Interference graph construction for D2D underlaying cellular networks and missing rate analysis

Abstract

This paper studies the interference graph construction problem for device-to-device (D2D) communications underlaying cellular networks. Firstly, an improved interference graph construction method compared to the previous work in Zhang et al. (IEEE Trans Vehicular Technol 66(4):3293–3305, 2017) is proposed. The difference is mainly that, in this work the BS allocates resources for transmitting probe packets for links in a centralized manner; while in the previous work the links select resources for transmitting probe packets in a random and autonomous manner. With this “BS-allocation” method, the BS can obtain more useful information about the interference graph than the previous “random allocating” method. Secondly, this work proposes a new theoretical analysis metric, i.e., the missing rate; while previous work analyzed the traditional convergence time. This difference is caused by that this work considers the dynamic scenario in which cellular and D2D links arrive to and leave the cell dynamically, while the previous work considered the static scenario. When considering dynamic scenario, it is possible that the interference graph has changed before the BS completes the graph construction. Hence, we must evaluate the accuracy of the constructed interference graph, i.e., the missing rate, for dynamic scenario. Simulation results validate the theoretical analysis and show that the proposed method outperforms existing methods. The impact of parameters on the missing rate is also investigated.

Appendices

Appendix A: Derivation of Eq. (6)

Assume the system is in the equilibrium state and recall that $\mu _{{\mathrm{cell}}, h}$ denote the average number of cellular links of priority h at the beginning of each frame. For these cellular links, some leave the cell during the tth frame. Then the average number of cellular links of priority h during the tth frame can be written as $\mu _{{\mathrm{cell}}, h} \cdot q_{{\mathrm{cell}}}$, where $q_{{\mathrm{cell}}}$ is the probability that a cellular link does not leave during one frame. On the other hand, some new cellular links arrive during the tth frame. According to the method, all newly arriving cellular links are assigned priority $H_{\max }$. Then the average number of cellular links of priority $H_{\max }$ during the tth frame can be written as $\mu _{{\mathrm{cell}}, H_{\max }} \cdot q_{{\mathrm{cell}}} + \Delta _{{\mathrm{cell}}}$. For convenience, let $\tilde{\mu }_{{\mathrm{cell}}, h}$ denote

$$\begin{aligned} {{\tilde{\mu }}}_{{\mathrm{cell}}, h} = \left\{ {\begin{array}{ll} {\mu _{{\mathrm{cell}}, h} \cdot q_{{\mathrm{cell}}},} &{} {h \ne H_{\max }} \\ {\mu _{{\mathrm{cell}}, h} \cdot q_{{\mathrm{cell}}} + \Delta _{{\mathrm{cell}}},} &{} {h = H_{\max }} \\ \end{array}} \right. . \end{aligned}$$

(29)

Then begins the FU allocation procedure as specified in Step 1a. For the proposed method, the BS sorts cellular links according to the priorities and selects the first $2 N_{{\mathrm{cell}}}$ cellular links to which FUs are allocated. Then we know there exists a $h_0$ so that for $h_0 + 1 \le h \le H_{\max }$ all ${{\tilde{\mu }}}_{{\mathrm{cell}}, h}$ cellular links are selected and allocated while for $0 \le h \le h_0 - 1$ no cellular link is selected and allocated. After FU allocation, the BS updates the priority according to the rule that, for cellular link which has been selected, its priority is cleared as 0; for cellular link which has not been selected, its priority is increased by 1 until reaching $H_{\max }$. Therefore, we can know

$$\begin{aligned} \mu _{{\mathrm{cell}}, h} = \left\{ {\begin{array}{ll} {2 N_{{\mathrm{cell}}},} &{} {h = 0} \\ {{{\tilde{\mu }}}_{{\mathrm{cell}}, h - 1},} &{} {1 \le h \le h_0} \\ {0,} &{} {h_0 + 2 \le h \le H_{\max }} \\ \end{array}} \right. . \end{aligned}$$

(30)

Iteratively executing Eqs. (30) and (29), we can have

$$\begin{aligned} \mu _{{\mathrm{cell}}, h} = \left\{ {\begin{array}{ll} {2 N_{{\mathrm{cell}}},} &{} {h = 0} \\ {2 N_{{\mathrm{cell}}} \cdot (q_{{\mathrm{cell}}})^h,} &{} {1 \le h \le h_0} \\ {0,} &{} {h_0 + 2 \le h \le H_{\max }} \\ \end{array}} \right. . \end{aligned}$$

(31)

There is still $\mu _{{\mathrm{cell}}, h_0 + 1}$ not determined. Actually, recalling that the average number of cellular links at the beginning of each frame is $\mu _{{\mathrm{cell}}}$, we have $\sum \nolimits _{h = 0}^{H_{\max } } {\mu _{{\mathrm{cell}},h} } = \mu _{{\mathrm{cell}}}$. Substituting Eq. (31), we can have

$$\begin{aligned} \mu _{{\mathrm{cell}}, h_0 + 1} = \mu _{{\mathrm{cell}}} - \sum \limits _{h = 0}^{h_0} {2 N_{{\mathrm{cell}}} \cdot (q_{{\mathrm{cell}}})^h}. \end{aligned}$$

(32)

Finally, there still $h_0$ not determined. We propose to determine the value of $h_0$ as the solution to the following inequality

$$\begin{aligned} \sum \limits _{h = 0}^{x} {2 N_{{\mathrm{cell}}} \cdot (q_{{\mathrm{cell}}})^h} < \mu _{{\mathrm{cell}}} \le \sum \limits _{h = 0}^{x + 1} {2 N_{{\mathrm{cell}}} \cdot (q_{{\mathrm{cell}}})^h}, \end{aligned}$$

(33)

where x is an integer. Additionally, if $2 N_{{\mathrm{cell}}} \ge \mu _{{\mathrm{cell}}}$, there is no solution to the above inequality and we simply have

$$\begin{aligned} \mu _{{\mathrm{cell}}, h} = \left\{ {\begin{array}{ll} {2 N_{{\mathrm{cell}}},} &{} {h = 0} \\ {0,} &{} {1 \le h \le H_{\max }} \\ \end{array}} \right. . \end{aligned}$$

(34)

Combining Eqs. (31) and (32), we can obtain the expression in Eq. (6).

