# Age of Information in an URLLC-enabled Decode-and-Forward Wireless
Communication System ††thanks: This work is funded by the Seed Funding program
of Sri Lanka Technological Campus, Sri Lanka grant no. RRSG/20/A7 and by the
Russian Foundation Basic Research grant № 19-37-90037.

Chathuranga M. Wijerathna Basnayaka  Centre for Telecommunications Research
Sri Lanka Technological Campus
Padukka, SRI LANKA
<EMAIL_ADDRESS>Dushantha Nalin K. Jayakody  School of Computer
Science and Robotics
National Research Tomsk Polytechnic University
Tomsk,RUSSIA
<EMAIL_ADDRESS>Tharindu D. Ponnimbaduge Perera  School of Computer
Science and Robotics
National Research Tomsk Polytechnic University
Tomsk, RUSSIA
<EMAIL_ADDRESS>Moisés Vidal Ribeiro  Department of Electrical
Engineering
Federal University of Juiz de Fora Campus Universitário
Plataforma 5, BRAZIL
<EMAIL_ADDRESS>
###### Abstract

Age of Information (AoI) measures the freshness of data in mission critical
Internet-of-Things (IoT) applications i.e., industrial internet, intelligent
transportation systems etc. In this paper, a new system model is proposed to
estimate the average AoI (AAoI) in an ultra-reliable low latency communication
(URLLC) enabled wireless communication system with decode-and-forward relay
scheme over the quasi-static Rayleigh block fading channels. Short packet
communication scheme is used to meet both reliability and latency requirements
of the proposed wireless network. By resorting finite block length information
theory, queuing theory and stochastic processes, a closed-form expression for
AAoI is obtained. Finally, the impact of the system parameters, such as update
generation rate, block length and block length allocation factor on the AAoI
are investigated. All results are validated by the numerical results.

###### Index Terms:

Age-of-Information, finite block length regime, latency, reliability, ultra-
reliable low latency communications (URLLC) and 5GB.

##  I Introduction

The concept of age of information (AoI) as a data freshness metric was
introduced in the early 90s by data scientists to evaluate data in terms of
temporal consistency in real-time databases. The term named “AoI” was
initially used by Kaul and Marco in their work on vehicular communication
networks[1]. With the rapid advancements of information technology, the idea
of AoI was spread to other research domains such as wireless communication,
since the dissemination of timely information became increasingly critical for
modern wireless communication applications[2]. On the other hand, wireless
cooperative communication technologies have been vigorously investigated in
the telecommunication domain due to its added advantages, i.e, improved
reliability of transmissions and latency requirement of the network.

Existing works in the literature related to AoI is concerned with single-hop
wireless networks assuming each communication nodes is capable of transmitting
data with an infinite block length regime [3].However, transmitting
information involving an infinite-block length regime is impractical in the
presence of URLLC-enabled wireless communication networks. Therefore, short-
packet data transmission is used to meet both reliability and latency
requirements of the wireless communication networks. Further, it is also
noteworthy that in short-packet communication scenario, the Shannon-Hartley
Capacity theorem is no longer applicable [4]. In addition, integrating finite-
block length regime into wireless communication networks provide a precise
framework to determine the relationship between the latency and the
reliability between communication nodes.

To the best of the authors knowledge, there is no existing work found that
studies the AAoI performance metric of a wireless decode-and-forward (DF)
relay scheme under finite block length regime. To fill this research gap, in
this paper studies the AAoI in a DF relay-assisted wireless communication
system considering a finite block length regime over a quasi-static Raleigh
fading channel conditions. Consequently, AAoI in wireless networks is analyzed
using queuing theory and stochastic processes, and then a closed-form
expression for AAoI is derived. Finally, in order to estimate AAoI in the
proposed system, a closed-form expression of the block error probability is
incorporated into derived closed-form expression of AAoI.

Our numerical simulation results proved that by selecting a small packet size
when the update size is equal or less than $10$ bits always help to maintain
the URLLC with low AAoI in the proposed system. However, if the update size is
greater than $10$ bits, optimal block length that achieved lowest AAoI is
approximately equals to $20$ channel uses. It is also noteworthy that the
lowest AAoI is achieved when the number of update generation rate at the
source is equals to $22$ updates per second when the each update size is
equals to $100$ bits.

