Ascending runs in dependent uniformly distributed random variables: Application to wireless networks

HAL Id: hal-00384027

https://hal.archives-ouvertes.fr/hal-00384027

Submitted on 14 May 2009

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Ascending runs in dependent uniformly distributed

random variables: Application to wireless networks

Nathalie Mitton, Katy Paroux, Bruno Sericola, Sébastien Tixeuil

To cite this version:

Nathalie Mitton, Katy Paroux, Bruno Sericola, Sébastien Tixeuil. Ascending runs in dependent

uni-formly distributed random variables: Application to wireless networks. Methodology and Computing

in Applied Probability, Springer Verlag, 2010, 12 (1), pp.51-62. �10.1007/s11009-008-9088-0�.

�hal-00384027�

a p p o r t

d e r e c h e r c h e

IS

S

N

0

2

4

9

-6

3

9

9

IS

R

N

IN

R

IA

/R

R

--0

1

2

3

4

5

6

7

8

9

--F

R

+

E

N

G

Thèmes COM et NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Ascending runs in dependent uniformly distributed

random variables: Application to wireless networks

Nathalie Mitton , Katy Paroux , Bruno Sericola , Sébastien Tixeuil

N° 0123456789

Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71

random variables: Appli ation to wireless networks

Nathalie Mitton

∗

, Katy Paroux

†

, Bruno Seri ola

‡

, Sébastien Tixeuil

§

Thèmes COM etNUM  Systèmes ommuni ants etSystèmes numériques

Projets Dionysos,Grand Large et Pops

Rapport de re her he n° 0123456789 February 2008 12 pages

Abstra t: We analyze in this paper the longest in reasing ontiguous sequen e ormaximal

as ending run of random variables with ommon uniform distribution but not independent.

Theirdependen eis hara terizedby thefa tthattwosu essiverandomvariables annottake

the same value. Using a Markov hain approa h, we study the distribution of the maximal

as ending run and we develop an algorithm to ompute it. This problem omes from the

analysis of several self-organizing proto ols designed for large-s ale wireless sensor networks,

and weshow howour results appliesto this domain.

Key-words: Markov hains, maximalas ending run, self-stabilization, onvergen e time.

∗

INRIA Lille-NordEurope/LIP6(USTL,CNRS), nathalie.mittoninria.fr

†

UniversitédeFran he-Comté,katy.parouxuniv-f omte.fr

‡

INRIA Rennes-BretagneAtlantique,bruno.seri olainria.fr

§

dépendantes uniformément distribuées: appli ation aux

réseaux sans l

Résumé : Nous analysonsdans et arti lela plus longue sous-suite roissante ontiguë d'une

suite de variables aléatoires de même distribution uniforme mais non indépendantes. Leur

dépendan e est ara térisée par le fait que deux variables su essives ne peuvent prende la

même valeur. En utilisant une appro he markovienne, nous étudions ladistribution de la plus

longue sous-suite roissante ontiguë et nous développons un algorithme pour la al uler. Ce

probème provient de l'analyse de plusieurs proto oles auto-organisants pour les réseaux de

apteurs sans-l à grande é helle, et nous montrons omment nos résultats s'appliquent à e

domaine.

Mots- lés : Chaînes de Markov, sous-suites roissantes ontiguës, auto-stabilisation, temps

1 Introdu tion

Let

X

= (X

n

)

n>1

be a sequen e of identi ally distributed random variables on the set

S

=

{1, . . . , m}

. As in [8℄, we dene an as ending run as a ontiguous and in reasing subse-quen e in the pro ess

X

. For instan e, with

m

= 5

, among the

20

rst following values of

X

:

23124342313451234341

, there are

8

as ending runs and the length of maximal as ending run is

4

. More formally, an as ending run of length

ℓ >

1

, starting at position

k >

1

, is a subsequen e

(X

k

, X

k+1

, . . . , X

k+ℓ−1

)

su h that

X

k−1

> X

k

< X

k+1

<

· · · < X

k+ℓ−1

> X

k+ℓ

,

where we set

X

0

= ∞

in order toavoid spe ial ases at the boundary. Under the assumption that the distribution is dis rete and the random variables are independent, several authors

havestudiedthebehaviourofthemaximalas endingrun, aswellasthelongestnon-de reasing

ontiguoussubsequen e. Themainresults on erntheasymptoti behaviourofthesequantities

when the number of random variables tends to innity, see for example [6℄ and [4℄ and the

referen es therein. Note that these two notions oin ide when the ommon distribution is

ontinuous. In this ase, the asymptoti behaviour is known and does not depend on the

distribution, as shown in[6℄.

We denote by

M

n

the length of the maximal as ending run among the rst

n

random variables. The asymptoti behaviourof

M

n

hardlydepends onthe ommondistributionof the randomvariables

X

k

, k >

1

. Someresults have been established forthe geometri distribution in[10℄ whereanequivalentofthe lawof

M

n

isprovided andpreviously in[1℄wherethe almost-sure onvergen e isstudied, aswell asfor Poisson distribution.

In [9℄, the ase of the uniform distribution on the set

{1, . . . , s}

is investigated. The au-thor onsiders the problemof the longest non-de reasing ontiguoussubsequen e and gives an

equivalentof itslawwhen

n

islargeand

s

isxed. The asymptoti equivalentof E

(M

n

)

isalso onje tured.

In this paper, we onsider a sequen e

X

= (X

n

)

n>1

of integer randomvariableson the set

S

= {1, . . . , m}

, with

m >

2

. The random variable

X

1

is uniformlydistributed on

S

and, for

n >

2

,

X

n

isuniformlydistributed on

S

with the onstraint

X

n

6= X

n−1

. This pro ess may be seen asthe drawing of balls, numbered from

1

to

m

inan urn where atea h step the lastball drawn iskept outside the urn. Thus wehave, for every

i, j

∈ S

and

n >

1

,

P(X

1

= i) =

1

m

and

P(X

n

= j|X

n−1

= i) =

1

{i6=j}

m

− 1

By indu tion over

n

and un onditioning, we get, for every

n >

1

and

i

∈ S

,

P(X

n

= i) =

1

m

Hen e the randomvariables

X

n

are uniformlydistributedon

S

but arenot independent. Using a Markov hain approa h, we study the distribution of the maximal as ending run and we

develop an algorithmto ompute it. This problem omes from the analysis of self-organizing

proto ols designed for large-s ale wirelesssensor networks, and weshow howour results apply

to this domain.

The remainder of the paper is organized as follows. In the next se tion, we use a Markov

hain approa h to study the behavior of the sequen e of as ending runs in the pro ess

X

. In Se tion 3, we analyze the hittingtimes of an as ending run of xed length and we obtain the

distribution ofthe maximalas ending

M

n

overthe

n

rst randomvariables

X

1

, . . . , X

n

usinga Markovrenewalargument. Analgorithmto omputethisdistributionisdevelopedinSe tion4

and Se tion5isdevoted tothe pra ti alimpli ationsofthis work inlarge-s alewirelesssensor

networks.

2 Asso iated Markov hain

The pro ess

X

isobviouslyaMarkov hain on

S

. Asobserved in[10℄,we ansee theas ending runs asa dis rete-timepro ess havingtwo omponents: the valuetaken by the rst elementof

the as ending run and its length. We denote this pro ess by

Y

= (V

k

, L

k

)

k>1

, where

V

k

is the value ofthe rst element of the

k

th

as ending runand

L

k

isitslength. Thestate spa e of

Y

is a subset

S

2

we shall pre isenow.

Only the rst as ending run an start with the value

m

. Indeed, as soon as

k >

2

, the random variable

V

k

takes its values in

{1, . . . , m − 1}

. Moreover

V

1

= X

1

= m

implies that

L

1

= 1

. Thus, for any

ℓ >

2

,

(m, ℓ)

is not a state of

Y

whereas

(m, 1)

is only an initialstate that

Y

will nevervisit again.

