Age Analysis of Correlated Information in Multi-Source Updating Systems with MAP Arrivals

Abstract

Information freshness in cyber-physical systems is essential for real-time applications that involve perception and control. Many of these applications rely on data generated by multiple correlated sources. To capture the inherent dependency in status update packets, we quantify the age of correlated information (AoCI) in a multi-source queueing system where each source generates packets following a Markovian arrival process (MAP) and these packets are subsequently served with exponentially distributed service times. Using the matrix geometric approach, we derive the sojourn time distribution in this MAP/M/1 queueing system. Furthermore , we obtain closed-form expressions for the average and peak AoCI in terms of the system parameters, considering both single and multiple source scenarios. Numerical results show the impact of correlated arrivals with different distribution properties on the achievable AoCI. Index Terms-Age of information, MAP/M/1 queueing model, correlated sources, matrix geometric approach, renewal processes.

Full text

1
Age Analysis of Correlated Information in
Multi-Source Updating Systems with MAP Arrivals
Muthukrishnan Senthil Kumar, Member, IEEE, Aresh Dadlani, Senior Member, IEEE,
Omid Ardakanian, Member, IEEE, Ioanis Nikolaidis, Member, IEEE, Janelle J. Harms
Abstract—Information freshness in cyber-physical systems is
essential for real-time applications that involve perception and
control. Many of these applications rely on data generated by
multiple correlated sources. To capture the inherent dependency in
status update packets, we quantify the age of correlated information
(AoCI) in a multi-source queueing system where each source gener-
ates packets following aMarkovian arrival process (MAP) and these
packets are subsequently served with exponentially distributed
service times. Using the matrix geometric approach, we derive the
sojourn time distribution in this MAP/M/1 queueing system. Fur-
thermore, we obtain closed-form expressions for the average and
peak AoCI in terms of the system parameters, considering both
single and multiple source scenarios. Numerical results show the
impact of correlated arrivals with different distribution properties
on the achievable AoCI.
Index Terms—Age of information, MAP/M/1 queueing model,
correlated sources, matrix geometric approach, renewal processes.
I. INTRODUCTION
THE advent of wireless technologies such as 5G and beyond
has enabled widespread adoption of networked sensors for
time-sensitive applications. Conventional performance metrics
like throughput and latency fail to quantify the freshness of re-
ceived information due to varying wireless conditions, prompt-
ing the introduction of age of information (AoI) as a destination-
centric metric [1], [2]. Assuming each status update packet con-
tains measurement data and a timestamp, AoI is formally defined
as ∆(t) = t−u(t), where u(t)is the timestamp of the most
recently received packet at the destination. Without fresher
updates, ∆(t)increases linearly with time. Upon receiving a
fresher status update, the destination updates u(t), reducing AoI.
Inspired by a real-time multi-view image processing appli-
cation, the age of correlated information (AoCI) was first
introduced in [3], where a complete scene image is reconstructed
by a remote monitor upon receiving overlapping field-of-view
images from multiple cameras. This concept proves valuable in
low-latency communication systems where the AoI at the des-
tination changes only upon receiving the fresher (partial) status
updates from correlated sources. In [4], an optimal scheduling
policy that jointly minimizes the average AoI and the energy
cost associated with two devices observing the same physical
process (hence being correlated) was proposed. A multi-armed
Muthukrishnan S. Kumar is with the Department of Applied Mathemat-
ics and Computational Sciences, PSG College of Technology, Coimbatore
641004, India (e-mail: msk.amcs@psgtech.ac.in).
Aresh Dadlani, Omid Ardakanian, Ioanis Nikolaidis, and Janelle J.
Harms are with the Department of Computing Science, University of Alberta,
Edmonton, AB T6G 2E8, Canada. (e-mails: {aresh, ardakanian, nikolaidis,
janelleh}@ualberta.ca).
bandit algorithm was explored in [5] to minimize the average AoI
in IoT networks with stochastically identical and non-identical
multi-channel setups, in the presence of correlated information
sources. It has been shown in [6] that in a non-collaborative
game setup, the competition among sources in an M/M/1 sensing
system can be influenced by the correlation of their content. To
minimize AoCI through adaptive scheduling, the challenge of
latent battery states was addressed in [7] by formulating the
dynamic update procedure as a partially observable Markov
decision process. In [8], novel scheduling policies were explored
to optimize the weighted-sum of average AoI in scenarios with
time-dependent correlations.
Given the ubiquity of data dependency that exists among mul-
tiple sources in practice, the analysis of AoCI through the lens
of queueing theory under generally-distributed update arrival
times becomes indispensable. Such an analytical framework is
essential for understanding the interplay among the number of
sources, their offered traffic, and the minimum achievable AoCI.
However, no such analysis, particularly within the context of a
multi-source queueing system, has been done yet.
In this letter, we investigate an updating system with correlated
sources, where the arrivals of status updates follow a Markovian
arrival process (MAP). This tractable subclass of Markov re-
newal process, which includes Poisson, Erlang-renewal, and PH-
renewal processes, is typically used to capture correlations in
packet inter-arrival times, reflecting various traffic patterns [9],
[10]. Our main contributions are: (i) derivation of a closed-form
expression for the sojourn time distribution of update packets in
a MAP/M/1 model, (ii) derivation of closed-form expressions for
the average and peak AoCI metrics in single and multiple-source
scenarios, and (iii) numerical evaluation of the average AoCI
under different arrival processesand varying system parameters.
II. SY ST EM MO DE L
Consider a system with a set of sources N(|N |=N) shown in
Fig. 1. These sources can be partitioned into multiple subsets
such that each subset contains correlated sources producing real-
time status information pertaining to a separate and independent
physical process. These status packets are then transmitted to
a receiver, where they are enqueued and processed in a first-
come-first-served (FCFS) manner, with service times following
an exponential distribution parameterized by µ. Packets from
each source i∈N arrive based on a continuous-time MAP, with
a mean arrival rate of λi>0. The symmetric matrix X=
(xi,j )∈{0,1}N×Nencodes the correlation relationship between
any ordered pair of sources (i, j). Specifically, xi,j=1 indicates
dependency between updates from iand jimplying they are in

