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1
Modeling Epidemic Routing: Capturing Frequently
Visited Locations While Preserving Scalability
Leila Rashidi, Amir Dalili-Yazdi, Reza Entezari-Maleki, Leonel Sousa, Senior Member, IEEE
and Ali Movaghar, Senior Member, IEEE
Abstract—This paper investigates the performance of epidemic
routing in mobile social networks considering several communities which are frequently visited by nodes. To this end, a
monolithic Stochastic Reward Net (SRN) is proposed to evaluate
the delivery delay and the average number of transmissions
under epidemic routing by considering skewed location visiting
preferences. This model is not scalable enough, in terms of
the number of nodes and frequently visited locations. In order
to achieve higher scalability, the folding technique is applied
to the monolithic model, and an approximate folded SRN is
proposed to evaluate performance of epidemic routing. Discreteevent simulation is used to validate the proposed models. Both
SRN models for predicting the performance of epidemic routing
exhibit high accuracy. We also propose an Ordinary Differential
Equation (ODE) model for epidemic routing and compare it with
the folded model. The obtained results show that the folded
model is more accurate than the ODE model. Moreover, it is
proved that the number of transmissions by the time of delivery
follows a uniform distribution, for a general class of networks,
where positions of nodes are always independent and identically
distributed.
Index Terms—Epidemic routing and performance evaluation.
I. I NTRODUCTION
ELAY tolerant and mobile social networks have drawn
the attention of the research community due to the everincreasing number of smart devices [1]. Under these network,
short-range wireless technologies of portable devices, such as
smart phones, tablets, and sensors in vehicles, can be used
by mobile users to share multimedia, data large-size files,
etc. [2]. One of the main characteristics of Mobile Social
Networks (MSNs) is that nodes have skewed location visiting
preferences [3]. In real world scenarios, people visit locations
with different frequencies. As an example, every employee
visits her/his work place each business day while she/he
D
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However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org
<mailto:pubs-permissions@ieee.org>.
This work was partially supported by the FCT (Fundação para a Ciência
e a Tecnologia, Portugal) through the project UIDB/50021/2020 and by the
Iran National Science Foundation.
L. Rashidi is with the Department of Computer Science, University of
Calgary, Calgary, AB, Canada. E-mail: leila.rashidi@ucalgary.ca
A. Dalili-Yazdi and A. Movaghar are with the Department of Computer Engineering, Sharif University of Technology, Tehran, Iran. E-mail:
dalili@ce.sharif.edu; movaghar@sharif.edu
R. Entezari-Maleki is with the School of Computer Engineering, Iran
University of Science and Technology, Tehran, Iran and INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal. E-mail:
entezari@iust.ac.ir
L. Sousa is with INESC-ID, Instituto Superior Técnico, Universidade de
Lisboa, Lisbon, Portugal. E-mail: las@inesc-id.pt
might prefer to go to a shopping center only once a week.
Specifically, we tend to spend most of our time at a few
frequently visited locations [3], [4]. We call such a location
community.
Despite various network models considered in the literature
to analyze the performance of routing in Delay Tolerant
Networks (DTNs), the performance of networks where nodes
have skewed location visiting preferences has not been wellstudied. Scalability is one of the most important challenges in
performance analysis of heterogeneous networks. In this paper,
we focus on evaluating the performance of epidemic routing
[5], in a scalable way, considering skewed location visiting
preferences for nodes in a heterogeneous network. The aim is
to compute the average and Cumulative Distribution Function
(CDF) of the delivery delay of a message, from the source to
the destination, and the average number of transmissions of
the message by time of delivery as in [6].
Analytical models have various advantages in comparison to
other evaluation methods, providing good insights into the network, less time to construct and analyze, and easy adaptation
[7], [8]. For example, extensive experimentations for different
network configurations are needed for measurement method,
which may not be feasible specially for real-world large-scale
networks, either with the large number of nodes/communities
or large areas in which nodes move. In this paper, we first
propose a monolithic Stochastic Reward Net (SRN) model [9]
to evaluate the performance of epidemic routing in a network
where nodes move in a large area, including some communities frequently visited by nodes. This proposed monolithic
model significantly differs from the presented in our prior work
[10] due to following reasons.
• In the network model considered in [10], it was assumed
that each node frequently visits only one specific community, and the number of local nodes in each community is
limited by 𝑀
𝑁 , where 𝑀 and 𝑁 are the number of nodes
and the number of communities, respectively. However,
in real-world, nodes usually frequently visit more than
one community. For example, a graduate student in a
university campus visits different communities such as
his/her office, dining center, and cafeteria. In this paper,
we disregard the two aforementioned assumptions, considering a more realistic scenario where each node can
frequently visit each community. The monolithic SRN
model proposed herein accounts for the possibility of
moving any number of local nodes in a community, to
evaluate the performance of epidemic routing.
• In [10], communities were homogeneous with respect to
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how much they are interesting for nodes, i.e., exactly
𝑀
𝑁 nodes visit each community when they are in local
mode. In this paper, we consider skewed location visiting
preferences, allowing to detail the interest of nodes to
frequently visit each community. We assume that nodes
select communities according to an arbitrary Probability
Density Function (PDF). This network model provides an
opportunity to analyze the performance of the network
also when some communities are more interesting for
nodes. Using the proposed models, we can also analyze
the effect of community selection PDF on the performance of the network.
Although the monolithic SRN model is able to evaluate
the performance of small networks, it faces the problem of
state space explosion when the network scales up in terms
of the number of nodes and communities. In order to solve
this problem, an approximate SRN model is proposed, by
applying the folding technique. Numerical results show that
the number of states in the underlying Markov chain of the
folded model is significantly less than that of the monolithic
model. The results of both monolithic and folded SRN models
are validated through discrete-event simulation. Analytical and
simulation results indicate that both monolithic and folded
models are accurate enough to evaluate the performance of
the epidemic routing in the target networks. In order to prove
the superiority of the proposed folding-based approach for
evaluating the performance of large-scale networks, we apply
the Ordinary Differential Equation (ODE) approach [6] to
model the target network, and then show that the proposed
folded model is more accurate than the ODE-based model.
The main contributions of this paper are as follows.
•
•
•
•
•
A monolithic SRN model is proposed to evaluate the
average value and CDF of the delivery delay, and the
average number of transmissions of epidemic routing in
a network consisting of some communities frequently
visited by nodes.
By applying the folding technique to the proposed monolithic SRN, a scalable approximate SRN is derived to
evaluate the performance of large-scale networks.
The validation is done by simulation, comparing the
results of both monolithic and folded SRN models. This
comparison indicates that the proposed models have a
good accuracy.
According to both analytical and simulation results, the
average number of transmissions is very close to half of
the number of nodes. In order to interpret this observation
and justify this number, it is proved that the average
number of transmissions is equal to half of the number
of nodes in any network, not only for the target network,
where positions of nodes are always independent and
identically distributed (i.i.d.).
In order to compare the proposed folded model with the
ODE approach, this latter approach is also applied to
model epidemic routing in the target networks. Comparison of the results of the folded SRN and ODE models
with the results obtained from simulation indicates the
practical interest of the folded model.
2
The rest of this paper is organized as follows. The related
state-of-the-art and the main differences to the current work are
introduced in Section II. Section III introduces the target network model and the assumptions made herein. Afterwards, a
monolithic SRN and an approximate folded SRN are proposed
for epidemic routing in the target network model, in Sections V
and VI, respectively. Section VII is dedicated to the figures of
merit and how to compute them applying the proposed models.
Numerical results obtained from the proposed models and by
simulation are provided in Section VIII. The proposed SRN
models are compared in terms of the scalability in Section IX.
Finally, Section X concludes the paper and provides some
directions for future work.
II. RELATED WORK
A. Delay tolerant networks
An ODE-based framework has been proposed to evaluate
the performance of epidemic routing and its variations [6].
The network in [6] consists of a set of nodes moving in a
closed area according to a common mobility model, such as
random direction or random waypoint models. The closedform expressions for some performance measures, such as
the average number of transmissions by the time of delivery,
were derived using the analytical solution of the proposed
ODE model. However, the ODE approach provides limits
to the Markov models when the number of nodes tends to
infinity [6]. Thus, it is not accurate to study the performance of
networks with a moderate number of nodes [11]. In particular,
the average number of infected nodes at the time of delivery,
including the destination node, was estimated to be half of the
number of nodes [6]. In this paper, we model the epidemic
routing in a more realistic way, considering skewed location
visiting preferences. Moreover, it is proved that the average
number of transmissions by time of delivery is equal to half of
the number of nodes, for a general class of networks, where
positions of nodes at any time are independent and follow
the same PDF. This class includes the network considered
in [6], and the exact expression for the average number of
transmissions, derived herein, is close to the approximate
expression derived in [6] considering one initial infected node.
In [12], a network consisting of two classes of nodes has
been considered, wherein the inter-meeting time of any two
nodes is exponentially distributed. Subsequently, three rates
were defined, one per each class as the meeting rate of any two
nodes belonging to that class, and another as the meeting rate
of any two nodes belonging to different classes. Afterwards,
epidemic routing was modeled as a Continuous Time Markov
Chain (CTMC), and then two ODE models were proposed
in order to evaluate the performance of large-scale networks.
One ODE model is an extension of the model proposed in [6],
while the other ODE model exploits the Kolmogorov forward
equation. The network studied in [13] is similar to [12],
but an arbitrary number of classes was considered. In order
to evaluate the performance of epidemic routing and some
variants of spray and wait routing, a framework that applies
ODE model was proposed [13].
In [14], asymptotic results and closed-form approximations
have been derived for epidemic spreading, considering a
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contact network with probabilistic meeting rates. Unlike [12]–
[17], the models proposed in this paper are not based on fixed
meeting rates/probabilities or probabilistic meeting rates; two
different movement modes are considered, and the meeting
rate of any two nodes changes as the movement mode of at
least one of them changes. In [3] and [18], a time-variant
community mobility model has been proposed. The ODE
model proposed in [6] was extended in [18] to evaluate the
performance of epidemic routing on a network consisting of
two communities. As shown in [18], although the average
number of infected nodes as a function of time, obtained
from the ODE model, follows a trend similar to that observed
in simulation results, the ODE model does not yield a good
accuracy. This paper adopts SRNs and the folding technique,
instead of the ODE approach, to model epidemic routing in
target large-scale networks, and it is shown that the proposed
folded model is more accurate than the ODE approach.
The network studied in [19] is similar to [13], but nodes
can move between classes with specific rates. In [19], meeting
times of all pairs of nodes are assumed to be independent
from each other. In [20], an edge-Markovian dynamic graph
model has been proposed for epidemic routing. In that model,
the states of the edges change independently from each other.
However, in real scenarios, the meeting times of some pairs
of nodes depend on each other. This dependency was not
considered in [19] and [20] although it is important to take
it into account when studying MSNs, as it is addressed in
this paper. A family of restricted epidemic routings has been
modeled in [21] by applying Discrete Time Markov Chains
(DTMCs). Those models are not scalable, the number of states
exceedingly grows when the number of nodes/communities
increases. Moreover, considering slotted time is a shortcoming
of the models proposed in [20] and [21], while SRNs are
based on continuous time which is more realistic. In [8],
two monolithic and folded SRNs have been proposed for the
epidemic content retrieval scheme in DTNs with restricted
mobility. In [8] and [21], each node is assumed to move
only within the community to which it belongs, while in
the networks targeted herein, nodes can freely move in a
common area and enter all communities. In [22], the delivery
delay under both multi-copy two-hop forwarding and direct
forwarding has been studied.
The effect of selfishness on the performance of epidemic routing was studied in a DTN including several nonoverlappling groups of nodes [23]. There, it was assumed that
the process of meeting between any two nodes is Poisson,
and two different rates were considered: one for inter-meeting
time of any two nodes belonging to the same group and one
for inter-meeting time of nodes belonging to different groups.
However, in this paper, we consider a mobility model for nodes
instead of accounting for the meeting processes in an abstract
way. Li et al. in [24] have studied the effect of selfishness on
multicasting according to the epidemic and two-hop relaying.
To this end, they proposed a 3-D CTMC for the delivery
process. Afterwards, they computed delay and cost as closedform formulas.
3
B. Mobile social networks
The optimality of a routing scheme was studied in [25],
assuming that the inter-meeting time of any two nodes and
the time between two consecutive visits of a node to its home
are exponentially distributed. Based on these assumptions, the
proposed routing scheme was modeled by a CTMC. In [25],
the next location of each node is randomly selected from the
set of its homes or the set of other cells independently from
the current location of that node, and the path a node should
traverse to reach the next location was ignored. However, we
consider that nodes move according to the random direction
mobility model both when they are in communities and outside
of communities. In [2], the single-copy routing problem was
studied considering a MSN with a certain number of locations
and slotted time. Three assumptions were made in [2]: i) the
time taken for each node to reach a frequently visited location
follows an exponential distribution; ii) there is a throwbox at
each frequently visited location; iii) nodes cannot transfer the
message to each other when they are outside of the frequently
visited locations. The existence of a throwbox at each frequently visited location and transmission only at frequently
visited locations are oversimplifications. The network model
adopted herein is more realistic than [2] and [25], where the
location of a node is considered as a discrete quantity. Also it
is assumed that time is continuous unlike [2].
The effect of various distributions of unselfishness on the
throughput and delay of MSNs was analyzed, based on four
human mobility datasets with community-biased and uniform
traffic patterns [26]. Hui et al. discovered that distribution of
unselfishness does not have a significant impact since there
are usually multiple paths. Moreover, information diffusion
was optimized with respect to the speed of diffusion [27].
The goal was the identification of a small set of nodes which
can spread the information over the network with minimum
delay. In [28], proactive placement of static relays to improve
the performance of MSNs was studied. The problem was
formalized as an optimization problem, and then a heuristic
based solution was proposed.
C. Vehicular ad hoc networks (VANETS)
In [29], a dynamic epidemic-based broadcast scheme has
been proposed to optimize the probability of message delivery
in VANETs, while some constraints about the number of
transmissions were satisfied. The performance metrics under
this scheme were analyzed using ODEs. A unicast scheme
has been proposed in [30], leveraging the cellular attractor
mechanism for relay selection. This scheme can adapt itself
to dynamic environments using transferring feedback packets.
The timely delivery of emergency messages to the interested
vehicles has been addressed in [31]. A position-based scheme
was proposed in [31] for message dissemination in large-scale
VANETs.
III. N ETWORK M ODEL
The mobility model considered in this paper is similar
to the model proposed in [3], which matches with reallife traces from several scenarios. The network consists of
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4
TABLE I: Notations adopted to define the network model
𝒄𝟏
𝒄𝟑
𝒄𝟒
𝒄𝟐
Fig. 1: A network with four communities.