We observe also that if

V

k

= 1

then ne essarily

L

k

>

2

, whi h implies that

(1, 1)

is not a state of

Y

. Moreover

V

k

= i

impliesthat

L

k

6

m

− i + 1

.

A ording tothis behaviour, we have

Y

1

∈ E ∪ {(m, 1)}

and for

k >

2

,

Y

k

∈ E,

where

E

= {(i, ℓ) | 1 6 i 6 m − 1

and

1 6 ℓ 6 m − i + 1} \ {(1, 1)}.

We denethe following useful quantities for any

i, j, ℓ

∈ S

and

k >

1

:

Φ

ℓ

(i, j) = P(V

k+1

= j, L

k

= ℓ|V

k

= i)

(1)

ϕ

ℓ

(i) = P(L

k

= ℓ|V

k

= i)

(2)

ψ

ℓ

(i) = P(L

k

>

ℓ|V

k

= i).

(3)

Theorem 1. The pro ess

Y

is ahomogeneousMarkov hain withtransitionprobabilitymatrix

P

, whi h entries are given for any

(i, ℓ) ∈ E ∪ {(m, 1)}

and

(j, λ) ∈ E

by

P

(i,ℓ),(j,λ)

=

Φ

ℓ

(i, j)ϕ

λ

(j)

ϕ

ℓ

(i)

.

Proof. We exploit the Markov property of

X

, rewriting events for

Y

asevents for

X

.

For every

(j, λ) ∈ E

and taking

k >

1

then for any

(v

k

, ℓ

k

), . . . , (v

1

, ℓ

1

) ∈ E ∪ {(m, 1)}

, we denote by

A

k

the event :

A

k

= {Y

k

= (v

k

, ℓ

k

), . . . , Y

1

= (v

1

, ℓ

1

)}.

We have to he k that

P(Y

k+1

= (j, λ)|A

k

) = P(Y

2

= (j, λ)|Y

1

= (v

k

, ℓ

k

)).

First, we observethat

and

A

2

= {Y

2

= (v

2

, ℓ

2

), Y

1

= (v

1

, ℓ

1

)}

= {X

1

= v

1

<

· · · < X

ℓ

1

> X

ℓ

1

+1

= v

2

<

· · · < X

ℓ

1

+ℓ

2

> X

ℓ

1

+ℓ

2

+1

}

= A

1

∩ {X

ℓ

1

+1

= v

2

<

· · · < X

ℓ

1

+ℓ

2

> X

ℓ

1

+ℓ

2

+1

}.

By indu tion, we obtain

A

k

= A

k−1

∩ {X

ℓ(k−1)+1

= v

k

<

· · · < X

ℓ(k)

> X

ℓ(k)+1

},

where

ℓ(k) = ℓ

1

+ . . . + ℓ

k

. Using this remark and the fa t that

X

is a homogeneous Markov hain, weget

P(Y

k+1

= (j, λ)|A

k

) = P(V

k+1

= j, L

k+1

= λ|A

k

)

= P(X

ℓ(k)+1

= j < · · · < X

ℓ(k)+λ

> X

ℓ(k)+λ+1

|X

ℓ(k−1)+1

= v

k

<

· · · < X

ℓ(k)

> X

ℓ(k)+1

, A

k−1

)

= P(X

ℓ(k)+1

= j < · · · < X

ℓ(k)+λ

> X

ℓ(k)+λ+1

|X

ℓ(k−1)+1

= v

k

<

· · · < X

ℓ(k)

> X

ℓ(k)+1

)

= P(X

ℓ

k

+1

= j < · · · < X

ℓ

k

+λ

> X

ℓ

k

+λ+1

|X

1

= v

k

<

· · · < X

ℓ

k

> X

ℓ

k

+1

)

= P(V

2

= j, L

2

= λ|V

1

= v

k

, L

1

= ℓ

k

)

= P(Y

2

= (j, λ)|Y

1

= (v

k

, ℓ

k

)).

