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Semantic Source Coding for Two Users with HeterogeneousSemantic Source Coding for Two Users with Heterogeneous
GoalsGoals
This paper was downloaded from TechRxiv (https://www.techrxiv.org).
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CC BY-NC-SA 4.0
SUBMISSION DATE / POSTED DATE
25-05-2022 / 25-05-2022
CITATION
Agheli, Pouya; pappas, nikolaos; Kountouris, Marios (2022): Semantic Source Coding for Two Users with
Heterogeneous Goals. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.19869334.v1
DOI
10.36227/techrxiv.19869334.v1
Semantic Source Coding for Two Users
with Heterogeneous Goals
Pouya Agheli∗, Nikolaos Pappas†, and Marios Kountouris∗
∗Communication Systems Department, EURECOM, Sophia Antipolis, France
†Dept. of Science and Technology, Link ¨
oping University, Norrk¨
oping Campus, Sweden
Email: pouya.agheli@eurecom.fr, nikolaos.pappas@liu.se, marios.kountouris@eurecom.fr
Abstract—We study a multiuser system in which an information
source provides status updates to two monitors with heterogeneous
goals. Semantic filtering is first performed to select the most useful
realizations for each monitor. Packets are then encoded and sent
so that each monitor can timely fulfill its goal. In this regard, some
realizations are important for both monitors, while every other
realization is informative for only one monitor. We determine the
optimal real codeword lengths assigned to the selected packet
arrivals in the sense of maximizing a weighted sum of semantics-
aware utility functions for the 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 semantic filtering and source coding.
I. INT RO DUC TIO N
Goal-oriented semantic communication is recently consid-
ered as a promising and timely research avenue towards
realizing the long-standing vision of Shannon and Weaver
[1] and incorporating the significance and the importance of
information into the existing theoretic edifice. Despite various
past endeavors [2]–[6], which remained at a conceptual level,
leading to hardly any or no practically relevant application, the
quest for such theory has recently gained new impetus [7]–[9],
fueled by the emergence of networked intelligent systems and
autonomous networks. This communication paradigm has the
potential to render various network processes more efficient and
effective, providing a parsimonious usage of communication
and computation resources. Key to this is the definition of a
concise yet insightful metric of the importance or the usefulness
of information. Age of information (AoI) [10], [11], which
measures information freshness in networks, and value of
information (VoI) [12], [13], which quantifies the information
utility or gain in decision making, can be viewed as simple,
quantitative surrogates for information semantics.
In this paper, we investigate a multiuser system in which
two monitors (users) receive status updates from a transmitter
observing an information source. The updates may correspond
to observations or measurements of a random phenomenon and
are generated from a known discrete distribution with finite
support. Semantic filtering is first performed as a means to se-
lect only the most important or valuable realizations according
to each monitor’s goal. Status updates are then encoded and
sent to the monitor(s) over packet erasure channels (PECs).
To harness packet transmission failures, an automatic repeat
request (ARQ) protocol is applied. Our objective is to design a
timely source coding scheme for two users with heterogeneous
goals. Specifically, we consider that only a fraction of the
“least” (“most”) frequent source realizations is important for
the first (second) monitor. This setting model for instance
the case in which one user is interested in regular/standard
information for monitoring purposes or typical actuation (nor-
mal 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). The notion
of semantics (importance) is captured here through a metric
of timeliness, which is a nonlinear function of AoI, for the
received updates at both monitors
This work falls within the realm of source coding problem
for status update systems in which the goal is to minimize the
average age of information, such as in [14]–[17]. In [18], we
proposed a semantics-aware encoding scheme for a single user
in an error-free point-to-point status update link. Our paper
extends prior work into multiuser systems with heterogeneous
goals, which could also be competing or diverging for certain
realizations. Specifically, we derive the optimal real codeword
lengths that maximize a weighted sum of the semantics-aware
utility functions for two heterogeneous monitors. Our analytical
and numerical results characterize the promising performance
gains by properly designing semantic filtering and source
coding for timely status update delivery.
II. SY S TE M MO DE L
We consider a multiuser network in which a transmitter
observes a physical phenomenon/event and sends status updates
(samples) to two monitors (named 1 and 2, c.f. Fig. 1).
