​​Performance Optimization in RSMA-assisted Uplink xURLLC IIoT Networks with Statistical QoS Provisioning

Although RSMA has demonstrated promise in enhancing the QoS in next-generation wireless networks, recent research has predominantly concentrated on downlink multi-antenna and multi-access scenarios under Shannon regimes with long blocklengths [15, 16, 17]. Studies reveal that uplink RSMA can realize the optimal rate region of a $2M$-user Gaussian Multiple Access Channel (MAC) by exploiting up to $4M-1$ virtual point-to-point Gaussian channels generated through message splitting, thereby providing superior reliability, sum-throughput, and lower latency compared to non-orthogonal multiple access (NOMA) [18] and orthogonal multiple access (OMA) [14]. Furthermore, most studies on uplink RSMA are conducted under Shannon regimes with long blocklengths [19, 20, 21], while the exploration of RSMA-assisted xURLLC under FBL regimes is still in its infancy, accompanied by various challenges.

On the one hand, the stringent low-latency demands of IIoT devices necessitate xURLLC to operate under FBL regimes, underscoring the need to investigate the DEP and total throughput of uplink RSMA-assisted xURLLC networks. However, RSMA’s flexible interference management, involving message splitting and combining, potentially introduces complexity to networks and poses challenges for seamless xURLLC integration [5, 12, 11]. On the other hand, previous research endeavors on xURLLC have predominantly focused on deterministic QoS provisioning (DQP) mechanisms, which are insufficient for analyzing the QoS requirements of extreme and rare events [4, 1, 3, 22, 2, 23]. Specifically, xURLLC demands $99.9999\%$ reliability, sub-1 ms latency, statistical QoS provisioning (SQP), and relies on short-blocklength packets with sporadic arrival traffic, presenting unparalleled challenges to the existing Industrial Internet [4, 1]. Moreover, given the highly time-varying nature of wireless fading channels, DQP fails to accurately capture the queueing behavior of xURLLC traffic [4, 3, 22]. This deviates from the fundamental design principles mandated by xURLLC, which involve analyzing the tail distribution of stringent QoS, underscores the need for a paradigm shift. Consequently, a thorough understanding of statistical metrics associated with rare and extreme events, known as the tail distribution, is essential to accommodate xURLLC’s stringent QoS requirements [4, 3, 22].

Although embracing the SQP mechanism for xURLLC appears more feasible, there is a dearth of relevant studies in this area, leaving the underlying principles enigmatic [2, 1, 3, 4, 23, 22]. Currently, SQP theory championed by stochastic network calculus (SNC), has gained extensive traction [24, 25, 26]. SNC has evolved into a prominent methodology providing dependable theoretical insights into the SQP performance of latency-sensitive services [4, 3, 22]. Nevertheless, current endeavors fall short of providing sufficient guidance for xURLLC’s SQP, and existing state-of-the-art solutions for xURLLC operate independently, lacking direct compatibility and seamless integration [27, 28, 29, 30, 31, 32]. Therefore, tailoring an innovative RSMA-assisted transmission framework for xURLLC looms large.

State-of-the-art research endeavors have extensively focused on uplink RSMA under Shannon regimes with long blocklengths [19, 20, 21, 33], and relatively little research has focused on uplink RSMA under FBL regimes [34], let alone considering the SQP-based tail analysis for uplink RSMA-assisted xURLLC networks. The authors in [19] have investigated a two-user uplink cooperative RSMA (C-RSMA) scheme, which jointly optimizes beamforming and device transmit power. In [20], authors have tackled the optimization problem of maximizing the minimum SE of uplink RSMA-assisted user-centric massive MIMO networks. To enhance SE and user fairness, the authors in [21] have proposed a novel C-RSMA scheme for uplink user cooperation. Additionally, the authors in [33] have studied the optimization problems of power allocation and decoding order, and have proposed a low-complexity user-pair-based resource allocation algorithm to minimize the maximum latency. To study the impact of target rate and blocklength on the DEP and throughput performance, the authors in [34] have investigated the performance of uplink RSMA under FBL regimes.

