# Optimization of Irregular Repetition Slotted ALOHA with Imperfect SIC in 5G CIoT

Saeed Alsabbagh, Cédric Adjih, Amine Adouane, Nadjib Aitsaadi

► To cite this version:

Saeed Alsabbagh, Cédric Adjih, Amine Adouane, Nadjib Aitsaadi. Optimization of Irregular Repetition Slotted ALOHA with Imperfect SIC in 5G CIoT. ICC 2025 - IEEE International Conference on Communications, Jun 2025, Montreal, Canada. pp.5234-5240, 10.1109/ICC52391.2025.11161616. hal-05472822

HAL Id: hal-05472822

https://hal.science/hal-05472822v1

Submitted on 23 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Optimization of Irregular Repetition Slotted ALOHA with Imperfect SIC in 5G CIoT

Saeed Alsabbagh*, Cédric Adjih†, Amine Adouane‡, Nadjib Aitsaadi*

*University of Paris-Saclay, UVSQ, DAVID Laboratory, F-78035, Versailles, France

‡University of Algiers, A-16000, Algiers, Algeria

†INRIA, Saclay, F-91120, Palaiseau, France

Emails: (firstname). (lastname)@{*uvsq.fr, ‡univ-alger.dz, †inria.fr}

***Abstract***—Irregular Repetition Slotted ALOHA (IRSA) is an effective grant-free random access scheme that is well-suited for managing the sporadic nature of IoT traffic, particularly in dense environments prone to collisions. In this paper, we evaluate the performance of IRSA under realistic conditions involving imperfect successive interference cancellation (SIC) and non-ideal physical layer environments. Specifically, we investigate the impact of various channel conditions and physical layer impairments on IRSA's performance. Previous studies on IRSA often assume ideal physical layer conditions or use simplified models for SIC errors, which fail to fully capture practical implementation complexities. To address this gap, we propose integration of practical factors, such as channel estimation imperfections, into our model of SIC failures using detailed baseband simulations. Based on that, we employ density evolution analysis to evaluate system throughput and optimize the degree distributions to enhance IRSA performance in the presence of imperfect SIC. Our results focusing on 5G CIoT demonstrate that optimizing IRSA parameters, while accounting for SIC errors, can significantly improve system performance, resulting in notable throughput gains.

***Index Terms***—IRSA, imperfect SIC, Density Evolution, 5G NR, Grant-Free, CIoT Networks, Random Access

## I. INTRODUCTION

The Internet of Things (IoT) is emerging as a fundamental component of modern society, connecting billions of devices across various sectors, such as smart homes, cities, industrial automation, and healthcare [1]. These applications require efficient and reliable communication, posing significant challenges to existing wireless networks, particularly due to the massive scale and sporadic traffic patterns of IoT devices. Grant-free access has been proposed as a promising solution to address these challenges by allowing devices to transmit data without prior scheduling, reducing latency and enabling efficient handling of sporadic traffic [2]. Thus, random access protocols, which allow multiple users to transmit data without prior coordination, are seen as key enablers to manage the massive access required in IoT systems. Among these protocols, Irregular Repetition Slotted ALOHA (IRSA) has gained significant attention as a promising random access solution to effectively manage collisions in highly dense IoT environments [3], [4]. Numerous variants have been proposed, such as applying packet erasure codes to enhance IRSA's performance [5], extending IRSA to support multi-packet reception [6], [7], or optimization through machine learning [8].

These studies have provided insights into the performance bounds of IRSA variants in ideal scenarios, but have largely overlooked the impact of physical layer imperfections and practical implementation challenges.

Physical layer imperfections, such as channel estimation errors and transmission errors, can significantly degrade the performance of IRSA. Previous studies have often employed simplified models to address these imperfections, such as the exponential model for Successive Interference Cancellation (SIC) errors [9]. However, these simplified models fail to fully capture the complexity of a realistic physical layer environment, which includes effects like channel fading, noise, and hardware imperfections. The impact of these factors on system performance can be substantial, as demonstrated in [10], [11], making it imperative to consider them for more accurate evaluations.

In this work, we aim to bridge this gap by evaluating the performance of IRSA under realistic physical layer conditions, incorporating practical aspects such as channel estimation, modulation, and coding schemes. Specifically, we study the effects of these imperfections on the performance of IRSA under a more realistic modeling of SIC failures that captures the system's dynamics more accurately, going up to detailed baseband simulations. Additionally, we analyse the system by adapting the classical *density evolution* tool to study the achieved IRSA throughput under practical constraints.

