# Distributed Backlog-Aware D2D Communication for Heterogeneous IIoT
Applications

Hossam Farag1, Čedomir Stefanović1 and Mikael Gidlund2  1Department of
Electronic Systems, Aalborg University, Denmark
2Department of Information Systems and Technology, Mid Sweden University,
Sweden
Email: hmf<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Delay and Age-of-Information (AoI) are two crucial performance metrics for
emerging time-sensitive applications in Industrial Internet of Things (IIoT).
In order to achieve optimal performance, studying the inherent interplay
between these two parameters in non-trivial task. In this work, we consider a
Device-to-Device (D2D)-based heterogeneous IIoT network that supports two
types of traffic flows, namely AoI-orientated. First, we introduce a
distributed backlog-aware random access protocol that allows the AoI-
orientated nodes to opportunistically access the channel based on the queue
occupancy of the delay-oriented node. Then, we develop an analytical framework
to evaluate the average delay and the average AoI, and formulate an
optimization problem to minimize the AoI under a given delay constraint.
Finally, we provide numerical results to demonstrate the impact of different
network parameters on the performance in terms of the average delay and the
average AoI. We also give the numerical solutions of the optimal parameters
that minimize the AoI subject to a delay constraint.

###### Index Terms:

Industrial IoT, AoI, D2D communication

##  I Introduction

The emerging technology of Internet of Things (IoT) enables ubiquitous
connectivity and innovative services spanning a wide range of applications,
such as industrial automation, healthcare and intelligent transportation [1].
The realization of such applications implies the interconnection of a large
number of heterogeneous and pervasive IoT devices continuously generating and
transmitting sensor data to their respective destinations. According to the
Global System for Mobile Communications Association (GSMA), it is expected
that the number of connected IoT devices will grow up to 75 billion by 2025
[2]. As a subset of the IoT, Industrial IoT (IIoT) has gained significant
attention as a key technology supporting machine type communication for time-
sensitive industrial applications [3]. For a vast number of IIoT applications,
the field network comprises a high number of sensors and actuators that sense
the physical environment and perform appropriate actions, respectively [4].
Different from the conventional star-network configuration with a static sink,
Device-to-device (D2D) communications provide a promising transmission
paradigm for sensor-actuator pairs which are in propinquity [5]. In such
setups, sensors directly transmit data to actuators without involving a
central node, resulting in improved throughput and energy efficiency.

Typically, the field network in IIoT applications supports transmissions of
heterogeneous traffic flows which are characterized by different performance
objectives [6]. In process monitoring and control scenarios, the distributed
sensor nodes either report special events to their corresponding actuators
that perform actions in response, or transmit either status updates about a
certain proces. In the former case, it is essential to maintain low latency
communication of event reporting from sensors to actuators in order to avoid
production inefficiency or safety-critical situations. An event could be any
incident happening, e.g., fire or leakage of gas, that should be delivered to
the corresponding actuator within a predefined deadline. In the latter case,
the freshness of the status updates is crucial for proper functioning of the
system as the actuation decisions are mainly influenced by the freshest
information available [7]. For instance, in oil refineries, valve actuators
should acquire timely monitoring of oil level to avoid spilling of oil tanks
[8]. Classical time-related performance metrics, such as throughput and delay,
have a transmitter-side view of the channel and are insufficient to reflect
information freshness at the receiver side. Age of Information (AoI) has been
proposed to characterize information freshness which measures the time elapsed
since the generation of the most recently received packet [9]. Although D2D
communication can improve throughput and energy efficiency [10], the high
number of deployed sensor-actuator pairs and the uncoordinated channel access
pose a challenge on guaranteeing the performance objectives (low latency and
low AoI) of heterogeneous traffic streams.

In this work, we consider a D2D-based heterogeneous IIoT network that supports
two traffic flows, delay-oriented traffic and AoI-oriented traffic. The delay-
oriented traffic is transmitted by a single node and is subject to a
predefined delay constraint, while the AoI-oriented traffic is transmitted by
a set of nodes distributed according to a stochastic point process. Our
contributions in this work can be summarized as follows:

  * •

We introduce a distributed backlog-aware random access protocol, in which the
AoI-oriented nodes access the shared channel based on the backlog status of
the delay-oriented node.

  * •

Using stochastic geometry, Discrete-Time Markov Chain (DTMC) and queueing
theory, we derive closed-form expressions for the average AoI, average delay,
and the average queue size of the delay-oriented node, and formulate an
optimization problem to minimize the average AoI under a given delay
constraint.

  * •

We evaluate the network performance using numerical results in order to
investigate the impact of different network parameters on the average AoI and
the average delay.

  * •

We solve the formulated optimization via a 2-D exhaustive search under
different network parameters.

##  II Related Work

AoI has been receiving a growing attention as a promising metric for time-
sensitive applications. Several works were conducted to study and analyze AoI
utilizing queuing theory [11, 12]. The AoI minimization problem has been
addressed through different approaches including packet management schemes
[13, 14], adjusting sampling rate [9] and introducing packet deadlines [15].
The queuing analysis in these works considers only the temporal traffic
dynamics associated with the transmitters and receivers based on the
assumption of an error-free channel, i.e., without considering transmission
failures due to interference and collisions that are inevitable in the
industrial environment. In [16, 17, 18, 19], the authors introduced a number
of scheduling protocols with the goal to minimize the average AoI considering
wireless networks with unreliable channels. All these works are based on a
centralized scheduling policy which is not applicable to the D2D communication
scenario in IIoT, as it would incur high communication overhead and cannot
scale with the network size.

The main research on scheduling in wireless D2D networks focuses on the
analysis and optimization of resource allocation [20], traffic density [21],
or user fairness [22], however, there are few works that focus on the study
and minimization of AoI The authors in in [23] statistically analyzed the peak
AoI and the conditional success probability in a D2D communication scenario
under preemptive and non-preemptive queues. Based on a stochastic geometry
approach, the authors in [24] proposed a distributed scheduling policy that
utilizes local observations, encapsulated in the concept of stopping sets to
make scheduling decisions that minimize the network-wide average AoI. The work
in [25] introduced a joint optimization problem of the transmit probability
and transmit power to maximize the energy efficiency of a covert D2D
communication under AoI constraint. The authors in [26] proposed an age-aware
scheduling algorithm where edge controllers assist each D2D node to select a
proper resource block and transmit power level to satisfy the AoI constraint.
All these works consider that the network supports only a single traffic flow,
which is age-oriented traffic and aim to minimize the average AoI without
considering the coexistence of other traffic flows. The works in [27, 28]
proposed scheduling policies to optimize the AoI while satisfying throughput
constraints. The authors in [29] proposed a link scheduling policy for a joint
optimization of AoI and throughput based on a deep learning approach. These
works assume that, at each timeslot, the same user transmit either AoI-
oriented packet or a throughput-oriented packet. The works in [30, 31] study
the AoI-delay tradeoff for multi-server queuing systems considering error-free
transmissions which is not applicable to real IIoT systems. In [32], the
authors studied the delay and AoI violation probabilities considering finite-
block length packets under noisy channels, however, the study was conducted
separately for the two parameters.

##  III System Model

Figure 1: The network topology with multiple $\mathrm{T_{AoI}}$ pairs and a
single $\mathrm{T_{D}}$ pair distributed in a circular region $\mathcal{C}$
with a radius $R$.

###  III-A Network Topology and Communication Model

We consider a heterogeneous IIoT network comprising a set of sensor-actuators
(D2D) pairs that are randomly distributed in a circular region $\mathcal{C}$
with a radius $R$ as depicted in Fig. 1. The network supports two traffic
flows, namely AoI-sensitive flow ($\mathrm{T_{AoI}}$) and delay-sensitive flow
($\mathrm{T_{D}}$). The $\mathrm{T_{AoI}}$ traffic represent the status
updates of an underlying time-varying process, where the $\mathrm{T_{D}}$
traffic denotes event reporting about a certain incident.

In our model, we consider a single D2D pair for transmitting the
$\mathrm{T_{D}}$ traffic which characterized by a strict delay performance.
The $\mathrm{T_{D}}$ transmitter is fixed in predefined location where the
$\mathrm{T_{D}}$ receiver is located at the origin of considered network
region with a distance $d_{D}$. The $\mathrm{T_{D}}$ transmitter has an
infinite output queue where the arrival process of the $\mathrm{T_{D}}$
packets follows a Poisson process with average rate $\lambda_{D}$
packets/slot, which captures the scenario where such traffic can be triggered
by the occurrence of some random incident [33]. The $\mathrm{T_{D}}$ packets
are transmitted in First-Come-First-Served (FCFS) basis, and a packet is
removed from the buffer if it is successfully transmitted, which is
acknowledged by the $\mathrm{T_{D}}$ receiver through an ACK feedback. If no
ACK is received within a predefined timeout, the packet is retransmitted. We
assume that the ACK transmission is instantaneous and error-free, which is
common assumption in the literature [34].

Further, we consider a set of $\mathrm{T_{AoI}}$ transmitters randomly
scattered according to a homogeneous Poisson Point Process (PPP) $\phi_{A}$
with spatial density $\lambda_{A}$, with
$\phi_{A}=\\{x_{i}=\mathbb{R}^{2},\forall i\in\mathbb{N^{+}},i\geq 1\\}$ [35],
where $x_{i}$ is the location of the $i$-th $\mathrm{T_{AoI}}$ transmitter.
Each $\mathrm{T_{AoI}}$ transmitter has a dedicated receiver which is located
at a distance $d_{A}$ in a random orientation. The $\mathrm{T_{AoI}}$ nodes
report status updates following the generate-at will model [9, 36], i.e., the
$\mathrm{T_{AoI}}$ transmitter is activated with a certain probability, and
once becoming active, it samples a fresh packet and transmits it to the
$\mathrm{T_{AoI}}$ receiver111The generate-at-will model is commonly used for
optimal AoI performance as it avoids unnecessary staleness to the sampled
information due to queuing [9, 36, 37, 38]. However, our analysis can be
extended to other arrival models.. After a transmission attempt, the
$\mathrm{T_{AoI}}$ discards the packet regardless it is a successful or failed
attempt222This scenario is based on the fact that retransmitting a stale
packet might be useless for the receiver. In that sense, it would be better to
transmit a fresh status update about the underlying industrial process instead
of consuming the communication resources for retransmitting outdated
information. Adopting retransmission-aided approach is out of the scope of
this paper and left as a future work..

