Performance Analysis and Blocklength Minimization of Uplink RSMA for Short
Packet Transmissions in URLLC

Awide range of real-time applications and services, such as autonomous vehicles, Industrial Internet of Things (IIoT), and augmented reality/virtual reality (AR/VR), are emerging at a fast speed in the upcoming sixth generation (6G) wireless networks [1]. Ultra-reliable and low-latency communications (URLLC), as the core service in the fifth generation (5G) communication networks, aims to provide end-to-end (E2E) delay of less than 1 ms for 32-bit packet transmission while ensuring packet error probability of less than $10^{-5}$ [2]. To meet the ultra-high quality of service (QoS) demands for various delay-sensitive services, more stringent requirement on delay is put forward as sub-millisecond level in 6G networks [3]. One of the key methods to meet the delay requirements of real-time applications is to use finite blocklength (FBL) code for short packet transmissions [4]. The traditional Shannon capacity, which is used in the infinite blocklength (IBL) for high-capacity-demanded services, is no longer applicable to short packet transmissions in URLLC. The authors derived a closed-form expression of the achievable rate and the decoding error probability in the FBL regime [5]. In addition, the design of future 6G networks also requires support for large-scale access to ensure stringent delay and reliability requirements from a large number of devices [6]. In the existing network architecture, delay timeout under multiple access (MA) conditions remains a difficult problem, where time-saving and reliable solutions are still very important for future networks.

Recently, rate splitting multiple access (RSMA) has been proposed as a promising technology to enhance spectral efficiency (SE), energy efficiency (EE), coverage, QoS, user fairness, and reliability while entailing lower delay, feedback overhead, and complexity [7]. RSMA relies on rate splitting (RS) and superposition coding (SC) at the transmitter, as well as successive interference cancellation (SIC) at the receiver. RSMA has been widely studied for downlink systems, showing the SE, EE, and delay enhancement of downlink RSMA [8, 9]. As for the uplink RSMA, it was first proposed in [10] for the single-input single-output (SISO) multiple access channel (MAC) to achieve every point of the Gaussian MAC capacity region without the need for time sharing and joint encoding-decoding among users. In this way, uplink RSMA can avoid high complexity and overhead, resulting in a relatively high rate and low delay. The throughput improvement of uplink RSMA has been studied in [11]. However, the delay performance with FBL code of uplink RSMA has not been studied, which means how to take advantage of RSMA in the URLLC system still needs to be addressed. To fully explore the delay performance improvement brought by RSMA to URLLC, the FBL and RSMA combined analysis and delay minimization with blocklength optimization are highly needed.

In order to solve this problem, in this paper we analyze the impact of FBL code on uplink RSMA to obtain the achievable rate region and the effective throughput of uplink RSMA in the FBL regime. Based on the above analysis, we propose the blocklength minimization problem with power allocation for uplink RSMA under constrained reliability and effective throughput. Then, we present an alternating optimization algorithm to solve this non-convex problem. Simulation results show that our proposed uplink RSMA-based blocklength minimization problem with power allocation can use a lower blocklength with a lower delay to achieve the same achievable rate compared to uplink non-orthogonal multiple access (NOMA), frequency division multiple access (FDMA), and time division multiple access (TDMA).

The rest of this paper is organized as follows. Section II introduces the two-user uplink RSMA system model. Section III analyzes the performance of uplink RSMA in the FBL regime. Section IV presents the uplink RSMA-based blocklength minimization problem with power allocation and the corresponding algorithm. Section V provides the numerical results. Finally, we conclude this paper in Section VI.

