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arXiv:2305.11549v1 [cs.IT] 19 May 2023
1
Semantic Filtering and Source Coding in
Distributed Wireless Monitoring Systems
Pouya Agheli, Student Member, IEEE, Nikolaos Pappas, Senior Member, IEEE,
and Marios Kountouris, Fellow, IEEE.
Abstract
The problem of goal-oriented semantic filtering and timely source coding in multiuser communica-
tion systems is considered here. We study a distributed monitoring system in which multiple information
sources, each observing a physical process, provide status update packets to multiple monitors having
heterogeneous goals. Two semantic filtering schemes are first proposed as a means to admit or drop
arrival packets based on their goal-dependent importance, which is a function of the intrinsic and extrinsic
attributes of information and the probability of occurrence of each realization. Admitted packets at each
sensor are then encoded and transmitted over block fading wireless channels so that served monitors
can timely fulfill their goals. A truncated error control scheme is derived, which allows transmitters
to drop or retransmit undelivered packets based on their significance. Then, we formulate the timely
source encoding optimization problem and analytically derive the optimal codeword lengths assigned
to the admitted packets which maximize a weighted sum of semantic utility functions for all pairs of
communicating sensors and monitors. Our analytical and numerical results provide the optimal design
parameters for different arrival rates and highlight the improvement in timely status update delivery
using the proposed semantic filtering, source coding, and error control schemes.
Index Terms
Goal-oriented semantic communication, semantic filtering, timely source coding, distributed moni-
toring systems.
P. Agheli and M. Kountouris are with the Communication Systems Department, EURECOM, Sophia-Antipolis, France, email:
{pouya.agheli, marios.kountouris}@eurecom.fr. N. Pappas is with the Department of Computer and Information
Science, Link¨oping University, Sweden, email: nikolaos.pappas@liu.se. Part of this work is presented in [1], [2].
2
I. INT RODU CTI ON
GOAL -ORI ENT ED semantic communication has recently attracted considerable attention and
constitutes an information handling paradigm that has the potential to render various
network processes more efficient and effective through a parsimonious usage of communication
and computation resources. The design and the evolution of communication systems to date have
been mainly driven by a maximalistic approach, which sets audacious yet often hard to achieve
goals and comes with inflated requirements in terms of resources, network over-provisioning,
and ineffective scalability. In sharp contrast, goal-oriented semantics communication could be
seen as a minimalist design approach (“less is more”), advocating a paradigm shift from extreme
performance to sustainable performance, where the effective performance is maximized while
significantly improving network resource usage, energy consumption, and computational effi-
ciency. This vision has a long history dating back to Weaver’s introduction of Shannon model of
communication [3]. Various attempts, from different angles and using diverse tools, have been
made in the past towards a semantic theory of communication [4]–[11]; the vast majority of these
endeavors remained at a conceptual level and did not lead to an elegant and/or insightful theory
with immediate practical applications. Nonetheless, the quest for such theory has recently gained
new impetus [12]–[14], fueled by the emergence of connected intelligence systems, real-time
cyber-physical systems, and interactive, autonomous multi-agent systems.
An indispensable element to unlock the potential of goal-oriented communication is a con-
cise, operational, and universal definition of the semantics of information (SoI), i.e., a set of
measures for the significance and the usefulness of messages with respect to the goal of data
exchange. Going beyond surrogate metrics, such as age of information (AoI) [15]–[17], quality
of information (QoI) [18], and value of information (VoI) [19]–[21], SoI can be leveraged
so that the communication process, together with key associated functionalities (e.g., sensing,
learning, processing), are adapted to the end-user goals/requirements. Empowered by networked
intelligence, in semantic communication, only valuable and relevant content with respect to a
goal is acquired, transported and reconstructed, leading to a drastic reduction in the number of
unnecessary bits processed and sent. Otherwise stated, following the mantra that not all bits are
equal, semantic communication may boost the “information efficiency” of future communication
systems, meaning that it could maximize the number of bits of useful information extracted and
delivered per resource consumed. This new communication paradigm has the potential to redefine
3
importance, timing, and effectiveness in future networked intelligent systems.
In this paper, we study a distributed monitoring system (DMS), in which multiple remote
monitors receive status updates from multiple smart devices (e.g., sensors), each observing
an information source. The updates generated by an information source may correspond to
observations or measurements of a random physical phenomenon (event) and are taken from a
known discrete distribution with finite support. Each status update is assigned a value reflecting
its importance based on its intrinsic features, such as probability of occurrence, and on its
extrinsic attributes, related to the sensor (source) at which it is generated. Semantic filtering is
first performed at the transmitter, as a means to admit or drop only the most relevant or useful
packets according to the associated monitor’s application-dependent goal. Admitted status updates
are then encoded and sent to connected monitors over orthogonal block fading channels. Different
error control protocols are employed to harness packet transmission failures due to fading. The
objective of this paper is to design a semantic source coding scheme for a multiuser system with
heterogeneous goals, considering the probability of occurrence of a realization at a sensor side,
the probability that a monitor successfully receives an update from its connected sensor, and the
rationale for which update packets are sent to the destination. Specifically, we consider that only
a fraction of the source realizations is important for the different monitors. A set of realizations
becomes more significant or relevant than the others for a certain application, where its elements
could potentially vary for different goals or over different time spans. A simple instance of
this model is a scenario where one decision maker is interested in regular/standard information
for monitoring purposes or typical actuation (normal mode), whereas the other monitor tracks
the outliers that could potentially represent some kind of threat to the system or a possibly
dangerous situation (alarm mode). In that case, only “most” (“least”) frequent source realizations
are important for the first (second) monitor, treating the remaining ones as not informative or
irrelevant. The SoI is captured here via a metric of timeliness for the received updates at the
monitor(s), which in turn is a nonlinear function of AoI [21].
A. Related Work
This work falls within the realm of timely source coding problem [1], [2], [22]–[27]. These
works study the design of lossless source codes and block codes that minimize the average AoI
in status update systems under different queuing theoretic considerations. References [25], [27]
consider a selective encoding mechanism at the transmitter for timely updates. The optimal real
4
codeword lengths that minimize the average age at the receiver are derived therein, whereas
in [26] an empty symbol is used to reset (or not) the age. In [22], the authors consider a
zero-wait update policy and find optimal source codes which achieve the minimum average age
up to a constant gap, using Shannon codes based on a tilted version of the original symbol
generating probability mass function (pmf). Semantic source coding is studied in [1], where
optimal real codeword lengths that maximize a semantics-aware utility function and minimize a
quadratic average length cost [28] are determined. The semantic encoding problem is extended
to a two-user system with heterogeneous, possibly conflicting or diverging, goals in [2]. This
paper extends our prior work into distributed monitoring systems (DMS) with multiple sensors
at the transmitter side and multiple monitors at the destination, which have heterogeneous and
dissimilar goals. We propose simple semantic value assessment schemes for status update packets
and an adaptive semantic filtering method. We then determine codeword lengths that maximize
a weighted sum of semantics-aware utility functions for all pairs of communicating sensors and
monitors, highlighting the performance gains of semantic filtering and source coding.
