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Age of Information in Multi-Source Updating

Systems: An M/G/1 Vacation Queueing Model

Muthukrishnan Senthil Kumar∗, Aresh Dadlani†, Masoumeh Moradian‡, Behrouz Maham†, and Theodoros A. Tsiftsis§

∗Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore, India

†Department of Electrical and Computer Engineering, Nazarbayev University, Astana, Kazakhstan

‡School of Computer Science, Institute for Research in Fundamental Sciences, Tehran, Iran

§Department of Informatics and Telecommunications, University of Thessaly, Lamia, Greece

Emails: msk.amcs@psgtech.ac.in, {aresh.dadlani, behrouz.maham}@nu.edu.kz, mmoradian@ipm.ir, tsiftsis@uth.gr

Abstract—Concurrent with the rise of real-time wireless systems

enabled by the Internet of Things, age of information (AoI) has

been widely perceived as a crucial destination-centric perfor-

mance metric to quantify the timeliness of data delivery. This

paper concerns itself with the analysis of information freshness

in a multi-source M/G/1 queueing system with utilization of idle

server time, referred to as server vacation, in an effective manner.

In particular, the status update packets in our model are gener-

ated independently and according to a Poisson process by each

source, while their service time follows a general random variable.

Using stochastic decomposition and Laplace-Stieltjes transform,

we derive the average AoI (AAoI) expression for the multi-source

M/G/1 queueing model with generally-distributed vacation time

in closed-form. Our numerical simulations validate the accuracy

of the derived AAoI expression and assess the impact of different

parameters on the system performance.

Index Terms—Age of information, multi-source queueing model,

server vacation, stochastic decomposition, internet of things.

I. INTRODUCTION

Driven by wireless technology advancements, the evolution

of Internet of Things (IoT) is progressively reshaping the perfor-

mance requirements of ubiquitous real-time systems. Dissimilar

to traditional network performance metrics such as throughput

and latency, status update systems mainly rely on the freshness

of data source information received at the destination. Widely

referred to as the age of information(AoI), this end-to-end metric

fully describes the timeliness of a monitor’s knowledge of a

stochastic process generated by a remote source over the wireless

medium. Having practical implications in wireless communica-

tion and cyber-physical networks, research in AoI has garnered

considerable attention in recent years [1]–[5].

In the realm of queueing systems, where information tranmsis-

sion over source-destination communication links are modeled

as queues, the authors of [6] were the ﬁrst to study the average AoI

(AAoI) in single-source M/M/1 ﬁrst-come ﬁrst-served (FCFS)

queues. Thenceforth, many efforts on queue-theoretic AoI anal-

ysis involving a single updating source have been reported [7].

For systems with multiple updating sources that share a common

queue however, the number of notable contributions is quite

scarce. In [8], a queue management technique has been proposed

to reduce the AAoI in multi-source M/M/1 systems under high

queue utilization. Closed-form expressions for the AAoI and the

average peak AoI (PAoI) of each stream in a preemptive multi-

stream M/G/1/1 queue have been derived in [9]. The authors of

[10] characterized the region of feasible average status ages for

simple M/M/1 queues with multiple updating sources and de-

rived the corresponding approximate AAoI. In [11], the authors

obtained closed-form expression for the AAoI of each source

in terms of the Laplace transform of the service time. The same

authors further derived exactand approximate expressions for the

AAoI in multi-source M/M/1 and M/G/1 systems, respectively, in

[12]. Recently, exact distributions for AoI and PAoI metrics have

been numerically obtained for different queueing disciplines by

employing a discrete-time quasi-birth-death type model [13].

Besides the typical queueing discipline, arrival process, and

service process, an important yet unexplored modeling feature in

real-time updating systems is the vacation process of the server.

Finding applications in energy-dependent systems such as smart

manufacturing [14], underwater acoustic networks [15], and re-

mote health monitoring [16], sensors with limited battery supply

can transition to a lower energy state rather than waiting idly until

data packets arrive at the buffer. Though systems with vacation

servers have been studied widely in terms of average waiting

time, queue length, and throughput, very few papers analyze

data freshness using AoI-related metrics. In [17], the authors

formulated a closed form for the average age of the stream of

interest that relates the interaction between its aging and vacation

processes. They however, assumed an M/M/1 vacation queueing

model. More recently, PAoI analysis in a vacation server system

modeled as M/G/1 priority queues with multiple buffers has been

reported in [18]. To the best of our knowledge, this paper is the

ﬁrst to analyze AoI in multi-source M/G/1 queueing models with

single buffer and server vacations.