The rest of the paper is organized as follows: Section II provides background on IRSA principles to set the stage for the rest of the paper. Section III describes the proposed model for the failures of the SIC process. In Section IV, we detail the density evolution analysis and the optimization of IRSA parameters. Section V presents the simulation results, and finally, Section VI concludes the paper.

## II. BACKGROUND ON IRSA PRINCIPLES

### A. IRSA Protocol

The Irregular Repetition Slotted ALOHA (IRSA) protocol is a grant-free random access scheme that allows User Equipment (UEs) to transmit data to a Base Station (BS) without the need for prior scheduling. This eliminates the complexity of coordination, making it well-suited to the sporadic nature of IoT traffic and efficient in handling large-scale deployments. In this work, we consider a typical framework in which *M*
<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of the IRSA protocol. Four users transmit their blocks in a frame of five slots. First, the BS managed to retrieve the block of UE 2 in slot 2. After that, the other replica of UE 2 in slot 5 is removed, allowing the recovery of UE 3 in slot 5. All the replicas of UE 3 are canceled in the next step, allowing the recovery of UE 1 in slot 1. Following the same process, the block of UE 4 in slot 4 is recovered, and the decoding process is completed.</figcaption></figure>

active UEs transmit their data to the BS over a common uplink channel, organized into fixed-duration frames.

Each frame consists of *N* equally spaced time slots, and UE's transport block occupies one slot. All UEs are assumed to know the structure of the frame and be synchronized with the BS. Each UE is assumed to have a single transport block to transmit per frame. The BS receives the superposition of all the signals transmitted by the UEs in a given slot. Successful decoding occurs if only one UE transmits in a slot. The average number of transport blocks per slot, known as the normalized traffic load, is defined as $G = M/N$.

In the conventional Slotted ALOHA (SA) protocol, UEs randomly select one slot per frame to transmit their transport block [12]. To improve the performance of SA, the Contention Resolution Diversity Slotted ALOHA (CRDSA) protocol was introduced [13]. In the original CRDSA, UE transmit two identical replicas of their transport block in multiple slots within the frame, but these replicas are transmitted in different slots. This repetition increases the likelihood of successful decoding at the BS. Every time one of the UE's replicas is successfully decoded, the BS performs *Successive Interference Cancellation* (SIC): it reconstructs the physical signal corresponding to the recovered users transport blocks and subtracts it from the slot where the other replica has been transmitted.

Liva [4] introduced IRSA as a generalization of the CRDSA scheme. In IRSA, each UE transmits *l* replicas of its transport block across different slots in the frame. The number of replicas, *l*, can differ among UEs and is selected based on a predefined probability distribution. It was shown that under classical random access hypothesis, IRSA can outperform other random access schemes achieving higher throughput and efficiency at high traffic loads. Figure 1 illustrates an example of IRSA protocol.

## B. Decoding Process of IRSA

The decoding process of the IRSA protocol can be understood through a graph-based representation (a *Tanner graph*, see Fig. 1, left), which provides a clear method for visualizing the interactions between transmitted replicas and time slots. This representation, inspired by the decoding of Low-Density Parity-Check (LDPC) codes, uses a bipartite graph connecting

user nodes to slot nodes. An edge between a user node and a slot node indicates that the user has transmitted a replica in that slot.

The decoding process begins by identifying slots with only one user node connected to them—these are referred to as degree-1 slots or singletons. These singletons are decoded first because there are no collisions in those slots. Once a user's transport block is successfully decoded, the base station can remove all its replicas from the other slots by canceling its contribution, effectively removing the corresponding edges. This cancellation enables the decoding of other blocks in subsequent iterations. The decoding process is iterative. After removing a decoded user's replicas from the graph, new singletons may appear. These newly revealed singletons are then decoded, and the cancellation process continues. The decoding proceeds until no more singletons are available, at which point the process may halt since no further decoding is possible. By studying the evolution of the graph during the decoding process, we can predict the likelihood of successful transmission recovery and optimize the degree distribution to maximize system throughput, as will be discussed in detail in Section IV.

