Resource Discovery

If at least one received device responds, the active device A will choose one amongst all the received devices which have responded and emit a probing signal with power P_D in the preferred RB to check if any CU is interfered. If CU’s SINR doesn’t meet the requirement, the active devices can not reuse the CU’s spectral resources. Therefore, we will analyze the interference from active devices to CU first [24].

Not every active device has a chance to probe to check whether there exist usable RBs or not. The realistic interference to CU is very complicated. We use another method to make the analysis tractable. In order to model the interference sources as PPP model (the interference analysis is tractable), we introduce two scenarios.

One is an approximate scenario, we will limit the intensity λ_a of active devices.

Following by the access the probing channel step, there are no active devices that the distance between one active device and another one smaller than r_a. Based on equation (2.16), we let probability that the distance between two active devices smaller than r_a bound by ε. Somehow we can use an approximate scenario to model the interference sources which are generated by PPP model.

F_D(r_a) = 1− exp(−λaπr²_a)≤ ε (5.12)

Therefore, the intensity λ_a of active devices needs to satisfy

λ_a ≤ −ln(1− ε)

πr_a² (5.13)

In addition to the limit of intensity of active device, another idea is that the interference is upper bounded by letting the interference sources at the same RB as a homogeneous PPP with intensity λ^_a= ^p_M^dλ_a (ignore the effect of step-1).

Interference Analysis under PPP

We will see the total interference I(y) at the location y from the other interferers positioned at the points by using the point process Φ. We define Φ as a d-dimensional homogeneous PPP with intensity λ. If we condition the y at the origin, that will yield the Palm distribution for other devices. The conditional distribution keeps the same as the original one by Slivnyak’s theorem [21]. By mapping the d-dimensional homogeneous PPP onto R⁺, it will become an inhomogeneous PPP Φ  {ri = xi} which is the distance between the device and the origin. Because Φ is a stationary PP, the intensity measure function becomes Λ(r) = λc_ddr^d−1.

The Poisson shot noise process [24] at location y as I(y) in Fig. 5.1 is defined as I(y) =

x∈Φ

P_Dg_xx − y^−α (5.14)

where g_x is the channel fading effect of the active device at location x.

By Slivnyak’s theorem [21], I(y) can also be defined as I.

I =

x∈Φ

PDgxx^−α (5.15)

By mapping the d-dimensional homogeneous PPP onto R⁺, it will become an inhomo-geneous PPP Φr  {ri = xi}. Therefore, the interference I can be defined in the following.

I = 

r∈Φr

P_Dg_rr^−α (5.16)

where g_r is the channel fading effect of the active device at distance r. g_r and g_x have the same distribution.

The received signal power Pc from the transmission of eNB at the CU and the corresponding SINR γc in stochastic channel are given respectively by

P_c = hP_T

d^α_eNB,CU (5.17)

and

γ_c = P_c

I + N₀ (5.18)

where d_eNB,CU is the distance between eNB and CU, h is the channel fading effect which follows a unit mean exponential distribution and I (random variable) is the interference from the active devices which reuse the same CU’s spectral resources.

The success probability ps(θu) of CU which received the SINR larger than the pre-determined threshold θ_u.

ps(θu) =P

P_Td^−α_eNB,CU and LI(s) is the Laplace transform of random variable I.

The Laplace transform of random interference I can be defined as below.

LI(s)  EΦr,g_r[e^−sI] =EΦr,g_r where (a) follows the independence between PPP and channel fading effect, and (b) is based on the intensity measure and PGFL of PPP.

We define the integral equation F as below.

F =

where (a) is derived from the definition of intensity measure Λ(r), (b)-(c) uses the change of variable, (d) is that we see X as an exponential random variable with unit mean, and (e) is based on Corollary 1. According to the (5.21), the equation (5.20) can be obtain by

LI(s) = exp(−λcdE[g^δ]Γ(1− δ)s^δP_D^δ) (5.22) If the channel fading gain g is followed by exponential distribution with unit mean, E[g^δ] will equal to Γ(1 + δ) which is from the Corollary 1. By using the Euler’s reflection formula, we can obtain the closed-form of equation (5.20). This equation also can be seen in [24]. Assume the responded received device randomly chooses one preferred RB from M RBs.

Therefore, based on the thinning property of PPP, the equivalent intensity λ^_a of Φ^_a for the active devices which have the responded received device with probability p_d and commonly transmit on the same RB with probability _M¹ will be defined in the following.

Namely, Φ^_a is the PPP which the active device will transmit signal to another received device by reusing the same CU’s spectral resources.

λ^_a= p_d

Mλ_a (5.24)

The scenario that we consider is a two-dimensional environment. Therefore, the Laplace transform of I can be shown.

LI(s) = exp

If the CU’s transmission is not outage (successful transmission), the active devices can reuse its spectral resources.

The success probability of CU:

From the equation (5.26), the success probability of CU is an inverse proportion to the active device intensity λ_a, active device transmit power P_D, SINR requirement θ_u of CU, noise power N₀, device discovery probability p_d, and the distance d_eNB,CU between eNB and CU. It is proportional to the eNB transmit power PT and the number of RBs M . It is very intuitive for our thought. The larger the values of λ_a and P_D are, the stronger the interference to CU will be. P_T and d_eNB,CU will affect the CU’s received power. θ_u is the SINR requirement of CU, so the larger of this value is, the lower the success probability of CU will be.