We nowhave toshow that

P(Y

k+1

= (j, λ)|Y

k

= (v

k

, ℓ

k

)) = P(Y

2

= (j, λ)|Y

1

= (v

k

, ℓ

k

)).

Using the previous result, wehave

P(Y

k+1

= (j, λ)|Y

k

= (v

k

, ℓ

k

)) =

P(Y

k+1

= (j, λ), Y

k

= (v

k

, ℓ

k

))

P(Y

k

= (v

k

, ℓ

k

))

=

k−1

X

i=1

X

(v

i

,ℓ

i

)∈E

P(Y

k+1

= (j, λ), Y

k

= (v

k

, ℓ

k

), A

k−1

)

k−1

X

i=1

X

(v

i

,ℓ

i

)∈E

P(Y

k

= (v

k

, ℓ

k

), A

k−1

)

=

k−1

X

i=1

X

(v

i

,ℓ

i

)∈E

P(Y

k+1

= (j, λ)|A

k

)P(A

k

)

k−1

X

i=1

X

(v

i

,ℓ

i

)∈E

P(A

k

)

= P(Y

2

= (j, λ)|Y

1

= (v

k

, ℓ

k

)).

We have shown that

Y

is a homogeneous Markov hain over its state spa e. The entries of matrix

P

are then given, for every

(j, λ) ∈ E

and

(i, ℓ) ∈ E ∪ {(m, 1)}

by

P

(i,ℓ),(j,λ)

= P{V

k+1

= j, L

k+1

= λ|V

k

= i, L

k

= ℓ)

= P{V

k+1

= j|V

k

= i, L

k

= ℓ)P{L

k+1

= λ|V

k+1

= j, V

k

= i, L

k

= ℓ)

= P{V

k+1

= j|V

k

= i, L

k

= ℓ)P{L

k+1

= λ|V

k+1

= j)

=

P(V

k+1

= λ, L

k

= ℓ|V

k

= i)

P(L

k

= ℓ|V

k

= i)

ϕ

λ

(j)

=

Φ

ℓ

(i, j)ϕ

λ

(j)

ϕ

ℓ

(i)

,

where the third equality follows fromthe Markov property.

We givethe expressions of

ϕ

λ

(j)

and

Φ

ℓ

(i, j)

forevery

i, i, ℓ

∈ S

inthe followinglemma. Lemma 2. For every

i, j, ℓ

∈ S

, we have

Φ

ℓ

(i, j) =

m − i

ℓ

− 1



(m − 1)

ℓ

1

{m−i>ℓ−1}

−

 j − i

ℓ

− 1



(m − 1)

ℓ

1

{j−i>ℓ−1}

ψ

ℓ

(i) =

m − i

ℓ

− 1



(m − 1)

ℓ−1

1

{m−i>ℓ−1}

ϕ

ℓ

(i) =

m − i

ℓ

− 1



(m − 1)

ℓ−1

1

{m−i>ℓ−1}

−

m − i

ℓ



(m − 1)

ℓ

1

{m−i>ℓ}

.

Proof. Forevery

i, j, ℓ

∈ S

,itiseasily he ked that

Φ

ℓ

(i, j) = 0

if

m < i+ℓ−1

. If

m > i+ℓ−1

, we have

Φ

ℓ

(i, j) = P(V

2

= j, L

1

= ℓ|V

1

= i)

= P(i < X

2

< . . . < X

ℓ

> X

ℓ+1

= j|X

1

= i)

= P(i < X

2

< . . . < X

ℓ

, X

ℓ+1

= j|X

1

= i)

−P(i < X

2

< . . . < X

ℓ

< X

ℓ+1

= j|X

1

= i)1

{j>i+ℓ−1}

.