xURLLC underscores extreme and rare events such as 99.9999% reliability and sub-1 ms latency [1, 35]. Currently, some relevant studies have explored the statistical QoS provisioning for xURLLC [4, 3, 22, 36, 37, 38]. In [4], we have proposed a NOMA-assisted uplink xURLLC network and analyzed the tail distributions of latency, age-of-information (AoI), and reliability of xURLLC leveraging SNC theory. To explore xURLLC’s fundamentals and performance tradeoffs, we have developed an xURLLC-enabled massive MIMO network and proposed a theoretical framework for statistical QoS provisioning analysis for xURLLC by leveraging and promoting SNC theory [3]. In [22], the authors have proposed a 360^∘ virtual reality (VR) streaming architecture with statistical QoS provisioning by developing an SNC-based SQP theoretical framework from delay and rate perspectives. To fulfill QoS guarantees for URLLC slicing, the authors in [36] have proposed an SNC-based URLLC slicing allocation scheme, which effectively ensures the QoS violation probability remains consistently lower than the target value over the long term with a certain probability. To investigate the violation probability of end-to-end (E2E) latency for target services in IIoT, the authors in [37] have introduced an SNC-based cascaded theorem to evaluate the service capability of xURLLC networks and studied the service processes across various wireless fading channels.

To effectively overcome the aforementioned challenges, we have developed an innovative RSMA-assisted uplink xURLLC IIoT (RSMA-xURLLC-IIoT) network architecture under FBL regimes and imperfect channel-state-information (CSI) scenarios. To accurately capture the queueing behavior of sporadic xURLLC traffic, we have proposed an SNC-based SQP (SNC-SQP) theoretical framework to analyze the tail distribution of xURLLC’s QoS requirements. Building upon this theoretical framework, we have formulated two SQP-driven optimization problems tailored for the developed RSMA-xURLLC-IIoT network architecture and proposed a low-complexity algorithm to tackle them. Extensive simulations demonstrate the dependability of the proposed theoretical framework, and comprehensive comparisons validate the superior performance of our developed RSMA-xURLLC-IIoT network architecture. The primary contributions of this paper are summarized as follows:

• •

  We have developed an innovative RSMA-xURLLC-IIoT network architecture, which seamlessly bridges the integration gaps between RSMA and xURLLC under FBL regimes and imperfect CSI scenarios, providing flexible interference management, robust multi-access, impeccable reliability, and resilient low latency.

• •

  Leveraging SNC theory, we have proposed a dependable SNC-SQP theoretical framework. In particular, a novel terminology named statistical delay violation probability (SDVP) is introduced to access the SQP performance of xURLLC’s delay. Subsequently, we derive a closed-form upper-bounded expression for SDVP termed UB-SDVP to quantify SDVP accurately. Additionally, we have derived various closed-form expressions for the probability density functions (PDFs) of wireless channels involved in our developed RSMA-xURLLC-IIoT network architecture.

• •

  Building upon the proposed theoretical framework, we have delved into two critical optimization problems. The first one addresses the SQP-driven short-packet size maximization problem, which holds extraordinary significance for the IIoT since it guarantees not only the execution of complex machine instructions but also the accomplishment of coherent mission-critical tasks. The second problem deals with the SQP-driven transmit power minimization, which is also of great importance, particularly with the application prospect of massive xURLLC networks and battery-constrained IIoT devices. Then, we have proposed a low-complexity three-step sequential optimization algorithm (TSOO) for their efficient resolutions.

• •

  Extensive simulations have reliably demonstrated the effectiveness of our developed RSMA-xURLLC-IIoT network architecture. Comprehensive comparisons with results from Monte-Carlo methods rigorously validate the dependability of the proposed SNC-SQP theoretical framework. Furthermore, through comparisons with prevalent NOMA and OMA, we have further substantiated the remarkable performance gains achieved by our developed network architecture.

The remainder of this paper is organized as follows. In Sec. II, the RSMA-xURLLC IIoT network architecture is investigated. In Sec. III, the SNC-SQP theoretical framework is developed. In Sec. IV, two optimal SQP-driven optimization problems are formulated and addressed, followed by the performance evaluation and comparisons. Finally, a conclusion of this paper is given.

At the transmitter side, the message from IIoT device-$u$ at time slot $t$ is represented as $M_{u}(t),u\!\in\!\mathcal{U}$. Following RSMA principles [12, 11], the message $M_{1}(t)$ from IIoT device-$1$ is split into two components, denoted as $M_{1,k}(t),k\!\in\!\left\{1,2\right\}$. The message $M_{q}(t)$ can be first encoded into the stream $x_{q}(t)$, where $q\in\mathcal{Q}\triangleq\{(1,1),(1,2),(2)\}$. These streams are mapped into finite-length codewords and transmitted over wireless fading channels. The received signal at the BS can be given as follows:

where $p_{q}$ is the uplink transmit power of $x_{q}$, $g_{u}(t)\!\!=\!\!\sqrt{\zeta_{u}(t)}h_{u}(t)$ represents the channel coefficient of IIoT device-$\!u\!\in\!\mathcal{U}$, and $n(t)\!\sim\!\mathcal{CN}\left(0,\sigma_{n}^{2}\right)$ is the Additive White Gaussian Noise. $\zeta_{u}(t)$ and $h_{u}(t)$ denote the large-scale fading coefficient and small-scale fading coefficient, respectively.