## III. IMPERFECT SIC ANALYSIS

Consider a slot where *l* packets collide. We define $\omega_l$ as the probability of successfully retrieving a packet after SIC, meaning the interference from the other $l-1$ colliding packets has been eliminated. In IRSA literature [5], [6], [14], [15], it is often assumed that $\omega_l = 1$ for all $l$, which implies perfect SIC (and no noise). Some studies have explored the impact of imperfect SIC on IRSA performance, with limited attention given to the physical layer implementation. The work of [9] considers a simplified exponential error model for SIC errors with $\omega_l = \gamma^{l-1}$. Here $\gamma$ represents the probability of correctly canceling the interference from a single packet in a slot, and this probability is independent of the other packets in the slot (with $\gamma = 1$ indicating perfect SIC). This model provides a simplified view of SIC errors and can be useful for deriving a mathematical model to evaluate IRSA performance (see [9]). However, it does not fully capture the actual behavior of the SIC process. In practice, $\gamma$ is unlikely to remain constant with varying *l*, as it may depend on the specific implementation and channel conditions. Thus, this approach serves as an idealized physical layer simulation, while in real hardware environments, more significant variations may be expected. Several prior works have investigated the impact of SIC errors on IRSA decoding performance. Studies such as [11], [16], [17] have explored the limitations of various channel estimation techniques. While these analyses derive complex estimation methods and mathematical expressions, they do not readily translate into a universal model for SIC failure probability, and their assumptions may not hold across different setups.

In this work, we adopt a more realistic approach to calculate the probabilities {$\omega_l$} by modeling the physical layer implementation of the SIC process, as outlined in Annex B of [4].
We consider a generic time slot k where $\ell$ UEs are transmitting simultaneously, resulting in a superimposed received signal. The received signal in slot k can be expressed as:

$$ \mathbf{y}_k = \sum_{i=1}^{\ell} h_{i,k} \mathbf{x}_i + \mathbf{n}_k, \qquad (1) $$

where $\mathbf{x}_i$ is the vector of symbols transmitted by user $i$. For simplicity, we assume unit power transmission, meaning that $\mathbf{E}[\|\mathbf{x}_i\|^2] = 1$. The channel coefficient for user $i$ in slot $k$, denoted by $h_{i,k}$, represents the channel gain. In this paper, we do not consider the capture effect, which implies that all UEs experience the same channel conditions within a given slot. This assumption is reflected by setting $h_{i,k} = \exp(j2\pi\phi_{i,k})$, where $\phi_{i,k}$ represents the phase shift of user $i$ in slot $k$ and is assumed to follow an uniform distribution $\phi_{i,k} \sim U[0, 2\pi)$. The noise $\mathbf{n}_k$ is modeled as complex AWGN with zero mean and variance $\sigma^2$, i.e., $\mathbf{n}_k \sim CN(0, \sigma^2\mathbf{I})$.

We assume that the receiver has already recovered the data from $\ell-1$ UEs and is now focusing on recovering the data of UE 1, without loss of generality. This implies that the receiver has full knowledge of all transmitted symbols $\mathbf{x}_i$ for users $i = 2, \dots, \ell$. To perform the IC, the receiver first needs to estimate the channel coefficients $h_{i,k}$ for all $\ell-1$ users in slot $k$.

The interference cancellation process proceeds in two steps. First, the phase of user $i$ is estimated. Here, since the BS has the complete knowledge of the transmitted symbols $\mathbf{x}_i$ this can used to provide a more accurate phase estimation. Therefore, using matched filtering, the receiver estimates the phase of user $i$ in slot $k$ as:

$$ \hat{\phi}_{i,k} = \arg (\mathbf{y}_k \mathbf{x}_i^H), \qquad (2) $$

Secondly, the contribution of user $i$ is removed from the received signal using the estimated channel coefficient $\hat{h}_{i,k} = e^{j2\pi\hat{\phi}_{i,k}}$. The updated signal, denoted as $\mathbf{y}_k^{(i)}$, is computed as:

$$ \mathbf{y}_k^{(i)} = \mathbf{y}_k - \mathbf{x}_i \hat{h}_{i,k}. \qquad (3) $$

This removal reduces the interference caused by user $i$, leaving the contributions from the remaining users and noise. The process is performed sequentially, user by user, until all $\ell-1$ interfering users have been addressed. After removing all interfering users, only the desired user's signal remains. However, due to imperfections in channel estimation, the contribution of other users may not be fully eliminated, resulting in residual interference. The final remaining signal is given by:

$$ \tilde{\mathbf{y}}_k = \mathbf{x}_1 h_{1,k} + \mathbf{i}_k + \mathbf{n}_k, \qquad (4) $$

where $\mathbf{i}_k$ represents the accumulated residual interference from imperfect cancellation. We define $\omega_\ell$ as the probability of successfully decoding $\mathbf{x}_1$ after removing $\ell-1$ interferers.