B. Contributions
The main contributions of this work can be summarized as follows.
•We perform semantic value assessment for the status update packets in two levels. At the
source level, importance of an arrival takes on the form of a meta-value and captures in a
flexible manner the interdependence between different intrinsic and extrinsic attributes. At
the link level, SoI is measured using a nonlinear function of AoI.
•We propose two semantic filtering mechanisms to control the flow of arrival packets,
drop unimportant arrivals, and reserve the channel for informative packets. In the first
(fixed-length) scheme, a set of unimportant arrivals is filtered using selective encoding.
In the second (adaptive-length) scheme, a smaller fraction of the least important packets
(considering the occurrence probability) are immediately dropped upon arrival, while the
remaining ones may pass the filter depending on their relative importance for a predefined
goal and its evolution. We show that the latter method performs better for any range of
arrival rates at the expense of a small increase in the channel load.
•For packet error control, we employ truncated forms of simple and hybrid automatic repeat
request (ARQ and HARQ) protocols. In contrast to conventional importance-agnostic pro-
tocols with a fixed number of retransmissions, we propose a scheme in which the maximum
5
...
...
PSfrag replacements
MM1
MM2
MMm
SSM1
SSM2
SSM3
SSM4
SSMkp(1)
1
p(1)
2
p(1)
3
Fig. 1: A goal-oriented, semantics-empowered distributed monitoring system.
number of a packet’s retransmissions is adapted to its importance with respect to the specific
goal for which it is generated. Evidently, a packet with higher importance is assigned with
more retransmission rounds in the event of multiple failures.
•We determine the optimal real codeword length of each admitted arrival packet, based
on its probability of occurrence, the observation probability related to the sensor tracking
the realization, and the served monitor, as well as its meta-value. For that, we cast an
optimization problem that maximizes a weighted sum of semantics-aware utility functions
for all pairs of sensors and monitors, and obtain the solution both analytically and using a
proposed simple algorithm.
Notations: R,R+,R+
0, and Z+denote the set of real, positive real, non-negative real, and
positive integer numbers, respectively. E[·]is the expectation operator, W0(.)denotes the principal
branch of Lambert Wfunction, and O(·)denotes the growth rate of a function.
II. SYST EM MOD EL
We consider a DMS, in which a set Kof smart sensoring modules (SSMs), with cardinality
|K| =K, provides timely status updates to a set Mof Mmonitor modules (MMs) (see Figure 1).
An SSM consists of a sensor, a semantic filtering module, a source encoder, and a module for
information transmission (PHY - physical layer) operations, such as modulation and channel
coding. An MM could be a display device or an actuator, which serves a specific goal and
performs tasks based on status updates and commands received from the SSMs.
We consider a model where the k-th SSM, k= 1,2,...,K, is connected to a subset of MMs,
denoted by M(k), based on a fixed topology. Therefore, the m-th MM, where m= 1,2,...,M,
receives timely status updates only from its serving SSMs, denoted as K(m) = {k:m∈ M(k)}.
6
An SSM tracks a physical phenomenon (event) with finite realizations and generates independent
and identically distributed (i.i.d.) status update packets. The realization set for the k-th SSM is
defined as X(k)={x(k)
i|i∈ Ink},Ink={1,2, ..., nk}, where each element has a probability
of occurrence ¯p(k)
i=PX(x(k)
i)with PX(·)being a known pmf. Depending on the task(s) at the
MM side or the goal specified by the application or the end-user, a realization x(k)
iat the k-th
SSM connected to the m-th MM is attributed a certain importance (value) v(k,m)
iat its source
level. The feature-based value assessment of each packet at an SSM is discussed in Section III.
In this work, we consider a continuous-time system where packet generation follows a Poisson
process with input rate λkfor the k-th SSM. Thus, the input arrival rate at one layer of an MM
connected to a set of identical sensors is equal to sum of the input rates from those SSMs.
The probability that the m-th MM observes an update packet from the k-th serving SSM among
all the sensors it is connected to, i.e., K(m), is denoted by p(m)
k, where Pk∈K(m)p(m)
k= 1,∀m.
Since the observation probability of an SSM from each of its connected MMs can be different, an
arbitrary realization at that SSM is supposed to occur with different probabilities for those served
MMs. In this regard, information processing transmission is performed in multiple layers and
different communication channels, respectively. For instance, semantic filtering, source encoding,
and PHY operations at the k-th SSM are performed in |M(k)|parallel layers, each of which
is reserved for packets transmitted to one served MM. From the monitoring perspective, the
sensors with the similar physical features, probabilities of realizations p(k,m)
i, and observation
probabilities p(m)
kare considered identical1, while the rest are called dissimilar smart sensors.
Like the SSMs, multiple layers are utilized at each MM to process packets arriving from its
dissimilar sensors, whereas packets from identical sensors are processed in the same layers. A
functional diagram for the k-th SSM and the m-th MM, ∀m∈ M(k), is depicted in Figure 2.
A. Packet Transmission
Assuming no buffer is employed at the transmitter of every SSM, a status update packet is
blocked when the channel is busy. Semantics-aware packet filtering is employed at every SSM
for each of the connected MMs as a means to transfer only the most valuable and important
packets for effectively serving the goal or purpose of the data exchange, as well as having the
least possible blockage of valuable arrivals due to heavy packet load. Specifically, at the k-th
1In practice, sensors with identical characteristics can monitor the same phenomenon or process from different angles, locations,
or time instants. In our analysis, a set of identical sensors can be treated as a unique sensor with shared properties.
7
SubmoduleSubmodule
Semantic
PSfrag replacements
Display
Actuator
Sensor
PHY PHY
modulemodule
Semantic
filter
Source Source
encoder decoder
Block
fading
channels
ACK/NACK
k-th SSM m-th MM
Transmitter
Application-dependent goals per layer
Fig. 2: The functional diagram of a communication link between the k-th SSM and the m-th MM.
SSM, a semantic filter admits the l(m)
kmost important realizations via a flow controller for the
m-th MM and discards the rest, i.e., nk−l(m)
k. Thus, the index set of the most valuable arrivals
at the k-th SSM from the perspective of the m-th MM is denoted by Il(m)
k
where Il(m)
k
⊆ Ink.