The network operates in a time-slotted fashion where the generation and the
transmission of a packet is aligned with the boundary of a timeslot and a
transmission attempt takes a constant time, which is assumed to be equal to
the duration of one timeslot. The $\mathrm{T_{D}}$ transmitter freely access
the channel as long as there are packets in its queue. The communication
between each D2D pair is performed via a shared communication channel in a
slotted ALOHA-like access using a slot-access probability that is subject to
optimization.In other words, in every slot there is a random number of
interfering users that are active, comprising the $\mathrm{T_{D}}$ and/or the
activated $\mathrm{T_{AoI}}$ transmitters. The collisions among simultaneously
active users are not necessarily destructive, due to the capture effect [39].
In that sense, a packet is decoded successfully at the end of a timeslot when
the Signal-to-Interference plus-Noise ratio (SINR) at the corresponding
receiver exceeds the capture ratio $\beta$. The received SINR at an arbitrary
receiving node $i$ is given as

$\text{SINR}_{i}=\frac{P_{i}|h_{i,i}|^{2}d_{i,i}^{-\alpha}}{\sigma^{2}+\sum_{j\in\mathcal{K}\setminus\\{i\\}}P_{j}|h_{j,i}|^{2}d_{j,i}^{-\alpha}},$
(1)

where $P_{i}$ is the transmitting power of node $i$, $h_{j,i}$ represents the
Raleigh fading of the channel between the transmitter $j$ and the receiver $i$
with $h_{j,i}\sim\mathcal{CN}(0,1)$, $d_{j,i}$ is the distance between the
transmitter $j$ and the receiver $i$, $\alpha$ is the path loss exponent,
$\mathcal{K}$ is the set of transmitting nodes at the same timeslot and
$\sigma^{2}$ denotes additive white Gaussian noise (AWGN) power.

Figure 2: The DTMC model representing the evaluation of the AoI of the
$\mathrm{T_{AoI}}$ traffic.

###  III-B Analysis of the Average AoI and the average Delay

In this section, we analyze the Average AoI of the $\mathrm{T_{AoI}}$ traffic
considering that the status updates from all $\mathrm{T_{AoI}}$ nodes are
equally important, and without loss of generality, we evaluate the average AoI
at an arbitrary $\mathrm{T_{AoI}}$ receiver in discrete time. In our model,
the AoI represents the number of time slots elapsed since the last received
$\mathrm{T_{AoI}}$ packet was generated. If $\Delta(t)$ denotes the AoI at the
end of time slot $t$, then we have

$\Delta(t+1)=\begin{cases}\Delta(t)+1&\text{unsuccessful transmission}\\\
1&\text{successful transmission}.\end{cases}$ (2)

Fig. 2 shows the Discrete-Time Markov Chain (DTMC) model of the AoI
corresponds to an arbitrary $\mathrm{T_{AoI}}$ node where each state
represents the value of the AoI at the tagged $\mathrm{T_{AoI}}$ receiver. The
DTMC transits from any state $j$ to 1 only upon a successful reception of a
$\mathrm{T_{AoI}}$ packet, otherwise it transits to state $j+1$. Let $X_{t}$
represent the value of $\Delta(t)$ at time slot $t$, then the transition
probability from state $j$ to state $k$ is
$P_{jk}=\mathrm{Pr}(X_{t+1}=k\,\,|X_{t}=j)$, and the transition matrix
$\mathbf{P_{AoI}}$ is written as

$\mathbf{P_{AoI}}=\begin{bmatrix}s_{A}&1-s_{A}&0&0&\dots\\\
s_{A}&0&1-s_{A}&0&\dots\\\ \vdots&\vdots&\vdots&\ddots&\ddots\end{bmatrix},$
(3)

where $s_{A}$ denotes the average successful update probability per timeslot,
which is defined as the probability that a $\mathrm{T_{AoI}}$ is delivered
successfully to its corresponding receiver node. The row vector
$\boldsymbol{\pi_{AoI}}=[\grave{\pi}_{1},\grave{\pi}_{2},......,\grave{\pi}_{l-1},\grave{\pi}_{l},...]$
represents the steady-state probability vector of the DTMC in Fig. 2, where
$\grave{\pi}_{l}=\lim_{t\to+\infty}\mathds{P}(X_{t})=l$ denotes the
probability that the AoI is equal to $n$ at the steady state.

Using the set of equations
$\boldsymbol{\pi_{AoI}}\mathbf{P_{D}}=\boldsymbol{\pi_{AoI}}$ and
$\sum_{l}\pi_{l}=1$, $\pi_{l}$ is obtained as

$\grave{\pi_{l}}=s_{A}(1-s_{A})^{(l-1)},\,\forall l.$ (4)

Accordingly, the average AoI $\overline{\Delta}$ is calculated as

$\begin{split}\overline{\Delta}&=\sum_{l=1}^{\infty}\pi_{l}l=\sum_{l=1}^{\infty}ls_{A}(1-s_{A})^{(l-1)}\\\
&=\frac{s_{A}}{1-s_{A}}\sum_{l=1}^{\infty}l(1-s_{A})^{l}.\end{split}$ (5)

Since we have $s_{A}<1$, (5) can be rewritten as

$\overline{\Delta}=\frac{s_{A}}{1-s_{A}}\frac{1-s_{A}}{s_{A}^{2}}=\frac{1}{s_{A}}.$
(6)

However, the average AoI cannot account for extreme AoI events occurring with
very low probabilities. As mentioned in Section I, the received updates of the
$\mathrm{T_{AoI}}$ traffic are used for control and actuation actions, which
implies certain requirements on tolerated values of AoI. In that context, we
also analyze the AoI violation probability, which is the probability that the
AoI exceeds a certain constraint. Denote $c$ as the target AoI constraint,
then we can express the AoI violation probability as

$\begin{split}&\mathrm{Pr}(\Delta>c)=1-\mathrm{Pr}(\Delta\leq
c)=1-\sum_{l=1}^{c}\pi_{l}\\\
&=1-\frac{s_{A}}{1-s_{A}}\sum_{l=1}^{c}(1-s_{A})^{l}\overset{(b)}{=}(1-s_{A})^{c},\end{split}$
(7)

where (b) is obtained by using $\sum_{i=0}^{y}x^{i}=\frac{x^{y+1}-1}{x-1}$
with $x\neq 1$.

We define the average delay $D_{\text{avg}}$ of a $\mathrm{T_{D}}$ packet as
the time elapsed from the instant the packet was generated until it is
successfully delivered to its destination. Therefore, $D_{\text{avg}}$
consists of the queueing time of the packet $\mathrm{T_{D}}$ packet and its
transmission time. According to Little’s law [40], the queuing time is
calculated as $\frac{Q_{\text{avg}}}{\lambda_{D}}$, where $Q_{\text{avg}}$
denotes the average queue size of the $\mathrm{T_{D}}$ source. Then, the
average delay $D_{\text{avg}}$ can be expressed as

$D_{\text{avg}}=\frac{Q_{\text{avg}}}{\lambda_{D}}+\frac{1}{s_{D}},$ (8)

where $s_{D}$ is the average probability of a successful transmission of a
$\mathrm{T_{D}}$ packet per timeslot (average service probability), which is
inversely proportional to the transmission time component of $D_{\text{avg}}$
[41]. As we can see from (6) and (8), $\overline{\Delta}$ and $D_{\text{avg}}$
depend on the values $s_{A}$, $s_{D}$ and $Q_{\text{avg}}$. In the next
sections, we derive the expressions of $s_{A}$, $s_{D}$ and $Q_{\text{avg}}$
based on our proposed distributed backlog-aware channel access.

##  IV Analysis of the Proposed Distributed Backlog-Aware Channel Access

In the conventional slotted Aloha-based D2D communication, all nodes compete
blindly and equally to transmit their data to their respective receivers via
the shared channel. For a network with dense $\mathrm{T_{AoI}}$ links (which
is the scenario of our considered network model), this means that the
$\mathrm{T_{D}}$ node will have a low chance to gain access to the channel.
This in turn results in a congestion problem where the buffered
$\mathrm{T_{D}}$ packets suffers from an extended queuing time and a poor
delay performance. However, such performance would be unacceptable in IIoT
applications where safety-critical events ($\mathrm{T_{D}}$ traffic), such as
fire detection or gas leakage, must be reported with very low delay in order
to perform the appropriate actuation in response.

On the other hand, low AoI is required for the $\mathrm{T_{AoI}}$ traffic in
order maintain fresh knowledge of the underlying industrial process. In that
case, it is inefficient to force the $\mathrm{T_{AoI}}$ nodes to remain
totally silent when the $\mathrm{T_{D}}$ node is active, in particular as the
channel offers a possibility for capture effect to take place. In this regard,
we propose a backlog-aware channel access to achieve a proper delay-AoI
tradeoff in dense IIoT networks. In our proposed scheme, the $\mathrm{T_{D}}$
node transmits a packet in each timeslot as long as it has a non-empty queue,
while the $\mathrm{T_{AoI}}$ nodes access the channel in an opportunistic,
slotted ALOHA fashion, with the slote-access probability optimized according
to the status of the queue size of the $\mathrm{T_{D}}$ node. Denoting $Q$ as
the random variable representing the queue size of the $\mathrm{T_{D}}$ node,
the introduced access model is defined by the following cases.

### Case 1: Empty Queue ($Q=0$)

When the queue of the $\mathrm{T_{D}}$ node is empty, i.e $Q=0$, the
$\mathrm{T_{AoI}}$ nodes access the channel with a probability $p_{1}$. Let
$\phi_{A_{0}}$ represent the locations of the active $\mathrm{T_{AoI}}$
transmitters when the $\mathrm{T_{D}}$ node is silent. Leveraging the thinning
property of Poisson processes [42], $\phi_{A_{0}}$ follows a homogeneous PPP
with intensity $p_{1}\lambda_{A}$ and we have $\mathcal{K}=\phi_{A_{0}}$.
According to Slivnyak’s theorem [42] and without loss of generality, the
following analysis focuses on a typical active $\mathrm{T_{AoI}}$ pair. For
the considered channel model and conditioned on the spatial realization
$\phi_{A_{0}}$, the successful decoding probability $p_{A_{0}}$ at the tagged
$\mathrm{T_{AoI}}$ receiver when $Q=0$ is obtained as

$\begin{split}p_{A_{0}}&=\mathrm{Pr(SINR}_{i}>\beta\,|\,\mathcal{K}=\phi_{A_{0}})\\\
&=\mathrm{Pr}\left(\frac{P_{2}|h_{i,i}|^{2}d_{A}^{-\alpha}}{\sigma^{2}+\sum_{j\in\phi_{A_{0}}\setminus\\{i\\}}P_{2}|h_{j,i}|^{2}d_{j,i}^{-\alpha}}>\beta\right)\\\
&\overset{(a)}{=}\mathrm{exp}\left(-\frac{\pi
p_{1}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right),\end{split}$
(9)

where $P_{2}$ is the transmitting power of the $\mathrm{T_{AoI}}$ nodes. The
result in ($a$) follows by leveraging the Rayleigh distribution of the channel
gain, i.e., $|h_{j,i}|^{2}\sim\mathrm{exp(1)}$ and the the probability
generating functional (PGFL) of the PPP [35].