Samples from the information source/process are generated
in the form of packets carrying different realizations from
a finite set X={xi|i∈ In},In={1,2, ..., n}, each
having a probability of occurrence ˜pi=PX(xi)where PX(·)
is a known probability mass function (pmf). The observation
sequence is independent and identically distributed (i.i.d.) and
packets arrivals are Poisson distributed with rate λ. Only the
most important status updates are transmitted, hence a semantic
filter admits the kmost important arrivals and discards the rest,
i.e., n−k, done via a flow controller. The difference between
the two monitors (receivers) lies in their significance/value
assessment policies, i.e., one observation can be essential for
one monitor while not relevant for the other. However, some
observations are important or valuable for both monitors. The
importance (semantics) of each arrival is assumed to be related
to its probability of occurrence. Specifically, we consider that
monitor 1 is only interested in the set of the k1least frequent
arrivals (Ik1), whereas the set of the k2most frequent ones
(Ik2) is highly important for monitor 2. Thus, the index
set of the admitted packets, i.e., Ik⊆ In, is defined as
Ik=Ik1∪ Ik2. In this regard, some arrivals are important
for both monitors with some non-zero probability. A source
encoder assigns codewords of length ℓito the admitted status
packets xi,∀i∈ Ik, using prefix-free coding, based on the
truncated distribution calculated as follows pi= ˜pi/qk,∀i∈ Ik
(and zero otherwise), where qk=Pi∈Ik˜pi.
Assuming no buffer is employed at the transmitter, an admit-
ted status arrival is blocked when the channel is busy. Besides,
it is assumed that each packet is delivered to both monitors
at the same time. Transmission occurs over a noisy network
which we model by two identical packet erasure channels
with erasure probability δ.1A simple ARQ protocol is used
for fixing potential transmission errors. In the event of failure
at either monitor, the transmitter, upon receipt of negative
feedback, retransmits the same packet to both monitors. The
propagation and decoding delays are considered negligible for
both transmission and feedback processes.
After successfully transferring each packet to its destination,
i.e., monitor 1, 2, or both, the transmitter waits for a new
admitted arrival. We define tr,j−1the time instant that the
j-th packet is received at both monitors concurrently but is
valuable for monitor 1 (r= 1) and/or monitor 2 (r= 2).
The update interval between the j-th successive and valuable
arrival and the next one at the same monitor is then modeled
as a random variable (r.v.) Yr,j =tr,j −tr,j−1. This interval
consists of the service time Sr,j and the waiting time Wr,j
such that Yr,j =Wr,j +Sr,j.Wr,j is the time between the
transmitted j-th packet and its previously delivered packet
at the same monitor, thus the waiting time is written as
Wr,j =PMr,j
m′=1 Zr,m′, where Mr,j is an r.v. of the number
of admitted arrivals that are generated before finding the
channel idle for the j-th packet. Zr,m′is the time between two
admitted arrivals and is exponentially distributed with mean
γr=1
λqkr, since the admitted arrivals for each monitor are
generated under a Poisson process with rate λqkr. Let us define
qkr=Pi∈Ikr˜pi≤qk. The transmission time is proportional
to the codeword length and the number of ARQ repetitions.
Thereby, the service time of realization xi,∀i∈ Ikr, which
is counted important for monitor 1 and/or 2 during the j-th
arrival becomes Sr,j =jℓitime units, where jand ℓiare
independent. Here, jshows the total number of transmissions
for the j-th packet and is geometrically distributed with success
probability 1−ǫ0, and first and second moments π1=1
1−ǫ0
and π2=1+ǫ0
(1−ǫ0)2, respectively.
1Our analysis can easily be extended to the case of different erasure
probabilities.
Source Semantic
filter
Packet
encoder
Transmitter
Monitors
Feedback
Monitor 1
Monitor 2
PECs
Fig. 1. Semantics-aware transmission over a multiuser network.
III. PROB LEM FO RMU LATIO N
In this section, we formulate the problem of optimal semantic
source coding for randomly arriving status updates.