To this end, the key contributions of this paper are: (i) closed-

form expression derivation for the Laplace-Stieltjes transform

(LST) of the waiting time of single source FCFS M/G/1 queueing

model with general vacation using stochastic decomposition

(SD) approach, (ii) AAoI formulation for the multi-source

FCFS M/G/1 queueing model with vacation, and (iii) numerical

assessment of the impact of arrival rates, server vacation, and

service time distributions on AAoI.

The remainder of this paper is organized as follows. Section II

formally describes the vacation queueing system model and

AAoI. Section III details the SD approach, which is followed

by AoI analysis in Section IV. Simulation results are discussed

in Section V. Finally, Section VI concludes the paper.

.

.

.

s1

sN

λ1

λN

D

Medium

Queue Server with

Vacation

Independent

Sources

Destination

Fig. 1. The multi-source M/G/1 vacation queueing model of interest.

II. SY ST EM MO DE L DESCRIPTION

We ﬁrst describe the multi-source M/G/1 vacation queueing

model, and then, we formally deﬁne AAoI in this section.

A. The Vacation Queueing Model

Consider a single server queueing model with a set of Nin-

dependent sources, denoted by S={s1, s2, . . . , sN}, and server

vacations as shown in Fig. 1. Each source sk∈ S generates

packets following a Poisson process with rate λk, where k=

1,2, . . . , N . Let the packet transmission times be independent

and identically distributed (i.i.d.) random variables (RVs) with

distribution function FSk(t), where Skdenotes the RV for

transmission (service time) of a packet generated by source sk.

Let the corresponding LST be S∗

k(a)=R∞

0e−atdFSk(t), and the

moment of order ibe given as β(k,i)=(−1)idi

daiS∗

k(a)|a→0+. If

the server is idle, it takes a vacation for random time and resumes

service whenever at least one packet enters the buffer for service.

Also, let Vkdenote the general server vacation time RV when a

packet is generated for update from source sk, and FVk(t)be the

distribution function of Vk. Subsequently, the LST of FVk(t)is

V∗

k(a) = R∞

0e−atdFVk(t)and the corresponding i-th moment

is given as α(k,i)=(−1)idi

daiV∗

k(a)|a→0+.

The arrival process of the packets, idle time of the server,

and service times of the packets are assumed to be mutually

independent. As deﬁned in [12] and [19], let tk,i denote the

time epoch at which the i-th status update packet of source skis

generated, and t′

k,i be the time epoch at which this packet arrives

at the destination D. At some time instant γ, the index of the most

recently received packet of source skis given as follows:

Nk(γ)=max{i′|t′

k,i′≤γ}.(1)

In addition, the time stamp of the recently received packet from

source skis Uk(γ) = tk,Nk(γ). As a result, the AoI of skat the

destination is deﬁned as the random process ∆k(t)≜t− Uk(t).

An instance of the time evolution of AoI is shown in Fig. 2.

B. Average AoI Deﬁnition

The most widely used metric for evaluating AoI from skat the

destination in interval (0, t)is AAoI and is given by ([10],[12]):

∆t,k =1

tZt

0

∆k(u)du. (2)

As depicted in Fig. 2, the area under ∆k(u)is the sum of the

disjoint areas determined by Bk,i, where i∈1,2,...,Nk(γ). As

γ→∞, we obtain the stochastic averages E[Bk,i]. Therefore, we

have ∆k=λkE[Bk,i]. The RV Xk,i ≜tk,i−tk,i−1represents the

i-th inter-arrival time of source sk. Let sk,i denote the starting

epoch of the (k, i)-th packet transmission and RV Ik,i ≜sk,i−tk,i

be the i-th idle time of the server for source sk. Also, let Tk,i ≜

t

∆(t)