### Case 2: Moderate Backlog ($1\leq Q\leq M)$

We denote $M$ as the backlog threshold that will determine the activity limit
for the $\mathrm{T_{AoI}}$ nodes. When the queue is non-empty, the
$\mathrm{T_{D}}$ node transmits in each timeslot, while the $\mathrm{T_{AoI}}$
nodes attempt to access the channel with a probability $p_{2}$. We assume that
$p_{2}<p_{1}$ , as the $\mathrm{T_{AoI}}$ nodes should cause, as well as
experience a lower interference level on the channel. Let $x_{D}$ represent
the location of the $\mathrm{T_{D}}$ node and $\phi_{A_{1}}$ denote the
locations of the active $\mathrm{T_{AoI}}$ nodes that follow a homogeneous PPP
with intensity $p_{2}\lambda_{A}$. Then we have
$\mathcal{K}=\\{x_{D}\cup\phi_{A_{1}}\\}$. In this case, the successful
decoding probability $p_{D_{1}}$ at the $\mathrm{T_{D}}$ receiver, conditioned
on the spatial realization $\phi_{A_{1}}$, is given as

$\begin{split}p_{D_{1}}&=\mathrm{Pr(SINR}_{D}>\beta\,|\,\mathcal{K}=\\{x_{D}\cup\phi_{A_{1}})\\\
&=\mathrm{Pr}\left(\frac{P_{1}|h_{D,D}|^{2}d_{D}^{-\alpha}}{\sigma^{2}+\sum_{j\in\phi_{A_{1}}}P_{2}|h_{j,D}|^{2}d_{j,D}^{-\alpha}}>\beta\right)\\\
&=\mathrm{exp}\left(-\frac{\pi
p_{2}\lambda_{A}d_{D}^{2}\left(\beta\frac{P_{2}}{P_{1}}\right)^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{D}^{\alpha}}{P_{1}}\right).\end{split}$
(10)

For the active $\mathrm{T_{AoI}}$ nodes, we derive the successful decoding
probability at an arbitrary $\mathrm{T_{AoI}}$ receiver using the following
proposition.

Proposition 1: When $1\leq Q\leq M$, the successful decoding probability
$p_{A_{1}}$ at an arbitrary $\mathrm{T_{AoI}}$ node is given by

$p_{A_{1}}=\mathrm{exp}\left(-\frac{\pi
p_{2}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\frac{\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{2}}{P_{2}}\right)}{1+\frac{d_{A}^{2}}{\mathbb{E}[d_{D,i}]^{2}}\left(\beta\frac{P_{1}}{P_{2}}\right)^{\frac{2}{\alpha}}}$
(11)

where we have

$\operatorname{\mathbb{E}}\left[d_{D,i}\right]=\int_{0}^{2\pi}\frac{1}{2\pi}\int_{0}^{R}\frac{2r}{R^{2}}\sqrt{r^{2}+d_{D,0}^{2}-2d_{0}d_{D,0}\cos{\theta}}\mathrm{d}r\mathrm{d}\theta.$

Proof: See Appendix A.

### Case 3: Congested Queue ($Q>M$)

When the queue size of the $\mathrm{T_{D}}$ node exceeds the backlog threshold
$M$, i.e., $Q>M$, then all the $\mathrm{T_{AoI}}$ nodes remain silent and only
the $\mathrm{T_{D}}$ node transmits. In this case, the successful decoding
probability $p_{D_{0}}$ at the $\mathrm{T_{D}}$ receiver is given by

$\begin{split}p_{D_{0}}&=\mathrm{Pr(SINR}_{D}>\beta\,|\,\mathcal{K}=x_{D})=\mathrm{Pr}\left(\frac{P_{1}|h_{D,D}|^{2}d_{D}^{-\alpha}}{\sigma^{2}}>\beta\right)\\\
&=\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{D}^{\alpha}}{P_{1}}\right).\end{split}$
(12)

Note that $p_{D_{0}}>p_{D_{1}}$. Exploiting (9), (10), (11), and (12), we can
express $s_{A}$ and $s_{D}$ as follows

$\begin{split}s_{A}&=p_{1}\mathrm{Pr(SINR}_{i\in\phi_{A_{0}}}>\beta\,|\,Q=0)\,\mathrm{Pr(Q=0)}\\\
&+p_{2}\mathrm{Pr(SINR}_{i\in\phi_{A_{1}}}>\beta\,|\,1\leq Q\leq
M)\,\mathrm{Pr(1\leq Q\leq M)}\\\
&=p_{1}p_{A_{0}}\,\mathrm{Pr(Q=0)}+p_{2}p_{A_{1}}\mathrm{Pr(1\leq Q\leq
M)}.\end{split}$ (13)

$s_{D}=\frac{\mathrm{Pr(1\leq Q\leq
M)}p_{D_{1}}+\mathrm{Pr(Q>M)}p_{D_{0}}}{\mathrm{Pr(1\leq Q\leq
M)}+\mathrm{Pr(Q>M)}}.$ (14)

As indicated by (13) and (14), both $s_{A}$ and $s_{D}$ depend on the
distribution of the random variable $Q$, which is analyzed in the next
section.

Figure 3: DTMC model of the queue at the $\mathrm{T_{D}}$ node.

##  V The Distribution of the Queue Size $Q$

In this section, we analyze the distribution of the queue size $Q$ to derive
$\mathrm{Pr}(Q=0)$, $\mathrm{Pr}(1\leq Q\leq M)$, and $\mathrm{Pr}(Q>M)$. The
evaluation of the queue size $Q$ can be modelled by the DTMC shown in Fig. 3,
where each state represents the queue size. The success probability is
$p_{D_{1}}$ when $1\leq Q\leq M$, while it is $p_{D_{0}}$ when $Q>M$.

Let $\boldsymbol{\pi_{D}}=\\{\pi_{0},\pi_{1},.....\\}$ represent the steady-
state probability vector of the queue size of the $\mathrm{T_{D}}$ node where
$\pi_{n}=\lim_{t\to\infty}\mathrm{Pr}(Q=n)$ is the steady-state probability of
having $n$ packets at the queue. The stationary distribution of the queue size
of the $\mathrm{T_{D}}$ node is given by the following lemma:

Lemma 1: According to the DTMC in Fig. 3, the steady-state probability
$\pi_{n}$ is given as

$\pi_{n}=\begin{cases}\frac{\lambda_{D}^{n}(1-p_{D_{1}})^{(n-1)}}{p_{D_{1}}^{n}(1-\lambda_{D})^{n}}\pi_{0},&1\leq
n\leq M\\\
\frac{\lambda_{D}^{n}(1-p_{D_{1}})^{M}(1-p_{D_{0}})^{(n-M-1)}}{p_{D_{1}}^{M}p_{D_{0}}^{(n-M)}(1-\lambda_{D})^{n}}\pi_{0},&n>M\end{cases}$
(15)

where $\pi_{0}$ is given as

$\pi_{0}=\begin{cases}\frac{(p_{D_{1}}-\lambda_{D})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right]^{M}(p_{D_{0}}-p_{D_{1}})}&\lambda_{D}\neq
p_{D_{1}}\\\
\frac{p_{D_{0}}-p_{D_{1}}}{p_{D_{1}}+(p_{D_{0}}-p_{D_{1}})\frac{M+1-p_{D_{1}}}{1-p_{D_{1}}}}&\lambda_{D}=p_{D_{1}}\end{cases}$
(16)

Proof: See Appendix B.

Note that the queue of the $\mathrm{T_{D}}$ node is stable under the condition
that $\lambda_{D}<p_{D_{0}}$ (See Appendix C).

Based on the results of the steady-state probabilities in Lemma 1, the
distribution of the queue size $Q$ can be given as follows:

Lemma 2: For a stable queue ($\lambda_{D}<p_{D_{0}}$), and if $\lambda_{D}\neq
p_{D_{1}}$333Our derived expressions also hold for $\lambda_{D}=p_{D_{1}}$ by
substituting $\pi_{0}$ with its corresponding expression in (16)., then the
steady-state probabilities for the queue size are

$\mathrm{Pr}(1\leq Q\leq
M)=\frac{\lambda_{D}(1-\psi^{M})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})},$
(17)

and we also obtain

$\mathrm{Pr}(Q>M)=\frac{\lambda_{D}\psi^{M}(p_{D_{1}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})},$
(18)

where we have

$\psi=\left(\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right).$

Proof: See Appendix D.

Based on the results obtained from Lemma 1 and Lemma 2, the average queue size
$Q_{\text{avg}}$ can be obtained from the following theorem:

Theorem 1: The average queue size of the $\mathrm{T_{D}}$ node is given by

$Q_{\text{avg}}=\sum_{n=1}^{\infty}n\pi_{n}=\frac{Q_{1}+Q_{2}}{p_{D_{0}}p_{D_{1}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})},$
(19)

where we have

$\begin{split}Q_{1}&=\lambda_{D}(1-\lambda_{D})p_{D_{1}}\frac{p_{D_{0}-\lambda_{D}}}{p_{D_{1}}-\lambda_{D}}\\\
&.\left(M\psi^{(M+1)}-\psi^{M}(M+1)+1\right),\\\
Q_{2}&=\psi^{M}\lambda_{D}(p_{D_{1}}-\lambda_{D})\left(M+\frac{p_{D_{0}}(1-\lambda_{D})}{p_{D_{0}}-\lambda_{D}}\right).\\\
\end{split}$ (20)

Proof: See Appendix E

$\displaystyle\begin{split}s_{A}=&p_{1}\mathrm{exp}\left(-\frac{\pi
p_{1}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right)\left(\frac{(p_{D_{1}}-\lambda_{D})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right]^{M}(p_{D_{0}}-p_{D_{1}})}\right)\\\
+&p_{2}\mathrm{exp}\left(-\frac{\pi
p_{2}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\frac{\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{2}}{P_{2}}\right)}{1+\frac{d_{A}^{2}}{\mathbb{E}[d_{D,i}]^{2}}\left(\beta\frac{P_{1}}{P_{2}}\right)^{\frac{2}{\alpha}}}\left(\frac{\lambda_{D}(1-\psi^{M})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\right).\end{split}$
(21)

$\displaystyle\begin{split}s_{D}&=\frac{\mathrm{exp}\left(-\frac{\pi
p_{2}\lambda_{A}d_{D}^{2}\left(\beta\frac{P_{2}}{P_{1}}\right)^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{D}^{\alpha}}{P_{1}}\right)\left(\frac{\lambda_{D}(1-\psi^{M})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\right)}{1-\left(\frac{(p_{D_{1}}-\lambda_{D})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right]^{M}(p_{D_{0}}-p_{D_{1}})}\right)}\\\
&+\frac{\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{D}^{\alpha}}{P_{1}}\right)\left(\frac{\lambda_{D}\psi^{M}(p_{D_{1}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\right)}{1-\left(\frac{(p_{D_{1}}-\lambda_{D})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right]^{M}(p_{D_{0}}-p_{D_{1}})}\right)}\end{split}$
(22)

Hence, based on (17) and (18), we can obtain $s_{A}$ and $s_{D}$ as shown in
(21) and (22). Then, by substituting $s_{A}$ in (6), we can obtain $\Delta$.
Similarly, by substituting $s_{D}$ and $Q_{avg}$ in (8), we can obtain
$D_{avg}$.