Once semantic filtering is performed, the source encoder at the k-th SSM assigns instantaneous
(prefix-free) codewords with lengths ℓ(k,m)
ito the status packets x(k)
iadmitted for the m-th MM,
∀m∈ M(k), based on the following truncated distribution
p(k,m)
i=
p(m)
k
¯p(k)
i
ql(m)
k
,∀i∈ Il(m)
k
0,∀i /∈ Il(m)
k
(1)
where ql(m)
k
:=Pi∈Il(m)
k
¯p(k)
i.
Processed packets are then forwarded to the PHY module, mapped using a binary modulation
scheme, and transmitted over a noisy orthogonal channel subject to block fading. Thus, channel
gains remain constant during each packet transmission period and change in an i.i.d. manner
among different periods.
B. Packet Reception
We define t(k,m)
jthe time instant that the j-th packet is received at the m-th MM from the
serving k-th SSM. The update interval between the j-th successive arrival and the next one at the
same layer is then modeled as random variable (r.v.) Y(k,m)
j=t(k,m)
j−t(k,m)
j−1. Alternatively, this
interval is formed as Y(k,m)
j=W(k,m)
j+S(k,m)
j, where W(k,m)
jand S(k,m)
jindicate the waiting and
service time variables, respectively. The waiting time denotes the span between the j-th packet
and the previously delivered one at the same layer, which is written as W(k,m)
j=PA(k,m)
j
a=1 Z(k)
a.
In this definition, A(k,m)
jfollows a geometric distribution with success probability (1−ψ(m)
k)ql(m)
k
8
and indicates the number of packets discarded until the arrival of the j-th packet from the
selected set. Herein, 0≤ψ(m)
k<1denotes the semantics-aware drop factor, which is analyzed
in Section IV-A. Furthermore, Z(k)
ais the time between two arrivals, which is exponentially
distributed with rate λk. Therefore, the admitted arrivals are generated under a Poisson process
with rate λk(1−ψ(m)
k)ql(m)
k
. Besides, the service time corresponds to the duration an update packet
spends in the DMS until being completely decoded at its destination. The service time analysis
and channel error control protocols are investigated in Section IV-B.
III. SEMAN TIC VALUE ASSE SS M ENT O F UPDATE PACKETS
In this section, we present the semantic value extraction and assessment of both arrival and
successfully decoded packets. This is performed at two different scales, namely a microscopic
one, which is related to the importance of the arrivals at the information source, and a mesoscopic
one, which captures the importance of received packets at a link level [13].
A. Microscopic Scale
At the source level, the relative importance of an arrival packet depends on extrinsic and
intrinsic features, which in turn could be either time-sensitive or time-tolerant. Extrinsic features
are related to the smart sensor characteristics, such as spatial location, resolution and measure-
ment quality, battery level, and reliability. Intrinsic features depend on the properties of a packet
related to goal/application requirements, e.g., probability of occurrence, urgency, and loss risk.
1) Meta-value assignment: The goal-dependent importance of each packet can take on the
form of a meta-value, which is characterized by a function of the above features. In general, the
meta-value of each packet results from knowledge fusion or aggregation of Aintrinsic and B
extrinsic features. To overcome the limitations of aggregation functions in the form of weighted
sums, we resort to knowledge fusion [29], which takes into account possible interdependence
between different features/criteria and provides commensurate scales representing the attributes
using the Choquet integral [30]. Specifically, using the discrete Choquet integral, a general meta-
value formula at one arrival interval for the i-th realization of the k-th SSM connected to the
m-th MM is given by
v(k,m)
i=
A
Y
a=1
(g(k,m)
a,i )α(k,m)
a
| {z }
Intrinsic features
B
X
b=1 U(k,m)
b−U(k,m)
b−1(h(k,m)
b)¯α(k,m)
b!
|{z }
Extrinsic features
(2)
9
where α(k,m)
aand ¯α(k,m)
bare constant factors. Furthermore, defining w(k,m)
b≥0as the weight of
the b-th extrinsic feature, U(k,m)
bas the weight of the b-th subset of extrinsic features is given by
U(k,m)
b=1+λ(k,m)
gw(k,m)
bU(k,m)
b−1+w(k,m)
b(3)
where U(k,m)
0= 0,U(k,m)
1≤U(k,m)
2≤... ≤1,U(k,m)
B= 1, and λ(k,m)
g≥ −1comes from the
Sugeno fuzzy measure [31], as 1 + λ(k,m)
g=QB
b=1 1+λ(k,m)
gw(k,m)
b. For λ(k,m)
g= 0, we have
U(k,m)
b=U(k,m)
b−1+w(k,m)
b, and the extrinsic part’s fusion form in (2) becomes a weighted sum.
Also, g(k,m)
a,i :R→R+
0and h(k,m)
b:R→R+
0in (2) denote value functions (VFs) of intrinsic
and extrinsic features, respectively. Here, every member of g(k,m)
a,i ,∀a, i, k, m, or h(k,m)
b,∀b, k, m,
with known parameters is presented via a general function VF : R→R+
0. Thus, for Aintrinsic
and Bextrinsic features, we have A+Bdifferent versions of VF. Conventionally, we model VF
as a sum of Gaussian functions with predefined critical points znc, for nc= 1,2, ..., Nc, relative
criticality nc≥0, and minimum importance VFmin ≥0. For a sample point z, we can write
VF(z) = c
VF(z)
max
znc
VF(z)o(4)
where, if zmin ≤z≤zmax, we have
c
VF(z) =
Nc
X
nc=1
nce−(z−znc)2
2σ2.(5)
In (5), the standard deviation σis derived such that all sample points of VF get higher importance
than a given threshold VFmin. For that, with given Nc,znc, and nc,σstarts from a small value
and increases with a fixed step over a finite number of iterations until VF(z)≥VFmin,∀z. An
illustrative example how to compute VF according to (4) and (5) is given in Appendix A.
The proposed value assessment scheme can operate under both pull-based and push-based
communication models. Assume a packet from realization x(k)
i,∀i∈ Il(m)
k
arrives at the j-th
interval and is assigned importance value v(k,m)
j. In the push-based model, the value of the
generated packet comes from (2), i.e., v(k,m)
j=v(k,m)
i, whereas the pull-based policy adds a
constraint such that v(k,m)
j=v(k,m)
ionly if the m-th MM has requested an update from the k-th
SSM at the j-th interval; otherwise, v(k,m)
j= 0.
10
B. Mesoscopic Scale
At the link level, the importance of a received packet at a destination is measured using the
semantics of information (SoI) [13], [32]. In this regard, we consider timeliness as a contextual
attribute of information, which is a non-increasing function fk:R+
0→Rof the freshness of
information from the k-th sensor. The instantaneous SoI provided by the k-th SSM, ∀k∈ K(m),
for the m-th MM at time tis modeled as
S(m)
k(t) = fk(∆(m)
k(t)) (6)
where ∆(m)
k(t) = t–u(t)denotes the instantaneous AoI at the m-th MM, which is defined as the
difference between the current time instant tand the timestamp u(t)of the most recently arrived
update from the considered k-th SSM or another identical sensor connected to that monitor.
