Congestion dispersion in device-to-device discovery for proximity-based services

Abstract

Proximity service (ProSe) enables the development of various applications that exploit proximity of the mobile devices. In ProSe, devices are expected to communicate with each other on device-to-device (D2D) links and make use of proximity information for the provision of a variety of services. An essential functionality in ProSe is the ability of a device to identify its neighbors. This process is referred to as D2D discovery. This paper investigates distributed D2D discovery based on greedy resource selection. In order to facilitate rapid and efficient distributed D2D discovery, we propose group-based peer discovery resource selection range restriction (GPSRR) scheme. GPSRR scheme disperses the devices concentrated at a certain discovery resource over multiple resources. This is done by limiting the resource selection of a device to a specified group of resources. We also propose group reselection (GR) scheme to resolve the unbalance of congestion among different groups of resources. Simulation results show that the proposed schemes can improve the speed of discovery and the number of discovered devices, especially in densely populated environments.

Appendix: Developing models for performance metrics in D2D discovery

In order to analyze the impact of various parameters on the performance of distributed D2D discovery, we develop simple analytical model to derive the number of discoverable UEs. The analysis is performed based on the previous studies reported in [10, 13, 14], and [21]. It has been assumed that UEs are deployed in an area according to HPPP with intensity λ (UEs/m ²) [13]. A probing UE (UE 0) deployed at the center (0, 0) of the area performs discovery for other UEs. Euclidean distance from UE 0 to other UE k positioned at (x _k , y _k ) in $\mathbb {R}^{2}$ is denoted by d _k (=$\sqrt {x{_{k}}{{~}^{2}}+y{_{k}}{{~}^{2}}}$). Free-space path loss model with exponent α, where α is an integer larger than 2, and Rayleigh fading with mean 1 are considered. In addition, it has been assumed that each UE broadcasts discovery message with transmission power P _tx [W]. In this case, the received power at UE 0 from another UE k is expressed as $P_{\text {tx}} h_{k} d_{k}^{-\alpha }$ [W], where the random variable h _k follows an exponential distribution with mean 1 [14]. For simplicity, we consider random PDR selection, where each UE randomly selects a PDR between [1, N _PDR] with uniform distribution.

We firstly find detection probability P _dct(d _k ) in order to derive the number of detectable UEs [13, 21]. P _dct(d _k ) is defined as the probability that UE 0 will successfully receive discovery message sent by UE k which is d _k [m] apart from the UE 0. The received SIR of the discovery message sent from UE k to UE 0 (SIR _k ) should be larger than T _SIR for successful decoding of the message. Thus, P _dct(d _k ) can be expressed as:

$$\begin{array}{@{}rcl@{}} P_{\text{dct}} (d_{k}) &=& \text{Pr}\left( \text{SIR}_{k} >T_{\text{SIR}} \right ) \\ &=& \text{Pr}\left( \frac{P_{\text{tx}} h_{k} d_{k}^{-\alpha}}{{\sum}_{j\in {\Phi}} P_{\text{tx}} h_{j} d_{j}^{-\alpha} }>T_{\text{SIR}} \right) \\ &=& \text{Pr}\left( \frac{h_{k} d_{k}^{-\alpha}}{{\sum}_{j\in {\Phi}} h_{j} d_{j}^{-\alpha} }>T_{\text{SIR}} \right), \end{array} $$

(10)

where Φ is a set of UE j, ∀j≠0 and k. Equation 10 is rewritten as follows:

$$\begin{array}{@{}rcl@{}} P_{\text{dct}} (d_{k}) &=& \text{Pr}\left( h_{k} > d_{k}^{\alpha}T_{\text{SIR}{\sum\limits_{j\in {\Phi} } h_{j} d_{j}^{-\alpha} }} \right) \\ &=& 1-\text{Pr}\left( h_{k} \leq d_{k}^{\alpha}T_{\text{SIR}{\sum\limits_{j\in {\Phi}} h_{j} d_{j}^{-\alpha} }} \right). \end{array} $$

(11)

Since cumulative distribution function of h _k ($\text {Pr}\left (h_{k} \leq b \right )$) is 1−e ^−b [14], Eq. (11) can be rewritten as follows:

$$\begin{array}{@{}rcl@{}} P_{\text{dct}} (d_{k}) &=& E \left[ \text{exp} \left( -d_{k}^{\alpha} T_{\text{SIR}{\sum\limits_{j\in {\Phi} } h_{j} d_{j}^{-\alpha} }} \right) \right ]. \end{array} $$

(12)

where h _j and d _j are independent random variables. Thus, Eq. (12) becomes:

$$\begin{array}{@{}rcl@{}} P_{\text{dct}} (d_{k}) &=& E \left [ \prod\limits_{j\in {\Phi}} E \left [ \text{exp} \left( - d_{k}^{\alpha}T_{\text{SIR}} d_{j}^{-\alpha} h_{j} \right) | d_{j} \right ] \right]. \end{array} $$

(13)

Through calculating the expectation for h _j which follows an exponential distribution with mean 1, Eq. (13) can be rewritten as follows:

$$\begin{array}{@{}rcl@{}} P_{\text{dct}} (d_{k}) &=& E \left [ \prod\limits_{j \in {\Phi}} \frac{1}{1+d_{k}^{\alpha}T_{\text{SIR}} d_{j}^{-\alpha}} \right ]. \end{array} $$

(14)