𝑁 (𝑁 > 1) 𝐿 𝑐 × 𝐿 𝑐 communities, denoted by 𝑐 1 , 𝑐 2 , . . . ,
𝑐 𝑁 , located in an 𝐿 × 𝐿 square, called common area. For
instance, a university campus and each department located
in that campus can be considered as the common area and
a community, respectively. As an example, a network with
four communities (𝑁 = 4) is represented in Fig. 1. 𝑀 nodes
move in common area such that they visit the communities
frequently. In contrast to the network model considered in [10],
nodes do not visit a specific community frequently, rather they
visit all communities frequently with different frequencies.
Initially, nodes are randomly placed within the common area
with a uniform distribution.
The communication range of nodes is fixed, and it is denoted
by 𝑅. We assume each node can move in two different modes:
local and roaming. A node moves within a community or
the common area when it is in local or roaming modes,
respectively. In each of these modes, a node moves according
to the random direction mobility model, with reflection when
hitting boundaries [32], which is more realistic than the torus
boundaries. The speed of a node is chosen from (𝑣 𝑚𝑖𝑛 , 𝑣 𝑚𝑎𝑥 ]
according to a uniform distribution. The time it takes for each
travel in local and roaming modes is distributed exponentially
with rates 𝛼 and 𝛽, respectively. When the movement mode
of a node is local and its travel ends, that node changes its
movement mode to roaming with probability 𝑃𝑟 . Moreover,
if a travel of a node ends, while it is in roaming movement
mode, the node decides to change its movement mode to local
with probability 𝑃𝑙 . Note that 𝑟 and 𝑙 in 𝑃𝑟 and 𝑃𝑙 refer to
the roaming and local modes, respectively. In case of changing
the movement mode to local, the node selects community 𝑐 𝑖
to move into during local mode with probability 𝑃𝑠𝑒𝑙_𝑖 .
Let us consider a roaming node that chooses local mode
and a community to move in. If it has just ended its travel
somewhere in the selected community, the mode is immediately changed; otherwise, it chooses a random position in
the selected community and begins to move towards that
position by the shortest straight path [18]. We call this movement transitional travel. Unlike [18], it is assumed that in a
transitional travel, a node moves with high speed, denoted
by 𝑣 𝑡𝑟 𝑎𝑛𝑠 , to reach the community soon. This change is
applied to the mobility model introduced in [18] in order to
make the mobility model theoretically more tractable. Once
a node reaches the previously chosen random position in the
community, it begins to move in local mode. The notations
introduced in this section are summarized in Table I.
There are two specific nodes called source and destination.
The source wishes to send a message using epidemic routing
to the destination. Adopting the terminology from the field
of Epidemiology [13], the nodes that have (have not) already
received the message are called infected (susceptible). More-
Notation
Description
𝑁
Number of communities
𝑀
Total number of nodes
𝐿
Edge length of common area
𝐿𝑐
Edge length of each community
𝑅
Communication range of each node
𝛼
Rate of the duration of a travel in local mode
𝛽
Rate of the duration of a travel in roaming mode
𝑃𝑟
Probability of changing local mode to roaming mode
𝑃𝑙
Probability of changing roaming mode to local mode
𝑃𝑠𝑒𝑙_𝑖
Probability of selecting community 𝑐 𝑖
𝑣 𝑚𝑖𝑛
Minimum speed in local/roaming mode
𝑣 𝑚𝑎𝑥
Maximum speed in local/roaming mode
𝑣 𝑡𝑟 𝑎𝑛𝑠
Speed in a transitional travel
over, roaming node and local node are used to refer to the
nodes that move in roaming and local modes, respectively. In
order to be able to use the benefits of analytical models for
analyzing the network, the following assumptions are made,
most of them come from the previous works in this area.
1) Communities, frequently visited locations, do not overlap each other [2], [21], [25].
2) Initially, the movement mode of all nodes is roaming.
3) Speed 𝑣 𝑡𝑟 𝑎𝑛𝑠 is high, the duration of a transitional travel
is very short. Based on this assumption, transitional
travels are neglected in the proposed models.
4) The communication range of nodes, 𝑅, is much less than
both the length of the edges of the communities and the
common area, 𝑅 ≪ 𝐿 𝑐 and 𝑅 ≪ 𝐿. This assumption is
very common in the literature [8], [15], [21].
5) The first meeting time of any two nodes moving in the
same fixed movement mode, starting from a random
time, is exponentially distributed. This assumption is
reasonable when 𝑅 ≪ 𝐿 𝑐 and 𝑅 ≪ 𝐿 [15], and has
been extensively used in recent years [8], [11].
6) The first meeting time of a node constantly moving in
roaming mode with a set of nodes constantly moving
in the same community in local mode is exponentially
distributed [10].
7) The delay of a message transmission, which corresponds
to a short time, is negligible [8], [17], [21], [33]–[35].
IV.
OVERVIEW OF PREVIOUS MODELS
This section presents an overview of the monolithic and
folded SRN models proposed in our previous work [10]. Due
to the strict limitation of space, we do not present details of
SRNs. The formal definition and structure of SRNs can be
found in [9], [36], [37].
The previous monolithic model consists of 𝑁 +1 submodels,
one per each community and one to represent the state of the
destination node. In the previous monolithic model, submodel
𝑆𝑢𝑏 𝑖 , 1 ≤ 𝑖 ≤ 𝑁, represents the situation of nodes frequently
visiting community 𝑐 𝑖 excluding the destination node, 𝑖 = 𝑁.
Submodel 𝑆𝑢𝑏 𝑖 provides four main places to represent the infected local nodes, the infected roaming nodes, the susceptible
local nodes, and the susceptible roaming nodes that frequently
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𝒕𝒔𝒖𝒔_𝒓𝒍
𝒅𝒆𝒔_𝟏
𝒔𝒖𝒔_𝒍
𝑻𝒆𝒏𝒅_𝒅𝒆𝒔_𝟏
#
[𝑮𝒍𝒊𝒏𝒇_𝒅𝒆𝒔_𝟏 ]
𝑻𝒍𝒊𝒏𝒇_𝒅𝒆𝒔_𝟏
𝑷𝒔𝒖𝒔_𝒍
𝒅𝒆𝒄_𝒅𝒆𝒔_𝟏
…
𝒔𝒖𝒔_𝒍𝒍
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𝒕𝒔𝒖𝒔_𝒍𝒓
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𝒕𝒔𝒖𝒔_𝒓𝒍
𝒅𝒆𝒔_𝑵
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𝒕𝒔𝒖𝒔_𝒍𝒍
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𝒅𝒆𝒔_𝑵
…
In this section, we describe the proposed monolithic SRN
to evaluate the average and the CDF of delivery delay and
the average number of transmissions of the epidemic routing
in the network model described in Section III. In addition to
𝑁, 𝑀, 𝛼, 𝛽, 𝑃𝑟 , 𝑃𝑙 , and 𝑃𝑠𝑒𝑙_𝑖 , 1 ≤ 𝑖 ≤ 𝑁, introduced in
Section III, the proposed monolithic model has the following
input parameters, that are the rates of the exponential functions
according to which first meeting times are distributed.
• 𝜆: The rate of the first meeting time of any two local
nodes moving in the same community
• 𝜇: The rate of the first meeting time of any two roaming
nodes
• 𝛾: The rate of the first meeting time of any roaming node
with any local node
• 𝜂: The rate of the first meeting time of a roaming node
with the set of other nodes that are in local mode and
move in the same community. Definition of this kind of
first meeting time can be found in [10].
In order to evaluate the measures of interest, the model
needs to capture the situation of nodes, in terms of being
susceptible or infected, and meetings between susceptible and
infected nodes. In order to represent message delivery in
the proposed model, the state of the destination node, being
susceptible or infected, should be captured by the model. As
mentioned in Section III, all nodes move according to the
same mobility model, and initially they are placed within
the common area with a uniform distribution. Thus, there is
no need to capture the identity of non-destination nodes. In
order to distinguish the destination node from other nodes, we
dedicate a specific submodel, called 𝑆𝑢𝑏 𝑑𝑒𝑠 and represented
in Fig. 2, to the destination node.
𝒔𝒖𝒔_𝒍
𝑷𝒅𝒆𝒔_𝟏
…
V. T HE P ROPOSED M ONOLITHIC M ODEL
𝑺𝒖𝒃𝒅𝒆𝒔
…
visit community 𝑐 𝑖 . Excluding the initial number of tokens,
submodels 𝑆𝑢𝑏 𝑖 , 1 ≤ 𝑖 ≤ 𝑁, have the same structures.
In order to be able to evaluate the performance of largescale networks, we have proposed a folded model, by folding
submodels 𝑆𝑢𝑏 1 , 𝑆𝑢𝑏 2 , . . . , 𝑆𝑢𝑏 𝑁 −1 together into a single
submodel, named 𝑆𝑢𝑏 𝑓 . In addition to the place holders for
nodes and transitions representing infections and decision of
nodes about the movement mode, there are another place
acting as a counter, named 𝑃𝑐𝑛𝑡 , and two other transitions
in submodel 𝑆𝑢𝑏 𝑓 . These elements enumerate the number of
communities, among 𝑐 1 , 𝑐 2 , . . . , 𝑐 𝑁 −1 , which are frequently
visited by at least one infected node.
In contrast to the monolithic model, the number of susceptible (infected) nodes frequently visiting each community,
except community 𝑐 𝑁 , cannot be captured from the folded
model. Moreover, the number of local infected nodes and
the number of local susceptible nodes in each community are
not represented in the folded model. However, the values of
these quantities are needed to precisely define guard and rate
functions of some timed transitions of the folded model. In
order to overcome this shortcoming, we used an approximation
which is based on the assumption that nearly the same number
of infected nodes and the same number of roaming infected
nodes frequently visit each community 𝑐 𝑗 , 1 ≤ 𝑗 ≤ 𝑁.
5
𝒊𝒏𝒇
𝑷𝒅𝒆𝒔
Fig. 2: Submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 of the proposed monolithic model.
The time of meetings between two susceptible and infected
nodes is essential to model the epidemic routing protocol. Two
main factors affect the time of meetings between nodes: i)
movement modes of nodes, ii) the community in which each
local node move. Thus, the monolithic model needs to capture
the states of nodes in terms of being susceptible or infected,
the movement modes of nodes, and the communities in which
local nodes move. Based on the aforementioned attributes,
the non-destination nodes can be classified into the following
classes.
• 𝑪 𝒔𝒖𝒔_𝒓 : The class of susceptible roaming nodes
• 𝑪 𝒊𝒏 𝒇 _𝒓 : The class of infected roaming nodes
𝒔𝒖𝒔_𝒍
• 𝑪𝒋
(1 ≤ 𝒋 ≤ 𝑵): The class of susceptible local nodes
moving community 𝑐 𝑗
𝒊𝒏 𝒇 _𝒍
• 𝑪𝒋
(1 ≤ 𝒋 ≤ 𝑵): The class of infected local nodes
moving in community 𝑐 𝑗
Unlike the destination node, the identities of other nodes are
not important. Capturing only the number of nodes belonging
to each above-mentioned classes is sufficient to model the
epidemic routing. To this end, each node of the network is
represented by a token in the proposed model, and for each
above-mentioned class, we consider a place which acts as a
repository of the nodes belonging to that class.
In addition to 𝑆𝑢𝑏 𝑑𝑒𝑠 , the proposed monolithic model has
𝑁 + 1 other submodels, named 𝑆𝑢𝑏𝑟 , 𝑆𝑢𝑏 𝑙_1 , 𝑆𝑢𝑏 𝑙_2 , . . . , and
𝑆𝑢𝑏 𝑙_𝑁 . Submodels 𝑆𝑢𝑏𝑟 , 𝑆𝑢𝑏 𝑙_1 , and 𝑆𝑢𝑏 𝑙_𝑁 are represented
in Fig. 3. Submodels 𝑆𝑢𝑏 𝑙_ 𝑗 , 1 < 𝑗 < 𝑁, have the same graphical representation as 𝑆𝑢𝑏 𝑙_1 and 𝑆𝑢𝑏 𝑙_𝑁 . The role of submodel
𝑆𝑢𝑏𝑟 is to capture the numbers of susceptible and infected
roaming nodes, excluding the destination node. Moreover,
submodel 𝑆𝑢𝑏 𝑙_ 𝑗 , 1 ≤ 𝑗 ≤ 𝑁, is used to capture the numbers of
susceptible and infected local nodes in community 𝑐 𝑗 . Places
𝑃 𝑠𝑢𝑠_𝑟 and 𝑃𝑖𝑛 𝑓 _𝑟 in submodel 𝑆𝑢𝑏𝑟 and places 𝑃 𝑠𝑢𝑠_𝑙
and
𝑗
𝑖𝑛 𝑓 _𝑙
𝑃𝑗
in submodel 𝑆𝑢𝑏 𝑙_ 𝑗 , 1 ≤ 𝑗 ≤ 𝑁, are used as repositories
for nodes belonging to classes 𝐶 𝑠𝑢𝑠_𝑟 , 𝐶 𝑖𝑛 𝑓 _𝑟 , 𝐶 𝑠𝑢𝑠_𝑙
, and
𝑗
𝑖𝑛 𝑓 _𝑙
, respectively. Transitions and other places in submodels
𝐶𝑗
𝑆𝑢𝑏𝑟 and 𝑆𝑢𝑏 𝑙_ 𝑗 represent the infections during the meetings,
0018-9545 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Vehicular Technology
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
𝑷𝒔𝒖𝒔_𝒍
𝟏
𝒕𝟏𝒔𝒖𝒔_𝒍𝒍
[𝑮𝒍𝒊𝒏𝒇_𝟏 ]
𝑻𝒍𝒊𝒏𝒇_𝟏
#
𝒕𝟏𝒔𝒖𝒔_𝒍𝒓
𝒔𝒖𝒔_𝒍
𝑻𝒆𝒏𝒅_𝟏
[𝑮𝒘 ]
𝑷𝒔𝒖𝒔_𝒍
𝒅𝒆𝒄_𝟏
𝒊𝒏𝒇_𝒍
𝑻𝒆𝒏𝒅_𝟏
[𝑮𝒘 ]
𝒊𝒏𝒇_𝒍𝒍
𝒕𝟏
𝒕𝟏𝒔𝒖𝒔_𝒓𝒍
𝒕𝒔𝒖𝒔_𝒓𝒍
𝑵
𝒊𝒏𝒇_𝒍𝒓
𝒕𝟏
𝒊𝒏𝒇_𝒍
𝑷𝒅𝒆𝒄_𝟏
𝒊𝒏𝒇_𝒍
𝑷𝟏
𝒕𝒔𝒖𝒔_𝒓𝒓
…
𝑷𝒔𝒖𝒔_𝒓
…
…
#
𝒍
𝑻𝒊𝒏𝒇_𝑵
# 𝑻𝒔𝒖𝒔_𝒍
𝒆𝒏𝒅_𝑵
[𝑮𝒘 ]
[𝑮𝒘 ]
𝒊𝒏𝒇_𝒍𝒍
𝒕𝑵
𝒊𝒏𝒇_𝒍
𝑷𝑵
𝒔𝒖𝒔_𝒓
𝑷𝒅𝒆𝒄
Section III, and given that initially only the source node
has the message, the initial numbers of tokens in places
𝑃 𝑠𝑢𝑠_𝑟 and 𝑃𝑖𝑛 𝑓 _𝑟 are 𝑀 − 2 and 1, respectively. Transition
𝑟
𝑇𝑖𝑛
𝑓 represents the infection of a susceptible roaming node.
