[latexfont] [todonotes] biblatexPatching footnotes failed AIFSshort=AIFS,
long=Arbitration Inter-Frame Spacing, foreign-plural=, AoIshort=AoI, long=Age
of Information, foreign-plural=, AWGNshort=AWGN, long=Additive White Gaussian
Noise, foreign-plural=, BSshort=BS, long=Base Station, foreign-plural=,
BSSshort=BSS, long=Basic Service Set, foreign-plural=, CAMshort=CAM,
long=Cooperative Awareness packet, foreign-plural=, CPMshort=CPM,
long=Collective Perception packet, foreign-plural=, CBFshort=CBF,
long=Contention-Based Forwarding, foreign-plural=, CBRshort=CBR, long=Channel
Busy Ratio, foreign-plural=, CDFshort=CDF, long=Cumulative Distribution
Function, foreign-plural=, CCDFshort=CCDF, long=Complementary Cumulative
Distribution Function, foreign-plural=, CDMAshort=CDMA, long=Code Division
Multiple Access, foreign-plural=, CCHshort=CCH, long=Control Channel, foreign-
plural=, CSMAshort=CSMA, long=Carrier-Sense Multiple Access, foreign-plural=,
C-ITSshort=C-ITS, long=Cooperative Intelligent Transportation System, foreign-
plural=, short-plural-form=C-ITS DENMshort=DENM, long=Decentralized
Environmental Notification packet, foreign-plural=, DIFSshort=DIFS, long=DCF
Inter-Frame Space, foreign-plural=, DSRCshort=DSRC, long=Dedicated Short-Range
Communication, foreign-plural=, DTMCshort=DTMC, long=Discrete Time Markov
Chain, foreign-plural=, FCFSshort=FCFS, long=First Come First Served, foreign-
plural=, GFMAshort=GFMA, long=Grant Free Multiple Access, foreign-plural=,
i.i.d.short=i.i.d., long=independent and identically distributed, foreign-
plural=, IoTshort=IoT, long=Internet of Things, foreign-plural=, ITSshort=ITS,
long=Intelligent Transportation System, foreign-plural=, long-plural-form=
IVCshort=IVC, long=Inter Vehicle Communication, foreign-plural=,
LCFSshort=LCFS, long=Last Come First Served, foreign-plural=,
LCFSwOshort=LCFSwO, long=Last Come First Served with Overwrite, foreign-
plural=, LDMshort=LDM, long=Local Dynamic Map, foreign-plural=, MPRshort=MPR,
long=Multi-Packet Reception, foreign-plural=, mMTCshort=mMTC, long=Massive
machine-type Communication, foreign-plural=, M2Mshort=M2M, long=Machine-to-
Machine, foreign-plural=, MACshort=MAC, long=Medium Access Control, foreign-
plural=, MCMshort=MCM, long=Maneuver Coordination packet, foreign-plural=,
MUSAshort=MUSA, long=Multi User Shared Access, foreign-plural=,
NOMAshort=NOMA, long=Non-Orthogonal Multiple Access, foreign-plural=,
NoBshort=NoB, long=No Buffering, foreign-plural=, OBUshort=OBU, long=On-Board
Unit, foreign-plural=, OMAshort=OMA, long=Orthogonal Multiple Access, foreign-
plural=, PDFshort=PDF, long=Probability Density Function, foreign-plural=,
RSUshort=RSU, long=Road Side Unit, foreign-plural=, SCHshort=SCH, long=Service
Channel, foreign-plural=, SCOVshort=SCOV, long=Squared Coefficient Of
Variation, foreign-plural=, SAshort=SA, long=Slotted ALOHA, foreign-plural=,
SICshort=SIC, long=Successive Interference Cancellation, foreign-plural=,
SNRshort=SNR, long=Signal to Noise Ratio, foreign-plural=, SNIRshort=SNIR,
long=Signal to Noise plus Interference Ratio, foreign-plural=, PDMAshort=PDMA,
long=Pattern Division Multiple Access, foreign-plural=, V2Ishort=V2I,
long=Vehicle-to-Infrastructure, foreign-plural=, V2Vshort=V2V, long=Vehicle-
to-Vehicle, foreign-plural=, V2Xshort=V2X, long=Vehicle-to-Everything,
foreign-plural=, VANETshort=VANET, long=Vehicular Ad Hoc Network, foreign-
plural=, VLCshort=VLC, long=Visible Light Communication, foreign-plural=,
UAVshort=UAV, long=Unmanned Aerial Vehicle, foreign-plural=,

# Asymptotic analysis of sum-rate under SIC
††thanks: This work was partially supported by the European Union under the
Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU,
partnership on Ã¢ÂÂTelecommunications of the FutureÃ¢ÂÂ (PE0000001 -
program Ã¢ÂÂRESTARTÃ¢ÂÂ).

Andrea Baiocchi1, Asmad
<EMAIL_ADDRESS>1Dept. of Information
Engineering, Electronics and Telecommunications (DIET), University of Rome
Sapienza, Italy

###### Abstract

Limitation of the cost of coordination and contention among a large number of
nodes calls for grant-free approaches, exploiting physical layer techniques to
solve collisions. SIC is becoming a key building block of multiple access
channel receiver, in an effort to support massive IoT. In this paper, we
explore the large-scale performance of SIC in a theoretical framework. A
general model of a SIC receiver is stated for a shared channel with $n$
transmitters. The asymptotic sum-rate performance is characterized as
$n\rightarrow\infty$, for a suitably scaled target SNIR. The probability
distribution of the number of correctly decoded packets is shown to tend to a
deterministic distribution asymptotically for large values of $n$. The
asymptotic analysis is carried out for any probability distribution of the
wireless channel gain, assuming that the average received power level is same
for all nodes, through power control.

###### Index Terms:

Multiple access; SIC; large network asymptotic regime;

##  I Introduction

IoT is transforming and bringing a deep impact in every aspect of human life
such as healthcare, education, Industry, agrifood chain, power grid,
autonomous driving, logistic services, and smart cities [1, 2]. The evolution
of IoT toward these applications opens up a paradigm of massive access that
aims at realizing efficient and reliable communication for a massive number of
IoT devices [3, 4]. The main characteristics of massive access include low
power, massive connectivity, short packets, and low signaling overhead, which
call for more robust and flexible wireless communication networks [5, 6].
Also, these requirements envision the 6G ecosystem as the playground of a
data-driven society, which involves instant and unlimited connectivity to
anything, ranging from tiny static sensors to autonomous objects, from
anywhere, and anytime. This paradigm calls for the design of innovative
multiple access techniques to provide massive connectivity, and low latency to
future massive IoT systems.

A generic scenario of massive multiple access in IoT involves a huge number of
connected devices that give rise typically to sporadic and random traffic.
Hence, the actual number of backlogged devices, contending for access to the
medium, is largely variable. A fundamental issue toward such a massive IoT
scenario involves reducing the cost of coordination of a varying and possibly
large number of devices in an efficient and timely manner. One promising
solution toward this paradigm is the grant-free random multiple access
approach, in which each active device directly transmits its data to the BS
without waiting for any permission from the access point. Traditional grant-
based access requires multiple signaling exchanges between the device and the
network to allocate resources, which can be inefficient for small and
infrequent data transmissions, especially in mMTC. Grant-free access minimizes
this overhead [7].

The key features of grant-free random multiple access include no prior
scheduling which reduces the latency and improves the system response,
scalability to support a huge number of connections, energy efficiency,
eliminating the need for devices to wait for a grant, and collision
management.

As for the last point, since multiple devices may attempt to transmit
simultaneously, leading to collisions, grant-free random access protocols
often incorporate mechanisms to handle these collisions, such as coding,
advanced signal processing techniques, or NOMA [8]. NOMA enables multiple
users to share the same time-frequency resources non-orthogonally [9], thus
enhancing spectral efficiency and addressing the massive connectivity demands
anticipated in future wireless networks [10, 11].

Several NOMA schemes have been proposed over the past few years, which can be
divided into two main categories code-domain NOMA and power-domain NOMA [12].
A few variants of code-domain NOMA includes PDMA [13], and MUSA [14]. In this
paper, we mainly focus on the power-domain NOMA, as recently implemented in
3GPP LTE [15]. It involves the superposition of signals from different users,
then exploiting SIC to decode them [9].

Recent researches highlight a significant interest in leveraging advanced
signal processing and machine learning techniques to address the challenges of
GFMA in modern wireless networks [16, 17]. For instance, [18] proposed a novel
GFMA scheme based on deep learning techniques, achieving superior user
detection accuracy and connectivity. A reinforcement learning-based approach
is introduced in [19] to address the challenge of massive connectivity. An
optimal power from a pool of powers is chosen, with the help multi-agent
framework, to optimize spectral efficiency and user fairness while mitigating
interference. A stochastic optimization-based adaptive grant-free scheme is
proposed in [20], where the transmission probability and the target SNIR is
adjusted based on the number of backlogged nodes. It is quite evident that all
these performance enhancements towards adaptive grant-free protocols come from
advances carried out at the physical layer, i.e. SIC [21, 22]. Hence, it
becomes a crucial aspect to understand the behavior of MPR-capable receivers,
like those based on SIC, under a massive number of users.

