Ad Hoc Networks 16 (2014) 197–209

Contents lists available at ScienceDirect

Ad Hoc Networks

journal homepage: www.elsevier .com/locate /adhoc

Modeling message diffusion in epidemical DTN

1570-8705/$ - see front matter � 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.adhoc.2013.12.013

⇑ Corresponding author. Tel.: +55 21 3820 4196; fax: +55 21 2546 7099.E-mail addresses: cabreu@vialink.com.br (C.S. de Abreu), salles@

ieee.org (R.M. Salles).

Claudio Sá de Abreu, Ronaldo Moreira Salles ⇑Dept. of Computer Engineering, Military Institute of Engineering (IME), Praça Gen Tibúrcio 80, 22290-270 Rio de Janeiro, Brazil

a r t i c l e i n f o

Article history:Received 16 October 2012Received in revised form 20 September 2013Accepted 23 December 2013Available online 4 January 2014

Keywords:Delay and disruption networksIntermittently connected networksSIR mathematical modelEpidemic routing

a b s t r a c t

A Delay and Disruption-Tolerant Network (DTN) is a fault-tolerant network where end-to-end connections are not required for message transmissions between nodes. Usually, a DTNis implemented as a wireless mobile ad hoc network that can be applied, for instance, torapidly build a basic telecommunication infrastructure in case of catastrophes and disas-ters, or to support communication in a disruptive military environment. It is importantto model DTN behavior to better understand system dynamics and related physical laws,which may impact network performance. An accurate model will be useful to supportthe design of the network in such challenging scenarios and may allow to test design ideasbefore actually building the real system. This work proposes a mathematical model formessage diffusion in epidemical DTN. Our approach is based on previous models for thespread of human epidemical diseases, namely SIR. Simulation results on message diffusiontimes in an epidemical DTN show that the model is accurate regarding expected values,however large deviations above and below average are also observed on diffusion times.We further study such deviations and provide insights on how to reduce and deal withthem, making the model useful for DTN applications.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

In a Delay and Disruption-Tolerant Network (DTN) mes-sages are transmitted hop-by-hop from one network nodeto another until the destination is reached. Such networksfollow the store-carry-and-forward paradigm [1]. DTN tech-nology is getting more importance nowadays since it man-ages to establish connectivity in challenging networkenvironments where other types of technology fail:

� Mobile Phone to Mobile Phone networks, where wire-less channel impairments and mobility cause discon-nections and message delays.� Car networks (VANETS), where there is no continuous

network service along all paths and places.� Disaster and war zones, where communication infra-

structure is partially or completely destroyed.

� Sensor Networks, where some devices are not alwaysreachable to the sensor network coverage.� Communication in remote zones with no fixed

infrastructure.� Interplanetary communications, where the line of sight

communication between stations is difficult and delaysare huge.

One of the most basic and important services providedin the above-mentioned scenarios is information diffusion.For instance, message diffusion can be used in VANETs tobuild up a car alert accident system to notify all cars thatenter the affected regions. In war zones, a commandermay want to issue an order to all his troops; in disasterzones, a message conveying the current state of the catas-trophe may be issued to all local units attending the area;in sensor networks, contaminations detected by one sensormay be diffused to all other sensors or collecting nodes.

Regarding the delivery time and communication effi-ciency, good results are obtained when methods of epi-demic diffusion of messages are used. Each node that

Fig. 1. Epidemical process of message propagation. (a) Start. (b) First transmission. (c) Second transmission. (d) Delivery.

198 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

carries the message transmits copies of it during an estab-lished time interval to all nodes that still do not have themessage, as initially proposed by Vahdat and Becker [2].This type of DTN is also the fastest to deploy, because:

� It does not require any previous knowledge of networktopology.� There is no restriction on the capacity of the propagat-

ing nodes.� It is not necessary to consider paths or history of

contacts.

Fig. 1 illustrates in a small scale example how an Epi-demical DTN works. Consider initially a single source anda single destination node. The arrows depicted in all foursubfigures (Fig. 1(a)–(d)) indicate the moving direction ofeach node. In Fig. 1(a), the source node (blue1 dot) gener-ates a message to the destination node (green dot).

1 For interpretation of color in Fig. 1, the reader is referred to the webversion of this article.

Fig. 1(b) and (c) show intermediate steps, where otherDTN nodes are contacted and receive a copy of the message.Those nodes continue to propagate the epidemic to all theircontacts until the message is eventually delivered to thedestination (delivery time), as shown in Fig. 1(d). At thispoint two nodes (red dots) still do not have a copy of themessage. In case of message diffusion to all network, theprocess continues until all nodes get a copy of the message(all nodes are blue), which constitutes the total message dif-fusion time in the network.

Epidemic diffusion of messages is also a useful trans-mission method when network topology is unknown orfrequently changes over time due to the movement of mo-bile nodes. However, in epidemic diffusion, for high trans-mission rates, both battery consumption and memoryusage are critical, given that each network node ends uptransmitting more frequently and keeping almost all cop-ies of messages in transit in their memory.

It is important to model epidemic behavior in order toassist the design and management of an epidemic DTN.With an appropriate model it is possible, for instance, to

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 199

test design ideas before actually building the real system,and also to derive and maintain the optimal parameter set-tings for the desired DTN operation in terms of communi-cation efficiency, memory usage and latency (messagedelivery time). Such model constitutes the main motiva-tion behind this work.

One major mechanism to control the lifetime of a mes-sage in an epidemical DTN is message TTL (Time-To-Live).DTN performance depends on message TTL since memoryusage, latency and communication efficiency are all af-fected by this parameter. TTL may also be used as a QoSparameter for some DTN applications and scenarios wherea delay bound to message delivery time is necessary. Forinstance, in a military scenario, commander messages toall troops have fixed lifetimes otherwise the whole opera-tion may be compromised by late deliveries. Hence, TTLplays an important role on the performance and modelingof message diffusion in DTNs.

Due to the similarity observed between the epidemictransmission of messages in a DTN and the spread of infec-tious diseases, several works, like [3,4], employed analyti-cal models of epidemic diffusion in populations as a basefor DTN modeling. Almost all those works were derivedfrom the SIR model [5,6], which is a three-state-modelfor epidemical disease where an individual may be in oneof three possible states, in relation to the contamination:Susceptible, Infected or Recovered.

Regarding epidemical models applied to the DTN area,the work in [3] introduced important analytical resultsfor node contact rates and for the time distribution of thesecontacts, under the assumptions that: network is homoge-neous and sparse, node transmission range is much smallerthan the surface area, transmission interference andimpairments can be neglected. Those assumptions werepreserved in our work for the different scenarios studied.

However, it was observed from simulations that theabove-mentioned models are only accurate for the averagecase, i.e. when results of epidemic evolution (e.g. number osinfected nodes over time) were averaged for a large numberof different scenario settings (e.g. different source nodes –initial conditions). In some cases the epidemic spread inone setting is much slower than in another more favorablesetting providing large deviations above and below themean, either in diffusion times, nodes contact times ornumber of infected nodes over time. Such deviations werenot taken into account on previous works limiting theapplicability of the existing models to average cases only.

This work starts studying how analytical models for thespread of epidemic diseases could be applied to the DTNscenario, similarly to what has been done in some previousworks. However, it goes further in the analysis of expectedresults and their deviations, creating a better frameworkfor the design of a practical model to study DTN dynamics.The proposed model is validated through simulationresults.

The contributions of this work are:

� The employment and validation of a mathematicalmodel for epidemical DTNs based on an epidemicalmodel for the spread of infectious diseases (for the aver-age case).

� The discussion and investigation about the reasonsbehind the large deviations observed in simulationresults. Such deviations were ignored or sometimeswere observed but not treated in previous works, like[7–9].� The study of important DTN performance parameters

from the proposed model: the message TTL, the nodecontact rate (b), the number of nodes and the work areadimensions.� The proposal of a simple framework to predict the

behavior of an epidemical DTN in both mean and vari-ance of message diffusion times (or number of avail-able/propagating nodes over time).� The suggestion of practical procedures to improve the

functionality of epidemical DTNs from the resultsobtained with the model.

The rest of the paper is organized as follows. Section 2gives a general view of recent developments on DTN math-ematical modeling. In Section 3, the Epidemical DTN BasicModel (EDTN) is investigated, while in Section 4 someimportant characteristics of epidemical DTNs are observedfrom the model. In Section 5, the model is validatedthrough simulations for the average case, however devia-tions in the number of available/propagating nodes (nodeswithout/with the message) over time are also observedfor all scenarios. Section 6 deals with such deviations anddiscusses how to calculate them in order to improve modelprecision. A statistical model is proposed to predict DTNbehavior regarding the initial node contact times and theirvariance. Section 7 presents a summary of the mainparameters derived from the model, and finally, the paperis concluded in Section 8.

2. Related work and historical review

In 1906, Hamer [10] postulated that the evolution of anepidemical disease depends on the contact rate betweeninfected and susceptible individuals. This principle, nowknown as the mass interaction principle, becomes one ofthe most important concepts of mathematical epidemiol-ogy. The main idea was that the spread of an epidemicaldisease in a population is proportional to the density ofsusceptible by the density of infected individuals. TheHamer principle was originally formulated using a discretetime model, but in 1908 Sir Ronald Ross, who discoveredthe vector of malaria transmission, generalized the modelfor continuous time in his work about malaria dynamics.

In 1927 and 1932, Kermack and McKendrick in [5,6]improved the continuous model created by Sir Ronald Rosswith the threshold principle, establishing that the intro-duction of infected individuals in a community may notcause an epidemic, unless the density of susceptible indi-viduals is above a certain critical value. This principle, inconjunction with the mass action principle, are the basisof a model called SIR (Susceptible, Infected, Recovered),which is one of the most important models in modernmathematical epidemiology [11].

During the following decades, the model of Kermackand McKendrick was extended and improved, leading to

200 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

more sophisticated models, which have applications inmany different types of spreading diseases. In recent years,mathematical modeling of diseases has become increas-ingly important in the control of current epidemics [12].Nevertheless, SIR based models have also been applied inother areas whenever the studied scenario has similaritieswith the dynamics involving the spread of diseases.

In 2000, Vahdat and Becker [2] developed techniques todeliver messages in partially connected networks, likeMANETs. Such work employed epidemical routing.

In 2002, Kevin Fall proposed the first model of what wenow call DTN [13]. In that work and the following [13,1],the major concerns were with the concept and necessarystructure for DTN implementation.

In 2005, Groenevelt et al. [3] proposed a mathematicalmodel for DTN based on previous models for epidemicspread of diseases in populations. One of the main contri-butions of that work was the presentation of a closed formexpression to estimate the parameter b (see Eq. (15) in Sec-tion 5), the contact rate among nodes. This results was ob-tained by demonstrating that contact times between nodesare exponentially distributed if some assumptions hold:network is homogeneous and sparse, node transmissionrange is much smaller than the surface area, transmissioninterference and impairments can be neglected. Theexpression still applies for different node movement mod-els, such as the Random Way Point, Random Direction andBrownian.

In 2006, Haas and Small [7] used differential equationsand concepts applied to Epidemiology to model a networkthat used whales as DTN data mules. That work did notmention the problems related to the delay between thefirst contamination (source data node) and the spread ofthe epidemic to the population. It also did not take into ac-count the conditions that support the modeling of a mes-sage diffusion in DTN as an epidemic process inmathematical epidemiology.

The work in [8] developed a model to study the perfor-mance of ad hoc networks according to the movement ofnodes. Such model can be extended to the DTN scenarioand is useful for obtaining the efficiency of communicationin this context.

Zhang et al. in [9] used the approach derived by Groene-velt et al. [3] as basis to propose and investigate a unifiedODE framework to study the performance of various for-warding and recovery DTN schemes. They derived theODE model as limiting processes of Markovian modelsand employed the equations to obtain a rich set ofclosed-form formulas regarding the packet-delivery delay,number of copies sent and buffer occupancy under variousschemes. Those equations were validated through simula-tions, and good matches were obtained between the modelprediction and simulations results. However, as done inprevious works, the results were obtained for the averagecase only, and the deviations in forwarding times were alsonot treated by their approach.

In 2008, Walker et al. [14] studied the performance ofDTNs considering nodes moving according to the randomwalk mobility model. Such work introduced the localizedrandom walk mobility model, derived some of its proper-

ties and used it to construct models for simple DTN scenar-ios. The authors further checked the predictions againstsimulation results.

In 2010, Lu and Hui in [15] proposed an energy-efficientn-epidemic routing protocol for DTN, which was based ona mathematical model similar to the one studied in [7].However, the work did not consider possible deviationsof the model for some specific scenarios.

In 2010, Jacquet et al. [16] derived a theoretical upperbound for the information propagation speed in twodimensional large scale networks, and multi-dimensionalnetworks. Such theoretical bounds are useful to improvethe understanding about the fundamental properties andperformance limits of DTNs, as well as to evaluate and/oroptimize the performance of specific routing algorithms.The model used in that analytical study encapsulates manypopular mobility models, such as random way-point, ran-dom walk and Brownian motion, and provides a generalframework for the derivation of analytical bounds on theinformation propagation speed in DTNs. It explicitly takesinto account the spatial dimension as well as nodes densityand mobility.

In 2012, Shahbazi et al. in [17] proposed ways to im-prove DTN performance using stationary nodes, howeverall their analysis came directly from some particular exper-imental results, they were not concerned with the con-struction of a more general mathematical model.

Khabbaz et al. in [18] made a complete survey on recentdevelopments and persisting challenges in DTN. In thatwork, among other important conclusions is the observa-tion that still there is no adequate analytical model forthe performance evaluation of DTNs. This topic persistsas an open problem.

Wang and Haas in [19] proposed a methodology wherea random graph was used to model the dynamic connectiv-ity among network nodes. The work derived three mainmetrics: the probability distribution of the number of in-fected nodes, the average number of infected nodes andthe probability of all nodes being infected, all three metricsbeing functions of time. Simulation results were very closeto the results predicted by their model for the averages ofthe three metrics. Despite their contribution, that work didnot take in account the standard deviation of each metricon each time instant.

In 2013, Venkadatri et al. in [20] proposed the use ofM2M communication (many-to-many communications asdefined by Moraes et al. in [21]) to reduce the routing delayin epidemic DTNs.

The works in [7–9] provided some interesting analyt-ical models that could be employed to evaluate the per-formance of DTNs. In different ways, those worksreached some of the same equations presented in Sec-tion 3 of our work, specially the ones in (8). Those equa-tions are important to better understand the dynamicsinvolving DTN transmissions as a whole, but are not en-ough to explain the differences observed on contacttimes when the model is compared with simulation re-sults obtained for some specific scenarios. The under-standing of such deviations is one of the main goals ofthis work.

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 201

3. The Epidemical DTN Basic Model

In this section, a mathematical model for epidemicalDTNs will be investigated based on previous models forthe epidemical spread of diseases in populations.

Regarding the human spread of diseases, it is wellknown that some diseases are transmitted from one indi-vidual to another by direct contact or proximity, like fluor cholera. In general, when a significant portion of thepopulation is infected and the rate of growth of the in-fected population is high, it can be said that an epidemicis happening. There are several mathematical studiesinvestigating the dynamics of epidemic spread of diseases,but the original one used as reference by almost all otherswas first proposed in [6], named SIR. That model is basedon the assumption that in an approximately constant pop-ulation, an individual in a given time can be only in one ofthree possible states: Susceptible (S), Infected (I) andRecovered (R). A set of differential equations show howthe states S, I and R vary over time.

The dynamics involving the transmission of a messagethrough nodes in an epidemical DTN (called message for-warding) and the spread of an epidemic disease, when aninfected individual infects others, enjoy several similarities.The agent in the first process is the message carried by anode to be transmitted to other nodes, while in the secondprocess is a virus or some other plague carried by a person.In both cases, the agent stays in a carrier for some time be-fore the carrier becomes unavailable (recovered). Becauseof this similarity, the SIR model is used as a basis for creat-ing a mathematical model for epidemical DTN, consideringthat one node, in a given time, can only be in one of threepossible states:

� available (D): The node is available to receive amessage;� propagating (P): The node has received a copy of the

message and is spreading it until TTL (Time To Live)expires;� unavailable (I): The node had already received the mes-

sage and it has been delivered, or its TTL has expired.

From the above discussion and following what has beendone before to model the spread of epidemical diseases [5–7,9], it is possible to derive a simple model that hopefully

Fig. 2. Three/two s

approximates the dynamics of message transmissions inepidemical DTNs.

One of the main assumptions of the model is that thenumber of nodes on each state (D; P and I) can be approx-imated by continuous and differentiable functions of time,t : DðtÞ; PðtÞ and IðtÞ. Other assumptions are listed below:

� The number of nodes N is constant during messagepropagation.� At any time instant all nodes are equally likely to meet,

i.e. the effects of the spatial dimension are neglected.� When a contact occurs between a node in P and a node

in D, the message transferring probability is 1.� When a message is received, the node is immediately

able to transfer it to all reachable available nodes duringmessage lifetime.� A node makes only one message forwarding at each

time.� A node makes an average of bN contacts per unit time,

where b is the contact rate of the network.� The nodes in propagating state go to unavailable state

with rate cP.

The diagram below shows the sequence of states.According to Fig. 2, each arrow indicates the transition

rates from one state to the following. Between D and P,as one node makes bN contacts per unit time, P nodesmake bNP contacts per unit time. Then, since D

N is the pro-portion of available nodes in a given time, we can concludethat, in each time unit, bNP � D

N ¼ bDP nodes are infected.Thus, bDP is the rate of state change for available nodesper unit of time.

Between P and I, the transition rate is cP, which is therate of depletion, i.e. the number of nodes that stop trans-mitting the message per time unit. So, 1=c is the averagetime that a node remains transmitting the message.

Given that N is the total number of nodes, we have foreach time t : N ¼ DðtÞ þ PðtÞ þ IðtÞ. To simplify notationwe use:

N ¼ Dþ P þ I ð1Þ

From model construction assumptions, it is possible toderive (1) over time, obtaining:

dDdtþ dP

dtþ dI

dt¼ 0 ð2Þ

tate models.

202 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

According to the transition rates presented in Fig. 2, wehave:

dDdt ¼ �bDPdIdt ¼ cP

(ð3Þ

From (3) and (2), we can conclude that:

dPdt¼ bDP � cP ð4Þ

If c ¼ 0 (such consideration will be justified later on), dPdt

is always greater than zero, except in the end of communi-cation (diffusion process). As no node becomes unavailable,the model can be simplified to:

Dþ P ¼ NdDdt þ bDP ¼ 0

(

or

D0 þ bND ¼ bD2 ð5Þ

This is a Bernoulli differential equation that can besolved for D as follows. Multiplying (5) by D�2 and makingS ¼ D�1; S0 ¼ �D�2D0, we have:

S0 � bNS ¼ �b

then,

S ¼ 1Nþ CebNt ð6Þ

The DTN communication process starts when a node(source) generates the message, then the initial conditionis that Dð0Þ ¼ N � 1 ) Sð0Þ ¼ 1

N�1. Which applied in (6),gives:

C ¼ 1NðN � 1Þ

then,

S ¼ 1Nþ 1

NðN � 1Þ ebNt ¼ N � 1þ ebNt

NðN � 1Þ ð7Þ

As D ¼ S�1, the model can be finally expressed by:

DðtÞ ¼ NðN�1ÞN�1þebNt

PðtÞ ¼ NebNt

N�1þebNt

N ¼ DðtÞ þ PðtÞ

8><>: ð8Þ

One of the main differences between the application ofthe SIR model in a DTN environment and the application ina human epidemic scenario relies on the determination ofparameters b and c. In the latter case, parameters are di-rectly obtained from the behavior observed on individualsof an infected population, while in a network they dependon message transmission dynamics.

The work in [3] derived an expression for b under ame-nable assumptions suitable for DTN environments, suchexpression will be used in Section 5 and compared to sim-ulation results. The other parameter c (rate of depletion)enjoys a direct relationship with message TTL as it willbe explained next, but first we clarify the role of messageTTL in a DTN.

The TTL (Time To Live) is one of the main mechanismsused in an epidemic DTN to discard messages, it can be de-fined in several different ways, such as: global TTL, localTTL, hop-based TTL and time-based TTL [22]. In most cases,TTL is used as a fixed deadline valid for the original mes-sage and all copies forwarded by intermediate nodes [7,9].

In common data networks, the TTL of a packet is usuallytreated as a hop count, which is decreased by one at eachhop a packet goes through. When the number of hops isequal to the original TTL of the packet, it is discarded. Suchapproach can also be used in DTNs, however given the par-ticular requirements and message transmission dynamicsobserved in DTNs, we believe the use of message TTL asmessage lifetime (or literally ‘‘time to live’’) is more adher-ent to the architecture.

Typically, in common data networks a message is slicedinto packets that are sent through the network until reachthe destination, where the slices are remounted. It is alsoexpected that such communication process is reliable andenjoys low packet delays. In this context, package TTL isused more as a network management mechanism to re-spond to bad routing decisions and avoid packets beingindefinitely forwarded inside the network.

DTNs usually operate in a completely different scenario,where end-to-end connections cannot be established, de-lays are expected to be huge, nodes operate under energyrestrictions, messages are not chopped into packets andcan be hold in a node’s buffer for a long period of time untila next contact occurs. In most cases, nodes in a real DTNare actually wireless devices and have limited autonomy.In these scenarios message TTL is an important informa-tion to be confronted to battery duration in order to driverouting decisions and buffer management procedures.Such information also saves the scarce resources of thenetwork in holding and transmitting a message after ithas expired. Therefore, in this work the TTL will be usedas a fixed time, literally, the total time-to-live of a message.The simulator TheOne, used to obtain the simulation re-sults presented in this work, also implements TTL in thisway.

Nodes remain transmitting the message until TTL ex-pires and then discard the message before becomingunavailable. However, for a very large TTL (say TTL! 1)upon receiving and storing the message no node will dropit and remains propagating it forever.

Since TTL is an external parameter and can be freely setaccording to the application, user or network operator, weconsider it large enough so that no node becomes unavail-able during the time window under study (from messagegeneration until complete diffusion), which justifies theassumption of c ¼ 0 taken before. Hence, the EpidemicalDTN Basic model is reduced to the two state model indi-cated in the gray area of Fig. 2 and represented by the sys-tem of equations in (8).

A similar set of equations could also have been ex-tracted from [7,8] or [9], however we presented a more di-rect derivation and used some of the intermediate steps inthe next sections of this paper.

It remains to be seen if such simplified model indeedapproximates the behavior of epidemical DTNs, but beforethat we use the model to study other important properties.

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 203

4. Some important epidemical DTN modelcharacteristics

4.1. The mean forwarding times and intervals

Based on (8) and keeping in mind that the kth messageforwarding will occur when D ¼ N � ðkþ 1Þ, the averagetime tk of the kth message forwarding is given by:

N � ðkþ 1Þ ¼ NðN � 1ÞN � 1þ ebNtk

N � 1þ ebNtk ¼ NðN � 1ÞN � k� 1

ebNtk ¼ NðN � 1ÞN � k� 1

� N þ 1 ¼ ðN � 1Þðkþ 1ÞðN � ðkþ 1ÞÞ

then:

tk ¼1

bNlnðN � 1Þðkþ 1ÞðN � ðkþ 1ÞÞ

� �; for k ¼ 1; . . . ;N � 2 ð9Þ

From (9) and letting Dk be the kth interval between for-wardings, i.e., Dk ¼ tk � tk�1, we have:

Dk ¼1

bNlnðN � 1Þðkþ 1ÞðN � ðkþ 1ÞÞ

� �� ln

ðN � 1ÞðkÞðN � kÞ

� �� �

Dk ¼1

bNlnðkþ 1ÞðN � kÞkðN � ðkþ 1ÞÞ

� �; for k ¼ 1; . . . ;N � 2 ð10Þ

It is important to observe the conditions in which Eqs.(9) and (10) apply. They are not valid for k ¼ N � 1, i.e.for the last message forwarding. Then, to derive an estima-tion for the time until the whole population has gotten themessage, we go further into the analysis in Section 4.2,such estimation will be given as a lower bound for TTL asit will be seen in Eq. (14).

4.2. Lower bound for TTL

To maximize the delivery probability of a message in aDTN, maintaining the message in the network for theshortest time possible, TTL should be large enough so thatit is expected that all nodes receive the message before theepidemic process ends, but not much larger than that.

Eq. (9) could not be applied to determine a lower boundfor the TTL, because it cannot be used when D ¼ 0 (allnodes infected). We are going to explore the symmetry ob-served in Eq. (10) to estimate such bound.

Expanding (10) for k ¼ a and for k ¼ N � ðaþ 1Þ it canbe seen that Da ¼ DN�ðaþ1Þ, which means that the statechange intervals are symmetrical in relation to the pointwhere D ¼ N

2, i.e. D1 ¼ DN�2;D2 ¼ DN�3 and, generalizing:

Dk ¼ DN�ðkþ1Þ ð11Þ

Back to (9), the average time for the first contact is:

t1 ¼1

bNln

2ðN � 1ÞN � 2

� �ð12Þ

If N is large enough, (12) can be reduced to

t1 ¼1

bNln � 0:69

bN

Because of the symmetry proved in (11), the total timeto deliver the message can be approximately calculated asbeing twice the time to deliver the message to half of thenodes, i.e., tN�1 � 2tN

2. Then, applying this to (9):

tN�1 �2

bNlnðN � 1ÞðN2 þ 1ÞðN � ðN2 þ 1ÞÞ

" #ð13Þ

Then,

TTL P2

bNlnðN � 1ÞðN þ 2ÞðN � 2Þ

� �ð14Þ

In average, when TTL > tN�1, it is expected that at leastabout N � 1 nodes have received the message, and so tomaximize the delivery probability TTL must be greaterthan tN�1. Therefore, the right hand side of (14) can be con-sidered as a practical bound for TTL. As it is not possible toassure that all nodes will receive the message, increasingTTL much above (13) may cause waste of network re-sources since the message stays in the DTN consumingmemory of all nodes for more time than necessary in mostcases.

The above formulas could also be used for network de-sign purposes. Suppose in a given DTN, messages should bedelivered before a fixed period of time, say one hour(TTL = 60 min.) or one day (TTL = 1440 min.). Hence, someDTN configurations would be expected to diffuse the mes-sage to all nodes before TTL expires and others not. Thoseexpected to deliver the message are the ones configuredwith some specific values of N and b, such that from(14): tN�1 < TTL.

5. Validation with simulation

In order to validate the model presented in (8) and theformulas derived thereafter, we compare them with the re-sults obtained with a well known DTN simulator: The One[23].

Some simulator modules were modified to facilitate thecomparison with the model, specially the generation ofspecific data reports. In addition to that, given the largeamount of experiments that were carried out, it was alsonecessary to develop scripts to automate the tasks of sce-nario generation, simulating, processing reports and out-put files. All the developed framework, includingadaptations on the simulator, scripts and results can befound at http://cabreu.vialink.com.br/mestrado/dtn/simulacoes.

In all simulations, the contact rate b was obtained andcompared to the estimation proposed in [3], which consid-ers the Random Way Point (RWP) and Random Direction(RD) movement models:

b � 2xcE½V �L2 ð15Þ

where:

� b is the contact rate between any two nodes, i.e, themean time between two given nodes meeting eachother is 1

b.

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Ava

ilabl

e N

odes

(D

)

Time (s)

Average of ResultsStandard Deviation

Expected Values

Fig. 4. Available nodes (D) over time for SC2: average number (simula-tion), standard deviation (simulation) and expected values (model)(average over 1000 runs).

204 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

� x is a constant related to the specific mobility modelused. For random waypoint model (RWP), x ¼ 1:3683.� L is the side of the area where nodes are placed and

move around.� c is the transmission range of each node. It is assumed

that only one transmission occurs at each node inter-meeting and that c is much smaller than L. Furthermore,if c is near L, the end-to-end direct communication ismuch more probable, degenerating the DTN to adirectly connected network.� E½V � is the average relative speed between two nodes,

and can be calculated by a numerical integration givenby Hyytia et al. [24].

Simulations were set for different values of vmin (mini-mum node speed), vmax (maximum node speed), transmis-sion ranges, number of hosts and TTLs. We used 1000different random seeds for each simulated scenario. TheRWP movement model was adopted to all scenarios. In to-tal, we ran over 7000 simulations for about 100 differentscenarios.

Considering a given scenario, the number of availablenodes (D) over time was obtained for different scenariosettings (different random seeds). Then, taking all settings,the average number of available nodes over time and thecorresponding standard deviation were computed. Theevolution of the average and standard deviations on thenumber of available nodes (D) over time were plottedand compared to the expected values given by the mathe-matical model, see Figs. 3 and 4.

In other words, for each Scenario SCi of a set of i scenar-ios, Si simulations were carried out (Si scenario settings),SCik ; k ¼ 1;2; . . . ; Si, each one with a different random seed.Then, a sufficiently small time interval, Dt, was used toevaluate the amount of available nodes over time Dik ðtÞ:for t ¼ 0; Dt; 2Dt; 3Dt; . . . The end of each simulation isreached when D ¼ 0, producing a sequence of values:Dik ð0Þ; Dik ðDtÞ; Dik ð2DtÞ; Dik ð3DtÞ; . . . Thus, the average ofD over time for scenario SCi; �Dið0Þ; �DiðDtÞ;�Dið2DtÞ; �Dið3DtÞ; . . ., is obtained from

0

10

20

30

40

0 1000 2000 3000 4000 5000 6000 7000

Ava

ilabl

e N

odes

(D

)

Time (s)

Average of ResultsStandard Deviation

Expected Values

Fig. 3. Available nodes (D) over time for SC1: average number (simula-tion), standard deviation (simulation) and expected values (model)(average over 1000 runs).

�DiðsDtÞ ¼PSi

k¼1Dik ðsDtÞSi

for s ¼ 0;1;2; . . . ð16Þ

The curve of average results for scenario SCi as plottedin Figs. 3 and 4, is given by the pairsfðsDt; �DiðsDtÞÞ; s ¼ 0;1;2; . . .g. The standard deviation inrelation to the number of available nodes D of each setfDik ðDtÞ; k ¼ 0;1; . . . ; Sig was also calculated and used toevaluate the deviations between the model and the simu-lated scenarios.

Table 1 illustrates two of these scenarios, SC1 and SC2.SC1 is the same scenario used in [3], while in SC2 severalsimulation parameters were changed (area size, transmis-sion range, number of nodes, node speeds).

Figs. 3 and 4 present a plot of the amount of availablenodes (D) over time. It can be seen that the theoreticalcurve given by the proposed model in Eq. (8) is very closeto the simulation curve obtained from the average of 1000runs in the DTN simulator for both scenarios. The coeffi-cient of determination obtained by the method of leastsquares, R2 ¼ 0:99561, indicates a good fit of the modelto predict average system behavior. That is, even a contin-uous model, which is essentially a fluid approximation,provides a good fit for the expected number of availablenodes D in a discrete scenario with a finite number ofnodes.

Although the average of simulation results for D pro-vided a close approximation to the proposed model, it isimportant to observe that the standard deviation of D onall simulations is very high.

Table 1Simulation scenarios.

Parameter SC1 SC2

Area shape Square SquareSide 2 km 6 kmRange 30 m 100 mMovement model RWP RWPNodes 40 140Node speeds 4–10 km/h 5–50 km/h

Table 2Most important parameters for the Epidemical DTN Basic Model.

Parameter Equation

Basic modelDðtÞ ¼ NðN � 1Þ

N � 1þ ebNt

PðtÞ ¼ NebNt

N � 1þ ebNt

N ¼ DðtÞ þ PðtÞ

8>>>>><>>>>>:

Expected time of kthforwarding

tk ¼1

bNlnðN � 1Þðkþ 1ÞðN � ðkþ 1ÞÞ

� �for

k ¼ 1; . . . ;N � 2if k ¼ 1 and N is big enough:

t1 ¼1

bNln

2ðN � 1ÞN � 2

� �� 0;69

bN

TTLTTL P

2bN

lnðN � 1ÞðN þ 2ÞðN � 2Þ

� �Expected message diffusion

time (te)te � t1ð1þ 2log2NÞ

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 205

It can be seen in Figs. 3 and 4 that the standard devia-tion is larger than the expected value after 3000 s of simu-lation for SC1 and 1000 s for SC2. The model cannotcapture such deviations since it only provides estimationfor expected values based on the continuity (system is con-tinuous) and uniformity (all nodes are equal and presentthe same behavior) assumptions and on fluidapproximation.

Fig. 5 shows two extreme examples of scattering, onemuch lower than the average (for simulation seed = 53)and the other far above (for simulation seed = 90). The sce-nario generated with seed = 53 illustrates a fast epidemicspread where message diffusion was completed just after3000 s. On the other hand, for scenario with seed = 90 theepidemic process was delayed until approximately 2000 swhen first contacts start to be established (the origin nodewas isolated from the others).

Another situation can also be observed from the curvesin Fig. 5, there is also a difference on the behavior of theepidemic near the end of the process at each case.Although SC1 with seed = 90 suffered a long initial delay,last contacts were faster (last available nodes were closerto the rest of propagating nodes) than for SC1 withseed = 53.

Such differences on the curves of Fig. 5 may explainwhy standard deviation on D over time tends to be high.

0

10

20

30

40

0 1000 2000 3000 4000 5000 6000 7000

Ava

ilabl

e N

odes

(D

)

Time (s)

SC1 with seed=90SC1 with seed=53

Expected Values

Fig. 5. Expected results for D (model) compared to results for seed = 90and seed = 53 (simulations).

The effects of the initial and final phases of the epidemic(‘‘border effect’’) plays an important role on the deviationsobserved in the process.

The model as it is cannot predict deviations, such issuesare discussed in Section 6.

6. Dealing with model deviations

No mathematical model for DTNs is really useful if thereis no way to predict and minimize large deviations that oc-cur in relation to the average case, as already showed in thelast section. Such behavior was also observed in [25], butwas not predicted nor treated in previous works, like [3,4].

We start investigating the relationship between modeldeviations and the border effect just mentioned. First ofall, the model is itself symetric as shown in Eq. (11). A sys-tem with x propagating nodes and y available nodes has thesame behavior of a system with y propagating nodes and xavailable nodes regarding to message forwardings. Fromthe law of mass action, the product xy is the same in bothcases. Hence, model behavior in the initial phase is similarto the final phase.

Another important aspect of the model is that the rateof change in D is lower at the borders (either initial or finalphase) than in middle points. The largest decrease rate onD is obtained when D0 is maximum, i.e. when D00 ¼ 0. Thiscan be solved looking again to Eq. (5), observing thatD00 ¼ 2bD� bN and making D00 ¼ 0:

2bD� bN ¼ 0) D ¼ N2

Then the largest transmission rates, or the fastest rates ofepidemic spread (‘‘epidemic speed’’), are obtained whenD ¼ N

2 whereas lowest rates occur at the beginning andend of the process. At the borders either x (the number ofpropagating nodes) or y (the number of available nodes) issmall.

One important assumption applied so far is that theanalysis does not take in account the spatial dimension.At any time instant, all nodes have the same probabilityto meet each other. That is probably a good assumptionfor large systems where one can work with averages (fluidflow approximation), but can cause serious deviations insmall finite systems specially at the borders where the epi-demic process depends on a single (few) node(s).

Hence, at the borders the particular situation of a (afew) node(s) may cause a great impact and dependencyon the message diffusion process as a whole, we call suchphenomenon as the ‘‘border effect’’. Fig. 6 can be used toillustrate different situations at initial and final phases ofthe process.

Assuming node A the source of a message, the DTNcommunication process will be delayed until node A even-tually encounters another node and forwards the messageto it. On the other hand, if B is the source node, epidemicspread will be very fast with very low delays in this initialphase. The probability of an encounter and contact ratesfor node A are much lower than for node B in the beginningof the process. Such situations resemble the simulationscenario with seed = 90 and seed = 53 as shown in Fig. 5,respectively. In both cases, deviations from what was

Fig. 6. Hypothetical scenario.

206 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

expected by the model were observed and suffered theinfluence of the border effect.

The border effect is specially important because, if theepidemical process is delayed by a slow starting node, oranticipated by a fast one, all subsequent forwardings areaffected.

To evaluate how much the first contact affect the over-all simulation, we studied the importance (weight) of thefirst contact times to the variance observed in the overallsimulation. To do so, we considered all the initial contacttimes for each scenario as new starting times for the sim-ulation. Fig. 7 illustrates such approach, where curve‘‘STDEV N-1’’ considers the standard deviation on the num-ber of available nodes D over time obtained for the resultsjust after the first contact time (new simulation startingpoint), ‘‘STDEV N-2’’ for the second contact time, and soon. Thus, any possible effect of first contact times is elimi-nated, and the resulting variance is due only to the rest ofsimulations.

It can be seen from the simulation results that the con-tribution of first contact times is relevant to the variance ofthe process as a whole. In fact, the first contact contributedapproximately with 1

4 of the total standard deviation. Fig. 7

0

1

2

3

4

5

6

7

8

9

10

11

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Std

Dev

of

Ava

ilabl

e N

odes

Time (s) relative to the start of DTN - scenario N - or relative to the 1st, 2nd and 3rd contact - scenarios N-1, N-2 and N-3

Std Dev NStd Dev N-1Std Dev N-2Std Dev N-3

Fig. 7. First contacts influence on the model STD deviation for SC1 –standard deviation of D computed from the beginning (Std Dev N) andafter the 1st (Std Dev N-1), 2nd (Std Dev N-2), 3rd contacts (Std Dev N-3).

for SC1 shows the first, second and third contacts influenceon the standard deviation.

Given the importance of first contact to the dynamics ofmessage forwarding in an Epidemic DTN, we rewrite ourmodel in (8) as a function of t1, as follows. From eq. (12),bN ¼ 1

t1ln 2N�2

N�2

� �, then applying it to (8), we have:

DðtÞ ¼ NðN � 1ÞðN � 1Þ þ 2N�2

N�2

� � tt1

ð17Þ

Assuming that first contact time is exponentially dis-tributed with mean t1 [3], it is possible from (17) to derivea simple Monte Carlo method to estimate average andstandard deviation results without resorting to specificDTN simulation codes and packages.

Let s1; s2; . . . ; sm be a sequence of samples from anexponentially distributed random variable of mean t1, rep-resenting samples of first contact times. For a given samplesk, (17) can be expressed as:

DðtÞk ¼NðN � 1Þ

ðN � 1Þ þ 2N�2N�2

� �tþt1�skt1

ð18Þ

which gives the number of available nodes as a function oftime assuming first contact occurred at sk. Hence, using allsamples one can plot estimations for average and standarddeviations results for the number of available nodes D asshown in Fig. 8(for SC1) and Fig. 9 (for SC2).

It can be seen that the maximum standard deviation re-sults obtained directly from the model (using simpleMonte Carlo calculations)2 and from DTN simulations(using The ONE simulator) are very close. The advantage ofthe former method is that it is much faster than the latterone, besides there is no need to set up a simulation modelnor to configure and run specific DTN simulation packages.

The whole process is indeed very fast, and so the use of(18) becomes an interesting way to predict the behavior ofan epidemical DTN with N nodes and expected first contacttime t1.

It may also be important to have an approximate esti-mation of the overall elapsed time to deliver a messagein an Epidemic DTN (message diffusion time). When N islarge enough, we have NðN � 1Þ � N2 and 2N�2

N�2 � 2, andthen from (17):

DðtÞ � N2

N þ 2t

t1

ð19Þ

As the model is continuous, when 0 < DðtÞ < 1 the lastavailable node has already received the message. In thisway, it will be considered that all nodes received the mes-sage when t ¼ te and D ¼ 1

k ; k > 1, in (19):

kN2 ¼ N þ 2tet1 ) 2

tet1 ¼ kN2 � N � kN2 ) te

t1

¼ log2kN2 � log2kþ 2log2N ) te

� t1 log2kþ 2log2Nð Þ ð20Þ

2 The code used to implement this procedure is dvar2:py available inhttp://cabreu.vialink.com.br/mestrado/dtn/simulacoes.

0

10

20

30

40

0 1000 2000 3000 4000 5000 6000 7000

Ava

ilabl

e N

odes

(D

)

Time (s)

Average of ResultsStd Dev - modelStd Dev - simul

Fig. 8. Standard deviation of D for SC1: model (Monte Carlo) vs.simulation.

0102030405060708090

100110120130140

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Ava

ilabl

e N

odes

(D

)

Time (s)

Average of ResultsStd Dev - modelStd Dev - simul

Fig. 9. Standard deviation of D for SC2: model (Monte Carlo) vs.simulation.

0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

STD

DE

V o

f Ava

ilabl

e N

odes

(D

)

Time (s)

STD DEV of SC1SD 15 km/hSD 20 km/hSD 30 km/h

Fig. 10. Standard deviation of D for SC1 with 1 fast source node withspeeds of 15, 20 and 30 km/h and the original deviation of SC1.

0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

STD

DE

V o

f Ava

ilabl

e N

odes

(D

)

Time (s)

STD Dev of SC1SD 15 km/hSD 20 km/hSD 30 km/h

Fig. 11. Standard deviation of D for SC1 with 2 fast source nodes withspeeds of 15, 20 and 30 km/h and the original deviation of SC1.

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 207

From simulation results, the best fit for te was obtainedmaking k ¼ 2. Then, (20) can be reduced to:

te � t1ð1þ 2log2NÞ ð21Þ

where t1 is the mean time to the first contact, which is aconstant given by (12).

For instance, in a DTN with 40 nodes, the expected mes-sage diffusion time (time to deliver a message to almost allnodes) can be calculated as a function its own first contactt140 as te � 11t140 . For 90 nodes, te � 12t190 and so forth. Eq.(21) was also compared to simulation results providingagain a good fit of the model.

Looking at (21), one may observe that the total timeto deliver a message is much more sensible to the firstcontact time than to the number of nodes. This means that,besides having a fundamental influence on the deviationsof the epidemic process, the first contact also has greatinfluence on the total time to deliver a message (time toinfect all nodes in an epidemic DTN).

With such analysis, we may conclude that all actionsthat anticipate first contact times bring improvement tothe epidemic process as a whole, and reduce its deviations.

It is important to note that in an Epidemic DTN, as op-posed to epidemiological studies in populations, the main

goal is to speed up message (virus) spreading in order toinfect all nodes in the shortest period of time. Then somepossible procedures can be envisaged in the effort to re-duce the border effect and the variance on message diffu-sion of the entire DTN: starting the DTN process with afast source node, because it is likely to meet more nodesin a shorter period of time; starting the DTN process witha group of k propagating nodes, because this has a similareffect of a DTN after k message forwardings; and startingthe DTN process with a group of k fast nodes for the samereasons before.

The above mentioned procedures were evaluatedthrough simulations in order to verify their efficacy. First,we run scenario 1 again for one fast source node withspeeds of 15, 20 and 30 km/h. Fig. 10 shows the standarddeviation of D for the three cases, it can be seen a signifi-cant reduction on those values as long as speed is in-creased. The reduction achieved up to 35% in thestandard deviation for the best case.

In Fig. 11 we repeated the same simulation scenario butnow considering two fast source nodes. The general behav-ior was the same, but a larger reduction in the standarddeviation was achieved, about 60%. Such results confirmed

208 C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209

our expectations on what can be done to enhance Epidem-ical DTN performance.

7. Summary of the main DTN model results

It follows below the main DTN model parameters de-rived so far (see Table 2) taking into account all assump-tions and simplifications made in this work.

These results are valid under the assumptions that:

� The number of nodes N is constant during messagepropagation, and is big enough for a good approxima-tion with the differentiable model equations.� When a contact occurs between a node in P and a node

in D, the message transferring probability is 1.� When a message is received, the node is immediately

able to transfer it to all reachable available nodes duringmessage lifetime.� A node makes only one message forwarding at each

time.� A node makes an average of bN contacts per unit time,

where b is the contact rate of the network.� No nodes becomes unavailable (c ¼ 0).� The transmission range (coverage) of each node is much

smaller than the scenario area.

8. Conclusion and future work

This work investigated modeling and analysis of mes-sage diffusion in Epidemical DTNs. As stated in a recentsurvey [18], analytical modeling and performance evalua-tion for DTN are major research challenges in the field.

Some recent and promising DTN models are the onesderived from well known SIR epidemiological models,given the similarity between message forwardings in DTNsand the spread of diseases in populations. This paper alsofollows this line of thought and started revisiting the basicSIR model and using a special case, where the recovery rateis zero (c ¼ 0) to better fit a specific DTN environment thatuses fixed message TTL. From Section 4 onwards some spe-cific formulations, conclusions and findings were reported.

One of our main findings is that SIR models are quiteaccurate for the average behavior of Epidemical DTN, evenin a discrete scenario with a finite and not very large num-ber of nodes (e.g. 40 in SC1). As it was shown in Section 5,simulation provided a good fit when average results on D(number of available nodes over time) were compared toexpected model results. However, simulation results alsoshowed that the model may deviate with significant stan-dard deviations for some specific cases, e.g. at some pointsin time during simulations, standard deviations on thenumber of available nodes were higher than averagevalues.

In Section 6 we studied such deviations and observedthat they are intrinsic to the system under study (Epidem-ical DTN) given its strong dependency to first and last con-tact times (border effect). Such initial transient behavior onepidemical spread has to be investigated and treated in or-der to make the model useful for practical DTN networkingdesign, evaluation and optimization purposes.

We proposed a simple Monte Carlo method to computestandard deviations without resorting to complex networkscenario simulations. It was also presented some possibleprocedures to mitigate the effects of deviations and speedup the epidemic process, such procedures were tested andvalidated by simulations confirming our expectations. Thisresult has important impact in real life scenarios, becauseit helps to choose, if possible, the best node or nodes to firsttransmit a message.

As a future work, we intend to evaluate and extend themodel to more practical scenarios where the problems ofchannel propagation, node coverage, spatial distributionand multi-path fading should be considered, as well asmemory and energy consumption.

References

[1] K. Fall, A delay-tolerant network architecture for challengedinternets, in: Proc. of the 2003 Conference on Applications,Technologies, Architectures, and Protocols for ComputerCommunications, SIGCOMM ’03, ACM, New York, NY, USA, 2003,pp. 27–34.

[2] A. Vahdat, D. Becker, Epidemic Routing for Partially Connected AdHoc Networks, Tech. Rep., Technical Report CS-2000-06, Departmentof Computer Science, Duke University, April, 2000.

[3] R. Groenevelt, P. Nain, G. Koole, The message delay in mobile ad hocnetworks, Perform. Eval. 62 (2005) 210–228. Elsevier B.V..

[4] G. Sharma, R. Mazumdar, N.B. Shroff, Delay and capacity trade-offs inmobile ad hoc networks: a global perspective, IEEE/ACM Trans.Netw. 15 (5) (2007) 981–992.

[5] W. Kermack, A. McKendrick, A contribution to the mathematicaltheory of epidemics, Proc. Roy. Soc. Lond. A 115 (1927) 700–721,http://dx.doi.org/10.1098/rspa.1927.0118.

[6] W. Kermack, A. McKendrick, Contributions to the mathematicaltheory of epidemics. II. The problem of endemicity, Proc. Roy. Soc.Lond. A 138 (1932) 55–83.

[7] Z.J. Haas, T. Small, A new networking model for biologicalapplications of ad hoc sensor networks, IEEE/ACM Trans. Netw. 14(1) (2006) 27–40.

[8] A. Al-Hanbali, A.A. Kherani, R. Groenevelt, P. Nain, E. Altman, Impactof mobility on the performance of relaying in ad hoc networks, in:IEEE Infocom, 2006.

[9] X. Zhang, G. Neglia, J. Kurose, D. Towsley, Performance modeling ofepidemic routing, Elsevier B.V. 51 (2006) 2867–2891.

[10] S. Hamer, William Heaton, Epidemic disease in England – theevidence of variability and of persistency of type, Lancet I 1 (1906)733–739.

[11] D.J. Daley, J. Gani, Epidemic Modeling: An Introduction, Studies inMathematical Biology, Cambridge University Press, 1999.

[12] H.W. Hethcote, Mathematical Understanding of Infectious DiseaseDynamics, Lectore Notes Series, vol. 16, Institute of MathematicalSciences, National University of Singapore, 2009.

[13] K. Fall, A Message-Switched Architecture for Challenged Internets,Intel Research Berkeley – (IRB-TR-02-010), 2002, pp. 1–14.

[14] B.D. Walker, T.C. Clancy, J.K. Glenn, Using localized random walks tomodel delay-tolerant networks, in: Military CommunicationsConference, IEEE MILCON, 2008, pp. 1–7.

[15] X. Lu, P. Hui, An energy-efficient n-epidemic routing protocol fordelay tolerant networks, in: Proc. IEEE Fifth Int. Networking,Architecture and Storage (NAS) Conf., 2010, pp. 341–347.

[16] Philippe Jacquet, B. Mans, G. Rodolakis, Information propagationspeed in mobile and delay tolerant networks, IEEE Trans. Inform.Theor. 56 (10) (2010) 5001–5015.

[17] S. Shahbazi, S. Karunasekera, A. Harwood, Improving performance indelay/disruption tolerant networks through passive relay points,Wirel. Netw. 18 (1) (2012) 9–31.

[18] M.J. Khabbaz, C.M. Assi, W.F. Fawaz, Disruption-tolerantnetworking: a comprehensive survey on recent developments andpersisting challenges, IEEE Commun. Surv. Tutorials 14 (2) (2012)607–640.

[19] Q. Wang, Z.J. Haas, Analytical model of epidemic routing for delay-tolerant networks, in: Proceedings of the 1st ACM Workshop onHigh Performance Mobile Opportunistic Systems, ACM, 2012, pp. 1–8.

C.S. de Abreu, R.M. Salles / Ad Hoc Networks 16 (2014) 197–209 209

[20] Giridhari Venkatadri, V. Mahendran, C.S.R. Murthy, Improving DTNrouting performance using many-to-many communication: aperformance modeling study, WiOpt (2013) 125–132.

[21] H.R.S. Renato, M. de Moraes, J.J. Garcia-Luna-Aceves, Many-to-manycommunication for mobile ad hoc networks, IEEE Trans. Wirel.Commun. 8 (5) (2009) 2388–2399.

[22] S. Kaveevivitchai, H. Esaki, Independent DTNs message deletionmechanism for multi-copy routing scheme, in: AINTEC ’10Proceedings of the Sixth Asian Internet Engineering Conference,2010, pp. 48–55.

[23] A. Keränen, J. Ott, T. Kärkkäinen, The one simulator for DTN protocolevaluation, in: SIMUTools ’09: Proceedings of the 2nd InternationalConference on Simulation Tools and Techniques, ICST, New York, NY,USA, 2009.

[24] E. Hyytia, J. Virtamo, P. Lassila, J. Kangasharju, J. Ott, When doescontent float? Characterizing availability of anchored information inopportunistic content sharing, in: Proc. of the IEEE INFOCOM, 2011,pp. 3137–3145.

[25] T. Spyropoulos, K. Psounis, C.S. Raghavendra, Performance analysisof mobility-assisted routing, in: Proc. the 7th ACM InternationalSymposium on Mobile ad hoc Networking, Computing (Eds.),MobiHoc ’06, 2006, pp. 49–60.

Claudio Abreu is president of VialinkInformatica Ltda., Rio de Janeiro, since 1997.He was graduated in electronic engineering in1987 and received his master degree inComputer Science in 2012 both from theMilitary Institute of Engineering (IME). Hismain research interests include performanceanalysis of computer networks, Internet traf-fic engineering, network resilience, graphstheory and delay and disruption tolerantnetworks.

Ronaldo Salles is appointed professor at theMilitary Institute of Engineering (IME), Rio deJaneiro, and currently serves the BrazilianArmy as a lieutenant colonel. Since 2007 heholds the position of postgraduate coursecoordinator in the Department of ComputerEngineering. He received his master degree inComputer Science from IME in 1998, and hisPh.D. degree in 2004 from the Imperial Col-lege London, U.K. His main research interestsinclude performance analysis of computernetworks, internet traffic engineering, delay

and disruption tolerant networks, network resilience and network secu-rity. He received best paper awards at the following conferences: ACMLANC’09, IEEE/IFIP LANOMS’09 and SBRC’08. He has served on the tech-

nical program committees of many conferences and has published morethan 70 papers in international journals and conference proceedings. Heis a member of the IEEE and SBC.