The average SoI over an observation interval (0, T ), assuming a stationary ergodic process,
is given by
¯
S(m)
k= lim
T→∞
1
TZT
0
fk(∆(m)
k(t))dt. (7)
From a monitor’s perspective, the SoI at a layer reserved for the k-th SSM decreases according
to fk(·)until a valuable status update for that layer arrives. Then, the SoI rises to the value of
the new update at that time. Therefore, dissimilar monitors potentially attain different SoIs over
similar time spans.
In this paper, we study the following forms for the SoI, namely exponential (E-), logarithmic
(L-), and reciprocal utility of timeliness (RUT) cases, i.e.,
fk(∆(m)
k(t)) =
exp(−ρ(m)
k(t)∆(m)
k(t)) + β(m)
kEUT case
ln(−ρ(m)
k(t)∆(m)
k(t)) + β(m)
kLUT case
(ρ(m)
k(t)∆(m)
k(t))−κ+β(m)
kRUT case
(8)
where κ∈Z+and β(m)
kare constant parameters, and ρ(m)
k(t)>0is an attenuation factor that
is fixed during each update interval of the k-th SSM and is reinitialized for a new arrival.2The
value of the attenuation factor at an update interval comes from the importance of that admitted
2The analysis is extendable for any (single or composite) non-increasing forms for the utility of timeliness.
11
arrival at the source level. Hence, we can define
ρ(m)
k(t):=ρ(k,m)
j
1
nt(k,m)
j−1+S(k,m)
j≤t≤t(k,m)
j+S(k,m)
j+1 o(9)
where ρ(k,m)
j=ρmin+max
i∈Il(m)
knv(k,m)
io−v(k,m)
jρmax−ρminfor the j-th admitted arrival. ρmin and
ρmax are the minimum and maximum values that the attenuation factors may attain, respectively.
Evidently, the higher the importance of an arrival, the lower attenuation factor it is assigned.
IV. SEMAN TIC FILTERI NG AND CHANNEL ER ROR CON TROL
In this section, we introduce a semantics-aware packet filtering mechanism and analyze two
asynchronous error control schemes.
A. Semantics-Aware Filtering
We consider two semantics-aware packet filtering schemes, which control the flow of update
packet arrivals through different levels of each SSM, namely fixed- and adaptive-length filtering.
In the light of assessed meta-values and freshness, the rationale of these filtering mechanisms is
to feed the transmitter with important arrivals, while keeping the blockage rate as low as possible.
The blockage rate quantifies the number of blockages occurring to the admitted packets due to
heavy channel load over the total number of arrival packets.
1) Fixed-length filtering: A fixed-length filter merely admits packets generated from the l(m)
k
most important realizations among all nkrealizations the k-th SSM shares with the m-th MM.
Packets of less important realizations, based on the given index set Il(m)
k
, are directly discarded,
independently of their freshness (e.g., most recent realization) or the channel idleness. Therefore,
even the most recent (highest freshness) yet less important packets are blocked.
2) Adaptive-length filtering: In the proposed adaptive-length filtering mechanism, packets’
importance at both microscopic and mesoscopic scales is taken into account, which gives a
chance to a subset of the least important packets, in addition to the most important ones, to be
carried over the DMS. Differently from the fixed-length filtering, only a fraction of the least
important packets are directly discarded in the adaptive-length filtering scheme, and a subset
of the fresh yet of lower importance arrivals may pass the filter in an adaptive manner. This
means that the number of admitted realizations in adaptive-length filtering is larger or equal to
that of the fixed-length one. Specifically, upon the arrival of a packet generated from one of the
12
admitted realizations when the channel is idle, the adaptive-length filter inspects whether that
packet can increase the SoI at the link level after being decoded, thanks to its freshness or not.
If the inspection outcome is positive, the packet passes the filter. Thus, adaptive-length filtering
brings flexibility owing to which packets with lower importance but higher freshness have a
chance to pass the filter.
3) Acceptance probability analysis: A key element of adaptive-length filtering is a parameter
that quantifies the probability that a packet generated from the selected realizations is rejected
(or accepted), considering previously observed updates at the link level. As mentioned in Sec-
tion II-B, we coin this as semantics-aware drop factor. Using this, a new arrival (status update)
that does not increase the SoI upon receipt at the monitor, despite being the most recent/fresh,
is rejected, hence not transmitted. To make it clear, Figure 3illustrates the shape of packet
drop by adaptive-length filtering at the k-th SSM connected to the m-th MM under three
possible conditions listed below, considering a non-increasing fk(·), and convex for the purpose
of semantic filtering.
I. The first and second arrival are of comparable importance, hence attenuation factors. There-
fore, the SoI curves provided by these arrivals intersect at infinity. In this case, the filter
does not discard the new arrival since it may offer a higher value upon receipt at its monitor.
II. The j-th arrival has higher importance than the (j+1)-th one, where the primary arrival
crosses the latter one before both reach β(m)
k. However, the cross point is after the service
time of the new arrival, i.e., t(k,m)
j+S(k,m)
j+1 . Therefore, the filter keeps the (j+1)-th packet
since it can increase S(m)
k(t)after its successful decoding.
III. The j′-th arrival has higher importance compared to the (j′+ 1)-th one; thus, the primary
arrival crosses the latter one at a point before the delivery of that new packet. In that case,
the filter discards the (j′+ 1)-th packet since it cannot bring a higher value at the link
level even after its successful delivery. Thanks to this mechanism, the system obtains better
S(m)
k(t), as highlighted in the figure.
The semantics-aware drop factor is denoted by ψ(m)
kfor the link between the k-th SSM and
m-th MM. Here, ψ(m)
k= 1 for the fixed-length filtering, while we can define
ψ(m)
k= lim
T→∞
1
N(m)
k(T)
N(m)
k(T)
X
j=1
dmax
X
d=1
1
(ρ(k,m)
j+d
ρ(k,m)
j
>τ (k,m)
d)
13
PSfrag replacements
S(m)
k(t)
t
0t(k,m)
1t(k,m)
2
t(k,m)
j−1
t(k,m)
j−1t(k,m)
j′−1
t(k,m)
j
t(k,m)
jt(k,m)
j′
S(k,m)
j+1 S(k,m)
j′+1
S(k,m)
2
W(k,m)
j
W(k,m)
j
Q(k,m)
j
Q(k,m)
j
ρ(k,m)
1ρ(k,m)
2ρ(k,m)
j+1
ρ(k,m)
jρ(k,m)
j′+1
ρ(k,m)
j′
β(m)
k
Cross
Cross
point
point
Fig. 3: Three conditions for packet drop via adaptive-length filtering.