(4) We introdu e the sets

G

1

(i, j, ℓ, m)

,

G

2

(i, j, ℓ, m)

,

G(i, ℓ, m)

and

H(ℓ, m)

dened by

G

1

(i, j, ℓ, m) = {(x

2

, . . . , x

ℓ+1

) ∈ {i + 1, . . . , m}

ℓ

; x

2

<

· · · < x

ℓ

6= x

ℓ+1

= j},

G

2

(i, j, ℓ, m) = {(x

2

, . . . , x

ℓ+1

) ∈ {i + 1, . . . , m}

ℓ

; x

2

<

· · · < x

ℓ

= x

ℓ+1

= j},

G(i, ℓ, m) = {(x

2

, . . . , x

ℓ

) ∈ {i + 1, . . . , m}

ℓ−1

; x

2

<

· · · < x

ℓ

},

H(ℓ, m) = {(x

2

, . . . , x

ℓ+1

) ∈ {1, . . . , m}

ℓ

; i 6= x

2

6= · · · 6= x

ℓ+1

}.

It is well-known, see for instan e [5℄, that

|G(i, ℓ, m)| =

m − i

ℓ

− 1



.

Sin e

|G

2

(i, j, ℓ, m)| = |G(i, ℓ − 1, j − 1)|

, the rst term in (4) an bewritten as

P(i < X

2

< . . . < X

ℓ

, X

ℓ+1

= j|X

1

= i) =

|G

1

(i, j, ℓ, m)|

|H(ℓ, m)|

=

|G(i, ℓ, m)| − |G

2

(i, j, ℓ, m)|

|H(ℓ, m)|

=

|G(i, ℓ, m)| − |G(i, ℓ − 1, j − 1)|

|H(ℓ, m)|

=

m − i

ℓ

− 1



−

j − i − 1

ℓ

− 2



1

{j−i>ℓ−1}

(m − 1)

ℓ

,

The se ondterm is given, for

j > i

+ ℓ − 1

, by

P(i < X

2

< . . . < X

ℓ

< X

ℓ+1

= j|X

1

= i} =

|G(i, ℓ, j − 1)|

|H(ℓ, m)|

=

j − i − 1

ℓ

− 1



(m − 1)

ℓ

.

Adding these two terms, we get

Φ

ℓ

(i, j) =

m − i

ℓ

− 1



1

{m−i>ℓ−1}

−

j − i − 1

ℓ

− 2



1

{j−i>ℓ−1}

−

j − i − 1

ℓ

− 1



1

{j−i>ℓ}

(m − 1)

ℓ

=

m − i

ℓ

− 1



1

{m−i>ℓ−1}

−

 j − i

ℓ

− 1



1

{j−i>ℓ−1}

(m − 1)

ℓ

,

whi h ompletes the proof of the rst relation.

The se ond relationfollows fromexpression (3) by writing

ψ

ℓ

(i) = P(L

1

>

ℓ|V

1

= i)

= P(i < X

2

< . . . < X

ℓ

|X

1

= i)

1

{m−i>ℓ−1}

=

|G(i, ℓ, m)|

|H(ℓ − 1, m)|

=

m − i

ℓ

− 1



(m − 1)

ℓ−1

1

{m−i>ℓ−1}

.

The thirdrelation follows from expression (2) by writing

ϕ

ℓ

(i) = ψ

ℓ

(i) − ψ

ℓ+1

(i)

. Notethat the matrix

Φ

dened by

Φ =

m

X

ℓ=1

Φ

ℓ

is obviously asto hasti matrix,whi h meansthat, for every

i

= 1, . . . , m

, we have

m

X

ℓ=1

ϕ

ℓ

(i) = 1.

m

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j) =

m

X

ℓ=1

ϕ

ℓ

(i) = ψ

(

i) = 1.

3 Hitting times and maximal as ending run

For every

r

= 1, . . . , m

,we denoteby

T

r

the hittingtime ofanas ending run of lengthatleast equal to

r

. More formally,we have

T

r

= inf{k > r ; X

k−r+1

<

· · · < X

k

}.