The importance of received packets at the monitor(s) is mea-
sured using a special case of the semantics of information (SoI)
[8], [19]. Specifically, we consider timeliness as a contextual
attribute under the form of a non-increasing utility function
f:R+
0→Rof information freshness, i.e., S(t) = f(∆(t)).
∆(t) = t−u(t)is the instantaneous AoI at the receiver, defined
as the difference between the current time instant and the
timestamp u(t)of the most recently arrived update. Thereafter,
∆r(t)with indices r= 1 and r= 2 is the AoI of monitor 1 and
2, respectively. The SoI at each monitor decreases according to
f(.)until a valuable arrival for that monitor is observed. Then,
the SoI rises to the value of the new update at that time. Hence,
monitors could potentially attain different SoIs over similar
monitoring time spans. The average SoI for an observation
interval (0, T ), assuming a stationary ergodic process, is given
by ¯
Sr= lim
T→∞
1
TZT
0
f(∆r(t))dt.
A. Semantics-Aware Source Coding
Our objective is to optimally assign codeword lengths ℓi,
∀i∈ Ik, for semantics-aware encoding as a means to maximize
the weighted sum of the monitors’ average SoI, i.e., ¯
S1and
¯
S2. For the convenience of analytical derivation and to ensure
positiveness, maximizing the average SoI can be turned into
minimizing the average penalty of lateness defined as
Lr(∆r) = lim
T→∞
1
TZT
0
g(∆r(t))dt(1)
where g:R+
0→Ris a non-decreasing function [11]. For
feasibility, codeword lengths should be positive integers, i.e.,
ℓi∈Z+, and for constructing uniquely decodable codes, the
Kraft-McMillan inequality [20] has to be satisfied. Thus, the
optimization problem is formulated as
P1:min
{ℓi}∈Z+w1L1(∆1) + w2L2(∆2)
s.t. X
i∈Ik
2−ℓi≤1,(2)
where w1, w2≥0are weight parameters. To solve P1, we
relax the integer constraint for ℓito allow for non-negative real-
valued codeword lengths.2To explicitly find Lr(∆r), hence
2After computing real-valued ℓi,∀i∈ Ik, its corresponding integer value
can be obtained by using the rounded-off operation, as ⌈ℓi⌉.
exp(ρ∆(t))
exp(ρ∆(t))
t
t
0
0
t1,2
t2,2
t1,n
t2,n
t1,j−1
t2,j−1
t1,j
t2,j
S1,j S1,j+1
W1,j
S2,j S2,j+1
W2,j
Q1,j
Q2,j
Q1,∞
Q2,∞
T
T
Valuable for
Valuable for
Valuable for
monitor 2
monitor 1
both monitors
Fig. 2. Sample evolution for the EDT case over time for ρ= 0.2.
solve (2), we need to define the penalty function. In this paper,
we propose three different forms of g(.)as follows.
g(∆r(t)) =
exp(ρ∆r(t)) EDT case
ln(ρ∆r(t)) LDT case
ρ(∆r(t))κPDT case
(3)
where ρ≥0and κ∈Z+are constant coefficients. The above
cases correspond to exponentially (E-), logarithmically (L-),
and polynomially decreasing timeliness (PDT), respectively.
The average penalty of lateness is computed for all cases
using (1) and (3). We divide the positive area below the curve
of g(∆r(t)) over the interval (0, T )into polygons of Qr,j,
j= 1,2, ..., Nr(T), and Qr,∞, as in Fig. 2 for the EDT case.
Herein, Nr(T)≤ N(T)is the number of admitted packets for
monitor 1 (r= 1) or monitor 2 (r= 2) by time T, where
N(T)is the number of all admitted packets. Thus, we have
Lr(∆r) = lim
T→∞
1
TNr(T)
X
j=1
Qr,j +Qr,∞=ηrE[Qr](4)
where ηr= lim
T→∞
Nr(T)−1
Tis the steady-state time average
arrival rate. From (2)–(4), after merging η1with w1and η2
with w2as all being positive constants, we have
P2:min
{ℓi}∈R+w1E[Q1] + w2E[Q2] := JSoI
s.t. X
i∈Ik
2−ℓi≤1.(5)
IV. SEM A NT I C MULT IU S ER CO DEW ORD DE SI G N
In what follows, we find the optimal semantics-aware real
codeword lengths by solving P2for the particular cases in (3).