# IV. OPTIMIZING DEGREE DISTRIBUTIONS

## A. Density Evolution Analysis

Pioneering work [4] has introduced and formalized the use of techniques from LDPC coding theory as a tool that can be

transposed to the analysis of the performance of IRSA. As for LDPC codes, the analysis is valid asymptotically; that is, for IRSA, when the frame size converges toward infinity.

1) Preliminary Notations: In IRSA modeling, one introduces degree distributions.

Instead of having a fixed number or replicas, each user draws at random (for each frame), the number of replicas they will send. It is drawn from a user-perspective degree distribution $\Lambda = \{\Lambda_1, \Lambda_2, \dots, \Lambda_D\}$, where $\Lambda_d$ is thus the probability that the user sends $d$ replicas.

Symmetrically, the slot-perspective degree distribution, denoted by $\Psi = \{\Psi_1, \Psi_2, \dots, \Psi_N\}$, describes the probability that a slot contains exactly $\ell$ transmissions. One can further introduce their generating functions as:

$$ \Lambda(x) \stackrel{\text{def}}{=} \sum_{\ell=1}^{D} \Lambda_{\ell} x^{\ell}, \quad \Psi(x) \stackrel{\text{def}}{=} \sum_{\ell=1}^{N} \Psi_{\ell} x^{\ell}. \qquad (5) $$

The average number of replicas transmitted per user is given by $\Lambda'(1) = \sum \ell \Lambda_\ell$. We define the IRSA rate, $R$, as $R = 1/\Lambda'(1)^1$. Similarly, for the slot-perspective degree distribution, the average number of transmissions per slot is $\Psi'(1) = \sum \ell \Psi_\ell$. The probability that a user transmits in a given slot is $\Lambda'(1)/N = \Psi'(1)/M$, which establishes that $G = \Lambda'(1)/\Psi'(1)$. To express $\{\Psi_\ell\}$ in terms of the input parameters $\{\Lambda_\ell\}$ and $G$, we use the fact that the slot-perspective distribution $\{\Psi_\ell\}$ follows a binomial distribution $B(M, \frac{G}{RM})$. For large $M$, this binomial distribution approximates a Poisson distribution with parameter $\frac{G}{R}$.

The edge-perspective degree distributions capture the probability that a randomly chosen edge in the IRSA graph connects to (either a user or a slot) of a certain degree. For user nodes, $\{\lambda_\ell\}$ represents the probability that a randomly chosen edge connects to a user transmitting exactly $\ell$ replicas, while for slot nodes, $\{\rho_\ell\}$ represents the probability that a randomly chosen edge connects to a slot containing $\ell$ transmissions. Still following [4], these probabilities are expressed as:

$$ \lambda_{\ell} = \frac{\ell \Lambda_{\ell}}{\sum_{\ell} \ell \Lambda_{\ell}}, \quad \rho_{\ell} = \frac{\ell \Psi_{\ell}}{\sum_{\ell} \ell \Psi_{\ell}} \qquad (6) $$

and the corresponding generating functions are:

$$ \lambda(x) \stackrel{\text{def}}{=} \sum_{\ell} \lambda_{\ell} x^{\ell-1}, \quad \rho(x) \stackrel{\text{def}}{=} \sum_{\ell} \rho_{\ell} x^{\ell-1}. \qquad (7) $$

Based on these definitions, we can also write:

$$ \lambda(x) = \frac{\Lambda'(x)}{\Lambda'(1)}, \quad \rho(x) = \frac{\Psi'(x)}{\Psi'(1)}. \qquad (8) $$

Given that $\Psi \approx \text{Poisson}(\frac{G}{R})$, we have $\rho(x) = \Psi(x)$. By comparing the polynomial representations of $\rho$ and $\Psi$, it follows that, asymptotically for large $M$:

$$ \rho_{\ell} = \Psi_{\ell-1} \approx \frac{1}{(\ell-1)!} \left(\frac{G}{R}\right)^{\ell-1} e^{-\frac{G}{R}}. \qquad (9) $$