As shown in Fig. 1, we consider a two-user uplink RSMA system consisting of one base station (BS) and two users U1 and U2. In this paper, 1-layer RSMA is adopted to serve U1 and U2. The message $W_{1}$ of U1 is split into two sub-messages $W_{11}$ and $W_{12}$, which can be interpreted as creating 2 virtual users at U1 [7]. The messages $W_{11}$ and $W_{12}$ are independently encoded into streams $s_{11}$ and $s_{12}$, which are then respectively allocated with certain powers $P_{11}$ and $P_{12}$. Thus, the transmit signal at U1 is given by

At U2, the message $W_{2}$ is directly encoded into $s_{2}$. By allocating a certain power $P_{2}$, the transmit signal at U2 is given by $x_{2}=\sqrt{P_{2}}s_{2}$. Thus, the signal received at the BS is given by

where $h_{i}\ (i=1,2)$ denotes the channel coefficient of U$i$ and $z$ denotes the additive white Gaussian noise (AWGN) at the BS with zero-mean and variance $\sigma_{n}^{2}$. We assume that the decoding order is $s_{11}\rightarrow s_{2}\rightarrow s_{12}$. The BS first regards $s_{12}$ and $s_{2}$ as interference to decode $s_{11}$. Thus, the signal to interference noise ratio (SINR) of the first decoded stream $s_{11}$, denoted by $\gamma_{11}$, can be expressed as

where $G_{i}=|h_{i}|^{2}\ (i=1,2)$ denotes the channel gain of U$i$. Assuming that $s_{11}$ is successfully decoded, the BS removes $s_{11}$ and decodes $s_{2}$ while treating $s_{12}$ as noise. Through SIC, the SINR of the second decoded stream $s_{2}$, denoted by $\gamma_{22}$, can be expressed as

Next, the BS removes $s_{2}$ and decodes $s_{12}$ when $s_{2}$ is successfully decoded. The SINR of the third decoded stream $s_{12}$, denoted by $\gamma_{12}$, can be expressed as

To satisfy stringent delay requirements in URLLC, we propose the uplink RSMA-based blocklength minimization problem with power allocation scheme for short packet transmissions under reliability and effective throughput demands.

In this section, we evaluate the achievable rate region performance and the proposed blocklength minimization of uplink RSMA. We set the channel gain $G_{1}=1$ for U1 and $G_{2}=0.7$ for U2 with $G_{1}>G_{2}$. Without loss of generality, we assume the noise variance $\sigma^{2}=1$. For NOMA, we set NOMA-12 with the decoding order $s_{1}\rightarrow s_{2}$ and NOMA-21 with the decoding order $s_{2}\rightarrow s_{1}$. We set the bandwidth fraction and time fraction as $\alpha^{\rm F}=\alpha^{\rm T}=\frac{P_{1}}{P_{1}+P_{2}}$ in FDMA and TDMA. To satisfy the reliability requirements in URLLC, the predefined error probability of each stream is set to $\varepsilon=10^{-6}$ to ensure the overall error probability of uplink RSMA is lower than $10^{-5}$. In addition, we set the maximum blocklength $N_{\max}=3000$ and the minimum blocklength $N_{\min}=100$ [13].

Figure 3 depicts the achievable rate region comparison of different MA strategies with different blocklengths as $n$ set to $500,1000,2000$, and infinite. The achievable rates of U1 and U2 both increase as blocklength increases, and therefore the achievable rate region also expands with the increase of blocklength. In the traditional IBL regime, uplink RSMA can achieve the Gaussian MAC capacity region, while NOMA without time sharing can achieve maximum rate only at one of the users. This is because by changing the transmit power allocation between two data streams $s_{11}$ and $s_{12}$, RSMA can bridge NOMA-12 (allocate all the transmit power to $s_{11}$) and NOMA-21 (allocate all the transmit power to $s_{12}$), which leads to a larger rate region without time sharing. FDMA can only reach the capacity region at one point with $\alpha^{\rm F}=\frac{P_{1}}{P_{1}+P_{2}}$, while TDMA cannot reach the capacity region without variable transmit power.