A. EDT Case
Using second-order Taylor expansion for the exponential
function, the area Qr,j for j≥2yields
Qr,j =Ztr,j+Sr,j+1
tr,j−1
eρ(t−tr,j−1)dt−Ztr,j +Sr,j+1
tr,j
eρ(t−tr,j)dt
≈ρ
2Y2
r,j +ρSr,j+1Yr,j +Yr,j .(6)
Therefore, E[Qr]is derived as
E[Qr]≈ρ
2E[Y2
r] + ρE[Sr]E[Yr] + E[Yr]
=ρ
2π2E[L2]r+ρπ2
1(E[L]r)2
+ (1+2ργr)π1E[L]r+ργ2
r+γr,(7)
using E[Yr] = π1E[L]r+γrand E[Y2
r] = π2E[L2]r+
2γrπ1E[L]r+ 2γ2
rwith γr= (λqkr)−1[17]. We also have
E[Sr] = π1E[L]rand E[S2
r] = π2E[L2]rwhere E[L]r=
Pi∈Ikrpiℓiand E[L2]r=Pi∈Ikrpiℓ2
iindicate the first and
second moments of the codeword lengths, respectively.
Applying (7), the objective function for the optimization
problem P2becomes
JSoI =ρ
2π2w1E[L2]1+w2E[L2]2
+ρπ2
1w1(E[L]1)2+w2(E[L]2)2
+w1(1+2ργ1)π1E[L]1+w1ργ2
1+w1γ1
+w2(1+2ργ2)π1E[L]2+w2ργ2
2+w2γ2.(8)
Proposition 1. The codeword length ℓi,∀i∈ Ik, that mini-
mizes (8) in P2for the EDT case is given by
ℓi=−ln2αpi
µ(ln(2))2W0
µ(ln(2))2
αpi
2β
α (9)
where µ≥0is the Lagrange multiplier, α=ρπ2(1+2),
β=2µρ ln(2)π2
1(1χ1+2χ2) + αθπ1
α+ 2ρπ2
1(1χ1+2χ2),(10)
θ=1(1+2ργ1) + 2(1+2ργ2), and W0(.)is the principal
branch of Lambert Wfunction. Moreover, 1:=w11and
2:=w22, with 1, 2∈ {0,1}being indicator parameters,
initialized as follows: (1= 1, 2= 0),(1=2= 1), and
(1= 0, 2= 1) if realization xibelongs to set A=Ik1−B,
B=Ik1∩ Ik2, and C=Ik2− B, respectively.
Before proceeding with the proof, we remark that the values
of χ1and χ2are calculated using Algorithm 1. First, we as-
sume uniformly distributed pi,∀i∈ Ik, and assign χ(0)
1=k1/k
and χ(0)
2=k2/k. Then, we find ℓiand compute new χ1and χ2.
Based on them, the new values for ℓiare found. This process
continues until the convergence criterion εis satisfied. The
algorithm converges to the final values of χ1,χ2,β, and ℓiwith
the rate of O((NnNm)−1)in which Nnand Nmdenote the
maximum numbers of inner and outer iterations, respectively.
Proof: By (8), we define the Lagrange function for P2as
L(ℓi;µ) = ρ
2π2w1X
i∈Ik1
piℓ2
i+w2X
i∈Ik2
piℓ2
i
+ρπ2
1w1X
i∈Ik1
piℓi
2+w2X
i∈Ik2
piℓi
2
+w1(1+2ργ1)π1X
i∈Ik1
piℓi+w2(1+2ργ2)π1X
i∈Ik2
piℓi
+w1ργ2
1+w2ργ2
2+w1γ1+w2γ2+µX
i∈Ik
2−ℓi−1
(11)
where µ≥0is the Lagrange multiplier. Then, we write the
Karush-Kuhn-Tucker (KKT) conditions for i∈ Ik, as follows:
∂L(ℓi;µ)
∂ℓi
=ρπ2(1+2)piℓi+ 2ρπ2
11X
i∈Ik1
piℓi
+2X
i∈Ik2
piℓi
pi+1(1+2ργ1)π1pi
+2(1+2ργ2)π1pi−µln(2)2−ℓi= 0 (12)
where 1:=w11and 2:=w22with 1, 2∈ {0,1}
being indicator parameters. Let us split set Ikinto three disjoint
sets of A,Ik1− B,B,Ik1∩ Ik2, and C,Ik2− B.