∆0

•

tk,1

∗

sk,1

•

tk,2

∗

t′

k,1=sk,2

∗

t′

k,2=sk,3

∗

t′

k,3=sk,4

•

tk,3

•

tk,4

Bk,1

Bk,2

Bk,3

Ik,1Ik,2Ik,3Ik,4

Xk,2Xk,3Xk,4

Tk,2

Fig. 2. Time evolution of AoI for source skin the vacation queueing model.

t′

k,i

−tk,i be the RV denoting the sojourn system time. This reﬂects

the time spent by the packet in the system and is given by the sum

of the waiting, service, and vacation times. In a classical queueing

model, the transmission completion epoch of the (k, i −1)-th

packet and transmission starting epoch of the (k, i)-th packet will

be the same if the packet generated at time tk,i lies within the

interval (sk,i−1, t′

k,i−1). Otherwise, the server will take vacation

for some random time period when idle. From [12], the AAoI of

skcan be determined as follows:

∆k=λkE[Bk,i] =λk1

2E(Xk,i +Tk,i )2−1

2EX2

k,i

=λk E[X2

k,i]

2+ E [Xk,i Tk,i ]!.(3)

Moreover,let the RVs Wk,i,Sk,i , and Vk,i denote, respectively,

the waiting and service times of packet (k, i), and the vacation

time of the server. Consequently, the sojourn time can be written

as Tk,i =Wk,i +Sk,i +Vk,i , and the AAoI is given as:

∆k=λk E[X2

k,i]

2+ E [Xk,i (Wk,i +Sk,i +Vk,i )]!.(4)

III. STOCHASTIC DECOMPOSITION OF THE MODEL

If an M/G/1 vacation queue is empty at the completion epoch of

service, the server begins a vacation of random length. When the

server returns from vacation and ﬁnds one or more unprocessed

packets, it resumes service until the system is completely empty.

If the server returns from vacation and ﬁnds no packets waiting,

it awaits the arrival of packets. This policy is referred to as the

single vacation policy. Contrarily, if the server ﬁnds no packets,

it will take a series of vacations until it ﬁnds at least one packet

awaiting service. This is referred to as the multiple vacation

policy. In this paper, we adopt the latter policy.

The SD condition is held by Poisson input queueing models

with vacation [20]. Let P(z)be the probability generating func-

tion (PGF) of the number of packets in the system at a departure

epoch in a classical M/G/1 queue without vacation. Using the

Poisson Arrivals See Time Averages (PASTA) property, the

packet distribution in the system at a random epoch, an arrival

epoch, or a departure epoch are the same. Moreover, let Q(z)

and V(z)be the PGFs of the number of packets in the system

at a departure epoch and a random epoch when the server is on

vacation. Using the SD property, we have:

Q(z) = P(z)V(z).(5)

This implies that the number of packets at a departure epoch

in an M/G/1 queue with vacations is the sum of two RVs, i.e.,

the number of packets at a departure epoch of the M/G/1 queue

without vacation and the number of packets at a random epoch

when the server is on vacation. Here, we consider a single source

and hence, the service time distribution FS(t) = FSk(t)and the

vacation time distribution FV(t)= FVk(t). Let S∗(a)and V∗(a)

be the LSTs of the service and vacation time distributions,

respectively. Now, P(z)can be calculated using the Pollaczek-

Khinchin formula [21] for an M/G/1 queueing system as follows:

P(z) = (1 −ρ)(1 −z)S∗(λ−λz)

S∗(λ−λz)−z,(6)

where ρis the probability that the server is busy. To derive V(z),

we deﬁne AVas the number of arrivals during a typical vacation

period of length V. Thus, the PGF of AV, denoted by α(z), is:

α(z)=

∞

X

n=0

Pr(AV=n)zn=

∞

X

n=0

znZ∞

0

e−λt(λt)n

n!dFV(t)

=V∗(λ−λz).(7)

By deﬁning Z(t)to be the residual lifetime of a vacation RV,

the limiting distribution FZ(t)of Z(t)as t→ ∞ is as below:

FZ(t) = Pr(Z≤t) = Rt

0(1 −FV(y)) dy

E[V],(8)

where E[V]is the expected vacation time. If vndenotes the prob-

ability that npacket arrivals occur during the residual vacation

time, we can use (7) to ﬁnd V(z)as:

V(z)=

∞

X

n=0

vnzn=

∞

X

n=0

znZ∞

0

e−λt(λt)n

n!dFZ(t)

=Z∞

0

eλt(1−z)(1 −FV(t))

E[V]dt =1−α(z)

(1 −z)λE[V].(9)

From [20] and [21], we learn that the LST of the sojourn

time distribution for an M/G/1 queue with vacations is W∗

T(a)=

W∗

1(a)V(1 −a

λ), where W∗

1(a)is the LST of the sojourn time

distribution of a standard M/G/1 queue without vacation. To

obtain W∗

T(a), we substitute a=λ(1 −z)into (5) as shown

below:

Q1−a

λ=P1−a

λ1−V∗(a)

aE[V].(10)

As a result of this, W∗

T(a)can be obtained as follows, where

W∗

1(a)=P(1 −a

λ):

W∗

T(a) = W∗

1(a)1−V∗(a)

aE[V]

=(1 −ρ)aS∗(a)

a−λ(1 −S∗(a))1−V∗(a)

aE[V].(11)

IV. AAOI AN ALYS IS F OR MU LTI-SOURCE M/G/1

VACATIO N QUEUEING MOD EL

In this section, we detail the steps involved in deriving (4) for

the multi-source M/G/1 vacation model. For the sake of analysis,

we focus on a two-source M/G/1 vacation queue. Derivation of

the ﬁrst term in (4) is straightforward. Since the inter-arrival time

of source s1is exponentially distributed with parameter λ1, we

have E[X2

1,i]=2/λ2

1. The second term in (4) can be written as:

E [X1,i(W1,i +S1,i +V1,i )] = E [X1,iW1,i ]+E[X1,i] E[S1,i]

+ E[X1,i] E[V1,i ],(12)

where the second and third terms in (12) arise from the indepen-

dence between the inter-arrival time, the service and vacation

times, respectively. However, evaluating the ﬁrst term in (12) is

cumbersome since the inter-arrival and waiting times are de-

pendent. To address this, we characterize W1,i by means of two

events, namely Ab

1,i and Al

1,i. Here, Ab

1,i is the event that the inter-

arrival time between packets (1, i −1) and (1, i)is brief, i.e., the

inter-arrival time of packet (1, i)is shorter than the system time

of packet (1, i−1).Al

1,i is the complementary event of Ab

1,i.

The components comprising W1,i in event Ab

1,i are: (i) the

remaining system time to complete serving packet (1, i −1), (ii)

the sum of the service times of packets from s2that arrived during

X1,i and must be served before packet (1, i)as per the FIFO rule,

and (iii) the sum of server vacation times before serving packet

(1, i). Similarly, the waiting time of packet (1, i)in an event of

Al

1,i includes: (i) the remaining service time of the packet from

s2that is under service at the arrival epoch of packet (1, i), (ii) the

sum of service times of s2packets served prior to packet (1, i)

according to FIFO, and (iii) the sum of server vacation times

before serving the (1, i)-th packet.

For event Ab

1,i, let the residual system time to complete service

of packet (1, i−1) be given as Rb

1,i =T1,i−1−X1,i. Also, let the

sum of service times of s2packets arriving during X1,i that must

be served before packet (1, i)be Sb

1,i =Pi′∈Ωb

2,i S2,i′, where Ωb

2,i

is the index set of queued packets originating from s2. Moreover,

let Vb

1,i =Pi′∈Ωb

2,i V2,i′be the sum of server vacation times prior

to the service of packet (1, i). In the same manner, Sl

1,i and Vl

1,i

can be deﬁned over the index set Ωl

2,i. Thus, the waiting time for

packet (1, i)can now be expressed as:

W1,i =(Sb

1,i +Vb

1,i +Rb

1,i,if event Ab

1,i occurs,

Sl

1,i +Vl

1,i +Rl

2,i,if event Al

1,i occurs,(13)

where Rl

2,i represents the residual service time RV of s2packet

that is being served at the arrival epoch of packet (1, i)condi-

tioned on the event Al

1,i. From (13), we arrive at the following

expectation, where Pb

1,i and Pl

1,i are the probabilities of events

Ab

1,i and Al

1,i, respectively:

E[X

1,i W

1,i ] = ERb

1,i X

1,i |Ab

1,i +E(Sb

1,i +Vb

1,i )X

1,i |Ab

1,i Pb

1,i

+ E(Sl

1,i +Vl

1,i +Rl

2,i)X1,i |Al

1,iPl

1,i .(14)

In what follows, Theorem 1 derives the probabilities Pb

1,i and

Pl

1,i given in (14). It is then succeeded by Theorems 2 to 4 that

deal with the conditional expectations in (14). Henceforth, we

assume that ∀i, k, λ =Pkλkand ρk=λkE[S], where S=Sk,i.

Theorem 1. The probabilities Pb

1,i and Pl

1,i given in (14) are

computed as follows, where the service times of all packets are

stochastically identical:

Pb

1,i =1−(1−ρ)S∗(λ1

)1−V∗(λ1

)

E[V]λ1−λ1−S∗(λ1), P l

1,i =1 −Pb

1,i .(15)

Proof. Since T1,i−1and X1,i are independent and the probability

density function (PDF) of X1,i is fX1,i (t)=λ1e−λ1t, we have:

Pb

1,i =Z∞

0

Pr(T1,i−1≥X1,i|T1,i−1=t)fT1,i−1(t)dt

= 1 −Z∞

0

e−λ1tfT1,i−1(t)dt = 1 −W∗

T(λ1),

where the sojourn times of different packets are stochastically

identical, i.e., ∀i, T1,i =T2,i =. . . =TN,i =T. Here, W∗

T(λ1)

is the LST of sojourn time distribution function Tat λ1. Since

Al

1,i is the complement of event Ab

1,i, we get Pl

1,i = 1 −Pb

1,i =

W∗

T(λ1), which completes the proof. ■

By substituting Mb

1,i =Sb

1,i +Vb

1,i and Ml

1,i =Sl

1,i +Vl

1,i into

(13), the last two conditional expectations in (14) can be re-

written as E[Mb

1,iX1,i |Ab

1,i]and E[(Ml

1,i +Rl

2,i)X1,i |Al

1,i].

From probability theory, we know that given the RVs X1,i and

T1,i−1, and the event Ab

1,i, the conditional joint PDF is given by:

f

X

1,i,T1,i−1

|Ab

1,i(s, t|Ab

1,i) = (λ1e

−λ1sfT1,i−1(t)

Pr(Ab

1,i),if s≤t,

0,if otherwise.

(16)

Using (16) in Theorem 2, we now derive the ﬁrst conditional

expectation stated in (14).

Theorem 2. The closed-form expression for the ﬁrst condi-

tional expectation in (14) is as follows, with W∗

T(λ1)derived in

(11),W∗′

T(λ1

)=−E[T e

−λ1T], and E[T]=E[W]+E[S]+E[V]:

E[Rb

1,iX1,i |Ab

1,i] = E[T]−W∗′

T(λ1

)

λ1Pb

1,i

+2 (W∗

T(λ1

)−1)

λ2

1Pb

1,i

.(17)

Proof. The ﬁrst conditional expectation in (14) can be written as:

E[Rb

1,iX1,i |Ab

1,i] = E[T1,i−1X1,i |Ab

1,i]−E[X2

1,i|Ab

1,i].(18)

The ﬁrst term on the right-hand side of (18) can be reduced to:

E[T1,i−1X1,i|Ab

1,i] = Z∞

0Z∞

0

xtfX1,i,T1,i−1|Ab

1,i

(x, t)dx dt

=1

Pb

1,i Z∞

0Zt

0

txλ1e−λ1xfT1,i−1(t)dx dt

=1

Pb

1,i Z∞

0h−t2e−λ1t−te−λ1t

λ1

+1

λ1ifT1,i−1(t)dt

=−E[T2e−λ1T]

Pb

1,i

+−E[T e−λ1T]

λ1Pb

1,i

+E[T]

λ1Pb

1,i

=−W∗′′

T(λ1)

Pb

1,i

+W∗′

T(λ1)

λ1Pb

1,i

+E[T]