However, in the FBL regime, uplink RSMA cannot achieve the Gaussian MAC capacity region and the achievable rate region of RSMA is not always larger than that of NOMA. On the one hand, the error probability no longer approaches $0$ with the FBL code, resulting in a decrease in the achievable rate. On the other hand, the signal $s_{1}$ is divided into two streams in uplink RSMA, which brings more channel dispersion terms than NOMA, resulting in a decrease in the total achievable rate. According to $D=\sqrt{\frac{V}{n}}Q^{-1}(\varepsilon){\log_{2}}e$, it can be observed that as the blocklength increases, the impact of channel dispersion gradually decreases. As shown in Fig. 3, the rate region of RSMA can gradually include NOMA with blocklength increasing.

Figure 4 shows the blocklength comparison of different MA strategies versus the maximum transmit power with different effective throughput requirements. The blocklength gradually decreases as the maximum transmit power increases. As shown in Fig. 4(a), although RSMA and NOMA can use the common blocklength without dividing it to two users, RSMA can achieve a smaller blocklength. This is because NOMA decodes $s_{1}$ first and removes $s_{1}$ when decoding $s_{2}$, which can guarantee a high rate of $s_{2}$ but result in a relatively low rate of $s_{1}$. When $T^{\rm th}_{1}$ is large, U1 needs a larger blocklength to reach $T^{\rm th}_{1}$ than that of RSMA. In FDMA and TDMA, U1 and U2 cannot share the common blocklength. As a result, the blocklength is divided into U1 and U2 in the frequency domain and time domain, resulting in a relatively large blocklength. It can be seen from Fig. 4(b) that the blocklengths of RSMA and NOMA are the same, which means NOMA can achieve the same rate as RSMA with the same blocklength. This is because when the effective throughput requirement is set as $T^{\rm th}_{1}<T^{\rm th}_{2}$, the guaranteed rate of $s_{2}$ can satisfy a larger requirement $T^{\rm th}_{2}$. However, RSMA can adapt to different throughput requirements and achieve the smallest blocklength, which means it is more suitable and flexible than NOMA for heterogeneous networks with different QoS requirements.

Figure 5 plots the blocklength comparison of different MA strategies versus the error probability with different maximum transmit powers. The blocklength decreases as the error probability increases for the four MA strategies, while the blocklength of RSMA is always lower than that of NOMA, FDMA, and TDMA. In addition, for vertical businesses with different reliability requirements in URLLC, as long as their reliability requirements are met, the minimum blocklength can be selected to achieve the minimum delay.

Figure 6 shows the sum of the achievable rates versus the blocklength. RSMA can achieve the highest rate in both IBL and FBL regimes. In the FBL regime, the sum of achievable rates increases with the blocklength increasing. Therefore, the minimum blocklength can be selected as long as the rate requirements are achieved to satisfy low-delay requirements in URLLC. In addition, for a fixed value of blocklength, the achievable rate of RSMA without power allocation (PA) is lower than those of RSMA, NOMA-12, and NOMA-21 with power allocation. This means that through power allocation can RSMA achieve the function of bridging NOMA-12 and NOMA-21, resulting in a higher rate with the same blocklength.

In this paper, we solved the problem of minimizing the blocklength with the power allocation for uplink RSMA in URLLC. In particular, we analyze the performance of uplink RSMA in the FBL regime, in terms of the achievable rate region and the effective throughput. On this basis, we proposed the uplink RSMA-based blocklength minimization problem under the reliability and effective throughput constraints to satisfy the low-delay and reliability requirements in URLLC. Furthermore, we developed an alternating optimization algorithm to solve this non-convex problem to obtain the optimal power allocation and the minimum blocklength. Numerical results demonstrated that uplink RSMA cannot achieve the Gaussian MAC capacity region in the FBL regime. However, with the help of our proposed blocklength minimization scheme, uplink RSMA can significantly reduce the blocklength to achieve a lower delay compared to uplink NOMA, FDMA, and TDMA, showing the potential of uplink RSMA for URLLC.

The achievable rate of U1 can be expressed as follows:

Thus, we have $R_{11}\left(n,\gamma_{11}\right)+R_{12}\left(n,\gamma_{12}\right)+R_{2}\left(n,\gamma_{22}\right)=C(\gamma_{\rm sum})-D_{11}-D_{12}-D_{22}$.