Several works have been done toward the design, dimensioning, and optimization
of scalable MPR systems, with respect to the number of transmitting nodes in
the range of thousands. The authors in [23] proposed a theoretical framework
to investigate the effect of several parameters, such as the capture threshold
and the multi-packet reception capability, with the help of SIC, with a
limited number of iterations involved in SIC decoding process. Enabling the
SIC mechanism brings significant performance gains in terms of throughput,
especially for low capture thresholds, however, the normalized throughput
increment rapidly diminishes with each iteration of SIC [23]. SIC is utilized
under the framework of ALOHA in [24], with varying numbers of users driven by
the duration of previous contention periods, to maximize the throughput along
with optimization of the age of information. In heavily loaded systems, the
optimum SNIR profile decreases in the SIC decoding order [25], contrasting
with uniform SNIR profiles found in previous studies [26] optimizing
asymptotic capacity or under different energy constraints.

There still exists a significant gap in understanding the performance of SIC
in large-scale systems. The role of SIC toward massive access needs great
attention, especially in view of the standardization of NOMA in 3GPP network
[15].

In this paper, we characterize the asymptotic behavior of sum-rate in a SIC-
based, grant-free multiple access for $n$ nodes, as $n\rightarrow\infty$ The
key contributions of the paper can be summarized as follows:

  * •

We characterize the asymptotic behavior of the probability distribution of the
number of correctly decoded packets utilizing SIC in large-scale systems.

  * •

We highlight the scaling of target SNIR that guarantees the asymptotically
maximum achievable sum-rate as the number of nodes grows. This analysis is
valid for any probability distribution of the channel random gain, given that
the average received power is the same for all nodes.

The paper is organized as follows. In Section II, we introduce the system
model and provide basic definitions. The key results are stated and discussed
in Section III, with reference to the special case of Rayleigh channel fading.
The model is generalized in Section IV to any possible distribution of channel
gain. Conclusions are drawn in Section V. Proof and details of mathematical
developments are given in the Appendices.

##  II System Model

We consider $n$ nodes sharing a common communication channel and sending
packets to a sink, referred to as the BS. The focus is on uplink, i.e.,
packets are sent by nodes and addressed to the BS.

The communication link is modeled as an AWGN channel with block Rayleigh
fading. Let $P_{j}$ denote the _average received_ power level at the BS from
node $j$. Let also $\mathbf{S}_{j}=[S_{j}(1),\dots,S_{j}(\ell)]$ the symbol
string transmitted by node $j$ and $h_{j}$ the channel gain coefficient of
node $j$. The coefficient $h_{j}$ accounts for Rayleigh fading, hence
$|h_{j}|^{2}$ is a negative exponential random variable. It is normalized so
that $\mathrm{E}[|h_{j}|^{2}]=1$. The received symbol string
$\mathbf{R}=[R(1),\dots,R(\ell)]$ at the BS is modeled as

$\mathbf{R}=\sum_{j=1}^{n}{\sqrt{P_{j}}h_{j}\mathbf{S}_{j}}+\mathbf{Z}$ (1)

where $\mathbf{Z}=[Z(1),\dots,Z(\ell)]$ is the additive Gaussian noise, and
the $Z(i)$’s are i.i.d. Gaussian random variables, i.e.,
$Z(i)\sim\mathcal{CN}(0,P_{N})$ for $i=1,\dots,\ell$. ‘

Assimilating interference to additive Gaussian noise, the SNIR of node $j$ is
given by

$\Gamma_{j}=\frac{P_{j}|h_{j}|^{2}}{P_{N}+I_{j}}$ (2)

where $I_{j}$ denotes the interference power level on the reception of packet
of node $j$. Node $j$ packet is correctly decoded if its average SNIR at the
BS is no less than a suitable threshold $\gamma$, i.e.,
$\Gamma_{j}\geq\gamma$. The target SNIR $\gamma$ is tied to the spectral
efficiency $\eta$ (information bit per symbol) of the communication channel
according to $\eta=\log_{2}(1+\gamma)$, consistently with the AWGN channel
assumption.

The time axis is divided in slots. Nodes make transmission attempts in each
slot. The slot size just fits a packet transmission time. Let $L$ be the
length of transmitted packets for all nodes and $W$ the channel bandwidth. The
time required to transmit a packet for a given target SNIR $\gamma$ is

$T=T(\gamma)=\frac{L}{W\log_{2}(1+\gamma)}$ (3)

In the following, we drop the subscript $j$ referring to the considered node
to discuss transmission power setting, since the same procedure applies to
each node. We assume that node $j$ estimates the long-term average path gain
of its channel to the BS, e.g., thanks to pilot tomes transmitted periodically
by the BS. Then, node $j$ adjusts its transmitted power level so that the
average received power level at the BS matches a prescribed value $P_{0}$,
same for all nodes. $P_{0}$ is set so that the probability of failing to
decode a packet sent by a single transmitting node is no more than $\epsilon$.
Decoding of a packet sent by a single transmitting node is successful, if the
SNR exceeds $\gamma$, i.e., $P_{\text{rx}}/P_{N}\geq\gamma$, where $P_{N}$ is
the background noise power level. Since the average received power level is
$P_{0}$, the requirements translates to

$\frac{G_{f}P_{0}}{P_{N}}\geq\gamma\qquad\text{w.p. }1-\epsilon$ (4)

Given that $G_{f}$ has a negative exponential PDF, $P_{0}$ is to be set so
that $\mathcal{P}(G_{f}P_{0}/P_{N}\geq\gamma)=e^{-\gamma
P_{N}/P_{0}}=1-\epsilon$. Hence, the target SNR level $S_{0}$ at the receiving
BS is set as follows:

$S_{0}=\frac{P_{0}}{P_{N}}=\frac{\gamma}{-\log(1-\epsilon)}=\frac{\gamma}{c}$
(5)

where we have introduced the constant $c=-\log(1-\epsilon)$.

We assume an ideal SIC receiver. Let $n$ packets be received simultaneously at
the BS in one slot. Channel gain coefficient of node $j$ is given by
$|h_{j}|^{2}$ and it is a negative exponential random variable with mean 1. We
denote the (descending) order statistics of the sequence
$|h_{1}|,\dots,|h_{n}|$ with $|\tilde{h}_{1}|,\dots,|\tilde{h}_{n}|$. By
definition, we have
$|\tilde{h}_{1}|\geq|\tilde{h}_{2}|\dots\geq|\tilde{h}_{n}|$ (ties are broken
at random). SIC works as follows. Provided decoding of packets $1,\dots,k-1$
be successful, packet $k$ is decoded successfully if and only if the following
inequality holds111In the following, for ease of notation, it is mean that
$\sum_{i=a}^{b}{}=0$ and $\prod_{i=a}^{b}{}=1$, if $a>b$.:

$\Gamma_{j}=\frac{P_{0}|\tilde{h}_{k}|^{2}}{P_{N}+\sum_{r=k+1}^{n}{P_{0}|\tilde{h}_{r}|^{2}}+\xi\sum_{r=1}^{k-1}{P_{0}|\tilde{h}_{r}|^{2}}}\geq\gamma$
(6)

where $\xi$ accounts for the fraction of residual interference due to
imperfect cancellation. Perfect SIC is obtained by setting $\xi=0$. On the
contrary, $\xi=1$ corresponds to a receiver not endowed with SIC capability,
where multi-packet reception is possible only thanks to capture effect222In
this special case, ordering is not relevant and decoding of individual packets
is done independently of one another.. Using Equation 5, Equation 6 can be re-
written as follows:

$\Gamma_{j}=\frac{|\tilde{h}_{k}|^{2}}{c/\gamma+\sum_{r=k+1}^{n}{|\tilde{h}_{r}|^{2}}+\xi\sum_{r=1}^{k-1}{|\tilde{h}_{r}|^{2}}}\geq\gamma$
(7)

To simplify notation, we let $Y_{j}=|h_{j}|^{2}$ and
$Y_{(j)}=|\tilde{h}_{j}|^{2}$ for $j=1,\dots,n$ in the following. The
$Y_{j}$’s are i.i.d. negative exponential random variables with mean 1, while
the $Y_{(j)}$’s form the associate descending order statistic. Successful
decoding of packet carried by the $m$-th strongest signal with SIC requires
that $\Gamma_{j}\geq\gamma$ for all $j=1,\dots,m$. Using the notation
introduced so far, the condition is stated as

$\Gamma_{j}=\frac{Y_{(j)}}{c/\gamma+\sum_{r=j+1}^{n}{Y_{(r)}}+\xi\sum_{r=1}^{j-1}{Y_{(r)}}}\geq\gamma$
(8)

for all $j=1,\dots,m$. In Equation 8 the residual interference coefficient
$\xi$ is assumed to satisfy $0\leq\xi<1$.

If SIC is not used at the receiver, successful decoding is still possible.
Each packet is decoded independently of all others, so that ordering does not
matter. The packet carried by the $j$-th signal is successfully decoded if

$\frac{Y_{j}}{c/\gamma+\sum_{r=1,r\neq j}^{n}{Y_{r}}}\geq\gamma$ (9)

We introduce the notation $S_{j}=S_{0}Y_{j}$, $j=1,\dots,n$, to denote the SNR
of node $j$, where $S_{)}$ is the average SNR level achieved by means of
transmission power control (see Equation 5), while $Y_{j}$ accounts for the
residual randomness of the communication channel.

