Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems

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(1)Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems Emmanuelle Anceaume, François Castella, Romaric Ludinard, Bruno Sericola. To cite this version: Emmanuelle Anceaume, François Castella, Romaric Ludinard, Bruno Sericola. Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems. [Research Report] 2010, pp.29. �inria-00485667�. HAL Id: inria-00485667 https://hal.inria.fr/inria-00485667 Submitted on 25 May 2010. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems Emmanuelle Anceaume — François Castella — Romaric Ludinard — Bruno Sericola. N° 1953. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--1953--FR+ENG. Mai 2010.

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(18) . Centre de recherche INRIA Rennes – Bretagne Atlantique IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71.

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(25) . # $ . 3 + . + - - ;&℄ - / + . 3 - . / , 0 = - - - - / # . - - / 0 - n + . + - / +. + + - - n . 0 . / > 0 1 2 Θn 0 - n . + / ,

(26) . - 0 + / ,

(27) 0 n - . + 2 0

(28) - Θn - + - n/ ! - - + 0 0 0 / - + 1 -0 / #

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(30) - + . - n . 0 / , +.

(31) - - 2 Θn 0 - n . + / # ' 0 0 0

(32) 0 0 - Θn

(33) / # ? 0 - + + - 1 - .

(34) - Θn n + 2 / , 0 0

(35) - - Θn - + - n/ & - 0 + 0 - + /. % , + . X = {Xk , k ≥ 0} 0 2 S - - B . + a/

(36) P - . P =. Q v 0 1. ,. 0 Q

(37) - |B| × |B| + 0 - B / # 0 v 0 |B| + - + / , B // {X0 ∈ B} = 1 0 α 0 - |B| + .

(38) ?.

(39)

(40) . // - i ∈ B . αi = È{X0 = i}.. , Θ1 B - + + 2 0 + a / , Θ1 = inf{k ≥ 0 | Xk = a}.. - - Θ1 - ;$)℄ ;@℄ È{Θ1 > k} = È{Xk ∈ B} = αQk ½, A$B 0 ½ - |B| 0 1. I

(41) - + / - B

(42) I − Q

(43) - Θ1 + E(Θ1 ) = α(I − Q)−1 ½. A)B 4 0 - n ≥ 1 n . X (1) , . . . , X (n) X // 0 S

(44) P α/ n . . + - π(n) = (p1,n , . . . , pn,n )/ > n . 0 0 . Y = {Yk , k ≥ 0} -0 / - Y S n (1) (n) Yk = (Xk , . . . , Xk )/ ! . Y . - . X (1) , . . . , X (n) + / . . 0 - π(n) 0 . X (ℓ) . 0 pℓ,n / , 0 - + - ℓ = 1, . . . , n 0 0 < pℓ,n < 1/

(45) - Y 0 0 R + - (i1 , . . . , in ) (j1 , . . . , jn ) ∈ S n . R((i1 , . . . , in ), (j1 , . . . , jn )) =. ⎧ n ⎪ ⎪ ⎪ pℓ,n Piℓ ,iℓ ⎪ ⎪ ⎪ ⎪ ⎨ ℓ=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩. pℓ,n Piℓ ,jℓ. - (i1 , . . . , in ) = (j1 , . . . , jn ) - ∃! ℓ iℓ = jℓ. 0 / # . Y - (i1 , . . . , in ) (i1 , . . . , in ) - . X (ℓ) . iℓ A + B (i1 , . . . , iℓ − 1, jℓ , iℓ + 1, . . . , in ) - . X (ℓ) . - iℓ jℓ A + B/ 0 - . Y

(46) (a, . . . , a) 0 +/ , β - Y 0 . X0(ℓ)9 β(j1 , . . . , jn ) =. 0. n. È{X0(ℓ) = jℓ }.. ℓ=1 .

(47) &.

(48) . β(j1 , . . . , jn ) =. n. {X0 = jℓ } =. ℓ=1. n. αjℓ .. A'B. ℓ=1. -0+ +

(49) - - Yk / > . ℓ ≥ 1 0 Sk,ℓ 2 . k≥0. Sk,ℓ = {k = (k1 , . . . , kℓ ) ∈ ℓ | k1 + · · · + kℓ = k}..

(50)

(51) k ≥ 0 n ≥ 1 (j1 , . . . , jn ) ∈ S n . . {Yk = (j1 , . . . , jn )} =. . k∈Sk,n. n. k! pkr {Xkr = jr }. k1 ! · · · kn ! r=1 r,n. A?B. - k/ - k = 0 - A'B/ A?B - + k − 1/ , . {Yk = (j1 , . . . , jn )} R((i1 , . . . , in ), (j1 , . . . , jn )){Yk−1 = (i1 , . . . , in )}. . =. (i1 ,...,in )∈S n. =. n . ph,n Pjh ,jh {Yk−1 = (j1 , . . . , jn )}. h=1. +. n . h=1. =. n . . ph,n . ph,n. h=1. Pih ,jh {Yk−1 = (j1 , . . . , jh−1 , ih , jh+1 , . . . , jn )}. ih ∈S\{jh }. Pih ,jh {Yk−1 = (j1 , . . . , jh−1 , ih , jh+1 , . . . , jn )}.. ih ∈S. C + 0 +. {Yk = (j1 , . . . , jn )} =. n . ph,n. ih ∈S. h=1. ×. . n. Pih ,jh. . k∈Sk−1,n. (k − 1)! kh p {Xkh = ih } k1 ! · · · kn ! h,n. r pkr,n {Xkr = jr }. r=1,r =h. =. n . . h=1 k∈Sk−1,n. (k − 1)! kh +1 p {Xkh+1 = jh } k1 ! · · · kn ! h,n. n. r pkr,n {Xkr = jr }.. r=1,r =h. > h = 1, . . . , n 0 Uh,k - Sk,n 2 Uh,k = {k ∈ Sk,n | km ≤ k − 1 0 m = h}. + kh := kh − 1 . {Yk = (j1 , . . . , jn )} =. n n kh (k − 1)! pkr {Xkr = jr }. k1 ! · · · kn ! r=1 r,n. h=1 k∈Uh,k. .

