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. Methodology and Computing in Applied Probability, Springer Verlag, 2013, 15 (2), pp.305–332. �10.1007/s11009-011-9239-6�. �hal00650081�. HAL Id: hal-00650081 https://hal.archives-ouvertes.fr/hal-00650081 Submitted on 9 Dec 2011. 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. N° 1953. apport de recherche. ISRN INRIA/RR--1953--FR+ENG. Mai 2010. ISSN 0249-6399. inria-00485667, version 1 - 25 May 2010. Emmanuelle Anceaume — François Castella — Romaric Ludinard — Bruno Sericola.

(3) inria-00485667, version 1 - 25 May 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) . inria-00485667, version 1 - 25 May 2010. # $ . 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 =. {X0 = i}.. , Θ1 B - + + 2 0 + a / , . inria-00485667, version 1 - 25 May 2010. Θ1 = inf{k ≥ 0 | Xk = a}.. - - Θ1 - ;$)< ;@< {Θ1 > k} = {Xk ∈ B} = αQk , A$B 0 - |B| 0 1. I

(41) - + / - B

(42) I − Q

(43) - Θ1 + E(Θ1 ) = α(I − Q)−1 . 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 () . 0 p,n / , 0 - + - = 1, . . . , n 0 0 < p,n < 1/

(45) - Y 0 0 R + - (i1 , . . . , in ) (j1 , . . . , jn ) ∈ S n . R((i1 , . . . , in ), (j1 , . . . , jn )) =. ⎧ n ⎪ ⎪ ⎪ p,n Pi ,i ⎪ ⎪ ⎪ ⎪ ⎨ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩. p,n Pi ,j. - (i1 , . . . , in ) = (j1 , . . . , jn ) - ∃! i = j. 0 / # . Y - (i1 , . . . , in ) (i1 , . . . , in ) - . X () . i A + B (i1 , . . . , i − 1, j , i + 1, . . . , in ) - . X () . - i j A + B/ 0 - . Y

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

(47) &.

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

(49) - - Yk / > . ≥ 1 0 Sk, 2 . k≥0. Sk, = {k = (k1 , . . . , k ) ∈ | k1 + · · · + k = 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. inria-00485667, version 1 - 25 May 2010. - 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 }. inria-00485667, version 1 - 25 May 2010. h=1. n. {X0 = jr }.. r=1,r=h. > k ∈ Vk 0 n n (k − 1)! kh (k − 1)! k! = , kh = k1 ! · · · kn ! k1 ! · · · kn ! k1 ! · · · kn !. h=1. h=1. 0 {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 = j} = p (1 − ph,n )k− {X = j}. h,n =0. 3 Xk(h) h + . ph,n - 0 + 0 . - - h = n/ C + $ 0 . .

(57) @.

(58) (n). {Xk = =. = j}. {Yk ∈ S × · · · × S × {j}} . ···. j1 ∈S. =. k k pkn (1 − pn,n )k−kn {Xkn = j} kn n,n . ×. k∈Sk−kn ,n−1. inria-00485667, version 1 - 25 May 2010. n−1. k! n pkr {Xkr = jr }pkn,n {Xkn = j} k1 ! · · · kn ! r=1 r,n. n−1. k! n r pkn,n {Xkn = j} pkr,n k1 ! · · · kn ! r=1. kn =0. =. . jn−1 ∈S k∈Sk,n. k∈Sk,n. =. . 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 {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 . {Θn > k} =. k∈Sk,n. n. k! pkr αQkr . k1 ! · · · kn ! r=1 r,n. A&B. > k ≥ 0 n ≥ 1 0 (1). (n). Θn > k ⇐⇒ Xk ∈ B, . . . , Xk. ∈ B.. , - $ A$B/ {Θn > k} = =. (1). (n). {Xk ∈ B, . . . , Xk ∈ B} {Yk = (j1 , . . . , jn )} (j1 ,...,jn )∈B n. =. k∈Sk,n. =. k∈Sk,n. =. k∈Sk,n. k! k1 ! · · · kn !. . n. r pkr,n {Xkr = jr }. (j1 ,...,jn )∈B n r=1. n. k! pkr {Xkr ∈ B} k1 ! · · · kn ! r=1 r,n n. k! pkr αQkr , k1 ! · · · kn ! r=1 r,n. 0 -/ 3