<footnote n="1">¹In IRSA, the rate $R$ does not directly correspond to throughput, as replicas removed by SIC do not consume additional slots. However, $R$ strongly impacts energy efficiency, with the average energy cost proportional to $1/R$.</footnote>
2) **Density Evolution Analysis with Ideal SIC:** This iterative decoding process, described in Section III is tracked by two quantitie at each step $i$: $p_i$, the probability that a random edge connects to a slot with an unresolved collision, and $q_i$, the probability that a random edge connects to a user whose packet has not yet been recovered. These quantities should be considered as ensemble averages over all frame instances generated with the same parameters, analogous to the averaging over code ensembles in LDPC code analysis [18]. In the absence of SIC errors, the evolution of $p_i$ and $q_i$ at each iteration $i$ is governed by recursive density evolution equations [4, Eqs. (1) and (2)]:

$$q_i = \sum_{\ell} \lambda_{\ell} p_{i-1}^{\ell-1} = \lambda(p_{i-1}), \quad (10)$$

$$p_i = \sum_{\ell} \rho_{\ell} (1 - (1 - q_i)^{\ell-1}) = 1 - \rho(1 - q_i), \quad (11)$$

starting with $q_1 = 1$. For given $\Lambda$ and $G$, we define $p_\infty(G, \Lambda) = \lim_{i \to \infty} p_i$, which always exists<fnref n="2" />. The Packet Loss Rate (PLR) of the system is then given by $P_L = \Lambda(p_\infty(G, \Lambda))$, and the throughput $\mathcal{T}(G, \Lambda)$ is calculated as:

$$\mathcal{T}(G, \Lambda) = G \cdot (1 - \Lambda(p_\infty(G, \Lambda))). \quad (12)$$

3) **Density Evolution Analysis with SIC Failures:** The density evolution equations can be adapted to systems affected by different types of errors, including SIC failures [9], [16] and transmission errors [20]. In the presence of SIC failures, these equations can be modified to incorporate the probabilities $\omega_\ell$, defined in Section III. Then we can adopt the general framework proposed in [4, Appendix A.] Precisely, $\rho(x)$ is updated as $\tilde{\rho}(x)$ to account for these SIC errors that, intuitively, results in lower decoding progress ($\tilde{\rho}(x) \le \rho(x)$):

$$p_i = \sum_{\ell} \rho_{\ell} (1 - \omega_{\ell} (1 - q_i)^{\ell-1}) = 1 - \tilde{\rho}(1 - q_i), \quad (13)$$

where $\tilde{\rho}(x) = \sum_{\ell} \rho_{\ell} \omega_{\ell} x^{\ell-1}$. Using the expression derived in Equation 9, $\tilde{\rho}(x)$ can be expressed as:

$$\tilde{\rho}(x) = e^{-\frac{G}{R}} \sum_{\ell} \frac{\omega_{\ell}}{(\ell-1)!} \left(\frac{G}{R}x\right)^{\ell-1}. \quad (14)$$

With the generating function $\tilde{\rho}(x)$ defined, the asymptotic throughput $\mathcal{T}(G, \Lambda, \omega)$ can be calculated using the recursive equations for $p_i$ and $q_i$ provided in Equations 10 and 13.

## B. Optimal Degree Distributions

From the definition of the asymptotic throughput $\mathcal{T}(G, \Lambda, \omega)$, it is possible to formulate an optimization problem. In presence of errors, it is known that its behavior differs from the Ideal SIC case: with Ideal SICs, for a fixed $\Lambda$, there is a load threshold $G^*(\Lambda)$ such that for every $G \le G^*$, $\mathcal{T}(G, \Lambda) = G$, i.e. there is no loss and $P_L = 0$. This is not the case when there are errors. Prior work as considered relaxed definitions such as load threshold $G^*_\alpha$ under hard

constraints of on the PLR $\le \alpha$, as in [20, Eq. (4)], or [9]. In this work, we advocate for a straightforward optimization of the objective function:

**Problem 1:** Given $L$, the maximum number of iterative decoding iterations, the probabilities of SIC success $(\omega_\ell)_{\ell=0,...,L}$, and a maximum degree distribution $D$. Find a degree distribution $\Lambda = (\Lambda_2, ..., \Lambda_D)$,

$$\underset{(\Lambda_i)}{\text{maximize}} \quad \mathcal{T}(G, \Lambda, \omega) \quad (15)$$

$$\text{subject to} \quad \sum_{i=0}^{i=D} \Lambda_i = 1 \quad \text{and} \quad 0 \le \Lambda_i \le 1 \quad \forall i$$

As commonly done in the IRSA literature, numerical heuristic methods such as differential evolution [21] can be used to find numerically optimal distributions (observe however that the function $p_\infty$ hence $\mathcal{T}$ can be discontinuous [19, Fig. 5]).

## V. NUMERICAL RESULTS

In this section, we present the results of our simulations, aimed at evaluating the performance of the IRSA protocol in the context of 5G Cellular IoT network. We start by outlining the key assumptions and setup used for our simulations, followed by a discussion of the obtained results.