≃E
i∈Ink
"1
dmax
dmax
X
d=1
Pr(i′∈ Il(m)
k
ρ(k,m)
i′
ρ(k,m)
i
>τ (k,m)
d)# (10)
for the adaptive-length filtering scheme. In (10), N(m)
k(T)is the number of all admitted packets
at the k-th SSM for the m-th MM by time T, and dmax ≪ N (m)
k(T)for T→ ∞. Also, τ(k,m)
d
denotes a threshold for the drop of order d, which is computed in the following lemma.
Lemma 1. The threshold τ(k,m)
din (10)is derived as
τ(k,m)
d= (d+1) + c
W(k,m)
d
ℓmax
(11)
where c
W(k,m)
dis an Erlang r.v. of order dwith rate λkql(m)
k
, and ℓmax indicates an upper bound
to which the size of a codeword length converges. Notably, we reach τ(k,m)
d=d+ 1 if ℓmax → ∞.
Proof. See Appendix B.
B. Channel Error Control and Service Time Analysis
We utilize two non-adaptive asynchronous error control schemes for handling transmission
errors over block fading channels, namely ARQ and truncated HARQ based on chase combing.
In this regard, an SSM connected to one of the served MMs is equipped with an individual
buffer to support ARQ and HARQ protocols. The successful delivery of each packet to an MM
is declared by an instantaneous and error-free acknowledgment (ACK) feedback to the serving
SSM (see Figure 2). However, in the event of failure at each monitor, the serving SSM retransmits
the packet only to the MM from which a negative ACK (NACK) message is received. Packets
are retransmitted either until successful reception or up to the maximum allowable number of
14
transmission rounds. After successful delivery to its destination or reaching the retransmission
limit (in which case the packet is dropped), the transmitter waits for a new admitted arrival.3
Consequently, the service time of packet x(k)
i, which is tagged important for the m-th MM
during the j-th arrival, is a function of its codeword length, the channel conditions, and the
number of transmission rounds according to the error control protocol. Thus, we consider
E[(S(k,m))c] = Eϕ(k,m)(L(k,m))cthe c-th moment of the service time for packets being delivered
to the m-th MM from the k-th SSM, where
Eϕ(k,m)(L(k,m))c= lim
T→∞
1
N(m)
k(T)
N(m)
k(T)
X
j=1
r(k,m)
max,j
X
r=1
ϕ(k,m)
j,r (ℓ(k,m)
j)c(12)
since fk(·)is non-increasing. Herein, ris the order of transmission and truncated by r(k,m)
max,j ≪
N(m)
k(T), where r(k,m)
max,j depends on the j-th arrival’s meta-value, i.e., v(k,m)
j. As a simple arbitrary
form, we can write
r(k,m)
max,j =
v(k,m)
j
1
l(m)
kPi∈Il(m)
k
v(k,m)
i
rmax (13)
for the j-th arrival. According to (13), higher sample/packet importance results in larger r(k,m)
max,j.
Besides, ϕ(k,m)
j,r is relevant to error control processes for the j-th arrival packet that can be
transmitted up to rtimes before reaching r(k,m)
max,j. In this regard, ϕ(k,m)
j,r is given by [33]
ϕ(k,m)
j,r =c(k,m)
j
1−θ(k,m)
r,j rθ(k,m)
r,j +
r
X
r′=1
r′θ(k,m)
r′−1,j −θ(k,m)
r′,j !(14)
where c(k,m)
j≥1indicates the reverse of the channel coding rate with c(k,m)
j= 1 for the ARQ
protocol, and θ(k,m)
r,j denotes the probability that the first rtransmissions of the j-th packet from
the k-th SSM to the m-th MM are performed with error. For the chase-combining HARQ, θ(k,m)
r,j ,
∀r, j, k, m ∈ M(k), is given by [34]
θ(k,m)
r,j =e
−
γ(k,m)
M,j
¯γ(m)
k×
∞
X
i=r
γ(k,m)
M,j
¯γ(m)
k!i
i!+
r−1
X
i=0
γ(k,m)
M,j
¯γ(m)
k!i
i!
r−i
Y
i′=1
1
1 + i′g(k,m)
j¯γ(m)
k
,(15)
3The computation and propagation delays are assumed negligible for both packet transmission and feedback processes.
15
while for ARQ we have
θ(k,m)
r,j =
1−g(k,m)
j¯γ(m)
k
1 + g(k,m)
j¯γ(m)
k
e
−
γ(k,m)
M,j
¯γ(m)
k
r
.(16)
Expressions (15) and (16) are obtained by applying an approximated expression for packet
error rates as a function of the signal-to-noise ratio (SNR) for Rayleigh fading channels [34],
[35], where ¯γ(m)
kis the average received SNR at the m-th MM from the k-th SSM. Fitting
the approximated expression to the exact formula, we find γ(k,m)
M,j and g(k,m)
junder a considered
modulation scheme.
V. SE MAN TIC SO URCE CODING
In this section, we formulate the problem of semantics-aware packet encoding, which aims to
maximize the semantic value of packets delivered at the served monitors, and whose solution
provides the optimal codeword lengths assigned to admitted packets at each sensor.
A. Problem Statement
The objective of optimal codeword length assignment is to maximize the weighted sum of
the average SoIs provided at all layers of MMs, subject to the following two constraints: (i)
codeword lengths should be positive integers, i.e., ℓ(k ,m)
i∈Z+(feasibility); (ii) existence of
a prefix-free (or uniquely decodable) code for a given set of codeword lengths at each SSM
(Kraft-McMillan inequality [36]). Therefore, the optimization problem is formulated as
P1:maximize
{ℓ(k,m)
i}
M
X
m=1
wmX
k∈K(m)
p(m)
k¯
S(m)
k:=
M
X
m=1 X
k∈K(m)
J(k,m)
SoI
subject to
C1:Pi∈Il(m)
k
2−ℓ(k,m)
i≤1,∀k, m
C2:ℓ(k,m)
i∈Z+,∀i, k, m
(17)
where wmdenotes a weight parameter.
