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

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6443--FR+ENG Thèmes COM et NUM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Ascending runs in dependent uniformly distrib uted random variables: Application to wireless netw orks Nathalie Mitton , Katy Paroux , Bruno Sericola , Sébastien T ixeuil N° 6443 February 2008 Centre de recherche INRIA Rennes – Bretagne Atlantique IRISA, Campus universitaire de Beaulieu, 3504 2 Rennes Cedex Téléphone : +33 2 99 84 71 00 — Téléco pie : +33 2 99 84 71 71 Ascendi ng runs in dep enden t uniformly dis tribute d random v ariables: Applicatio n to wirel e ss net w orks Nathal ie Mitton ∗ , Ka ty Paroux † , Br uno Ser icola ‡ , S´ ebastien T i xeuil § Th ` emes COM et NUM — Syst ` emes comm unican ts et Sys t ` emes n um ´ eriques ´ Equipes-Pro jets D ion ysos, Grand Large et P ops Rapp ort de rec herc he n ° 6443 — F ebruary 200 8 — 12 pages Abstract: W e analyze in this pa p er the longest increasing con tiguous sequence or maximal ascending run of random v ariables with common uniform distribution but not inde p enden t. Their dep endence is c haracterized b y the fact that t w o success iv e random v aria bles cannot tak e the same v alue. Using a Mark o v c hain a ppro ac h, w e study the distribution o f the maximal ascending run and w e dev elop an algorithm to compute it. This problem comes from the analysis of sev eral self-organizing prot o cols designed for large-scale wireless sensor net w orks, and w e sho w how our results apply to this domain. Key-w ords: Mark o v ch ains, maximal ascending run, self-stabilization, con v ergence time. ∗ INRIA Lille - Nor d Eur op e/LIP6(USTL,CNRS), nathalie.mitton@ inria.fr † Univ ers it´ e de F r anche-Com t´ e, k at y .paroux@ univ-fcomte.fr ‡ INRIA Rennes - Br etagne Atlan tique, br uno .sericola @inria.fr § INRIA Saclay - ˆ Ile-de-F rance/LIP6 , sebastien.tixeuil@lri.fr Sous-s u ites croi ssan tes contigu ¨ es de v ariables al ´ eatoires d ´ ep end antes uniform ´ emen t dis tribu ´ ees: appli c ation aux r ´ ese aux sans fil R´ esum ´ e : Nous analysons dans cet article la plus longue sous-suite croissan te con tigu¨ e d’une suite de v a riables al ´ eato ires de m ˆ eme distribution uniforme mais non ind ´ ep endan tes. Leur d ´ ep endance est caract´ eris ´ ee par le fait que deux v ar ia bles successiv es ne p euve n t prendre la m ˆ eme v aleur. En utilisan t une a ppro c he mark ov ienne, nous ´ etudions la distribution de la plus longue sous-suite croissan te con tigu ¨ e et nous d´ ev elopp ons un a lg orithme p our la calculer. Ce probl ` eme provien t de l’analyse de plusieurs proto coles auto-organisan ts p our les r ´ eseaux de capteurs sans fil ` a grande ´ ec helle, et nous montrons commen t nos r ´ esultats s’appliquen t ` a ce domaine. Mots-cl ´ es : Cha ˆ ınes de Mark o v, sous-suites croissan tes contigu ¨ es, auto- stabilisation, temps de con v ergence. Asc ending runs in de p endent uniformly di stribute d r ando m vari a bles 3 1 In tro duction Let X = ( X n ) n > 1 b e a sequence of iden tically distributed random v a riables on the set S = { 1 , . . . , m } . As in [8], w e define an ascending run as a contiguous and increasing subse - quence in the pro cess X . F or instance, with m = 5, among the 20 first following v alues of X : 23124342 31345123 4341, there are 8 ascending runs and the length of maximal ascending run is 4. More fo rmally , an ascending run of length ℓ > 1, starting at p osition k > 1, is a subseque nce ( X k , X k +1 , . . . , X k + ℓ − 1 ) suc h that X k − 1 > X k < X k +1 < · · · < X k + ℓ − 1 > X k + ℓ , where w e set X 0 = ∞ in order to av oid sp ecial cases at the b oundary . U nder the assumption that the distribution is discrete and the random v ar ia bles are independen t, sev eral authors ha v e studied the b ehaviour of the maximal ascending run, as w ell as the longest non-decreasing con tiguous subsequence. The main results concern the asymptotic b eha viour of these quan tities when the num b er of random