The remaining part of this paper is organized as follows. In Section II, the
system model is presented with the derivations for closed-form expression of
the block error probability for the proposed system. In the section III, AAoI
of the proposed system is derived. In Section IV, numerical simulation results
are provided to verify the mathematical derivations derived in Section II and
III. Finally, paper concludes in Section V.

##  II System Model

Figure 1: System model for URLLC-enabled relay system

In this paper, a URLLC-enable DF cooperative relaying system is proposed as
illustrated in Figure1, where the source $(S)$ sends its newly generated
updates to the destination $(D)$ via aid of the relay $(R)$. In this proposed
cooperative communication system, each transmission block is split into two
separate time slots. The source sends data to the relay in the first time
slot. Then, relay decodes and re-transmits the received data to the
destination during the second time slot. It is assumed that there is no direct
transmission between the source and the destination due to unfavourable line-
of-sight (LoS) connection between them. The transmitted signal by the source,
the received signal at the relay, the transmitted signal by the relay, and the
received signal at the destination are denoted by $X_{1}$, $Y_{1}$, $X_{2}$
and $Y_{2}$, respectively. The received signal at each communication node can
be written as

$Y_{1}=\sqrt{P_{S}}H_{SR}X_{1}+W_{SR},$ (1)
$Y_{2}=\sqrt{P_{R}}H_{RD}X_{2}+W_{RD},$ (2)

where $H_{ij}$ is the channel coefficient of the channel between node $i$ to
node $j$ where $i\in\left\\{S,R\right\\}$ and $j\in\left\\{R,D\right\\}$. The
symbol $W_{ij}$ denotes the independent and identically distributed Addictive
white Gaussian noise (AWGN) of the channel with zero mean and $\sigma^{2}$
variance. In this work, transmit power of the communication node $i$ is
modelled as

$P_{i}=\varphi_{i}P,$ (3)

where $P$ is the total transmission power constraint of the system and
$0<\varphi_{i}\leq 1$ is the power allocation factor of each communication
node. The distance between the source and the destination is considered as $d$
and the distance between the source and the relay is represented by $d\tau$,
where $0<\tau<1$. The small scale channel gain is denoted as
$g_{ij}=|h_{ij}^{2}|$, where $h_{ij}\sim\mathcal{C}\mathcal{N}(0,1)$ denotes
the Rayleigh fading channel coefficient. The probability density function
(PDF) of the small scale channel gain $g_{ij}$ can be expressed as

$f_{g_{ij}}(z)=e^{-z}.$ (4)

The large scale channel gain $\alpha_{ij}$ between the communication nodes can
be written as

$-10log(\alpha_{\textit{i}\textit{j}})=20log(d_{\textit{i}\textit{j}})+20log\bigg{(}\frac{4\pi
f_{c}}{C}\bigg{)},$ (5)

where $f_{c}$ and $d_{ij}$ are the carrier frequency and distance between the
communication nodes $i$ and $j$ respectively. The variable $C$ denotes the
speed of light in the space. Considering small and large scale channel gains,
the channel coefficient between the communication nodes can be written as

$H_{ij}=\sqrt{\alpha_{ij}g_{ij}}.$ (6)

The normalized received signal-to-noise ratio (SNR) $\gamma_{j}$ at the
receiving node $j$ can be written as

$\gamma_{\textit{j}}=\frac{\alpha_{\textit{i}\textit{j}}g_{\textit{i}\textit{j}}P_{\textit{i}}}{\sigma_{ij}^{2}}.$
(7)

The total block length $n$ allocated for the proposed system is is allocated
for each transmission block according to the predefined allocation factor
$\eta_{ij}$ between two communication nodes. The block length of each
transmission block can be calculated as

$n_{ij}=\eta_{ij}n.$ (8)

###  II-A Block Error Probability in the Finite-Block Length Regime

Considering DF relay protocol, the overall decoding error probability can be
written as