Thus, we have (1=w1, 2= 0),(1=w1, 2=w2), or
(1= 0, 2=w2)if xibelongs to A,B, or C, respectively.
The complementary slackness condition is
µX
i∈Ik
2−ℓi−1
= 0.(13)
There exist two conditions, one of which meets (13): (i) µ= 0,
Pi∈Ik2−ℓi<1; or (ii) µ6= 0,Pi∈Ik2−ℓi= 1. Under
(i), the right hand side of (12), in which all terms are non-
negative, equals to zero, resulting in ℓi= 0,∀i∈ Ik. Since
negative codeword lengths are not meaningful, by contradic-
tion, condition (ii) must satisfy (13). Summing (12) over all
codeword indices, it is hard to find a closed-form expression
based on E[L],E[L]1and E[L]2. To this end, we introduce
two parameters χ1and χ2, which captures the nonlinear
relationship with k1and k2, respectively. Specifically, we define
χ1=E[L]1
E[L]and χ2=E[L]2
E[L], with 0≤χ1≤1and 0≤χ2≤1.
Now, for given χ1and χ2, one can derive E[L],E[L]1, and
E[L]2. After some calculations, we reach
µ(ln(2))2
αpi
2−ℓiexpµ(ln(2))2
αpi
2−ℓi
=µ(ln(2))2
αpi
2β
α(14)
where α=ρπ2(1+2), and βis given in (10). The form
of (14) is equal to xexp(x) = yfor which the solution is
x=Wm(y), where m= 0 for y≥0.
In order to find the optimal codeword lengths, we start from
a value of µthat satisfies Pi∈Ik2−ℓi= 1. Then, its value is
updated by the use of E[L],E[L]1, and E[L]2.
B. LDT Case
In this case, the area Qr,j for j≥2yields
Qr,j =Ztr,j+Sr,j+1
tr,j−1
ln(ρ(t−tr,j−1))dt
−Ztr,j+Sr,j +1
tr,j
ln(ρ(t−tr,j))dt
≈ρY 2
r,j + 2ρSr,j+1Yr,j −2Yr,j ,(15)
Algorithm 1: Solution for deriving χ1and χ2
Input: Fixed parameters Ik,Ik1,Ik2, and pi,∀i∈ Ik.
Stopping accuracy ε. Initial parameters µ(0),
χ(0)
1,χ(0)
2,β(0), and ℓ(0)
i,∀i.
Output: Final-form parameters χ1=χ(n)
1,χ2=χ(n)
2,
β=β(n),ℓi=ℓ(n)
i,∀i, and µ=µ(m).
1Iteration m:
2Iteration n:
3Update β(n)and ℓ(n)
iusing (10) and (9), respectively.
4Compute E[L] = Pi∈Ikpiℓ(n)
i,E[L]1=Pi∈Ik1piℓ(n)
i,
and E[L]2=Pi∈Ik2piℓ(n)
i.
5Update χ(n)
1and χ(n)
2based on 4.
6if Criterion
χ(n)
1−χ(n−1)
1
>ε or
χ(n)
2−χ(n−1)
2
>ε
then set n=n+1, and goto 2.
7Compute β(n)from (10), and derive ℓ(n)
ifrom (9).
8if Pi∈Ikpiℓ(n)
i=1 then stop the process, and goto 11.
9else if Pi∈Ikpiℓ(n)
i<1then decrease µ(m), set
m=m+1, and goto 1.
10 else increase µ(m), set m=m+1, and goto 1.