In next Section we analyze the asymptotic regime for $n\rightarrow\infty$ of
SIC reception, under the assumption of Rayleigh fading, i.e., that $Y_{j}$ be
i.i.d. negative esponential random variable with mean 1. In Section IV this
assumption is relaxed, by allowing the $Y_{j}$’s to have any probability
distribution.

##  III Model Analysis

The first step is to transform the inequalities in Equation 8 into a simpler
form. This is the purpose of next theorem.

######  Theorem 1.

Let $V_{j},\,j=1,\dots,n$, be a non-negative random variables defined by

$V_{j}=\sum_{k=j}^{n}{b_{jk}X_{k}}$ (10)

where $X_{k},\,k=1,\dots,n$, are i.i.d. negative exponential random variables
with mean 1 and

$b_{jk}=\begin{cases}-\gamma\xi&k=1,\dots,j-1,\\\
\frac{1+j\gamma-(j-1)\gamma\xi}{k}-\gamma&k=j,\dots,n.\end{cases}$ (11)

for $j=1,\dots,n$. The packet carried by the $m$-th strongest signal is
decoded successfully if and only if $V_{j}\geq c$ for $j=1,\dots,m$.

###### Proof.

See Appendix A.∎

The random variables $V_{j},\,j=1,\dots,n$ are not independent of one another.
The joint condition for successful decoding stated in Theorem 1 is not easy to
evaluate.

In case $\xi=0$, the probability $p_{V}(j)=\mathcal{P}(V_{j}\geq c)$, can be
evaluated starting from the Laplace transform of the PDF of $V_{j}$

$\varphi_{V_{j}}(s)=\mathrm{E}[e^{-sV_{j}}]=\frac{1}{\prod_{k=j}^{n}{(1+b_{jk}s)}}$
(12)

This transform can be inverted in closed form, but the numerical evaluation of
the obtained formulas is highly unstable, due to sums of alternating sign
terms. Numerical error propagation forbids a direct evaluation of the
probabilities $p_{V}(j)$. Numerical experience suggests that it is much more
reliable and efficient to use numerical inversion of the Laplace transform
directly, e.g., exploiting the Fourier algorithm in [27].

Figure 1 illustrates $p_{V}(j)=\mathcal{P}(V_{j}\geq c)$ as a function of $j$
for four values of the number of nodes $n$, with $\xi=0$ and $\gamma=1/(\alpha
n)$ for a given constant $\alpha>0$ ($\alpha=0.32$ in the left plot,
$\alpha=0.38$ in the right one). The realizations of the random variable $V$
on the $x$ axis are normalized as $j/n$. The number of nodes is stretched to
possibly unrealistically large values for two reasons. A first reason is to
highlight the asymptotic behavior of the probabilities $p_{V}(j)$ as $n$ gets
large. The second reason derives from a system point of view: massive IoT
calls for the possibility of even very large set of concurrently transmitting
nodes. Even if this may not be feasible with today’s technology, it is
interesting to understand what happens in such a regime.

(a) $\alpha=0.32$

(b) $\alpha=0.38$

Figure 1: Probability that the random variable $V_{j}$ exceeds the threshold
$c$ for $j=1,\dots,n$. The abscissa is normalized as $j/n$. The required SNIR
threshold is set to $\gamma=1/(\alpha n)$, with $\alpha=0.32$ in the plot on
the left, $\alpha=0.38$ in the plot on the right.

The probabilities $p_{V}(j)$ form a decreasing sequence with $j$, which
matches intuition, given that higher values of $j$ are related to a higher
number of successfully decoded packets. Hence, $p_{V}(j)\approx 1$ for lower
values of $j$, while $p_{V}(j)$ becomes negligible for $j$ approaching $n$. In
fact, for $j=n$ it is easy to verify that $p_{V}(n)=e^{-nc}=(1-\epsilon)^{n}$.
The transition from 1 to 0 as $j$ is increased is apparent in Figure 1. The
striking result is that the transition gets steeper as $n$ grows, i.e., SIC
decoding in our model moves towards a phase transition phenomenon as the
number of concurrent transmissions gets large. The convergence is rather slow,
since very large values of $n$ have to be considered to appreciate a sharp
transition. The transition corresponds to a decoding threshold, highlighted as
$\zeta$ in the plots. Up to a fraction $\zeta$ of the $n$ transmissions are
decoded successfully as $n\rightarrow\infty$. In terms of random variables,
Figure 1 tells us that the random variables $V_{j}$ tend to become
deterministic quantities. This is formally proved in the following theorem.

######  Theorem 2.

Let $\mu_{j}(n)$ and $\sigma^{2}_{j}(n)$ denote the mean and the variance of
$V_{j}$ respectively, for $n$ concurrent transmissions and for $j=1,\dots,n$.
Let $\gamma=1/(\alpha n)$ for a positive constant $\alpha$. Let also

$f_{\alpha,\xi}(x)=-\left(1+\frac{(1-\xi)x}{\alpha}\right)\log
x-\frac{1-(1-\xi)x}{\alpha}$ (13)

for $x\in(0,1]$. Then

$\displaystyle\lim_{n\rightarrow\infty}{\frac{1}{n}\sum_{j=1}^{n}{\left|\mu_{j}(n)-f_{\alpha,\xi}\left(\frac{j}{n}\right)\right|}}=0$
(14)
$\displaystyle\lim_{n\rightarrow\infty}{\frac{\sigma_{j}(n)}{\mu_{j}(n)}}=0$
(15)

Hence, for any $\epsilon>0$, we have

$\lim_{n\rightarrow\infty}{\mathcal{P}\left(\left|\frac{V_{j}}{\mu_{j}(n)}-1\right|>\epsilon\right)}=0$
(16)

###### Proof.

See Appendix B.∎

Note that the function $f_{\alpha,\xi}(x)$ can be written as

$f_{\alpha,\xi}(x)=-\frac{1}{\alpha}+g_{\frac{\alpha}{1-\xi}}(x)$ (17)

where

$g_{\beta}(x)=-\left(1+\frac{x}{\beta}\right)\log x+\frac{x}{\beta}$ (18)

This representation of $f_{\alpha,\xi}(x$ shows that, apart from a constant,
it depends only on the parameter $\beta=\alpha/(1-\xi)$.

Theorem 2 says that the mean values of the random variables $V_{j}$ tend to
lie on the curve $f_{\alpha,\xi}(x)$, with $x=j/n$, as $n$ grows, while the
coefficient of variation (the ratio of the standard deviation to the mean)
shrinks to 0. Hence, for each $j$ with $1\leq j\leq n$, $V_{j}$ tends to a
deterministic random variable that takes the value $\mu_{j}(n)$ as
$n\rightarrow\infty$. This asymptotic regime sets on because of the crucial
condition on the required SNIR threshold, namely, $\gamma=1/(\alpha n)$.
Intuitively, if the required SNIR scales with the number $n$ of concurrent
transmissions, then the mean of $V_{j}$ tend to a positive value. The
implications on the number of correctly decodable packets are addressed by
next theorem. Before stating it, numerical evidence of the convergence of the
random variables $V_{j}$ to a deterministic quantity is given in curve plots,
from which we appreciate _how_ the asymptotes are achieved.

The sequence of means $\mu_{j}(n)$ (discrete stems ending with a circle) is
compared with $f_{\alpha,\xi}(x)$ (thick red line) in Figure 2 for
$\alpha=0.32$, $\xi=0$ and several values of $n$, by placing the mean values
at points $j/n,\,j=1,\dots,n$. It is apparent that mean values match more and
more closely as $n$ grows. It is also evident that convergence is rather slow.
It appears to be very accurate for $n=500$, but it is still not perfect for
$n=50$.

(a) $n=50$

(b) $n=100$

(c) $n=500$

(d) $n=1000$

Figure 2: Sequence of mean values $\mu_{j}(n)$ (discrete stems ending with a
circle) compared with function $f_{\alpha,\xi}(x)$ (thick red line) as a
function of $x=j/n$ for several values of $n$ and $\xi=0$ and $\alpha=0.32$.

Figures 3 and 4 show the sequence of means $\mu_{j}(n)$ as a function of
$x=j/n$ (solid line), as well as the asymptotic curve $f_{\alpha,\xi}(x)$
towards which the mean values approach as $n$ grows. The shaded area
corresponds to one standard deviation from the mean, i.e., it is the area
within the limits $\mu_{j}(n)\pm\sigma_{j}(n)$. Plots are shown for several
values of $n$, for two values of $\alpha$ and for $\xi=0$. Studying the
function $f_{\alpha,\xi}(x)$ for $x\in(0,1]$, it can be verified that its
derivatives has two zeros, $x_{1}$ and $x_{2}$, with $x_{1}<1/e<x_{2}$, if and
only if $\alpha/(1-\xi)<1/e\approx 0.37$. Otherwise, the function
$f_{\alpha,\xi}(x)$ is monotonously decreasing.