### A. Simulation Setup

The simulations are carried out under the system model described in Section III. Specifically, we focus on 5G CIoT scenario where a large number of IoT nodes, each generating small traffic, attempt to transmit data simultaneously to a BS on the uplink. This scenario captures the essence of massive Machine-Type Communication (mMTC) as envisioned in 5G. To model physical layer functions, we adopt an OFDM-based system similar to that used in the 5G New Radio (5G NR) standard. The IRSA frame structure is chosen to match the 5G NR frame configuration, consisting of 10 subframes [22]. A subcarrier spacing of 15 kHz is used, resulting in having one slot per subframe. Each slot, is considered as a time slot in the IRSA protocol and consist of 14 OFDM symbols. Each IoT node (UE) transmits a packet that fits within one resource grid of (subcarriers, OFDM symbols) of size $12 \times 14$. Two of these OFDM symbols are allocated to carry Demodulation Reference Signals (DMRS), which are used for channel estimation. The rest of the resource elements are used for transmitting the data. We assume that the base station and all UEs are equipped with a single antenna. For both data and pilot transmissions, UEs employ QPSK modulation, which provides a robust balance between spectral efficiency and resilience against noise and interference. The channel estimation is performed using the conventional Least Squares (LS) estimator, while the equalization of the received signal is achieved through a Linear Minimum Mean Square Error (LMMSE) equalizer. Furthermore, channel decoding is performed using LDPC codes, which are also part of the 5G NR standard. To simulate the physical layer functions, we used the open-source 5G NR physical layer library sionna [23]. This tool allows us to simulate the entire signal processing

<footnote n="2">²as proven in [19, Th. 3] by following LDPC literature, i.e. [18, Sect. 3.10] and specifically [18, Lemma 3.55] "Monotonicity with Respect to Iteration"</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Packet Error Rate (PLR) as a function of the number of UEs in the slot for different SNR values.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Probability of successful decoding $\omega_{\ell}$ as a function of the number of interfering UEs $\ell$ for different code rates.</figcaption></figure>

chain, including the channel estimation, equalization, and decoding, in a realistic manner.

## B. Analysis of the Inter-Slot SIC

The first part of our simulations focuses on evaluating the efficiency of the inter-slot SIC mechanism in the presence of imperfect channel estimation. The goal is to see how well we can recover the transmitted packets after canceling the interference from the other UEs. We study the impact of having imperfect SIC under different channel conditions by varying the Signal-to-Noise Ratio (SNR). The performance is evaluated in terms of the Packet Loss Rate (PLR), which is defined as the ratio of the number of packets that are not correctly decoded to the total number of transmitted packets. PLR is calculated by averaging over multiple Monte Carlo simulations, where each simulation consists of large number of IRSA slots. We use the PLR as a performance metric since it provides a direct measure of the reliability of the communication link. Fig. 2 shows the PLR as a function of the number of UEs in the slot for different SNR values. The results indicate that PLR increases as the number of UEs in a slot grows, which is an expected outcome due to the cumulative interference from these additional UEs. Notably, when the number of interfering UEs within the slot exceeds a specific threshold, the PLR experiences a significant rise, regardless of the channel quality (i.e., the SNR level). This is because the residual interference becomes the dominant source of errors during the decoding process. In other words, when $\ell$ becomes large, we observe that $\|\mathbf{i}_k\| \gg \|\mathbf{n}_k\|$ in Equation (4).

In the following, to effectively assess the efficiency of the SIC mechanism, we assume a relatively good channel quality is (i.e., when PLR is small when $\ell = 0$). Once PLR is obtained, the probabilities of successful decoding, denoted as $\omega_{\ell}$ and introduced in Section III, can be calculated using $\omega_{\ell} = 1 - \text{PLR}$. Fig. 3 illustrates $\omega_{\ell}$ as a function of the number of UEs $\ell$ for various code rates. As anticipated, the probability

of successful decoding declines as the number of UEs per slot increases. The values of $\omega_{\ell}$ are influenced by specific aspects of the physical layer implementation, such as channel estimation and equalization techniques, modulation schemes, and overall channel conditions. Consequently, these results are tied to the system model described in Section V-A and may vary under different configurations. This analysis emphasizes the impact of code rate on decoding performance, with higher code rates generally enhancing decoding probability. Contrary to the exponential decay pattern proposed in [9], the probabilities $\omega_{\ell}$ exhibit a more intricate behavior, shaped by the system's physical layer characteristics. For smaller values of $\ell$, $\omega_{\ell}$ remains close to 1 but decreases more rapidly as $\ell$ increases. Overall, these results provide a realistic perspective on the efficiency of the interference cancellation, considering the specific physical layer implementation and channel conditions.