It is easy to he k that we have

T

1

= 1

and

T

r

>

r

. The distribution of

T

r

is given by the following theorem.

Theorem 3. For

2 6 r 6 m

, we have

P(T

r

6

n|V

1

= i) =



















0

if

1 6 n 6 r − 1

ψ

r

(i) +

r−1

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j)P(T

r

6

n

− ℓ|V

1

= j)

if

n > r.

(5)

Proof. Sin e

T

r

>

r

, we have, for

1 6 n 6 r − 1

,

P(T

r

6

n|V

1

= i) = 0

Let usassume fromnow that

n > r

. Sin e

L

1

>

r

impliesthat

T

r

= r

, we get

P(T

r

6

n, L

1

>

r|V

1

= i) = P(L

1

>

r|V

1

= i) = ψ

r

(i).

(6) We introdu e the random variable

T

(p)

r

dened by hitting time of an as ending run length at least equal to

r

when ounting fromposition

p

. Thuswe have

T

_r

(p)

= inf{k > r ; X

p+k−r

<

· · · < X

p+k−1

}.

We then have

T

r

= T

(1)

r

. Moreover,

L

1

= ℓ < r

implies that

T

r

= T

(L

1

+1)

r

+ ℓ

, whi h leads to

P(T

r

6

n, L

1

< r|V

1

= i) =

r−1

X

ℓ=1

P(T

r

6

n, L

1

= ℓ|V

1

= i)

=

r−1

X

ℓ=1

P(T

(L

1

+1)

r

6

n

− ℓ, L

1

= ℓ|V

1

= i)

=

r−1

X

ℓ=1

m

X

j=1

P(T

(L

1

+1)

r

6

n

− ℓ, V

2

= j, L

1

= ℓ|V

1

= i)

=

r−1

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j) P(T

r

(L

1

+1)

6

n

− ℓ|V

2

= j, L

1

= ℓ, V

1

= i)

=

r−1

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j) P(T

r

(L

1

+1)

6

n

− ℓ|V

2

= j)

=

r−1

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j) P(T

r

6

n

− ℓ|V

1

= j),

(7)

wherethefthequalityfollowsfromtheMarkovpropertyandthelastonefromthehomogeneity

of

Y

. Putting together relations(6) and (7), we obtain

P(T

r

6

n|V

1

= i) = ψ

r

(i) +

r−1

X

ℓ=1

m

X

j=1

Φ

ℓ

(i, j)P(T

r

6

n

− ℓ|V

1

= j).

Forevery

n >

1

, we dene

M

n

asthe maximalas ending run lengthover the

n

rst values

X

1

, . . . , X

n

. Wehave

1 6 M

n

6

m

∧ n

and

M

n

>

r

⇐⇒ T

r

6

n,

whi h implies E

(M

n

) =

m∧n

X

r=1

P(M

n

>

r) =

m∧n

X

r=1

P(T

r

6

n) =

1

m

m∧n

X

r=1

m

X

i=1

P(T

r

6

n|V

1

= i).

4 Algorithm

For

r

= 1, . . . , m

,we denote by

ψ

r

the olumnve tor of dimension

m

whi h

i

th entry is

ψ

r

(i)

. For

r

= 1, . . . , m

,

n >

1

and

h

= 1, . . . , n

, we denote by

W

r,h

the olumn ve tor of dimension

m

whi h

i

thentry is dened by

W

h,r

(i) = P(T

r

6

h|V

1

= i) = P(M

h

>

r|V

1

= i),

and wedenoteby 1the olumnve torofdimension

m

withallentriesequal to

1

. Analgorithm for the omputation of the distribution and the expe tation of

M

n

is given inTable 1.

input :

m

,

n

output : E

(M

h

)

for

h

= 1, . . . , n

.