2
Physical Processes
D
Destination
MAP arrivals Server
Sources
Fig. 1. Schematic of the MAP/M/1 system with MAP update arrivals from N
sources, grouped by correlated data sensed from distinct physical processes.
the same subset, and xi,j = 0 indicates no dependency, thus
updates from both sources are not required to reconstruct the
real-time status of the respective processes.
A. Time-Average AoCI Definition
Let ti,k denote the generation timestamp of the k-th update
from each i∈N , and t′
i,k be its arrival time at destination D.
Fig. 2 illustrates the contrast between the ages under independent
and correlated packets in a two-source system. For uncorre-
lated sources (Fig. 2a), the AoI of source iat D is given by
∆i(tk) = t′
i,k −ti,k, where tk∈[t′
i,k, ti,k +1)[3]. When the
queue is empty at t= 0, the age is ∆i(0). To quantify the
AoI for any arbitrary pair of correlated sources (i, j), one must
account for the temporal difference between the update received
at the destination and the generation time of the first packet in
the pair. As depicted in Fig. 2b, assuming ti,k < tj,k, the AoCI
for source iat time tkis computed as ∆i(tk) = t′
j,k −ti,k, where
tk∈[t′
j,k, ti,k +1)and t′
j,k > t′
i,k. In either case, the status age
at D increases linearly as packets arrive and the limiting average
age for the updates from source iover the interval (0, T )is given
as [11], [12]:
¯
∆i= lim
T→∞
1
TZT
0
∆i(t)dt=E[Z W ]+E[Z2]/2/E[Z],(1)
where Zdenotes the inter-arrival time between consecutive
updates from source i, and Wrepresents the sojourn (waiting)
time, which includes both the queueing time and the subsequent
service time for such update packets.
B. Single-Source MAP/M/1 Model Formulation
Consider a two-dimensional stochastic process {Np(t), A(t)}
on the state space Ω={(k, i); k≥0,1≤i≤m}where Np(t)
is the number of update packet arrivals from the single source in
(0, t]and A(t)represents the phase of the arrival process. For
any phase i∈A(t), the Poisson arrival rate is γi=Pm
j=1 γi,j . A
state transition occurs in two cases: (i) with rate γi,j , there is a
transition from state (k, i)to state (k+ 1, j)due to an update
arrival and possibly a phase shift, and (ii) with rate αi,j, there is
a transition from state (k, i)to state (k, j )(where i=j) due to
a phase shift. The phase process is a continuous-time Markov
chain (CTMC) with an irreducible generator D=D0+D1,
where D0= (d(0)
i,j )and D1= (d(1)
i,j )are m-order matrices
defined as [10]:
d(0)
i,j =(−γi−Pj=iαi,j ,if j=i;
αi,j ,if j=i, for 1≤i, j ≤m, (2)
d(1)
i,j =γi,j ,for 1≤i, j ≤m. (3)
time
(a)
time
(b)
Fig. 2. Age evolution for source 1in a two-source system with (a) uncorre-
lated and (b) correlated update packets generated at times {t1,1, t2,1}.
Matrix D0contains negative diagonal elements and non-
negative off-diagonal elements, reflecting phase transitions with-
out packet arrivals, while D1comprises non-negative elements
corresponding to arrival rates in the mstates. The infinitesimal
generator matrix Q∗for the quasi-birth-death (QBD) process
of this MAP/M/1 model exhibits the following structure, where
A0=D1,B0=D0,A1=D0−µI,A2=µI, and Iis the
identity matrix:
Q∗=