As we can see from (13) and (6), when the $\mathrm{T_{D}}$ link is inactive,
i.e., $Q=0$, the age of the $\mathrm{T_{AoI}}$ nodes mainly depends on the
access probability $p_{1}$ for a given value of the transmission power
$P_{2}$. In that case, if we consider $\Delta$ as a function of $p_{1}$, then
there is an optimal value $p_{1}^{*}$ that minimizes the average age $\Delta$,
which is equivalent to
$p_{1}^{*}=\underset{p_{1}\in[0,1]}{\mathrm{arg\,max}}\,\,s_{A}$. So we have
the following optimization problem

$\begin{split}&\underset{p_{1}}{\mathrm{min}}\,\,\,p_{1}\mathrm{exp}\left(-\frac{\pi
p_{1}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right)\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right)\\\
&\mathrm{subject\,to}\,\,\,\,\,0<p_{1}\leq 1.\end{split}$ (23)

The optimization problem in (23) can be solved via the first-order optimality
condition [43], which yields

$\begin{split}1-&p_{1}\frac{\pi\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}=0\\\
&\Rightarrow
p_{1}=\frac{\mathrm{sinc}(\frac{2}{\alpha})}{\pi\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}\\\
&p_{1}^{*}=\mathrm{min\,\left\\{\frac{\mathrm{sinc}(\frac{2}{\alpha})}{\pi\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}},1\right\\}}.\end{split}$
(24)

By setting the obtained optimal access probability $p_{1}^{*}$ in (13), and by
fixing the arrival rate $\lambda_{D}$ and the transmission power $P_{1}$, we
can see that the average AoI $\Delta$ mainly depends on the access probability
$p_{2}$ and the transmission power $P_{2}$.

Remark 1: The average delay $D_{avg}$ is an independent function of $p_{1}$
given a certain value of $\lambda_{D}$. On the other hand, increasing $p_{2}$
and/or $P_{2}$ would in turn improve $\Delta$ for the $\mathrm{T_{AoI}}$ nodes
but at the cost of increased delay $D_{avg}$ for the $\mathrm{T_{D}}$ node due
to the increased level of interference. This is because the probability
$p_{D_{1}}$ is decreased accordingly leading to increased queuing time at the
$\mathrm{T_{D}}$ node. Thus, the average delay $D_{avg}$ is an increasing
function of $p_{2}$ and $P_{2}$.

For a given delay constraint $D_{max}$ of the $\mathrm{T_{D}}$ traffic, there
exists an optimal pair $(p_{2}^{*},P_{2}^{*})$ that minimizes the average AoI
$\Delta$. Considering the delay constraint $D_{max}$ and the queue stability
condition ($\lambda_{D}<p_{D_{0}}$) of the $\mathrm{T_{D}}$ node, the optimal
pair $(p_{2}^{*},P_{2}^{*})$ lies within a feasible region $F_{R}$ that can
defined as

$F_{R}=\left\\{(p_{2},P_{2})\,:\,D_{avg}<D_{max},\,\lambda_{D}<p_{D_{0}}\right\\}.$
(25)

In order to find the optimal pair $(p_{2}^{*},P_{2}^{*})$, we formulate the
following optimization problem

$\begin{split}\underset{(p_{2},P_{2})}{\mathrm{min}}&\,\,\,\Delta\\\
\mathrm{subject\,to}\,\,\,\,&\,D_{avg}(p_{2},P_{2})<D_{max}\\\ &0<p_{2}\leq
1\\\ &0<P_{2}\leq P_{max},\end{split}$ (26)

where $P_{max}$ denotes the maximum transmission power of the
$\mathrm{T_{AoI}}$ nodes. Deriving closed form solutions for the optimal
values $(p_{2}^{*},P_{2}^{*})$ is a hard problem due to the complexity of the
corresponding expressions of $s_{D}$, $s_{A}$ and $D_{avg}$. Therefore, in the
next section we do a numerical analysis of the solution to the optimization
problem in (26).

##  VI Results and Discussion

TABLE I: Evaluation parameters. Parameter | Value
---|---
$\lambda_{A}$ | $2\times 10^{-4}$
$d_{D}$ |  $100$ m
$d_{A}$ |  $50$ m
$R$ |  $300$ m
$\alpha$ | $3$
$\beta$ |  $0$ dB
$\sigma^{2}$ |  $-90$ dBm
$P_{1}$ |  $100$ mW
$P_{max}$ |  $0.02$ mW
$D_{max}$ |  $5$ time slots

In this section, we present numerical results to evaluate the performance of
the proposed backlog-aware method in terms of the average delay ($D_{avg}$)
and the average AoI ($\Delta$). We also introduce numerical solutions to the
optimization problem (26) under different values of $\lambda_{D}$ and $M$.
Unless stated otherwise, we adopt the evaluation parameters described in Table
I.

Figure 4: Evaluation of the successful decoding probabilities with varying
$p_{1}$ and $p_{2}$ with $P_{2}=0.01$ mW.  Figure 5: The average AoI
performance versus $p_{2}$ under different values of $\lambda_{D}$ and $M$
with $P_{2}=0.01$ mW.

Fig. 4 shows the successful decoding probabilities in (9), (10), (11) and (12)
as a function of the access probabilities $p_{1}$ and $p_{2}$. The value of
$p_{D0}$ is independent of the access probabilities $p_{1}$ and $p_{2}$, thus
showing a constant value of 0.996. The successful decoding probability
$p_{A0}$ depends only on $p_{1}$ while the values of $p_{D1}$ and $p_{A1}$
depend on $p_{2}$. Increasing $p_{1}$ implies higher collision probability
between the $\mathrm{T_{AoI}}$ nodes, hence decreasing $p_{A0}$. Also, when
the $\mathrm{T_{D}}$ node is active ($1\leq Q\leq M$), increasing $p_{1}$
would in turn decrease both $p_{A1}$ and $p_{D1}$. In order to satisfy the
queue stability condition $\lambda_{D}<p_{D_{0}}$, we utilize
$\lambda_{D}<0.996$ for all the following results.

Fig. 5 presents the average AoI $\Delta$ versus the access probability $p_{2}$
under different values of $\lambda_{D}\in\\{0.2,0.6\\}$ and $M\in\\{1,3\\}$.
Besides the numerical results obtained from (6), we also plot the simulation
results obtained via discrete-time simulations in MATLAB to support our
analytical analysis. Each simulation run lasts for $10^{4}$ timeslots, and we
plot the average results over 100 runs. First, we can see that the simulation
results match well with the numerical results, which validates our analytical
analysis presented in Section *****. Moreover, we note that $\Delta$ is not a
monotonic function of $p_{2}$. At very low values of $p_{2}$, the tagged
$\mathrm{T_{AoI}}$ node is likely to be inactive for excessive period pf time.
For a certain values of $\lambda_{D}$ and $M$, the average AoI first decreases
as $p_{2}$ increases where the tagged $\mathrm{T_{AoI}}$ node tends to
transmit updates to its corresponding actuator more frequently. However, after
a certain value of $p_{2}$, the average AoI at the corresponding actuator
starts to significantly increase due to consecutive failed transmission
attempts from the tagged source as a result of increased level of contention
among the $\mathrm{T_{AoI}}$ nodes. We also observe that increasing $M$ would
improve the AoI as it implies a higher chance for the $\mathrm{T_{AoI}}$
source to be active. For the same $M$, higher $\lambda_{D}$ degrades the AoI
performance as the probability $\mathrm{Pr(Q>M)}$ increases, which implies
that the $\mathrm{T_{AoI}}$ nodes remain silent (Case 3 in Section IV).

Figure 6: The average delay performance versus $p_{2}$ under different values
of $\lambda_{D}$ and $M$ with $P_{2}=0.01$ mW.

Fig. 6 shows the evaluation of the average delay $D_{avg}$ against $p_{2}$
under different values of $\lambda_{D}\in\\{0.2,0.6,0.8\\}$ and
$M\in\\{1,3,9\\}$. For a certain $M$, we can observe that the average delay
increases rapidly as $p_{2}$ increases and tends to saturate at a certain
pointat high values of $\lambda_{D}$, which can be explained as follows.
Increasing $p_{2}$ will in turn decrease $p_{D1}$ due to the increased
interference from the $\mathrm{T_{AoI}}$ nodes, which leads to extended
average queueing time at the $\mathrm{T_{D}}$ node. When $\lambda_{D}$ is
high, it is likely that the $\mathrm{T_{AoI}}$ nodes remain inactive as the
probability $\mathrm{Pr(Q>M)}$ is high, hence the $\mathrm{T_{D}}$ node would
probably have interference-free transmission with a stable delay. This also
explains the counter-intuitive observation in Fig. 6 where the
$\mathrm{T_{D}}$ node may experience a lower delay at high value of
$\lambda_{D}$ than the one at low value of $\lambda_{D}$. The figure also
shows that, for the same $\lambda_{D}$, increasing $M$ degrades the average
delay, because the backlog-aware strategy becomes weaker leading to lower
$s_{D}$ and higher $D_{avg}$ accordingly.

Figure 7: The feasible region $F_{R}$ under different values of $\lambda_{D}$
and $M$.

Fig. 7 plots the boundary of the feasible region $F_{R}$ defined in (25) with
$\lambda_{D}\in\\{0.2,0.6\\}$ and $M\in\\{1,3\\}$. The area below each curve
comprises the possible pairs $(p_{2},P_{2})$ that solve the optimization
problem in (26) and satisfy the delay constraint $D_{max}$. As we can see from
the figure, higher $M$ implies narrower $F_{R}$ where the average delay
becomes higher, hence narrowing down the feasible region of $(p_{2},P_{2})$
that satisfies the delay constraint $D_{max}$.

