Analysis of per-node traffic load in multi-hop wireless sensor networks

958 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 2, FEBRUARY 2009

Analysis of Per-Node Traffic Load inMulti-Hop Wireless Sensor Networks

Quanjun Chen, Student Member, IEEE, Salil S. Kanhere, Member, IEEE,and Mahbub Hassan, Senior Member, IEEE

Abstract—The energy expended by sensor nodes in datacommunication makes up a significant quantum of their totalenergy consumption. Consequently, a mathematical model thatcan accurately predict the communication traffic load of asensor node is critical for designing efficient sensor networkprotocols. In this paper, we present an analytical model forestimating the per-node traffic load in a multi-hop wirelesssensor network. We consider a typical scenario wherein, thesensor nodes periodically sense the environment and forwardthe collected samples to a sink using greedy geographic routing.The analysis incorporates the idealistic circular coverage radiomodel as well as a realistic model, log-normal shadowing. Ourresults confirm that irrespective of the radio model, the trafficload generally increases as a function of the node’s proximityto the sink. However, in the immediate vicinity of the sink, thetwo radio models yield quite contrasting results. The ideal radiomodel reveals the existence of a volcano region near the sink,where the traffic load drops significantly. On the contrary, withthe log-normal shadowing model, the opposite effect is observed,wherein the traffic load actually increases at a much higherrate as one approaches the sink, resulting in the formation of amountain peak. The results from our analysis are validated byextensive simulations.

Index Terms—Traffic load, geographic routing protocols, sen-sor networks, multi-hop wireless networks.

I. INTRODUCTION

W IRELESS Sensor Networks (WSN) are being increas-ingly used in a variety of applications ranging from

health care, environmental monitoring to industrial automation[1]. Ensuring energy efficient operation is critical, especiallygiven that a typical WSN is deployed in remote and unac-cessible areas and sensor nodes are equipped with a limitedbattery source. A typical WSN consists of a large number ofnodes deployed over a large area. Hence, packets generatedat nodes that are outside the communication range of the sinkhave to be relayed by other nodes. It has been well acceptedthat the energy expended in transmission and reception ofpackets forms a significant component of the total energybudget of a sensor node [1, 2]. Consequently, an analyticalmodel that can accurately estimate the traffic load incurredat each sensor node is instrumental in predicting the energyconsumption of the nodes and thus, the operational lifetime of

Manuscript received January 8, 2008; revised July 2, 2008 and September2, 2008; accepted September 6, 2008. The associate editor coordinating thereview of this paper and approving it for publication was W. Liao.

The authors are with the School of Computer Science and Engineering, TheUniversity of New South Wales, Sydney, Australia (e-mail: {quanc, salilk,mahbub}@cse.unsw.edu.au).

Digital Object Identifier 10.1109/TWC.2009.080008

the entire WSN. In addition, the traffic load characterizationcan provide important insights for designing and configur-ing energy efficient network protocols. As an example, theinformation about the traffic load of nodes in a collisiondomain can be used to tune MAC parameters such as slotassignments in TDMA-based schemes and also for configuringthe wake-up and sleep durations in duty-cycling protocolswhere nodes periodically go to sleep to conserve energy. Inaddition, knowledge of the energy expenditure of nodes can beuseful in planning deployment and maintenance of the WSN.The network designer can deploy redundant nodes in regionswhere nodes are expected to expend their energy at a higherrate. Maintenance cycles can also be planned for replacingor charging depleted nodes, thus preventing the formation ofcoverage holes in the network.

The traffic load of a given sensor node depends on severalfactors. The first and foremost is the relative distance of thenode to the sink. In general, the closer the sensor is to the sink,the greater is the traffic load. This is because the nodes closerto the sink have to relay data packets transmitted by other far-off nodes. The traffic load also depends on the routing protocolemployed in the network as it determines the selection ofthe next hop node for relaying the data towards the sink.Lastly, the characteristics of the environment, which affectsthe radio communication behavior of the sensor nodes alsohas an impact on the traffic load. [1]–[5]

Analytical models which can accurately characterize thetraffic load of nodes in a WSN are rare. The available models,e.g., those presented in [3, 4, 5], are very simplistic and donot incorporate the specifics of the routing protocol used bythe sensor nodes. As such, these models can only be used toprovide a rough estimate of the mean traffic load over a relativelarge geographic area. In [6], Esa Hyytia et. al. have analyzedthe traffic load of nodes in a very dense wireless multi-hop network with randomly selected communication pairs.However, their measure of traffic load per unit radio coveragecan only provide for a coarse-grained characterization of thetraffic load. In addition, they have assumed that the shortestpath from any node to the sink can be approximated by astraight line segment, which is true only for extremely densedeployments. Furthermore, this work and in fact all otheraforementioned efforts have adopted the idealistic circularcoverage radio model, which is known to be a poor abstractionof the real communication environment.

In this paper, we take a first step towards developing adetailed and precise analytical model for estimating the per-

1536-1276/09$25.00 c© 2009 IEEE

CHEN et al.: ANALYSIS OF PER-NODE TRAFFIC LOAD IN MULTI-HOP WIRELESS SENSOR NETWORKS 959

node traffic load in a WSN. Given that the target domainof WSN is quite broad in terms of the type of applications,networking protocols (routing, MAC, etc) employed and char-acterization of the communication environment, developing ageneralised model is extremely challenging. Thus, we havechosen to focus on a important sub-set within this vastdomain. In terms of the application load, we focus on periodicmonitoring applications, wherein the sensor nodes sample theenvironment periodically and forward the collected data (e.g.temperature, humidity) to the sink. A significant portion ofWSN deployed today fall into this category (e.g.: RedwoodForest [7], Great Duck Island [8]). With regards to the routingstrategy, which is important for selecting the next-hop node,we have considered the popular greedy routing forwardingscheme [9-13]. In greedy routing, a sensor node forwards itspackets to a neighbor, which is geographically closest to thesink amongst possible neighbors. In doing so, greedy routingcan approximately find the shortest path in terms of hops [9]between a sensor node and the sink. Moreover, greedy routingprovides a scalable solution for large WSN, because it requiresonly local (i.e. one hop neighborhood) information for makingforwarding decisions. For developing an analytical model ofthe traffic load it is necessary to use an appropriate modelthat abstracts the wireless communication characteristics of arealistic environment. In our analysis we have used both theidealistic radio model and the realistic log-normal shadowingmodel, thus enabling us to compare the impact of the twoon the results. To the best of our knowledge, this work isthe first attempt at developing a comprehensive model forcharacterizing the per-node traffic load in a WSN.

The analytical model is validated by a rich set of simula-tions. Our results quantitatively confirm that the traffic loadof a node increases with the proximity to the sink. Moreimportantly, we observe a peculiar difference in the trafficload in the immediate vicinity of the sink for the two radiomodels. For the idea radio model, the traffic load declinessignificantly after a certain knee point as one moves closer tothe sink, resulting in the formation of a volcano shape. On thecontrary, with the more realistic log-normal shadowing radiomodel, we observe the opposite effect, i.e., the traffic load ofnodes very close to the sink increases quite significantly asa function of the proximity to the sink, which results in amountain peak pattern. For simplifying our analysis we havehad to make several assumptions. We have also investigatedthe impact of relaxing some of these assumptions and observedthat our analytical results are still valid in these circumstances.

The rest of the paper is organized as follows. In SectionII, we provide an overview of our model. In Section III, wepresent the analysis of the traffic load for both the ideal andlog-normal radio models. Section IV validates the analyticalmodel by comparing the results with those from simulations.Finally, Section V concludes the paper.[9]–[13]

II. OVERVIEW OF THE SYSTEM MODEL

For mathematical tractability, we make the following sim-plifying assumptions:

• The sensor nodes are randomly deployed in an infiniteplane area. The node distribution follows a homogenous

Poisson point process with a density of ρ sensors perunit area, which can approximate uniform distributionfor large area. This assumption has been widely usedin analyzing multi-hop wireless ad-hoc networks [14, 15,16]. [14]–[16]

• The sensor nodes are deployed in a circular region withthe sink (base station) situated at the centre. While thisassumption simplifies the closed-form expressions in ouranalysis, the results can be readily applied to otherdeployment scenarios (e.g., a sink at the edge of thenetwork) with minor modifications. It should be notedthat similar assumptions are made in prior work [3, 4,5].

• All sensors have identical transceivers and the wirelesslinks are assumed to be symmetric.

• Each sensor node periodically generates a data packetcontaining the relevant sensed data and routes the packetto the sink. This clock driven data generation modelis typical for many monitoring applications (e.g., mon-itoring of temperature, soil moisture, etc.) [17]. Ourmodel can be readily extended for an event-driven datageneration model, wherein the sensor nodes generatepackets in response to certain events of interest.

• All sensors generate packets with the same periodicity.This is indeed the case with several realistic sensornetwork deployments. For example, the WSN deploy-ments in Redwood Forest [7] and Great Duck Island [8]conform with this assumption.

• The network is dense enough such that greedy routingalways succeeds in finding a next hop node that advancesthe data packet towards the sink.

• Sensor nodes forward packets towards the sink withoutany data aggregation.

• Packet loss rate is q and it is uniform for all datatransmissions. Therefore, a sender needs to transmit av-eragely 1

1−q times to successfully deliver one packet tothe receiver.

In the first part of our analysis we consider an idealradio model, wherein the signal attenuation between any twonodes is a function of the Euclidean distance separating thenodes. Consequently, in this idealistic environment, the radiocoverage of a sensor node is a perfect circular disc withthe radius equal to its radio range. However, in reality, thesignal attenuation is not solely dependent on the distance. Forexample, signal reflection or signal noise can also attenuatethe signal. To make our analytical results more realistic, weextend our analysis and incorporate the log-normal shadowingradio model. This model adds a random signal loss componentto the purely distance-dependent signal attenuation. As willbe elaborated later, we have observed significant differencesin the analytical results with the two models. Note that, byemploying these two radio models, we implicitly assume thatsignal attenuation over different link are independent. For thesake of mathematical tractability, we do not consider signalcorrelation among different links. This link independent log-normal shadowing model has been widely used to approximatethe real environment [14, 15, 16].

Excluding packets that are generated by sensor nodes whichcan directly communicate with the sink, packets generated by


Fig. 1. Example of state transition (from state i to state j).

all other node are relayed by intermediate nodes as determinedby the routing strategy employed. The behavior of a packets’progress towards the destination is therefore important indetermining the traffic load at each sensor. We use a discreteMarkov chain to model the hop-by-hop progress of a packetfrom the source to the destination. The state of the Markovchain is defined as the Euclidean distance (measured in someconsistent metric unit, e.g. meters) between the current for-warding node that holds the packet and the destination. Ideally,this distance should be modeled as a continuous randomvariable. However, to simplify our model, we use a discretestate space to approximately represent the continuous distancevalues. We quantize the distances resulting in a state space of(0, ε, 2ε, ..., nε, ...), where the parameter ε is the interval of thestate space (i.e. the quantization coefficient). When the intervalε is small enough, the discrete state space approximates theoriginal continuous distance metric.

We elaborate on the state transition of the Markov chainusing the example illustrated in Fig. 1, Assume that a packet iscurrently held by node X as it makes its way towards the sink,node D. Since node X is at a distance i from the destination,the current state for this packet is i. Assume that the next hopnode chosen by node X using greedy forwarding is node N,which is at a distance of j from the destination. The packetforwarding operation thus results in a state transition from ito j for the packet. In general, the hop-by-hop progress madeby a packet towards the destination can be represented by aseries of states that the packet transitions through, eventuallyculminating in state 0 when the packet reaches the sink.

Intuitively, the sensors closer to the sink will have morepackets to transmit, because they have to relay the packetsoriginated from other sensors that are distant. Therefore thetraffic load of a sensor is dependent on its distance to thesink. We refer the packet generating interval as “time unit".The goal of our analysis is to determine the traffic load at asensor, which is defined as the total packets transmitted bythe sensor in one time unit. Let f(d) represent the averagetraffic load incurred by sensor nodes located at a Euclideandistance of d units from the sink. Note that, f(d) includes thepacket generated by the sensor and those forwarded on behalfof other sensors. Consider a sensor node located at distance jfrom the sink. Since greedy forwarding is employed, this nodecan only receive packets from its one-hop neighbors that arefurther from the base station than j. In other words, any other

node at distance i, where i > j, could possibly forward itspackets to this particular node. If we denote the state transitionprobability from state i to j as Pi,j , then the traffic load of asensor at distance j is dependent on the traffic load of sensorsat distance i and the state transition probability Pi,j .

Therefore, in our model, we first analyze the state transitionprobability of the packets. Using the transition probabilities,we can then recursively calculate the traffic load of theindividual sensors. The main symbols used in the rest of thispaper are listed in Table I.

TABLE ILIST OF MAIN SYMBOLS USED IN THE ANALYSIS

Symbol Definition

l Radius of circular network

R Average radio range of sensors

ρ Node density, i.e. the number of nodes per unit area

ε Quantization interval

i, j, d Euclidean distance from a sensor to the centrally located sink

ξ Signal randomness parameter in log-normal shadowing radiomodel

q packet loss rate

P∧(s) The probability that two nodes separated by distance s cancommunicate with each other

Pi,j State transition probability. i.e. the probability that a sensor atdistance i from the sink can forward its packets to the sensorat distance j

f(d) Traffic load, i.e. the average number of data packets transmit-ted by a sensor at distance d during one time unit

III. ANALYTICAL MODEL OF FORWARDING OVERHEAD

We first evaluate the state transition probabilities assumingthe ideal circular disc radio model. Next we extend this toinclude the log-normal shadowing model. Finally, in the lastsub-section, we use these transition probabilities to evaluatethe traffic load.

A. Evaluating the State Transition Probability for the IdealRadio Model

For the ideal radio model, a node can only communicatewith other nodes that are located within the circular cover-age region of this node. We employ an approach that usesgeometric computations and probability theory to prove thefollowing,

Theorem 1: In the context of an ideal radio model, thetransition probability of a packet from state i to j whenemploying greedy routing is,

Pi,j =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1 if i ≤ R and j = 0,

exp(−ρAi,j) − exp(−ρAi,j+ε)if i > R and i − R ≤ j < i,

0 others,(1)

where Ai,j is,

Ai,j = R2 arccos i2+R2−j2

2iR + j2 arccos i2+j2−R2

2ij

−√

(R+i+j)(R+i−j)(R−i+j)(i+j−R)

2

(2)


Fig. 2. Illustration used to prove Theorem 1.

Proof: Assume that a packet is currently at node X as itmakes its way towards the sink. Let node X be at a distancei from the sink as illustrated in Fig. 2. Consequently thepacket is currently in state i. The probability that the packetis forwarded to a sensor at distance j and thus resulting ina transition to state j is the probability that node X finds aneighbor at distance j as the next hop.

We start with a simple case, where i ≤ R, i.e. thedestination node is within radio coverage of the current nodeX . Now, since the next hop is the destination, the state i musttransition to state 0. Consequently, we have,

Pi,j =

{1 if i ≤ R and j = 0,0 if i ≤ R and j > 0.

(3)

Now let us consider the situation where i > R. Recall that,we have assumed that greedy routing can always succeed infinding a next hop node, which is closer to the sink. Thus, thenext hop of node X must be located at a distance that is lessthan i from the sink. In other words, the probability that thenext hop node lies outside distance region [i − R, i), is zero.We have,

Pi,j = 0, if i > R and (j < i − R or j ≥ i) (4)

Now, we discuss the more complicated and plausible casewhere, i−R ≤ j < i. In greedy routing, if the next hop of nodeX is at distance j, it implies that at least one neighbor of nodeX is at distance j and none of its other neighbors are closerto the destination than j. Thus, the transition probability is theprobability that at least one neighbor of node X lies on theperimeter of the curve of radius j centered at the destination(see Fig. 2) with no neighbors located to the right of thiscurve. Since, we assume a discrete state space with ε as theinterval of the state space, we can approximate the curve as aring of thickness ε, as illustrated in Fig. 2. Let Ri,j representthe region of this thin ring that intersects with the radio rangeof node X (narrow dark region in Fig. 2). We also denote Ai,j

as the area of the light shaded region in Fig. 2, which is theintersecting region between the radio coverage of node X anda circle of radius j centered at the destination. Ri,j and Ai,j

are also used to represent the area of each region referred.Now, the transition probability Pi,j is the probability that

at least one node lies inside region Ri,j and that no nodes arewithin Ai,j . Let P1 be the probability that at least one node is

within Ri,j , and P2 be the probability that no nodes lie withinAi,j .

Recall that, we have assumed that the node distributionfollows a homogenous Poisson point process with density ρ.Thus, the number of nodes in region Ri,j follows a Poissondistribution with mean ρRi,j , and the number of nodes inregion Ai,j has a Poisson distribution with mean of ρAi,j .Note that, the area of Ri,j can be computed as Ai,j+ε −Ai,j .Consequently, we have,

P1 = 1 − Prob(no node in Ri,j) = 1 − exp(−ρRi,j)= 1 − exp(ρAi,j − ρAi,j+ε)

(5)P2 = Prob(no node in Ai,j) = exp(−ρAi,j) (6)

In the Poisson point process, the distribution of the numberof nodes in any two disjoint region is independent. Thus P1

and P2 are independent and we have,

Pi,j = P1 · P2 = exp(−ρAi,j) − exp(−ρAi,j+ε) (7)

The area Ai,j can be computed using simple geometriccalculations and expressed as Equation (2). The details areomitted due to space limitations. Finally, combining Equations(3), (4), (7) and (2), the theorem is proved.

B. Evaluating the State Transition Probability for the Log-normal Shadowing Radio Model

Next, we study a more realistic radio model. In the log-normal shadowing radio model, the signal attenuation betweentwo nodes is dependent not only on the distance separating thetwo nodes, but also a random signal loss. More formally, givena distance s that separates two nodes, the signal attenuation(in dB) from one node to another one is,

β(s) = α log10(s

reference distance) + β1 (8)

where α is a path loss rate, and β1 is a random variable thatfollows a normal distribution with zero mean and a standarddeviation of σ,

f(β1) =1√2πσ

exp(− β21

2σ2) (9)

Now two nodes are one-hop neighbors, i.e. they havea direct link between them, only if the signal attenuationbetween them is less than or equal to a pre-defined attenuationthreshold βth. Thus, for two nodes separated by a distance s,the probability that they have a direct link, denoted as P∧(s),is given by,

P∧(s) = Prob(β(s) < βth) (10)

The above equation has been solved by Bettstetter in [14]and the result can be represented by,

P∧(s) =12

[1 − erf(

10√2ξ

log10s

R)], ξ = σ/α (11)

where R = 10βthα·10 , is referred to as the average radio range,

which is the maximum distance that permits the existence of alink between two nodes in the absence of signal randomness.The function erf(.) is defined as follows,

erf(z) =2√π

∫ z

0

exp(−x2)dx (12)


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200

link

prob

abili

ty

distance (meters)

ideal radiolog-normal shadowing radio (ξ=1)log-normal shadowing radio (ξ=2)log-normal shadowing radio (ξ=3)

Fig. 3. Link probability with different radio models (R = 50).

As an illustrative example, Fig. 3 plots the link probabilityfor the log-normal shadowing model for R = 50m anddifferent values of the random parameter ξ. Note that, thecurve has a longer tail for increasing values of ξ, which impliesthat a node’s radio may cover a larger area for larger ξ. Basedon the aforementioned characteristics of the log-normal model,we can have the following theorem,

Theorem 2: In the context of the log-normal shadowingradio model, the transition probability of a packet state i to jwhen employing greedy routing is,

Pi,j =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

P∧(i) if j = 0 and i > 0,0 if j > 0 and j ≥ i,[1 − P∧(i)

] · exp(−πρj2P∧(Ai,j)) ·[ − πρε(2j + ε)

· exp(1 − (j+ε)2P∧(Ai,j+ε)−j2P∧(Ai,j)(2j+ε)ε )

]others,

(13)Where P∧(i) is defined in equation (11), and

P∧(Ai,j) =∫ i+j

i−j

P∧(s)fi,j(s)ds (14)

fi,j(s) =1

πj2(j2θ

′+2sφ− i sin(φ)+

s2 + j2 − i2

2φ

′) (15)

θ′ = 2s√4i2j2−(i2+j2−s2)2

φ = arccos( s2+i2−j2

2is )

φ′= is

i√

4i2s2−(i2+s2−j2)2( i2−j2

s2 − 1)(16)

Proof: We start with the simple case when j = 0 andi > 0. Assume that a packet is currently in state i, whilelocated at a certain node X . The transition probability of thepacket from state i to zero is the probability that there isa direct link between node X and the sink. Thus we havePi,j = P∧(i) when j = 0 and i > 0.

Now let us consider the situation where j > 0 and j ≥ i.Since the next hop of node X must has a distance that is lessthan i from the sink, the probability that the next hop nodelies outside distance region [0, i), is zero. Therefore, we havePi,j = 0, when j > 0 and j ≥ i.

In other cases where the next state j > 0 and j < i, thetransition probability Pi,j is the multiplication of the followingthree independent probabilities,

Fig. 4. Illustration used to prove Theorem 2.

• The probability that no direct link exists between nodeX and the sink (otherwise the packet can be forwardedto the destination directly), which is 1 − P∧(i).

• the probability that the node X can find at least oneneighbor at distance j, denoted as 1 − P1, where P1 isthe probability that there is no neighbor at distance j.

• The probability that no neighbor is within the region thatis closer to the sink than j, which is denoted as P2.

Thus, we have,

Pi,j =[1 − P∧(i)

](1 − P1)P2, if j > 0 and j < i (17)

Similar to the previous sub-section, we use a ring ofthickness ε to represent the curve that is located at distancej from the sink, as illustrated in Fig. III-B. Let Ri,j denotethe ring area that has distance j to the destination, and LetAi,j represent the shaded disc shaped region that is closerto the sink than j but does not include the location of thesink itself. Note that, the area included under Ri,j and Ai,j

with the log-normal model is much larger as compared withthe ideal radio model in section III-A. The reason being thatwith the realistic log-normal model the one-hop neighbors ofX can located anywhere in the network. On the contrary, inthe case of the ideal radio model, the one-hop neighbors arerestricted in the circular coverage area of node X . Thus, P1

is the probability that no direct link exists between X and anynode in region Ri,j , and P2 is the probability that no directlink exists between X and any node in region Ai,j .

We first calculate P1. Since the number of nodes within Ri,j

is a random variable, according to the law of total probability,we have, [18] [19]

P1 =∞∑

k=0

{Prob(k nodes in Ri,j)

·Prob(no direct link to any one of those k nodes)} (18)

According to the Poisson point process, the number ofnodes within Ri,j has a Poisson distribution with mean ρRi,j .Also, given that there are k nodes in area Ri,j , these k nodesare independently distributed [18, 19]. Therefore the existenceof a direct link between node X and every node in area Ri,j isindependent of each other. Let P∧(Ri,j) be the probability thatthere exists a direct link between node X and a node within


area Ri,j . Equation (18) can be rewritten as,

P1 =∑∞

k=0(ρRi,j)

k

k! exp(−ρRi,j)(1 − P∧(Ri,j))k

= exp(−ρRi,jP∧(Ri,j))= exp(−πρε(2j + ε)P∧(Ri,j))

(19)

Similar, for P2, we can have,

P2 = exp(−ρAi,jP∧(Ai,j)) = exp(−πρj2P∧(Ai,j)) (20)

Combining Equations (17) (19) and (20), we have,

Pi,j =[1 − P∧(i)

] · exp[ − πρε(2j + ε)P∧(Ri,j)

]· exp

[ − πρj2P∧(Ai,j)] (21)

Next we compute P∧(Ri,j) and P∧(Ai,j). According to thedefinition of Ai,j , the combined region of Ri,j and Ai,j can berepresented by Ai,j+ε. Therefore P∧(Ri,j) can be representedby Ai,j and Ai,j+ε. In the Poisson point process distribution,given that a node is present within Ai,j+ε, the node isuniformly distributed in the region and it is either inside regionRi,j or Ai,j . Based on the law of total probability, we have,

P∧(Ai,j+ε) = P∧(Ri,j)Prob(the node is within Ri,j)+P∧(Ai,j)Prob(the node is within Ai,j)

= P∧(Ri,j)(2j+ε)ε(j+ε)2 + P∧(Ai,j) j2

(j+ε)2

(22)Equivalently, we have,

P∧(Ri,j) = (j+ε)2P∧(Ai,j+ε)−j2P∧(Ai,j)(2j+ε)ε

(23)

Combining Equations (21) and (23), we have,

Pi,j =[1 − P∧(i)

] · exp(−πρj2P∧(Ai,j)) ·[ − πρε

·(2j + ε) exp(1 − (j+ε)2P∧(Ai,j+ε)−j2P∧(Ai,j)(2j+ε)ε )

] (24)

Finally, we compute the last unknown variable P∧(Ai,j),i.e., given there exists a node within region Ai,j , the proba-bility that this node has a direct link with node X. Let fi,j(s)represent the probability that this node is located at distances from the node X. By the law of total probability, we have,

P∧(Ai,j) =∫ i+j

i−j

P∧(s)fi,j(s)ds (25)

Given a node is within region Ai,j , the distance from thisnode to node X varies from (i − j) to (i + j). Thus theintegral in Equation (25) represents the conditional probability.Following rigorous geometric calculations, fi,j(s) is computedas indicated in Equation (15). The detailed derivation isomitted here due to the limited space.

Finally, combining Equations (24), (25) and (15), the theo-rem is proved.

We now provide an example to illustrate the state transitionprobability, Pi,j . In this example, we assume the followingset of parameters, R = 50m, ε = 1m, ρ = 0.0019, the currentstate of a packet is i = 100 and the next state varies from100 to 0. Fig. 5 illustrates the distribution of the transitionprobability from state i to the next state j for both radiomodels under consideration. Note that, the log-normal modelreduces to the ideal circular coverage model when the randomparameter ξ is equal to zero. For the ideal radio model the peakof the distribution is around j = 57 and it reduces to zero for

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20 40 60 80 100

tran

sitio

n pr

obab

ility

from

i=10

0

state of j (distance to destination)

ideal radio (ξ=0), analysis resultslog-normal radio (ξ=1), analysis resultslog-noraml radio (ξ=2), analysis results

simulation results

Fig. 5. State transition probability from i = 100 (R = 50).

all states beyond 50. This is because of the circular coverageassumption (recall that R = 50m). With the more realisticlog-normal model the distribution is more spread out over theentire range and the peak shifts towards the right, i.e., closerto the sink. This effect is more pronounced as the randomparameter ξ increases. This is because higher the randomnessin the signal, the greater is the chance that a node closer tothe sink is chosen as the next hop.

C. Analysis of the Per-node Traffic Load

Based on the state transition probability Pi,j , we nowproceed to calculate the traffic load incurred at the individualsensors. The analysis is independent of the radio model underconsideration. One simply has to substitute the appropriatestate transition probability equations as derived in the previoustwo sub-sections for the radio model under consideration.Recall that, the traffic load of a sensor refers to the averagenumber of data packets transmitted by the sensor duringone time unit (see definition in Section II). By recursivecalculation, we have,

Theorem 3: Consider a circular shaped network of radius lwith the sink located at the centre. The traffic load of a sensorlocated at a distance d from the sink, f(d), is given by,

f(d) =St(d)

π(2d + ε)ερ· 11 − q

(26)

where St(d) is given by,

St(d) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

π(2d + ε)ερ if d = l,

π(2d + ε)ερ +∑

i∈(d,l]

Pi,dSt(i)

if 0 < d < l,

(27)

and Pi,d denotes the state transition probability for the radiomodel under consideration. Pi,d for the ideal and log normalradio model have been derived in theorem 1 and theorem 2respectively.

Proof: Let n(d) be the average number of sensors locatedat distance d, and St(d) be the total number of differentpackets collectively transmitted by these n(d) nodes. Notethat, to successfully deliver one data packet, we actually need


11−q transmissions due to packet lost rate of q. We have:

f(d) =St(d)n(d)

· 11 − q

(28)

Recall that, we approximate distance as a discrete state spacewith an interval of ε. As a result, the number of sensors locatedat distance d are actually the number of sensors within thethin ring of thickness ε located between the two concentriccircles of radius d and d + ε. Since the area of the thin ringis π(2d + ε)ε, the average number of sensors at distance d isgiven by,

n(d) = π(2d + ε)ερ (29)

The sensors at distance d transmit all the correctly receivedpackets plus the packets originated by these n(d) sensors. LetSr(d) be the average number of different packets received byall sensors at distance d in one time unit. We have,

St(d) = n(d) + Sr(d) (30)

We now proceed to calculate Sr(d). When d = l, thesensors at distance l are the furthest sensors from the sinkand therefore they are not involved in forwarding packets forother sensors. Therefore Sr(d) = 0 and

St(d) = n(d) if d = l (31)

For d < l, as discussed in Section II, any sensor at distancefarther than d can forward packets to distance d. In otherwords, the sensors at distance i, where i ∈ (d, l] can forwardpackets to sensors at distance d. For a particular distancei, the average number of different packets transmitted fromthe nodes at i to those at d is St(i) multiplied with thecorresponding state transition probability Pi,d. By summingup all cases of i, we have,

Sr(d) =∑

i∈(d,l]

Pi,dSt(i) if 0 < d < l (32)

Combining Equations (30) and (32), we have,

St(d) = n(d) +∑

i∈(d,l]

Pi,dSt(i) if 0 < d < l (33)

Finally, combining Equations (28, 29), (31) and (33), thetheorem is proved.

Note that, St(d) is a recursive function. Since, we knowSt(l), the initial value of St(d), we can calculate St(l − ε)according to Equation (27). Similarly, St(l−2ε) and so on canbe derived. Finally, for any given d, St(d) can be computed,following which, we can calculate f(d) using Equation (26).Note that, theorem 3 derives the traffic load of all sensorsexcluding the sink. Recall that, the sink exclusively acts as areceiver. Hence, the traffic load of the sink is zero.

IV. SIMULATION RESULTS

In this section, we present results from an extensive set ofsimulations. The first goal of our evaluations is to validatethe results of our analysis. For this, we developed a customC++ simulator, which conforms to the assumptions listed inSection II. We investigate if the state transition probabilitiesand traffic load derived in Section III conforms with thesimulation results. Our second objective, is to investigate if our

analytical results are valid in more realistic scenarios wherethe simplifying assumptions made in our analysis are relaxed.To study this, we used the NS2 simulator and incorporated apopular geographic routing protocol, GPSR and a CSMA/CAMAC protocol to make the simulations more realistic.

Recall that, our analysis assumes that nodes are deployed inan infinite plane and that we focus on a circular sub-region ofarea πl2 (l being the radius). To realize this we simulate a largesquare network to approximate the infinite network, and selecta smaller circular network at the center of this infinite networkas the target network for our simulations. A similar approachis also used in [14]. We consider a circular region of radius l =200m, and assume that sensor nodes are deployed with a nodedensity of ρ = 0.0019 (resulting in a total of 240 nodes). Theaverage radio range of each node, R, is assumed to be 50m.We simulate three values of the signal randomness parameterξ, namely, 0,1, and 2, where 0 represents the ideal radio modeland other values represent the log-normal shadowing radiomodel. For each case of ξ, we run 5000 simulations and theresults presented are averaged over all runs.

For an individual run of the simulation, we randomly deploynodes according to homogenous Poisson point process with adensity of ρ = 0.0019. Once the nodes are placed, we usethe appropriate radio model with the particular value of ξ togenerate link connectivity over all pairs of nodes. Then foreach node in the selected circular network, we employ greedyrouting to find the routing path to the centrally located sink.Once the routing paths are established, we can identify thenext hop node for each individual sensor. This enables usto determine the next hop state j for each current state i.Grouping the transitions from all nodes located at distancei from the sink gives us the distribution of the transitionprobabilities from state i.

Recall that, each sensor node generates one data packetduring one time unit. In other words, each routing path in thenetwork carries one data packet in one time unit. Therefore, foreach node, we count the number of routing paths that traversethrough the node, which effectively represent the traffic loadat this node. Finally, we group all the nodes that are locatedat the same distance i from the sink and calculate the averageper-node traffic load for that particular state, i.

Fig. 5 compares the results from the simulations with theanalytical results for the state transition probability when theinitial state, i = 100, R = 50m and l = 200m. Thegraph illustrates that our model is quite accurate in predictingthe one-hop transition probabilities. The slight inconsistencyis caused by the fact that all forwarding paths lead to acommon destination, the sink. As a result, the next hopneighbors of two nodes that are relaying two distinct packetstowards the sink are likely to overlap (i.e. include commonneighbors). Note that, this contradicts our assumption in theanalysis that the neighbors of a particular forwarder arecompletely independent of the neighbors of other forwarders.The resulting correlation causes the slight deviation from theanalytically results, as observed in Fig. 5. Note that, thisslight inconsistency was not observed in additional simulations(not included) wherein packets were forwarded to differentdestinations placed far apart from each other, thus confirmingthe above hypothesis.


0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160 180 200

traffi

c lo

ad (p

kts/

unit

time)

distance to sink (meters)

ideal radio (ξ=0), analysis resultslog-normal radio (ξ=1), analysis results log-normal radio (ξ=2), analysis results

simulation results

Fig. 6. Traffic load as a function of distance to sink (l = 200, R = 50,q = 0).

Fig. 6 plots the average per node traffic load as a functionof distance to the sink for a network of radius l = 200mand R = 50m (assuming packet loss rate q = 0). The closematch between the simulation and analytical results confirmsthe validity of the analytical model. As intuitively expected,the closer the node is to the sink, the greater is its trafficload. However, the interesting result revealed by the graph isthat the traffic load pattern of nodes close to the sink variessignificantly with the radio model. A 3-D plot of the trafficload, as depicted in Fig. 7, illustrates this effect more clearly.In these graphs, the z-axis represents the traffic load and the xand y axes represent the two dimensional cartesian coordinatesystem with the sink located at the origin (0,0). For the idealradio model, the traffic load declines significantly if nodesare too close to the sink, giving rise to a volcano shape asillustrated in Fig. 7(a). We observe that the traffic volcanois clearly contained within a circular area (volcano zone) ofradius R, the radio range of the sensor nodes, around the sink.The crater signifies a safe area within the volcano zone wherethe sensor nodes do not experience the full force of the trafficvolcano.

The volcano shape of the traffic load in Fig. 7(a) is a directconsequence of the state transition probability distribution (seeSection III-B). Since, the crater in Fig. 7(a) is observed in theradio range of the sink, we focus on nodes in that region. Thetraffic load of a node is largely dependent on the forwardingprobabilities (i.e. state transition probabilities) of its one-hopneighbors, which could potentially select this node as the next-hop relay. Fig. 8 above illustrates the forwarding probabilitiesof three nodes, which are located at a distance of 60m, 55mand 51m, respectively from the sink. As seen from this graph,the forwarding probability initially increases up to a knee pointas the distance to the sink reduces. However, after the kneepoint, the probability reduces in the immediate vicinity of thesink. The general shape of the forwarding probability curvesimplies that the nodes, which are very close to the sink arerarely chosen as forwarders (Note that the nodes, located at adistance less than 50m, do not need any forwarders). Hence,the traffic load decreases in the immediate neighborhood ofthe sink leading to the formation of the crater in Fig. 7(a).

As evident from Fig. 7(b), the situation is quite differentwith the log-normal shadowing radio. In contrast with the idealradio model, the traffic load of the nodes increases continu-

(a) ideal radio

(b) log-normal radio (ξ = 1)

Fig. 7. Traffic load as a function of position relative to the sink (l = 200,R = 50, q = 0).

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

01020515560

forw

ardi

ng (

tran

sitio

n) p

roba

bilit

y


forwarding probability from distance 60mforwarding probability from distance 55mforwarding probability from distance 51m

Fig. 8. Forwarding probability (state transition probability) of nodes as afunction of the distance to the sink.

ously as we move closer to the sink creating a mountain peakeffect. This pattern is caused by the radio signal randomnessthat is prevalent in the log-normal shadowing radio. Recallthat, due to this two nodes that are separated by a distancegreater than the average radio range R may still be able tocommunicate with each other. As a result, the nodes closer tothe sink tend to receive a large number of packets from otherdistant nodes. At the same time the signal randomness doesnot guarantee that two nodes, which may be very close to eachother will always be able to communicate. Consequently, eventhe nodes that are close to the sink may still require other


0

10

20

30

40

50

60

70

0 20 40 60 80 100 120 140 160 180 200

traf

fic lo

ad (

pkts

/uni

t tim

e)


ideal radio (ξ=0), q=0ideal radio (ξ=0), q=0.2ideal radio (ξ=0), q=0.4

log-normal radio (ξ=1), q=0 log-normal radio (ξ=1), q=0.2log-normal radio (ξ=1), q=0.4

Fig. 9. The effect of packet loss on traffic load (l = 200, R = 50).

nodes to relay their packets. This effect further compoundsthe traffic load in the vicinity of the sink, thus resulting in themountain peak effect.

The effect of packet loss on the traffic load is depicted inFig. 9, where the packet loss rate varies from 0 to 0.4. Fig. 9shows that traffic load with the presence of packet loss has asimilar characteristics as those of no-loss situation. However,the presence of packet loss increases the height of the volcanoand the mountain peak, therefore further deteriorates the un-even distribution of the traffic load over network area.

In addition to the above topology-driven simulation, wehave also carried out simulations using NS-2. Unlike theprevious simulations, the network now actively carries trafficand we have also relaxed several of the assumptions usedin our analysis. This enables us to evaluate if the analyticalresults are applicable in more realistic scenarios. We nowuse a bounded circular network of radius, l = 200m with acentrally located sink and incorporate GPSR, a popular greedyforwarding routing strategy. The node density is again 0.0019and the average radio range is 50m. Further, we do away withthe assumption of uniform packet loss rate and simulate aCSMA/CA MAC protocol. We wanted to investigate if thesefactors significantly affect the observed traffic load. For thiswe considered two different traffic loads - (i) low traffic loadwith sensors transmitting a new packet to the sink every 10seconds and (ii) high traffic load with the packet generationperiodicity reduced to 2 seconds. The results presented areaveraged over 2000 runs, each of which lasts 60 seconds. Wecompare the simulation results to the analytical results wherethe packet loss rate is set to 0.1. As seen from Fig. 10, for lowload, the observed traffic load is very close to the analyticalresults. At higher load, there is a slight deviation from theanalytical results. This difference can be attributed to theretransmissions caused due to the frequent packets collisions.However, in both cases, the analytical results still provide antight approximation. Similar results are also observed for idealradio model (not included due to the space constraints).

V. CONCLUSION

We have proposed an accurate mathematical model thatanalyzes the per-node communication traffic load, the dom-inant source of energy consumption, in a multi-hop wireless

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100 120 140 160 180 200

traf

fic lo

ad (

pkts

/uni

t tim

e)


Analysis results (q=0.1)Simulation results (τ=10s)

Simulation results (τ=2s)

Fig. 10. Traffic load simulation results using NS-2, log-normal radio model(l = 200, R = 50, ξ = 1).

sensor network. Our results confirm that the traffic load of anode increases with the proximity to the sink. In addition, wediscover that the radio characterization model has a significantimpact on the traffic load pattern of sensors in the immediatevicinity of the sink. The ideal model leads to a volcano effectwhereas the log-normal model causes a mountain peak shape.Our analytical model is validated by extensive simulations.Furthermore, the simulations demonstrate that our results arealso valid in realistic scenarios where the assumptions madefor the analysis have been relaxed.

Models that accurately predict the traffic load of eachsensor node make valuable contributions towards designingenergy efficient sensor networks. For example, the per-nodetraffic load can be used to estimate the energy consumptionof each sensor node, given that the energy expended intransmitting packets makes up a significant portion of a nodes’energy budget. The latter can then be used to predict theoperational lifetime of the network, which is an importantcriteria for designing sensor networks. Further, the per-nodeenergy consumption derived from our analysis can be usefulin identifying hotspots in the network, i.e., regions wheresensor nodes are expected to drain their energy at a faster rate(due to higher traffic load). Additional sensor nodes can bedeployed at these hotspots to extend the operational lifetimeof the network. In fact, the number of redundant nodes tobe deployed at each of the hotspots can also be estimatedusing the knowledge of the per-node energy consumption.The per-node traffic load derived in our analysis can alsobe incorporated in MAC protocol design. In particular, theduty-cycle of each sensor node can be adjusted in accordancewith the estimated traffic load, thereby improving the energyefficiency. We intend to investigate these applications in ourfuture work.

REFERENCES

[1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wirelesssensor networks: a survey," Computer Networks, vol. 38, pp. 393-422,2002.

[2] W. R. Heinzelman, A. Chandrakasan, and H. Balakrishnan, “Energy-efficient communication protocol for wireless microsensor networks,"in Proc. 33rd Annual Hawaii International Conf. Syst. Sciences, vol. 8,p. 8020, Hawaii, USA, Jan. 2000.


[3] K. Sha and W. Shi, “Revisiting the lifetime of wireless sensor networks,"in SenSys 2004: Proc. 2nd International Conf. Embedded NetworkedSensor Systems, pp. 299-300, Baltimore, MD, USA, Nov. 2004.

[4] J. Li and P. Mohapatra, “An analytical model for the energy hole problemin many-to-one sensor networks," in VTC-2005-Fall: Proc. IEEE Veh.Technol. Conf., vol. 4, pp. 2721-2725, Dallas, Texas, USA, Sept. 2005.

[5] D. Wang, Y. Cheng, Y. Wang, and D. P. Agrawal, “Lifetime enhancementof wireless sensor networks by differentiable node density deployment,"in MASS 2006: Proc. IEEE International Conf. Mobile Adhoc SensorSystems, pp 546-549, Vancouver, BC, Canada, Oct. 2006.

[6] E. Hyytiä and J. Virtamo, “On traffic load distribution and load balancingin dense wireless multihop networks," EURASIP J. Wireless Commun.Networking, 2007:Article ID 16932, 15 pages, 2007.

[7] G. Tolle, J. Polastre, R. Szewczyk, D. Culler, N. Turner, K. Tu,S. Burgess, T. Dawson, P. Buonadonna, D. Gay, and W. Hong, “Amacroscope in the redwoods," in SenSys 2005: Proc. 3rd InternationalConf. Embedded Networked Sensor Systems, pp. 51-63, San Diego, CA,USA, Nov. 2005.

[8] R. Szewczyk, A. Mainwaring, J. Polastre, J. Anderson, and D. Culler,“An analysis of a large scale habitat monitoring application," in SenSys2004: Proc. 2nd International Conf. Embedded Networked SensorSystems, pp. 214-226, Baltimore, MD, USA, Nov. 2004.

[9] B. Karp and H. T. Kung, “GPSR: greedy perimeter stateless routingforwireless networks," in Proc. 6th Annual International ConferenceMobile Computing Networking, pp. 243-254, Boston, MA, USA, Aug.2000.

[10] Y. Yu, R. Govindan, and D. Estrin, “Geographical and energy awarerouting: a recursive data dissemination protocol for wireless sensornetworks," UCLA Computer Science Department Technical Report,UCLA-CSD TR-01-0023, May 2001.

[11] F. Kuhn, R. Wattenhofer, and A. Zollinger, “Worst-case optimal andaverage-case efficient geometric ad-hoc routing," in MobiHoc ’03:Proc. 4th ACM International Symposium Mobile Ad Hoc NetworkingComputing, pp. 267-278, Annapolis, Maryland, USA, June 2003.

[12] S. Ratnasamy, B. Karp, L. Yin, F. Yu, D. Estrin, R. Govindan, andS. Shenker, “GHT: a geographic hash table for data-centric storage," inWSNA 2002: Proc. 1st ACM International Workshop Wireless SensorNetworks Applications, pp. 78-87, Atlanta, GA, USA, Sept. 2002.

[13] H. Luo, F. Ye, J. Cheng, S. Lu, and L. Zhang, “TTDD: two-tierdata dissemination in large-scale wireless sensor networks," WirelessNetworks, vol. 11, pp. 161-175, Mar. 2005.

[14] C. Bettstetter, “Connectivity of wireless multihop networks in a shadowfading environment," Wireless Network, vol. 11, no. 5, pp. 571-579,2005.

[15] R. Hekmat and P. V. Mieghem, “Connectivity in wireless ad-hoc net-works with a log-normal radio model," Mobile Networks Applications,vol. 11, no. 3, pp. 351-360, 2006.

[16] D. Miorandi and E. Altman, “Coverage and connectivity of ad hocnetworks presence of channel randomness," in Proc. INFOCOM 2005,vol. 1, pp. 491-502, Mar. 2005.

[17] Y. Chen and Q. Zhao, “On the lifetime of wireless sensor networks,"IEEE Commun. Lett., vol. 9, no. 11, pp. 976-978, 2005.

[18] C. Lantuejoul, Geostatistical Simulation: Models and Algorithms.Springer, page 121, 2001.

[19] A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, CaseStudies in Spatial Point Process Modeling. ‘Springer, page 18, 2006.

Quanjun Chen is a Ph.D. student in the Schoolof Computer Science and Engineering at the Uni-versity of New South Wales (UNSW), Sydney,Australia. He received the BSc degree in computerscience from the Civil Aviation University of China(CAUC), Tianjin, China, in 1999. He worked fouryears as a software engineer in Beijing. His currentresearch focuses on performance analysis and rout-ing protocol design in wireless ad-hoc networks.

Salil S. Kanhere received the BE degree in elec-trical engineering from the University of Bombay,Bombay, India in 1998 and the MS and Ph.D.degrees in electrical engineering from Drexel Uni-versity, Philadelphia, USA in 2001 and 2003, re-spectively. He is currently a senior lecturer with theSchool of Computer Science and Engineering at theUniversity of New South Wales, Sydney, Australia.His current research interests include wireless sensornetworks, vehicular communication, mobile comput-ing and network security. He is a member of the

IEEE and the ACM.

Mahbub Hassan (M’91 - SM’00) is a full Professorin the School of Computer Science and Engineering,University of New South Wales, Sydney, Australia,where he leads a research program on mobile andwireless systems. He earned his PhD in Com-puter Science from Monash University, Melbourne,Australia (1997), MSc in Computer Science fromUniversity of Victoria, Canada (1991), and BScin Computer Engineering (with High Honor) fromMiddle East Technical University, Turkey (1989).He has co-authored and co-edited three books on

computer networking. His recent book titled High Performance TCP/IPNetworking (Prentice Hall, 2004) has been widely adopted across the worldwith translations published in foreign languages. He serves in the EditorialAdvisory Board of COMPUTER COMMUNICATIONS (Elsevier Science). In1999-2001, he served as Associate Technical Editor for IEEE COMMUNICA-TIONS MAGAZINE, and was a Guest Editor of the magazine’s feature topicon TCP Performance in Future Networking Environments (April 2001 issue),and Wireless Mesh Networks (November 2007 issue). He was organizer andchair of IEEE ICON Wireless Workshop held in Sydney, 2003, and co-chair for the SPIE conference on Internet Quality of Service in 2001-2003.He was the recipient of the 2000 School of Computer Science (MonashUniversity) Teaching Excellence Award and nominee for the 2004 Facultyof Engineering (UNSW) Research Excellence Award. Professor Hassan haswritten several successful Australian Research Council (ARC) grants, workedon collaborative R&D projects with large industrial research laboratories(Canon and Vodafone), and developed industry short courses (for Telstra,Lucent, and NEC) on leading edge networking topics. His other recentappointments include Invited Professor at the University of Nantes, France(April-May 2005) and Project Leader and Principal Researcher at the NationalICT Australia (NICTA) in 2005-2006. His current research interests includevehicle-to-vehicle communications and mobile computing for high-speedmobility.

Analysis of per-node traffic load in multi-hop wireless sensor networks

Documents

Transcript of Analysis of per-node traffic load in multi-hop wireless sensor networks