To derive ¯
S(m)
k, we use different forms of fk(·)defined in (8), considering the type and
functionality of the k-th sensor. Thus, from (7) and (8), the average SoI can be computed for
the proposed cases. To do so, we divide the non-negative area below the curve of fk(∆(m)
k(t))
16
PSfrag replacements
S(1)
1(t)
S(1)
2(t)
t
0t(1,1)
1
t(2,1)
1
t(1,1)
j−1
t(2,1)
j−1
t(1,1)
j
t(2,1)
j
S(1,1)
j
S(2,1)
j
W(1,1)
j
W(2,1)
j
Q(1,1)
j
Q(2,1)
j
ρ(1,1)
j
ρ(2,1)
j
Q(1,1)
∞
Q(2,1)
∞
T
SSM1SSM3
β(1)
1
β(1)
2
(a)
PSfrag replacements
S(1)
1(t)
S(1)
2(t)
t
0
t(1,1)
1
t(2,1)
1
t(1,1)
j−1
t(2,1)
j−1
t(1,1)
j
t(2,1)
j
S(1,1)
j
S(2,1)
j
W(1,1)
j
W(2,1)
j
Q(1,1)
j
Q(2,1)
j
ρ(1,1)
j
ρ(2,1)
j
Q(1,1)
∞
Q(2,1)
∞
T
SSM2
β(1)
1
β(1)
2
(b)
Fig. 4: Sample evolution for the EUT case, where (a) MM1receives updates from identical SSM1and SSM3, and (b) MM2
receives packets from SSM2.
over interval (0, T )into polygons of Q(k,m)
j,j= 1,2, ..., N(m)
k(T), and Q(k,m)
∞, as depicted in
Figure 4for the EUT case. Thus, we can rewrite (7) as
¯
S(m)
k= lim
T→∞
1
T
N(m)
k(T)
X
j=1
Q(k,m)
j+Q(k,m)
∞
=η(m)
kE[Q(k,m)](18)
where η(m)
k= lim
T→∞
1
TN(m)
k(T)−1denotes the steady-state time average arrival rate. Importing
(18) into (17), we reach the following problem.
P2:maximize
{ℓ(k,m)
i}
M
X
m=1
wmX
k∈K(m)
p(m)
kη(m)
kE[Q(k,m)]
subject to
C1:Pi∈Il(m)
k
2−ℓ(k,m)
i≤1,∀k, m
ˆ
C2:ℓ(k,m)
i∈R+,∀i, k, m.
(19)
where constraint ˆ
C2is a relaxation of C2in (17) to allow for non-negative real-valued codeword
lengths. The solution of P2gives real-valued codeword lengths, whose corresponding integer
values can be found using a rounded-off operation.
B. Semantics-Aware Encoding Design
To solve the problem P2, in what follows, we find E[Q(k,m)]based on different forms of
fk(∆(m)
k(t)) defined in (8).
1) EUT case: Considering the exponential utility function of timeliness, we propose the
following lemma.
17
Lemma 2. The expected form of polygons Q(k,m)
jfor the EUT case is approximately derived as
E[Q(k,m)]≈1+β(m)
k−¯ρ(m)
kγ(m)
k−Eρ(k,m)ϕ(k,m)L(k,m)Eϕ(k,m)L(k,m)−1
2¯ρ(m)
kEϕ(k,m)(L(k,m))2
−γ(m)
kEρ(k,m)ϕ(k,m)L(k,m)−¯ρ(m)
k(γ(m)
k)2+ (1+β(m)
k)γ(m)
k(20)
where γ(m)
k:= 1/(λkψ(m)
kql(m)
k
)and ¯ρ(m)
k:=E[ρ(k,m)] = lim
T→∞
1
T
N(m)
k(T)
X
j=1
ρ(k,m)
j=X
i∈Il(m)
k
p(k,m)
iρ(k,m)
i.
Proof. See Appendix C.
Importing (20) into (19), P2becomes a convex problem and can be solved using standard
solvers, e.g., MOSEK in CVX. Alternatively, we propose a heuristic solution to compute closed-
form expressions for the codeword length, as follows.
Proposition 1. The codeword lengths ℓ(k,m)
i,∀i∈ Il(m)
k
, which maximize (19)in P2for the EUT
case are computed as
ℓ(k,m)
i=−log2 ¯ρ(m)
kp(k,m)
iϕ(k,m)
i
µ(m)
k(ln(2))2W0 µ(m)
k(ln(2))2
¯ρ(m)
kp(k,m)
iϕ(k,m)
i
2ξ(m)
k!! (21)
where µ(m)
k≥0is a constant multiplier,
ξ(m)
k:=2χ(m)
kµ(m)
kln(2) + ¯ρ(m)
kγ(m)
k1+χ(m)
k−1+β(m)
k
¯ρ(m)
k1+2χ(m)
k¯ϕ(m)
k,(22)
¯ϕ(m)
k:=E[ϕ(k,m)] = lim
T→∞
1
TPN(m)
k(T)
j=1 ϕ(k,m)
j, and
χ(m)
k=Eρ(k,m)ϕ(k,m)L(k,m)
¯ρ(m)
kE[ϕ(k,m)L(k,m)].(23)
Proof. See Appendix D.
The optimal codeword lengths of delivered packets from the k-th SSM to the m-th MM can
be found via (21) for (µ(m)
k, χ(m)
k)pair, subject to µ(m)
k>0and Pi∈Il(m)
k
2−ℓ(k,m)
i= 1. The
values of χ(m)
kand µ(m)
kcan be obtained using Algorithm 1, through its inner and outer loops,
respectively. We first assume similar importance, hence attenuation factors, for all packets and
initialize χ(m)
k= 1. Then, given a small value of µ(m)
k, we compute ℓ(k,m)
iand new χ(m)
kthrough
the inner loop. Thereafter, based on the found χ(m)
k, we derive new values for ℓ(k,m)
iin the
18
Algorithm 1: Solution for deriving µ(m)
kand χ(m)
k
Input: Fixed parameters Il(m)
k
,p(k,m)
i,ρ(k,m)
i, and ϕ(k,m)
i,∀i∈ Il(m)
k
, and β(m)
k. Stopping
accuracy ε. Initial parameters (µ(m)
k)(0),(χ(m)
k)(0),(ξ(m)
k)(0), and (ℓ(k,m)
i)(0),
∀i∈ Il(m)
k
.
Output: Computed parameters χ(m)
k= (χ(m)
k)(b),ξ(m)
k= (ξ(m)
k)(b),ℓ(k,m)
i= (ℓ(k,m)
i)(b),
∀i∈ Il(m)
k
, and µ(m)
k= (µ(m)
k)(a).
1Iteration a:⊲Outer loop
2Iteration b:⊲Inner loop
3Compute (ξ(m)
k)(b)and (ℓ(k,m)
i)(b),∀i∈ Il(m)
k
, using (22) and (21), respectively.
4Calculate Eρ(k,m)ϕ(k,m)L(k,m)and ¯ρ(m)
kEϕ(k,m)L(k,m).
5Update (χ(m)
k)(b)from (23) based on 4.
6if Criterion (χ(m)
k)(b)−(χ(m)
k)(b−1)> ε then set b=b+ 1, and goto 2.