(52) D.

(53)

(54) . # + Vk - Sk,n 2 Vk = {k ∈ Sk,n | km ≤ k − 1. 0 0 . - m = 1, . . . , n},. Uh,k = Vk ∪ {uh },. 0 uh = (0, . . . , 0, k, 0, . . . , 0) 0 - n 0 h k 0/ , +. {Yk = (j1 , . . . , jn )}. n n kh (k − 1)! kr p {Xkr = jr } k1 ! · · · kn ! r=1 r,n. =. k∈Vk h=1. +. n . pkh,n {Xk = jh }. h=1. > k ∈ Vk 0 . n. {X0 = jr }.. r=1,r =h. n n k! kh (k − 1)! (k − 1)! = , kh = k1 ! · · · kn ! k1 ! · · · kn ! k1 ! · · · kn !. h=1. 0 . h=1. {Yk = (j1 , . . . , jn )}. =. . k∈Vk. +. n. k! pkr {Xkr = jr } k1 ! · · · kn ! r=1 r,n. n . h=1. =. . k∈Sk,n. pkh,n {Xk = jh }. n. {X0 = jr }. r=1,r =h n. k! pkr {Xkr = jr }. k1 ! · · · kn ! r=1 r,n. / C + 0 -0+ 0 + - . X (h) k - + Y / .

(55)

(56) h = 1, . . . , n k ≥ 0 j ∈ S . . (h) {Xk. k k ℓ = j} = p (1 − ph,n )k−ℓ {Xℓ = j}. ℓ h,n ℓ=0. 3 Xk(h) h + ph,n - 0 + 0 . - - h = n/ C + $ 0 . .

(57) @.

(58) . È{Xk(n) = j} = È{Yk ∈ S × · · · × S × {j}} =. . ···. j1 ∈S. =. . jn−1 ∈S k∈Sk,n. n−1. k! n pkr È{Xkr = jr }pkn,n È{Xkn = j} k1 ! · · · kn ! r=1 r,n. n−1. k! n r pkn,n È {Xkn = j} pkr,n k1 ! · · · kn ! r=1. . k∈Sk,n. =. . k k pkn (1 − pn,n )k−kn È{Xkn = j} kn n,n. kn =0. ×. . k∈Sk−kn ,n−1. =. k . kn =0. kr n−1 pr,n (k − kn )! k1 ! · · · kn−1 ! r=1 1 − pn,n. k pkn (1 − pn,n )k−kn È{Xkn = j}, kn n,n. . 0 -/ -0+ - 2 Θn. 0 - n . X (1), . . . , X (n) + / - Θn 2. (r) Θn = inf{k ≥ 0 | ∃r Xk = a}.

(59)

(60) k ≥ 0 n ≥ 1 . È{Θn > k} =. . k∈Sk,n. . n. k! pkr αQkr ½. k1 ! · · · kn ! r=1 r,n. A&B. > k ≥ 0 n ≥ 1 0 (1). (n). Θn > k ⇐⇒ Xk ∈ B, . . . , Xk. ∈ B.. , - $ A$B/. È{Θn > k}. = =. È{Xk(1) ∈ B, . . . , Xk(n) ∈ B} È{Yk = (j1 , . . . , jn )} (j1 ,...,jn )∈B n. =. . k∈Sk,n. =. . k∈Sk,n. =. . k∈Sk,n. k! k1 ! · · · kn !. . n. r pkr,n È{Xkr = jr }. (j1 ,...,jn )∈B n r=1. n. k! pkr È{Xkr ∈ B} k1 ! · · · kn ! r=1 r,n n. k! pkr αQkr ½, k1 ! · · · kn ! r=1 r,n. 0 -/ 3

(61) - - È{Θn > k} + A&B

(62) / ! -0+ / .

(63) E. &.

(64)

(65) .

(66) . , - -

(67) - Θn / , $ - + . Y .

(68) (a, . . . , a) 0 +/

(69) E(Θn ) 2 + - n ≥ 1 E(Θn ) =. ∞ . È{Θn > k}.. ADB. k=0. -0+ +

(70) - Θn /

(71)

(72) k ≥ 0 n ≥ 2 . È{Θn > k} =. k k ℓ=0. ℓ. k−ℓ. pℓn,n (1 − pn,n ). αQℓ ½È{Θn−1 > k − ℓ},. A@B.

(73)

(74)

(75) π(n − 1) = (p1,n−1 , . . . , pn−1,n−1 ) Θn−1

(76) r = 1, . . . , n − 1 pr,n−1 = . pr,n . 1 − pn,n. > k ≥ 0 n ≥ 2 0 - 3 '. È{Θn > k} =. n. k! k (pr,n ) r αQkr ½ k1 ! · · · kn ! r=1. . k∈Sk,n. =. k . kn. (pn,n ). kn =0. =. k . kn =0. =. k . kn =0. ×. =. kn =0. ×. . =. kn =0. k∈Sk−kn ,n−1. n−1. k! k (pr,n ) r αQkr ½ k1 ! · · · kn−1 ! r=1. . k∈Sk−kn ,n−1. n−1 (k − kn )! (pr,n )kr αQkr ½ k1 ! · · · kn−1 ! r=1. k (pn,n )kn (1 − pn,n )k−kn αQkn ½ kn. . . . kr n−1 (k − kn )! pr,n αQkr ½ k1 ! · · · kn−1 ! r=1 1 − pn,n. k k k−k (pn,n ) n (1 − pn,n ) n αQkn ½ kn. . . k∈Sk−kn ,n−1 k . . k (pn,n )kn αQkn ½ kn. . k∈Sk−kn ,n−1 k . αQkn ½ kn !. n−1 (k − kn )! k (pr,n−1 ) r αQkr ½ k1 ! · · · kn−1 ! r=1. k k k−k (pn,n ) n (1 − pn,n ) n αQkn ½È{Θn−1 > k − kn }, kn. . .

