DTMR: delay-aware traffic admission, mode selection, and resource allocation for D2D communication

Abstract

In device-to-device (D2D) communication underlaying a cellular network, the resource control involves traffic admission, mode selection, orthogonal channel assignment, and power control. Traffic admission limits queueing delay, mode selection exploits the proximity gain, and resource allocation guarantees user performance. Jointly optimizing these factors is highly challenging due to the stochastic nature of the system and the coupled control actions. Many previous works only consider a subset of these factors. In this paper, we tackle the joint optimization problem for delay-aware D2D communication. In particular, considering both dynamic traffic arrival and time-varying channel fading, we aim to maximize the time-average sum-rate of the network subject to the time-average throughput guarantee of users and resource allocation constraints. Thus, through presenting a Lyapunov optimization framework, we design an optimal delay-aware traffic admission, mode selection, and resource allocation (DTMR) strategy with polynomial time complexity based on dual optimization and ellipsoid search. We also analytically derive lower bound of the time-average sum-rate achieved by the proposed DTMR strategy. Further, we develop a fast heuristic strategy by decoupling the binary constraints of mode selection and channel assignment. Finally, simulation results demonstrate the superiority of the DTMR strategy against alternative strategies.

Appendices

Appendix 1: Proof of Proposition 1

If there exists $\lambda _{n,b,k}+\lambda _{b,n,k}\ne Q_{n}(t)$ for any n, k, then $g(\varvec{\lambda })=-\infty$ because ${\mathbf {Z}}$ is an unbounded vector. In this case, $\lambda _{n,b,k}+\lambda _{b,n,k}=Q_{n}(t), \forall n, k$ is required to maximize $g(\varvec{\lambda })$ in the dual problem of ${\mathbf {P}}4$. Further, (35)–(38) are proved by applying the KKT conditions [30]. To avoid redundancy, we take $W_{m,k}^{\mathcal {C}}(t)$, $\forall m\in \mathcal {M}, \forall k\in \mathcal {K}$, as an illustration. Based on the KKT conditions of $g(\varvec{\lambda })$ in (33), we can derive that

$$\begin{aligned}&W_{m,k}^{\mathcal {C}}(t)=\left[ \frac{Q_{m}(t)w\tau }{(\lambda _{m}-\theta _{m,k}^{\mathcal {C}})\ln 2}-\frac{\varGamma \sigma ^{2}}{h_{m,k}(t)}\right] \cdot x_{m,k}^{\mathcal {C}}(t). \end{aligned}$$

(55)

$$\begin{aligned}&\theta _{m,k}^{\mathcal {C}}W_{m,k}^{\mathcal {C}}(t)=0\end{aligned}$$

(56)

$$\begin{aligned}&\lambda _{m} \ge 0, \theta _{m,k}^{\mathcal {C}}\ge 0 \end{aligned}$$

(57)

where $\theta _{m,k}^{\mathcal {C}}$ is the dual variable for $W_{m,k}^{\mathcal {C}}\ge 0$ in (27). Based on (55)–(57), if $W_{m,k}^{\mathcal {C}}(t)>0$, then $\theta _{m,k}^{\mathcal {C}}=0$ and $W_{m,k}^{\mathcal {C}}(t)=\left[ \frac{Q_{m}(t)w\tau }{\lambda _{m}\ln 2}-\frac{\varGamma \sigma ^{2}}{h_{m,k}(t)}\right] \cdot x_{m,k}^{\mathcal {C}}(t)$. Otherwise, if $W_{m,k}^{\mathcal {C}}(t)=0$, then $\theta _{m,k}^{\mathcal {C}}\ge 0$ and $\frac{Q_{m}(t)w\tau }{\lambda _{m}\ln 2}<\frac{\varGamma \sigma ^{2}}{h_{m,k}(t)}$. Hence, we can express the optimal relation between $W_{m,k}^{\mathcal {C}}(t)$ and $x_{m,k}^{\mathcal {C}}(t)$ with respect to $g(\varvec{\lambda })$ in (33) as

$$\begin{aligned} W_{m,k}^{\mathcal {C}^{*}}(t)&=\left[ \frac{Q_{m}(t)w\tau }{\lambda _{m}\ln 2}-\frac{\varGamma \sigma ^{2}}{h_{m,k}(t)}\right] ^{+}\cdot x_{m,k}^{\mathcal {C}}(t). \end{aligned}$$

(58)

where $[x]^{+}=\max \{x,0\}$. In addition, with a similar mathematical structure, (36)–(38) can be proved in the same way.

Appendix 2: Proof of Proposition 2

We take $\lambda _{m}$ as an example and separate the optimal ${\mathbf {W}}_{\mathcal {C}}^{*}(t)\,\triangleq\, [W_{m,k}^{\mathcal {C}^{*}}(t)]_{m\in \mathcal {M},k\in \mathcal {K}}$ into two categories: ${\mathbf {W}}_{\mathcal {C}}^{*}(t)={\mathbf {0}}_{M\times K}$ and ${\mathbf {W}}_{\mathcal {C}}^{*}(t)\ne {\mathbf {0}}_{M\times K}$. In the former case, we have $\sum _{k\in \mathcal {K}}W_{m,k}^{\mathcal {C}}(t)-P_{\max }^{m}=P_{\max }^{m}$. Based on the complementary slackness condition [30], we have $\lambda _{m}=0$. While in the latter case, from (35), we can see that if $\lambda _{m}^{*} \ge \frac{Q_{m}(t)w\tau }{\varGamma \sigma ^{2}\ln 2}\cdot \max _{k\in \mathcal {K}}\left\{ h_{m,k}(t)\right\}$, all $W_{m,k}^{\mathcal {C}^{*}}(t)$ are equal to zero, which contradicts ${\mathbf {W}}_{\mathcal {C}}^{*}(t)\ne {\mathbf {0}}_{M\times K}$.