11 Save χ(n)
1,χ(n)
2,β(n),ℓ(n)
i,∀i, and µ(m).
which results in
E[Qr]≈ρπ2E[L2]r+ 2ρπ2
1(E[L]r)2
+ 2(2ργr−1)π1E[L]r+ 2ργ2
r−2γr.(16)
Consequently, we obtain the following objective function.
JSoI =ρπ2w1E[L2]1+w2E[L2]2
+ 2ρπ2
1w1(E[L]1)2+w2(E[L]2)2
+ 2w1(2ργ1−1)π1E[L]1+ 2w1ργ2
1−2w1γ1
+ 2w2(2ργ2−1)π1E[L]2+ 2w2ργ2
2−2w2γ2.(17)
Putting (17) into P2and following the same procedure as (11)–
(14), we obtain the optimal codeword length ℓi,∀i∈ Ik, as
ℓi=−ln2α′pi
µ′(ln(2))2W0
µ′(ln(2))2
α′pi
2β′
α′ (18)
where µ′≥0,α′= 2ρπ2(1+2),
β′=4µ′ρln(2)π2
1(1χ1+2χ2) + 2α′θ′π1
α′+ 4ρπ2
1(1χ1+2χ2),(19)
and θ′=1(2ργ1−1) + 2(2ργ2−1). For i∈ A and i∈ C,
we initialize 2= 0 and 1= 0, respectively. Otherwise, 1=
2= 1. The values of χ1and χ2are found using Algorithm 1
and replacing the parameters of ℓi, such as βwith β′.
C. PDT Case
Setting κ= 1 for exposition convenience, we obtain
Qr,j ≈ρ
2Y2
r,j +ρSr,j+1Yr,j ,(20)
for which the expected value is derived as
E[Qr]≈ρ
2π2E[L2]r+ρπ2
1(E[L]r)2
+ 2ργrπ1E[L]r+ργ2
r.(21)
Similar to IV-A and IV-B, we get JSoI and solve P2. The
solution for optimal ℓi,∀i∈ Ik, is
ℓi=−ln2αpi
µ′′(ln(2))2W0
µ′′(ln(2))2
αpi
2β′′
α (22)
where µ′′ ≥0, and if θ′′ = 21ργ1+ 22ργ2, we have
β′′ =2µ′′ρln(2)π2
1(1χ1+2χ2) + αθ′′π1
α+ 2ρπ2
1(1χ1+2χ2).(23)
We set 2= 0 and 1= 0 for i∈ A and i∈ C, respectively, or
otherwise 1=2= 1. The values of χ1and χ2are obtained
from Algorithm 1 by substituting the parameters of ℓi.
D. Asymptotic Expansions
Using W0(y)
yξ= exp(−ξW0(y)), for y≥e, and W0(y) =
ln(y)−ln(ln(y)) + ≀(1) for large y, we can write ℓi∝
c/pi−ln(c/pi), for pi≤c
e≤1, with c:=µ(ln(2))2
α2β
α. This
expression, which is monotonically decreasing, corroborates
that the higher the probability of occurrence of a realization, the
shorter the assigned codeword length, and vice versa. pi→0
yields ℓi≫1.
Furthermore, consider a uniform pmf for the realizations, i.e.,
pi= 1/n,∀i. In that case, we have codewords of equal size,
i.e., ℓ1=ℓ2=... =ℓn, with ℓi∝n−ln(n). Moreover, n≫1
results in ℓi=ℓmax,∀i∈ Ik, where ℓmax is an upper bound
to which the size of a codeword length converges, and ℓmax
remains almost fixed for the large enough n. As the number of
realizations increases, the assigned codeword lengths become
longer at the expense of longer service time.
V. NUM ERI CA L RE SU LTS
In this section, we provide numerical results for the opti-
mal number of selected status updates (packets) in different
scenarios, highlighting the advantages of semantic filtering
and source coding. We use Zipf(n, s) distribution with pmf
PX(x) = 1/xs
Pn
j=1 1/js,n=|X| = 100, and s= 0.4. The
parameter sallows us to vary from a uniform distribution
(s= 0) to a “peaky distribution”. The packet error rate
ǫ0is initialized according to its upper bound Φδ
√δ(1−δ)
[21], where δand Φ(.)denote the erasure probability and
the cumulative distribution function of the standard normal
Gaussian distribution, respectively. Unless otherwise stated, we
set ρ= 0.2,δ= 0.5,w1=w2= 1, and T= 10 [sec].