v ariables tends to infinity , see for example [6] and [4] and the references therein. Note that these t w o notions coincide when the common distribution is con tin uous. In this case, the asy mptotic b eha viour is kno wn and do es not depend on the distribution, as show n in [6]. W e denote b y M n the length of the max imal a scending run among the first n ra ndom v ariables. The asymptotic behav iour of M n hardly de p ends on the common distribution of the random v ariables X k , k > 1. Some results ha v e b een established for the g eometric distribution in [10] where an equiv alen t o f t he la w of M n is pro vided and previously in [1] where the almost- sure con v ergence is studied, a s w ell as for P oisson distribution. In [9], the case of the uniform distribution on the set { 1 , . . . , s } is inv estigated. The au- thor considers the problem of the longest non-decreasing con tiguous subseque nce and giv es an equiv alen t of its law when n is large and s is fixed. The asymptotic equiv alen t o f E ( M n ) is also conjectured. In t his pap er, w e consider a sequence X = ( X n ) n > 1 of integer r a ndom v ariables on the set S = { 1 , . . . , m } , with m > 2. The ra ndom v ar ia ble X 1 is uniformly distributed on S and, fo r n > 2, X n is uniformly distributed on S with the constrain t X n 6 = X n − 1 . This pro cess may b e seen as the dra wing of balls, n um b ered from 1 to m in an urn where at eac h step the last ball dra wn is k ept o utside the urn. Th us we hav e, for ev ery i, j ∈ S and n > 1, P ( X 1 = i ) = 1 m and P ( X n = j | X n − 1 = i ) = 1 { i 6 = j } m − 1 . By induction ov er n and unconditioning, w e get, for ev ery n > 1 and i ∈ S , P ( X n = i ) = 1 m . Hence the random v ariables X n are uniformly distributed on S but a re no t indep enden t. Using a Mark ov c hain approa c h, w e study t he distribution of the maximal ascending run and w e dev elop an algorithm to compute it. This problem comes from the analysis of self-o r g anizing proto cols designed for lar ge-scale wireless sensor net w orks, and w e sho w ho w our results apply to this domain. The remainder of the pap er is or g anized as follows . In the next section, w e use a Mark ov c hain approac h to study the b ehavior o f the sequence of a scending runs in the pro cess X . In Section 3, w e a nalyze the hitting times of a n ascending run of fixed length and w e obtain the RR n ° 6443 4 N. Mitton, K. Par oux, B. Se ric ola & S. Tixeuil distribution of the maximal ascending M n o v er the n first random v ariables X 1 , . . . , X n using a Mark o v renew al argumen t. An a lgorithm to compute this distribution is dev elop ed in Section 4 and Section 5 is dev oted to the practical implications of this w ork in large-scale wireless sensor net w orks. 2 Asso ciated Mark o v c hain The pro cess X is ob viously a Marko v c hain on S . As o bserv ed in [10 ], w e can see the ascending runs as a discrete-time pro cess having t w o comp onents: the v alue take n b y the first elemen t of the a scending run and its length. W e denote this pro cess b y Y = ( V k , L k ) k > 1 , where V k is the v alue of the first elemen t of the k th ascending run and L k is its length. The state space of Y is a subset S 2 w e shall pr ecise now . Only the first ascending run can start with the v alue m . Indeed, as so on as k > 2, the random v ariable V k tak es its v alues in { 1 , . . . , m − 1 } . Moreo v er V 1 = X 1 = m implies that L 1 = 1. Th us, for any ℓ > 2, ( m, ℓ ) is not a state of Y whereas ( m, 1) is only an initial state that Y will nev er visit aga in. W e observ e also that if V k = 1 then necess arily L k > 2, which implies that (1 , 1) is not a state of Y . Moreov er V k = i implies that L k 6 m − i + 1. According to t his b eha viour, we hav e Y 1 ∈ E ∪ { ( m, 1) } and for k > 2, Y k ∈ E , where E = { ( i, ℓ ) | 1 6 i 6 m − 1 a nd 1 6 ℓ 6 m − i + 1 } \ { (1 , 1) } . W e define the 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 pr o c ess Y is a homo gene ous Markov chain with tr ansition pr ob ability