(77) %.

(78) . 0 -/ 0 - È{Θn > k} +. 0

(79) / , 0 -0+ 0 K . - i = 1, . . . , n È{Θi > k} ≤ ε - k ≥ K 0 ε / , - + 0 pr,n . p1,n ≥ p2,n ≥ · · · ≥ pn,n .. -0+ /. !.

(80)

(81) i = 1, . . . , n − 1 k ≥ 0 k. (pi,i Q + (1 − pi,i )I). ½ ≤ (pi+1,i+1 Q + (1 − pi+1,i+1 )I)k ½,.

(82)

(83)

(84) ! . - k = 0/ > ? 0 . pi,i − pi+1,i+1 =. pi,i+1 − pi+1,i+1 + p2i+1,i+1 pi,i+1 − pi+1,i+1 = ≥ 0. 1 − pi+1,i+1 1 − pi+1,i+1.

(85) Q + 0 Q½ − ½ ≤ 0 0 - F 0 + / # + Qi = pi,iQ + (1 − pi,i)I 0 + Qi ½ − Qi+1 ½ = (pi,i − pi+1,i+1 )(Q½ − ½) ≤ 0,. 0 - k = 1/ 0 - + k − 1 0 k ≥ 2/ Qi Qi+1 0 (Qi )k ½ = Qi (Qi )k−1 ½ ≤ Qi (Qi+1 )k−1 ½ = (Qi+1 )k−1 Qi ½ ≤ (Qi+1 )k ½.. -/ ".

(86)

(87) n ≥ 1

(88)

(89) ε ∈ (0, 1) max. i=1,...,n.

(90) K1 = inf. . È{Θi > k} ≤ ε

(91)

(92) k ≥ K1,. k

(93) k. ℓ k−ℓ ℓ k≥0 p (1 − pn,n ) αQ ½ ≤ ε .. ℓ n,n ℓ=0.

(94) $*.

(95)

(96) . . > i = 1, . . . , n 0 . È{Θi > k}. =. ≤. k k ℓ p (1 − pi,i )k−ℓ αQℓ ½È{Θi−1 > k − ℓ} ℓ i,i ℓ=0 k k ℓ p (1 − pi,i )k−ℓ αQℓ ½ ℓ i,i ℓ=0. = ≤ =. α (pi,i Q + (1 − pi,i )I). k. ½. α (pn,n Q + (1 − pn,n )I) ½ k k ℓ p (1 − pn,n )k−ℓ αQℓ ½. ℓ n,n. A- 4 &B. k. ℓ=0. 5

(97) pn,nQ + (1 − pn,n )I / / (pn,n Q + (1 − pn,n )I)½ ≤ ½ 0 - / α (pn,n Q + (1 − pn,n )I)k ½ + 0 k lim α (pn,n Q + (1 − pn,n )I). k. k−→∞. ½ = 0,. - 2

(98) ε ∈ (0, 1) 2 - + K1 0 - i = 1, . . . , n È{Θi > k} ≤ ε, - k ≥ K1 , 0 -/ . - - -

(99) E(Θi ) - 0 - ADB / #.

(100)

(101) n ≥ 1

(102)

(103) ε ∈ (0, 1) 0 ≤ max. i=1,...,n. E(Θi ) −.

(104) K2 = inf . K 2 −1. È{Θi > k}. k=0. . ≤ ε,.

(105) k 1 k ℓ. k≥0 p (1 − pn,n )k−ℓ α(I − Q)−1 Qℓ ½ ≤ ε . pn,n ℓ n,n ℓ=0. , . ei = E(Θi ) −. K 2 −1 . È{Θi > k}.. k=0. .

(106) $$.

(107) . , - i = 1, . . . , n ei. ∞ . =. È{Θi > k}. k=K2 ∞ k k ℓ p (1 − pi,i )k−ℓ αQℓ ½È{Θi−1 > k − ℓ} ℓ i,i. =. k=K2 ℓ=0. ∞ k k ℓ p (1 − pi,i )k−ℓ αQℓ ½ ℓ i,i. ≤ = ≤. k=K2 ℓ=0 ∞ . k. α (pi,i Q + (1 − pi,i )I). k=K2 ∞ . ½. k. α (pn,n Q + (1 − pn,n )I). ½. A - 4 &B. k=K2. = = =. −1. K2. α (I − (pn,n Q + (1 − pn,n ) I)) (pn,n Q + (1 − pn,n ) I) 1 K α(I − Q)−1 (pn,n Q + (1 − pn,n ) I) 2 ½ pn,n K2 K2 ℓ 1 pn,n (1 − pn,n )K2 −ℓ α(I − Q)−1 Qℓ ½ pn,n ℓ. ½. ℓ=0. ≤. ε. 2 - + K2 .. 0 maxi=1,...,n ei ≤ ε/ - - Θn + A$B A@B/ #- - π(n) - // pr,n = 1/n - r = 1, . . . , n . - π(i) - - i ≤ n/ # 0 . - Θn - Θi9 - i ≤ n + + + D/ #-

(108) Θn 0 + + @/. '

(109) - Θn 0 n +/ + - + 0 .

(110) / > - -0+ - ?/ $.

(111)

(112) n ≥ 1 E(Θn ) ≤. E(Θn−1 ) , 1 − pn,n. E(Θn ) ≤. . E(Θ1 ) . p1,n.

(113) $).