for

ℓ

= 1

to

m

do Compute the matrix

Φ

ℓ

endfor

for

r

= 1

to

m

do Compute the olumnve tors

ψ

r

endfor for

h

= 1

to

n

do

W

h,1

=

1 endfor for

r

= 2

to

m

∧ n

do for

h

= 1

to

r

− 1

do

W

h,r

= 0

endfor for

h

= r

to

n

do

W

h,r

= ψ

r

+

r−1

X

ℓ=1

Φ

ℓ

W

h−ℓ,r

endfor endfor for

h

= 1

to

n

do E

(M

h

) =

1

m

m∧h

X

r=1

1

t

_W

h,r

endfor

Table 1: Algorithmfor the distribution and expe tation omputation of

M

n

.

5 Appli ation to wireless networks : fast self-organization

Our analysis has important impli ations in fore ast large-s ale wireless networks. In those

networks, the number of ma hines involved and the likeliness of fault o urren es prevents

any entralizedplani ation. Instead, distributed self-organizationmust bedesigned toenable

proper fun tioning of the network. A useful te hnique to provide self-organization is

self-stabilization [2, 3℄. Self-stabilizationis a versatile te hnique that an make awireless network

withstand any kindof fault and re onguration.

A ommondrawba k withself-stabilizingproto olsisthatthey werenot designedtohandle

re overfromany possibledisaster) ould berelated tothe a tualsize of the network. Inmany

ases, this high omplexity was due to the fa t that network-wide unique identiers are used

to arbitrate symmetri situations [13℄. However, there exists a number of problems appearing

in wireless networks that need only lo allyunique identiers.

Modeling the network as a graph where nodes represent wireless entities and where edges

represent the ability to ommuni ate between two entities (be ause ea h is within the

trans-mission range of the other), a lo al oloringof the nodes at distan e

d

(i.e. havingtwo nodes atdistan e

d

orless assignedadistin t olor) an beenoughtosolveawiderange ofproblems. Forexample,lo al oloringatdistan e

3

an beusedtoassignTDMA timeslotsinanadaptive manner[7℄, andlo al oloringatdistan e

2

hassu essivelybeenusedtoself-organizeawireless network into more manageable lusters [12℄.

In the performan e analysis of both s hemes, it appears that the overall stabilization time

is balan ed by a tradeo between the oloring time itself and the stabilization time of the

proto ol using the oloring (denoted in the following as the lient proto ol). In both ases

(TDMA assignment and lustering), the stabilization time of the lient proto ol is related to

the height of the dire ted a y li graph indu ed by the olors. This DAG is obtained by

orienting an edge from the node with the highest olor to the neighbor with the lowest olor.

As a result, the overall height of this DAG is equal to the longest stri tly as ending hain of

olors a rossneighboringnodes. Of ourse,alargerset of olorsleads toashorterstabilization

timeforthe oloring(duetothehigher han eofpi kingafresh olor),butyieldstoapotential

higher DAG, that ould delay the stabilization time of the lientproto ol.

In [11℄, the stabilization time of the oloringproto ol was theoreti ally analyzed while the

stabilizationtimeofaparti ular lientproto ol(the lusterings hemeof[12℄)wasonlystudied

by simulation. The analysis performed in this paper gives a theoreti al upper bound on the

stabilization time of all lient proto ols that use a oloring s heme as an underlying basis.

Togetherwith the results of[11℄, our study onstitutes a omprehensive analysis of the overall

stabilizationtimeof a lassofself-stabilizingproto olsusedforthe self-organizationof wireless

sensor networks. In the remaining ofthe se tion, weprovidequantitative resultsregarding the

relative importan eof the numberof used olors with respe t to othernetwork parameters.

Figure 1 shows the expe ted length of the maximal as ending run over a

n

-node hain for dierent values of

m

.

Resultsshowseveralinteresting results. Indeed,self-organizationproto ols relyingona

ol-oring pro ess a hievebetterstabilizationtime whenthe expe ted lengthof maximalas ending

run isshort but a oloringpro ess stabilizes faster when the numberof olors is high [11℄.