[𝑮𝒘 ]
#
𝟏
𝒕𝒊𝒏𝒇_𝒓𝒓
𝒊𝒏𝒇_𝒓
𝑻𝒆𝒏𝒅
[𝑮𝒘 ]
𝒊𝒏𝒇_𝒓
𝑷𝒅𝒆𝒄
𝑖𝑛 𝑓 _𝑟
𝒕𝒔𝒖𝒔_𝒍𝒓
𝑵
𝑷𝒔𝒖𝒔_𝒍
𝒅𝒆𝒄_𝑵
𝒊𝒏𝒇_𝒍
𝑷𝒅𝒆𝒄_𝑵
𝒊𝒏𝒇_𝒓𝒍
𝒕𝟏
…
…
#
𝒊𝒏𝒇_𝒍
𝑻𝒆𝒏𝒅_𝑵
#
𝒔𝒖𝒔_𝒓
[𝑮𝒘 ] 𝑻𝒆𝒏𝒅
𝑻𝒓𝒊𝒏𝒇
𝑷𝒊𝒏𝒇_𝒓
𝒔𝒖𝒔_𝒍𝒍
𝒕𝑵
[𝑮𝒍𝒊𝒏𝒇_𝑵 ]
𝑴
−𝟐
#
𝑺𝒖𝒃𝒍_𝑵
𝒔𝒖𝒔_𝒍
𝑷𝑵
Notation
Description
inf
Infection of a susceptible node
dec
Deciding about the next movement mode
Subscript
end
Ending of the current travel of a node
des
Destination node
sus
Susceptible node
inf
Infected node
Superscript
𝑙
Local node / Local movement mode
𝑟
Roaming node / Roaming movement mode
…
…
#
#
TABLE II: Notations adopted in naming the elements
𝑺𝒖𝒃𝒓
𝑺𝒖𝒃𝒍_𝟏
6
𝒊𝒏𝒇_𝒍𝒓
𝒕𝑵
𝒊𝒏𝒇_𝒓𝒍
𝒕𝑵
Fig. 3: Submodels 𝑆𝑢𝑏𝑟 , 𝑆𝑢𝑏 𝑙_1 , . . . , 𝑆𝑢𝑏 𝑙_𝑁 .
changes in the movement modes, and community selections.
In the following, first, the role of elements of each submodel
is described, and then the guard and rate functions of the
transitions are introduced. The range of 𝑗 in the rest of this
paper is from 1 to 𝑁 (1 ≤ 𝑗 ≤ 𝑁). This notation in the
subscripts of the names of places and transitions refers to the
ID of the corresponding community.
A. Elements of submodels
In order to facilitate understanding the model, Table II
shows the notation used in the elements’ names, as subscripts
𝑖𝑛 𝑓 _𝑙
or superscripts. Place 𝑃 𝑠𝑢𝑠_𝑙
(𝑃 𝑗
) of submodel 𝑆𝑢𝑏 𝑙_ 𝑗 con𝑗
tains the tokens representing the susceptible (infected) local
𝑙
nodes that are moving in community 𝑐 𝑗 . Transition 𝑇𝑖𝑛
𝑓 _𝑗
represents the infection of a susceptible local node while
𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
moving in community 𝑐 𝑗 . Transition 𝑇𝑒𝑛𝑑_
𝑗 (𝑇𝑒𝑛𝑑_ 𝑗 ) represents
the ending of travels of susceptible (infected) local nodes
𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
moving in community 𝑐 𝑗 . When transition 𝑇𝑒𝑛𝑑_
𝑗 (𝑇𝑒𝑛𝑑_ 𝑗 )
𝑖𝑛 𝑓 _𝑙
fires, a token is removed from place 𝑃 𝑠𝑢𝑠_𝑙
(𝑃 𝑗
) and put
𝑗
𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
into place 𝑃 𝑑𝑒𝑐_
(𝑃
).
As
soon
as
a
token
is
put
in place
𝑗
𝑑𝑒𝑐_ 𝑗
𝑖𝑛 𝑓 _𝑙
𝑖𝑛 𝑓 _𝑙𝑙
𝑠𝑢𝑠_𝑟
Moreover, transition 𝑇𝑒𝑛𝑑
(𝑇𝑒𝑛𝑑 ) models the end of travel
of a susceptible (infected) roaming node. As soon as a token
𝑖𝑛 𝑓 _𝑟
𝑠𝑢𝑠_𝑟
is put in the places 𝑃 𝑑𝑒𝑐
and 𝑃 𝑑𝑒𝑐 , it is removed upon
firing of an immediate transition. Both transitions 𝑡 𝑠𝑢𝑠_𝑟𝑟 and
𝑡 𝑖𝑛 𝑓 _𝑟𝑟 fire with the probability 1 − 𝑃𝑙 , which represents
remaining in the roaming mode during the next travel. If a
𝑖𝑛 𝑓 _𝑟
𝑠𝑢𝑠_𝑟
token is in place 𝑃 𝑑𝑒𝑐
(𝑃 𝑑𝑒𝑐 ), with probability 𝑃𝑙 , one of
𝑖𝑛 𝑓 _𝑟𝑙 𝑖𝑛 𝑓 _𝑟𝑙
𝑠𝑢𝑠_𝑟𝑙 𝑠𝑢𝑠_𝑟𝑙
the transitions 𝑡1
, 𝑡2
, . . . , 𝑡 𝑠𝑢𝑠_𝑟𝑙
(𝑡1
, 𝑡2
, ...,
𝑁
𝑖𝑛 𝑓 _𝑟𝑙
𝑖𝑛 𝑓 _𝑟𝑙
𝑠𝑢𝑠_𝑟𝑙
𝑡𝑁
) fires. Specifically, transitions 𝑡 𝑗
and 𝑡 𝑗
signal
that the node that has just finished its travel in roaming mode
selects the local mode and community 𝑐 𝑗 to move in. Thus,
these transitions fire with probability 𝑃𝑙 · 𝑃𝑠𝑒𝑙_ 𝑗 .
As it can be observed in Fig. 2, there is one initial token in
𝑠𝑢𝑠_𝑟
place 𝑃 𝑑𝑒𝑠
of submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 . This token represents the
destination node and circulates among places of this submodel
𝑖𝑛 𝑓
𝑠𝑢𝑠_𝑙
until it is put in place 𝑃 𝑑𝑒𝑠 . When this token is in place 𝑃 𝑑𝑒𝑠_
𝑗
𝑠𝑢𝑠_𝑟
(𝑃 𝑑𝑒𝑠 ), the destination node is in community 𝑐 𝑗 (common
area) and moves in local (roaming) mode. Depositing this
𝑖𝑛 𝑓
token in place 𝑃 𝑑𝑒𝑠 represents the delivery of message to the
𝑠𝑢𝑠_𝑙
destination node. Roles of place 𝑃 𝑑𝑒𝑐_𝑑𝑒𝑠_
𝑗 and transitions
𝑠𝑢𝑠_𝑙𝑙
𝑠𝑢𝑠_𝑙𝑟
𝑠𝑢𝑠_𝑟𝑙
𝑠𝑢𝑠_𝑙
𝑙
𝑇𝑖𝑛 𝑓 _𝑑𝑒𝑠_ 𝑗 , 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠_ 𝑗 , 𝑡 𝑑𝑒𝑠_ 𝑗 , 𝑡 𝑑𝑒𝑠_ 𝑗 , 𝑡 𝑑𝑒𝑠_ 𝑗 are similar to
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑙𝑙
𝑙
those of place 𝑃 𝑑𝑒𝑐_
,
𝑗 and transitions 𝑇𝑖𝑛 𝑓 _ 𝑗 , 𝑇𝑒𝑛𝑑_ 𝑗 , 𝑡 𝑗
𝑠𝑢𝑠_𝑟𝑙
𝑡 𝑠𝑢𝑠_𝑙𝑟
,
𝑡
of
submodel
𝑆𝑢𝑏
,
respectively.
Moreover,
𝑙_ 𝑗
𝑗
𝑗
𝑠𝑢𝑠_𝑟
𝑠𝑢𝑠_𝑟
𝑠𝑢𝑠_𝑟𝑟
𝑟
, and 𝑇𝑖𝑛
, 𝑡 𝑑𝑒𝑠
place 𝑃 𝑑𝑒𝑐_𝑑𝑒𝑠
and transitions 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠
𝑓 _𝑑𝑒𝑠
𝑠𝑢𝑠_𝑟
can be presented in a similar manner to the place 𝑃 𝑑𝑒𝑐 and
𝑠𝑢𝑠_𝑟 𝑠𝑢𝑠_𝑟𝑟
𝑟
transitions 𝑇𝑒𝑛𝑑
,𝑡
, and 𝑇𝑖𝑛
𝑓 of submodel 𝑆𝑢𝑏 𝑟 , respectively. The only difference of the aforementioned elements of
submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 with those of submodels 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟
is that the elements of 𝑆𝑢𝑏 𝑑𝑒𝑠 represent the situation of the
destination node exclusively, while corresponding elements of
𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 are used to model the situation of all other
susceptible nodes.
𝑖𝑛 𝑓 _𝑙𝑟
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑙𝑙
𝑃 𝑑𝑒𝑐_
or 𝑡 𝑠𝑢𝑠_𝑙𝑟
(𝑡 𝑗
or 𝑡 𝑗
)
𝑗
𝑗 (𝑃 𝑑𝑒𝑐_ 𝑗 ), transition 𝑡 𝑗
fires with probabilities 1 − 𝑃𝑟 or 𝑃𝑟 , respectively. Transitions
𝑖𝑛 𝑓 _𝑙𝑟
𝑖𝑛 𝑓 _𝑙𝑙
) represent choosing local
and 𝑡 𝑗
𝑡 𝑠𝑢𝑠_𝑙𝑙
and 𝑡 𝑠𝑢𝑠_𝑙𝑟
(𝑡 𝑗
𝑗
𝑗
and roaming modes, respectively, by the susceptible (infected)
node, that has just finished its travel in local mode.
Places 𝑃 𝑠𝑢𝑠_𝑟 and 𝑃𝑖𝑛 𝑓 _𝑟 are containers for tokens representing the susceptible and infected roaming nodes, respectively. According to the second assumption provided in
B. Guard functions
As mentioned in Section V-A, the existence of a token in
𝑖𝑛 𝑓
place 𝑃 𝑑𝑒𝑠 indicates that the message is delivered to the destination. Since the average delivery delay is one of our measures
of interest, the proposed monolithic model is designed to be
absorbed when the token representing the destination is put in
𝑖𝑛 𝑓
place 𝑃 𝑑𝑒𝑠 . To this end, a guard function satisfying condition
0018-9545 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Authorized licensed use limited to: b-on: Universidade de Lisboa Reitoria. Downloaded on February 11,2021 at 14:52:19 UTC from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2021.3057541, IEEE
Transactions on Vehicular Technology
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
7
𝑖𝑛 𝑓
#𝑃 𝑑𝑒𝑠 == 0 should be associated with each timed transition in
submodels 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 . Symbol "#" shows the number
𝑖𝑛 𝑓
of tokens inside a place (mark of a place), where #𝑃 𝑑𝑒𝑠 refers
𝑖𝑛 𝑓
to the number of tokens in place 𝑃 𝑑𝑒𝑠 .
We associate the guard function 𝐺 𝑤 , defined in Eq. (1),
to all timed transitions in submodel 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 except
𝑙
transitions 𝑇𝑖𝑛
𝑓 _𝑗.
𝑖𝑛 𝑓
𝐺 𝑤 = (#𝑃 𝑑𝑒𝑠 == 0)
(1)
𝑖𝑛 𝑓
In addition to condition #𝑃 𝑑𝑒𝑠 == 0, there is another condition
𝑙
which should be satisfied before firing the transition 𝑇𝑖𝑛
𝑓 _𝑗.
Each local node has a chance to meet only roaming nodes and
other local nodes of the community in which it moves. Thus,
we associate the guard function 𝐺 𝑙𝑖𝑛 𝑓 _ 𝑗 , defined in Eq. (2), to
𝑙
transition 𝑇𝑖𝑛
𝑓 _ 𝑗 to guarantee that there is at least one infected
local node in community 𝑐 𝑗 or at least one infected roaming
node in the common area.
𝑖𝑛 𝑓
𝑖𝑛 𝑓 _𝑙
𝐺 𝑙𝑖𝑛 𝑓 _ 𝑗 = #𝑃 𝑑𝑒𝑠 == 0 ∧ (#𝑃 𝑗
+ #𝑃𝑖𝑛 𝑓 _𝑟 ) > 0
(2)
Guard function 𝐺 𝑙𝑖𝑛 𝑓 _ 𝑗 is also associated with transition
𝑙
𝑇𝑖𝑛
𝑓 _𝑑𝑒𝑠_ 𝑗 in submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 .
If the destination moves in a community during local mode, it
meets each infected local node moving in that community with
rate 𝜆. Moreover, the destination meets each infected roaming
𝑙
node with rate 𝛾. Thus, the rate of transition 𝑇𝑖𝑛
𝑓 _𝑑𝑒𝑠_ 𝑗 is
computed by Eq. (5).
𝑖𝑛 𝑓 _𝑙
𝑙
𝑅𝑖𝑛
𝑓 _𝑑𝑒𝑠_ 𝑗 = #𝑃 𝑗
· 𝜆 + #𝑃𝑖𝑛 𝑓 _𝑟 · 𝛾
(5)
Each susceptible roaming node meets an infected roaming
node with rate 𝜇. Moreover, the rate with which time taken for
each susceptible roaming node to meet the first infected local
node in community 𝑐 𝑗 is distributed equals value of function
𝑖𝑛 𝑓 _𝑙
𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑛) at 𝑛 = #𝑃 𝑗
. The number of susceptible roaming
𝑟
nodes is equal to #𝑃 𝑠𝑢𝑠_𝑟 . Therefore, the rate of transition 𝑇𝑖𝑛
𝑓
is computed by Eq. (6).