TABLE II: Numerical solutions to the optimization problem in (26). $\lambda_{D}$ | $M$ | $p_{2}^{*}$ | $P_{2}^{*}(mW)$ | $\Delta^{*}$
---|---|---|---|---
0.2 | 1 | 0.275 | 0.0122 | 8.22
3 | 0.303 | 0.0185 | 6.65
6 | 0.282 | 0.0161 | 6.01
0.6 | 1 | 0.145 | 0.0091 | 13.28
3 | 0.192 | 0.0132 | 11.22
6 | 0.184 | 0.0121 | 10.08
0.8 | 1 | 0.125 | 0.0063 | 16.81
3 | 0.132 | 0.0072 | 15.01
6 | 0.108 | 0.0055 | 14.09

In Table II, we present the numerical solutions ($p_{2}^{*}$, $P_{2}^{*}$) to
the optimization problem in (26) under different values of $\lambda_{D}$ and
$M$ and give the minimum achievable average AoI $\Delta^{*}$. The optimal
solutions presented in this table are obtained via a 2-D numerical exhaustive
search. As we can see, for the same value of $\lambda_{D}$, the minimum
average AoI ($\Delta^{*}$) can be improved by increasing $M$ as the
probability that the $\mathrm{T_{AoI}}$ node to be active is increased
accordingly, i.e., higher $p_{2}^{*}$. However, higher $M$ increases the
average delay as shown by Fig. 6, and tighten the feasible region $F_{R}$ in
order to satisfy the constraint $D_{max}$. For instance, when
$\lambda_{D}=0.2$, increasing $M$ from 1 to 3 would improve $\Delta^{*}$ when
the $p_{2}^{*}$ is increased from 0.275 to 0.303. However, increasing $M$
further to 6 would imply decreasing $p_{2}^{*}$ to 0.282 in order to keep
$p_{D1}$ sufficiently high to comply with the delay constraint of $D_{max}=5$
timeslots. In addition, for a given value of $M$, the optimal access
probability $p_{2}^{*}$ and the optimal transmission power $P_{2}^{*}$ are
lower for higher $\lambda_{D}$ in order to reduce the interference from the
$\mathrm{T_{AoI}}$ nodes and satisfy $D_{max}$.

##  VII Conclusion and Future Work

In this paper, the inherent interplay between delay and AoI is investigated
for a D2D-based heterogeneous IIoT network. We proposed a distributed backlog-
aware access protocol and developed closed-form expressions for the average
AoI and average delay using stochastic geometry and queuing theory. We
evaluated the network performance using numerical results under different
parameters and obtained optimal solutions that minimize the AoI via 2-D
exhaustive numerical search. As a future work, this work can be extended to
adopt queue management approaches and adaptive retransmission to introduce
extra enhancement to the AoI. Moreover, we consider using machine learning-
based techniques to set the optimal network parameters to adapt to the dynamic
industrial environment.

## Acknowledgement

This paper has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No. 883315.

###  -A Proof of Proposition 1

Figure 8: Geometric representation of the distribution of the $\mathrm{T_{D}}$
transmitter and an arbitrary $\mathrm{T_{AoI}}$ receiver.

The successful update probability $p_{A_{1}}$ of an arbitrary
$\mathrm{T_{AoI}}$ pair when the $\mathrm{T_{D}}$ node is active is given as

$\begin{split}&p_{A_{1}}=\mathrm{Pr(SINR}_{i}>\beta\,|\,\mathcal{K}=\phi_{A_{1}}\cup
x_{D})\\\
&=\mathrm{Pr}\left(\frac{P_{2}|h_{i,i}|^{2}d_{A}^{-\alpha}}{\sigma^{2}+\sum_{j\in\phi_{A_{1}}\setminus\\{i\\}}P_{2}|h_{j,i}|^{2}d_{j,i}^{-\alpha}+P_{1}|h_{D,i}|^{2}d_{D,i}^{-\alpha}}>\beta\right).\end{split}$
(27)

Since we have $|h_{i,i}|^{2}\sim\mathrm{exp(1)}$ and
$|h_{D,i}|^{2}\sim\mathrm{exp(1)}$, so (27) can be rewritten as

$\begin{split}p_{A_{1}}&=\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right)\operatorname{\mathbb{E}}\left[\int_{0}^{\infty}\mathrm{exp}\left(-\frac{P_{1}}{P_{2}}d_{D,i}^{-\alpha}x\right)e^{-x}\mathrm{d}x\right]\\\
&\cdot\underbrace{\operatorname{\mathbb{E}}\left[\mathrm{exp}\left(-s\sum_{j\in\phi_{A_{1}}\setminus\\{i\\}}|h_{j,i}|^{2}d_{j,i}^{-\alpha}\right)\right]}_{L_{S}}\\\
&=\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right)\operatorname{\mathbb{E}}\left[\frac{1}{1+\frac{P_{1}}{P_{2}}\beta
d_{A}^{\alpha}d_{D,i}^{-\alpha}}\right]\\\
&\cdot\underbrace{\operatorname{\mathbb{E}}\left[\mathrm{exp}\left(-s\sum_{j\in\phi_{A_{1}}\setminus\\{i\\}}|h_{j,i}|^{2}d^{-\alpha}_{j,i}\right)\right]}_{L_{S}},\end{split}$
(28)

where the term $L_{S}$ denotes the Laplace transform of the interference
component of the nodes in $\phi_{A_{1}}$ which can be expressed as [44]

$\operatorname{\mathbb{E}}\left[\mathrm{exp}\left(-s\sum_{j\in\phi_{A_{1}}\setminus\\{i\\}}|h_{j,i}|^{2}d^{-\alpha}_{j,i}\right)\right]=\mathrm{exp}\left[-\frac{\pi
p_{2}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right].$
(29)

Based on [45], we can also have

$\operatorname{\mathbb{E}}\left[\frac{1}{1+\frac{P_{1}}{P_{2}}\beta
d_{A}^{\alpha}d_{D,i}^{-\alpha}}\right]\simeq\frac{1}{1+\frac{d_{A}^{2}}{\operatorname{\mathbb{E}}\left[d_{D,i}\right]^{2}}\left(\beta\frac{P_{1}}{P_{2}}\right)^{\frac{2}{\alpha}}}.$
(30)

Then, with (29) and (30), $p_{A_{1}}$ can be expressed as

$p_{A_{1}}\simeq\mathrm{exp}\left[-\frac{\pi
p_{2}\lambda_{A}d_{A}^{2}\beta^{\frac{2}{\alpha}}}{\mathrm{sinc}(\frac{2}{\alpha})}\right]\frac{\mathrm{exp}\left(-\frac{\beta\sigma^{2}d_{A}^{\alpha}}{P_{2}}\right)}{1+\frac{d_{A}^{2}}{\operatorname{\mathbb{E}}\left[d_{D,i}\right]^{2}}\left(\beta\frac{P_{1}}{P_{2}}\right)^{\frac{2}{\alpha}}}$
(31)

The level of interference experienced by an arbitrary $\mathrm{T_{AoI}}$
receiver from the $\mathrm{T_{D}}$ transmitter depends on the distance
$d_{D,i}$. Therefore, the term $\operatorname{\mathbb{E}}\left[d_{D,i}\right]$
is calculated by taking the expectation over all the possible locations of
that $\mathrm{T_{AoI}}$ receiver within the considered region $\mathcal{C}$.
Since the locations of the $\mathrm{T_{AoI}}$ transmitters follow a
homogeneous PPP model, we can consider that the locations of the
$\mathrm{T_{AoI}}$ receivers approximately follow a uniform distribution in
the region $\mathcal{C}$ with a radius $R$. Accordingly, the distance $d_{0}$
of an arbitrary $\mathrm{T_{AoI}}$ receiver from origin of $\mathcal{C}$
follow the probability distribution function $f_{d_{0}}(r)$ which is given by

$f_{d_{0}}(r)=\begin{cases}\frac{2r}{R^{2}}&\mathrm{if}\,0\leq r\leq R\\\
0&\mathrm{otherwise.}\end{cases}$ (32)

If we consider the region $\mathcal{C}$ depicted in Fig. 8, then we have

$d_{D,i}=\sqrt{d_{0}^{2}+d_{D,0}^{2}-2d_{0}d_{D,0}\cos{\theta}},$ (33)

where $d_{D,0}$ is the distance from the $\mathrm{T_{D}}$ transmitter to the
origin of the region $\mathcal{C}$, and $\theta\in\\{0,2\pi\\}$ is a uniformly
distributed random variable. Then, the expectation
$\operatorname{\mathbb{E}}\left[d_{D,i}\right]$ can be obtained as

$\operatorname{\mathbb{E}}\left[d_{D,i}\right]=\int_{0}^{2\pi}\frac{1}{2\pi}\int_{0}^{R}\frac{2r}{R^{2}}\sqrt{r^{2}+d_{D,0}^{2}-2d_{0}d_{D,0}\cos{\theta}}\mathrm{d}r\mathrm{d}\theta.$
(34)

Based on (31) and (34), we can obtain the probability $p_{A_{1}}$, which
concludes the proof.

###  -B Proof of Lemma 1

The balance equations for the DTMC in Fig. 3 can be obtained as

$\begin{split}\lambda_{D}\pi_{0}=(1-\lambda_{D})p_{D_{1}}\pi_{1}&\Leftrightarrow\pi_{1}=\pi_{0}\frac{\lambda_{D}}{(1-\lambda_{D})p_{D_{1}}}\\\
\pi_{1}\left(\lambda_{D}(1-p_{D_{1}})+(1-\lambda_{D})p_{D_{1}}\right)&=\lambda_{D}\pi_{0}+\pi_{2}p_{D_{1}}(1-\lambda_{D})\\\
\Leftrightarrow&\pi_{2}=\pi_{0}\frac{\lambda_{D}^{2}(1-p_{D_{1}})}{(1-\lambda_{D})^{2}p_{D_{1}}^{2}}.\end{split}$
(35)

Hence, for $1\leq n\leq M$ we have

$\pi_{n}=\frac{\lambda_{D}^{n}(1-p_{D_{1}})^{(n-1)}}{p_{D_{1}}^{n}(1-\lambda_{D})^{n}}\pi_{0},$
(36)

while for $n>M$ we get

$\pi_{n}=\frac{\lambda_{D}^{n}(1-p_{D_{1}})^{M}(1-p_{D_{0}})^{(n-M-1)}}{p_{D_{1}}^{M}p_{D_{0}}^{(n-M)}(1-\lambda_{D})^{n}}\pi_{0}$
(37)

Since we have $\sum_{n=0}^{\infty}\pi_{n}=1$, with $\lambda_{D}\neq
p_{D_{1}}$, the empty queue probability $\pi_{0}$ is expressed as

$\pi_{0}=\frac{(p_{D_{1}}-\lambda_{D})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right]^{M}(p_{D_{0}}-p_{D_{1}})}.$
(38)

When $\lambda_{D}=p_{D_{1}}$, (38) is no longer valid ($\pi_{0}=\frac{0}{0}$).
In this case we apply L’Hôspital’s rule and get $\pi_{0}$ as

$\pi_{0}=\lim_{\lambda_{D}\to
p_{D_{1}}}\frac{f^{\prime}(\lambda_{D})}{h^{\prime}(\lambda_{D})}=\frac{p_{D_{0}}-p_{D_{1}}}{p_{D_{1}}+(p_{D_{0}}-p_{D_{1}})\frac{M+1-p_{D_{1}}}{1-p_{D_{1}}}},$
(39)

where $f(\lambda_{D})$ and $h(\lambda_{D})$ represent the nominator and
denominator of (38), respectively. Combining (38) and (39), we get (16) which
concludes the proof of Lemma 1.