More in depth, the derivative of $f^{\prime}_{\alpha,\xi}(x)$ is
$=-1/x-[(1-\xi)/\alpha]\log x$. Hence $x_{1}$ and $x_{2}$ are the roots of

$\frac{\alpha}{1-\xi}+x\log x=0$ (19)

for $x\in(0,1]$. These roots, when they exist, depend only on the parameter
$\beta=\alpha/(1-\xi)$. When existing, the smaller zero $x_{1}$ corresponds to
a local minimum, while the larger $x_{2}$ to a local maximum. Hence, in
Figures 3 and 4 two values of $\alpha$ across the critical value $1/e$ have
been picked. The zero $x_{1}$ is a function of $\beta$, defined implicitly
by333The function $x_{1}(\beta)$ is connected to Lambert’s function, but we
will not pursue this connection further here. $\beta+x_{1}(\beta)\log
x_{1}(\beta)=0$. Deriving both sides, it follows that
$x_{1}^{\prime}(\beta)=-1/(1+\log x_{1}(\beta))$, for $\beta\in(0,1/e)$. Since
$x_{1}(\beta)<1/e$, it must be $x_{1}^{\prime}(\beta)>0$ for
$\beta\in(0,1/e)$, i.e., $x_{1}(\beta)$ is an increasing function of $\beta$,
its maximum being attained for $\beta=1/e$, where it is $x_{1}=x_{2}=1/e$. We
note that the smallest positive root of $f_{\alpha,\xi}(x)=c$ is a continuous
function of $\beta$ if and only if
$f_{\alpha,\xi}(x_{1}(1/e))=f_{\alpha,\xi}(1/e)\leq c$. Since for $\beta=1/e$
we have $f_{\alpha,\xi}(1/e)=3-e/(1-\xi)$, the condition becomes $\xi\geq
1-e/(3-c)$. For $\epsilon=0.1$, it is $c\approx 0.1054$, hence the condition
is $\xi\geq 0.0609$. For smaller values of $\xi$ the smallest positive root of
$f_{\alpha,\xi}(x)=c$ as a function of $\alpha$ has a jump.

The plots in these figures highlight the convergence of the random variables
$V_{j}$ to deterministic quantities, as the shaded area shrinks for increasing
$n$. It is apparent that convergence is not uniform. The range of $x$ around
the local minimum of the curve $f_{\alpha,\xi}(x)$ appears to be the most
critical region for convergence.

(a) $n=50$

(b) $n=100$

(c) $n=500$

(d) $n=1000$

Figure 3: Sequence of mean values $\mu_{j}(n)$ as a function of $x=j/n$ for
several values of $n$ and $\xi=0$. The shaded area corresponds to one standard
deviation, i.e., $\mu_{j}(n)\pm\sigma_{j}(n)$. The dashed line is the function
$f_{\alpha,\xi}(x)$ defined in Equation 13. The required SNIR threshold is set
to $\gamma=1/(\alpha n)$ with $\alpha=0.32$

(a) $n=50$

(b) $n=100$

(c) $n=500$

(d) $n=1000$

Figure 4: Sequence of mean values $\mu_{j}(n)$ as a function of $x=j/n$ for
several values of $n$ and $\xi=0$. The shaded area corresponds to one standard
deviation, i.e., $\mu_{j}(n)\pm\sigma_{j}(n)$. The dashed line is the function
$f_{\alpha,\xi}(x)$ defined in Equation 13. The required SNIR threshold is set
to $\gamma=1/(\alpha n)$ with $\alpha=0.38$

The same data is diaplyed in log scale in Figures 5 and 6. This scale provides
a better appreciation of the non-uniform convergence. Convergence is more
critical for smaller values of $\alpha$, i.e., for $\alpha<1/e$, when local
minimum and maximum exist.

(a) $n=50$

(b) $n=100$

(c) $n=500$

(d) $n=1000$

Figure 5: Sequence of mean values $\mu_{j}(n)$ as a function of $x=j/n$ for
several values of $n$ and $\xi=0$. The shaded area corresponds to one standard
deviation, i.e., $\mu_{j}(n)\pm\sigma_{j}(n)$. The dashed line is the function
$f_{\alpha,\xi}(x)$ defined in Equation 13. The required SNIR threshold is set
to $\gamma=1/(\alpha n)$ with $\alpha=0.32$

(a) $n=50$

(b) $n=100$

(c) $n=500$

(d) $n=1000$

Figure 6: Sequence of mean values $\mu_{j}(n)$ as a function of $x=j/n$ for
several values of $n$ and $\xi=0$. The shaded area corresponds to one standard
deviation, i.e., $\mu_{j}(n)\pm\sigma_{j}(n)$. The dashed line is the function
$f_{\alpha,\xi}(x)$ defined in Equation 13. The required SNIR threshold is set
to $\gamma=1/(\alpha n)$ with $\alpha=0.38$

The main result stated below concerns the asymptotic behavior of the mean
number of correctly decoded packets in a slot as $n\rightarrow\infty$. Let
$m_{n}(\gamma)$ denote the mean number of correctly decoded packets in a slot
with target SNIR equal to $\gamma$, given that $n$ nodes transmit in that
slot. We have the following.

######  Theorem 3.

Let $\gamma_{n}=1/(\alpha n)$ for a positive constant $\alpha$. Let also
$f_{\alpha,\xi}(x)$ denote the function in Equation 13. It is
$m_{n}(\gamma_{n})\sim\zeta n$ where $\zeta$ is defined as444Note that the
condition $f_{\alpha,\xi}(z)\geq c$ is equivalent to $g_{\beta}(z)\geq
c+1/\alpha$, where $g(\cdot)$ is defined in Equation 18.

$\zeta=\sup\left\\{x\in(0,1]:f_{\alpha,\xi}(z)\geq c,\forall
z\in(0,x)\right\\}$ (20)

for given values of $\alpha>0$ and $\xi\in[0,1)$.

###### Proof.

See Appendix C.∎

The meaning of this result is that it is possible to successfully decode a
substantial (i.e., proportional to $n$) amount of packets among those
transmitted simultaneously by the $n$ nodes, provided that SIC is used and the
target SNIR is set to a sufficiently small value, namely a value inversely
proportional to the number of concurrent transmissions.

The achieved sum-rate is defined as the spectral efficiency of the channel.
The sum-rate gives the achieved spectral efficiency of the multiple access
channel in bit/s/Hz [22]. Given that $n$ nodes transmit in one slot, the sum-
rate is evaluated as follows:

$\displaystyle U_{n}(\gamma)$ $\displaystyle=\frac{\mathrm{E}[\text{ \\#
delivered bits per unit time }|\,n,\gamma]}{\text{ Channel bandwidth }}$
$\displaystyle=\frac{L\,\mathrm{E}[\text{ \\# delivered packets per slot
}|\,n,\gamma]/T(\gamma)}{W}$
$\displaystyle=\log_{2}(1+\gamma)\,\mathrm{E}[\text{ \\# delivered packets per
slot }|\,n,\gamma]$ $\displaystyle=\log_{2}(1+\gamma)m_{n}(\gamma)$

Setting $\gamma=\gamma_{n}=1/(\alpha n)$, we have in the limit for large $n$:

$U_{n}(\gamma_{n})=\log_{2}\left(1+\frac{1}{\alpha
n}\right)m_{n}(\gamma_{n})\sim\frac{1}{\alpha n\log 2}\cdot\zeta n$ (21)

hence

$\lim_{n\rightarrow\infty}{U_{n}(\gamma_{n})}=\frac{\zeta}{\alpha\log
2}=U_{\infty}$ (22)

This result gives a pathway to choosing a value for $\alpha$. Specifically,
$\alpha$ can be set to that value that maximizes the asymptotic value of the
sum-rate in the limit for large $n$. More in depth, according to the statement
of Theorem 3, $\zeta$ is a function of $\alpha$, i.e., $\zeta=\zeta(\alpha)$,
where $\zeta$ is defined in Equation 20. Then, in view of Equation 22, we set
$\alpha$ to the value that maximizes the ratio $\zeta(\alpha)/\alpha$.

Figure 7 shows the asymptotic value of the sum-rate $U_{\infty}$ as a function
of $\alpha$ for two values of $\xi$, namely, $\xi=0$ on the left and $\xi=0.1$
on the right. In both cases, an optimal configuration of the parameter
$\alpha$ is identified, that maximizes the asymptotic value of the sum-rate.
However, it is apparent that even a relatively small residual interference
(10% for $\xi=0.1$) makes the achievable asymptotic sum-rate drop from almost
4 bit/s/Hz to about 2 bit/s/Hz.

(a) $\xi=0$

(b) $\xi=0.1$

Figure 7: Asymptotic value of the maximum sum-rate $U_{\infty}$ as a function
of $\alpha$ for two values of $\xi$. The required SNIR threshold is set to
$\gamma=1/(\alpha n)$.

The behavior of the optimal value of $\alpha$, denoted with $\alpha^{*}$, and
the corresponding maximum value of the asymptotic sum-rate, denoted with
$U_{\infty}^{*}$, are shown as a function of $\xi$ in Figure 8.

