Resource Allocations for Ultra-Reliable Low-Latency Communications

Abstract

Ultra-reliable low-latency communications (URLLC) is a new feature to be considered for the fifth generation (5G) cellular systems. This feature is essential for the support of envisioned mission-critical applications, particularly in the realm of machine-type communications. These applications require that the messages, which are generally short-length packets, to be exchanged between a source and a destination with the high level of reliability and within a short period of time. The characteristics of URLLC do not fit directly in the conventional communication models. For instance, most of the existing communication models are developed considering moderate levels of reliability, neglecting the small effects of the feedback errors. However, even such small errors cannot be ignored for URLLC. This paper proposes a communication model for URLLC considering the reliabilities of both data and control channels. Then, the optimal and sub-optimal resource allocations are derived. We show that the proposed sub-optimal resource allocations have lower computational complexities with a negligible performance degradations compared to that of the optimal solutions. The results reveal that the possibility of performing only one retransmission can significantly reduce the required radio resources needed for data delivery compared to the case of performing a single transmission round.

Appendix

We prove the convexity of the average number of channel uses for non-adaptive scenario given in (5) provided $\tilde{n}$ and $\hat{n}$ ( $\tilde{n}\le n \le \hat{n}$). Since it has been shown in (4) that $E_{x}$ given in (2) is a convex and decreasing function of x, we have $E_{\lambda a+(1-\lambda )b}\le \lambda E_{a} + (1-\lambda ) E_{b}, \forall \lambda \in [0,1]$.

Let $\overline{N}_n$ denote $\overline{N}(n,k,\gamma )$. In the following we prove the convexity when $\varDelta _{\epsilon } \triangleq 1-\epsilon _{\text {N}} -\epsilon _{\text {A}}\ge 0$. We first rewrite (5) as

$$\begin{aligned} \overline{N}_{\lambda \tilde{n} + (1-\lambda )\hat{n}}= \left( \lambda \tilde{n} + (1-\lambda )\hat{n}\right) \left[ 1+ \epsilon _{\text {A}} + \varDelta _{\epsilon } E_{\lambda \tilde{n} + (1-\lambda )\hat{n}} \right] . \end{aligned}$$

In addition,

$$\begin{aligned} \lambda \overline{N}_{\tilde{n}} + (1-\lambda )\overline{N}_{\hat{n}}&= \lambda \tilde{n} \left[ 1+ \epsilon _{\text {A}} +\varDelta _{\epsilon } E_{\tilde{n}}\right] + (1-\lambda )\hat{n} \left[ 1+ \epsilon _{\text {A}} +\varDelta _{\epsilon } E_{\hat{n}}\right] \\&= (1+ \epsilon _{\text {A}}) \left[ \lambda \tilde{n} + (1-\lambda )\hat{n}\right] + \varDelta _{\epsilon } \left[ \underbrace{\tilde{n}\lambda E_{\tilde{n}} + \hat{n}(1-\lambda ) E_{\hat{n}}}_{\mathcal {A}} \right], \end{aligned}$$

and

$$\begin{aligned} \overline{N}_{\lambda \tilde{n} + (1-\lambda )\hat{n}} = (1+ \epsilon _{\text {A}}) \left[ \lambda \tilde{n} + (1-\lambda )\hat{n}\right] + \varDelta _{\epsilon } \left[ \underbrace{\left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] E_{\lambda \tilde{n} + (1-\lambda )\hat{n}}}_{\mathcal {B}} \right] . \end{aligned}$$

In the following, we prove that $\mathcal {A}\ge \mathcal {B}$. Let $\alpha \triangleq \frac{\tilde{n}\lambda }{\left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] }\in [0,1]$ given $\lambda \in [0,1]$.

$$\begin{aligned} \mathcal {A}&= \left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] \left[ \frac{\tilde{n}\lambda }{\left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] } E_{\tilde{n}} + \frac{\hat{n}(1-\lambda )}{\left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] } E_{\hat{n}}\right] \\&= \left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] \left[ \alpha E_{\tilde{n}} + (1-\alpha ) E_{\hat{n}}\right] \\&\ge \left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] E_{\alpha \tilde{n} + (1-\alpha )\hat{n}}, \quad \forall \alpha \in [0,1]. \end{aligned}$$

Hence, substituting $\alpha$ by $\lambda$ we obtain,

$$\begin{aligned} \mathcal {A} \ge \left[ \tilde{n}\lambda + \hat{n}(1-\lambda )\right] E_{\lambda \tilde{n} + (1-\lambda )\hat{n}}\ge \mathcal {B}. \end{aligned}$$

Consequently,

$$\begin{aligned} \lambda \overline{N}_{\tilde{n}} + (1-\lambda )\overline{N}_{\hat{n}} \ge \overline{N}_{\lambda \tilde{n} + (1-\lambda )\hat{n}}, \end{aligned}$$

which leads that $\overline{N}_n, \; \tilde{n}\le n \le \hat{n}$ is convex. Therefore, we could use bisection searching method in Sect. 3.1 for the non-adaptive retransmission scenario.