In realistic industrial scenarios, channel estimation is essential since IIoT devices do not have perfect prior knowledge of channel states, and the actual channel coefficient $h_{u}(t)$ can not be perfectly known. We consider each IIoT device-$u$ transmits a known orthogonal pilot sequence of length $N_{p,u}$ to the BS at the beginning of each time slot. As a result, the BS can obtain the minimum mean squared error (MMSE) channel estimation $\hat{h}_{u}(t)$ for IIoT device-$u$. Given the obtained $\hat{h}_{u}(t)$, the $h_{u}(t)$ can be given as

where $\tilde{e}_{u}(t)\!\sim\!\mathcal{CN}\left(0,\sigma_{e,u}^{2}\right)$, with $\sigma_{e,u}^{2}\!=\!\frac{1}{1+\bar{\gamma}_{p,u}(t)N_{p,u}}$, and $\bar{\gamma}_{p,u}(t)$ denotes the actual signal-noise-ratio (SNR) of IIoT device-$u$ during the training phase, which is constant and known at both transmitter and receiver. The channel estimation $\hat{h}_{u}$ is Gaussian distributed as $\hat{h}_{u}\!\sim\!\mathcal{CN}\left(0,\rho_{u}^{2}\right)$ with $\rho_{u}^{2}\!=\!\bar{\gamma}_{p,u}N_{p,u}/\left(1+\bar{\gamma}_{p,u}N_{p,u}\right)$. Then, the actual SNR $\Gamma_{q}=\bar{\gamma}_{q}|h_{u}|^{2}$ can be given as follows:

where operators $\Re\{\cdot\}$ and $\angle\{\cdot\}$ represent the real part and phase of the complex number, respectively. The term $\Re\left\{e^{-j\angle\left(\hat{h}_{u}\right)}\tilde{e}_{u}\right\}$ follows a Gaussian distribution with variance $\sigma_{e,u}^{2}/2$, and the term of estimation error $\bar{\gamma}_{q}|\tilde{e}_{u}|^{2}$ becomes quite small after sufficient pilot training, i.e., $|\tilde{e}_{u}|\ll|\hat{h}_{u}|$, which can be reasonably neglected in (3). In this case, the distribution of $\Gamma_{q}$ can be approximated as $\Gamma_{q}\sim\mathcal{N}\left(\widehat{\gamma}_{q},\widehat{\sigma}_{q}\right)$, where $\widehat{\gamma}_{q}=\bar{\gamma}_{q}|\hat{h}_{u}|^{2}$, $\widehat{\sigma}_{q}=2\bar{\gamma}_{u}^{2}|\hat{h}_{u}|^{2}\sigma_{e,u}^{2}$, and $\bar{\gamma}_{q}=p_{q}\zeta_{u}/\sigma_{n}^{2}$ denotes the average SNR of stream $x_{q}$. If $q\in\big{\{}(1,1),(1,2)\big{\}}$, then $u=1$; otherwise, $u=2$. The blocklength used for short-packet communications and channel estimation is typically extremely limited. We consider that the blocklength of each time slot is $N_{0}$, then only $N_{d}\!=\!N_{0}-N_{p,1}-N_{p,2}$ channel uses (CUs) available for short-packet data transmission.

At the receiver side, we consider that the streams are decoded in the order of $x_{1,1}\!\rightarrow\!x_{1,2}\!\rightarrow\!x_{2}$²²2The decoding order can be categorized into six cases, including (1) $x_{1,1}\!\rightarrow\!x_{2}\!\rightarrow\!x_{1,2}$, (2) $x_{1,2}\!\rightarrow\!x_{2}\!\rightarrow\!x_{1,1}$, (3) $x_{1,1}\!\rightarrow\!x_{1,2}\!\rightarrow\!x_{2}$, (4) $x_{1,2}\!\rightarrow\!x_{1,1}\!\rightarrow\!x_{2}$, (5) $x_{2}\!\rightarrow\!x_{1,1}\!\rightarrow\!x_{1,2}$, (6) $x_{2}\!\rightarrow\!x_{1,2}\!\rightarrow\!x_{1,1}$. Without loss of generality, we select the decoding order (3) in this paper.. Accordingly, $x_{1,1}(t)$ is first decoded by treating $x_{1,2}(t)$ and $x_{2}(t)$ as noise. Then, the SINR of $x_{1,1}(t)$ is represented as