(a) $\xi=0$

(b) $\xi=0.1$

Figure 8: Optimal configuration for maximum asymptotic sum-rate as a function
of the coefficient $\xi$. (a) Optimal value of $\alpha$. (b) Maximum
asymptotic sum-rate.

Finally, Theorem 4 gives an explicit form of the mean number of successfully
decoded packets in case there is no SIC (see Equation 9). This result is not
new (e.g., see [CITATION]). We report it here for ease of the reader, along
with the asymptotic regime.

######  Theorem 4.

Let us consider $n$ concurrent transmissions towards a single receiver that
can only exploit capture (no SIC) with a target SNIR $\gamma$. The mean number
of successfully decoded packets $m_{n}(\gamma)$ is given by

$m_{n}(\gamma)=e^{-\gamma
P_{N}/P_{0}}\frac{n}{(1+\gamma)^{n-1}}=\frac{(1-\epsilon)n}{(1+\gamma)^{n-1}}$
(23)

in view of the setting $P_{0}/P_{N}=-\gamma/\log(1-\epsilon)$. With
$\gamma=1/(\alpha n)$, it is

$U_{\infty}=\lim_{n\rightarrow\infty}{U_{n}}=\frac{(1-\epsilon)}{\log
2}\frac{1}{\alpha}e^{-1/\alpha}$ (24)

Setting $\gamma=1/(\alpha n)$, it is easy to derive the asymptotic expansion
$m_{n}(1/(\alpha n))\sim n(1-\epsilon)\exp(-1/\alpha)$. The asymptotic sum-
rate in this case is maximized for $\alpha=1$. The maximum asymptotic sum-rate
is $U_{\infty}^{*}=(1-\epsilon)/(e\log 2)$.

The model with no SIC applies to those multiple access schemes that mitigate
interference, yet they do not cancel it actively, e.g., Code Division Multiple
Access [REF] or Golden Modulation based on Zadoff-Chu sequences [REF]. As a
matter of example, in the former case $\gamma=\gamma^{\prime}/PG$, where
$\gamma^{\prime}$ is the target narrowband SNR of a node (it is typically
$\gamma^{\prime}\gg 1$) and $PG$ is the processing gain (typically $PG\gg 1$).

##  IV Generalization

Let us now generalize the model of the random variable $S_{i}$, that
represents the SNR of node $i$. Let us write $S_{i}=S_{0}Y_{i}$, where $S_{0}$
is the SNR level required to guarantee successful reception of a single
transmitter with probability $1-\epsilon$. In the analysis developed in
previous Sections it has been assumed that $Y_{i}$ be a negative exponential
random variable with mean 1, representing the Rayleigh fading gain. We now
generalize the analysis, by allowing $Y_{i}$ to be a general non-negative
random variable, with continuous PDF, still constrained to have mean equal to
1. Sources of randomness in the description of $Y_{i}$ come from node path
loss statistics and from the selected transmission power level.

Given $n$ nodes transmitting, we assume $Y_{1},\dots,Y_{n}$ are i.i.d. random
variables with CDF $F(t)=\mathcal{P}(Y\leq t)$, CCDF $G(t)=1-F(t)$, and PDF
$f(t)$.

The target average received SNR $S_{0}$ is set so that
$\mathcal{P}(S_{0}Y\geq\gamma)\geq 1-\epsilon$. Since $\mathcal{P}(Y>t)=G(t)$,
we have $G(\gamma/S_{0})=1-\epsilon$, i.e., $S_{0}=\gamma/G^{-1}(1-\epsilon)$.
To keep notation uniform with the one used in previous section, we let
$c=G^{-1}(1-\epsilon)$, so that we still wrtie $S_{0}=\gamma/c$. In case of
Rayleigh fading it is $G(t)=e^{-t}$. hence, $G^{-1}(y)=-\log y$, and we
recover $c=-\log(1-\epsilon)$ (see Equation 5).

The order statistics associated with $Y_{1},\dots,Y_{n}$ is denoted with
$Y_{(1)},\dots,Y_{(n)}$, where labeling is assigned so that
$Y_{(1)}\geq\dots\geq Y_{(n)}$, i.e., in descending order. It is known that
[28]

$f_{Y_{(h)}}(u)=\frac{n!}{(n-h)!(h-1)!}\left[G(u)\right]^{h-1}\left[1-G(u)\right]^{n-h}f(u)$
(25)

for $u\geq 0$ and $h=1,\dots,n$. The mean value of $Y_{(h)}$ is

$\displaystyle\mu_{h}$
$\displaystyle=\mathrm{E}[Y_{(h)}]=\int_{0}^{\infty}{uf_{Y_{(h)}}(u)\,du}$
$\displaystyle=n\int_{0}^{\infty}{u\binom{n-1}{h-1}\left[G(u)\right]^{h-1}\left[1-G(u)\right]^{n-h}f(u)\,du}$
$\displaystyle=n\int_{0}^{1}{G^{-1}(u)\binom{n-1}{h-1}u^{h-1}(1-u)^{n-h}\,du}$
(26) $\displaystyle=\int_{0}^{1}{G^{-1}(u)\beta_{h,n-h+1}(u)\,du}$ (27)

where $\beta_{a,b}(u)$ is the Beta PDF with parameters $a$ and $b$, for
$u\in[0,1]$ and $a,b>0$.

Packet from node $h$ is decoded correctly if all packets sent by nodes whose
signal was stronger (nodes from 1 to $h-1$) have been decoded correctly and if
the SNIR of node $h$ exceeds the threshold $\gamma$, i.e.,

$\frac{S_{0}Y_{(h)}}{1+\sum_{i=h+1}^{n}{S_{0}Y_{(i)}}}\geq\gamma$ (28)

This inequality can be re-arranged as follows:

$Y_{(h)}-\gamma\sum_{i=h+1}^{n}{Y_{(i)}}\geq c$ (29)

where we have used the equality $S_{0}=\gamma/c$.

For large $n$, we expect that the random variable on the left-hand side
collapses to a deterministic distribution, hence it is equal to its mean with
probability 1. Let us therefore turn to the study of the inquality obtained by
replacing the random variable on the left-hand side with its _mean_.

$\mu_{h}-\gamma\sum_{i=h+1}^{n}{\mu_{i}}\geq c$ (30)

We consider the asymptotic regime where $h\sim nx$, for a fixed $x\in(0,1)$,
and $\gamma=1/(\alpha n)$, with $\alpha>0$. Then, we have

$\displaystyle\mu_{n}$
$\displaystyle=\int_{0}^{1}{G^{-1}(u)\beta_{nx,n(1-x)+1}(u)\,du}$
$\displaystyle\sim\int_{0}^{1}{G^{-1}(u)\beta_{nx,n(1-x)}(u)\,du}$ (31)

where we account for the fact that $n(1-x)+1\sim n(1-x)$ for large $n$ and
fixed $x$.

We use the following asymptotic property of the beta probability distribution:

$\beta_{na,nb}(u)\sim\mathcal{N}\left(\frac{a}{a+b},\frac{1}{n}\,\frac{ab}{(a+b)^{2}}\right)$
(32)

Therefore, in our case we get

$\beta_{nx,n(1-x)}(u)\sim\mathcal{N}\left(x,\frac{x(1-x)}{n}\right)$ (33)

This shows that as $n\rightarrow\infty$, the scaled Beta distribution tends to
a Dirca impulse centered in $x$. As a consequence, we find as $n$ tends to
infinity:

$\mu_{n}\rightarrow\int_{0}^{1}{G^{-1}(u)\delta(u-x)\,du}=G^{-1}(x)$ (34)

As for the sum appearing on the right-hand side of Equation 30, we have, for
$\gamma=1/(\alpha n)$,

$\gamma\sum_{i=h+1}^{n}{\mu_{i}}=\frac{1}{\alpha}\int_{0}^{1}{G^{-1}(u)\sum_{i=h}^{n-1}{B_{u,n-1}(i)}\,du}$
(35)

where

$B_{u,n-1}(i)=\binom{n-1}{i}u^{i}(1-u)^{n-1-i}\,,\quad i=0,1,\dots,n-1,$ (36)

is a Binomial probability distribution with parameters $u$ and $n-1$. Let $Z$
denote the binomial random variable with probability distribution in Equation
36. Then, we have

$\sum_{i=h}^{n-1}{B_{u,n-1}(i)}=\mathcal{P}(Z\geq h)$ (37)

In the considered asymptotic regime, it is $h\sim nx$. Moreover, we use the
following asymptotic property of the Binomial probability distribution:

$\lim_{n\rightarrow\infty}{\mathcal{P}\left(\frac{Z-nu}{\sqrt{nu(1-u)}}\leq
t\right)}=\Phi(t)$ (38)

where $\Phi(t)$ is the CDF of the standard Gaussian random variable:

$\Phi(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{t}{e^{-u^{2}/2}\,du}$ (39)

This stems from the Central Limit Thoerem and from the fact that

$Z=\sum_{k=1}^{n-1}{B_{k}}$ (40)

where $B_{k}$ is a Bernoulli random variable that equals 1 with probability
$u$ and it is equal to 0 with probability $1-u$.