## C. Optimization of the IRSA Throughput

In the second part of our simulations, we use the findings from the previous section to optimize the IRSA protocol's throughput. The methodology is to select a code rate and an SNR level, then to compute the associated $\omega_{\ell}$ (up to some maximum value $L$). Given this input, the optimization problem (15) is numerically solved through differential evolution method with maximum degree $D = 5$, and a distribution is obtained, namely $\text{Opt}_5 = 0.0038x^2 + 0.2396x^3 + 0.0009x^4 + 0.7557x^5$. In Fig. 4, the compare the throughput $T(G, \Lambda)$ for three distributions, with perfect SIC and imperfect SIC. The distributions are $L_3(x) = 0.5x^2 + 0.28x^3 + 0.22x^8$, from [13, named $\Lambda_3(x)$], CRDSA with 3 repetitions [13], and $\text{Opt}_5$.

We observe that first the performance of $\text{Opt}_5$ is the indeed the better (even if it has only 5 coefficients); but also that its performance with imperfect SIC is close to its performance with perfect SIC (within 10%, showing a good adaptation to imperfect SIC). This is in contrast with $L_3$ with a dramatic
<figure><img src="image_5.png" /><figcaption>Fig. 4. The asymptotic throughput of IRSA with and without errors as a function of the number of UEs in the slot for different distributions Λ.</figcaption></figure>

drop of 40% of maximum throughput<fnref n="3" />. Of all the distributions, we tried, CRDSA-3 was found to be perhaps surprisingly robust and good compared to most others, showing that proper accounting of errors in necessary when performing distribution optimization.

## VI. CONCLUSION

This paper has presented a detailed evaluation of the Irregular Repetition Slotted ALOHA (IRSA) protocol under realistic physical layer conditions, particularly focusing on the impact of imperfect Successive Interference Cancellation (SIC) caused by channel estimation errors. By incorporating practical physical layer impairments into our density evolution analysis, we highlighted how these factors can alter system performance across varying conditions.

Our findings reveal that optimizing IRSA parameters while accounting for SIC errors leads to significant throughput improvements. The optimized degree distribution we propose maintains performance levels close to the ideal case, even in the presence of imperfect SIC. Additionally, our analysis based on 5G CIoT demonstrates that the proposed distribution is robust, offering better performance compared to traditional schemes like CRDSA under realistic settings. These results underscore the necessity of considering physical layer imperfections to enhance the efficiency of uncoordinated access mechanisms in dense 5G CIoT environments.

## ACKNOWLEDGEMENTS

This work was supported 1) by the 5G-mMTC project, funded by the French government as part of the "Plan de Relance et du Programme d'investissements d'avenir", 2) by the ANR under the France 2030 program, grant "NF-PERSEUS: ANR-22-PEFT-0004". The authors also thank S. Sharma and K. Deka for insightful discussions in the framework of the Inria MAGICO Team.

## REFERENCES

[1] M. Vaezi, A. Azari, S. R. Khosravirad, M. Shirvanimoghaddam, M. M. Azari, D. Chasaki, and P. Popovski, "Cellular, Wide-Area, and Non-Terrestrial IoT: A Survey on 5G Advances and the Road Toward 6G," *IEEE Communications Surveys & Tutorials*, vol. 24, no. 2, pp. 1117-1174, 2022.

[2] M. B. Shahab, R. Abbas, M. Shirvanimoghaddam, and S. J. Johnson, "Grant-Free Non-Orthogonal Multiple Access for IoT: A Survey," *IEEE Communications Surveys & Tutorials*, vol. 22, no. 3, pp. 1805-1838, 2020.

[3] F. Clazzer, A. Munari, G. Liva, F. Lazaro, C. Stefanovic, and P. Popovski, "From 5G to 6G: Has the Time for Modern Random Access Come?" Mar. 2019.

[4] G. Liva, "Graph-Based Analysis and Optimization of Contention Resolution Diversity Slotted ALOHA," *IEEE Transactions on Communications*, vol. 59, no. 2, pp. 477-487, Feb. 2011.

[5] E. Paolini, G. Liva, and M. Chiani, "Coded Slotted ALOHA: A Graph-Based Method for Uncoordinated Multiple Access," *IEEE Transactions on Information Theory*, vol. 61, no. 12, pp. 6815-6832, Dec. 2015.

[6] F. Clazzer, E. Paolini, I. Mambelli, and Č. Stefanović, "Irregular repetition slotted ALOHA over the Rayleigh block fading channel with capture," in *2017 IEEE International Conference on Communications (ICC)*, May 2017, pp. 1-6.

