Modeling the Extinction in the InformationDiffusion Process in Wireless Sensor Networks

Kenji LeibnitzOsaka University, Graduate School ofInformation Science and Technology

1-5 Yamadaoka, Suita, Osaka, 565-0871, JapanEmail: leibnitz@ist.osaka-u.ac.jp

Marie-Ange RemicheFUNDP - The Unversity of Namurand Universite libre de Bruxelles,

Faculty of Computer ScienceRue Grandgagnage 21, B-5000-Namur, Belgium

Email: mre@info.fundp.ac.be

Abstract—Wireless sensor networks (WSNs) are useful forgathering information from the environment, either periodicallyfor monitoring purposes or when certain events need to bedetected. Since the wireless communication range of each sensornode is limited, data is propagated in a multi-hop manner untilit reaches one or more specific sink nodes. However, sensornodes usually employ a sleep scheduling mechanism for powersaving to prolong the operational lifetime of the network and aretherefore not always available for forwarding messages receivedfrom neighboring nodes. In this paper, we analytically investigatethe diffusion behavior of information in a WSN modeled as atwo-dimensional spatial branching process. Our analysis permitsthe derivation of the extinction probability of information beingdiffused from a source node. Our model permits observing howthe activity pattern of each node influences the probability ofmaintaining time-dependent information in the network.

I. INTRODUCTION

The recent trends in miniaturization of technology is leadingtoward more sophisticated development of micro-sensors [1]that can be embedded in future ubiquitous and ambient infor-mation environments. With growing ubiquitous connectivitythere will be also an increased diversity in devices that needto interconnect and cooperate among each other. To satisfythe wide range of user requirements and to support our dailylife by user-centric information networks in many aspects,a variety of devices including PCs, servers, home electronicappliances, and information kiosk terminals will be distributedin the environment and interconnected among each other.Furthermore, mobile devices that are attached to persons andvehicles, as well as small and scattered devices, such as RFIDtags and sensors, will require the support of mobility fromthe network. Especially, sensor devices will play a growinglyimportant role in the future for monitoring applications inhealthcare, environment, or public safety.

While current research on wireless sensor networks (WSNs)mainly focuses on designing energy-efficient protocols takinginto account routing and coverage, e.g., [2], our focus is onthe study of the diffusion process of information. In relatedwork, Intanagonwiwat et al. [3] proposed directed diffusionas a framework for data-centric information diffusion. Itsrouting mechanism variants were analyzed by Krishnamachariet al. [4]. In [5], data gathering in sensor networks was inves-tigated from an information theoretic viewpoint to consider

how much the node density and coverage affect the amountof gathered information in a spatially correlated node layout.

In this paper we take a slightly different approach fromabove-mentioned work by considering uncoordinated nodesin a sensor network that may independently be in an activeor sleep state. In this varying environment, we are interestedin the dissemination process of a specific information thatoriginates at a certain node and is propagated throughout thenetwork. Information can get lost when a node goes to sleepstate or when it expires its time of validity. This problemresembles that of handling churn in peer-to-peer (P2P) net-works, where the goal is to maintain the information withinthe network, despite peers arbitrarily joining or leaving thesystem [6]. However, in the case of P2P, the spatial relationshipoften becomes negligible due to hash functions, whereas in thispaper our goal is to apply the theory of Markovian branchingprocesses to model the spatial dissemination of information.

This paper is organized as follows. In Section II wesummarize the features of the considered wireless sensornetwork. This is followed in Section III where we describethe analytical model for evaluating the information diffusionprocess. Numerical results are presented in Section IV and thispaper is concluded in Section V.

II. SYSTEM DESCRIPTION

In our scenario, wireless sensor nodes are randomly de-ployed within the monitoring area. Since the sensor nodesneed to operate in an energy-efficient manner or the monitoringobjective does not require them to be active all the time, theyadopt a sleep scheduling scheme for power saving. In otherwords, nodes are not active all the time, but perform dutycycling between an active and sleep state. In the active state,the sensory unit of the node provides certain measurement datadepending on the type of sensor (temperature, light, location,accelerometer, etc.), and the RF transceiver unit sends thisdata to neighboring nodes and listens to the radio channelfor incoming messages from neighboring nodes. On the otherhand, in the sleep state, the node switches off its sensorunit and RF transceiver, and enters a power saving modethat consumes only a fraction of the power that is neededin active state. Our scenario considers heterogeneous devicesthat operate with the same information, but where the nodes

978-1-4244-7116-4/10/$26.00 ©2010 IEEE

R

Fig. 1. Sensor network scenario with information being disseminated fromthe (blue) node detecting an event. Inactive nodes are shown in grey.

are otherwise independent in their duty cycle scheduling, e.g.,different types of household devices reacting to the user’spresence. However, for the sake of tractability, we assume thatall nodes have the same stationary probability of being active.