From Equation 38, we derive that

$\mathcal{P}(Z\leq nx)\sim\Phi\left(\frac{nx-
nu}{\sqrt{nu(1-u)}}\right)=\Phi\left(\sqrt{n}\frac{x-u}{\sqrt{u(1-u)}}\right)$
(41)

As $n$ gets large, it is seen that
$\Phi\left(\sqrt{n}\frac{x-u}{\sqrt{u(1-u)}}\right)\rightarrow 0$, if $x<u$,
whereas $\Phi\left(\sqrt{n}\frac{x-u}{\sqrt{u(1-u)}}\right)\rightarrow 1$, if
$x>u$. Therefore, the probability of the event $Y\leq h=nx$ tends to a step
function with jump located at $x$ as $n$ grows.

Summing up, we have

$\sum_{i=nx}^{n-1}{B_{u,n-1}(i)}=\mathcal{P}(Y\geq
nx)\rightarrow\begin{cases}1&u>x\text{ },\\\ 0&u<x.\end{cases}$ (42)

Inserting this asymptotic result in the integral in Equation 35, we have as
$n$ tends to infinity:

$\gamma\sum_{i=nx+1}^{n}{\mu_{i}}\rightarrow\frac{1}{\alpha}\int_{x}^{1}{G^{-1}(u)\,du}$
(43)

In the end, the inequality on the mean values in Equation 34 approaches the
following limiting inequality in the asymptotic regime

$G^{-1}(x)-\frac{1}{\alpha}\int_{x}^{1}{G^{-1}(u)\,du}\geq c$ (44)

for a given $x\in(0,1)$. The quantity $x$ represents the fraction of correctly
decoded packets. The asymptotic sum rate is
$U_{\infty}=\frac{x^{*}(\alpha)}{\alpha\log 2}$, where $x^{*}(\alpha)$ is the
smallest value of $x\in(0,1)$ such that the inequality in Equation 44 is
satisfied.

In the special case where $Y$ is a negative exponential random variable with
mean 1, it is $G(t)=e^{-t}$, hence $G^{-1}(u)=-\log u$. It can be verified
that Equation 44 reduces to Equation 13.

Equation 44 can be transformed as follows:

$y-\frac{1}{\alpha}\int_{0}^{y}{uf(u)\,du}\geq c=G^{-1}(1-\epsilon)$ (45)

where $y=G^{-1}(x)$, hence it is $x=G(y)$. The fraction of transmission that
can be correctly decoded is $x^{*}=G(y^{*})$, with $y^{*}$ equal to the
largest value of $y$ such that the left-hand side of Equation 45 is less than
$c$. Formally,

$y^{*}=\inf\left\\{y:z\geq c+\int_{0}^{z}{uf(u)du},\forall
z\in(y,\infty)\right\\}$ (46)

Since $f(u)\geq 0$ and the mean of the PDF $f(u)$ is constrained to be 1, it
is easy to verify that it must be $c\leq y^{*}\leq c+1/\alpha$. Since
$x^{*}=G(y^{*})$ and $G(\cdot)$ is monotonously decreasing, we have
$G\left(G^{-1}(1-\epsilon)+1/\alpha\right)\leq x^{*}\leq
G\left(G^{-1}(1-\epsilon)\right)=1-\epsilon$.

To provide a numerical example, we assume $Y$ has a Gamma PDF, one of the
simplest probability distributions that provides two degrees of freedom. We
set the mean of $Y$ to 1 and let $\eta$ denote the reciprocal of the SCOV of
$Y$, i.e., $\eta$ is the ratio of the mean of $Y$ squared to the variance of
$Y$. As $\eta$ gets smaller, the random variable $Y$ exhibits more
variability, which is deemed to be favorable to the success of SIC decoding.

Formally, for any $\eta>0$, we have

$f(t)=\frac{\eta^{\eta}t^{\eta-1}}{\Gamma(\eta)}e^{-\eta t}\,,\quad t>0.$ (47)

where $\Gamma(a)$ is the Euler Gamma function, defined by
$\Gamma(a)=\int_{0}^{\infty}{u^{a-1}e^{-u}\,du}$ for $a>0$. With this PDF,
Equation 45 specializes into

$y-\frac{1}{\alpha}\frac{\Gamma(\eta+1,\eta y)}{\Gamma(\eta+1)}\geq
c=\frac{\Gamma^{-1}\left(\eta,\epsilon\Gamma(\eta)\right)}{\eta}$ (48)

where $\Gamma(a,x)$ is the incomplete Euler Gamma Function, given by
$\Gamma(a,x)=\int_{0}^{x}{u^{a-1}e^{-u}\,du}$ for $a,x>0$, and
$\Gamma^{-1}(a,\cdot)$ is its inverse with respect to the second variable,
$x$. Note $\Gamma(a)=\Gamma(a,\infty)$ and $\Gamma(a,x)$ is a monotonously
increasing function of $x$ for any positive $a$. We have also used the
equality $\Gamma(\eta+1)=\eta\Gamma(\eta)$, holding for any $\eta>0$.

Exploiting the equation $U_{\infty}=x^{*}/(\alpha\log(2))$, it is possible to
find numerical values of the asymptotic sum-rate $U_{\infty}$ as a function of
$\alpha$, for several values of the SCOV of $Y$ (see Figure 9a).

(a)

(b)

Figure 9: Analysis of the asymptotic sum-rate in case of generalized fading
with Gamma PDF. (a) Asymptotic sum-rate as a function of $\alpha$ for several
values of the SCOV of $Y$. (b) Maximum achievable asymptotic sum-rate as a
function of the SCOV of $Y$.

It appears that, for any fixed value of the SCOV, there is an optimal value of
$\alpha$ that maximizes the asymptotic sum-rate. The behavior of the maximum
achievable asymptotic sum-rate as a function of the SCOV is shown in Figure
9b. This results confirm the intuition that suggests a higher variability of
$Y$ (hence of the received power level) is beneficial to the success of SIC
receiver.

In General, we can write $Y=XZ$, where $X$ represents the random component of
the block fading channel and $Z$ stems from a randomization of the
transmission power level. We remind that it must be $\mathrm{E}[Y]=1$. If we
assume Rayleigh fading, so that $\mathcal{P}(X>u)=e^{-u}$, then it must be
$\mathrm{E}[Z]=1$. With a two level choice for transmission power, we have
$Z=1/b$ with probability $b/(1+b)$ and $Z=b$ with probability $1/(1+b)$, for
any $b\geq 1$. It can be verified that
$\mathrm{E}[Y^{2}]=2\mathrm{E}[Z^{2}]=2\left(b+1/b-1\right)$. Hence, the SCOV
of $Y$ is $2\left(b+1/b-1\right)-1$, which can be made as large as desired
with growing values of $b$. To ease transmission power setting, the nodes
could be instructed to opt for the largest transmission power level in case
their average channel gain is above a suitable threshold, for the lower value
otherwise. Setting the threshold properly assigns probability of using either
level as desired (i.e., so that the average of $Z$ be 1).

##  V Conclusions

We have characterized the asymptotic behavior of the mean number of correctly
decoded packets and the ensuing sum-rate of a SIC receiver with $n$
transmitting nodes and a target SNIR that scales in inverse proportion to $n$
The considered SIC is quite general, encompassing any kind of random fading.
The average received power level is assumed to be same for all nodes. It is
found that the random variable representing the number of correctly decoded
packets converges to a deterministic quantity as $n\rightarrow\infty$. The
asymptotic regime is fully characterized.

Two major points for further work stem out of the result of this paper. First,
the generalization to systems where nodes play with transmission power levels,
so as to obtain different average received power levels, are of interest. In
fact, SIC performance are expected to improve, if superposed signals have well
separated reception power levels. Secondly, practical techniques for realizing
SIC should be evaluated. Referring to specific SIC algorithms allows the
identification of a refined reception model. It could also provide evidence of
the feasibility of large scale interference cancellation receivers.

##  Appendix A Proof of Theorem 1

We first characterize the order statistics of a set $\\{Y_{1},\dots,Y_{n}\\}$
of $n$ i.i.d. negative exponential random variables with mean 1. Let
$Y_{(1)},Y_{(2)},\dots,Y_{(n)}$ denote the corresponding order statistics in
descending order, i.e., we have $Y_{(1)}\geq Y_{(2)}\geq\dots\geq Y_{(n)}$.
The main difficulty with this set of random variables is that they are not
independent. It is possible however to find a simpler representation for the
order statistics, namely, we can write:

$Y_{(k)}=\sum_{j=k}^{n}{\frac{X_{j}}{j}}$ (49)

where $X_{1},\dots,X_{n}$ are i.i.d. negative exponential random variable with
mean 1.