(114)

(115) . . C + ? 0 +. E(Θn ) =. ∞ . È{Θn > k}. k=0. =. = = = ≤. ∞ k k ℓ k−ℓ αQℓ ½È{Θn−1 > k − ℓ} p (1 − pn,n ) ℓ n,n k=0 ℓ=0 ∞ k k k−ℓ ℓ (1 − pn,n ) αQk−ℓ ½È{Θn−1 > ℓ} p ℓ n,n k=0 ℓ=0 ∞ ∞ k k−ℓ k−ℓ (1 − pn,n )ℓ È{Θn−1 > ℓ} p αQ ½ ℓ n,n ℓ=0 k=ℓ ∞ ∞ k+ℓ k ℓ (1 − pn,n ) È{Θn−1 > ℓ} pn,n αQk ½ ℓ ℓ=0 k=0 ∞ ∞ k+ℓ k ℓ (1 − pn,n ) È{Θn−1 > ℓ} pn,n ℓ ℓ=0. =. ∞ . k=0. ℓ. (1 − pn,n ). È{Θn−1 > ℓ} (1 − pn,n )−(ℓ+1). ℓ=0. =. E(Θn−1 ) . 1 − pn,n. 0 + E(Θn ) ≤. E(Θn−1 ) E(Θn−2 ) E(Θn−2 ) = ≤ 1 − pn,n (1 − pn,n )(1 − pn−1,n−1 ) 1 − pn,n − pn−1,n. E(Θn ) ≤. E(Θ1 ) . p1,n. 0 -/ , π(n) - // 0 pi,n = 1/n 0 E(Θn ) ≤. nE(Θn−1 ) ≤ nE(Θ1 ). n−1. , -0+ -/ > - Fn (x) 2 Fn (x) =. n ≥ 1. . x ∈. Ê 0. ∞ xk È{Θn > k}. k! k=0. - Fn 2 - x ∈ Ê

(116)

(117) + -0+ / %

(118)

(119) n ≥ 1 x ∈ Ê Fn (x) =. n. αeQxpℓ,n ½,. AEB. ℓ=1 .

(120) $'.

(121)

(122)

(123) k ∈ Æ. È{Θn > k} = Fn(k) (0),. A%B.

(124) Fn(k) k

(125) Fn

(126) x! . Fn (x). > A@B 0 + + - È{Θ1 > ℓ} = αQℓ ½ = =. ∞ k xk k ℓ p (1 − pn,n )k−ℓ È{Θ1 > ℓ}È{Θn−1 > k − ℓ} ℓ n,n k! k=0 ∞ ℓ=0. =. ∞. xk−ℓ x ℓ k−ℓ pn,n È{Θ1 > ℓ} (1 − pn,n ) È{Θn−1 > k − ℓ} ℓ! (k − ℓ)! k=ℓ. ∞ xℓ ℓ=0. =. ℓ=0. ℓ. ℓ!. pℓn,n È{Θ1 > ℓ}. ∞ xk k=0. k!. (1 − pn,n )k È{Θn−1 > k}. F1 (xpn,n )Fn−1 (x(1 − pn,n )).. - i = 1, . . . , n Fn (x). = = =. F1 (xpn,n )F1 (xpn−1,n )Fn−2 (x(1 − pn,n − pn−1,n )) ·· · n. F1 (xpℓ,n ) Fn−i (x(1 − pn,n − . . . − pn−i+1,n )), ℓ=n−i+1. 0 0 . pr,n−1 =. , - i = n Fn (x) =. pr,n . 1 − pn,n. n. F1 (xpℓ,n ).. ℓ=1. 50 F1 (x) =. ∞ xk k=0. 0 . È{Θ1 > k} = k! Fn (x) =. n. ∞ xk. k=0. k!. αQk ½ = αeQx ½,. αeQxpℓ,n ½.. ℓ=1. > - 0 0 2 - - Fn. Fn(h) (x) =. ∞ ∞ xk−h xk È {Θn > k} = È{Θn > k + h}, (k − h)! k!. k=h. k=0. 0 + .+ x = 0/ 0 È{Θn > 0} = 1

(127) - n ≥ 1 0 . È{Θn > 1} = αQ½.. .

(128) $?.

(129)

(130) . .

(131) . , 0 - π(n) -/ -0+ + - Θn / & " pℓ,n = 1/n

(132)

(133) ℓ = 1, . . . , n

(134)

(135) x ∈ Ê lim Fn (x) = eαQ½x .. n−→∞. . #- pℓ,n = 1/n - ℓ = 1, . . . , n 0 - AEB n Fn (x) = αeQx/n ½ ,. 0 0 . Fn (x) = e. „ « Qx/n n ln αe ½. = en(αQ½x/n+ε(1/n)/n)) = eαQ½x+ε(1/n) ,. 0 ε - -+ limn−→∞ ε(1/n) = 0/ -/ ++ - È{Θn > k} 0 n + 2 (αQ½)k / ! - pℓ,n = 1/n - ℓ = 1, . . . , n 0 + - AEB A%B. È{Θn > k} =. Fn(k) (0). + - 0 n ≥ 4 Fn(1) (x) Fn(2) (x). Fn(3) (x). Fn(4) (x). n dk αeQx/n ½ =. dxk. . x=0. n−1 αeQx/n ½ αQeQx/n ½, 2 n − 1 Qx/n n−2 αe αQeQx/n ½ = ½ n 1 Qx/n n−1 + αe ½ αQ2 eQx/n ½, n 3 (n − 1)(n − 2) Qx/n n−3 Qx/n αe αQe = ½ ½ n2 3(n − 1) Qx/n n−2 αe + ½ αQeQx/n ½αQ2 eQx/n ½ n2 n−1 1 + 2 αeQx/n ½ αQ3 eQx/n ½, n 4 (n − 1)(n − 2)(n − 3) Qx/n n−4 Qx/n αe αQe = ½ ½ n3 2 6(n − 1)(n − 2) Qx/n n−3 Qx/n + αe ½ αQe ½ αQ2 eQx/n ½ n3 3(n − 1) Qx/n n−2 2 Qx/n 2 αe αQ e + ½ ½ n3 4(n − 1) Qx/n n−2 αe + ½ αQeQx/n ½αQ3 eQx/n ½ n3 n−1 1 + 3 αeQx/n ½ αQ4 eQx/n ½, n =. . .