$\varepsilon=\varepsilon_{R}+\varepsilon_{D}(1-\varepsilon_{R}),$ (9)

where $\varepsilon_{R}$ is the block error probability at the relay,
$\varepsilon_{D}$ is the block error probability at the destination. Due to
the static nature of the communication channels, it is assumed that the fading
coefficients stay constant over the duration of each transmission block.
Following Polyanskiy’s results on short packet communication [4] and assuming
that the receiver has perfect channel state information, the expectation of
the block error probability of a given block length can be written as

$\varepsilon_{j}=\mathbb{E}\left[Q\left(\frac{n_{ij}C(\gamma_{j})-k}{\sqrt{n_{ij}V(\gamma_{j})}}\right)\right],$
(10)

where $\mathbb{E}\left[.\right]$ is the expectation operator,
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{t_{2}}{2}}dt$ and
$V(\gamma_{\textit{j}})$ is the channel dispersion, which can be written
$V(\gamma_{j})=\frac{{\log_{2}}^{2}e}{2}(1-\frac{1}{(1+\gamma_{j})^{2}})$. The
variable $C(\gamma_{j})$ denotes the channel capacity of a complex AWGN
channel and it is given by $C(\gamma_{j})=\log_{2}(1+\gamma_{j})$. The number
of bits per block represents by $k$. Moreover, under the Rayleigh fading
channel conditions, $\varepsilon_{j}$ can be formulated as

$\varepsilon_{j}=\int_{0}^{\infty}f_{\gamma_{j}}(z)Q\left(\frac{n_{ij}C(\gamma_{j})-k}{\sqrt{n_{j}V(\gamma_{j})}}\right)dz,$
(11)

where $f_{\gamma_{j}}(z)$ denotes the PDF of the received SNR $\gamma_{j}$ at
the $j^{t}h$ communication node. Due to the complexity of the Q-function, it
is difficult to get a closed-form expression for the overall decoding error
probability. Thus, using the approximation technique given in [5] and [6],
(11) can be approximated as

$\varepsilon_{\textit{j}}\approx\int_{0}^{\infty}f_{\gamma_{\textit{j}}}(z)\Theta_{\textit{j}}(z),$
(12)

where $\Theta_{j}(z)$ denotes the linear approximation of
$Q\left(\frac{n_{ij}.C(\gamma_{j})-k}{\sqrt{n_{ij}.V(\gamma_{j})}}\right)$,
this can be expressed as [6]

$\Theta_{j}(z)=\left\\{\begin{matrix}1,&\gamma_{j}\leq\phi_{j},&\\\
\frac{1}{2}-\beta_{j}\sqrt{n_{ij}}(\gamma_{j}-\psi_{j}),&\phi_{j}<\gamma_{j}<\delta_{j},&\\\
0,&\gamma_{j}\geq\delta_{j},&\end{matrix}\right.$ (13)

where
$\beta_{j}=\frac{1}{2\pi\sqrt{2^{\frac{2k}{n_{ij}}}-1}},\psi_{j}=2^{\frac{k}{n_{ij}}}-1,\phi_{j}=\psi_{j}-\frac{1}{2\beta_{j}\sqrt{n_{ij}}}$
and $\delta_{j}=\psi_{j}+\frac{1}{2.\beta_{j}\sqrt{n_{ij}}}$. Using (12) and
(13), a closed-form expression for the block error probability can be derived
as

$\varepsilon_{ij}\approx
1-\left(\frac{\beta_{j}\sqrt{n_{ij}}P_{i}\alpha_{ij}}{\sigma^{2}}\right)\left(e^{-\frac{\phi_{i}\sigma^{2}}{\alpha_{ij}P_{i}}}-e^{-\frac{\delta_{j}\sigma^{2}}{\alpha_{ij}P_{i}}}\right).$
(14)

Finally, substituting (14) in (9), a closed-form expression for the overall
error probability can be obtained as