B0A0· · · · · ·
A2A1A0· · ·
.
.
........
.
.


.(4)
Let A=A0+A1+A2be the stochastic matrix and vbe
the steady-state probability vector of the generator Asatisfying:
v A = 0; v e = 1,(5)
where eis the unit column vector. The system is stable if and only
if vA0e<vA2e. Moreover, the rate of update arrivals per unit
time is given by λ=v D1e. Consequently, the invariant prob-
ability vector for Q∗, represented as π= (π0,π1,...,πm), is
the unique positive solution to the following system [13]:



π0(B0+RA2)=0,
πi=π0Ri, i ≥1,
π0(I−R)−1e= 1,
(6)
where Ris the minimal non-negative solution to the matrix
equation R2A2+RA1+A0= 0 and πi, for i≥0, is a
stationary vector of dimension m.
C. Multi-Source MAP/M/1 Model Formulation
The traffic originating from Nsources, with dependency ma-
trix X, can be effectively modeled using an extended MAP
characterized by matrices (D0,D1,D2,...,DN), each of order
m. Here, D0governs transitions for no arrivals, while Digoverns
transitions for update arrivals from source i∈ N . The associated
CTMC has D=D0+PN
i=1 Diand the packet arrival rate per
unit time from source iis λi=vDie, with the total rate
λ=PN
i=1 λi. The infinitesimal generator Q∗maintains a
similar structure as in (4), with sub-matrices A0=PN
i=1 Di,
A2=µI,A1= (D0−µ)I, and B0=D0. Moreover, the
stochastic matrix A=A0+A1+A2and vsatisfy (5). Solving
the same system as in (6), the steady-state probability vector π
can be determined.
III. AVE RAG E AOCI A NALYSI S
Building on the structural findings from the previous section,
we now derive the sojourn time distribution and average AoCI
in closed form for both single and multiple source scenarios.