(136) $&.

(137) . - x = 0. È{Θn > 1} È{Θn > 2}. = =. È{Θn > 3}. =. È{Θn > 4}. =. αQ½, n−1 1 2 (αQ½) + αQ2 ½, n n (n − 1)(n − 2) 3(n − 1) 1 (αQ½)3 + αQ½αQ2 ½ + 2 αQ3 ½, n2 n2 n (n − 1)(n − 2)(n − 3) 6(n − 1)(n − 2) 4 2 (αQ½) + (αQ½) αQ2 ½ 3 n n3 3(n − 1) 2 2 4(n − 1) 1 + αQ ½ + αQ½αQ3 ½ + 3 αQ4 ½. 3 3 n n n. !

(138)

(139) - Fn(k) (x) + > 7 " G - - ;'℄ ;$?℄/ # f g

(140) . '() *+ ,- .. . mℓ k k (ℓ) . g (x) dk f (j) (g(x)) j! f (g(x)) = k! , dxk j! m1 ! · · · mk ! ℓ! j=1 m∈Tj,k. ℓ=1.

(141) Tj,k Tj,k =. k . m = (m1 , . . . , mk ) mℓ = j. . ℓ=1. .+ g(x) = αeQx/n ½ f (x) = xn 0 + f (j) (x) =. n! xn−j 1{j≤n} (n − j)!. k . ℓmℓ = k. ℓ=1.

(142). .. A$*B. g(ℓ) (x) = n1ℓ αQℓ eQx/n ½,. 0 - n ≥ 1 k ≥ 1 Fn (x) = f (g(x)) Fn(k) (x) =. mℓ k∧n k . αQℓ eQx/n ½ k! n Q nx n−j j! ½ . αe nk j=1 j m1 ! · · · mk ! ℓ! m∈Tj,k. ℓ=1. .+ x = 0 0 . È{Θn > k} = nk!k. k∧n j=1. n j. . m∈Tj,k. mℓ k . αQℓ ½ j! . m1 ! · · · mk ! ℓ!. A$$B. ℓ=1. 5 .+ g(x) = ex/n f (x) = xn 0 f (g(x)) = ex - n ≥ 1 0 k∧n k! n j! = 1. k m 1 n j=1 j m1 !(1!) m2 !(2!)m2 · · · mk !(k!)mk m∈Tj,k. + - È{Θn > k} + -0+ / . A$)B.

(143) $D.

(144)

(145) . . . " pℓ,n.

(146)

(147) . = 1/n. . k ≥ 0. . k. lim E(Θn ) =. n−→∞.

(148)

(149) . {Θn > k} = (αQ). lim. n−→∞. . ℓ = 1, . . . , n. ⎧ ⎨. 1 1 − αQ ⎩ ∞. αQ < 1. . αQ = 1.. ,

(150) 2 - - Fn 0

(151) . / > z ∈ 0 . ∞ zk Fn (z) = {Θn > k}, k! k=0. 0 C = {z ∈ | |z| = 1}/ - Fn (z) +. 0 - p ≥ 0 . C. Fn (z) dz z p+1. 2π. . =. i. =. 0 ∞ k=0. =. . . 2π. 2iπ. iei(k−p)η dη =. 0. , - p ≥ 0 {Θn > p} =. Fn (eiη )e−ipη dη {Θn > k} k!. . 2π. iei(k−p)η dη. 0. {Θn > p} , p! . -. 0 2iπ. p! 2iπ. . C. k = p k = p.. Fn (z) dz. z p+1. , 0 $* Fn (z) + 0 z eαQ½z 0 n + 2 / 0 |Fn (z)| ≤ e|z|. . . C. e|z| |dz| = e |z|p+1. . |dz| = e. C. - + 0 + lim. n−→∞. {Θn > p}. = =. p! 2iπ. . C. . 2π. dη = 2eπ.. 0. eαQ½z dz z p+1. ∞ p! (αQ)k 2π i(k−p)η ie dη 2iπ k! 0 k=0 p. = (αQ) ,. 0 2 - -/. .

(152)

(153) . $@. > 0 . E(Θn ) =. ∞ . È{Θn > k}.. k=0. , 2 0 αQ½ = 1/ > - - 0 - k ≥ 0 lim. n−→∞. È{Θn > k} = 1.. C + > 9 - 0 + lim inf n−→∞. ∞ . È{Θn > k} ≥. k=0. ∞ . k=0. lim. n−→∞. È{Θn > k} = ∞,. limn−→∞ E(Θn ) = ∞ 0 - - 0 αQ½ = 1/ , 0 0 αQ½ < 1/ C + A$$B A$)B 0 . È{Θn > k} =. ≤ ≤. k∧n k ℓmℓ. k! n j! ℓ 1/ℓ (αQ ½ ) nk j=1 j m1 !(1!)m1 m2 !(2!)m2 · · · mk !(k!)mk m∈Tj,k ℓ=1 k max (αQℓ ½)1/ℓ ℓ=1,...,k. k sup(αQℓ ½)1/ℓ . ℓ≥1. #- 0 0 supℓ≥1 (αQℓ ½)1/ℓ < 1 -0 + + - / 4 0 supℓ≥1 (αQℓ ½)1/ℓ < 1/ > - αQ½ < 1 0 αQℓ ½ < 1 - ℓ ≥ 1 ℓ 1/ℓ αQ ½ < 1 - ℓ ≥ 1.. A$'B. , 2 - 2 -

(154) M ||M || = sup i. |Mi,j |.. j. , ||α|| = 1 ||½|| = 1 - ℓ ≥ 1. 1/ℓ (αQℓ ½)1/ℓ ≤ ||α||||½||||Qℓ || = ||Qℓ ||1/ℓ .. A$?B. # 0.0 - ;$$℄ ρ(Q) -

(155) Q 2 ρ(Q) = lim ||Qℓ ||1/ℓ < 1. A$&B ℓ−→∞. > A$?B A$&B 0 lim sup(αQℓ ½)1/ℓ < 1. ℓ−→∞. . A$DB.

