, y ≥ 0}, −pLS(θx + (1− θ)y) ≤ −pLS(y)≤ max{(−pLS(x),−pLS(y))}.
2. If {x, y ≥ PD∗}, −pLS(θx + (1− θ)y) ≤ −pLS(x)≤ max{(−pLS(x),−pLS(y))}.
3. If {PD∗ ≥ x ≥ 0} and {y ≥ PD∗}, we can find the θ to let θx + (1− θ)y = PD∗. Therefore, for any x, y ∈ dom −pLS(PD),−pLS(θx+(1−θ)y) ≤ max{(−pLS(x),−pLS(y))} for 0≤ θ ≤ 1.
According to Definition 5, −pLS(PD) is a quasiconvex function. Similarly, pLS(PD) is a quasiconcave function.
Because the solution of f (z) = 0 is hard to find by solving it directly, we will show how another method to find the optimal solution PD∗ to make ∂pLS∂P(PD)
D = 0. Divide (5.29)
into two equations as following (f (z) = A(z)− B(z)):
As the discriminant of equation A(z) is always positive, there exist two real-number solutions.
The bisection method is executed between 0 and PDupper. By letting PD(k) is the midpoint of the interval in the kth step, the difference between PD(k) and an optimal solution PD∗ is bounded by
|PD(k)− PD∗| ≤ PDupper
2k (5.35)
The active device will execute this algorithm. The eNB will broadcast the network conditions to devices. According to these information, active devices find the common optimal transmit power PD respectively.
Difference at Interference Sources
In the upper bound scenario, the transmit power PD of active devices is unbounded.
But in approximate scenario, the intensity of active device is relative to PD. Therefore, we will fix PD in approximate scenario.
Table 5.1: Simulation Parameters under PPP
Parameter Value
Path loss exponent (α) 4
Received power at CU
Pr = dαPT
eNB,CU
5, 10, 15, 20dB
Noise power (N0) 1
Tolerable interference for accessing (θa) 5dB SINR requirement of CU (θu) 10dB SNR requirement of received device (θd) 15dB Radius of sensing range (ra) 20
Approximate threshold () 0.1
Received device intensity (λd) 0.005
Number of RBs (M ) 10
Busy probability of received device (β) 0.3
5.4 Analytical Results
In this section we examine the analytical results of SDLS performance under PPP model. The parameters of our simulation are shown in Table 5.1. The radius refers to the transmit power of active devices (PD = rαaθa, the corresponding λa to approximate the interference sources as PPP is 0.0001). Fig. 5.4 which is based on upper bound scenario shows the solution which is from the bisection method is also the optimal solution.
Because the SINR requirement of CU, the received power at CU will affect the SDLS probability. When the CUs have very reliable links (more tolerable interference), the SDLS probability also increased. The gap is very evident. Compare Fig. 5.4 (a) with (b), when path loss exponent α is larger, the path loss will more severe. The interference and coverage range of active device also decreased. The total effect of α leads to the better performance under large α. When the received power at CU is 5dB (the channel gain between the CU and eNB needs to be very large to meet the SINR requirement), the SDLS probability is below 0.02.
The following Figures are obtained using the approximate scenario. Fig. 5.5 shows
the effect of busy probability β of received device. The parameter β means the busy state of the network. In the intuitive sense, the D2D link probability is a monotonic decreasing function of β. We verify this idea in the Fig. 5.5. Even the received signal power at CU is large, the link setup probability is below 0.15 under high β. Fig. 5.6 show the effect of the intensity of received device. The parameter λd means the number of received device stochastically. When λdbecomes larger, the D2D link is easy to set up.
It means the active device has more choices to set up a D2D link with another received device. Fig. 5.7 shows that the value of noise power affects the D2D link and CU’s SINR. Even when the received power at CU is 20dB, the SDLS probability is very low under high noise power level. At low received power at CU, the link setup probability approaches to zero. Fig. 5.8 shows the effect of the number of RBs. M affects the probability that the active device chooses the same RB. Therefore, the interference at the common RB will decrease when M becomes larger. But when M is large enough, the SDLS probability becomes flat. This phenomenon can be explained by the success probability of CU under large M . The noise power will dominate the success probability of CU. The effect of M can be ignored when M is large enough.
0 5 10 15 20
(a) α=4. The black solid circle represents the solution by bisection method.
0 5 10 15 20
(b) α=6. The black solid circle represents the solution by bisection method.
Figure 5.4: SDLS probability (pLS) performance as a function of the active device trans-mit power PD in the upper bound scenario.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 5.5: SDLS probability (pLS) performance as a function of busy probability β of received device in approximate scenario.
1 2 3 4 5 6 7 8 9 10
Figure 5.6: SDLS probability (pLS) performance as a function of received device intensity λd in approximate scenario.
1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 5.7: SDLS probability (pLS) performance as a function of noise power N0 in approximate scenario.
Figure 5.8: SDLS probability (pLS) performance as a function of the number of RB M in approximate scenario.
Chapter 6
Conclusions and Future Works
In this thesis, we review the concept of D2D communications and some stochastic ge-ometric related point processes. We study the SDLS probability under two device/mobile terminal location distributions characterized by Binomial point process (BPP) and Pois-son point process (PPP), respectively. We propose a D2D link setup protocol and analyze the corresponding link setup performance.
For the BPP model, we consider both deterministic path loss and stochastic channel fading in our analysis. For the deterministic case, we obtain closed-form expression of the SDLS probability. But for the latter scenario, we evaluate the SDLS probability performance through numerical computing. When the coverage radius of an active device increases, so is the interference to CUs. Our analysis indicates that a judicious choice of the device transmit power is needed to maximize the SDLS probability. The optimal transmit power depends on the network condition, i.e., the parameters of the associated BPPs.
By constraining the peak transmit device power and the minimum distance between two active devices which use the shared macrocell spectrum in their D2D links, we obtain a proper PPP parameter to characterize the location distribution of active devices who have successfully establish D2D links and evaluate the associated average sum interference power. An upper bound for the latter power is obtained by removing the minimum distance constraint. We refer to these two scenarios as the constrained device
separation and the worst case scenarios.
The SDLS probabilities for both scenarios are functions of the active device’ average transmit power PD. We prove that the corresponding functions are quasiconcave and apply the bisection search to find the optimal solution. We present a scheme to set an upper limit of this approach. The optimal transmit power PD∗ is a function of the param-eters of the communication environment (e.g., the intensity of the associated PPP and the devices’ busy probability) and proper power control is needed to maximize the SDLS probability. We examine how system parameters like device intensity, average transmit power, noise power, allocated resource blocks,affect the SDLS probability through either analysis or computer evaluations.
Our analysis is based on a simplified wireless environment. There is no consider-ation of the shadowing effect and only single cell system is considered. Furthermore inhomogeneous PPPs may be more appropriate in characterizing the device/terminal locations. A protocol which allows multiple devices to multiple devices communications that minimizes the collision probability and solve the device identification problem is called for. Finally, our investigation can be regarded as a first step in an autonomous piconet setup protocol which partitions devices into groups (local networks) so that intra-net D2D communication becomes much easier. In forming such groups, a device discovery process is definitely needed.
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