(61) - - {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 ) =. ∞ . ADB. {Θn > k}.. k=0. -0+ +

(70) - Θn /

(71)

(72) k ≥ 0 n ≥ 2 {Θn > k} =. k k. inria-00485667, version 1 - 25 May 2010. =0. . k−. pn,n (1 − pn,n ). αQ {Θn−1 > k − },. 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 ' {Θn > k} =. n. k! k (pr,n ) r αQkr k1 ! · · · kn ! r=1. k∈Sk,n. =. k kn =0. =. k kn =0. =. k kn =0. kn. (pn,n ) . . . . =. kn =0. . k∈Sk−kn ,n−1. =. k kn =0. k∈Sk−kn ,n−1. n−1 (k − kn )! (pr,n )kr αQkr k1 ! · · · kn−1 ! r=1. kr n−1 pr,n (k − kn )! αQkr k1 ! · · · kn−1 ! r=1 1 − pn,n. k k k−k (pn,n ) n (1 − pn,n ) n αQkn kn. . ×. k∈Sk−kn ,n−1. n−1. k! k (pr,n ) r αQkr k1 ! · · · kn−1 ! r=1. k (pn,n )kn (1 − pn,n )k−kn αQkn kn. k∈Sk−kn ,n−1 k . . k (pn,n )kn αQkn kn. . ×. αQkn kn !. . n−1 (k − kn )! k (pr,n−1 ) r αQkr k1 ! · · · kn−1 ! r=1. k k k−k (pn,n ) n (1 − pn,n ) n αQkn {Θn−1 > k − kn }, kn. .

(77) %.

(78) . 0 -/ 0 - {Θn > k} +. 0

(79) / , 0 -0+ 0 K . - i = 1, . . . , n {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. inria-00485667, version 1 - 25 May 2010. (pi,i Q + (1 − pi,i )I). ≤ (pi+1,i+1 Q + (1 − pi+1,i+1 )I)k ,.

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

(86)

(87) n ≥ 1

(88)

(89) ε ∈ (0, 1) max. i=1,...,n.

(90) . K1 = inf. {Θi > k} ≤ ε. k

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

(92)

(93) k ≥ K1,.

(94) $*.

(95)

(96) . > i = 1, . . . , n 0 {Θi > k} = ≤. k k p (1 − pi,i )k− αQ {Θi−1 > k − } i,i =0 k k p (1 − pi,i )k− αQ i,i =0. = ≤ =. α (pi,i Q + (1 − pi,i )I). k. . α (pn,n Q + (1 − pn,n )I) k k p (1 − pn,n )k− αQ . n,n. A- 4 &B. k. inria-00485667, version 1 - 25 May 2010. =0. 5

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

(98) ε ∈ (0, 1) 2 - + K1 0 - i = 1, . . . , n {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.

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

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

(106) $$.