λ1Pb

1,i

,(19)

where W∗′′

T(λ1) = E[T2e−λ1T]. Similarly, the second term on

the right-hand side of (18) results in the following, where ˆ

W∗

T=

(λ1W∗′′

T(λ1)−2W∗′

(λ1))/λ1+ 2W∗(λ1)/λ2

1:

E[X2

1,i|Ab

1,i] = Z∞

0

x2fX1,i|Ab

1,i (x)dx

=Z∞

0

t2λ1e−λ1t(1−FT1,i−1(t))

Pb

1,i

dt

=1

Pb

1,i

Z∞

0

t2λ1e

−λ1tdt−λ1Z∞

0

t2e

−λ1tFT

1,i−1(t)dt

=2

λ2

1Pb

1,i

−λ1

ˆ

W∗

T(λ1)

Pb

1,i

,(20)

By substituting (19) and (20) into (18), and reaaranging the

terms, we arrive at (17), which completes the proof. ■

Theorem 3. The closed-form expression for the second condi-

tional expectation in (14) is as follows, where δ2=ρ2

+λ2E[V]:

E[Mb

1,i X1,i|Ab

1,i ]= 2δ2W∗′

T(λ1

)

λ1Pb

1,i

−δ2W∗′′

T(λ1

)

Pb

1,i

+2δ2(1−W∗

T(λ1

))

λ2

1

.

(21)

Proof. The second conditional expectation in (14) can be ex-

pressed as follows, where Ωb

2,i is the index set of packets that have

arrived from s2and must be served before packets from s1:

E[Mb

1,iX1,i |Ab

1,i]

=Z∞

0

xEhX

i′∈Ωb

2,i

(S2,i′+V2,i′)|Ab

1,i, X1,i =xifX1,i|Ab

1,i (x)dx.

Since Ωb

2,i is independent of the inter-arrival time of packets

generated by s1, it is also independent of T1,i−1. Hence, we get:

E[Mb

1,iX1,i |Ab

1,i]

=λ2(E[S]+E[V])

Z∞

0

xE[Ωb

2,i|X1,i =x]fX1,i|Ab

1,i(x)dx

=δ2

Pb

1,i Z∞

0

x2λ1e−λ1x(1 −FT1,i−1(x)) dx.

After some simple mathematical manipulations, we obtain (21).

This completes the proof. ■

Theorem 4. The closed-form expression for the third condi-

tional expectation in (14) is as follows:

E[(Ml

1,i +Rl

2,i)X1,i|Al

1,i ]= δ2W∗′′

T(λ1)

Pl

1,i

−δ2W∗′

T(λ1)

λ1Pl

1,i

.(22)

Proof. The expression for the third conditional expectation in

(14) can be deduced as shown below:

E[(Ml

1,i+Rl

2,i)X1,i |Al

1,i]

=δ2

Pl

1,i Z∞

0Z∞

t

xtλ1e−λ1xfT1,i−1(t)dx dt

=δ2

Pl

1,i Z∞

0ht2e−λ1t+te−λ1t

λ1ifT1,i−1(t)dt, (23)

which then reduces to (22) straightforwardly. This completes the

proof. ■

(a) Exponential service time distribution (b) Erlang-2 service time distribution (c) H2service time distribution

Fig. 3. The AAoI of source 1 as a function of λ1for different values of Nunder service times following (a) exponential (b) Erlang-2 and (c) hyper-exponential

distributions with E[S]=0.2,E[V]=0.5,λi(i= 1) = 0.6, and p= 0.58.

(a) Exponential service time distribution (b) Erlang-2 service time distribution (c) H2service time distribution

Fig. 4. The AAoI of source 1 as a function of E[S]for different values of Nunder service times following (a) exponential (b) Erlang-2 and (c) hyper-

exponential distributions with λ1= 0.12,E[V]=0.5,λi(i= 1)=0.02, and p= 0.4.