matrix P , which entries ar e given for any ( i, ℓ ) ∈ E ∪ { ( m, 1) } and ( j, λ ) ∈ E b y P ( i,ℓ ) , ( j,λ ) = Φ ℓ ( i, j ) ϕ λ ( j ) ϕ ℓ ( i ) . Pro of. W e exploit the Mark o v pro p ert y o f X , rewriting ev en ts for Y as ev en ts for X . F or ev ery ( j, λ ) ∈ E and taking k > 1 then fo r an y ( v k , ℓ k ) , . . . , ( v 1 , ℓ 1 ) ∈ E ∪ { ( m, 1) } , w e denote b y A k the ev en t : A k = { Y k = ( v k , ℓ k ) , . . . , Y 1 = ( v 1 , ℓ 1 ) } . W e ha v e to che c k that P ( Y k +1 = ( j, λ ) | A k ) = P ( Y 2 = ( j, λ ) | Y 1 = ( v k , ℓ k )) . First, w e observ e that A 1 = { Y 1 = ( v 1 , ℓ 1 ) } = { X 1 = v 1 < · · · < X ℓ 1 > X ℓ 1 +1 } , INRIA Asc ending runs in de p endent uniformly di stribute d r ando m vari a bles 5 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 induction, w e obta in 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 a nd the fact that X is a homogeneous Mark o v c hain, w e get 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 )) . W e no w hav e to sho w 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, w e hav e 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 )) . W e hav e sho wn that Y is a ho mo g eneous Mark ov c hain ov er its state space. The en tries of matrix P are t hen given, fo r eve ry ( j, λ ) ∈ E and ( i, ℓ ) ∈ E ∪ { ( m, 1) } b y 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 ) , RR n ° 6443 6 N. Mitton, K. Par oux, B. Se ric ola & S. Tixeuil where the third equalit y follows f r o m the Marko v prop erty . W e giv e the expressions of ϕ λ ( j ) and Φ ℓ ( i, j ) for ev ery i, i, ℓ ∈ S in the fo llo wing lemma. Lemma 2. F or every i, j, ℓ ∈ S , we h ave 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 } . Pro of. F or ev ery i, j, ℓ ∈ S , it is easily c hec ked that Φ ℓ ( i, j ) = 0 if m < i + ℓ − 1. If m > i + ℓ − 1, w e hav e 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) W e in tro duce the sets G 1 ( i, j, ℓ, m ), G 2 ( i, j, ℓ, m ), G ( i, ℓ, m ) and H ( ℓ, m ) defined b y 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 w ell-kno wn, see for instance [5], that | G ( i, ℓ, m ) | =  m − i ℓ − 1  . Since | G 2 ( i, j, ℓ, m ) | = | G ( i, ℓ − 1 , j − 1) | , the first term in (4) can b e written 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 , INRIA Asc ending runs in de p endent uniformly di stribute d r ando m vari a bles 7 The second term is giv en, fo r j > i + ℓ − 1, b y 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 t w o 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 , whic h completes t he pro of of the first relation. The second r elat io n follow s from expression (3) b y 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 third relation follows from definition (2 ) by writing ϕ ℓ ( i ) = ψ ℓ ( i ) − ψ ℓ +1 ( i ). Note that the matrix Φ defined b y Φ = m X ℓ =1 Φ ℓ is ob viously a sto chastic matrix, which means that, f or every i = 1 , . . . , m , w e hav e 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 asc e nding run F or ev ery r = 1 , . . . , m , we denote b y T r the hitting time of an ascending run of length at least equal to r . More formally , we hav e T r = inf { k > r ; X k − r +1 < · · · < X k } . It is easy to c hec k that w e ha ve T 1 = 1 and T r > r . The distribution of T r is giv en by t he follo wing theorem. RR n ° 6443 8 N. Mitton, K. Par oux, B. Se ric ola & S. Tixeuil Theorem 3. F or 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) Pro of. Since T r > r , we hav e, for 1 6 n 6 r − 1 , P ( T r 6 n | V 1 = i ) = 0 Let us assume from now that n > r . Since L 1 > r implies that T r = r , w e get P ( T r 6 n, L 1 > r | V 1 = i ) = P ( L 1 > r | V 1 = i ) = ψ r ( i ) . (6) W e in tro duce the random v aria ble T ( p ) r defined by hitting time o f an ascending run length a t least equal to r when coun ting fro m p osition p . Thu s we hav e T ( p ) r = inf { k > r ; X p + k − r < · · · < X p + k − 1 } . W e then hav e T r = T (1) r . Moreo v er, L 1 = ℓ < r implies that T r = T ( L 1 +1) r + ℓ , which 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 ( 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 ( L 1 +1) r 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) where the fifth equality follo ws from the Mark ov prop erty and the last one fr o m the homogeneity of Y . Putting t o gether relations (6) and (7), w e 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 ) . INRIA Asc ending runs in de p endent uniformly di stribute d r ando m vari a bles 9 F or ev ery n > 1, we define M n as the maximal ascending run length o v er the n first v alues X 1 , . . . , X n . W e hav e 1 6 M n 6 m ∧ n and M n > r ⇐ ⇒ T r 6 n, whic 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 Algorith m F or r = 1 , . . . , m , we denote b y ψ r the column v ector o f dimension m whic h i th entry is ψ r ( i ). F or r = 1 , . . . , m , n > 1 and h = 1 , . . . , n , w e denote b y W r,h the column v ector of dimension m whic h i th en try is defined b y W h,r ( i ) = P ( T r 6 h | V 1 = i ) = P ( M h > r | V 1 = i ) , and w e denote by 1 the column v ector of dimension m with all en tries equal to 1. An algorithm for the computation of the distribution and the exp ectatio n of M n is giv en in T able 1. input : m , n output : E ( M h ) for h = 1 , . . . , n . for ℓ = 1 to m do Compute the matr ix Φ ℓ endfor for r = 1 t o m do Compute the column v ectors ψ r endfor for h = 1 t o n do W h, 1 = 1 endfor for r = 2 t o 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 t o n do E ( M h ) = 1 m m ∧ h X r =1 1 t W h,r endfor T able 1: Algorithm for the distribution and exp ectation computation o f M n . 5 Application to w ireless net w o rks : fa st self-or g anization Our analysis has imp ortant implications in forecast large-scale wireless net w orks. In those net w orks, the num b er of mac hines in v olv ed and the likeline ss of fault o ccurrences prev ents an y cen tralized planification. Instead, distributed self-org anization m ust b e designed to enable prop er functioning of the net w ork. A useful techniq ue to pro vide self-organizatio n is self- stabilization [2, 3]. Self-stabilization is a vers atile tec hnique that can make a wireless net w ork withstand an y kind of fault and reconfigurat io n. A common dra wbac k with self-stabilizing proto cols is that they w ere not designed to handle prop erly la rge-scale net w orks, as the stabilizing time ( t he maxim um a moun t of time needed to RR n ° 6443 10 N. Mitton, K. Par oux, B. Se ric ola & S. Tixeuil reco v er from any p ossible disaster) could b e related to the actual size of the netw ork. In many cases, this high complexit y w as due to the fact that net w ork-wide unique identifiers a re used to ar bit r a te symmetric situatio ns [13 ]. How ev er, t here exists a n um b er of problems app earing in wireless net w orks that need only lo cally unique identifiers . Mo deling the net w ork as a graph where no des represen t wireless entities and where edges represen t the abilit y to comm unicate b et w een t w o en tities ( b ecause eac h is within the tr a ns- mission range o f the other), a lo cal coloring of the no des at distance d ( i.e. ha ving t w o no des at distance d or less assigned a distinct color) can b e enough to solv e a wide range of problems. F or example, lo cal coloring at distance 3 can b e used to assign TDMA time slots in an adaptive manner [7], and lo cal coloring at distance 2 has succ essiv ely b een used to self-organize a wireles s net w ork into more manageable clusters [12]. In the p erfor ma nce analysis of b oth sc hemes, it app ears that the ov erall stabilization time is balanced b y a tradeoff b et w een the coloring time itself and the stabilization time of the proto col using the coloring (denoted in the following as the client proto col). In bo t h cases (TDMA assignmen t and clustering), the stabilization time of t he c lien t pro to col is related to the heigh t of the directed acyclic graph induced b y the colors. This DA G is obtained b y orien ting an edge fro m the no de with the highest colo r to t he neighbor with the lo w est color. As a