Since the wireless transmission power of each node islimited, a node can only reach the nodes that are locatedwithin the wireless transmission range. While other papersoften assume that there is a single sink node to which alldata is propagated, we consider the case that all other nodesin the monitoring region are interested in receiving the datathat is diffused from a single node. For instance, a node mayregister some certain event that needs to be propagated to asmany active nodes as possible, see Fig. 1. Furthermore, weassume that this information is only valid for a limited timeand becomes obsolete after a random time period.

Our objective in this paper is to model this disseminationprocess to quantify the proportion of nodes in a given area thathave had the key information at any time, but also the timeneeded for the process to reach a given number of nodes.We are also interested in computing the extinction probabilityof this dissemination process, i.e., the probability that thisinformation does not exist anymore in the system.

III. ANALYTICAL MODEL

We assume that nodes are randomly located according toa planar point Poisson process of rate λ (see [7, Chapter 2]for a formal definition). This implies that given the number ofnodes in an area, these would be uniformly scattered in thatarea. As a result, the probability p(k), k ∈ {1, 2, . . .} that thek-th nearest neighbor is located within a distance R from anysource node is

p(k) = P [N(C(0, R)) ≥ k]

= 1 −k−1∑i=0

(πR2λ

)i

i!exp

{−πR2λ}

,

given that R is the transmission range of any node of thesystem, N(A) is a random measure counting the number ofnodes in a set A and C(0, R) denotes a circle of radius Rcentered at (0, 0).

A node experiences during its lifetime cyclic status changes.These may be modeled with the help of a Markov processresulting in a stationary behavior. We call pon the stationary

probability that a node is active at time t. With the comple-mentary probability, the node is either inactive or sleeping.Information remains in a node’s message buffer only for alimited amount of time until losing its validity and will be alsolost due to the node’s status change from active to inactive.We assume here as approximation that this message lifetimeis random and exponentially distributed with parameter μ.

Due to the random access to the radio channel and transmis-sion errors, the transmission process for a message is randomand we assume here that it takes an exponential amount oftime (with parameter δ) for an error-free reception of theinformation of interest at an active node.

We are interested in studying the number of nodes that havethe targeted information in their buffer at any time instant. Thisis expressed by the process

{(N(t), φ0(t), φ1(t), . . . , φm(t)) ; t ∈ R

+}

,

where N(t) denotes the total number of nodes with theinformation and φi(t), i ∈ {0, 1, . . . ,m} describes the numberof nodes in phase i in the system. Remember that a node inphase m can not share the information anymore. Clearly, itfollows that

m∑i=0

φi(t) = N(t),

for all t ∈ R+. This process is a branching process, in

particular a Markovian binary tree. This family of randomprocesses has been extensively analytically characterized byHautphenne et al. [8] among others. Two different techniquesmay be used to analyze this particular class of branching pro-cesses. One way is to apply branching process theory itself andanother is the structured Markov chain approach, as previouslysuccessfully applied in queueing theory (see Latouche andRamaswami [9] among many others). In Sections III-A andIII-B, we use these both techniques to analyze our model.

A. Branching Process Approach

In our particular setting, the branching process is fullydescribed by two different matrices, namely D1 and P , andby a column vector d. We propose to identify them accordingto the previously described system behavior and to explain therelated performance metrics accordingly. The proofs of theserelations are given in [8].

The matrix D1 is a square matrix of size m + 1. It recordsthe rate at which a node succeeding a transmission changesits status. As mentioned earlier, a node in phase i can onlymove to phase i + 1. A node in phase m can not transmit theinformation. Accordingly, D1 is a sparse matrix where only(D1)i,i+1, i ∈ {0, 1, . . . ,m − 1} are non-zero entries and areequal to

(D1)i,i+1 = δ pon p(i + 1),

since only active nodes can receive the transmitted informationand this happens at rate δ. Let us note that due to ourcontinuity assumption, the probability that two nodes receivethe information exactly at the same time is zero. Moreover, it is

worth noting here that the model overestimates this transitionrate by not taking into account that i nodes already had theinformation. However, the bigger i, the smaller pi and thusthe smaller the transition rate.

The matrix P has size (m + 1) × (m + 1)2, composedof elements Pj|ik, i, j, k ∈ {0, 1, . . . ,m} that correspond tothe conditional probabilities that the node receiving the infor-mation starts in phase j, knowing that the node transmittingwas in phase i and enters phase k just after the transmission.In our scenario, P is a sparse matrix where only the blockP0|ik is non-zero and entries only exist for k = i + 1 andi = 0, 1, . . . ,m − 1.