This representation is derived by considering $n$ negative exponential timers,
starting off at the same time. The first timer to expire marks the value of
$Y_{(n)}$. The residual time to expiry for all remaining timers is still
distributed according to a negative exponential random variable, thanks to the
memoryless property of the negative exponential random variable. In general,
let $T_{k}$ denote the time it takes for the next timer to go, when $k$ timers
are left. Thanks to the memoryless property, $T_{k}$ has the same probability
distribution as the minimum of $k$ i.i.d. negative exponential random variable
of mean 1, i.e., $\mathcal{P}(T_{k}>t)=e^{-kt},\,t\geq 0$. Let $t_{0}$ be the
time when all timers are triggered. The time $E_{j}$ when the $j$-th timer
expires and only $j-1$ survive is $E_{j}=t_{0}+T_{n}+\dots+T_{n-j+1}$. This is
the $j$-th smallest value among all timer lifetimes, hence
$Y_{(n-j+1)}=E_{j}-t_{0}$ for $j=1,\dots,n$ (remember the order statistics
$\\{Y_{(k)}\\}_{1\leq k\leq n}$ that we consider is in _descending_ order). We
conclude then

$Y_{(k)}=T_{n}+T_{n-1}+\dots+T_{k+1}+T_{k}$ (50)

where $T_{j}$ is a negative exponential random variable with mean $1/j$ and
the $T_{j}$’s are independent. Therefore, we can finally write the
representation in Equation 49

Starting from Equation 8 and replacing $|\tilde{h}_{k}|^{2}$ with the
descending order statistics $Y_{(k)}$ of a set of $n$ i.i.d. negative
exponential random variables with mean 1, we get the following condition for
the successful reception of the packet associated with the $m$-th strongest
signal:

$Y_{(j)}\geq
c+\gamma\sum_{r=j+1}^{n}{Y_{(r)}}+\gamma\xi\sum_{r=1}^{j-1}{Y_{(r)}}\,,\quad
j=1,\dots,m$ (51)

Using Equation 49 into Equation 51, we re-write the $j$-th condition above as
follows:

$\sum_{k=j}^{n}{\frac{X_{k}}{k}}\geq
c+\gamma\sum_{r=j+1}^{n}{\sum_{k=r}^{n}{\frac{X_{k}}{k}}}+\gamma\xi\sum_{r=1}^{j-1}{\sum_{k=r}^{n}{\frac{X_{k}}{k}}}$
(52)

Swapping the two last double summations, we get

$\sum_{r=j+1}^{n}{\sum_{k=r}^{n}{\frac{X_{k}}{k}}}=\sum_{k=j+1}^{n}{\sum_{r=j+1}^{k}{\frac{X_{k}}{k}}}=\sum_{k=j+1}^{n}{X_{k}\,\frac{k-j}{k}}$
(53)

and

$\displaystyle\sum_{r=1}^{j-1}{\sum_{k=r}^{n}{\frac{X_{k}}{k}}}$
$\displaystyle=\sum_{r=1}^{j-1}{\left(\sum_{k=r}^{j-1}{\frac{X_{k}}{k}}+\sum_{k=j}^{n}{\frac{X_{k}}{k}}\right)}$
(54)
$\displaystyle=\sum_{k=1}^{j-1}{\sum_{r=1}^{k}{\frac{X_{k}}{k}}}+\sum_{k=j}^{n}{\sum_{r=1}^{j-1}{\frac{X_{k}}{k}}}$
(55)
$\displaystyle=\sum_{k=1}^{j-1}{X_{k}}+\sum_{k=j}^{n}{X_{k}\,\frac{j-1}{k}}$
(56)

Plugging those expressions back into Equation 52, we find

$\sum_{k=j}^{n}{\frac{X_{k}}{k}}\geq
c+\gamma\sum_{k=j}^{n}{X_{k}\,\frac{k-j}{k}}+\gamma\xi\sum_{k=1}^{j-1}{X_{k}}+\gamma\xi\sum_{k=j}^{n}{X_{k}\,\frac{j-1}{k}}$
(57)

that is

$\left[1+j\gamma-(j-1)\gamma\xi\right]\sum_{k=j}^{n}{\frac{X_{k}}{k}}-\gamma\sum_{k=j}^{n}{X_{k}}-\gamma\xi\sum_{k=1}^{j-1}{X_{k}}\geq
c$ (58)

Let us define the coefficients

$b_{jk}=\begin{cases}-\gamma\xi&k=1,\dots,j-1,\\\
\frac{1+j\gamma-(j-1)\gamma\xi}{k}-\gamma&k=j,\dots,n.\end{cases}$ (59)

for $k=1,\dots,n$. Then, we can re-write Equation 58 as follows

$V_{j}=\sum_{k=1}^{n}{b_{jk}X_{k}}\geq c$ (60)

where we have defined the random variable $V_{j}$. The conditions for decoding
the $m$ strongest packet successfully can be stated concisely as $V_{j}\geq c$
for $j=1,\dots,m$.

Since the mean and variance of $X_{k}$ is equal to 1 for all $k$ and the
$X_{k}$’s are i.i.d. random variables, we have

$\displaystyle\mu_{j}(n)=\mathrm{E}[V_{j}]=\sum_{k=j}^{n}{b_{jk}}$ (61)
$\displaystyle\sigma^{2}_{j}(n)=\mathrm{E}[(V_{j}-\mu_{j})^{2}]=\sum_{k=j}^{n}{b_{jk}^{2}}$
(62)

##  Appendix B Proof of Theorem 2

###  B-A Asymptotic regime for large $n$: mean value of $V_{j}$

We recall that we are studying the asymptotic regime as $n\rightarrow\infty$
with $\gamma=1/(\alpha n)$. The expression of the mean of $V_{j}$ can be
derived explicitly from Equations 59 and 61:

$\mu_{j}(n)=A_{j}(n)\left[H_{n}-H_{j-1}\right]-B_{j}(n)$ (63)

where

$\displaystyle A_{j}(n)$ $\displaystyle=1+\frac{\xi+(1-\xi)j}{\alpha n}$ (64)
$\displaystyle B_{j}(n)$ $\displaystyle=\frac{n-(1-\xi)(j-1)}{\alpha n}$ (65)
$\displaystyle H_{n}$ $\displaystyle=\sum_{k=1}^{n}{\frac{1}{k}}$ (66)

for $j=1,\dots,n$.

The number $H_{n}$ is known as the $n$-th harmonic number. For ease of
notation, we let also $H_{0}=0$. It is well known that $H_{n}\sim\log(n+1)+E$,
where $E$ is the Euler-Mascheroni constant, $E\approx 0.5772$. As for the
difference of harmonic numbers, it is easy to verify that

$\log\left(\frac{n}{j}\right)+\frac{1}{n}\leq
H_{n}-H_{j-1}=\sum_{k=j}^{n}{\frac{1}{k}}\leq\log\left(\frac{n}{j}\right)+\frac{1}{j}$
(67)

This is proved by starting from the inequalities
$\frac{1}{k+1}\leq\frac{1}{x}\leq\frac{1}{k}$, holding for $k\leq x\leq k+1$
and for any $k\geq 1$. Then, integrating over $x$ and summing over $k$ leads
to the inequalities in Equation 67

Let us define the function $f_{\alpha,\xi}(x)$ for $x\in(0,1)$, given by

$f_{\alpha,\xi}(x)=-\left(1+\frac{(1-\xi)x}{\alpha}\right)\log
x-\frac{1-(1-\xi)x}{\alpha}$ (68)

After a long yet simple calculation, it is verified that

$\frac{1}{n}+\epsilon_{j}(n)\leq\mu_{j}(n)-f\left(\frac{j}{n}\right)\leq\frac{1}{j}+\delta_{j}(n)$
(69)

where

$\displaystyle\delta_{j}(n)=\frac{\xi}{\alpha
n}\left(\log\left(\frac{n}{j}\right)+\frac{1}{j}\right)$ (70)
$\displaystyle\epsilon_{j}(n)=\frac{\xi}{\alpha
n^{2}}-\frac{(1-\xi)(1-j/n)}{\alpha n}+\frac{\xi}{\alpha
n}\log\left(\frac{n}{j}\right)$ (71)

for $j=1,\dots,n$. Since $\delta_{j}(n)$ for fixed $n$ is a monotonously
decreasing function of $j$ and a monotonously increasing function of
$\xi\in[0,1]$, for all $j=1,\dots,n$ we have

$\delta_{j}(n)\leq\xi\frac{\log n+1}{\alpha n}\leq\frac{\log n+1}{\alpha n}$
(72)

As for $\epsilon_{j}(n)$, it is a monotonously increasing function of
$\xi\in[0,1]$. Setting $\xi=1$ in Equation 71, we get for all $j=1,\dots,n$:

$\epsilon_{j}(n)\leq\frac{\log(n/j)+1/n}{\alpha n}\leq\frac{\log n+1/n}{\alpha
n}$ (73)

For $\xi=0$, and for all $j=1,\dots,n$, we have also:

$\epsilon_{j}(n)\geq-\frac{1-j/n}{\alpha n}\geq-\frac{1}{\alpha n}$ (74)

Summing up, for any fixed $n$ and for all $j=1,\dots,n$ and $\xi\in[0,1]$, we
have

$\displaystyle 0\leq\delta_{j}(n)\leq\frac{\log n+1}{\alpha n}$ (75)
$\displaystyle|\epsilon_{j}(n)|\leq\max\left\\{\frac{\log n+1/n}{\alpha
n}\,,\frac{1}{\alpha n}\right\\}\leq\frac{\log n+1}{\alpha n}$ (76)