(156) $E.

(157)

(158) . > A$'B A$DB 0 + supℓ≥1 (αQℓ ½)1/ℓ < 1/ G + - 0 lim E(Θn ) =. n−→∞. ∞ . (αQ½)k =. k=0. 1 . 1 − αQ½. -/ # 0 - A$$B/ / 0 0 - + - Θn - + - n - k - . X / > 1 ≤ k ≤ n A$$B 0 . È{Θn > k} = nk!k 0 uj,k (Q) =. . m∈Tj,k. k n uj,k (Q), j j=1. A$@B. k. ℓ mℓ j! . αQ ½ m1 !(1!)m1 m2 !(2!)m2 · · · mk !(k!)mk ℓ=1. # . j O(1/nk−j ) // k! n uj,k (Q) = O(1/nk−j ). nk j. 5

(159) k + - - k/ 0

(160) 0 5 - A$@B/ G 2 - Tj,k 0 - k ≥ 1 Tk,k Tk−1,k. = =. {(k, 0, . . . , 0)}, {(k − 2, 1, 0, . . . , 0)},. Tk−2,k Tk−3,k. = =. {(k − 4, 2, 0, . . . , 0), (k − 3, 0, 1, . . . , 0)}, {(k − 6, 3, 0, . . . , 0), (k − 5, 1, 1, 0, . . . , 0), (k − 4, 0, 0, 1, 0 . . . , 0)},. Tk−4,k. =. {(k − 8, 4, 0, . . . , 0), (k − 7, 2, 1, 0, . . . , 0), (k − 6, 1, 0, 1, 0, . . . , 0), (k − 6, 0, 2, 0, . . . , 0), (k − 5, 0, 0, 0, 1, 0 . . . , 0)},. .

(161) $%.

(162) . 0 + + 0 0 - + / , - k ≥ 1 uk,k (Q) = (αQ½) , k−1 k−2 (αQ½) uk−1,k (Q) = αQ2 ½1{k≥2} , 2 2 (k − 2)(k − 3) (αQ½)k−4 αQ2 ½ 1{k≥4} uk−2,k (Q) = 8 k−2 k−3 (αQ½) + αQ3 ½1{k≥3} , 6 3 (k − 3)(k − 4)(k − 5) k−6 (αQ½) αQ2 ½ 1{k≥6} uk−3,k (Q) = 48 (k − 3)(k − 4) (αQ½)k−5 αQ2 ½αQ3 ½1{k≥5} + 12 k−3 (αQ½)k−4 αQ4 ½1{k≥4} , + 24 4 (k − 4)(k − 5)(k − 6)(k − 7) k−8 (αQ½) uk−4,k (Q) = αQ2 ½ 1{k≥8} 384 2 (k − 4)(k − 5)(k − 6) k−7 (αQ½) + αQ2 ½ αQ3 ½1{k≥7} 48 (k − 4)(k − 5) k−6 (αQ½) + αQ2 ½αQ4 ½1{k≥6} 48 2 (k − 4)(k − 5) k−6 (αQ½) + αQ3 ½ 1{k≥6} 72 k−4 k−5 (αQ½) + αQ5 ½1{k≥5} . 120 k. 5 -+ Q 0

(163) - H - uj,k (Q) 0 + uj,k (I) 0 αQℓ ½ = 1 0 Q = I / k k! n È{Θn > k} = nk uj,k (Q) + en,k , j j=k−4. 0 en,k 2 + A$)B en,k. k−5 k−5 k k! n k! n k! n = k uj,k (Q) ≤ k uj,k (I) = 1 − k uj,k (I). j n j=1 j n j=1 j n j=k−4. , bn,k // bn,k. k k! n =1− k uj,k (I). j n j=k−4. , . 0 ≤ È{Θn > k} −. k k! n uj,k (Q) ≤ bn,k . j nk j=k−4. .

(164) )*.

(165)

(166) . # 0 bn,k + 0 k - 2

(167) ε - n 0

(168) k ∗ - k . ε // k ∗ = sup{k ∈ Æ | bn,k ≤ ε}.. >+ $ + - k∗ - = - ε n/ # 0 - - k ≤ 53 0 0 ≤ È{Θ1000 > k} −. k 1000 k! uj,k (Q) ≤ 10−2 . j 1000k j=k−4. ε n = 102 n = 103 n = 104 n = 105. 10−2 18 53 162 508. 10−3 15 41 124 387. 10−4 12 32 97 301. >+/ $ I - k∗ - = - ε n/ + 1 / > . + - + 0 + / αQℓ ½ + + 0 ℓ - 1 0 0 u + + αQu ½ = 1 / αQℓ ½. u = sup{ℓ ≥ 0 | αQℓ ½ = 1}.. G - ρ(Q) - Q 2 ρ(Q) <. .+ ε ρ(Q) < ρ(Q) + ǫ < 1

(169) . Cε > 0 - ℓ ≥ 0 0 αQℓ ½ ≤ Cε (ρ(Q) + ε)ℓ / - + + 0 + q ρ(Q) < q < 1 + ℓ0 . αQℓ ½ ≤ 1 0 0 ≤ ℓ ≤ ℓ0 , αQℓ ½ ≤ q ℓ 0 ℓ ≥ ℓ0 + 1.. 1. 4 0 È{Θn > k} + A B / C + 3 ' 0 - k ≥ 0 n ≥ 1 n. 1 k! È{Θn > k} = nk αQkr ½ ≤ An,k , k1 ! · · · kn ! r=1 k∈Sk,n. 0 An,k. =. =. n. 1 k! 1{kr ≤ℓ0 } + q kr 1{kr ≥ℓ0 +1} nk k1 ! · · · kn ! r=1 k∈Sk,n n k . 1 q k! 1 + 1 {kr ≤ℓ0 } {kr ≥ℓ0 +1} . nk k1 ! · · · kn ! r=1 q kr k∈Sk,n. .