[7] Č. Stefanović, E. Paolini, and G. Liva, "Asymptotic Performance of Coded Slotted ALOHA With Multipacket Reception," *IEEE Communications Letters*, vol. 22, no. 1, pp. 105-108, Jan. 2018.

[8] I. Hmedoush, P. Gu, C. Adjih, P. Mühlethaler, and A. Serhrouchni, "DS-IRSA: A Deep Reinforcement Learning and Sensing Based IRSA," in *GLOBECOM 2023 - 2023 IEEE Global Communications Conference*, Dec. 2023, pp. 2790-2795.

[9] C. Dumas, L. Salaün, I. Hmedoush, C. Adjih, and C. S. Chen, "Design of Coded Slotted ALOHA With Interference Cancellation Errors," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 12, pp. 12742-12757, Dec. 2021.

[10] A. Mengali, R. De Gaudenzi, and Č. Stefanović, "On the Modeling and Performance Assessment of Random Access With SIC," *IEEE Journal on Selected Areas in Communications*, vol. 36, no. 2, pp. 292-303, Feb. 2018.

[11] C. R. Srivatsa and C. R. Murthy, "On the Impact of Channel Estimation on the Design and Analysis of IRSA Based Systems," *IEEE Transactions on Signal Processing*, vol. 70, pp. 4186-4200, 2022.

[12] L. G. Roberts, "ALOHA packet system with and without slots and capture," *SIGCOMM Comput. Commun. Rev.*, vol. 5, no. 2, pp. 28-42, Apr. 1975.

[13] E. Casini, R. De Gaudenzi, and O. Del Rio Herrero, "Contention Resolution Diversity Slotted ALOHA (CRDSA): An Enhanced Random Access Scheme for Satellite Access Packet Networks," *IEEE Transactions on Wireless Communications*, vol. 6, no. 4, pp. 1408-1419, Apr. 2007.

[14] Y. Takahashi, G. Song, T. Kimura, and J. Cheng, "Irregular Repetition Slotted ALOHA Over Rayleigh Block Fading Channels: Bounds and Threshold Saturation via Spatial Coupling," *IEEE Access*, vol. 11, pp. 106528-106543, 2023.

[15] E. E. Khaleghi, C. Adjih, A. Alloum, and P. Mühlethaler, "Near-far effect on coded slotted ALOHA," in *2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC)*, Oct. 2017, pp. 1-7.

[16] C. R. Srivatsa and C. R. Murthy, "Throughput Analysis of PDMA/IRSA under Practical Channel Estimation," in *2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Jul. 2019, pp. 1-5.

[17] K. Zidane, J. Lacan, M.-L. Boucheret, C. Poulliat, M. Gineste, D. Roques, C. Bes, and A. Deramecourt, "Effect of Residual Channel Estimation Errors in Random Access Methods for Satellite Communications," in *2015 IEEE 81st Vehicular Technology Conference (VTC Spring)*, May 2015, pp. 1-5.

[18] T. Richardson and R. Urbanke, *Modern Coding Theory*. Cambridge: Cambridge University Press, 2008.

[19] I. Hmedoush, C. Adjih, K. Khawam, and P. Mühlethaler, "A Game Theoretic Approach to Irregular Repetition Slotted Aloha," *IEEE Access*, vol. 10, pp. 4600-4614, 2022.

[20] Z. Sun, Y. Xie, J. Yuan, and T. Yang, "Coded Slotted ALOHA for Erasure Channels: Design and Throughput Analysis," *IEEE Transactions on Communications*, vol. 65, no. 11, pp. 4817-4830, Nov. 2017.

<footnote n="3">³Like many others. In general, the closer the throughput is to 1, the higher the rate of the distribution must be, and the larger the number of SICs must be done, and the higher the probability of SIC errors.</footnote>
<footnote>[21] R. Storn and K. Price, "Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces," *Journal of Global Optimization*, vol. 11, no. 4, pp. 341–359, Dec. 1997.</footnote>

<footnote>[22] “Release 15 Description; Summary of Rel-15 Work Items,” 3rd Generation Partnership Project (3GPP), Technical Report (TR) TR 21.915, Oct. 2019.</footnote>

<footnote>[23] J. Hoydis, S. Cammerer, F. A. Aoudia, A. Vem, N. Binder, G. Marcus, and A. Keller, “Sionna: An Open-Source Library for Next-Generation Physical Layer Research,” Mar. 2023.</footnote>