Appendix B: Derivation of Eq. (14)

Let $B_j$ denote the number of D2D neighbors of a D2D link j. It can be approximately calculated as

$$\begin{aligned} B_j \approx \left\lfloor {\frac{\pi r_j^2}{\pi R^2} \cdot \mu _{{\mathrm{D2D}}}} \right\rfloor = \left\lfloor \left( {\frac{r_j}{R}} \right) ^2 \cdot \mu _{{\mathrm{D2D}}} \right\rfloor . \end{aligned}$$

(35)

Therefore, we can approximately calculate

$$\begin{aligned} B = {\mathrm{E}}\left[ {B_j} \right] \approx \left\lfloor {{\mathrm{E}}\left[ \left( {\frac{r_j}{R}} \right) ^2 \right] \cdot \mu _{{\mathrm{D2D}}}} \right\rfloor , \end{aligned}$$

(36)

where ${\mathrm{E}}\left[ {\cdot } \right]$ represents the expectation of a random variable. It can be calculated that

$$\begin{aligned} {\mathrm{E}}\left[ \left( {\frac{r_j}{R}} \right) ^2 \right] = \int \limits _{x = 0}^{ + \infty } {\left( {\frac{r_j}{R}} \right) ^2 \cdot f_{r_j}(x) dx}, \end{aligned}$$

(37)

where $f_{r_j}(x)$ is the the probability distribution function of random variable $r_j$. Assume all D2D links are uniformly distributed in the cell and then $d_j$ is uniformly distributed from 0 to and $d_{{\mathrm{max}}}$ where $d_{{\mathrm{max}}}$ is the maximum distance of D2D links. Since $r_j = w \cdot d_j$, we know $r_j$ is uniformly distributed from 0 to and $w \cdot d_{{\mathrm{max}}}$ and can write the probability distribution function of $r_j$ as

$$\begin{aligned} f_{r_j} (x) = \left\{ {\begin{array}{lc} {\frac{1}{{w \cdot d_{\max } }},} &{} {0< x < w \cdot d_{\max } } \\ {0,} &{} {x > w \cdot d_{\max } } \\ \end{array}} \right. . \end{aligned}$$

(38)

Substituting this equality and we can have the expression in Eq. (14).

Appendix C: Other methods

1.1 The raw method

For this method, the following procedures are used to determine the three types of edges.

1.1.1 Type 1 edges

For fair comparison, we also use the same procedure of the proposed method as presented in Section III-A to determine Type 1 edges.

1.1.2 Type 2 and 3 edges

In each frame, each D2D transmitter randomly select a FU in ${{\mathbb {N}}}_{{\mathrm{D2D}}}$ to broadcast a message. The information carried by the message is the identification of the D2D link. Simultaneously, the receiver of each D2D or cellular link k monitors each FU in ${{\mathbb {N}}}_{{\mathrm{D2D}}}$ to see whether there are messages being transmitted by the transmitters of other D2D links. For each FU, if the message transmitted on this FU is correctly decoded, then the BS knows there is an edge from g to k, where g is the decoded D2D link identification.

1.2 The method in [45]

The method proposed in [45] is a “random allocating” method, in which the D2D links select resources for transmitting probe packets in a random and autonomous manner. The following procedures are used by this method to determine the three types of edges.

1.2.1 Type 1 edges

For Type 1 edges, we use the same procedure of the proposed method as presented in Section III-A. Here is the reason. The work in [45] focused on the D2D overlay case. Thus, Type 1 edge is not considered by the method in [45]. For fair comparison, we use the same procedure of the proposed method to determine Type 1 edges.

1.2.2 Type 2 and 3 edges

In each frame, each D2D transmitter randomly select a FU in ${{\mathbb {N}}}_{{\mathrm{D2D}}}$ to broadcast a message. The information carried by the message is the identification of the D2D link. Let $n_j$ denote the FU selected by link j. Simultaneously, the BS monitors each FU in ${{\mathbb {N}}}_{{\mathrm{D2D}}}$ to see whether there are messages being transmitted by D2D links. For each FU, if the message transmitted on this FU is correctly decoded, the BS knows there is an edge from g to cellular links, where g is the decoded D2D link identification. The receiver of each D2D link l also monitors each FU in ${{\mathbb {N}}}_{{\mathrm{D2D}}}$. For each FU, there are two cases.

• Case 1: the message transmitted on this FU is correctly decoded. Let g denote the decoded D2D link identification, which is reported to the BS. Then the BS knows there is an edge from g to l.

• Case 2: a collision is detected on this FU. Let $U_l$ denote the set of all such FUs observed by link l.

Then, the receiver of each D2D link l broadcasts a message on each FU in $U_l$. The information carried by the message is also the identification of the D2D link. Simultaneously, transmitter of D2D link j monitors the FU $n_j$ to see whether there are messages being transmitted by other D2D links. If the message transmitted on this FU is correctly decoded, then the BS knows there is an edge between g and j, where g is the decoded D2D link identification.