The column vector d is of size m + 1 recording the rate atwhich a node may leave the system while in phase i. In oursetting, it consists of entries equal to μ.

The mean progeny matrix M gives, for entry i, j ∈{0, . . . , m}, the mean number of direct nodes of type j thathave recorded the information transmitted by a node of typei. Clearly, in our setting, a node can transmit the informationto at most one node of type 1 at a time.

Let the operator ⊕ be defined as follows

a ⊕ b = a ⊗ I + I ⊗ b,

where a and b are two n× 1 vectors, I is the identity matrixof size n and ⊗ the Kronecker product symbol. It has beenestablished in [8] that the matrix M respects the followingidentity:

M = Ψ(1 ⊕ 1)

with Ψ = (−D0)−1

B and D0 is a diagonal matrix such that

(D0)ii = −⎛⎝

m∑j=0

(D1)ij + μ

⎞⎠ ,

B is an (m + 1) × (m + 1)2 matrix composed of elementBi,jk = (D1)ikPj|ik, and 1 is a vector of ones.

Let us define the column vector q of size m + 1 where qi

gives the probability that the branching process experiencesextinction starting with one node in phase i. We call q theextinction probability vector. Given the spectral radius ρ of M ,we use the following theorem proven by Mode [10, Chapter1, Theorem 7.1].

Theorem 1: If ρ < 1, then q = 1, and we say that thebranching process is subcritical.

If ρ = 1, then q = 1, and we say that the branching processis critical.

If ρ > 1, then we say that the branching process is super-critical and in terms of an element-by-element comparison,we have that q ≤ 1,q �= 1.

Moreover, it has been established that the extinction prob-ability vector q is the minimal nonnegative solution of thevector equation

q = θ + Ψ (q ⊗ q)

with θ = (−D0)−1d.

In [8], three different algorithms are proposed to numeri-cally evaluate q. This vector gives us the probability that thedissemination process may eventually stop at a certain time.

The next performance measures can also be computed using[11]. The mean number N(t) of nodes with the informationat time t is

N(t) = α exp {Ω t}1,

where α = (1, 0, 0, 0) records the probability to start thesystem with only one node in phase 0, Ω is defined as

Ω = D0 + D1P (1 ⊕ 1) .

The mean total number of nodes T e that had the informationduring the dissemination process, given that extinction occurs,is directly obtained with

T e = α (−Θ Φ)−1 d,

where Φ is a diagonal matrix with Φii = qi and

Θ = D0 + D1P (q ⊕ q) .

B. Structured Markov Chain Approach

This approach is complementary to the previous one inSection III-A and permits us to compute the mean time neededfor the dissemination process to reach in such conditions thesek nodes before it ends. A Markovian binary tree is in facta particular Markov process whose resulting generator has aparticular structure. Namely, in this setting, the process

{(N(t), φ0(t), φ1(t), . . . , φm(t)) ; t ∈ R

+}

is completely characterized in the state space M of possiblevalues for (N(t), φ0(t), φ1(t), . . . , φm(t)) such that N(t) ≥ 0,∑

0≤i≤m φi(t) = N(t) and φi(t) ≥ 0. We partition this statespace into

M =⋃k≥0

L(k),

where L(k) is called the level k and is composed of statesL(k) = {(φ0, φ1, . . . , φm);

∑0≤i≤m φi = k, φi ≥ 0}. Ac-

cording to this partition, the generator Q has a block structureas follows

Q =

⎡⎢⎢⎢⎢⎢⎢⎣

A(0)0 A

(0)1 0 0 . . .

A(1)−1 A

(1)0 A

(1)1 0 . . .

0 A(2)−1 A

(2)0 A

(2)1 . . .

0 0 A(3)−1 A

(3)0 . . .

......

......

. . .

⎤⎥⎥⎥⎥⎥⎥⎦

.

We avoid the details of block contents as they can be easilydetermined. The block structure of the generator simplifiescomputing the mean first passage time to a particular numberK of nodes in the system, given it occurs before extinction.We avoid the details of the algorithm (proposed in [12]) thatis an extension of the one presented in [13].

0 0.2 0.4 0.6 0.8 110

−2

10−1

100

node active probability pon

prob

abili

ty o

f ext

inct

ion

λ = 0.01

μ−1 = 50

μ−1 = 10

μ−1 = 5

Fig. 2. Extinction probability over pon

0 5 10 15 2010

−1

100

101

102

μ−1 = 10, λ = 0.01

time t

mea

n nu

mbe

r of

nod

es w

ith in

form

atio

n N

(t)

pon

= 0.3

pon

= 0.2

pon

= 0.1

Fig. 3. Mean number of nodes with information N(t)

IV. NUMERICAL EVALUATION

In this section we numerically evaluate the model wedescribed in Section III. Let us assume that the average trans-mission rate δ is given for a specific application and packetsize. Therefore, all units of time are in the following expressedwith relation to δ−1. We have the following system parametersto consider: node density λ, node activity probability pon, andmessage lifetime μ−1. Obviously, the lifetime of a messagemust be μ−1 > δ−1, otherwise it would not be propagatedbeyond the first neighboring node. If not stated otherwise, weassume other parameters to be λ = 0.01 and R = 10.

Figure 2 shows the extinction probability over the activeprobability pon. We can see that for μ−1 = 5, 10, 50, the longerthe validity of the data is, the smaller the extinction probabilityis, even for small pon values. Due to the finite lifetime ofa message, the diffusion process always faces extinction atsome time. However, the diffusion process might stop evensooner depending on the nodes’ locations. We see that certainextinction might be possible even if the message is still valid.Moreover, this figure shows us that if the message validity ison average 5, 10, or 50 times longer than the average messagetransfer duration, the nodes should be active for at least pon of0.29, 0.14, or 0.03, respectively, to avoid certain extinction.

The mean number of nodes that have received the informa-

0 0.2 0.4 0.6 0.8 110

0

101

102

103

node active probability pon

cond

ition

al m

ean

tota

l num

ber

of n

odes

Te

μ−1 = 5

μ−1 = 50μ−1 = 10

Fig. 4. Conditional number of nodes T e under extinction

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

node active probability pon

first

pas

sage

tim

e to

K−

th n

ode

μ−1 = 10, λ = 0.01

K = 50

K = 5

K = 10

Fig. 5. Alternative computation with structured Markov chain model

tion message, N(t) is shown over time in Fig. 3 for the case ofμ−1 = 10. For pon = 0.1, the number of nodes N(t) decreasesover time, whereas for pon > 0.1 the successful diffusion ofthe information is achieved and N(t) grows rapidly.

If we consider the total number of nodes T e involvedin the diffusion process conditioned on the occurrence ofextinction, we obtain the behavior as observed in Fig. 4. Thevertical dashed lines indicate the points where the switchingbetween extinction and possible successful diffusion takesplace for μ−1 = 5, 10, 50 corresponding to the pon shownin Fig. 2. When the node activity probability becomes large,T e approaches 1. This means that the only way to observeextinction of the diffusion process corresponds to those caseswhere the source node was not able to reach its neighbors.Thus, in total, only one node had the information duringthe diffusion process. Since this happens less and less withincreasing pon, we see a decrease in the extinction probabilityas already observed in Fig. 2. Accordingly, if a source nodesucceeds to transmit the information to at least one of itsneighbors, then there exists a good chance that extinction ofthe diffusion process will not occur.

Figure 5 depicts the computation using the structuredMarkov chain model given in Section III-B. Recalling that forμ−1 = 10 and λ = 0.01, the probability pon must be greater

5 10 15 200

2

4

6

8

10

12

14

16

18

K−th node

first

pas

sage

tim

e to

K−

th n

ode

pon

= 0.1

pon

= 0.2

pon

= 0.3

Fig. 6. First passage time for reaching K-th node

than 0.14 to have an extinction probability less than 1, we cansee in Fig. 5 that the first passage time for reaching the K-thneighbor strictly increases until the same peak is reached atabout pon = 0.21 independent of K.

This behavior is shown in Fig. 6. As seen before, for pon =0.1 extinction occurs, so that the first passage time approaches0 for large K, as the nodes are not active long enough tosupport the diffusion process. A value of pon = 0.2 requiresthe longest time to reach the nodes and larger pon assists thediffusion for reaching nodes faster. We can also see that forlarge K values, the curves flatten and due to the spreading ofinformation more nodes are reached at nearly the same time.

V. CONCLUSION

In this paper we provided analytical models for the infor-mation diffusion process in a wireless sensor network using aMarkovian branching process. Following the detection of anevent, we determined the probability that this information willbe eventually lost in the network either due to sleeping nodesor expiration of the validity. In branching process setting, thisis called the extinction probability. We assume that nodes aredistributed according to a planar Poisson process and show thattwo complementary techniques for analyzing the branchingprocess are possible. Our results show that it is possible toderive the threshold value between extinction and successfuldiffusion which can be helpful in designing optimal duty cyclepatterns and retransmission update intervals. These findingshelp us to clarify the parameter setting in which a simulationcould now be performed in order to avoid certain extinctionand to concentrate more on the geographical diffusion only.

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