7Update (ξ(m)
k)(b)and (ℓ(k,m)
i)(b)according to 5.
8if Pi∈Il(m)
k
2−(ℓ(k,m)
i)(b)= 1 then stop the process, and goto 11.
9else if Pi∈Il(m)
k
2−(ℓ(k,m)
i)(b)<1then decrease (µ(m)
k)(a), set a=a+ 1, and goto 1.
10 else increase (µ(m)
k)(a), set a=a+ 1, and goto 1.
11 Save (χ(m)
k)(b),(ξ(m)
k)(b),(ℓ(k,m)
i)(b),∀i∈ Il(m)
k
, and (µ(m)
k)(a).
next iteration. This process continues until we reach the stopping accuracy ε. Once reached,
the outer loop checks the Kraft-McMillan condition and resets µ(m)
kif the condition is not
satisfied. Subsequently, the inner loop starts again, considering the new value of the constant
multiplier. Finally, the algorithm converges to the final amounts of χ(m)
kand µ(m)
kwith the rate of
O((NaNb)−1)in which Naand Nbdenote the maximum numbers of outer and inner iterations,
respectively.
2) LUT case: Under a logarithmic utility function of timeliness, E[Q(k,m)]is found as follows.
Lemma 3. The expected value of Q(k,m)
jfor the LUT case is approximately derived as
E[Q(k,m)]≈ −2Eρ(k,m)ϕ(k,m)L(k,m)Eϕ(k,m)L(k,m)−¯ρ(m)
kEϕ(k,m)(L(k,m))2
−2γ(m)
kEρ(k,m)ϕ(k,m)L(k,m)+β(m)
k−1−2¯ρ(m)
kγ(m)
kEϕ(k,m)L(k,m)
−2¯ρ(m)
k(γ(m)
k)2+ (β(m)
k−1)γ(m)
k.(24)
Proof. See Appendix E.
19
Inserting (24) into (19), P2is a convex problem, and the optimal codeword lengths ℓ(k,m)
ican
be obtained using either standard solvers or the following expression
ℓ(k,m)
i=−log2 ¯ρ(m)
kp(k,m)
iϕ(k,m)
i
ˆµ(m)
k(ln(2))2W0 ˆµ(m)
k(ln(2))2
¯ρ(m)
kp(k,m)
iϕ(k,m)
i
2ˆ
ξ(m)
k!!,(25)
which is derived with the same method as in (21). Herein, ˆµ(m)
k≥0indicates a constant. We
also have ˆ
ξ(m)
k:=4ˆχ(m)
kˆµ(m)
kln(2) + 2¯ρ(m)
kγ(m)
k1+ ˆχ(m)
k−β(m)
k−1
2¯ρ(m)
k1+2 ˆχ(m)
k¯ϕ(m)
k. The optimal codeword
lengths are computed in (25) based on the known pair of (ˆµ(m)
k,ˆχ(m)
k), subject to ˆµ(m)
k>0and
Pi∈Il(m)
k
2−ℓ(k,m)
i= 1. Similar to the EUT case, we can find the values of ˆµ(m)
kand ˆχ(m)
kusing
Algorithm 1replacing its parameters with those for the LUT case.
3) RUT case: Following the same steps as for deriving (20) and (24), the expected form of
polygons Q(k,m)
jfor the RUT case is obtained as
E[Q(k,m)]≈ −κEρ(k,m)ϕ(k,m)L(k,m)Eϕ(k,m)L(k,m)−κ
2Eρ(k,m)ϕ(k,m)(L(k,m))2
−2κγ(m)
kEρ(k,m)ϕ(k,m)L(k,m)+κ
κ+1 +β(m)
k−κ¯ρ(m)
kγ(m)
kEϕ(k,m)L(k,m)
−2κ¯ρ(m)
k(γ(m)
k)2+ ( κ
κ+1 +β(m)
k)γ(m)
k.(26)
Consequently, the optimal codeword lengths are given as
ℓ(k,m)
i=−log2 ¯ρ(m)
kp(k,m)
iϕ(k,m)
i
ˆ
ˆµ(m)
k(ln(2))2W0 ˆ
ˆµ(m)
k(ln(2))2
¯ρ(m)
kp(k,m)
iϕ(k,m)
i
2ˆ
ˆ
ξ(m)
k!! (27)
where ˆ
ˆµ(m)
k≥0, and ˆ
ˆ
ξ(m)
k:=2κˆ
ˆχ(m)
kˆ
ˆµ(m)
kln(2) + κ¯ρ(m)
kγ(m)
k1+2 ˆ
ˆχ(m)
k−κ
κ+1 +β(m)
k
κ¯ρ(m)
k1+2 ˆ
ˆχ(m)
k¯ϕ(m)
k. The
values of ˆ
ˆµ(m)
kand ˆ
ˆχ(m)
kare calculated using Algorithm 1with parameters for the RUT case.
C. Asymptotic Expansions for Codeword Lengths
We provide here two asymptotic expansions for the derived closed-form codeword lengths in
(21), (25), and (27), for the EUT, LUT, and RUT cases, respectively.
1) Dependency on occurrence probability and transmission rounds: As shown in Section V-B,
the codeword length assigned to a realization (update packet) depends - among others - on its
probability of occurrence and the number of transmission rounds. Using
W0(y)
yz
= exp−zW0(y),∀y≥e, (28)
20
from the Laurent series of order z, and W0(y) = ln(y)−ln(ln(y))+ O(1) from the second-order
Taylor expansion for large enough y, we can write
ℓ(k,m)
i∝
max
inp(k,m)
iϕ(k,m)
io
p(k,m)
iϕ(k,m)
i
−ln
max
inp(k,m)
iϕ(k,m)
io
p(k,m)
iϕ(k,m)
i
.(29)
From the asymptotic expression in (29), we deduce that the codeword length monotonically
decreases (increases) by an increase in the occurrence probability, the number of transmissions,
or both when p(k,m)
iϕ(k,m)
i≤1(p(k,m)
iϕ(k,m)
i>1). Besides, p(k,m)
iϕ(k,m)
i→0yields ℓ(k,m)
i→ℓmax,
∀i∈ Il(m)
k
, where ℓmax denotes the upper bound for the size of a codeword length. Furthermore, if
ϕ(k,m)
i=ϕ(k,m)
i′,∀i6=i′, the codeword length of a packet becomes proportional to the reverse of
its probability. The higher the probability of occurrence of a realization, the shorter its assigned
codeword length, and vice versa.