(170) )$.

(171) . 4 0 An,k / 8 0 -0+ 0 An,k ≥. qk k! = qk . nk k1 ! · · · kn ! k∈Sk,n. 8 0 0 0 -0+ An,k. ≤. n. qk 1 k! k n k1 ! · · · kn ! r=1 q ℓ0 k∈Sk,n. =. q. k−nℓ0. nk. = q. k−nℓ0. . k∈Sk,n. k! k1 ! · · · kn !. .. 0 An,k 0 2 0 q k ≤ An,k ≤ q k−nℓ0 .. > + - n // 0 n −→ ∞ 0 - 0 0 . È{Θn > k} −→ αQk ½. 5 An,k / > + - k 0 . È{Θn > k} ≤ qk−nℓ , 0. 0 È{Θn > k} +

(172) - 1 k −→ ∞/ 0 - - / . . - -

(173) - Θn + ? - - - - i - 1 n/ + 0

(174) O(n) 0 . - + - n/ # -0+ 0 + 1 - ? 0 + - n/

(175)

(176) k ≥ 0 n ≥ 2 . È{Θn > k} =. k k ℓ s (1 − su,n )k−ℓ È{Θu > ℓ}È{Θ′n−u > k − ℓ} ℓ u,n. A$EB. ℓ=0.

(177)

(178)

(179) Θ′n−u $ Θn−u

(180)

(181) . . pu+1,n pn,n ,..., 1 − su,n 1 − su,n. . su,n =. u r=1. pr,n ..

(182) )).

(183)

(184) . 4 n ≥ 2 u 1 ≤ u ≤ n − 1/ > r = 1, . . . , u pr,u = pr,n /su,n / > k ≥ 0 0 - 3 ' . 0 . È{Θn > k} =. . k∈Sk,n. =. n. k! k (pr,n ) r αQkr ½ k1 ! · · · kn ! r=1. k . . ℓ=0 k∈Sℓ,u (ku+1 ,...,kn )∈Sk−ℓ,n−u. =. k k. ℓ. ℓ=0. ×. k∈Sℓ,u. u. ℓ! (pr,n )kr αQkr ½ k1 ! · · · ku ! r=1. . (ku+1 ,...,kn )∈Sk−ℓ,n−u. =. k ℓ=0. n. k! (pr,n )kr αQkr ½ k1 ! · · · kn ! r=1. n. (k − ℓ)! k (pr,n ) r αQkr ½ ku+1 ! · · · kn ! r=u+1. u . k ℓ ℓ! k (pr,u ) r αQkr ½ su,n (1 − su,n )k−ℓ ℓ k1 ! · · · ku ! r=1 k∈Sℓ,u. . ×. (ku+1 ,...,kn )∈Sk−ℓ,n−u. kr n. pr,n (k − ℓ)! αQkr ½ ku+1 ! · · · kn ! r=u+1 1 − su,n. k k ℓ = s (1 − su,n )k−ℓ È{Θu > ℓ}È{Θ′n−u > k − ℓ} ℓ u,n ℓ=0. 0 -/. " pr,n = 1/n

(185)

(186) r = 1, . . . , n

(187)

(188) k ≥ 0 u 1 ≤ u ≤ n − 1 . È{Θn > k} =. k ℓ k u k−ℓ u 1− È{Θu > ℓ}È{Θn−u > k − ℓ} ℓ n n ℓ=0.

(189)

(190) m ≥ 1 . È{Θ2. m. k 1 k > k} = k È{Θ2m−1 > ℓ}È{Θ2m−1 > k − ℓ}. ℓ 2. A$%B. ℓ=0. . #- - su,n = u/n .+ n = 2m . Θ′n−u = Θn−u / u = 2m−1 /. A$%B + - + - n/ #

(191) - - -

(192) - Θn 0 O(log2 n)/ 5 2 + ℓ = k/2 0 k / .

(193)

(194) . )'. (

(195) 4 + - + / . + + - - 0 / - ) / ) 0. - 0./ 5 - . + + + - + + 0. A/+/ # 0. B/ . + + + 2 +/ + - - + // + + + / ) A/+/ J . K !B / 8 A " F A"F BB -+ 1 + A/+/ B/ -+ 1 + - 2 0+ 2 + - / A/+/ ;D % $* $' $D $&℄B + - - H . 0 + 0 . A// . + - + 0

(196) + + 0 1B/ F0 - - L + - ++ + + + . H / : + + . + + + - + 4 / ;E℄/ ! = 0. 0 A/+/ ;$ ) ? E℄B/ # 0 + +

(197) + + / - - +/ 3 1 0 / 0 - - A- . 0 + 3f + 1 0 G1 + f B/ O(logN ) 0 N - / - H 0 + + / 8 . 1 0 / F + + 1 + / 3 0 2 - "F - - 0 / + - / . X + - >+ $ 0 q = 1 − p p ∈ (0, 1)/ p 0 0 q / .

(198) )?.