If $\!x_{1,1}(t)\!$ is decoded successfully, it can be reconstructed into $\!\widehat{M}_{1,1}(t)$, and then removed from the received signal $y(t)$. Secondly, $x_{1,2}(t)$ is decoded by treating $\!x_{2}(t)\!$ as noise. Thus, the SINR of $x_{1,2}(t)$ can be expressed as

If $x_{1,2}(t)\!$ is decoded successfully, it can also be reconstructed into $\!\!\widehat{M}_{1,2}(t)$, and removed from $y(t)$. Using a message combiner, the estimated message of IIoT device-$1$ is $\!\widehat{M}_{1}(t)\!=\!\widehat{M}_{1,1}(t)\!+\!\widehat{M}_{1,2}(t)$. Finally, the SINR of $x_{2}(t)$ can be given as follows:

If $x_{2}(t)$ is decoded successfully, it can be reconstructed into $\widehat{M}_{2}(t)$.

Considering the short-packet size of $x_{q}$ to be $B_{q}$ bits, typically $B_{q}\!\leq\!500$ bits, $q\!\in\!\mathcal{Q}$. Then, the achievable coding rate can be expressed as $R_{q}=\frac{B_{q}}{N_{d}}$ (bpcu). Following the finite blocklength coding (FBC) theory [3, 4, 9], the decoding error probability can be given by

where $Q(\cdot)$ denotes the Q-function, which can be expressed by $Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\mathrm{d}x$. According to RSMA principles [5, 12, 11], since the rate of the IIoT device-$1$ $R_{1}$ is split, the achievable data rate of $x_{1,1}(t)$ and $x_{1,2}(t)$ can be represented as $R_{1,1}(t)\!=\!\alpha R_{1}(t)$ and $R_{1,2}(t)\!=\!(1-\alpha)R_{1}(t)$, respectively, where $0\leq\alpha\leq 1$ denotes the rate allocation ratio [12, 11].

To facilitate system-level analysis, we consider a statistical QoS-driven queueing system that employs the first-come-first-serve (FCFS) policy [3, 22, 26]. The cumulative arrival, service, and departure processes of $x_{q}$ within the time interval $[s,t)$ are given as $A_{q}(s,t)\!\triangleq\!\sum_{i=s}^{t-1}a_{q}(i)$, $S_{q}(s,t)\!\triangleq\!\sum_{i=s}^{t-1}s_{q}(i)$, and $D_{q}(s,t)\!\triangleq\!\sum_{i=s}^{t-1}d_{q}(i)$, respectively. Here, $a_{q}(i)$ and $s_{q}(i)$ represent the amount of short-packet data generated by $x_{q}(i)$ and transmitted over wireless channels at time slot $i$, respectively, while $d_{q}(i)$ indicates the amount of short-packet data of $x_{q}(i)$ that is successfully received at time slot $i$. Then, to facilitate the analysis of SQP performance, we introduce a succinct and novel operator by leveraging moment generating function (MGF)-based SNC theory, which is termed min-deconvolution $\widehat{\oslash}$ as follows:

Based on the operator min-deconvolution, we have the following Theorem 1:

The promotion of MGF-SNC theory in Theorem 1 provides a crucial theoretical foundation for further deriving the closed-form expression of UB-SDVP for the developed RSMA-xURLLC-IIoT network architecture. We consider $A_{q}(s,t)$, $S_{q}(s,t)$, and $D_{q}(s,t)$ have independent and identically distributed (i.i.d.) increments. Then, the MGF and the inverse-MGF of $A_{q}(s,t)$ and $S_{q}(s,t)$ can be respectively expressed as follows:

Combining Definition 1 with (10a) and (10b), we can derive the closed-form expression of UB-SDVP, as stated in Theorem 2.

Theorem 2 refers that further derivation of the inverse-MGF of the service process increment $s_{q}$ (i.e., $\overline{\mathbb{M}}_{s_{q}}\left(\theta_{q}\right)$) and the MGF of the arrival increment $a_{q}$ (i.e., $\mathbb{M}_{a_{q}}\left(\theta_{q}\right)$) is essential to determine the closed-form expression for UB-SDVP. In this case, we consider the arrival process follows a Poisson distribution, as follows:

where $\!\lambda_{q}^{{\dagger}}\!$ denotes the average arrival rate of $\!A_{q}\!\left(s,t\right)$.