$\displaystyle\varepsilon\approx$ $\displaystyle
1-\bigg{[}\left(\frac{\beta_{R}\beta_{D}\sqrt{n\eta_{SR}\eta_{RD}}\times\varphi_{S}\varphi_{R}(P)^{2}\alpha_{SR}\alpha_{RD}}{\sigma^{4}}\right)$
(15)
$\displaystyle\bigg{(}e^{-\frac{\phi_{R}\sigma^{2}}{\alpha_{SR}\varphi_{S}P}}-e^{-\frac{\delta_{R}\sigma^{2}}{\alpha_{SR}\varphi_{S}P}}\bigg{)}\bigg{(}e^{-\frac{\phi_{D}\sigma^{2}}{\alpha_{RD}\varphi_{R}P}}-e^{-\frac{\delta_{D}\sigma^{2}}{\alpha_{RD}\varphi_{R}P}}\bigg{)}\bigg{]}.$

##  III  Average Age of Information of the proposed URLLC-enabled relaying
scheme

If the generation time of the freshest update received at time stamp $t$ is
$g(t)$, then AoI can be defined as a random process as

$\Delta(t)=t-g(t).$ (16)

Figure 2:  Evolution of Age of Information $(\Delta(t))$ with the time

As illustrated in the Figure 2, it is assumed that at $t=0$ the measurements
of the AoI starts and the AoI at the destination is set to
$\Delta(0)=\Delta_{0}$. The information source generate updates at time stamps
$\mathit{g_{1},g_{2},...,}$ and the destination receive these updates at time
stamps $\mathit{D_{1},D_{2},...,}$. As illustrated in Figure 2, data update
$i$ is transmitted from the source at time stamp $t=g_{i}$ and it is
successfully delivered to the destination at time stamp $D_{i}=g_{i}+Y_{i}$.
Therefore, the AoI at the destination can be calculated as

$\Delta(D_{i})=D_{i}-g_{i}.$ (17)

AoI increases linearly until the next update is delivered to the destination.
Similarly, AoI just before $i+1$ update is successfully delivered, can be
written as

$\Delta(D_{i+1}^{-})=Y_{i}+X_{i}.$ (18)

Then, at the time stamp $D_{i+1}$, the AoI drops as follows

$\Delta\left(D_{i+1}\right)=Y_{i+1}.$ (19)

Considering the observation time interval as $[0,T_{n}]$ for the proposed
system, upper bound of the time interval can be given as a function of
$X_{i}$, which can be written as

$T_{n}=\sum_{i=0}^{n-1}X_{i}.$ (20)

For the considered time period, time average AoI can be computed using the
area under $\Delta(t)$ and it can be treated as a sequence of trapezoids
$Q_{i}$ when $0\leq i\leq n-1$ as shown in the Figure 2. Similarly, the time
average age of the proposed system can be given as

$\Delta_{{T_{n}}}=\frac{1}{T_{n}}\int_{0}^{T_{n}}\Delta(t)d(t)=\frac{\sum_{i=0}^{n-1}Q_{i}}{T_{n}},$
(21)

where

$Q_{i}=\int_{D_{i}}^{D_{i}+X_{i}}\Delta(t)d(t)=Y_{i}X_{i}+\frac{1}{2}X_{i}^{2}.$

Similar to the work presented in [7], the time average age
($\Delta_{{T_{n}}}$) tends to ensemble average age when $n\rightarrow\infty$,
i.e., which can be expressed as

$\Delta_{a}=\lim_{n\rightarrow\infty}\Delta_{{T_{n}}}.$ (22)

Since $\mathbb{E}[Y_{i}]=\mathbb{E}[Y_{i+1}]$, $X_{i}$ and update generation
time difference have the same distribution. Thus, substituting (20) and (21)
to (22), the AAoI can be written as

$\textrm{AAoI}=\Delta_{a}=\frac{\frac{\mathbb{E}[X^{2}]}{2}+\mathbb{E}[XY]}{\mathbb{E}[X]}.$
(23)

In (23), $\mathbb{E}[XY]$ is difficult to estimate without proper
approximation since $Y$ and $X$ are dependent random variables. Thus, it is
assumed that the communication node’s data buffers are equipped with $M/G/1$
queue, where the source generate updates according to a Poisson process at the
rate $\lambda$ while the service time of the update is geometrically
distributed. The system delay $(Y)$ can be calculated as a summation of the
service time $(s)$ and waiting time $(w)$ of the queue, which can be written
as

$Y=s+w.$ (24)

Since $X$ and $s$ are i.i.d random variables, AAoI is given by the (23) can be
re-formulate as

$\Delta_{a}=\frac{\frac{1}{2}\mathbb{E}\left[X^{2}\right]}{\mathbb{E}\left[X\right]}+\mathbb{E}\left[s\right]+\frac{}{}\frac{\mathbb{E}\left[wX\right]}{\mathbb{E}\left[X\right]}.$
(25)

Since $w$ and $X$ are dependent random variables, Pollaczek–Khinchine [8, 9]
formula is used to further simplify (25). Considering M/G/1 queue policy, the
relationship between expectation of the variables $w$ and $s$ have been
formulated as

$\mathbb{E}[w]=\frac{\mathbb{E}[s^{2}]}{2(\mathbb{E}\left[X\right]-\mathbb{E}[s])},$
(26)

The correlation term $\mathbb{E}[wX]$ can be evaluated as [10, 11]

$\displaystyle\mathbb{E}[wX]=$
$\displaystyle\frac{\mathbb{E}[X](\mathbb{E}[X]-\mathbb{E}[s])}{\mathbb{E}[e^{-\lambda
s}]}+\frac{\mathbb{E}[s^{2}]\mathbb{E}[X]}{2(\mathbb{E}[X]-\mathbb{E}[s])}$
(27) $\displaystyle-\frac{\mathbb{E}[X^{2}]}{2},$

where $\lambda$ is the update generation rate, where
$\mathbb{E}[X]=\frac{1}{\lambda}$ and
$\mathbb{E}[X^{2}]=\frac{2}{\lambda^{2}}$. Then, using (27) and (25), AAoI can
be written as

$\Delta_{a}=\mathbb{E}[s]+\frac{\mathbb{E}[s^{2}]}{2(\mathbb{E}[X]-\mathbb{E}[s])}+\frac{\mathbb{E}[X]-\mathbb{E}[s]}{\mathbb{E}[e^{-\lambda
s}]}.$ (28)

The variable $s$ in (28) depends on number of re-transmissions between the
communication nodes. The total number of transmission $R$ needed for the
reliable transmission of an update is geometrically distributed with
transmission success rate and its probability mass function can be given by

$P_{R(m)}=(1-\varepsilon)\varepsilon^{m-1};m=1,2....$ (29)

where $\varepsilon_{ij}$ denotes the block error probability. The service time
for an update, which is transmitted using one transmission block is given by

$s=(nT+\upsilon)R,$ (30)

where $n$ and $R$ denote block length and number of transmission needed for
reliable communication, respectively, $\upsilon$ is the channel induced delay
and $T$ is the symbol duration. Since all of these variables are constants and
the service time is a geometric random variable, thus, we have

$E(s)=\frac{nT+\upsilon}{1-\varepsilon},$ (31)
$E(s^{2})=\frac{(nT+\upsilon)^{2}(1+\varepsilon)}{(1-\varepsilon)^{2}},$ (32)
$E(e^{-\lambda
s})=\frac{(1-\varepsilon)e^{-(nT+\upsilon)\lambda}}{1-\varepsilon
e^{-(nT+\upsilon)\lambda}}.$ (33)

Finally, by using (15), (28), (31), (32) and (33) the AAoI of the proposed
system can be derived as

$\displaystyle{\Delta_{a}}=\frac{(nT+\upsilon)}{1-\varepsilon}+\frac{\frac{(nT+\upsilon)^{2}{(1+\varepsilon)}}{(1-\varepsilon)^{2}}}{2(\frac{1}{\lambda}-\frac{(nT+\upsilon)}{1-\varepsilon})}+\frac{(\frac{1}{\lambda}-\frac{(nT+\upsilon)}{1-\varepsilon})}{\frac{(1-\varepsilon)e^{-(nT+\upsilon)\lambda}}{1-\varepsilon
e^{-(nT+\upsilon)\lambda}}}.$ (34)