Using the inequalities above in Equation 69, we can bound the absolute
deviation between the sequences of the mean values and of the approximating
function:

$\displaystyle\Delta_{j}(n)$
$\displaystyle=\left|\mu_{j}(n)-f\left(\frac{j}{n}\right)\right|$
$\displaystyle\leq\max\left\\{\frac{1}{n}+|\epsilon_{j}(n)|,\frac{1}{j}+\delta_{j}(n)\right\\}$
$\displaystyle\leq\max\left\\{\frac{1}{n}+\frac{\log n+1}{\alpha
n},\frac{1}{j}+\frac{\log n+1}{\alpha n}\right\\}$
$\displaystyle=\frac{1}{j}+\frac{\log n+1}{\alpha n}$ (77)

We consider now the mean absolute error between the sequences
$\\{\mu_{j}(n)\\}_{1\leq j\leq n}$ and $\\{f(j/n)\\}|_{1\leq j\leq n}$ In view
of Equation 77, we have

$\displaystyle\Delta(n)$
$\displaystyle=\frac{1}{n}\sum_{j=1}^{n}{\left|\mu_{j}(n)-f\left(\frac{j}{n}\right)\right|}$
$\displaystyle\leq\frac{1}{n}\sum_{j=1}^{n}{\frac{1}{j}}+\frac{\log
n+1}{\alpha n}\sim O\left(\frac{\log n}{n}\right)$ (78)

where the last (asymptotic) equality is a consequence of
$\sum_{j=1}^{n}{1/j}=H_{n}\sim\log n$ for $n\rightarrow\infty$. This proves
that $\lim_{n\rightarrow\infty}{\Delta(n)}=0$, i.e., the sequence of mean
values $\mu_{j}(n)$, $j=1,\dots,n$, approaches the continuous curve
$f_{\alpha,\xi}(x)$, i.e., the points $\mu_{j}(n)$ move towards the curve
$f_{\alpha,\xi}(x),\,x\in(0,1]$ as $n$ grows. The result proved above
determines also the rate of convergence, which appears to be rather slow.

###  B-B Asymptotic regime for large $n$: variance of $V_{j}$

From Equation 62, the expression of the variance of $V_{j}$ is as follows:

$\sigma^{2}_{j}(n)=a^{2}_{j}(n)\sum_{k=j}^{n}{\frac{1}{k^{2}}}-2\frac{a_{j}(n)}{\alpha
n}\sum_{k=j}^{n}{\frac{1}{k}}+\frac{n-j+(j-1)\xi^{2}}{\alpha^{2}n^{2}}$ (79)

where

$a_{j}(n)=1+\frac{\xi+(1-\xi)j}{\alpha n}$ (80)

We distinguish two cases:

  1. 1.

Fixed $j$, so that the fraction $j/n$ becomes vanishingly small as $n$ grows.

  2. 2.

$j=nx$, with fixed $x$, so that the fraction $j/n$ is fixed and positive as
$n$ grows.

In the first case, for fixed $j$ we have for $n\rightarrow\infty$:

$\displaystyle a_{j}(n)\rightarrow 1$
$\displaystyle\sum_{k=j}^{n}{\frac{1}{k^{2}}}\rightarrow\frac{\pi^{2}}{6}-\sum_{k=1}^{j-1}{\frac{1}{k^{2}}}$
$\displaystyle\sum_{k=j}^{n}{\frac{1}{k}}=H_{n}-H_{j-1}\sim\log\left(\frac{n}{j}\right)$
$\displaystyle\frac{n-j+(j-1)\xi^{2}}{\alpha^{2}n^{2}}\rightarrow 0$

where the second inequality stems from the known series sum
$\sum_{k=1}^{\infty}{1/k^{2}}=\pi^{2}/6$. Using these limits in Equation 79,
we have for any given fixed $j$:

$\lim_{n\rightarrow\infty}{\sigma^{2}_{j}(n)}=\frac{\pi^{2}}{6}-\sum_{k=1}^{j-1}{\frac{1}{k^{2}}}$
(81)

On the other hand, from Equation 63, it is apparent that for any given fixed
$j$ we have $\lim_{n\rightarrow\infty}{\mu_{j}(n)}=\infty$. Hence, for any
given fixed $j$, we have

$\lim_{n\rightarrow\infty}{\frac{\sigma_{j}(n)}{\mu_{j}(n)}}=0$ (82)

In the second case, when the fraction $j/n=x$ is fixed as $n$ grows, we have

$\displaystyle\sigma^{2}_{j}(n)$ $\displaystyle\leq\left(1+\frac{j}{\alpha
n}\right)^{2}\sum_{k=j}^{n}{\frac{1}{k^{2}}}+\frac{n-1}{\alpha^{2}n^{2}}$
$\displaystyle\leq\left(1+\frac{j}{\alpha
n}\right)^{2}\frac{n-j}{j^{2}}+\frac{1}{\alpha^{2}n}$
$\displaystyle=\left(1+\frac{x}{\alpha}\right)^{2}\frac{n-nx}{n^{2}x^{2}}+\frac{1}{\alpha^{2}n}$
$\displaystyle=\left[\frac{1-x}{x^{2}}\left(1+\frac{x}{\alpha}\right)^{2}+\frac{1}{\alpha^{2}}\right]\frac{1}{n}$

The upper bound in the first line is obtained as a consequence of the
following facts.

  1. 1.

The coefficients $a_{j}(n)$ are decreasing functions of $\xi\in[0,1]$ for all
$j$, hence $a_{j}(n)\leq a_{j}(n)|_{\xi=0}=1+j/(\alpha n)$.

  2. 2.

The second term on the right hand side of Equation 79 is negative.

  3. 3.

The last term is a monotonously increasing function of $\xi$, hence it is
maximized for $\xi=1$.

The expression in the third line is obtained by replacing $j$ with $nx$.

This proves that, for any fixed $x$, we have

$\lim_{n\rightarrow\infty}{\sigma^{2}_{j}(n)|_{j/n=x}}=0$ (83)

As for the mean, substituting $j$ with $nx$ in Equation 63, and taking the
limit for $n\rightarrow\infty$ with fixed $x>0$, it is not difficult to verify
that

$\lim_{n\rightarrow\infty}{\mu_{j}(n)|_{j/n=x}}=f_{\alpha,\xi}(x)$ (84)

where $f_{\alpha,\xi}(x)$ is defined in Equation 13.

Putting together the limits of variance and mean of $V_{j}$ for $j/n=x$ for
fixed $x>0$, we get:

$\lim_{n\rightarrow\infty}{\left.\frac{\sigma_{j}(n)}{\mu_{j}(n)}\right|_{j/n=x}}=0$
(85)

Applying Chebichev inequality, we can write

$\mathcal{P}\left(\left|\frac{V_{j}}{\mu_{j}(n)}-1\right|>\epsilon\right)\leq\frac{\sigma^{2}_{j}(n)}{\mu^{2}_{j}(n)\epsilon^{2}}$
(86)

Taking the limit for $n\rightarrow\infty$, we have for any given $\epsilon>0$:

$\lim_{n\rightarrow\infty}{\mathcal{P}\left(\left|\frac{V_{j}}{\mu_{j}(n)}-1\right|>\epsilon\right)}=0$
(87)

which proves that $V_{j}\sim\mu_{j}(n)$ almost surely for large $n$, i.e.,
$V_{j}$ tends to become a deterministic variable, equal to its mean, with its
coefficient of variation (the ratio of the standard deviation to the mean)
shrinking to 0.

##  Appendix C Proof of Theorem 3

Note that $\lim_{x\rightarrow 0+}{f_{\alpha,\xi}(x)}=+\infty$ and
$f_{\alpha,\xi}(1)=-\xi/\alpha\leq 0$. Hence, the set
$\left\\{x:x\in(0,1)\text{ and }f_{\alpha,\xi}(x)=c\right\\}$ is nonempty for
any positive $c$.

As $n$ gets large, the sequence of random variables $V_{j}$ for $j=1,\dots,n$
tend to a series of deterministic random variables equal to $\mu_{j}(n)\sim
f_{\alpha,\xi}(j/n)$.

On the other hand, the condition for the packet carried by the $m$-th
strongest signal to be decoded successfully is that $V_{j}\geq c$ for
$j=1,\dots,m$. Asymptotically, this happens if $\mu_{j}(n)\geq c$, i.e.,
$f_{\alpha,\xi}(j/n)\geq c$ for $j=1,\dots,m$.

The overall number of correctly decoded packets will be $m$ if and only if
$f_{\alpha,\xi}(j/n)\geq c$, for $j=1,\dots,m$, and
$f_{\alpha,\xi}((m+1)/n)<c$. Reminding the definition of $\zeta$, in the limit
for $n\rightarrow\infty$, it is recognized that $m/n\sim\zeta$. Hence, the
average number of correctly decoded packets out of $n$ transmitted behaves as
$m_{n}(\gamma_{n})\sim\zeta n$ as $n$ tends to infinity.

## Acknowledgment

This work was partially supported by the European Union under the Italian
National Recovery and Resilience Plan (NRRP) of Next Generation EU,
partnership on “Telecommunications of the Future” (PE00000001 - program
“RESTART”).

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