2) Uniform sources and equal semantic value: For a uniform source, i.e., p(k,m)
i=p(m)
k/nk,
∀i, with equal goal-oriented importance, e.g., v(k ,m)
i= 1, and the same number of transmissions,
i.e., ϕ(k,m)
i=ϕ(k,m)
i′,∀i6=i′, we end up with codewords of equal size, i.e., ℓ(k ,m)
i=ℓ(k,m)
i′, where
ℓ(k,m)
i∝nk
p(m)
k
−ln nk
p(m)
k!,∀i∈ Il(m)
k
.(30)
Thus, nk≫1results in ℓ(k,m)
i→ℓmax, which remains almost fixed for very large nk. Therefore,
as the number of realizations increases, the assigned codeword lengths become longer to satisfy
the Kraft-McMillan condition at the expense of higher service time. On the other hand, (30)
indicates that the codeword lengths of packets transferred to a monitor with a higher probability
of observation from a serving sensor are shorter than those with a lower probability of observation
from the same sensor. That is, ℓ(k,m)
iof packets sent to the m-th SSM from the k-th SSM become
shorter with the increase of p(m)
k.
VI. SI MULATIO N RE SU LTS
In this section, we present simulation results that corroborate our analysis and show the
performance gains by properly designing semantic filtering and source coding for timely status
update delivery in DMSs.
21
-5 -2.5 0 2.5 5
-5
-2.5
0
2.5
5
PSfrag replacements
X-axis [km]
Y-axis [km]
MM4
MM11
MM13
MM8SSM65
SSM82
SSM31
SSM11
SSM51
SSM25
SSM74
SSM44
SSM40
SSM18
SSM2
Monitor module
Sensor module
Z8
Z11
Z13
Fig. 5: The distribution of SSMs and MMs in the assumed area.
A. Setup and Assumptions
We consider a DMS with K= 100 randomly distributed SSMs and M= 16 fixedly positioned
MMs in an area of 10 ×10 [km2]divided into sixteen identical subareas (see Figure 5). The k-th
SSM transmits update packets to its four closest MMs, i.e., |M(k)|= 4,∀k, over Rayleigh
block fading channels. The m-th MM observes packets from its serving SSMs with the same
probabilities, i.e., p(m)
k=1
|K(m)|,∀m, k ∈ Il(m)
k
. The information source follows a Zipf(n(k), s)
distribution with pmf
PX(x(k)
i) = 1/is
Pn(k)
l=1 1/ls(31)
for nk=|X (k)|realizations at the k-th sensor. The parameter sis an exponent characterizing
PX(·),varying from uniform distribution (s= 0) to peaky ones.
The intrinsic and extrinsic features considered in our simulation scenario are shown in Table I
for a fictitious air pollution monitoring platform that collects status packets from sixteen zones
(Z1to Z16), with different levels of importance. Without loss of generality, we consider Z8,Z11,
and Z13 the most crucial zones, and the average Euclidean distance of a sensor from the centers of
these three zones indicates its spatial importance. Furthermore, we assume that rare occurrences
are monitored, where the importance of an update packet increases as its carrying pollution rate
approaches 100%. From Table I, we first find λ(k,m)
g= 0.37, and then, the corresponding U(k,m)
n
for each w(k,m)
bis calculated. The parameters used in the simulations are summarized in Table II.
The values of γ(k,m)
M,j and g(k,m)
jdepend on the codeword length of the j-th arrival according to
22
TABLE I: I NTR IN SIC A ND EX TRI NS IC FE ATU RE S W IT H THE IR P ROPO SE D P RO PE RTIE S.
Feature w(k,m)
bU(k,m)
nLimits Critical points
Probability – – (0,1] {0}
Usefulness – – [1, nk]Ten random x(k)
i
Loss risk* – – [0,100]% {100}%
Average distance 0.4 0.4 [3.66,8.68) [km]{3.66}[km]
Resolution 0.2 0.6 (0, Rmax]{Rmax}
Tolerance 0.2 1 [0,100]% {100}%
Circuit power 0.2 0.63 [Pmin,∞){Pmin }
Battery state 0.3 1 [0,100]% {0}%
*The criticality for all critical points is equal to 1.
TABLE II: PARA ME T ER S FOR S IM ULATI ON RE SU LTS .
Parameter Symbol Value Parameter Symbol Value
Number of the SSMs K100 Minimum attenuation factors ρmin 0.1
Number of the MMs M16 Maximum attenuation factors ρmax 5
Size of realizations set nk,∀k100 Maximum packet drop times dmax 10
Constant exponent for Zipf(., .)s0.4Maximum transmission rounds rmax 3[35]
Update packet input rate λk,∀k0.5Reverse of channel coding rate c(k,m)
j,∀k, b, j 2[35]
Number of intrinsic features A3Constant weight parameter wm,∀m1
Number of extrinsic features B3Upper bound of codeword lengths ℓmax 100
Exponent for intrinsic features α(k,m)
a,∀a, k, m 0.5Monitoring time length of arrivals T1000 [sec]
Exponent for extrinsic features ¯α(k,m)
b,∀b, k, m 1Minimum circuit power Pmin 0.1 [W]
Minimum importance VFmin 0.1Maximum sensing resolution Rmax 5 [m]
Bias value for the utility forms β(m)
k,∀k, m 5Average received SNR ¯γ(m)
k,∀k, b 12 [dB]
Exponent for the RUT case κ2Packet error rate’s approx. factors
for the ARQ (HARQ) protocol
γ(k,m)
M,j ,∀k, b, j 17.19
(10.1) [dB]
g(k,m)
j,∀k, b, j 0.1(0.96)
(15) and (16). We consider a binary phase shift keying (BPSK) modulation scheme at the PHY
module. Initializing the codeword length ℓjfrom zero to ℓmax as the worst case, the value of
γ(k,m)
M,j varies from 5.53 to 5.63 with almost constant g(k,m)
j≈0.1under the ARQ protocol. We
consider the average value of γ(k,m)
M,j for all codeword lengths, as it results in negligible difference.
Applying (2)–(5), and Tables Iand II, meta-values are computed. Unless otherwise specified, the
EUT case, fixed-length filtering, ARQ protocol, and Algorithm 1are used to obtain the results.
23
100
50
0
10-1
100
101
102
100
101
10-2
10-3
10-1
PSfrag replacements
J(k,m)
SoI
λkl(m)
k
EUT case
LUT case
RUT case
Fig. 6: The interplay between the SoI, the arrival rate, and the
admission size for the EUT, LUT, and RUT cases.
10 20 30 40 50 60 70 80 90 100
10-3
10-2
10-1
100
PSfrag replacements
J(k,m)
SoI
λk
l(m)
k
λk= 10
λk= 5
λk= 2
λk= 1
λk= 0.5
λk= 0.2
λk= 0.1
Fig. 7: SoI versus the size of the admitted realizations for the
EUT case and different arrival rates.
B. Results and Discussion
Figure 6shows the objective function J(k,m)
SoI versus the input rate λkand the number of
selected realizations