By using the probability generating function for PPP [22], Eq. (14) is approximated to:

$$\begin{array}{@{}rcl@{}} &&P_{\text{dct}} (d_{k}) \cong\\ &&\text{exp} \left (- \lambda {\int}^{\infty}_{-\infty} {\int}^{\infty}_{-\infty} \left (1-\frac{1}{1 + d_{k}^{\alpha} T_{\text{SIR}} \left( {x_{j}^{2}} + {y_{j}^{2}}\right)^{-\alpha/2} } \right ) dx_{j} dy_{j}\right ). \end{array} $$

(15)

Let us denote $x_{j} = d_{k} T_{\text {SIR}}^{1/ \alpha } r_{j} \cos \theta _{j}$ and $y_{j} = d_{k} T_{\text {SIR}}^{1/ \alpha } r_{j} \sin \theta _{j}$. Then, we can find $dx_{j} dy_{j} = {d^{2}_{k}} T^{2/\alpha }_{\text {SIR}} r_{j} dr_{j} d\theta _{j}$ and ${x^{2}_{j}}+{y^{2}_{j}}={d^{2}_{k}} T^{2/\alpha }_{\text {SIR}} {r_{j}^{2}}$. By substituting these into Eq. (15), we can obtain:

$$ P_{\text{dct}} (d_{k}) \cong \exp \left (- \lambda {\int}^{2\pi}_{0} {\int}^{\infty}_{0} \left (1-\frac{1}{1 + r_{j}^{-\alpha}} \right ) {d^{2}_{k}} T^{2/\alpha}_{\text{SIR}} r_{j} d r_{j} d \theta_{j} \right ). $$

(16)

Through solving the integration for 𝜃 _j , Eq. (16) is simplified as follows:

$$ P_{\text{dct}} (d_{k}) \cong \exp \left (- 2 \pi \lambda {d^{2}_{k}} T^{2/\alpha}_{\text{SIR}} {\int}^{\infty}_{0} \left (\frac{r_{j}}{r_{j}^{\alpha} + 1} \right ) d r_{j} \right ). $$

(17)

By using the techniques of complex analysis, we can find that the integration ${\int }^{\infty }_{0} {r_{j}}/\left ({r_{j}^{\alpha } + 1}\right ) d r_{j}$ equals to $\pi / \left (\alpha \cdot \sin (2\pi / \alpha ) \right )$. Finally, the detection probability for UE k is obtained as follows:

$$ P_{\text{dct}} (d_{k}) \cong \exp \left (- \frac{2\pi^{2} \cdot \lambda \cdot {d^{2}_{k}} \cdot T_{\text{SIR}}^{2/\alpha}}{\alpha \cdot\text{sin} \left (2 \pi / \alpha \right ) } \right ). $$

(18)

Detectable area of UE k is defined as an area where the UE 0 can successfully receive discovery message from the UE k [13]. Mean detectable area for UE k ($\tilde {A}_{k} \left ({\Phi }, T_{\text {SIR}} \right )$ [m ²]) is expressed as follows:

$$\begin{array}{@{}rcl@{}} \tilde{A}_{k} \left ({\Phi}, T_{\text{SIR}} \right ) = \qquad \qquad \qquad \qquad \qquad \qquad \qquad\\ \qquad \left \{ (x_{k}, y_{k}) \in {\mathbb{R}}^{2} : \frac{h_{k} \left(\sqrt{{x_{k}^{2}} + {y_{k}^{2}}}\right)^{-\alpha}}{{\sum}_{j\in {\Phi} } h_{j} \left(\sqrt{{x_{j}^{2}} + {y_{j}^{2}}}\right)^{-\alpha} } > T_{\text{SIR}} \right\}. \end{array} $$

(19)

Averaging $\tilde {A}_{k} \left ({\Phi }, T_{\text {SIR}} \right )$ for all possible positions of UE k, we can obtain the mean volume of the detectable area for UE intensity λ as

$$\begin{array}{@{}rcl@{}} \tilde{A} \left( \lambda \right ) &=& E \left[ \tilde{A}_{k} \left ({\Phi}, T_{\text{SIR}} \right ) \right] \\ &=& {\int}^{\infty}_{-\infty} {\int}^{\infty}_{-\infty} 1\cdot P_{\text{dct}} \left( \sqrt{{x_{k}^{2}} + {y_{k}^{2}}} \right) dx_{k} dy_{k} \\ &=& \frac{\alpha \cdot \text{sin} (2\pi /\alpha)}{2\pi\cdot \lambda \cdot T_{\text{SIR}}^{2/\alpha}}. \end{array} $$

(20)

Let us denote the intensity of UEs occupying a PDR i as λ _i [UEs/m ²]. The mean number of discoverable UEs in a PDR i can be obtained by multiplying $\tilde {A} \left (\lambda _{i} \right )$ [m ²] with λ _i [UEs/m ²] as:

$$\begin{array}{@{}rcl@{}} \tilde{M}_{i} &=& \lambda_{i} \cdot \tilde{A}\left( \lambda_{i} \right ) \\ &=& \frac{\alpha \cdot \text{sin} (2\pi /\alpha)}{2\pi\cdot T_{\text{SIR}}^{2/\alpha}}. \end{array} $$

(21)

Finally, the mean number of discoverable UEs in a discovery period consisting of N _PDR PDRs is obtained as:

$$\begin{array}{@{}rcl@{}} \tilde{M} &=& \sum\limits_{i=1}^{N_{\text{PDR}}} \tilde{M}_{i} \\ &=&N_{\text{PDR}} \cdot \frac{ \alpha \cdot \text{sin} (2\pi /\alpha)}{2\pi\cdot T_{\text{SIR}}^{2/\alpha}}. \end{array} $$

(22)