𝑟
𝑠𝑢𝑠_𝑟
𝑅𝑖𝑛
· #𝑃𝑖𝑛 𝑓 _𝑟 · 𝜇 +
𝑓 = #𝑃
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
)
𝑅ˆ 𝑚𝑒𝑒𝑡 (#𝑃 𝑗
𝑗=1
(6)
𝑟 , the rate of transiIn a similar manner to transition 𝑇𝑖𝑛
𝑓
𝑟
tion 𝑇𝑖𝑛 𝑓 _𝑑𝑒𝑠 is obtained from Eq. (7).
𝑟
𝑖𝑛 𝑓 _𝑟
𝑅𝑖𝑛
·𝜇+
𝑓 _𝑑𝑒𝑠 = #𝑃
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
𝑅ˆ 𝑚𝑒𝑒𝑡 (#𝑃 𝑗
)
(7)
𝑗=1
C. Rate functions
Simulation results in [10] show that the first meeting time
of a roaming node with a set of 𝑛 local nodes moving in
the same community can be assumed to be exponentially
distributed. Let 𝑅𝑚𝑒𝑒𝑡 (𝑛) denote the rate of this exponential
distribution. 𝑅𝑚𝑒𝑒𝑡 (𝑛), 1 ≤ 𝑛 ≤ 𝑀, is needed to precisely
𝑙
𝑟
compute the rates of transitions 𝑇𝑖𝑛
𝑓 _ 𝑗 and 𝑇𝑖𝑛 𝑓 . However,
as mentioned earlier in this section, input parameters of the
monolithic model include the values of function 𝑅𝑚𝑒𝑒𝑡 only
at 𝑛 = 1 and 𝑛 = 𝑀, respectively denoted by 𝛾 and 𝜂, in order
to avoid complication and simplify the formulation. Like in
[10], we approximate 𝑅𝑚𝑒𝑒𝑡 (𝑛), 1 < 𝑛 < 𝑀, using 𝛾 and 𝜂.
As 𝑛 increases, the chance of the roaming node to meet a local
node also increases. Thus, 𝑅𝑚𝑒𝑒𝑡 (𝑛) is an increasing function
of 𝑛. We approximate this increasing function as the following
linear increasing function of 𝛾 and 𝜂. This approximation is
exact for 𝑛 = 1 and 𝑛 = 𝑀.
𝛾𝑛,
𝑛 = 0, 1
ˆ
(3)
𝑅𝑚𝑒𝑒𝑡 (𝑛) =
𝜂−𝛾
,
𝑛
>1
𝛾 + (𝑛 − 1) · 𝑀
−2
Each susceptible local node in community 𝑐 𝑗 meets an
infected local node in that community with rate 𝜆. As men𝑖𝑛 𝑓 _𝑙
tioned earlier, the number of tokens in place 𝑃 𝑠𝑢𝑠_𝑙
(𝑃 𝑗
)
𝑗
represents the number of susceptible (infected) local nodes
in community 𝑐 𝑗 . The meeting rate of these susceptible and
𝑖𝑛 𝑓 _𝑙
infected nodes is #𝑃 𝑠𝑢𝑠_𝑙
× #𝑃 𝑗
× 𝜆. Moreover, value of
𝑗
𝑠𝑢𝑠_𝑙
function 𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑛) at 𝑛 = #𝑃 𝑗
is the rate with which time
taken for each infected roaming node to meet the first susceptible local node that moves in community 𝑐 𝑗 is distributed.
The number of infected roaming nodes is given by #𝑃𝑖𝑛 𝑓 _𝑟 .
𝑙
Therefore, the rate of transition 𝑇𝑖𝑛
𝑓 _ 𝑗 can be computed by
Eq. (4).
𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
𝑙
𝑅𝑖𝑛
· #𝑃 𝑗
𝑓 _ 𝑗 = #𝑃 𝑗
· 𝜆 + #𝑃𝑖𝑛 𝑓 _𝑟 · 𝑅ˆ 𝑚𝑒𝑒𝑡 (#𝑃 𝑠𝑢𝑠_𝑙
) (4)
𝑗
The duration of the travel is exponentially distributed with
rates 𝛼 and 𝛽 in local and roaming modes, respectively. Thus,
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑟
transitions 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠_
𝑗 and 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠 fire with rates 𝛼 and 𝛽,
𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
respectively. Moreover, the rates of transitions 𝑇𝑒𝑛𝑑_
𝑗 , 𝑇𝑒𝑛𝑑_ 𝑗 ,
𝑖𝑛 𝑓 _𝑟
𝑇𝑒𝑛𝑑
𝑠𝑢𝑠_𝑟
𝑇𝑒𝑛𝑑
and
are computed as #𝑃 𝑠𝑢𝑠_𝑙
· 𝛼,
𝑗
𝑠𝑢𝑠_𝑟
𝑖𝑛
#𝑃
· 𝛽, and #𝑃 𝑓 _𝑟 · 𝛽, respectively.
𝑖𝑛 𝑓 _𝑙
#𝑃 𝑗
· 𝛼,
VI. T HE P ROPOSED F OLDED A PPROXIMATE M ODEL
Containing at least four places per each community among
which tokens representing local nodes circulate, the monolithic model is not scalable in terms of 𝑁, the number of
communities, and 𝑀, the number of nodes. To overcome this
limitation, we propose in this section a folded approximate
model to evaluate the performance of epidemic routing in
target mobile social networks. In contrast with the monolithic
model, in the folded model there is a single submodel, named
𝑆𝑢𝑏 𝑓 , instead of submodels 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 . The subscript 𝑓
in the notations refers to application of the folding technique.
As mentioned in Section V, submodels 𝑆𝑢𝑏 𝑙_1 , 𝑆𝑢𝑏 𝑙_2 , . . . ,
𝑆𝑢𝑏 𝑙_𝑁 of the monolithic model have the same structure.
Thus, in order to prevent rapid growth of the state space, we
fold submodels 𝑆𝑢𝑏 𝑙_1 , 𝑆𝑢𝑏 𝑙_2 , . . . , 𝑆𝑢𝑏 𝑙_𝑁 all together. Since
𝑖𝑛 𝑓 _𝑙
places 𝑃 𝑠𝑢𝑠_𝑙
(𝑃 𝑗
) are folded into a single place, we need
𝑗
𝑖𝑛 𝑓 _𝑟𝑙
to fold also transitions 𝑡 𝑠𝑢𝑠_𝑟𝑙
(𝑡 𝑗
) of submodel 𝑆𝑢𝑏𝑟 into
𝑗
a single transition. Submodel 𝑆𝑢𝑏 𝑓 is represented in Fig. 4.
Table III provides details of submodel 𝑆𝑢𝑏 𝑓 elements that
result from folding.
𝑖𝑛 𝑓 _𝑙
In submodel 𝑆𝑢𝑏 𝑓 , only places 𝑃 𝑠𝑢𝑠_𝑙
and 𝑃 𝑓
are
𝑓
used to represent the susceptible and infected local nodes,
respectively. Thus, this submodel does not capture the number
of local infected (susceptible) nodes moving in a specific
community. Since the source is the only initial infected node
in the network, capturing the community in which it moves
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Transactions on Vehicular Technology
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
𝒕𝒔𝒖𝒔_𝒓𝒍
𝒇
[𝑮𝒘 ]
𝑷𝒇𝒔𝒖𝒔_𝒍
#
𝒊𝒏𝒇_𝒓𝒍
𝒕𝒇
[𝑮𝒘 ]
𝒊𝒏𝒇_𝒍
𝑻𝒆𝒏𝒅_𝒇
𝒊𝒏𝒇_𝒍𝒍
𝒕𝒇
𝒊𝒏𝒇_𝒍
𝑷𝒅𝒆𝒄_𝒇
𝑻𝒔𝒖𝒔_𝒓
𝒆𝒏𝒅
𝑷𝒔𝒖𝒔_𝒓
#
[𝑮𝒘 ]
𝒊𝒏𝒇_𝒓
𝑻𝒆𝒏𝒅
𝒊𝒏𝒇_𝒍𝒓
𝒕𝒇
[𝑮𝒘 ]
𝒓
𝑻𝒊𝒏𝒇
𝒍𝒍
𝒕𝒔𝒓𝒄_𝑵
𝟏
𝑷𝒊𝒏𝒇_𝒓
𝑻𝒍𝒆𝒏𝒅_𝒔𝒓𝒄_𝑵
Element of
Corresponding
Initial Mark /
Submodel 𝑺𝒖𝒃 𝒇
Folded Elements in
Rate Function /
𝑖𝑛 𝑓 _𝑟
the Monolithic Model Firing Probability
𝑃 𝑠𝑢𝑠_𝑙
(1 ≤ 𝑗 ≤ 𝑁)
𝑗
0
𝑖𝑛 𝑓 _𝑙
𝑃𝑗
𝑠𝑢𝑠_𝑙
𝑃 𝑑𝑒𝑐_
𝑗
𝑖𝑛 𝑓 _𝑙
𝑃 𝑑𝑒𝑐_ 𝑗
𝑠𝑢𝑠_𝑙
𝑇𝑒𝑛𝑑_
𝑗
𝑠𝑢𝑠_𝑟𝑙
𝑡𝑗
𝑡 𝑠𝑢𝑠_𝑙𝑙
𝑗
𝑡 𝑠𝑢𝑠_𝑙𝑟
𝑗
𝑙
𝑇𝑖𝑛
𝑓 _𝑗
𝑖𝑛 𝑓 _𝑙
𝑇𝑒𝑛𝑑_ 𝑗
𝑖𝑛 𝑓 _𝑟𝑙
𝑡𝑗
𝑖𝑛 𝑓 _𝑙𝑙
𝑡𝑗
𝑖𝑛 𝑓 _𝑙𝑟
𝑡𝑗
(1 ≤ 𝑗 ≤ 𝑁)
0
(1 ≤ 𝑗 ≤ 𝑁)
0
(1 ≤ 𝑗 ≤ 𝑁)
0
(1 ≤ 𝑗 ≤ 𝑁)
𝑠𝑢𝑠_𝑙
𝑅𝑒𝑛𝑑_
𝑓
(1 ≤ 𝑗 ≤ 𝑁)
𝑃𝑙
(1 ≤ 𝑗 ≤ 𝑁)
1 − 𝑃𝑟
(1 ≤ 𝑗 ≤ 𝑁)
𝑃𝑟
(1 ≤ 𝑗 ≤ 𝑁)
𝑙
𝑅𝑖𝑛
𝑓 _𝑓
(1 ≤ 𝑗 ≤ 𝑁)
𝑅𝑒𝑛𝑑_ 𝑓
(1 ≤ 𝑗 ≤ 𝑁)
𝑃𝑙
(1 ≤ 𝑗 ≤ 𝑁)
1 − 𝑃𝑟
(1 ≤ 𝑗 ≤ 𝑁)
𝑃𝑟
𝑖𝑛 𝑓 _𝑙
[𝑮𝒘 ]
𝒍
𝑷𝒅𝒆𝒄_𝒔𝒓𝒄_𝑵
𝒕𝒍𝒓
𝒔𝒓𝒄_𝑵
𝒓
𝑷𝒔𝒓𝒄
𝟏
𝒕𝒓𝒓
𝒔𝒓𝒄
[𝑮𝒘 ]
𝒓
𝑷𝒅𝒆𝒄_𝒔𝒓𝒄
𝒓
𝑻𝒆𝒏𝒅_𝒔𝒓𝒄
𝑖𝑛 𝑓 _𝑟
𝑃 𝑑𝑒𝑐 and transitions 𝑇𝑒𝑛𝑑 and 𝑡 𝑖𝑛 𝑓 _𝑟𝑟 of submodel 𝑆𝑢𝑏𝑟
of the monolithic model, respectively. The only difference
between the aforementioned elements of submodel 𝑆𝑢𝑏 𝑠𝑟 𝑐
and those of submodels 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 is that elements of
𝑆𝑢𝑏 𝑠𝑟 𝑐 represent the situation of the source node exclusively,
but corresponding elements of 𝑆𝑢𝑏 𝑙_ 𝑗 and 𝑆𝑢𝑏𝑟 model the
situation of all infected nodes except the destination node. It
𝑙𝑙
𝑙𝑟
𝑟𝑙
is worth mentioning that transitions 𝑡 𝑠𝑟
𝑐_ 𝑗 , 𝑡 𝑠𝑟 𝑐_ 𝑗 , 𝑡 𝑠𝑟 𝑐_ 𝑗 , and
𝑟𝑟
𝑡 𝑠𝑟 𝑐 fire with probabilities 1 − 𝑃𝑟 , 𝑃𝑟 , 𝑃𝑙 · 𝑃𝑠𝑒𝑙_ 𝑗 , and 1 − 𝑃𝑙 ,
respectively.
In addition to submodels 𝑆𝑢𝑏 𝑓 and 𝑆𝑢𝑏 𝑠𝑟 𝑐 , the proposed
folded model has another submodel, named 𝑆𝑢𝑏 𝑑𝑒𝑠 , to represent the situation of the destination node as the monolithic
model. Submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 of the folded model has the same
graphical representation as the submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 of the monolithic model that is represented in Fig. 2. Thus, the elements
of 𝑆𝑢𝑏 𝑑𝑒𝑠 are not described herein.
𝑖𝑛 𝑓 _𝑙
𝑖𝑛 𝑓 _𝑙𝑙
similar to those of place 𝑃 𝑑𝑒𝑐_ 𝑗 and transitions 𝑇𝑒𝑛𝑑_ 𝑗 , 𝑡 𝑗
𝑖𝑛 𝑓 _𝑙𝑟 𝑖𝑛 𝑓 _𝑟𝑙
𝑡𝑗
, 𝑡𝑗
𝒕𝒓𝒍
𝒔𝒓𝒄_𝑵
A. Approximating the number of local nodes in a community
𝑖𝑛 𝑓 _𝑙
during local mode is important to achieve a good accuracy
when the probabilities of selecting communities are not equal,
and there are a few nodes in the network. Under these
conditions, the community in which the source node moves
during local mode has a significant effect on the average
time at which the first infection occurs. Thus, we model the
situation of the source node in a specific submodel, named
𝑆𝑢𝑏 𝑠𝑟 𝑐 , represented in Fig. 5.
There exists an initial token in place 𝑃𝑟𝑠𝑟 𝑐 that represents
the source node and circulates among places of submodel
𝑆𝑢𝑏 𝑠𝑟 𝑐 . The existence of this token in place 𝑃𝑙𝑠𝑟 𝑐_ 𝑗 (𝑃𝑟𝑠𝑟 𝑐 )
means that the source node is in community 𝑐 𝑗 (common
area) and moves in local (roaming) mode. Roles of place
𝑙
𝑙𝑙
𝑙𝑟
𝑟𝑙
𝑃𝑙𝑑𝑒𝑐_𝑠𝑟 𝑐_ 𝑗 and transitions 𝑇𝑒𝑛𝑑_𝑠𝑟
𝑐_ 𝑗 , 𝑡 𝑠𝑟 𝑐_ 𝑗 , 𝑡 𝑠𝑟 𝑐_ 𝑗 , 𝑡 𝑠𝑟 𝑐_ 𝑗 are
𝑖𝑛 𝑓 _𝑙
𝒍𝒓
𝒕𝒔𝒓𝒄_𝟏
Fig. 5: Submodel 𝑆𝑢𝑏 𝑠𝑟 𝑐 of the proposed folded model.
TABLE III: Elements of the proposed folded submodel
𝑖𝑛 𝑓 _𝑙
𝑃𝑓
𝑠𝑢𝑠_𝑙
𝑃 𝑑𝑒𝑐_
𝑓
𝑖𝑛 𝑓 _𝑙
𝑃 𝑑𝑒𝑐_ 𝑓
𝑠𝑢𝑠_𝑙
𝑇𝑒𝑛𝑑_
𝑓
𝑠𝑢𝑠_𝑟𝑙
𝑡𝑓
𝑡 𝑠𝑢𝑠_𝑙𝑙
𝑓
𝑡 𝑠𝑢𝑠_𝑙𝑟
𝑓
𝑙
𝑇𝑖𝑛
𝑓 _𝑓
𝑖𝑛 𝑓 _𝑙
𝑇𝑒𝑛𝑑_ 𝑓
𝑖𝑛 𝑓 _𝑟𝑙
𝑡𝑓
𝑖𝑛 𝑓 _𝑙𝑙
𝑡𝑓
𝑖𝑛 𝑓 _𝑙𝑟
𝑡𝑓
𝑷𝒍𝒅𝒆𝒄_𝒔𝒓𝒄_𝟏
𝑷𝒍𝒔𝒓𝒄_𝑵
Fig. 4: Submodel 𝑆𝑢𝑏 𝑓 of the proposed folded model.
𝑃 𝑠𝑢𝑠_𝑙
𝑓
𝒕𝒓𝒍
𝒔𝒓𝒄_𝟏
𝒍𝒍
𝒕𝒔𝒓𝒄_𝟏
𝑻𝒍𝒆𝒏𝒅_𝒔𝒓𝒄_𝟏
[𝑮𝒘 ]
#
𝒕𝒊𝒏𝒇_𝒓𝒓
𝒊𝒏𝒇_𝒓
𝑷𝒅𝒆𝒄
𝒍
𝑷𝒔𝒓𝒄_𝟏
𝑴
−𝟐
…
#
[𝑮𝒘 ]
𝒕𝒔𝒖𝒔_𝒍𝒓
𝒇
𝑷𝒔𝒖𝒔_𝒍
𝒅𝒆𝒄_𝒇
𝑺𝒖𝒃𝒔𝒓𝒄
𝑺𝒖𝒃𝒇
…
𝒍
𝑻𝒊𝒏𝒇_𝒇
𝒊𝒏𝒇_𝒍
𝑷𝒇
𝒕𝒔𝒖𝒔_𝒍𝒍
𝒇
[𝑮𝒍𝒊𝒏𝒇_𝒇 ]
#
𝒔𝒖𝒔_𝒍
𝑻𝒆𝒏𝒅_𝒇
𝒕𝒔𝒖𝒔_𝒓𝒓
…
#
𝑷𝒔𝒖𝒔_𝒓
𝒅𝒆𝒄
8
,
of submodel 𝑆𝑢𝑏 𝑙_ 𝑗 of the monolithic model,
𝑟
respectively. Moreover, place 𝑃𝑟𝑑𝑒𝑐_𝑠𝑟 𝑐 and transitions 𝑇𝑒𝑛𝑑_𝑠𝑟
𝑐
𝑟𝑟
and 𝑡 𝑠𝑟 𝑐 can be described in a similar manner with place
Places 𝑃 𝑠𝑢𝑠_𝑙
and 𝑃 𝑓
act as repositories for tokens
𝑓
representing susceptible and infected local nodes, respectively.
Thus, in contrast with the monolithic model, the number
of susceptible and infected local nodes in each community
cannot be captured in the folded model. However, the values
of these quantities are needed to precisely define guard and
rate functions of the timed transitions of submodels 𝑆𝑢𝑏 𝑓 and
𝑆𝑢𝑏 𝑑𝑒𝑠 that represent infection of nodes. To overcome this
difficulty, we apply the approximation in Algorithm 1. Let
𝑖𝑛 𝑓 _𝑙
𝑁ˆ 𝑖
( 𝑁ˆ 𝑖𝑠𝑢𝑠_𝑙 ), 1 ≤ 𝑖 ≤ 𝑁, denote an approximated number
of infected (susceptible) local nodes moving in community 𝑐 𝑖 .
Algorithm 1 can be used to compute both approximated
𝑖𝑛 𝑓 _𝑙
values 𝑁ˆ 𝑖
and 𝑁ˆ 𝑖𝑠𝑢𝑠_𝑙 based on the number of tokens in
𝑖𝑛 𝑓 _𝑙
, 𝑃 𝑠𝑢𝑠_𝑙
, 𝑃𝑙𝑠𝑟 𝑐_𝑖 , and 𝑃𝑠𝑒𝑙_𝑖 (1 ≤ 𝑖 ≤ 𝑁). Since
places 𝑃 𝑓
𝑓
𝑖𝑛 𝑓 _𝑙
the procedures to compute 𝑁ˆ 𝑖
and 𝑁ˆ 𝑖𝑠𝑢𝑠_𝑙 are similar,
Algorithm 1 is written in a generic form where 𝑥 can be
substituted by inf or sus.
𝑖𝑛 𝑓 _𝑙
In each marking of the folded SRN, there are #𝑃 𝑓
𝑠𝑢𝑠_𝑙
and #𝑃 𝑓
tokens that represent the infected and susceptible
𝑖𝑛 𝑓 _𝑙
local nodes, respectively. Algorithm 1 distributes the #𝑃 𝑓
(#𝑃 𝑠𝑢𝑠_𝑙
) tokens representing the infected (susceptible) local
𝑓
nodes among communities 𝑐 1 , . . . , 𝑐 𝑁 based on the probabilities 𝑃𝑠𝑒𝑙_1 , . . . , 𝑃𝑠𝑒𝑙_𝑁 , according to which prospective local
𝑖𝑛 𝑓 _𝑙
nodes select communities. 𝑃𝑠𝑒𝑙_𝑖 × #𝑃 𝑓
(𝑃𝑠𝑒𝑙_𝑖 × #𝑃 𝑠𝑢𝑠_𝑙
)
𝑓
can be a good indicator of the approximate number of infected
0018-9545 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Vehicular Technology
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
9
Algorithm 1: Approximation of the number of infected/susceptible local nodes in each community 𝑐 𝑖
(1 ≤ 𝑖 ≤ 𝑁), excluding the destination node
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Data: #𝑃 𝑥_𝑙
, 𝑃𝑠𝑒𝑙_𝑖 , #𝑃𝑙𝑠𝑟 𝑐_𝑖 (1 ≤ 𝑖 ≤ 𝑁)
𝑓
𝑥_𝑙
Result: 𝑁ˆ 𝑖 (1 ≤ 𝑖 ≤ 𝑁)
⌉ (1 ≤ 𝑖 ≤ 𝑁);
𝑁ˆ 𝑖𝑥_𝑙 = ⌊𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
𝑓
Í 𝑁 ˆ 𝑥_𝑙
𝑥_𝑙
𝑑 = 𝑖=1 𝑁𝑖 − #𝑃 𝑓 ;
for 𝑖 = 1 𝑡𝑜 𝑁 do
> 𝑁ˆ 𝑖𝑥_𝑙 then
if 𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
𝑓
add 𝑐 𝑖 to 𝑄 + .
else
add 𝑐 𝑖 to 𝑄 − .
+
Sort 𝑄 and 𝑄 − based on measure |𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
− 𝑁ˆ 𝑖𝑥_𝑙 | in
𝑓
descending order.
while 𝑑 > 0 do
𝑐 𝑘 = remove front element of 𝑄 − ;
𝑁ˆ 𝑘𝑥_𝑙 − −;
𝑑 − −;
while 𝑑 < 0 do
𝑐 𝑘 = remove front element of 𝑄 + ;
𝑁ˆ 𝑘𝑥_𝑙 + +;
𝑑 + +;
if 𝑥 == 𝑖𝑛 𝑓 then
for 𝑖 = 1 𝑡𝑜 𝑁 do
if #𝑃𝑙𝑠𝑟 𝑐_𝑖 == 1 then
𝑁ˆ 𝑖𝑥_𝑙 + +;
(susceptible) local nodes in community 𝑐 𝑖 . However, this
indicator may not be an integer. If that is the case, we round
the indicator to the nearest integer, and then we assign as many
nodes as that integer to each community. Let ⌊𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
𝑓 ⌉
𝑥_𝑙
denote the integer nearest to 𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑓 . We define 𝑑 as
follows:
𝑁
Õ
𝑥_𝑙
(8)
𝑑=
⌊𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
𝑓 ⌉ − #𝑃 𝑓 ,
𝑖=1
𝑖𝑛 𝑓 _𝑙
where #𝑃 𝑥_𝑙
denotes the number of tokens in place 𝑃 𝑓
or
𝑓
𝑃 𝑠𝑢𝑠_𝑙
if
𝑥
=
𝑖𝑛
𝑓
or
𝑥
=
𝑠𝑢𝑠,
respectively.
In
the
case
𝑑
is
zero,
𝑓
we need to revise the node assignments. In the following, we
show how to revise the node assignments for each cases of
𝑑 > 0 and 𝑑 < 0.
1) 𝒅 > 0: We deallocate 𝑑 nodes from the communities,
for which their indicators were rounded-up, such that
at most one node from each of these communities
can be deallocated. We call this part of the algorithm
deallocation phase.
2) 𝒅 < 0: We assign −𝑑 more nodes to communities, for
which their indicators were rounded-down, such that at
most one more node to each of these communities can be
allocated. We call this part of the algorithm reallocation
phase.
Deallocation and reallocation phases are performed using two priority queues of communities, denoted by 𝑄 −
and 𝑄 + , respectively. 𝑄 − (𝑄 + ) is a queue of communities, for which their indicators were rounded-up (rounded
down), sorted in descending order based on measure
𝑥_𝑙
|(𝑃𝑠𝑒𝑙_𝑖 · #𝑃 𝑥_𝑙
𝑓 ) − ⌊𝑃 𝑠𝑒𝑙_𝑖 · #𝑃 𝑓 ⌉| for each community 𝑐 𝑖 .
During deallocation (reallocation) phase, we start from the
head of 𝑄 − (𝑄 + ) and deallocate (allocate) one node from (to)
each community until |𝑑| nodes are deallocated (allocated). At
the end, if the source node is in the local mode and moves in
𝑖𝑛 𝑓 _𝑙
community 𝑐 𝑖 , we increase 𝑁ˆ 𝑖
by one.
B. Guard and Rate Functions
𝑖𝑛 𝑓 _𝑙
𝑖𝑛 𝑓 _𝑟
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑟
Transitions 𝑇𝑒𝑛𝑑_
represent the
𝑓 , 𝑇𝑒𝑛𝑑 , 𝑇𝑒𝑛𝑑_ 𝑓 , and 𝑇𝑒𝑛𝑑
end of travels of the susceptible local nodes, susceptible roaming nodes, infected local nodes, and infected roaming nodes,
respectively, excluding the source and destination nodes. Since
the duration of each travel of local (roaming) nodes is exponentially distributed with rate 𝛼 (𝛽), the rate functions of tran𝑖𝑛 𝑓 _𝑙
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑙
𝑠𝑢𝑠_𝑙
sitions 𝑇𝑒𝑛𝑑_
·𝛼
𝑓 and 𝑇𝑒𝑛𝑑_ 𝑓 are defined as 𝑅𝑒𝑛𝑑_ 𝑓 = #𝑃 𝑓
𝑖𝑛 𝑓 _𝑙
𝑖𝑛 𝑓 _𝑙
and 𝑅𝑒𝑛𝑑_ 𝑓 = #𝑃 𝑓
𝑠𝑢𝑠_𝑟
𝑇𝑒𝑛𝑑
· 𝛼, respectively. Note that rate of tran-
𝑖𝑛 𝑓 _𝑟
𝑇𝑒𝑛𝑑
sitions
and
are computed in a similar manner as
𝑙
the monolithic model. Moreover, rates of transition 𝑇𝑒𝑛𝑑_𝑠𝑟
𝑐_ 𝑗
𝑠𝑢𝑠_𝑙
of submodel 𝑆𝑢𝑏 𝑠𝑟 𝑐 and transition 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠_
of
submodel
𝑗
𝑟
𝑆𝑢𝑏 𝑑𝑒𝑠 are 𝛼 while transition 𝑇𝑒𝑛𝑑_𝑠𝑟
𝑐 of submodel 𝑆𝑢𝑏 𝑠𝑟 𝑐
𝑠𝑢𝑠_𝑟
and transition 𝑇𝑒𝑛𝑑_𝑑𝑒𝑠
of submodel 𝑆𝑢𝑏 𝑑𝑒𝑠 fire with rate 𝛽.
Similarly to the corresponding transitions in the monolithic
model, the guard function 𝐺 𝑤 , defined in Eq. (1), is associated
with all timed transitions of submodels 𝑆𝑢𝑏 𝑠𝑟 𝑐 and 𝑆𝑢𝑏 𝑓 ,
𝑙
except transition 𝑇𝑖𝑛
𝑓 _ 𝑓 to construct an absorbing model.
In addition to guard function 𝐺 𝑤 , we need to define other
𝑙
guard functions to be associated to transitions 𝑇𝑖𝑛
𝑓 _ 𝑓 and
𝑙
𝑙
𝑇𝑖𝑛 𝑓 _𝑑𝑒𝑠_ 𝑗 . Guard function 𝐺 𝑖𝑛 𝑓 _ 𝑓 , defined by Eq. (9), is
𝑙
𝑙
associated with transition 𝑇𝑖𝑛
𝑓 _ 𝑓 . Transition 𝑇𝑖𝑛 𝑓 _ 𝑓 models
meetings of the local susceptible nodes, excluding the destination node, with infected nodes. Such a meeting is possible
iff at least one infected node moves in the roaming mode, or
movement modes of one susceptible node, except the destination node, and one infected node are local, and they move in
Í
𝑖𝑛 𝑓 _𝑙
the same community. Condition 𝑁𝑗=1 𝑁ˆ 𝑠𝑢𝑠_𝑙
· 𝑁ˆ 𝑗
> 0 in
𝑗
Eq. (9) enforces moving of one susceptible node, except the
destination node, and one infected node in the local mode at
the same community. Moreover, condition #𝑃𝑖𝑛 𝑓 _𝑟 +#𝑃𝑟𝑠𝑟 𝑐 > 0
indicates the presence of at least one infected roaming node.
It is worth mentioning that like 𝐺 𝑤 , guard function 𝐺 𝑙𝑖𝑛 𝑓 _ 𝑓
should guarantee that the destination has not received the
𝑖𝑛 𝑓
message. So, condition #𝑃 𝑑𝑒𝑠 == 0 is included.
𝑖𝑛 𝑓
𝐺 𝑙𝑖𝑛 𝑓 _ 𝑓 = (#𝑃 𝑑𝑒𝑠 == 0) ∧
(
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
𝑁ˆ 𝑠𝑢𝑠_𝑙
· 𝑁ˆ 𝑗
> 0) ∨ (#𝑃𝑖𝑛 𝑓 _𝑟 + #𝑃𝑟𝑠𝑟 𝑐 > 0)
𝑗
𝑗=1
(9)
In a similar way, we associate guard function 𝐺 𝑙𝑖𝑛 𝑓 _𝑑𝑒𝑠_ 𝑗 ,
𝑙
defined by Eq. (10), to transition 𝑇𝑖𝑛
𝑓 _𝑑𝑒𝑠_ 𝑗 . The first and
second conditions in Eq. (10) represent the presence of at least
one infected local node in community 𝑐 𝑗 and one infected
roaming node, respectively.
𝑖𝑛 𝑓 _𝑙
𝐺 𝑙𝑖𝑛 𝑓 _𝑑𝑒𝑠_ 𝑗 = ( 𝑁ˆ 𝑗
> 0) ∨ (#𝑃𝑖𝑛 𝑓 _𝑟 + #𝑃𝑟𝑠𝑟 𝑐 > 0) (10)
Note that roaming susceptible nodes have chance to meet
each infected node, and there is at least one infected node,
the source, always in the network. Thus, function 𝐺 𝑤 is an
𝑟 , and we do
appropriate guard function for transition 𝑇𝑖𝑛
𝑓
0018-9545 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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10
not need to include any condition regarding the numbers of
infected and susceptible nodes.
Any two susceptible and infected local nodes in community 𝑐 𝑗 meet with rate 𝜆. Therefore, the rate at which the
non-destination local nodes in community 𝑐 𝑗 get infected by
the infected non-destination local nodes in that community
𝑖𝑛 𝑓 _𝑙
can be computed as 𝑁ˆ 𝑠𝑢𝑠_𝑙
· 𝑁ˆ 𝑗
· 𝜆. Subsequently, the
𝑗
summation of this rate over 𝑗 = 1, 2, . . . , 𝑁 results in the
total rate of infecting the non-destination susceptible local
nodes by the infected local nodes. Places 𝑃𝑖𝑛 𝑓 _𝑟 and 𝑃𝑟𝑠𝑟 𝑐
are dedicated to contain the tokens representing the infected
roaming nodes. Therefore, the number of these nodes is equal
to #𝑃𝑖𝑛 𝑓 _𝑟 + #𝑃𝑟𝑠𝑟 𝑐 . Distribution rate of the time taken for
each infected roaming node to meet one of the susceptible
local nodes in community 𝑐 𝑗 equals the value of function
𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑛) at 𝑛 = 𝑁ˆ 𝑠𝑢𝑠_𝑙
, represented as 𝑅ˆ 𝑚𝑒𝑒𝑡 ( 𝑁ˆ 𝑠𝑢𝑠_𝑙
). Thus,
𝑗
𝑗
𝑖𝑛
𝑓
_𝑟
𝑟
)
refers
to
the
rate
of
infect(#𝑃
+ #𝑃𝑠𝑟 𝑐 ) × 𝑅ˆ 𝑚𝑒𝑒𝑡 ( 𝑁ˆ 𝑠𝑢𝑠_𝑙
𝑗
ing the local nodes moving in community 𝑐 𝑗 by the infected
𝑙
roaming nodes. Therefore, the rate of transition 𝑇𝑖𝑛
𝑓 _ 𝑓 is
computed with Eq. (11).
𝑙
𝑅𝑖𝑛
𝑓 _𝑓 =
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
𝑁ˆ 𝑠𝑢𝑠_𝑙
𝑁ˆ 𝑗
𝜆
𝑗
(11)
𝑗=1
+ (#𝑃
𝑖𝑛 𝑓 _𝑟
+
#𝑃𝑟𝑠𝑟 𝑐 )
×
)
𝑅ˆ 𝑚𝑒𝑒𝑡 ( 𝑁ˆ 𝑠𝑢𝑠_𝑙
𝑗
If the movement mode of the destination node is local, it meets
each infected roaming node with rate 𝛾 and each infected local
node in the community, in which the destination node moves,
𝑙
with rate 𝜆. Thus, the rate of transition 𝑇𝑖𝑛
𝑓 _𝑑𝑒𝑠_ 𝑗 is obtained
from Eq. (12).
𝑖𝑛 𝑓 _𝑙
𝑙
𝑖𝑛 𝑓 _𝑟
𝑅𝑖𝑛
+ #𝑃𝑟𝑠𝑟 𝑐 ) · 𝛾 + 𝑁ˆ 𝑗
· 𝜆,
𝑓 _𝑑𝑒𝑠_ 𝑗 = (#𝑃
(12)
𝑖𝑛 𝑓 _𝑙
where (#𝑃𝑖𝑛 𝑓 _𝑟 + #𝑃𝑟𝑠𝑟 𝑐 ) · 𝛾 and 𝑁ˆ 𝑗
· 𝜆 are equal to the rate
of infecting the destination node by the infected roaming nodes
and the infected local nodes in community 𝑐 𝑗 , respectively.
Any two roaming nodes meet with rate 𝜇. Thus, the rate
of infecting a single susceptible roaming node by infected
roaming nodes can be computed as (#𝑃𝑖𝑛 𝑓 _𝑟 + #𝑃𝑟𝑠𝑟 𝑐 ) · 𝜇.
Moreover, the rate of the time taken for a susceptible roaming
node to meet one of infected local nodes, in a community, is
𝑟
estimated as 𝑅ˆ 𝑚𝑒𝑒𝑡 . Thus, the rate of transition 𝑇𝑖𝑛
𝑓 _𝑑𝑒𝑠 can
be computed by Eq. (13).
𝑟
𝑖𝑛 𝑓 _𝑟
𝑅𝑖𝑛
+ #𝑃𝑟𝑠𝑟 𝑐 ) · 𝜇 +
𝑓 _𝑑𝑒𝑠 = (#𝑃
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
𝑅ˆ 𝑚𝑒𝑒𝑡 ( 𝑁ˆ 𝑖
) (13)
𝑖=1
The tokens in place 𝑃 𝑠𝑢𝑠_𝑟 represent the susceptible roaming
nodes, excluding the destination node. The infection rate of a
𝑟
single roaming node is equal to 𝑅𝑖𝑛
𝑓 _𝑑𝑒𝑠 . Thus, multiplying
𝑟
#𝑃 𝑠𝑢𝑠_𝑟 by 𝑅𝑖𝑛
,
we
get
the
total
infection rate of non𝑓 _𝑑𝑒𝑠
destination susceptible roaming nodes. Therefore, the rate of
𝑟 , which represents the infection of roaming
transition 𝑇𝑖𝑛
𝑓
nodes, can be computed by Eq. (14).
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
𝑟
𝑠𝑢𝑠_𝑟
𝑖𝑛 𝑓 _𝑟
𝑟
ˆ
ˆ
𝑅𝑖𝑛
=
#𝑃
·
(#𝑃
+
#𝑃
)
·
𝜇
+
𝑅
(
𝑁
)
𝑚𝑒𝑒𝑡
𝑠𝑟 𝑐
𝑓
𝑖
𝑖=1
(14)
VII. P ERFORMANCE M EASURES
In this section, the performance measures and the way
to compute them, by applying the proposed models, are
presented:
Average Delivery Delay- As mentioned in Sections V and
VI, the proposed monolithic and folded models are absorbed
𝑖𝑛 𝑓
when a token is put in place 𝑃 𝑑𝑒𝑠 , representing the delivery
of the message to the destination. Thus, the Mean Time To
Absorption (MTTA) in both monolithic and folded models
represents the average delivery delay.
Average Number of Transmissions- This measure can be
computed from the proposed models after an appropriate
reward rate is assigned to each tangible marking of SRNs.
According to Section V-A, tokens representing the infected
𝑖𝑛 𝑓 _𝑙
nodes, except the destination, circulate among places 𝑃 𝑗
and 𝑃𝑖𝑛 𝑓 _𝑟 of the monolithic model. Thus, in each marking of
the monolithic SRN, the sum of the numbers of tokens in these
places represents the number of infected nodes, excluding the
destination but including the source, that is equal to the number
of transmissions. If we assign the reward rate represented in
Eq. (15) to each marking of the monolithic SRN, the average
reward rate at time 𝑡 is equal to the average number of
transmissions until time 𝑡.
𝑁
Õ
𝑖𝑛 𝑓 _𝑙
#𝑃 𝑗
+ #𝑃𝑖𝑛 𝑓 _𝑟
(15)
𝑟𝑚 =
𝑗=1
As 𝑡 increases, the average reward rate at time 𝑡 converges
to the average number of transmissions by the delivery time.
Thus, if 𝑡 is large enough, the average number of transmissions
by time of delivery is obtained. Similarly, the average number
of transmissions can be obtained from the folded model.
Tokens representing the infected nodes excluding the source
𝑖𝑛 𝑓 _𝑙
and destination nodes are hold in places 𝑃 𝑓
and 𝑃𝑖𝑛 𝑓 _𝑟 .
Thus, the appropriate reward rate, to be assigned to markings
of the folded SRN, is obtained from Eq. (16). Note that
addition of 1 in this equation is due to counting also the
transmission of the message to the destination node.
𝑖𝑛 𝑓 _𝑙
𝑟 𝑓 = #𝑃 𝑓
+ #𝑃𝑖𝑛 𝑓 _𝑟 + 1
(16)
CDF of the Delivery Delay- The probability of delivery of
the message no later than time 𝑡, 𝑡 > 0, can be computed using
transient analysis of the proposed SRNs. To this end, we need
𝑖𝑛 𝑓
to assign the reward rate #𝑃 𝑑𝑒𝑠 to each marking of the SRNs.
The CDF of the delivery delay at time 𝑡 is equal to the average
reward rate at time 𝑡.
After computing the parameters 𝜆, 𝜇, 𝛾, and 𝜂, we can
numerically solve the proposed models, and compute performance metrics using the Stochastic Petri Net Package (SPNP)
[38]. This tool automatically converts an SRN to its underlying
CTMC and facilitates computing the measures of interest.
VIII. P ERFORMANCE EVALUATION
This section presents the results obtained with the monolithic and folded models, which are validated. Moreover, in
this section, we propose an ODE model for epidemic routing
in the target network, and compare it with the folded model
in terms of accuracy.
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𝑵 Community (𝑐 𝑖 ) Coordinate of Center
𝑐1
(250, 250)
3
𝑐2
(250, 750)
𝑐3
(750, 250)
𝑐1
(𝐿/5 + 50, 𝐿/5 + 50)
4
𝑐2
(𝐿/5 + 50, 4𝐿/5 − 50)
𝑐3
(4𝐿/5 − 50, 𝐿/5 + 50)
𝑐4
(4𝐿/5 − 50, 4𝐿/5 − 50)
𝑐1
(250, 250)
𝑐2
(250, 750)
5
𝑐3
(750, 250)
𝑐4
(750, 750)
𝑐5
(500, 500)
𝑐1 , 𝑐2
(150, 150), (150, 500)
𝑐3 , 𝑐4
(150, 850), (500, 150)
10
𝑐5 , 𝑐6
(500, 350), (500, 650)
𝑐7 , 𝑐8
(500, 850), (850, 150)
𝑐 9 , 𝑐 10
(850, 500), (850, 850)
𝑷𝒔𝒆𝒍_𝒊
0.2
0.4
0.4
0.2
0.4
0.3
0.1
0.2
0.4
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
Network Setting: We set the network parameters as in [3]
and [18]. Specifically, the values of parameters are as follows,
unless otherwise noted. Parameters 𝐿, 𝐿 𝑐 , 𝑃𝑙 , and 𝑃𝑟 are set
to 1000 𝑚, 100 𝑚, 0.8, and 0.2, respectively. Moreover, the
average duration of a travel in local and roaming modes is 80 𝑠
and 520 𝑠, (𝛼 = 1/80 and 𝛽 = 1/520), respectively. According
to [3] and [18], these setting matches the MIT trace. It is worth
noting that parameters 𝑅, 𝑣 𝑚𝑖𝑛 , 𝑣 𝑚𝑎𝑥 , and 𝑣 𝑡𝑟 𝑎𝑛𝑠 are 10 𝑚,
5 𝑚/𝑠, 15 𝑚/𝑠, and 20 𝑚/𝑠, respectively. Table IV shows the
locations of communities and the probabilities at which they
are selected by the prospective local nodes for different values
of 𝑁. Considering the left-lower corner of common area as the
origin of a coordinate system, communities are centered in the
coordinates established in Table IV.
Rates of Meeting Time Distributions: Before using the
proposed models to evaluate the performance of epidemic
routing, we need to compute some input parameters, namely
𝜆, 𝜇, 𝛾, and 𝜂, by simulation. In order to obtain 𝜆 (𝜇), in each
run of the simulation, two nodes are uniformly placed in an
𝐿 𝑐 × 𝐿 𝑐 (𝐿 × 𝐿) square, and then nodes are moved until they
meet each other. Similarly, in order to compute 𝛾, one node is
uniformly placed in an 𝐿 × 𝐿 square, and one node is placed
uniformly in an 𝐿 𝑐 ×𝐿 𝑐 square within the aforementioned 𝐿×𝐿
square. Then, the former and latter nodes move in the 𝐿 × 𝐿
and 𝐿 𝑐 × 𝐿 𝑐 squares, respectively, until they meet each other.
Parameter 𝜂 is obtained in a similar way when 𝛾 is computed.
Validation Method: In order to validate the proposed models, the results of the monolithic and folded models are comparatively assessed against the simulation results. To achieve
this end, the network under-analysis is simulated by applying
the discrete-event simulation developed in Java. Although the
transmission delays are not considered in the proposed models,
they are considered in the simulation. We assume that the
transmission speed of each node and the message size are
2.5 MBps and 25 KB, respectively. This transmission speed
could be provided by Bluetooth technology.
1
0.9
CDF of the Delivery Delay
TABLE IV: Locations of communities and the probabilities of
choosing them by prospective local nodes.
0.8
Monolithic Model
Simulation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
300
600
900
1,200
1,500
Time, t (s)
Fig. 6: The CDF of the delivery delay for 𝑁 = 4 and 𝑀 = 15.
A. Validation of the proposed models
The Monolithic Model: Fig. 6 represents the CDF of the
delivery delay obtained with the monolithic model and by simulation for the aforementioned network with four communities
(𝑁 = 4) and 15 nodes (𝑀 = 15). The maximum error of
the analytical results reported in Fig. 6 is 6.32% indicating
high accuracy of the monolithic model to predict CDF of the
delivery delay.
Before continuing to present the numerical results, it is
worth mentioning that abbreviations Mono., Fold., and Sim.
used in tables and figures, stand for the monolithic model, the
folded model, and simulation, respectively. Table V presents
the average delivery delay and the average number of transmissions obtained with the proposed monolithic model and by
simulation. The Percent Errors of the results computed from
the monolithic model with respect to the corresponding results
computed by simulation are represented in columns PE. In
Table V, for each result obtained from simulation, the 95%
confidence interval is also presented. Due to memory shortage,
the monolithic model cannot be solved for some configurations. Notation "-" in Table V shows these configurations,
where the monolithic model encounters a scalability problem
in terms of the number of communities, 𝑁, and the number
of nodes, 𝑀. As an example, for networks consisting of 4
communities (𝑁 = 4), a system with 64 GB memory space
cannot solve the monolithic model even when there are only
20 nodes in the network (𝑀 = 20). As it can be observed
in Table V, the results obtained with the monolithic model
and by simulation are close indicating that the monolithic
SRN accurately models the network. Moreover, the average
number of transmissions does not depend on the number
of communities, 𝑁, and consequently the location visiting
preference. According to the results represented in Table V,
the average number of transmissions is nearly 𝑀/2.
The Folded Model: In order to evaluate the performance of
epidemic routing on a large scale network, the folded model
was adopted. Fig. 7(a) represents the CDF of the delivery
delay obtained with the folded model and by simulation for
a network with four communities (𝑁 = 4) and 100 nodes
(𝑀 = 100). As it can be observed in Fig. 7(a), simulation and
analytical results are very close to each other, indicating high
accuracy of the folded model to predict CDF of the delivery
delay.
Fig. 7(b) represents the average number of transmissions of
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TABLE V: Comparison of the results obtained with the monolithic model and by simulation.
3
4
5
1
Folded Model
Simulation
CDF of the Delivery Delay
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
30
70 110 150 190 230 270 310 350
Time, t (s)
(a)
Delivery Delay (𝑠)
Number of Transmissions
Confidence Interval
Average
Confidence Interval
PE Lower Bound Upper Bound Mono. Sim. PE Lower Bound Upper Bound
3.16
808.18
831.44
4.95 5.01 1.06
4.95
5.06
4.58
633.75
650.06
7.39 7.49 1.40
7.41
7.58
6.61
528.20
541.40
9.82 9.94 1.30
9.82
10.07
1.92
863.21
887.78
4.96 5.02 1.21
4.96
5.07
5.39
655.91
672.87
7.39 7.51 1.65
7.43
7.60
550.01
563.83
9.99
9.87
10.11
3.34
888.40
914.06
4.96 5.01 0.96
4.95
5.06
3.77
689.35
707.23
7.39 7.57 2.27
7.48
7.66
561.03
575.01
9.99
9.87
10.11
110
100
90
80
70
60
50
40
30
20
10
0
4,000
Folded Model
Simulation
3,000
2,500
2,000
1,500
1,000
500
0
0
1,200
Fold., 𝑃𝑟 = 0.2, 𝑃𝑙 = 0.8
Sim., 𝑃𝑟 = 0.2, 𝑃𝑙 = 0.8
Fold., 𝑃𝑟 = 0.8, 𝑃𝑙 = 0.2
Sim., 𝑃𝑟 = 0.8, 𝑃𝑙 = 0.2
3,500
20 40 60 80 100 120 140 160 180 200
𝑀
2
3
4
5
7
6
8
800
600
400
200
0
10
9
20
30
40
50
60
70
80
90 100
𝑀
(b)
(a)
Fig. 8: The average delivery delay obtained with the folded
SRN and by simulation when 𝑁 = 10.
2,500
Sim., 𝑅 = 5
Mono., 𝑅 = 5
Fold., 𝑅 = 5
Sim., 𝑅 = 10
Mono., 𝑅 = 10
Fold., 𝑅 = 10
2,000
1,500
Average Delivery Delay (𝑠)
Average Delivery Delay (𝑠)
2,500
the message obtained with the folded model and the simulation
when the network under-analysis consists of four communities
(𝑁 = 4), and 𝑀 varies from 10 to 200. As it can be observed
in Fig. 7(b), the folded model is very accurate to be used for
predicting the average number of transmissions. The percent
errors corresponding to the results represented in Fig. 7(b) is
less than 2% for 𝑀 > 20. As shown in Fig. 7(b), the average
number of transmissions changes approximately linearly as 𝑀
increases. Similarly to Table V, the results obtained from the
analytical model and the simulation represented in Fig. 7(b)
indicates that 𝑀/2 is an accurate estimation for the average
number of transmissions. Fig. 8 represents the average delivery
delay obtained from the folded model and by simulation for a
large-scale network with 10 communities (𝑁 = 10). As it can
be seen in that figure, the folded model can accurately evaluate
the average delivery delay when the number of communities
is large. Moreover, as observed in Fig. 8(a), increasing 𝑃𝑟 and
decreasing 𝑃𝑙 yield higher average delivery delay given that
the average time taken for the meeting of two roaming nodes is
greater than two local nodes moving in the same community.
Comparison of the Proposed Models: Fig. 9 represents
the average delivery delay obtained with the monolithic and
the folded models and by simulation for different values of 𝑀,
when the network under-analysis consists of four communities
(𝑁 = 4). Since the monolithic model cannot be solved in
practice for 𝑀 > 19 due to the state space explosion, the
maximum value of 𝑀 is 19 in Fig. 9. In addition to the
default setting of parameters, Figs. 9(a) and 9(b) represent the
results of the proposed models and simulation for 𝑅 = 5 𝑚
Folded Model
Simulation
1,000
𝑀
(b)
Fig. 7: Numerical results obtained from the folded model and
by the simulation for 𝑁 = 4.
The Average Delivery Delay (𝑠)
Average
Mono. Sim.
10 845.69 819.81
15 671.34 641.91
20 570.16 534.80
10 892.31 875.50
15 700.22 664.39
20
556.92
10 931.34 901.23
15 724.65 698.29
20
568.02
The Average Delivery Delay (𝑠)
𝑴
The Average Number of Transmissions
𝑵
1,000
500
0
10
11
12
13
14
15
16
𝑀
(a) 𝐿 = 1000 𝑚
17
18
19
Sim.
Mono.
Fold.
2,000
1,500
1,000
500
0
10
11
12
13
14
15
16
17
18
19
𝑀
(b) 𝐿 = 500 𝑚 and 𝑅 = 10 𝑚
Fig. 9: The average delivery delay for 𝑁 = 4.
and 𝐿 = 500 𝑚, respectively. As it can be seen in Fig. 9, the
results of the monolithic model are very close to the results
of the simulation indicating high accuracy of the monolithic
model. Moreover, the results of the folded model is close to the
results of the simulation and the monolithic model. It shows
that the folded model accurately approximates the monolithic
model.
B. Analysis of the number of transmissions by delivery time
In order to justify the linear behaviour of the average
number of transmission observed in Fig. 7(b) and Table V,
we present the following theorem about the average number
of transmission in a more general network model.
Theorem 1: The number of transmissions by the time of
delivery, including the forwarding to the destination node,
under epidemic routing follows a uniform distribution, in any
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arbitrary network, where at any time 𝑡 ≥ 0 positions of nodes
are independent and have the same PDF.
Proof: Label the transmissions of the message up to the
delivery of the message to the destination node, 𝑇1 , 𝑇2 , . . . ,
𝑇𝑛𝑢𝑚 , where 𝑛𝑢𝑚 denotes the number of transmissions, and
𝑇𝑖 does not occur after 𝑇 𝑗 iff 𝑖 < 𝑗. Due to the i.i.d. positions
of nodes, the probability of forwarding the message to an arbitrary susceptible node during transmission 𝑇𝑖 , 1 ≤ 𝑖 ≤ 𝑛𝑢𝑚,
is 1/(𝑀 − 𝑖). Initially, there exist 𝑀 − 1 susceptible nodes in
the network. Thus, the destination node receives the message
during transmission 𝑇1 with the probability 1/(𝑀 − 1).
If only transmissions 𝑇1 , 𝑇2 , . . . , 𝑇𝑖−1 , 2 ≤ 𝑖 ≤ 𝑛𝑢𝑚, have
occurred, 𝑀 −𝑖 nodes are still susceptible. Thus, if the message
has not been forwarded to the destination node in one of the
transmissions 𝑇1 , 𝑇2 , . . . , 𝑇𝑖−1 , the destination node receives
(does not receive) the message during transmission 𝑇𝑖 with
probability 1/(𝑀 − 𝑖) ((𝑀 − 𝑖 − 1)/(𝑀 − 𝑖)). As a result, the
probability of forwarding the message to the destination node
during transmission 𝑇𝑖 is obtained by Eq. (17).
𝑀 − (𝑖 − 1) − 1
1
𝑀 −2 𝑀 −3
×
×···×
×
𝑀 −1 𝑀 −2
𝑀 − (𝑖 − 1)
𝑀 −𝑖
1
=
.
𝑀 −1
(17)
𝑝(𝑖) =
Note that this proof is valid even if some transmissions occur
simultaneously.
Corollary 1: The average number of transmission under
epidemic routing by the delivery time in any arbitrary network
with 𝑀 nodes, where at any time 𝑡 ≥ 0 positions of nodes are
independent and have the same PDF, is 𝑀/2.
Proof: According to Theorem 1, the probability of forwarding the message 𝑖 times, 1 ≤ 𝑖 ≤ 𝑀−1, by time of delivery
including the forwarding to the destination node, is 1/(𝑀 −1).
Thus, the average number of transmissions is computed as,
𝑛𝑢𝑚 =
1
𝑀
· (1 + 2 + · · · + (𝑀 − 1)) = .
𝑀 −1
2
(18)
Given that initially, nodes are randomly placed within the
common area with a uniform distribution, and they select the
communities according to the same PDF, as mentioned in
Section III, the herein target network satisfies the condition
given in Theorem 1. That is not the case of the network
considered in our previous work [10] where the community
each node frequently visits is different from the communities
some other nodes frequently visit. However, the results of
Fig. 8 in [10] indicate that the average number of transmissions
can be estimated as a linear function of the total number
of nodes. As the number of transmissions is an important
performance measure, it is worth to characterize that when
tendencies of nodes to visit a community differ. However, it
is challenging, and we leave it for future work.
C. Comparison of the folded model with the ODE approach
To the best of our knowledge, there is no analytical approach
in the literature, considering exactly the same network model,
so we cannot entirely compare the proposed models with the
previous approaches. Modeling as ODEs is the main approach
to evaluate the performance of DTNs, it was extensively used
in the literature [6], [12], [13], [39]. Hence, we derive an
ODE model for epidemic routing in the defined target network,
and then compare the accuracy of the proposed folded model
with the proposed ODE model. ODE models are derived from
CTMCs [6]. A CTMC modeling the epidemic routing in the
target network should have at least 2𝑁 + 1 state variables, in
order to represent the number of nodes belonging to each of
𝑖𝑛 𝑓 _𝑙
, 1 ≤ 𝑗 ≤ 𝑁, which
, 𝐶𝑗
classes 𝐶 𝑠𝑢𝑠_𝑟 , 𝐶 𝑖𝑛 𝑓 _𝑟 , 𝐶 𝑠𝑢𝑠_𝑙
𝑗
is essential to capture the message propagation. In the ODE
approach, the derivative of the average value of each state
variable is estimated basically using the rates of incoming and
outgoing transitions of states.
In order to model the epidemic routing with ODEs, we
consider a CTMC having 2𝑁+1 state variables which represent
𝑖𝑛 𝑓 _𝑙
, and
the number of nodes belonging to classes 𝐶 𝑖𝑛 𝑓 _𝑟 , 𝐶𝑖
𝑠𝑢𝑠_𝑙
𝑙
𝑟
𝐶𝑖
, 1 ≤ 𝑖 ≤ 𝑁. We use functions 𝐼 (𝑡), 𝐼𝑖 (𝑡), and 𝑆𝑖𝑙 (𝑡)
to represent the average value of the above-mentioned state
variables at time 𝑡. Specifically, 𝐼𝑖𝑙 (𝑡) and 𝑆𝑖𝑙 (𝑡) denote the
average number of infected and susceptible local nodes in
community 𝑐 𝑖 at time 𝑡, respectively, and 𝐼 𝑟 (𝑡) denotes the
average number of infected roaming nodes at time 𝑡. Thus,
the average number of susceptible roaming nodes at time 𝑡
Í𝑁 𝑙
(𝐼𝑖 (𝑡) + 𝑆𝑖𝑙 (𝑡)). Using
can be computed as 𝑀 − 𝐼 𝑟 (𝑡) − 𝑖=1
functions 𝐼𝑖𝑙 (𝑡), 𝑆𝑖𝑙 (𝑡), and 𝐼 𝑟 (𝑡) and Eq. (20), we model the
network described in Section III by 2𝑁 + 1 ODEs represented
by equations (19) to (22), 1 ≤ 𝑖 ≤ 𝑁. Eqs. (19), (21), and (22)
represent the derivatives of 𝐼𝑖𝑙 (𝑡), 𝑆𝑖𝑙 (𝑡), and 𝐼 𝑟 (𝑡), respectively.
Three events change the number of infected local nodes
in community 𝑐 𝑖 : i) change of one of those nodes to the
roaming mode; ii) selecting community 𝑐 𝑖 by an infected node
which changes to local mode; iii) infection of a susceptible
node in community 𝑐 𝑖 . The travel of each local node is
exponentially distributed with rate 𝛼, and the node goes to the
roaming mode with probability 𝑃𝑟 after its travel. Thus, the
first above-mentioned event happens with rate 𝐼𝑖𝑙 (𝑡) · 𝛼 · 𝑃𝑟 .
The travel of an infected roaming node is distributed with
rate 𝛽. After finishing the travel, the node chooses the local
mode with probability 𝑃𝑙 , and selects community 𝑐 𝑖 with
probability 𝑃𝑠𝑒𝑙_𝑖 . Thus, the second event happens with rate
𝐼 𝑟 (𝑡) · 𝛽 · 𝑃𝑙 · 𝑃𝑠𝑒𝑙_𝑖 . Moreover, meeting of a susceptible
local node in community 𝑐 𝑖 with a local infected node in
that community or a roaming infected node happen with rate
𝑆𝑖𝑙 (𝑡) · 𝐼𝑖𝑙 (𝑡) · 𝜆 or 𝐼 𝑟 (𝑡) · 𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑆𝑖𝑙 (𝑡)), respectively. Thus, the
derivative of 𝐼𝑖𝑙 (𝑡) can be computed as (19).
𝑑𝐼𝑖𝑙 (𝑡)
= −𝐼𝑖𝑙 (𝑡)𝛼𝑃𝑟 + 𝐼 𝑟 (𝑡) 𝛽𝑃𝑙 𝑃𝑠𝑒𝑙_𝑖 + 𝑆𝑖𝑙 (𝑡)𝐼𝑖𝑙 (𝑡)𝜆
𝑑𝑡
+ 𝐼 𝑟 (𝑡) 𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑆𝑖𝑙 (𝑡)) ,
(19)
Note that in order to use function 𝑅ˆ 𝑚𝑒𝑒𝑡 to define ODEs,
we rewrite it as follows,
𝜂−𝛾
𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑛) = 𝛾+𝜃 (1−𝑛)·(𝑛−1)·𝛾+𝜃 (𝑛−1)·(𝑛−1)·
, (20)
𝑀 −2
where 𝜃 is the unit step function.
Similar to the way the derivative of 𝐼𝑖𝑙 (𝑡) is computed,
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Transactions on Vehicular Technology
1,300
1,200
1,100
1,000
900
800
700
600
500
400
300
200
100
10
14
TABLE VI: Number of states of the proposed SRN models.
210
Simulation
Folded Model
ODE Model
20
30
40
50
60
70
80
The Average Delivery Delay (𝑠)
The Average Delivery Delay (𝑠)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
90 100
Simulation
Folded Model
ODE Model
190
𝑵
𝑴
5
10
15
170
150
130
110
90
70
100
120
𝑀
140
160
180
200
𝑀
(a)
(b)
Fig. 10: The average delivery delay obtained with the folded
SRN, the ODE model, and by simulation when 𝑁 = 4.
3
Mono.
1,475
56,100
578,000
4
Fold.
400
3,300
11,200
Mono.
3,870
287,430
4,884,780
Fold.
600
4,950
16,800
TABLE VII: Number of states of the proposed monolithic
model and the previous monolithic model proposed in [10],
when 𝑁 = 4.
𝑴 Proposed Monolithic SRN Previous Monolithic SRN
8
66,660
8,400
12
999,570
192,000
16
7,821,768
2,205,000
derivatives of 𝑆𝑖𝑙 (𝑡) and 𝐼 𝑟 (𝑡) can be computed as follows.
𝑁
Õ
𝑑𝑆𝑖𝑙 (𝑡)
(𝑆 𝑙𝑗 (𝑡) + 𝐼 𝑙𝑗 (𝑡)) · 𝛽 · 𝑃𝑙 · 𝑃𝑠𝑒𝑙_𝑖
= 𝑀 − 𝐼 𝑟 (𝑡) −
𝑑𝑡
𝑗=1
− 𝑆𝑖𝑙 (𝑡) · 𝛼 · 𝑃𝑟 − 𝑆𝑖𝑙 (𝑡) · 𝐼𝑖𝑙 (𝑡) · 𝜆 − 𝐼 𝑟 (𝑡) · 𝑅ˆ 𝑚𝑒𝑒𝑡 (𝑆𝑖𝑙 (𝑡))
(21)
𝑁
Õ
𝑑𝐼 𝑟 (𝑡)
𝐼𝑖𝑙 (𝑡) · 𝛼 · 𝑃𝑟 + 𝑀 − 𝐼 𝑟 (𝑡)
= −𝐼 𝑟 (𝑡) · 𝛽 · 𝑃𝑙 +
𝑑𝑡
𝑖=1
−
𝑁
Õ
𝑖=1
𝑁
Õ
(𝑆𝑖𝑙 (𝑡) + 𝐼𝑖𝑙 (𝑡)) · 𝐼 𝑟 (𝑡) · 𝜇 +
𝑅ˆ 𝑚𝑒𝑒𝑡 (𝐼𝑖𝑙 (𝑡))
𝑖=1
𝐼𝑖𝑙 (0)
(22)
𝑆𝑖𝑙 (0),
In the network described in Section III,
and
1 ≤ 𝑖 ≤ 𝑁, are 0, whereas 𝐼 𝑟 (0) is 1. Once the system of equations represented in Eqs. (19), (21), and (22) is numerically
solved with the aforementioned initial conditions, the average
delivery delay, E(𝐷), is computed by Eq. (23), as follows [13].
∫ 𝑡𝑚𝑎𝑥
Í𝑁 𝑙
𝐼𝑖 (𝑡) − 1)𝑑𝑡
(𝐼 𝑟 (𝑡) + 𝑖=1
0
,
(23)
E(𝐷) = 𝑡 𝑚𝑎𝑥 −
𝑀 −1
Í𝑁 𝑙
𝐼𝑖 (𝑡 𝑚𝑎𝑥 )
where 𝑡 𝑚𝑎𝑥 is a large time such that 𝐼 𝑟 (𝑡 𝑚𝑎𝑥 ) + 𝑖=1
is close to 𝑀.
Fig. 10 represents the average delivery delay obtained with
the folded SRN, the ODE model, and by simulation when the
network under-analysis consists of four communities (𝑁 = 4),
and 𝑀 varies from 10 to 200. As it can be observed in
Fig. 10, the folded model is accurate in evaluating the average
delivery delay. Particularly, in Fig. 10(a), as the number of
nodes increases, the accuracy improves such that the error
is less than 3% for 𝑀 > 80. As it can be seen in Fig. 10,
the folded model is significantly more accurate than the ODE
model. According to Fig. 10(a), when the number of nodes
is not very large, the ODE model yields a significant error
since ODE approach is rather inaccurate for networks with a
moderate number of nodes [11]. Results represented in Fig. 10
indicate the superiority of the folding technique in terms of
accuracy when compared against the ODE approach, in what
concerns the performance of both networks with a moderate
number of nodes and large-scale networks.
IX. S CALABILITY A NALYSIS
In this section, we investigate the scalability of the proposed
monolithic and folded models and the previously presented
monolithic and folded models [10], in terms of the number of
states in the underlying Markov chains. Table VI represents
the number of states in the underlying Markov chains of the
proposed monolithic and folded models in columns Mono. and
Fold., respectively. As it can be seen in this table, the number
of states in the underlying Markov chain of the monolithic
model grows too fast as the number of communities, 𝑁, or
the number of nodes, 𝑀, increases. For instance, for a small
network with four communities and 15 nodes (𝑁 = 4 and
𝑀 = 15), the Markov chain has about 5 million states. Too
much memory is needed to save this large state space, while
the underlying Markov chain of the folded model for the
aforementioned setting has only 16,800 states. Therefore, the
folded model is scalable enough, it significantly reduces the
state space.
Table VII represents the number of states in underlying
Markov chains of the proposed monolithic model and the
previous monolithic model proposed in [10]. As it can be
observed in this table, the scalability problem of the proposed
monolithic model is severer than the previously proposed
monolithic model. This is due to the fact that the proposed
monolithic SRN models a more realistic network. In the
network model considered in [10], it was assumed that each
node frequently visits only one specific community, and the
maximum number of local nodes in each community is 𝑀/𝑁.
However, the models proposed in this paper account for
the possibility of moving any number of local nodes in a
community. Thus, the total number of tokens in places 𝑃 𝑠𝑢𝑠_𝑙
𝑗
𝑖𝑛 𝑓 _𝑙
and 𝑃 𝑗
of submodel 𝑆𝑢𝑏 𝑗 of the previous monolithic is at
𝑖𝑛 𝑓 _𝑙
most 𝑀/𝑁 tokens; while places 𝑃 𝑠𝑢𝑠_𝑙
and 𝑃 𝑗
of submodel
𝑗
𝑆𝑢𝑏 𝑙_ 𝑗 of the proposed monolithic model can have in total up
to 𝑀 − 2 tokens, which makes the state space larger.
Table VIII represents the number of states in the underlying
Markov chains of the folded models when the network consists
of 5 communities and 𝑀 nodes. As it can be seen in this
table, although the folded model previously published has less
number of states than the folded model proposed in this paper
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Transactions on Vehicular Technology
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
15
TABLE VIII: Number of states of the proposed folded model
and the folded model previously proposed in [10], when 𝑁 = 5.
𝑴 Proposed Folded Model Previous Folded Model
10
6,930
3,552
20
55,860
133,080
30
188,790
1,186,080
40
447,720
5,796,720
50
874,650
20,211,840
1,200
30
25
Previous Folded Model
Previous Simulation
Proposed Folded Model
Current Simulation
Average Delivery Delay (𝑠)
Average Number of Transmissions
35
20
15
10
5
0
10
20
30
𝑀
40
(a)
50
Previous Folded Model
Previous Simulation
Proposed Folded Model
Current Simulation
1,000
800
600
400
200
0
10
20
30
𝑀
40
50
(b)
Fig. 11: Comparing the results obtained from the proposed
folded model, the previously presented folded model [10], and
by simulation.
for the smallest case of 𝑀 = 10, its number of states radically
increases as the number of nodes increases up to five times.
For example, for a network with 50 nodes (𝑀 = 50), the
underlying Markov chain of the previous folded model has
about 20 million states while that of the folded model proposed
herein is less than 1 million states. It is worth mentioning that
the number of states of the proposed folded model even for
a three times larger network (𝑀 = 150) does not reach 20
million. Moreover, even when there are four communities and
60 nodes in the network, the previous folded model could not
be solved on a system with a 64 GB memory. However, using
that computing system, the results reported in Figs. 7(b) and
10(b) are computed. These figures present the results obtained
with the folded model for a network with four communities
and 200 nodes. In conclusion, the results presented in Table
VIII indicate that the current folded model is much more
scalable than the previous one in what concerns the number
of nodes (𝑀).
Fig. 11 represents the average number of transmissions and
the average delivery delay obtained from the proposed folded
model, the folded model previously proposed in [10], and
by simulation of the networks modeled by each folded SRN.
Parameters 𝑁 and 𝑀 in Fig. 11 are set to the values considered
in Table VIII. Fig. 11(a) indicates higher accuracy of the
proposed folded model for evaluation of the average number of
transmissions in comparison with the previous folded model.
In Fig. 11(b), we observe that although the current folded
model has less accuracy than the previous folded model,
when 𝑀 = 10 and 𝑀 = 20, it becomes more accurate upon
increasing 𝑀, such that it is strongly more accurate than the
previous folded model when 𝑀 = 50. The results presented
in Table VIII and Fig. 11 indicate that the proposed folded
model have better scalability than the previous folded model
while achieving higher accuracy in large-scale networks.
X. C ONCLUSIONS AND FUTURE WORK
The goal of this paper was to evaluate of the performance of
epidemic routing in MSNs. Its main contribution is analysis
of the performance of epidemic routing, considering a network model which is more realistic than those considered in
the state-of-the-art, while providing scalability. This network
model is based on the skewed location visiting preferences of
nodes, one of the main characteristics of MSNs. A monolithic
SRN model was proposed to evaluate the delivery delay and
the average number of transmissions by time of delivery
under epidemic routing. Although the monolithic model is
accurate to predict the measures of interest in MSNs, it suffers
from the state space explosion for networks with a large
number of nodes/communities. In order to overcome this issue,
an approximate model was proposed, applying the folding
technique to the monolithic model. This model can be used
to evaluate the performance of large-scale networks without
significant loss of accuracy. We also proposed an ODE model
for epidemic routing and compared the obtained results with
the ones for the folded model.
All the proposed models were validated against discreteevent simulation. The obtained results show that the folded
model is more accurate than the ODE model. Moreover, we
proved that the number of transmissions by the time of delivery
follows a uniform distribution, for a general class of networks, where the positions of nodes are always i.i.d. Finally,
we investigated the scalability of the proposed monolithic
and folded models, contrasting with the previously presented
monolithic and folded models [10], in terms of the number
of states in the underlying Markov chains. The current work
can be extended in different ways. Other characteristics of
MSNs, such as dependency of the next visited community
to the currently/previously visited community and the timedependency property of mobility [18], can be added to the
current models. It is also worth to evaluate and analyze the
performance of other routing or content retrieval schemes. As
discussed in Section VIII-B, another future research direction
can be further analysis of the number of transmissions when
tendencies of nodes to visit a community differ. Analysis of the
number of transmissions provides good insights for designing
efficient recovery schemes.
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Leila Rashidi is a postdoctoral associate at University of Calgary, Canada. She received her Ph.D.
in Computer Engineering from Sharif University
of Technology, Iran in 2019, and the B.S. degree
in Computer Engineering from the University of
Tehran, Iran, in 2014. Her main research interests
are performance evaluation and network security.
Amir Dalili-Yazdi obtained his M.Sc. degree from
the Department of Computer Engineering, Sharif
University of Technology, Iran in 2020. He received
his B.Sc. in Computer Engineering, Software Field,
from Islamic Azad University, Tehran, Iran in 2017.
His main research interests are performance modeling and computer networks.
Reza Entezari-Maleki received the B.S. and M.S.
degrees from the Iran University of Science and
Technology (IUST) in 2007 and 2009, and the Ph.D.
degree from the Sharif University of Technology
in 2014, all in computer engineering. He was a
postdoctoral researcher at Institute for Research in
Fundamental Sciences, Iran, before joining IUST as
an assistant professor in 2018. His main research
interest is performance and dependability modeling.
Leonel Sousa received the PhD degree from the
Instituto Superior Tecnico (IST), Universidade de
Lisboa, in 1996. Since 1996, he has been with IST,
where he is currently the chair of the Department of
Electrical and Computer Engineering. His research
interests include computer architectures, high performance computing, and multimedia systems.
Ali Movaghar is a Professor in the Department
of Computer Engineering at Sharif University of
Technology. He received his M.S. and Ph.D. degrees
in Computer, Information, and Control Engineering
from the University of Michigan, in 1979 and 1985,
respectively. His research interests include performance/dependability modeling and formal verification.
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