Figure1 learlyshowsthateven ifthenumberof olorsishigh omparedto

n

(

n << m

),the expe tedlengthofmaximalas endingrunremainsshort,whi hisagreatadvantage. Moreover,

even ifthe numberofnodes in reases,the sizeof themaximalas ending runremainsshortand

in reases very slowly. This observation demonstrates the s alability properties of a proto ol

relying ona lo al oloringpro ess sin e itsstabilization time isdire tly linked to the lengthof

this as ending run [11℄.

Figure 2 shows the expe ted length of maximal as ending run over a

n

-node hain for dierent values of

n

.

Results shows that for a xed number of nodes

n

, the expe ted length of the maximal as ending run onverges to a nite value, depending of

n

. This implies that using a large numberof olors does not impa t the stabilizationtime of the lient algorithm.

1

1.5

2

2.5

3

3.5

4

4.5

5

0

10

20

30

40

50

60

70

80

90

100

Maximal ascending run size

Number of nodes n

m = 5

m = 10

m = 20

m = 30

m = 40

m = 50

m = 60

m = 70

m = 80

m = 90

m = 100

m = 110

m = 200

Figure1: Expe ted lengthofthe maximalas ending run asafun tion ofthe numberof nodes.

2

2.5

3

3.5

4

4.5

5

0

20

40

60

80

100

120

140

160

180

200

Maximal ascending run size

Number of colors m

n = 5

n = 10

n = 50

n = 100

Referen es

[1℄ E. Csaki and A. Foldes. On the length of the longest monotone blo k. Stud. S i. Math.

Hungarian,31:3546, 1996.

[2℄ E. W. Dijkstra. Self-stabilizing systems in spite of distributed ontrol. Commun. ACM,

17(11):643644,1974.

[3℄ S.Dolev. Self Stabilization. MIT Press, 2000.

[4℄ S. Eryilmaz. A note on runs of geometri ally distributed random variables. Dis rete

Mathemati s, 306:17651770,2006.

[5℄ D. Foata and A. Fu hs. Cal ul des probabilités. Masson, 1996.

[6℄ A.N. Frolov and A.I. Martikainen. On the length of the longest in reasing run inr d ???

Statisti s and Probability Letters, 41:153161, 1999.

[7℄ T. Herman and S. Tixeuil. A distributed tdma slot assignment algorithm for wireless

sensornetworks. InPro eedings of the First Workshopon Algorithmi Aspe ts of Wireless

SensorNetworks (AlgoSensors'2004),number3121inLe tureNotesinComputerS ien e,

pages4558, Turku, Finland, July2004. Springer-Verlag.

[8℄ G.Lou hard. Runsofgeometri allydistributedrandomvariables: aprobabilisti analysis.

J.Comput. Appl. Math., 142(1):137153,2002.

[9℄ G.Lou hard. Monotone runs of uniformlydistributed integer randomvariables: a

proba-bilisti analysis. Theoreti al Computer S ien e,346(23):358387, 2005.

[10℄ G.Lou hard and H. Prodinger. As ending runs of sequen es of geometri ally distributed

randomvariables: aprobabilisti analysis. Theoreti alComputerS ien e,304:5986,2003.

[11℄ N. Mitton, E. Fleury, I. Guérin-Lassous, B. Seri ola, and S. Tixeuil. Fast onvergen e

in self-stabilizing wireless networks. In 12th International Conferen e on Parallel and

Distributed Systems (ICPADS'06),Minneapolis, Minnesota, USA, July 2006.

[12℄ N.Mitton,E.Fleury,I.Guérin-Lassous,andS.Tixeuil. Self-stabilizationinself-organized

multihopwireless networks. In WWAN'05, Columbus, Ohio, USA, 2005.

[13℄ S. Tixeuil. Wireless Ad Ho and Sensor Networks, hapter Fault-tolerant distributed

Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes

4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

http://www.inria.fr