##  IV Numerical Simulation Results

In this section, numerical simulation results are presented to verify the
theoretical analysis presented in this paper. First, the effect of the update
generation rate on the AAoI is investigated. Unless otherwise stated, the
simulation parameters are set as: $d=1$ km, $P=23$ dBm, $n=300$,
$\eta_{SR}=\eta_{RD}=0.5$, $\varphi_{S}=\varphi_{R}=0.5$, $\tau=0.5$,
$\sigma^{2}=-167$ dBm, $f_{c}=6$ GHz, $C=3\times 10^{8}$ m/s, $k=100$ and
$T=1\times 10^{-4}$ s. It is assumed that the channel induced delay
($\upsilon$) is negligible compared to $nT$.

Figure 3: AAoI as a function of update generate rate when $K=100$ bits.

In Figure 3, AAoI is plotted as a function of update generation rate when
$k=100$ bits. It can be observed from Figure 3 that the network has a higher
AAoI when the updates generation rate is either lower or greater than the
optimal value of $22$ updates per second.This phenomena can be easily
explained as follows. When the update generation rate is lower than the
optimal value, it increases time difference between the reception of two
consecutive updates causing an increase in AAoI. On the other hand, if the
update generation rate significantly larger than the optimal value, it is
impossible to stabilize the queue. Thus, packets are waiting for a longer
period of time in the queue leading to a higher AAoI. Furthermore, it can be
clearly seen from the figure that the simulation results are approximately
similar to their analytical counterparts.

Figure 4:  AAoI as a function total block length $n$

Figure 4 illustrates the effect of the total block length $n$ on the AAoI
performance of the proposed system. It is observed from the figure that when
the update size $(k)$ is significantly small, the AAoI increase with the
increases in block length. This is mainly due to the fact that when the block
length is increased, it causes increment in the service time for an update. On
the other hand, an update consists of large number of bits has a higher AAoI
when it is transmitted using block length less than $20$ channel uses due to
the high block error probability. It is also noteworthy that when the update
size is larger than $10$ bits, increase in block length towards its optimal
value $20$ channel uses decreases the AAoI due to the decease in error
probability. However, further increase in block length after the optimal value
has led to increase in AAoI due to the impact of the service time on AAoI is
dominant compared to the decrease in block error probability. This result
proved that transmitting small size update less than $10$ bits always help to
maintain minimum AAoI for mission critical application. However, if the update
size is greater than $10$ bits, lowest AAoI is achieved only when the block
length is equals to $20$ channel uses.

Figure 5:  AAoI as a function of block length allocation factor $\eta_{SR}$
for different power allocation factors at the $S$

In Figure 5, AAoI is plotted as a function of the block length allocation
factor of the $S$-$R$ link $(\eta_{SR})$ for different power allocation
factors. It is shown that when the block length allocation factor increases,
the AAoI decreases until it reaches the optimal value. This is mainly due to
the fact that increase in block allocation factor decreases the block error
probability at the $R$ compared to the block error probability at $D$. Thus,
overall block error probability is reduced, which can be easily understood by
referring to (9).When the block length allocation factor between $0.3$ \-
$0.7$, the relay scheme experience lower AAoI, due to the moderate block error
probabilities and stability of the queue. However, after the value $0.7$ the
AAoI increases rapidly since the small block length allocation for a any
channel increase the overall error probability. In addition, it is also shown
that equal power allocation in communication nodes results in better AAoI
regardless of the block length.

##  V CONCLUSIONS

This work developed a new model to measure the AAoI in a DF wireless relay
operating under ultra-reliable and low-latency constraints. The relationship
between the block error probability and AAoI is formulated within the finite
block length regime. In addition, the impact of update size on the AAoI is
investigated and it has been shown that maintaining a small update size (short
packet communication) minimizes the AAoI in wireless relay networks. In
addition, the impact of the update generate rate, the block length allocation
factor and the power allocation factor on the AAoI were also investigated.

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