###  -C Stability Condition of the $\mathrm{T_{AoI}}$ Node Queue

Our basic definition for the queue stability is based on Loynes’ theorem [46].
The theorem states that, a queue is stable if the average service probability
is greater than average arrival probability with the condition that the
arrival process and the service process are strictly jointly stationary and.
Hence, in order for the queue of the $\mathrm{T_{D}}$ to be stable, we need to
prove that the DTMC in Fig. 3 irreducible Markov chain. Then, the average
$\lambda_{D}$ in (38) must satisfy the condition $0<\pi_{0}<1$. We consider
the following cases of the values of $\lambda_{D}$ relative to the values of
$p_{D_{1}}$ and $p_{D_{0}}$:

  * •

When
$\lambda_{D}<p_{D_{1}}\Rightarrow\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}<1$,
then $h(\lambda_{D})>p_{D_{0}}(p_{D_{1}}-\lambda_{D})$. Hence, we have
$\pi_{0}<\frac{p_{D_{0}}-\lambda_{D}}{p_{D_{0}}}=1-\frac{\lambda_{D}}{p_{D_{0}}}<1$.

  * •

When $\lambda_{D}=p_{D_{1}}$, based on (39), it is obvious that $0<\pi_{0}<1$

  * •

When
$p_{D_{1}}<\lambda_{D}<p_{D_{0}}\Rightarrow\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}>1$,
then we have $f(\lambda_{D})<0$. Also, as from the following inequality

$\left(\frac{1-\lambda_{D}}{1-p_{D_{1}}}\right)^{M}\frac{p_{D_{0}}-\lambda_{D}}{p_{D_{0}}-p_{D_{1}}}<1<\left(\frac{\lambda_{D}}{p_{D_{1}}}\right)^{M+1}$
$\begin{split}&\Rightarrow
p_{D_{1}}(p_{D_{0}}-\lambda_{D})<\lambda_{D}\left(\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right)^{M}(p_{D_{0}}-p_{D_{1}})\\\
&\Rightarrow h(\lambda_{D})<0.\end{split}$

Thus, we have $\pi_{0}=\frac{f(\lambda_{D})}{h(\lambda_{D})}>0$. Since
$\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}>1$, we also have
$h(\lambda_{D})<p_{D_{0}}(p_{D_{1}}-\lambda_{D})<0$, so we obtain
$\pi_{0}<1-\frac{\lambda_{D}}{p_{D_{0}}}<1$

From the discussion of the above cases, we can see that the condition
$0<\pi_{0}<1$ is verified, and we can conclude that the queue of
$\mathrm{T_{D}}$ node is stable if $\lambda_{D}<p_{D_{0}}$.

###  -D Proof of Lemma 2

The probability $\mathrm{Pr}(1\leq Q\leq M)$ is obtained as

$\mathrm{Pr}(1\leq Q\leq M)=\sum_{n=1}^{M}\pi_{n}.$ (40)

Based on (15) and when $\lambda_{D}<p_{D_{0}}$ and $\psi\neq 1$, we get

$\begin{split}\mathrm{Pr}(1\leq Q\leq
M)&=\frac{\pi_{0}}{1-p_{D_{1}}}\sum_{n=1}^{M}\left(\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right)^{n}\\\
&=\frac{\pi_{0}}{1-p_{D_{1}}}\frac{\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}-\left(\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}\right)^{n+1}}{1-\frac{\lambda_{D}(1-p_{D_{1}})}{(1-\lambda_{D})p_{D_{1}}}}\\\
&=\frac{\pi_{0}\lambda_{D}(1-\psi^{M})}{p_{D_{1}}-\lambda_{D}}\\\
&=\frac{\lambda_{D}(1-\psi^{M})(p_{D_{0}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}.\end{split}$
(41)

For $n>M$, we obtain $\mathrm{Pr}(Q>M)$ as

$\begin{split}\mathrm{Pr}(Q>M)&=\sum_{n=M+1}^{\infty}\pi_{n}=1-\pi_{0}-\sum_{n=1}^{M}\pi_{n}\\\
&=\frac{\lambda_{D}\psi^{M}(p_{D_{1}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}.\end{split}$
(42)

###  -E Proof of Theorem 1

The average queue size $Q_{avg}$ of the $\mathrm{T_{AoI}}$ node can be given
as

$\begin{split}Q_{avg}&=\sum_{n=1}^{\infty}n\pi_{n}=\sum_{n=1}^{M}n\pi_{n}+\sum_{n=M+1}^{\infty}n\pi_{n}\\\
&=\sum_{n=1}^{M}n\pi_{n}+\sum_{n=1}^{\infty}(M+n)\pi_{M+n}\\\
&=\underbrace{\sum_{n=1}^{M}n\pi_{n}}_{R_{1}}+\underbrace{M\sum_{n=1}^{\infty}\pi_{M+n}}_{R_{2}}+\underbrace{\sum_{n=1}^{\infty}n\pi_{M+n}}_{R_{3}}\end{split}$
(43)

With $\lambda_{D}<p_{D_{0}}$ and $\psi\neq 1$, the first term $R_{1}$ can be
obtained as

$\begin{split}\sum_{n=1}^{M}n\pi_{n}&=\sum_{n=1}^{M}n\frac{\lambda_{D}^{n}(1-p_{D_{1}})^{(n-1)}}{p_{D_{1}}^{n}(1-\lambda_{D})^{n}}\pi_{0}\\\
&=\frac{\pi_{0}\lambda_{D}}{(1-\lambda_{D})p_{D_{1}}}\sum_{n=1}^{M}n\left[\frac{\lambda_{D}(1-p_{D_{1}})}{p_{D_{1}}(1-\lambda_{D})}\right]^{(n-1)}\\\
&=\frac{\pi_{0}\lambda_{D}}{(1-\lambda_{D})p_{D_{1}}}{\left(\sum_{n=1}^{M}\left[\frac{\lambda_{D}(1-p_{D_{1}})}{p_{D_{1}}(1-\lambda_{D})}\right]^{n}\right)}^{\prime}\\\
&=\frac{\pi_{0}\lambda_{D}}{(1-\lambda_{D})p_{D_{1}}}\frac{M\psi^{(M+1)}-\psi^{M}(M+1)+1}{\left(1-\frac{\lambda_{D}(1-p_{D_{1}})}{p_{D_{1}}(1-\lambda_{D})}\right)^{2}}\\\
&=\frac{\lambda_{D}(1-\lambda_{D})p_{D_{1}}\frac{p_{D_{0}}-\lambda_{D}}{p_{D_{1}}-\lambda_{D}}\left(M\psi^{(M+1)}-\psi^{M}(M+1)+1\right)}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\end{split}$
(44)

Based on the results of (18), the term $R_{2}$ is obtained as

$\begin{split}M\sum_{n=1}^{\infty}\pi_{M+n}&=M\,\mathrm{Pr}(Q>M)\\\
&=\frac{\lambda_{D}\psi^{M}(p_{D_{1}}-\lambda_{D})}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\end{split}$
(45)

Finally, the term $R_{3}$ can be obtained using the results from (37)

$\begin{split}\sum_{n=1}^{\infty}n\pi_{M+n}&=\frac{\pi_{0}\lambda_{D}\psi^{M}}{p_{D_{0}}(1-\lambda_{D})}\sum_{n=1}^{\infty}n\left[\frac{\lambda_{D}(1-p_{D_{0}})}{p_{D_{0}}(1-\lambda_{D})}\right]^{(n-1)}\\\
&=\frac{\pi_{0}\lambda_{D}\psi^{M}}{p_{D_{0}}(1-\lambda_{D})}\frac{1}{\left(1-\frac{\lambda_{D}(1-p_{D_{0}})}{p_{D_{0}}(1-\lambda_{D})}\right)^{2}}\\\
&=\frac{\lambda_{D}(1-\lambda_{D})p_{D_{0}}\psi^{M}\frac{p_{D_{1}}-\lambda_{D}}{p_{D_{0}}-\lambda_{D}}}{p_{D_{1}}p_{D_{0}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})}\end{split}$
(46)

Substituting (44), (45) and (46) in (43), we get

$Q_{avg}=\frac{Q_{1}+Q_{2}}{p_{D_{0}}p_{D_{1}}-\lambda_{D}p_{D_{1}}-\lambda_{D}\psi^{M}(p_{D_{0}}-p_{D_{1}})},$
(47)

where we have

$\begin{split}Q_{1}&=\lambda_{D}(1-\lambda_{D})p_{D_{1}}\frac{p_{D_{0}-\lambda_{D}}}{p_{D_{1}}-\lambda_{D}}\\\
&.\left(M\psi^{(M+1)}-\psi^{M}(M+1)+1\right),\\\
Q_{2}&=\psi^{M}\lambda_{D}(p_{D_{1}}-\lambda_{D})\left(M+\frac{p_{D_{0}}(1-\lambda_{D})}{p_{D_{0}}-\lambda_{D}}\right).\\\
\end{split}$ (48)

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# Anomalous transport regime in non-Hermitian,
Anderson-localizing hybrid systems

Himadri Sahoo  Nano-optics and Mesoscopic Optics Laboratory, Tata Institute of
Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India  Nanophotonics
Laboratory, Department of Physics and Nanotechnology, SRM Institute of Science
and Technology, Kattankulathur, Tamil Nadu 603203, India  R Vijay  Department
of Condensed Matter Physics and Materials Science, Tata Institute of
Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India  Sushil
Mujumdar<EMAIL_ADDRESS>Nano-optics and Mesoscopic Optics Laboratory,
Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005,
India

###### Abstract

In a disordered environment, the probability of transmission of a wave reduces
with increasing disorder, the ultimate limit of which is the near-zero
transmission due to Anderson localization. Under localizing conditions,
transport is arrested because the wave is trapped in the bulk of the sample
with decaying-exponential coupling to the boundaries. Any further increase in
disorder does not modify the overall transport properties. Here, we report the
experimental demonstration of a hitherto-unrealized anomalous transport of
hybrid particles under localizing disorder in a non-Hermitian setting. We
create hybrid polariton-photon states in a one-dimensional copper sample with
a comb-shaped periodic microstructure designed for microwave frequencies.
Metallic dissipation realizes the necessary non-Hermiticity. Disorder is
introduced by deliberate alterations of the periodic microstructure. Direct
measurement of wave-functions and phases was achieved by a near-field probe.
At a particular disorder, We observe the onset of Anderson localization of the
hybrid states endorsed by exponential tails of the wavefunction. However, at
stronger disorder and under conditions that support localization, an
unexpected enhancement in the transmission was facilitated by an emergent
mini-band. The transmission was traced to the hopping of the hybrid particle
over multiple co-existing localized resonances that exchange energy due to the
non-orthogonality. These emergent states are manifested in all configurations
under strong disorder, suggesting the formation of a novel transport regime.
This is verified by measuring the averaged conductance which endorses an
anomalous transport regime in the hybrid, non-Hermitian environment under
strong disorder. These experimental observations open up new unexplored
avenues in the ambit of disorder under non-Hermitian conditions.