3
A. AoCI Derivation for Single-Source Model
To obtain the sojourn time, we define Las the number of
update packets in the system just before the arrival of a new
packet, and Skas the exponential service time of the k-th packet.
The packet currently in service has a residual service time instead
of an exponential service time with a mean of 1/µ. As a result,
Skare identically distributed independent random variables and
the sojourn time random variable (W)includes the service time
of the packet in service. Hence, Wfollows the Erlang-(L+ 1)
distribution with mean 1/µ, i.e., W=PL+1
k=1 Sk. Conditioning
on Land using the independence between Skand L, the density
of the sojourn time distribution can be expressed in terms of the
following matrix exponential function:
f
W(t)=
∞
X
n=0
f
W|L=n
(t) Pr(L=n)=
∞
X
n=0
µ(µt)n
n!e
−(µt)(π0Rne)
=π0µ e−µ(I−R)te.(7)
The cumulative distribution function (CDF) and probability
density function (PDF) of the inter-arrival time (Z)for a packet
in a MAP/M/1 queue are respectively given as [14]:
(FZ(t) = v[I−eD0t](−D0)−1D1e;for t≥0,
fZ(t) = veD0tD1e;for t≥0.(8)
Using (8), the moments for Zare obtained to be:







E[Z] = R∞
−∞ zfZ(t)dt =vD1(−D0)−1e
vD1e=1
λ,
E[Z2] = R∞
−∞ z2fZ(t)dt =2v(−D0)−1e
λ.
(9)
Considering the service times as i.i.d. exponential random
variables with an average of 1/µ, the sojourn time for update
packet kis Wk=WQ
k+Sk, where WQ
kand Skdenote the
waiting and service times for packet k, respectively. If WQ
k=0
then packet k−1was served before packet kwas generated. At
the arrival of the k-th packet, k−1packets are in queue or being
served, making WQ
k=(Wk−1−Zk)+. Thus, the conditional
expected waiting time for the k-th packet, given Zk=x, is:
E[WQ
k|Zk=x] = E[(Wk−1−x)+|Zk=x] = E(W−x)+
=Z∞
x
(t−x)fW(t)dt
=1
µπ0e−µ(I−R)x(I−R)−2e.(10)
From (9), we derive the expression for E[ZkWk], under the
assumption that the service time Skand Zkare independent:
E[ZkWk] = E[Zk(WQ
k+Sk)] = E[ZkWQ
k+ZkSk]
=E[ZkWQ
k] + E[Zk]E[Sk].(11)
Theorem 1. The closed-form expression for E[ZkWQ
k]is:
E[ZkWQ
k] = 1
µ2π0RvD1(I−R−D0
)
−2(I−R)
−2e.(12)
Proof. Using the PDF of the inter-arrival time given in (8) and
(10), E[ZkWQ
k]can be reformulated as:
E[ZkWQ
k]=Z∞
0
tE[WQ
i|Zi=t]fZi(t)dt
=Z∞
0
t1
µπ0e
−µ(I−R)t(I−R)
−2eveD0tD1edt,
which after integration yields (12).
Applying Theorem 1, we now articulate the average and peak
AoCI (¯
∆p)for the single-source scenario in Theorem 2 below.
Theorem 2. The closed-form expressions for the average and
peak AoCI for a single-source MAP/M/1 model are given as:







¯
∆= λ
µ2π0RvD1(I−R−D0)
−2(I−R)
−2e+1
µ+v(−D0)
−1e,
¯
∆p=1
λ1 + π0R(I−R)−2e.
(13)
Proof. Upon substituting (11) in (1), we obtain:
¯
∆ = λ(E[ZkWQ
k] + E[Zk]E[Sk] + E[Z2
k]/2).
Then using (9) and (12), along with E[Sk]=1/µ, leads to the
closed-form expression for ¯
∆. Peak AoCI is the maximum value
of the age achieved before the latest update delivery. From (4)
and Little’s law, the average waiting time is computed as follows,
where Npdenotes the number of packets in the system:
E[Wk] = E[Np]
λ=P∞
n=1 nπn
λ=π0R(I−R)−2e
λ.(14)
Summation of E[Zk]from (9) and E[Wk]from (14) yields ¯
∆p,
which completes the proof.
B. AoCI Derivation for Multi-Source Model
To calculate the sojourn time under correlated sources, we de-
fine Si,k as the exponential service time of the k-th packet from
source i. For any two sources i, j ∈ N ,xi,j essentially signifies if
the current update at icontains information dependent on updates
from jduring Si,k. Given this dependency matrix Xand the fact
that Si,k’s are i.i.d., the sojourn time (Wi)includes the service
time of the update from i. Consequently, Wiconforms to the
Erlang-(L+ 1) distribution with a mean of 1/(µPN
j=1 xj,i):
Wi=
L+1
X
k=1
N
X
j=1
Si,k xj,i , i ∈ N ,(15)
and the conditional density of Wigiven Lis derived to be:
fWi(t) =
∞
X
n=0
N
X
j=1
µ(µPN
j=1 xi,j t)n
n!e
−(µPN
j=1 xi,j t)(π0Rne)
=π0µ
N
X
j=1
xi,j e−µPN
j=1 xi,j (I−R)te.(16)
Moreover, the inter-arrival time of an update packet from
source i(Yi)has the following moments:
E[Yi]= vDi(−D0)−1e
vDie=1
λi
,E[Y2
i]= 2v(−D0)−1e
λi
.(17)
By applying (15), (16), and (17), the closed-form expressions
for ¯
∆iand ¯
∆p
ifor any arbitrary source iare derived to be:

4



















¯
∆i=λiπ0R vDi(I−R−D0)−2(I−R)−2e
µPN
j=1 xj,i2
+1
µPN
j=1 xji
+v(−D0)−1e,
¯
∆p
i=1
λi1 + π0R(I−R)−2e).
(18)
IV. NUMERICAL RES ULT S AN D DISCUSSIONS
We validate our results for the average AoCI and show the im-
pact of correlated sources under varying model parameters. For
the arrival process, we examine the following five distinct sets
of values for D0and D1, as considered in [9], [15] and [16]:
•Exp.: The classical Poisson process with parameter λ, such
that D0= [−λ]and D1= [λ].
•Erl.: The Erlang inter-arrival time distribution of order 2
with parameter λ, such that D0= [−λ, λ; 0,−λ]and D1=
[0,0; λ, 0].
•H2: The combination of two exponential distributions (hy-
perexponential), such that D0= [−1.9λ, 0; 0,−0.19λ]
and D1= [1.71λ, 0.19λ; 0.171λ, 0.019λ], with mixing
probability p= 0.9.
•Neg.: A MAP exhibiting negative correlation be-
tween consecutive inter-arrival times, with D0=
λ[−1.0024,1.0024,0; 0,−1.0024,0; 0,0,−225.797] and
D1=λ[0,0,0; 0.01002,0,0.9924; 223.539,0,2.258].
•Pos.: A MAP with positive correlation between
consecutive inter-arrival times, where D0=
λ[−1.0024,1.0024,0; 0,−1.0024,0; 0,0,−225.797] and
D1=λ[0,0,0; 0.9924,0,0.01002; 2.258,0,223.539].
At λ=1, the five arrival processes exhibit standard devia-
tions of 1,0.5,2.245,1.4095, and 1.4095, respectively. The
corresponding 1-lag correlation coefficients for successive inter-
arrival times are 0,0,0,−0.4889, and 0.4889.
Fig. 3 shows the impact of data dependency in a two-source
system by comparing the average AoCI of source 1 under
both uncorrelated (x1,2=0) and correlated (x1,2=1) scenarios
across various inter-arrival time distributions, with µ=1 and
λ2={0.2,0.8}. As evident in Fig. 3a, Erl. yields the lowest aver-
age age among the three renewal distributions, with λ∗
1=0.53 for
uncorrelated and λ∗
1=0.47 for correlated sources, attributed to its
low variability in inter-arrival times. Moreover, H2, with a coeffi-
cient of variation greater than 1, consistently yields higher AoCI
values. To account for traffic models with statistically significant
correlations across large time scales, Fig. 3b contrasts Neg.
and Pos., characterized by dependent packet inter-arrival times
with the same variance. Under Neg. arrivals and for λ2=0.2,
the average AoCI decreases to a minimum of 0.95 at λ∗
1=0.6
when the two sources generate correlated updates, marking a
2.7% increase compared to scenarios where the sources update
independently. This is due to the formation of clusters of short
and long inter-arrival times in Pos. arrivals, resulting in higher
AoCI, while Neg. arrivals exhibit the opposite trend. That is to
say, Pos. arrivals tend to increase AoCI due to short inter-arrival
times being followed by another short one, leading to abrupt
spikes, whereas Neg. arrivals show a reduction in average AoCI,
as short intervals are typically followed by longer ones.
(a) Comparison of ¯
∆1among the three renewal processes.
(b) Comparison of ¯
∆1between Neg. and Pos. processes.
Fig. 3. The average AoCI of source 1 against λ1in a two-source system,
where λ2={0.2,0.8}and µ=1, illustrating the impact of correlated sources
under (a) independent and (b) dependent inter-arrival time distributions.
Fig. 4 compares the peak AoCI for the same two-source model,
highlighting the maximum AoCI achievable within the stable
range of λ1(i.e., Pi∈N λi<µ) for low and high traffic rates from
source 2. Both Fig. 4a and Fig. 4b show a substantial increase in
the peak AoCI when x1,2=1 for Exp. and H2distributions under
λ2=0.8. Specifically, in Fig. 4a, these peak age values reach
1235 and 3682, respectively (not shown). In Fig. 4b, with Pos.
arrivals, the minimum peak AoCI reaches 3388 at λ∗
1=0.1for
λ2=0.2, signifying a 98% increase compared to the uncorrelated
scenario. Similar to Fig. 3b, ¯
∆p
1becomes almost negligible for
higher λ1rates.
Fig. 5 illustrates the impact of data dependency for larger N
values, comparing average AoCI for N={4,6}. This compari-
son is based on matrix X, where sources {1,2},{2,4,6}, and
{3,4}are interdependent, while 5is independent of the others.
As seen in Fig. 5a, the minimum average age for source 1
increases by nearly 49% as Ngoes from 4 to 6 in the Erl. dis-
tribution. In line with findings from Fig. 3a, the H2distribution
shows the largest increase, approximately 60.9%, for the same
rise in N. An evident trend emerges as Nincreases, showing a
corresponding rise in dependency between source pairs. Hence,
the arrival rates conducive to achieving the minimum AoCI also
decreases. For instance, in Erl., increasing Nfrom 4 to 6 reduces
λ∗
1from 0.25 to 0.22. In Fig. 5b, a slight increase in the minimum
achievable AoCI for Neg. is observed as Nincreases. This is
consistent with the trend observed for i.i.d. arrivals within the
system stability range. For Pos., however, this increase is about

5
(a) Comparison of ¯
∆p
1among the three renewal processes.
(b) Comparison of ¯
∆p
1between Neg. and Pos. processes.
Fig. 4. The peak AoCI of source 1 against λ1in a two-source system, where
λ2={0.2,0.8}and µ=1, illustrating the impact of correlated sources under
(a) independent and (b) dependent inter-arrival time distributions.
9.2% which occurs at a relatively lower value of λ1=0.1as
compared to all other distributions.
V. CONCLUSION
In this article, we examined the average and peak AoCI for a
multi-source MAP/M/1 queueing system with correlated packet
arrivals and data correlation among sources, deriving the closed-
form stationary and sojourn time distributions using the matrix
geometric approach. We validated the analytical results and com-
pared the minimum achievable average AoCI across correlated
sources, considering updates with different inter-arrival time
distributions. A potential future direction is the analysis of AoCI
under time-varying data dependency between sources in mobile
and energy-harvesting updating systems.
ACKNOWLEDGMENTS
This research is supported in part by funding from the Natural
Sciences and Engineering Research Council of Canada.
REFERENCES
[1] M. A. Abd-Elmagid, N. Pappas, and H. S. Dhillon, “On the role of age of
information in the Internet of Things,” IEEE Commun. Mag., vol. 57,
no. 12, pp. 72–77, Dec. 2019.
[2] R. D. Yates, Y. Sun, D. R. Brown, S. K. Kaul, E. Modiano, and
S. Ulukus, “Age of information: An introduction and survey,” IEEE J.
Sel. Areas Commun., vol. 39, no. 5, pp. 1183–1210, May 2021.
[3] Q. He, G. D´
an, and V. Fodor, “Joint assignment and scheduling for
minimizing age of correlated information,” IEEE/ACM Trans. Netw.,
vol. 27, no. 5, pp. 1887–1900, Oct. 2019.
(a) Comparing ¯
∆1across three renewal processes with varying N.
(b) Comparing ¯
∆1between Neg. and Pos. processes for varying N.
Fig. 5. The average AoCI of source 1 against λ1for |N | ={4,6}, where
∀i∈ N \1, λi= 0.3/(i−1) and µ=1 under (a) independent and (b) depen-
dent inter-arrival time distributions. Here, x1,2=x2,4=x2,6=x3,4= 1.
[4] B. Zhou and W. Saad, “On the age of information in Internet of Things
systems with correlated devices,” in Proc. IEEE GLOBECOM, 2020, 6
pages.
[5] J. Tong, L. Fu, and Z. Han, “Age-of-information oriented scheduling for
multichannel IoT systems with correlated sources,” IEEE Trans. Wireless
Commun., vol. 21, no. 11, pp. 9775–9790, Nov. 2022.
[6] L. Badia and L. Crosara, “Correlation of multiple strategic sources
decreases their age of information anarchy,” IEEE Trans. Circuits Syst.
Express Briefs, 2024.
[7] C. Xu, X. Zhang, H. H. Yang, X. Wang, N. Pappas, D. Niyato, and
T. Q. S. Quek, “Optimal status updates for minimizing age of correlated
information in IoT networks with energy harvesting sensors,” IEEE
Trans. Mob. Comput., 2024.
[8] V. Tripathi and E. Modiano, “Optimizing age of information with cor-
related sources,” in Proc. MobiHoc, 2022, pp. 41–50.
[9] S. R. Chakravarthy, “The batch Markovian arrival process: a review and
future work,” Adv. Prob. Theory Stoch. Process., vol. 1, pp. 21–49, 2001.
[10] O. C. Ibe, Markovian Arrival Processes. Elsevier, 2013, pp. 349–376.
[11] R. D. Yates and S. K. Kaul, “The age of information: Real-time status
updating by multiple sources,” IEEE Trans. Inf. Theory, vol. 65, no. 3,
pp. 1807–1827, Mar. 2019.
[12] M. Moltafet, M. Leinonen, and M. Codreanu, “On the age of information
in multi-source queueing models,” IEEE Trans. Commun., vol. 68, no. 8,
pp. 5003–5017, Aug. 2020.
[13] N. Akar and K. Sohraby, “An invariant subspace approach in M/G/l and
G/M/l type Markov chains,” Communications in Statistics. Stoch. Mod-
els, vol. 13, no. 3, pp. 381–416, Jan. 1997.
[14] M. F. Neuts, Matrix-geometric solutions in stochastic models: an algo-
rithmic approach. Johns Hopkins University Press Baltimore, 1981.
[15] S. R. Chakravarthy, Shruti, and R. Kulshrestha, “A queueing model with
server breakdowns, repairs, vacations, and backup server,” Oper. Res.
Perspect., vol. 7, p. 100131, 2020.
[16] M. S. Kumar, A. Dadlani, M. Moradian, A. Khonsari, and T. A. Tsiftsis,
“On the age of status updates in unreliable multi-source M/G/1 queueing
systems,” IEEE Commun. Lett., vol. 27, no. 2, pp. 751–755, 2023.