Fig. 3 depicts the value of JSoI versus the number of
selected (important) realizations for the EDT case. We observe
that increasing the arrival rate reduces JSoI, as well as the
optimal k1and k2values. In addition, the transmitter filters a
considerably higher number of frequent arrivals than infrequent
ones. However, no filtering (k1, k2→n) results in performance
degradation due to spending time for sending insignificant
Fig. 3. The interplay between JSoI and the number of selected realizations
k1and k2for the EDT case with n= 100 and w1=w2= 1.
TABLE I
OPT IM AL N UM BE R O F S EL EC TE D PACK ET S F OR n= 100.
EDT case LDT case PDT case
Arrival rate k1k2k1k2k1k2
λ= 0.1 100 94 100 91 100 89
λ= 0.5 100 45 100 42 100 41
λ= 1 45 14 46 13 46 12
λ= 2 45 10 46 8 46 8
Fig. 4. The interplay between JSoI and the number of selected realizations
k1and k2for the EDT case with n= 100,λ= 1, and δ= 0.5.
realizations. Likewise, the derived optimal values of k1and
k2for the LDT and PDT cases with different arrivals rates
are listed in Table I. We see that the exponential penalty
results in the lowest and highest values for optimal k1and
k2, respectively, compared to the other forms.
Fig. 4 shows the effects of weight parameters in the objective
function JSoI and the number of selected realizations for the
TABLE II
OPT IM AL N UM BE R O F S EL EC TE D PACK ET S F OR n= 100 AND λ= 1 .
Weight parameters (w1, w2)
(1,1) (10,1) (1,10)
Erasure probability k1k2k1k2k1k2
δ= 0.25 47 17 100 60 33 34
δ= 0.5 45 14 50 5 33 33
Fig. 5. The objective function JSoI versus arrival rate λfor the EDT case
with n= 100 and w1=w2= 1.
EDT case, with λ= 1, and δ= 0.5. Varying each weight
parameter alters the optimal values of k1and k2, hence JSoI.
Notably, giving ten times more weight to the arrivals of monitor
2 compared to those of monitor 1 equalizes the optimal k1
and k2. The transmitter equally filters around 33% of frequent
and infrequent arrivals. The obtained information from Fig. 4
and from its extension for δ= 0.25 is given in Table II. We
observe that higher erasure probability results in fewer selected
packets since the transmitter spends more time to retransmit the
unsuccessful packets. The same conclusions hold for the LDT
and PDT cases.
Fig. 5 presents the objective function JSoI versus the arrival
rate λfor different numbers of selected packets and the EDT
case. Increasing the arrival rate decreases JSoI, which in
turn diminishes and saturates at higher rates. Furthermore,
increasing both or one of k1or k2means that the arrival
rate required to decrease the objective function diminishes. For
instance, the minimum values of arrival rates (hence, JSoI) for
k1=k2=n/4,k1=k2=n/2, and k1=k2=nare around
11(14),17(11),20(2), respectively. Thus, for large k1and k2,
the objective function takes high values for any arrival rates.
According to the analytical expressions derived for the EDT
case, we find the global optimal values of λ∗≃10.27,17.63,
and 19.12 sequentially for k1=k2=n/4,k1=k2=n/2,
and k1=k2=n. Likewise, Fig. 5 can be plotted for the LDT
and PDT cases, resulting in similar interpretations.
VI. CONCLUSION
We studied the timely source coding problem in a two-user
status update system, where observations of an information
source are filtered and sent to two monitors depending on their
importance for achieving each user’s goal. Optimizing the code-
word length according to semantics-aware utility functions,
the amount of status updates communicated is significantly
reduced. Our analytical and numerical results show that there
is an optimal value of realizations to send, which depends
on the source distribution, the arrival rate, and the weight
of each monitor to maximize the value of the communicated
information through the network.
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