We now can calculate the AAoI of s1for a two-source M/G/1

vacation queueing model using the above theorems as follows:

∆1= E[W] + 2(E[S]+E[V]) + 2(1−ρ2−λ2E[V]) W∗

T(λ1

)

λ1

+2(ρ2+λ2E[V])−1

λ1

+(ρ2+λ2E[V]−1) W∗′

T(λ1

).(24)

In general, the AAoI of source s1in a multi-source M/G/1

queueing model with server vacations can be expressed as:

∆1= E[W] + 2(E[S] + E[V]) + 2W∗

T(λ1)

λ1

−W∗′

T(λ1)−1

λ1

+(E[S]+E[V]) X

j∈X |{1}

λj 2

λj

+W∗′

T(λj)−2W∗

T(λ1

)

λ1!.

(25)

V. NUMERICAL RE SU LTS AND DISCUSSIONS

For our simulations, we consider exponential, Erlang-2, and

hyper-exponential of order 2 (H2) service time distributions. The

PDF of H2is given as follows:

f(t) = pγ1e−γ1t+ (1 −p)γ2e−γ2t, t ≥0,(26)

where p= (1+ p(c2−1)/(c2+1))/2,γ1= 2p/m1,γ2= 2(1 −

p)/m1,c2≥1is the squared coefﬁcient of variation,and m1is the

mean. When applicable, we compare our results with the multi-

source M/G/1 baseline model reported in [12]. Unless explicitly

speciﬁed, λ1=0.12,E[S]=0.2, and E[V]=0.5.

Fig. 3 plots the AAoI of source s1in terms of the packet arrival

rate λ1for the three service time distributions and varying N

values. As evident in all three ﬁgures, the AAoI resulting from the

vacation model is substantially higher than the baseline model

under light trafﬁc conditions. The gap however, reduces with

increase in trafﬁc of incoming packets from s1under which the

system is stable. It should also be noted that the AAoI of s1

increases with the number of sources in the system. Such behavior

is expected as the server becomes more busy in processing the

packets arriving from multiple sources. Therefore, in the vacation

model, the mean vacation time of the server would decrease as the

number of sources increases. By comparing the curves in Fig. 3a,

Fig. 3b and Fig. 3c, we observe similar trends under the different

service time distributions for the M/G/1 vacation model.

The impact of the expected service time (E[S]) on the AAoI

of s1for varying number of sources is depicted in Fig. 4. We

observe that with increase in the service time of the single server

while serving s1packets, the number of packets arriving from

other sources for updates is delayed and therefore, the AAoI

increases proportionally with the service time. Moreover, adding

(a) Exponential service time distribution (b) Erlang-2 service time distribution (c) H2service time distribution

Fig. 5. The AAoI of source 1 as a function of E[V]for different values of Nunder service times following (a) exponential (b) Erlang-2 and (c) hyper-

exponential distributions with λ1= 0.12,E[S]=0.2,λi(i= 1)=0.2, and p= 0.4.

more sources yields higher AAoI values in the vacation model,

in comparison to the baseline model, as the mean service time

grows to 2. Similar patterns under the three different types of

distributions are also apparent in this ﬁgure, leading to the same

inference.

Finally, Fig. 5 plots the AAoI of s1with respect to the mean

vacation time (E[V]) in the vacation model for differentNvalues

and service time distributions. We see that update packets arriving

from multiple sources are delayed as the vacation time of the

server increases while serving s1packets in all three distributions.

This arises due to the fact that the number of packets awaiting

service in the queue grows with increase in the vacation time

of the server which eventually, results in delayed service. It is

worth mentioning that in Fig. 5b and Fig. 5c, the AAoI achieved

becomes exponentially large beyond E[V]= 1, thus making the

system impracticable for time-sensitive applications.

VI. CONCLUSION

This work reports on average AoI analysis of a multi-source

M/G/1 queueing system in which updates generated from mul-

tiple sources arrive as independent Poisson processes and are

served by the vacating server. Employing the stochastic decom-

position approach, we derived the LST expression for the waiting

time of single source FCFS M/G/1 vacation model in closed form.

We then formulated average AoI for the extended multi-source

vacation model. Under varyingtrafﬁc loads and number of update

sources, the simulation results showed that the vacation model

resulted in higher average AoI in contrast to the multi-source

model without server vacation. As future work, the signiﬁcance

of AoI as a metric in energy-driven control systems will be inves-

tigated for queueing models with server breakdowns and repairs.

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