The inverse-MGF $\overline{\mathbb{M}}_{s_{q}}\!(\theta_{q})$ depends on $\widehat{\Gamma}_{q}(t)$, which varies from time slots, thus $\overline{\mathbb{M}}_{s_{q}}\!(\theta_{q})$ can be given by

where $\mathbb{E}_{\widehat{\Gamma}_{q}}\left[\cdot\right]$ denotes the expectation operator of $\widehat{\Gamma}_{q}(t)$. From (14), the expression of p.d.f. for $\widehat{\Gamma}_{q}(t)$ is indispensable for determining $\overline{\mathbb{M}}_{s_{q}}\!\!\left(\theta_{q}\right)$, which motivates Lemma 1 as follows:

Combining Theorem 1, Theorem 2, and Lemma 1, the closed-form expression of UB-SDVP can be finally derived.

Given the statistical QoS requirements $\left(\xi_{q}^{th},w_{q}^{th},\varepsilon_{q}^{th}\right)$, where $\xi_{q}^{th}$ denotes the threshold od SDVP, $w_{q}^{th}$ indicates the threshold of target delay, and $\varepsilon_{q}^{th}$ represents the threshold of decoding error probability. As a result, the SQP-driven short-packet size maximization problem can be formulated as

where $\bm{\mathrm{B}}\!\triangleq\!\big{[}B_{1,1},B_{1,2},B_{2}\big{]}$ and $\bm{\mathrm{p}}\!\triangleq\!\big{[}p_{1,1},p_{1,2},p_{2}\big{]}$, and (16a) defines the objective function and optimization variables; (16b) imposes a limit of $\xi_{q}^{th}$ on the SDVP of $x_{q}$, where $\xi_{q}^{th}\!$ is the SDVP threshold; (16c) indicates the constraint on the decoding error probability of $x_{q}$, with $\varepsilon_{q}^{th}$ as the decoding error probability threshold; (16d) restricts the transmit power $p_{u}$ of IIoT device-$u$ to a maximum value of $p_{max}$; (16e) specifies the constraint on the short-packet size of IIoT device-$u$. Finally, (16f) defines the range of values for the rate-splitting ratio.

Given $\left(\xi_{q}^{th},w_{q}^{th},\varepsilon_{q}^{th}\right),q\!\in\!\mathcal{Q}$, the SQP-driven transmit power minimization problem can be formulated as

The intertwined co-channel interference [12, 11] and the highly-varying wireless fading channels [3, 4] in RSMA-xURLLC-IIoT network architecture render the analytical expression for SDVP in (16b) is inaccessible [26, 22]. Additionally, the closed-form expression for the decoding error probability of short-packet data transmission becomes exceedingly complex in FBL regimes [22]. As a result, $\mathcal{P}1$ and $\mathcal{P}2$ become exceedingly intractable and cannot be tackled analytically. Fortunately, by leveraging the proposed SNC-SQP theoretical framework, the inaccessible SDVP can be converted into the manageable UB-SDVP, which not only provides insightful theoretical guidance but also guarantees relatively conservative information bit transmission and power allocation schemes. Consequently, (16b) in $\mathcal{P}1$ and $\mathcal{P}2$ can be reformulated as follows:

where (18a) imposes a limit of $\xi_{q}^{th}$ on the UB-SDVP for stream $x_{q}$, and $(\ref{e16}b)$ represents the stability condition of UB-SDVP.

Due to the complex constraints, solving $\mathcal{P}1$ and $\mathcal{P}2$ remains non-trivial. Therefore, we focus on exploring the intrinsic properties of $\mathcal{P}1$ and $\mathcal{P}2$.

From Corollary 1, $\left[0,\theta_{q}^{max}\!\left(\bm{\mathrm{p}},\bm{\mathrm{B}},\alpha\right)\right]$ can be easily determined by the one-dimensional search method. According to the properties of the convex function, we can easily determine $\theta_{q}^{\ast}\!\left(\bm{\mathrm{p}},\bm{\mathrm{B}},\alpha\right)$ through stochastic gradient descent (SGD) method⁴⁴4Due to space limitations, we have omitted detailed descriptions of one-dimensional search method and SGD method here since they are the fundamental content of optimization theory..

In addition, we also investigate the intrinsic properties of UB-SDVP as follows:

From Corollary 2 and Corollary 3, Theorem 3 can be derived as follows: