International audienceLes réseaux de capteurs sans fils sont constitués d'un grand nombre de noeuds assujettis `à de sévères contraintes en terme d'énergie, de capacité de traitement et de communication. Dans ce contexte, afin de réduire la complexité, un des défis majeurs rencontrés dans ce type de réseau est le calcul des routes et la mise en oeuvre de schémas de routage efficaces tout en minimisant la quantité d'information    utilisée sur l' état du système. De nombreux travaux ont ́etudié ce compromis de façon qualitative ou grâce à des simulations. Nous proposons un modèle basé sur la théorie de la marche aléatoire pour estimer analytiquement ce compromis en considérant plus particulièrement l'influence du degré de connaissance de l'état du système que posséde un noeud sur le temps moyen de collecte dans un réseau de capteurs sans fils

Froc, Gwillerm

Lagrange, Xavier

Mabrouki, Issam

English

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Biased Random Walk Model to estimate Routing
Performance in Wireless Sensor Networks
Issam Mabrouki, Gwillerm Froc, Xavier Lagrange
To cite this version:
Issam Mabrouki, Gwillerm Froc, Xavier Lagrange. Biased Random Walk Model to estimate Routing
Performance in Wireless Sensor Networks. 9ème rencontres francophones sur les aspects algorith-
miques de télécommunications, May 2007, Ile D’Oléron, France. ￿hal-02164516￿
Biased Random Walk Model to Estimate
Routing Performance in Sensor Networks
Issam Mabrouki, Gwillerm Froc and Xavier Lagrange
Mitsubishi Electric TCL - e-mails:{mabrouki, froc}@tcl.ite.mee.com
RSM, ENST Bretagne - email: xavier.lagrange@enst-bretagne.fr
Les réseaux de capteurs sans fils sont constitués d’un grand nombre de nœuds assujettis à de sévères contraintes en
terme d’énergie, de capacité de traitement et de communication. Dans ce contexte, afin de réduire la complexité, un
des défis majeurs rencontrés dans ce type de réseau est le calcul des routes et la mise en œuvre de schémas de routage
efficaces tout en minimisant la quantité d’information utilisée sur l’état du système. De nombreux travaux ont étudi
ce compromis de façon qualitative ou grâce à des simulations. Nous proposons un modèle basé sur la théorie de la
marche aléatoire pour estimer analytiquement ce compromis en considérant plus particulièrement l’influence du degré
de connaissance de l’état du système que possède un noeudsur le temps moyen de collecte dans un réseau de capteurs
sans fils.
Keywords: Réseaux de capteurs, routage, marche aléatoire, entropie, performance
1 Introduction
A wireless sensor network (WSN) is formed by a large number ofsensor nodes deployed over an extended
region. Sensor nodes, usually battery-operated, are simple and cheap, and cannot offer plentiful resource.
The routing problem is an important issue to be considered here b cause it is significantly sensitive to the
previous constraints for two reasons. First, searching a large space of possible routes –derived from having
a large number of nodes– may prove computationally prohibitive for low complexity devices such as sen-
sor nodes. Second, performing explicit route discovery/repair computations and maintaining explicit state
information about available routes at the nodes is costly interms of complexity and energy. Thus, the selec-
tion of a routing scheme depends on theknowledgeavailable at the network nodes and the communication
overhead that can be tolerated. Such knowledge provides nodes with a picture of the network that can be ex-
ploited to make decisions. There exist different routing approaches in this respect. Incentralizedapproach,
each node is provided withfull topology information, then the shortest path algorithm canbe applied. Large
networks with reasonably stable nodes over time, where autonom us nodes do not know the full network
graph (e.g., the Internet), require adistributedrouting approach. However, this approach induces signif-
icant communication overhead, which is problematic for large scale networks with high constraints (e.g.,
WSN). A localizedapproach, where nodes make decisions solely based onpartial information available
from neighbors is rather suitable. These approaches leverage the intuitive idea that the more knowledge is
available at the network nodes, the more efficient is the routing scheme but against the complexity and the
energy expenditure.
Many research works have addressed this tradeoff but only from a qualitative view or by means of simu-
lations [SB02]. In this paper we use the random walk theory toquantitatively estimate the influence of the
requisite knowledge on the routing efficiency in WSN. This ismotivated by the fact that making appropriate
decisions to forward data depends readily on the amount of state information a node holds. Thus, without
any state information, nodes wouldblindly forward data. This results in a packet wandering from node to
node until reaching its destination. As analyzed in [FML07], the routing problem becomes then a problem
of an unbiasedrandom walk taking place on a graph. However, if some state information is available at
nodes, the induced random walk would bebiasedwith a favored direction to enhance performance.
Issam Mabrouki, Gwillerm Froc and Xavier Lagrange
0 1 · · · N−1
0
1
··
·
N−1
~e1
~e2
1
4
1
4
(1−ε)
2
ε
2
(a) Torus latticeT
0 2 4 6 8 10 12 14
0
20
40
60
80
100
Positionr1
ε = 0.5
ε = 0.6
ε = 0.7
ε = 0.8
ε = 0.9
ε = 1
M
e
a
n
D
a
ta
G
a
th
e
rin
g
D
e
la
y
(b) Impact of the bias
Figure 1: (a) Torus latticeT is formed by connecting the opposite ends of the square lattice. Hops are represented by
red arrows. (b) Mean data gathering delay as a function of thepositionr1 for different values ofε and forN = 15.
We consider in this paper the case of a periodic lattice network of finite square cells, each containing
two kind of nodes: sensor and sink nodes. We choose this simple structure because it fits well the actual
properties of some WSN (e.g., mesh or grid networks) while being useful to incorporate theoretical elements
to modelize the impact of the requisite knowledge on the routing efficiency.
2 The Model: Definitions and Notation
Let C denote a finite square cell of sizeN×N. The envisioned network is then induced by replicatingC
by rigid translations specified by vectorsN~m, where~m ∈ Z2. With a large number of cells, we can assume
that the formed network is infinite and hence, it is equivalent to a torus latticeT formed by connecting the
opposite ends of cellC. Every node~r ∈ T is labeled with(r1, r2), wherer1 andr2 are integers such that
0 ≤ r1, r2 ≤ N−1. At a given node~r, let p(~r,~s) be a nonnegative real such that∑~s∈T p(~r,~s) = 1. This
defines a transition probability distribution over the nodes of T at fixed~r. When a packet reaches node~r,
the next hop to node~s occurs with probabilityp(~r,~s). The random sequence of nodes selected this way is
a random walkon T, andp(~r,~s) is called thetransition functionof the random walk [Hug95]. By making
different assumptions on the topology of the underlying network and on constraints imposed onp(~r,~s), we
are able to explore a large space of possible routing schemes. In particular, let us considerN sink nodes
corresponding to the setC of nodes~s j = (0, j) where j = 0, · · · ,N−1. SetC is called the gathering border.
The other nodes are members ofS , the set of sensor nodes. The data generated by all sensor nodes are
collected by the sink nodes without any specific mapping betwe n sensor nodes and sink nodes. At a given
sensor node~r, we assume that the next hop occurs only to the 4-nearest neighbors with probability12 (1−ε)
to the right neighbor,12 ε to the left neighbor (0≤ ε ≤ 1), and
1
4 to either top or bottom neighbor (Fig. 1(a)).
Key probabilities from which main results can be derived arePn(~r,~s) the probability of being at node
~s after n hops, given that a packet has been issued at node~r, andFn(~r,~s) the probability of arriving at
node~s for thefirst time on thenth hop, given that the packet started at node~r. They are called thenode
occupation probabilityand thefirst-passage probabilityrespectively. We make use in this paper of the
generating functionformalism to deal with a sequence{cn}n∈N by capturing all these coefficients into a
formal infinite series defined asC(z) = ∑∞n=0 cnzn where the complex variablez is small enough to ensure
the convergence of this series.C(z) is called the generating function associated with{cn}n∈N. Thus, we
denote the generating functions associated withPn(~r,~s) andFn(~r,~s) asP(~r,~s|z) andF(~r,~s|z) respectively.
P(~r,~s|z) andF(~r,~s|z) are related to each other according to a classical relation extensively used in random
walk theory
P(~r,~s |z) = δ~r~s +F(~r,~s |z)P(~s,~s |z), ~r,~s ∈ T. (1)
The proof of this relation is based on the law of total probability and can be found in [Hug95].
Biased Random Walk Model to Estimate Routing Performance inWSN
3 Requisite Knowledge vs. Routing Efficiency
Before estimating analytically the influence of the requisite knowledge on the routing efficiency, let us
give an interpretation of parameterε. If Hε denotes the ntropyassociated with the transition probabilities
at a given sensor node, we can then writeHε = 32 −
1
2
(
ε log2(ε) + (1− ε) log2(1− ε)
)
. However, from
information theory,Hε describes how much uncertainty (or knowledge) is carried bya single transition.
Therefore, parameterε can be considered as an estimator of the amount of requisite knowledge available
at a given sensor node. Note that forε = 12, Hε is maximal which corresponds to an equally likely hops.
Otherwise, the uncertainty is lower, which corresponds to transitions with afavoreddirection leading to
performance enhancement.
3.1 Mean Data Gathering Delay: a Routing Efficiency Indicator
Thedata gathering delayof a packet issued from sensor node~r is the time or the number of hops it takes
to reachC where it is trapped. This time is a random variable denoted byDε(~r). We propose in the
following to evaluateE
(
Dε(~r)
)
, themeandata gathering delay, as an indicator of the routing efficiency. For
that, letGn(~r) be the probability that a packet issued from sensor node~r will be trapped atC on thenth
hop. Therefore, in terms of probability notation we havePr{Dε(~r) = n} = Gn(~r). If G(~r|z) denotes the
generating function associated withGn(~r), E
(
Dε(~r)
)
can be then expressed as
E
(
Dε(~r)
)
=
∞
∑
n=0
nGn(~r) = ∂G(~r|z)/∂z |z=1· (2)
Let us now makeG(~r|z) explicit. Using the law of total probability, it can be possible to decompose
the event that a packet issued from sensor node~r will be trapped atC on thenth hop, which has the
probabilityGn(~r), into theN mutually exclusive events that the packet arrives at the sink node~s j for the
first time on thenth hop, which has the probabilityFn(~r,~s j ). Thus, we obtainGn(~r) = ∑N−1j=0 Fn(~r,~s j ). The
n-dependence can be eliminated from this equation by multiplying both sides byzn and summing over all
n. Therefore, we obtainG(~r|z) = ∑N−1j=0 F(~r,~s j |z). However, from (1),F(~r,~s j |z) = P(~r,~s j |z)/P(~s j ,~s j |z).
Since a packet starting from sink node~s j never leaves, we havePn(~s j ,~s j) = 1 for all n, which leads to
P(~s j ,~s j |z) = 1/(1−z). Hence, we obtain
G(~r|z) = (1−z)
N−1
∑
j=0
P(~r,~s j |z). (3)
Montroll et al. [MS73] evaluated the discrete Fourier transform of the sequence{P(~r,~s j |z)} j at points
0 ≤ k ≤ N− 1. Remarking that the sum involved in (3) is nothing but the value of this discrete Fourier
Transform at the origin (k = 0), (3) can be then rewritten as
G(~r|z) =
ρ1(z)r1 −ρ2(z)r1
ρ1(z)N −ρ2(z)N
+
(
ρ1(z)ρ2(z)
)r1 ρ1(z)N−r1 −ρ2(z)N−r1
ρ1(z)N −ρ2(z)N
, (4)
where
{
ρ1(z)
ρ2(z)
=
1±
√
1−4A(z)B(z)
2A(z)
and
{
A(z) = z(1− ε)/(2−z)
B(z) = zε/(2−z).
By plugging (4) into (2), the mean data gathering delay can besimplified as
E
(
Dε(~r)
)
=
2
(2ε−1)
(
r1−N
1− ( ε1−ε)
r1
1− ( ε1−ε)
N
)
, for ε 6=
1
2
, 1. (5)
By expanding (5) close toε = 1/2 andε = 1, it can be deduced thatE
(
Dε(~r)
)
is continuous for all 0≤ ε≤ 1.
We retrieve thatE
(
D1/2(~r)
)
= 2r1 (N− r1) andE
(
D1(~r)
)
= 2r1 respectively. Moreover, note that the value
of G(~r|z) at pointz= 1 equals to unit, which represents the probability that a packet issued from sensor node
~r is ever trapped byC . This means that the data gathering ensured by the random walk process iscertain.
Remark also thatE
(
D1−ε(~r)
)
= E
(
Dε(N~e1−~r)
)
, which means that the plots ofE
(
Dε(~r)
)
andE
(
D1−ε(~r)
)
as functions ofr1 are symmetric about the axisX = N/2. It suffices then to assume that 1/2≤ ε ≤ 1.
Issam Mabrouki, Gwillerm Froc and Xavier Lagrange
3.2 Analysis
Let us now give a physical interpretation of (5). Indeed, if we denote the displacement of a packet issued
from sensor node~r after n hops by~Mn(~r), then ~Mn(~r) = ∑nj=1~H j(~r) where the random vector~H j(~r)
represents the displacemento the jth hop. Since we have the same transition probability distribu ion over
all sensor nodes, themeandisplacement aftern hops is given byE
(
~Mn(~r)
)
= n~µ where~µ denotes themean
displacement on a single hop. Vector~µ epresents also the bias of the random walk and can be interpre d
as themeanvelocity of propagation of a packet. We have~µ = 12 (1−2ε)~e1. Therefore, we deduce that for
1/2≤ ε ≤ 1 the negative direction of theX-axis is favored by the walk. In other word, for 1/2≤ ε ≤ 1
packets are attracted by the random walk towards the gathering border. A packet issued from sensor node
~r travels then the distancer1 before being trapped by the gathering border with a speed|~µ |= 12 (2ε−1),
hence, the time required by this packet before being trappedis simply the distance divided by the speed,
that is, 2r1/(2ε−1). This result could be retrieved from the first term of (5).
The impact of the requisite knowledge via the bias on the meandata gathering delay is illustrated in Fig.
1(b). We see that the higher are the values ofε (lower entropy), the lower is the delay. Moreover, it turns out
that an attractive bias (i.e., 1/2< ε ≤ 1) towards the gathering border accelerates the data gathering process
whereas a repulsive bias (i.e., 0≤ ε < 1/2) backwards the gathering border decelerates the data gathering
process. The case ofε = 1/2 (maximum entropy) corresponds to anu biasedwalk where no direction is
favored. This represents the worst case for the performanceof th walk. Note also that for fixedε such that
1/2< ε < 1, the mean data gathering delay increases withr1 until it reaches a maximum value from which
it decreases asr1 increases. This observation can be explained by the fact that with the periodicity property
of the torus lattice, the gathering border (X = 0) is replicated to infinity by rigid translations specified by
vectorsmN~e1 wherem is an integer. In particular, the axisX = N corresponds to a line of sink nodes.
Hence, whenr1 increases, the attractive effect of the bias is compensatedby the farness from the gathering
borderX = 0 until the nearness from the axisX = N takes away and therefore, the mean data gathering
delay goes down again.
4 Conclusion
In this paper we have related quantitatively the degree of knwledge to the routing performance and we
have studied to what extend the state information availablet network nodes can be minimized to reduce
the complexity while ensuring an efficient routing scheme. This paradigm arises especially in the design of
WSN where the localized approach is extensively embraced. With the aid of random walk theory, we have
confirmed analytically the intuitive result that the largerthe amount of state information, the more efficient
the routing scheme. All details of this model will appear in the full version of this paper.
Acknowledgments
Authors would like to thank Stéphane Paquelet for helpful discussions about the reduction of complexity in
large scale systems.
References
[FML07] G. Froc, I. Mabrouki, and X. Lagrange. Design and performance of wireless data collection
networks based unicast random routing. 2007. Submitted.
[Hug95] B. D. Hugues.Random Walks and Random Environments, volume 1, pages 119–121. Oxford
University Press, New York, 1995.
[MS73] E.W. Montroll and H. Scher. Random walks on lattices.continuous-time walks and influence of
absorbing boundaries.Journal of Statistical Physics, 9(2):101–135, 1973.
[SB02] S. Servetto and G. Barrenechea. Constrained random walks on random graphs: Routing algo-
rithms for large scale wireless sensor networks. InProceedings of the 1st ACM International
Workshop on Wireless Sensor Networks and Applications, Atlanta, GA, April 2002.


Biased Random Walk Model to estimate Routing Performance in Wireless Sensor Networks

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Biased Random Walk Model to estimate Routing Performance in Wireless Sensor Networks

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