result, t he ov erall heigh t of this DA G is equal to the long est strictly a scending c hain of colors across neighboring no des. Of course, a larger set o f colors leads to a shorter stabilization time for the coloring (due to the higher c hance of pick ing a fresh color), but yields to a p otential higher D A G, tha t could delay the stabilizatio n time of the clien t proto col. In [11], the stabilization time of the coloring proto col was theoretically analyzed while the stabilization time of a particular clien t proto col (the clustering sche me of [12]) w as only studied b y sim ula tion. The analysis p erformed in this pap er g ives a theoretical upp er b ound on the stabilization time of all clien t proto cols that use a coloring s c heme as an underlying basis. T ogether with the results of [11], our study constitutes a comprehensiv e analysis of the ov erall stabilization time of a class of self-stabilizing pro t o cols used for the self-organizatio n of wireless sensor net w orks. In the remaining of the section, w e provide quan titative results regarding the relativ e imp ort a nce o f the num b er of used colors with resp ect to other net w ork parameters. Figure 1 sho ws the exp ected length of the maximal a scending run ov er a n -no de chain for differen t v alues of m . Results sho w sev eral in teresting b eha viors. Indeed, self-organization pro to cols relying on a coloring pro cess achie v e b etter stabilizatio n time when the exp ected length of maximal as- cending run is short but a coloring pro cess stabilizes faster when the n um b er o f colors is high [11]. Figure 1 clearly sho ws that ev en if the num b er o f colors is high compared to n ( n << m ), the exp ected length of maximal ascending run remains short, whic h is a great adv a n tage. Moreov er, ev en if the num b er of no des increases, the exp ected length of the maximal a scending run remains short and increases very slow ly . This observ ation demonstrates the scalability prop erties of a proto col relying on a lo cal coloring pro cess since its stabilization t ime is directly link ed to the length of this ascending run [11]. Figure 2 sho ws the exp ected length of maximal as cending run ov er a n -node chain for differen t v alues of n . Results shows that for a fixed n umber of no des n , the exp ected length of the maximal ascending r un con v erges to a finite v alue, dep ending of n . This implies that using a large n um b er o f colors do es not impact the stabilization time of the clien t algorithm. INRIA Asc ending runs in de p endent uniformly di stribute d r ando m vari a bles 11 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 Figure 1 : Exp ected length o f the maximal ascending run as a function of the n um b er o f no des. 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 Figure 2 : Expected length of the maximal ascending run as a function of the n um b er of colors. RR n ° 6443 12 N. Mitton, K. Par oux, B. Se ric ola & S. Tixeuil References [1] E . Csaki and A. F o ldes. On the length of the longest monotone blo c k. Stud. Sci. Math. Hungarian , 3 1:35–46, 1996. [2] E . W. D ij kstra. Self-stabilizing systems in spite of distributed control. Commun. A CM , 17(11):643 –644, 1974. [3] S . Do lev. Self Stabilization . MIT Press, 2000. [4] S . Eryilmaz. A note on runs of geometrically distributed ra ndom v ariables. Discr ete Mathematics , 3 06:1765– 1 770, 2006. [5] D. F oat a and A. F uc hs. C alcul des pr ob abilit´ es . Mass on, 1996. [6] A . N. F rolo v and A. I. Martik ainen. On the length of the longest increasing run in Rd. Statistics and Pr ob ability L etters , 4 1(2):153–1 61, 1999. [7] T . Herman and S. Tixeuil. A distributed TDMA slot assignmen t a lgorithm for wireless sensor netw orks. In Pr o c e e ding s of the F irst Workshop on Algorithmic Asp e cts of Wir ele ss Sensor Networks (A lgoSen s ors’2004) , num b er 3121 in Lecture Notes in Computer Science, pages 45–58, T urku, Finland, July 2004. Springer-V erlag. [8] G. Louc hard. 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