††preprint: APS/123-QED

One of the most exotic phenomena in mesoscopic physics of disorder is Anderson
localization (AL), which is synonymous to quasiparticle trapping facilitated
by destructive self-interference of quantum waves associated with the
quasiparticle. AL occurs at the limit of strongest disorder, with subsidiary
regimes of transport such as weakly localized, diffusive, quasi-ballistic and
ballistic manifesting with systematic reduction in disorder. Regardless of
whether the quasiparticle is electronic[1, 2], photonic[3, 4, 5, 6, 7, 8, 9,
10, 11], phononic[12] or of quantum matter[13], the degree of disorder places
the transport in one of the above regimes. Of late, however, novel and
unexpected behavior has been reported from non-Hermitian photonic systems that
belies the underlying strength of disorder[14, 15].

The advent of deliberate non-Hermiticity has led to striking developments in
photonics in recent years. Engineered non-Hermitian structures have revealed a
plethora of exciting photonic behaviors, such as the striking revelation of
exceptional points associated with P-T symmetry implemented through
simultaneous loss andgain[16, 17, 18]. In the domain of complex systems,
random lasers were the first non-Hermitian systems[19, 20, 21, 22], although
non-Hermiticity was not invoked in the physics thereof. Deliberate non-
Hermiticity introduced theoretically in topological disordered lattices
exhibited a gapless energy spectrum in the metallic phase, in stark contrast
to the gapped spectrum in the Hermitian case[23]. Furthermore, in two-
dimensional lattices, non-Hermiticity was shown to modify the level-spacing
statistics[24], and in fact for three-dimensional lattices, resulted in
converting the Anderson transition into a cross-over[25]. Apart from inclusion
of gain (negative imaginary refractive index), systems with pure dissipation
also exhibit unusual transport behavior. For instance, measurements on
diffusive transport in open systems[26, 27] helped in identifying specific
effects caused by non-Hermiticity, namely, increased eigenfunction correlation
accompanied by strong modal anticorrelation that facilitates normalized
transport[28]. Further, the presence of non-Hermiticity erases the equivalence
among total excited energy, dwell time and density of states(DoS), that
otherwise exists in the Hermitian system[29]. Recently, a surprising
development was reported in which a localized system was shown to exhibit
transport by a novel mechanism through sudden jumps between distant sites[30],
a result which has also been experimentally verified[31, 32]. Overall, one can
expect non-Hermiticity to continue unravelling novel effects in quasiparticle
transport in mesoscopic systems.

On a parallel note, ongoing studies in mesoscopic transport are exploring an
entity of particular interest, namely, composite or hybrid quasiparticles.
These particles are created in systems wherein strong interactions between two
forms of energy are induced by the underlying structure. They react to a
disordered environment with novel emergent behavior. For instance, novel
topological phases were discovered in photon-phonon composites in cavity
optomechanical systems[33], which were further shown to exhibit a nontrivial
frequency dependence of localization length in a disordered environment[34].
Interestingly, Anderson localization was demonstrated to be a potential
resource to enhance the coupling between phononic and photonic components to
manifest strongly hybridized composites[35]. The challenges in colocalization
of the two components has been overcome in cleverly-designed GaAs/AlAs random
superlattices[36].

Given the intricacies that non-Hermiticity and hybridization bring to
mesoscopic transport, one can expect exciting consequences of simultaneous
presence of both. Precisely such a scenario is addressed in this paper. We
experimentally unravel a novel anomalous transport regime at localizing
disorder in a hybrid non-Hermitian system. Spoof surface plasmon polaritons
(SSPPs)[37] realized in a corrugated metal structure were employed as the
composite particles. The SSPPs are generated due to the hybridization of the
cavity resonant mode with the polariton mode. We have confirmed the occurrence
of the anomalous transport regime (ATR) by measuring the generalized
conductance in the disordered system. In corroboration of our earlier
theoretical work, we find that the transport is effected via the manifestation
of necklace states[38, 39, 40, 41, 42, 43, 44, 45, 46], as identified from
direct measurements of phase-jumps occurring in an emergent miniband at high
disorder. Concomitant simulations reveal the constituent localized resonances
in the measured necklace states. A statistical study of the necklace states
reveals the high-probability of their occurrence, and hence the certainty of
the ATR in such systems.

Figure 1: Experimental setup and SSPP structure. (a) Schematic of the near-
field microwave experimental setup comprising the SSPP sample and a vector
network analyzer for source and detection. (b) Unit cell of the SSPP structure
with dimensions d = 6 mm (lattice constant); m = 2 mm (metal corrugation
width); h = 7 mm (height) and t=35$\mu$m (metal film thickness). (c) Images of
the fabricated samples using standard PCB fabrication technique: periodic
(top) and disordered (bottom)

Our experimental samples comprise the well-established spoof surface plasmon
polariton (SSPP) structures in the form of 1D arrays of microwave resonators
cast into comb shaped corrugated metal strips as shown in Fig.1(a). In a
single unit cell of periodicity $d=6$ mm, the groove depth and width are
denoted by $h$ and $a$, respectively, and the strip width and thickness are
$m$ and $t$, respectively fig.1(b). The structure is sculpted, using a
commercial circuit board milling machine, onto an FR4 dielectric substrate (a
commercial printed circuit board (PCB)) with $\epsilon_{r}=4.4$ and total
thickness of 1.45 mm. Several periodic samples of metallic corrugated strips
with varying $h$ were fabricated using the milling machine and experimentally
tested at designated microwave frequencies. Disorder is introduced into the
system by randomly displacing the position of each resonator by an amount
determined by a uniformly distributed random variate $\Sigma\in[-\delta\cdot
a/2,\delta\cdot a/2]$, where $0\leq\delta\leq 1$ represents the disorder
strength. Multiple configurations of disordered samples with $\delta=0.3,0.5$
(referred to as moderate disorder, hereafter) and $0.7$ and $0.9$ (strong
disorder), were fabricated. Fig.1(c) shows a fabricated periodic and a
disordered sample. We built impedance-matching structures at the output and
input ends of the array to ensure efficient launch of the microwave signal
into the sample. This was accomplished by constructing metal teeth with
gradually increasing heights over a few unit cells at the two ends[47]. The
experimental setup consisted of an Agilent E5071C Vector Network Analyzer
(VNA), an SMA connector to input the microwave signal at one end and a
monopole antenna that acts as the detector. The other end was terminated with
a $50~{}\Omega$ resistor. The detector, attached onto an XY translation stage,
is numerically controlled by a stepper motor. The sample was mounted taut
horizontally such that the probe distance from the surface of the sample
remained $\sim 1.5$ mm at any point along the X axis (axis of translation).
The scanning step was 1 mm, to obtain high-resolution images. The probe picked
up the complex $E_{z}$ field during the scan, which was recorded by the VNA.
In support of the experimental measurements, finite element computations were
carried out using the RF module of COMSOL Multiphysics software. Eigenmode
analysis was carried out in the simulation for the same structures as in the
experiments, as well as transmission profiles were calculated by implementing
a lumped port as a source of excitation.

Figure 2: Measured spatial intensity distribution I(x) maps for three
structures.(a) periodic ($\delta=0.0$), (b)moderate disorder ($\delta=0.3$)
and (c)high disorders ($\delta=0.7$). A high-transmission mode close to $4.5$
GHz seen in (c).

The experimentally measured spatial intensity distribution, $I(x)$, is shown
in Fig.2. Image 2(a) represents the intensity in the periodic structure, while
2(b) and 2(c) show the same for two particular configurations at moderate
disorder ($\delta=0.3$) and high disorder($\delta=0.7$) respectively. The
sample is excited from one end at $x/L_{sys}=0$, where $L_{sys}$ is the total
system length. The spectrally-resolved field is recorded at each $x$-position,
at which the source signal is swept through a frequency range of 3.5 GHz to 6
GHz. For the periodic system (2(a)), extended modes are formed across the
system for all frequencies upto the bandedge at 4.77 GHz. The bandgap (beyond
4.77 GHz) is completely devoid of modes. These frequencies were in excellent
agreement with the finite-element calculations, as elaborated in Supplementary
Information S1. Images 2(b) and 2(c) depict the behavior at intermediate and
high disorder, namely $\delta=0.3~{}\&~{}0.7$ respectively. Beyond 4 GHz, the
transport decays substantially due to the strong disorder. Nonetheless,
extended modes are observed around the frequency of 4.5 GHz despite the strong
disorder, leading to high transmission. That these extended modes were of the
character of necklace states was directly inferred from two diagnostics,
namely, the experimentally-measured phase characteristics and the computed
eigenmode analysis.

Figure 3: Experimentally measured dispersion for the finite SSPP structures.
(a) Dashed cyan line Non-dispersive resonant mode (5.35 GHz) for an infinite
sample. Dotted white curve: Hybrid mode at the bandedge $\sim 4.77$GHz.
Measured experimental results (Viridis colormap) show the hybrid band in
excellent agreement with the predicted band. Arrows demarcate the FWHM of the
$k$-space peaks. Highly disordered structures: (b) $\delta=0.7$ and (c)
$\delta=0.9$. (d) $\Delta k$ as a function of frequency, measured at the three
values of disorder. The band-edge region (vertical purple band) shows a drop
in $\Delta k$. (e) Measured DoS as a function of frequency. The band-edge
region (purple band) shows an enhanced DoS, a consequence of frequency pinning
in hybrid systems.