(199)

(200) p Smin + 1. p Smin + 2. q. p Smax − 2. ··· q. p. q. q. Smax − 1 q. p a. >+ $ . - / - Smin + 1 +

(201) 1 Smin + 0 / # 0 - Smax − 1 +

(202) 1 Smax 0 / α 0 0 ej - j = Smin + 1, . . . , Smax − 1/ α = ej . X0 = j 0 1/

(203) Q 0 + 0 X +

(204) 0 1 Qi,i+1 = p Qi,i−1 = q = 1 − p/ , - π(n) - // pi,n = 1/n - i = 1, . . . , n/ , + -

(205) - Θn + - $) - k ≥ 0 ⎧ pk - X0 = Smin + 1 ⎨ - X0 = j, - Smin + 2 ≤ j ≤ Smax − 2 1 lim {Θn > k} = n−→∞ ⎩ (1 − p)k - X0 = Smax − 1. ⎧ 1 ⎪ ⎪ ⎪ ⎨ 1−p ∞ lim E(Θn ) = n−→∞ ⎪ ⎪ 1 ⎪ ⎩ p. -. X0 = Smin + 1 X0 = j,. - Smin + 2 ≤ j ≤ Smax − 2. X0 = Smax − 1.. > 0 p = 1/2/ , 0 + 0 α = ej E(Θ1 ) = (j − Smin )(Smax − j).. , Smin = 4 Smax = 16 0 11/ >+ ) 0 - Θn - = - n 0 X0 - Smin + 1 = 5/ # 0 + + 1 {Θn > k} = k . n−→∞ 2 lim. # 0 0 + + . - - n/ .

(206) )&.

(207) 1. 0.8. 0.6. 0.4. 0.2. 0 0. 10. 20. 30. 40. 50. >+ ) > {Θn > k} 0 X0 = Smin + 1 = 5 - n = 1, 2, 3, 4, 5, 10, 15, 20, 25, ∞ - - k / # 0 >+ ' 0 - Θn - = - n 0 X0 (Smin + Smax )/2 = 10/ # + + lim. n−→∞. {Θn > k} = 1.. 2+ 0 0/ 5 + = 0 k

(208) >+ ) '/ >+ ? 0

(209) - Θn - = - - n - 1 25/ , + - E(Θn ) 0 2 0 X0 = 5 ∞ 0 X0 = 6, 7, 8, 9, 10/ F. + . 0 X0 = 5/ >+ & 0 - Θn - = + - n 0 X0 - (Smin + Smax )/2 = 10/ 2+ + 3 $?/ >+ D 0

(210) - Θn - = - - n = 1, 2, 4, 8, . . . , 215 = 32768/ F + 0 + - E(Θn ) 0 ∞ 0 X0 = 5, 6, 7, 8, 9, 10/ - E(Θn ) 0 X0 = 5 x

(211) / # + - E(Θn ) 2/. .

(212)

(213)

(214) . )D. 1. 0.8. 0.6. 0.4. 0.2. 0 0. 50. 100. 150. 200. 250. 300. >+ ' > {Θn > k} 0 X0 = (Smax + Smin )/2 = 10 - n = 1, 2, 3, 4, 5, 10, 15, 20, 25 - - k /. 140 120 100 80 60 40 20 0 1. 5. 10. 15. 20. 25. >+ ? > E(Θn ) - X0 = 5, 6, 7, 8, 9, 10 - - n/. .

(215)

(216) . )@. 1. 0.8. 0.6. 0.4. 0.2. 0 10000. 20000. 30000. 40000. 48000. >+ & > {Θn > k} 0 X0 = (Smax + Smin )/2 = 10 - n = 210 , 211 , 212 , 213 , 214 , 215 = 32768 - - k / 32768 30000 25000 20000 15000 10000 5000 0. 8 76 32 0 00 30. 0. 00 25. 0. 00 20. 0. 00 15. 0. 00 10. 00 50. 1. >+ D > E(Θn ) - X0 = 5, 6, 7, 8, 9, 10 - - n - n = 1, 2, 4, 8, . . . , 215 /. .

(217)

(218)

(219) . )E. ) ;$℄

(220)

(221)

(222) A)**EB/ ) + / # "%%% &

(223) "

(224)

(225) '

(226) ( )'*/ ;)℄.

(227)

(228) A)**?B/ J + ! - / # &

(229) "

(230) &

(231)

(232) )" &*/. ;'℄

(233) A)**&B/ C 9

(234)

(235) / > $$DA$B/ ;?℄. ! A)**&B/ .+ 1 . . / #

(236) )%*/. &

(237) %

(238) .

(239) "! A)**EB/ " . + - / # &

(240) . ;&℄. + "

(241)

(242) , % ),%-./0* > $?@$&D/ ;D℄. #

(243) $ % !

(244) &

(245) ! '. A)**'B/ . ! 0. 0 . / # &

(246) 1 2%3"4 "

(247) - )2"-*/. ;@℄ (

(248) &$

(249) / ! /. # ! A$%E$B/. ! 3 . ;E℄

(250) & ')

(251) A)**DB/ M ! . 0 / # &

(252) "

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(256) )&5&*/ ;%℄. !* + %

(257) ! , * + A)**'B/ I - N 0. - / # &

(258) &

(259)

(260) )&'6 */. ;$*℄ !* - ! A)**'B/ " + 0/ # &

(261) 1 2%3"4 "

(262) - )2"-*/ ;$$℄. !$

(263) + A)***B/

(264) ! ! 4 !+ / #!/ % ! A$%E$B/

(265) J . ;$)℄ ! !+ ! / O F. C /. $

(266) #$ ! (

(267) . *

(268) A)**$B/ ! 0./ # &

(269) . ;$'℄. "7 '. / ;$?℄.

(270) A$%D@B/ ! # 3 ! / ,/ .

(271)

(272) . )%. ;$&℄

(273) +

(274) A)**$B/ + - + / # &

(275) . "

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(280) */. / "% + !

(281)

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(286) .

(287) Centre de recherche INRIA Rennes – Bretagne Atlantique IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence Cedex Centre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier Centre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’Ascq Centre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex Centre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay Cedex Centre de recherche INRIA Sophia Antipolis – Méditerranée : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex. Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).

(288). ISSN 0249-6399.

(289)