The hybridization occurring in this system(see Supplementary information S2)
was experimentally verified by measuring the dispersion diagram of the finite-
sized periodic and disordered structures, and comparing it with the expected
resonance characteristics. Fig.3(a), 3(b) and 3(c) show the said dispersion.
The white dotted curve in 3(a) illustrates the dispersion behavior computed
using the eigenmode solver of COMSOL Multiphysics after applying periodic
boundary conditions. The band arises from the hybridisation of SPP dispersion
with the resonant modes of the individual cavities. The latter are non-
dispersive, and are shown as a horizontal cyan dashed line, whose resonant
frequency was calculated to be 5.35 GHz. The inset shows the corresponding
resonant mode for a single resonator. The experimental modes are in excellent
agreement with the computed dispersion, confirming successful excitation of
spoof SPP’s formed over the structured metal surface. The band-edge lay at
4.77 GHz. The cyan arrows indicate the width $\Delta k$ at any frequency,
which is inversely proportional to the spatial extent of the mode at the
frequency. Fig.3(b) and 3(c) show the measured dispersion of two particular
configurations at high disorder, $\delta=0.7$ and $0.9$ respectively. Disorder
clearly broadens the $\Delta k$ at a few frequencies, as labeled by the
arrows, reflecting the spatial localization of modes. The violet arrows
represent localized modes as inferred from the spatial studies (discussed
further), while the cyan arrows indicate persisting extended states in the
structure even under strong disorder. The co-existence of localized and
extended states is revealed in subplot 3(d), which illustrates the systematic
variation of $\Delta k$ with frequency. Blue circles represent the periodic
structure, with narrow and almost uniform $\Delta k$ across the frequency
axis. Orange and green circles ($\delta=0.7~{}\&~{}0.9$, respectively)
illustrate a rise in the $\Delta k$ at frequencies where localization sets in.
Both the curves show a fall in the shaded frequency region, where the extended
states manifest.

Theoretically, it is expected that these extended states lie in a miniband
formed by the coalescence of eigenvalues close to the bandedge. Interestingly,
we note that a similar phenomenon called “eigenvalue condensation” was
theoretically observed for strong randomness in imaginary potentials[30]. In
the experimental data, such a miniband can be directly inferred from the
increased DoS shown in the same frequency (shaded) region, as illustrated in
(e). In the weak disorder limit($\delta=0.0,~{}0.3$), the DoS shows a typical
behavior where it consistently increases upto the bandedge. The miniband
manifests at $\delta=0.7$, and is strongly conspicuous at $\delta=0.9$.

Figure 4: Measurement of phase and necklace state order in samples with high
disorder. (a) Experimentally measured transmission spectrum (green solid line)
and the corresponding extracted phase (pink solid line) for a single
configuration at high disorder. A drop in phase by a magnitude of $4\pi$
endorses a $4^{th}$ order necklace state. The simulated transmission spectrum
(shaded green) is in excellent agreement with the experimental measurements
and indeed reveal four resonances in the necklace (See Supplementary
Information S4). (c) shows the necklace order $m$ distribution measured over
49 modes. (c) Spatial intensity distributions measured for different orders of
necklace states. See Supplementary information S6.

In the hybrid plasmonic system under high disorder, the eigenvalues accumulate
in the vicinity of the hybridization frequency. Our simulations detected this
behavior in the eigenvalues, as discussed in the Supplementary information S3.
The non-Hermiticity lifts the orthogonality of the eigenmodes and enables
energy transport between them. In this scenario, whenever multiple localized
eigenmodes overlap even minimally in the tails, the spoof polariton can be
transmitted through the chain of eigenmodes, that is, the necklace states.
Such necklace states, essentially coupled localized resonances, can be
analyzed by observing the phase jumps in the constituent phase profile [42]
measured from complex field at the output end of the samples. Every component
resonance effects a phase shift of $\pi$. Fig.4 discusses the phase behavior
of a particular necklace state observed in a sample with high disorder. The
green curve depicts the measured transmission spectrum (left Y-axis). The peak
at around 3.7 GHz represents the photonic bands wherein localization hasn’t
yet set in. The drop in transmission at $\sim 4~{}$GHz represents incipient
localization. Next, a broad anomalous peak is seen in the vicinity of the
bandedge at around $4.5~{}$GHz which represents the miniband formed due to the
coalescence of eigenvalues. The peak at $\sim$4.2 GHz is not a consequence of
the structure, as ascertained from various configurations. The phase, as
extracted from the complex $E_{z}$ measurement, is shown in the pink curve
(right Y-axis). A clear jump of 4$\pi$ is observed over the frequency range
pertinent to the miniband. Corresponding simulations for this particular
disorder configuration provided the theoretical transmission spectrum shown by
the overlapping shaded green region. Eigenvalue analysis of this spectrum
clearly revealed four constituent coupled resonances creating the necklace
state. See Supplementary information S4 for details. Over the measured 49
states in 21 samples, the distribution of localized modes and necklace states
is shown in the bar graph in 4(b). The measured phase jump of 4$\pi$ endorses
this coupling. Several such necklace states of various orders were observed,
and a few representative states are shown in Fig.4(c). The top panel shows an
order-1, i.e, essentially a conventional Anderson localized mode, with a
localization length $\xi/L_{sys}=0.3$. The lower panels exhibit necklace
states with orders 2, 3 and 4 respectively, as characterized by the measured
phase jumps. As seen, almost 50% localized states couple to form necklace
states of some order. This fact distinguishes the hybrid non-Hermitian system
from conventional systems wherein the necklace states are statistically rare
and do not contribute to average transport. In contrast, the high probability
of necklace states in this system induces anomalous transport.

Figure 5: Effect of disorder on generalized conductance. Inset: Measured
$\langle g^{\prime}\rangle$ from intensity distributions averaged over 21
configurations as a function of frequency. The dashed line at $\langle
g^{\prime}\rangle=1$ demarcates localization regime. Main plot:
$P(g^{\prime})$ measured at two regions as marked by pink and orange
rectangles in (a): (I)Localized region ($\langle g^{\prime}\rangle<1$): Long-
tailed distribution, best fit with a log-normal function (solid line) with
mean $\langle g^{\prime}\rangle$ = 0.6 and width $\sigma_{g}=0.07$. (II)
Anomalous transmission regime ($\langle g^{\prime}\rangle\geq 1$): The
distribution exhibits a Gaussian profile with mean $\langle
g^{\prime}\rangle=1.2$ and width $\sigma_{g}=0.2$.

The physics of disordered systems is completely described by the behavior of
conductance. Therefore, in order to demarcate the novel regime of transport
uncovered in this work, we discuss the conductance of the samples in the
presence of the necklace states. Figure 5 depicts the behaviour of generalized
conductance $g^{\prime}$, a parameter derived from intensity statistics that
can faithfully characterize localization even in dissipative systems[48].
Since our system is inherently non-Hermitian, $g^{\prime}$ provides the sole
unambiguous characterisation of transport, which is illustrated in subplot
(a). The $g^{\prime}$ is provided by $2/3$var$(I)$, where $I$ is either the
normalized total transmission, or in-plane spatial intensity in a one-
dimensional sample[7, 49]. We extracted the generalized conductance $\langle
g^{\prime}\rangle$ over 21 highly disordered configurations. Fig.5(a) shows
the average generalized conductance $\langle g^{\prime}\rangle$ vs frequency,
with $\langle g^{\prime}\rangle=1$ delimiting the onset of the Anderson
localized regime. In the vicinity of $4.5$ GHz, the $\langle
g^{\prime}\rangle$ is clearly seen to rise above 1, endorsing the anomalous
transport. The error bars indicate the standard deviation of the $g^{\prime}$
values indicating that the majority of the samples were close to, or in the
anomalous transport regime. For lower frequencies, $\langle
g^{\prime}\rangle<1$) for the localized regime. The distribution of
conductances $P(g^{\prime})$ follows a stipulated behavior in disordered
systems. The rectangles (I) & (II) in 5(a) demarcate the regions over which
$g^{\prime}$ values were chosen to create the $P(g^{\prime})$, shown in
Fig.5(b). In the localized regime (magenta dots from rectangle (I)), the
distribution is asymmetric and long-tailed, and nicely fit by a log-normal
function with mean $0.6$, as seen from solid the magenta curve. That
conductances in the localized regime are distributed log-normally is a well-
established fact, also confirmed experimentally[10, 11]. On the other hand, in
the anomalous transport regime(ATR) (green squares, data from rectangle (II)),
the distribution is symmetric albeit with a longer tail. The data are
excellently fit by a Gaussian distribution, as is expected in the metallic
regime. The mean $\langle g^{\prime}\rangle=1.2$ from the Gaussian fit.

These data lie at the boundary of localization-delocalization. For this
regime, we apply the theory established in [50], where the authors
theoretically examine the conductance distributions in one-dimensional
disordered systems. In a particular situation where localized and extended
states co-exist, the $P(g^{\prime})$ for the metallic states is shown to
follow a Gaussian function
$P(g^{\prime})\propto\exp({{-\frac{15}{2}{(g^{\prime}-\langle
g^{\prime}\rangle)^{2}})}}$. Clearly, the experimental data in Fig 5(b) shows
an excellent fit to this equation, which provides a value of $\langle
g^{\prime}\rangle=1.16$, whereas the experimental value was measured to be
1.2, once again in excellent agreement. These data conclusively endorse the
existence of the system in a properly conducting regime, despite insulating
disorder. The insulating nature of disorder is certified by the fact that, at
frequencies away from the bandedge where necklaces are not yet forged, the
conductances are consistently sub-unity. The probability distributions
($P(I/\bar{I})$)of normalized intensity measured at localized and necklace
state regimes is provided in Supplementary information S5.

In conclusion, we have successfully uncovered an anomalous transport regime in
hybrid, non-Hermitian systems. One-dimensional structures supporting spoof
surface plasmon polaritons were investigated for the same. Conventional
Anderson localization leading to arrested transport was observed at moderate
disorders. However, at high disorder, an enhancement in transmission is
facilitated by formation of necklace states. The hitherto reported necklace
states in localizing systems are rare and do not contribute to the average
transport. In contrast, in the hybrid non-Hermitian system, the necklace
states manifest persistently for all configurations at high disorder. The
anomalous transmission manifests in the vicinity of the hybridization
frequency, where frequency-pinning of modes occurs due to constrained
migration. The resulting emergent mini-band has been experimentally
demonstrated in our experiments. Direct mode-mapping using a near-field
spatial probe allowed us to image the necklace states, as well as identify the
order thereof via phase jumps. The latter provided a direct measure of the
component localized modes within the state, which was in excellent agreement
with our numerical calculations on the various configurations. The transport
regime is labeled via the measurement of generalized conductance $\langle
g^{\prime}\rangle$, which was consistently above 1, endorsing high
transmission even under localizing conditions. Comparing the distribution of
$g^{\prime}$ with theoretical expressions for one-dimensional disordered
systems close to the localization transition, we found that our system was
indeed in a ‘metallic’ state despite the strong disorder.

###### Acknowledgements.

We thank M. Balasubrahmaniyam and Krishna Chandra Joshi for useful discussions
and helpful comments. We thank Meghan Patankar for his help in sample making.
We acknowledge funding from the Department of Atomic Energy, Government of
India (12-R&D-TFR-5.02-0200). S. Mujumdar acknowledges the Swarnajayanti
Fellowship from the Department of Science and Technology, Government of India.

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