(107) . , - i = 1, . . . , n ei. =. ∞ . {Θi > k}. k=K2. =. ∞ k k p (1 − pi,i )k− αQ {Θi−1 > k − } i,i. k=K2 =0. ≤ =. inria-00485667, version 1 - 25 May 2010. ≤. ∞ k k p (1 − pi,i )k− αQ i,i. k=K2 =0 ∞ . k. α (pi,i Q + (1 − pi,i )I). k=K2 ∞ . k. α (pn,n Q + (1 − pn,n )I). k=K2. = = = ≤. . −1. A - 4 &B 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 pn,n K2 K2 1 pn,n (1 − pn,n )K2 − α(I − Q)−1 Q pn,n . . =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 ) =. ∞ . {Θn > k}. k=0. = = =. inria-00485667, version 1 - 25 May 2010. = ≤. k ∞ k k− αQ {Θn−1 > k − } p (1 − pn,n ) n,n k=0 =0 k ∞ k k− (1 − pn,n ) αQk− {Θn−1 > } p n,n k=0 =0 ∞ ∞ k k− k− (1 − pn,n ) {Θn−1 > } p αQ n,n =0 k= ∞ ∞ k+ k (1 − pn,n ) {Θn−1 > } pn,n αQk =0 k=0 ∞ ∞ k+ k (1 − pn,n ) {Θn−1 > } pn,n =0. =. ∞ . k=0. . (1 − pn,n ). −(+1). {Θn−1 > } (1 − pn,n ). =0. =. E(Θn−1 ) . 1 − pn,n. 0 + E(Θn ) ≤. E(Θn−2 ) E(Θn−1 ) 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 ∈. 0. ∞ xk {Θn > k}. k! k=0. - Fn 2 - x ∈

(116)

(117) + -0+ / %

(118)

(119) n ≥ 1 x ∈ Fn (x) =. n. =1. αeQxp,n ,. AEB .

(120) $'.

(121)

(122)

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

(124) Fn(k) k

(125) Fn

(126) x!. > A@B 0 + + - . Fn (x). = = =. ∞ k xk k p (1 − pn,n )k− n,n k! k=0 ∞ =0 ∞ =0. inria-00485667, version 1 - 25 May 2010. =. =0. {Θ1 > } = αQ . {Θ1 > } {Θn−1 > k − }. ∞. xk− x k− pn,n {Θ1 > } (1 − pn,n ) ! (k − )! x p {Θ1 > } ! n,n. k= ∞ . k=0. xk (1 − pn,n )k k!. {Θn−1 > k − }. {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 (xp,n ) Fn−i (x(1 − pn,n − . . . − pn−i+1,n )), =n−i+1. 0 0 pr,n−1 =. , - i = n Fn (x) =. pr,n . 1 − pn,n. n. F1 (xp,n ).. =1. 50 F1 (x) =. ∞ xk k=0. k!. {Θ1 > k} =. ∞ xk k=0. 0 Fn (x) =. n. =1. k!. αQk = αeQx ,. αeQxp,n .. > - 0 0 2 - - Fn Fn(h) (x) =. ∞ ∞ xk−h xk {Θn > k} = {Θn > k + h}, (k − h)! k!. k=h. k=0. 0 + .+ x = 0/ 0 {Θn > 0} = 1

(127) - n ≥ 1 0 {Θn > 1} = αQ.. .

(128) $?.

(129)

(130) . .

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

(132)

(133) = 1, . . . , n

(134)

(135) x ∈ lim Fn (x) = eαQ½x .. n−→∞. #- p,n = 1/n - = 1, . . . , n 0 - AEB n Fn (x) = αeQx/n ,. inria-00485667, version 1 - 25 May 2010. 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/ -/ ++ - {Θn > k} 0 n + 2 (αQ)k / ! - p,n = 1/n - = 1, . . . , n 0 + - AEB A%B {Θn > k} =. Fn(k) (0). + - 0 n ≥ 4 Fn(1) (x) Fn(2) (x). Fn(3) (x). Fn(4) (x). n dk αeQx/n =. dxk. . x=0. n−1 αeQx/n αQeQx/n , 2 n − 1 Qx/n n−2 αQeQx/n αe = n 1 Qx/n n−1 αe + αQ2 eQx/n , n 3 (n − 1)(n − 2) Qx/n n−3 Qx/n αe αQe = n2 3(n − 1) Qx/n n−2 αe + αQeQx/n αQ2 eQx/n n2 n−1 1 + 2 αeQx/n αQ3 eQx/n , n 4 (n − 1)(n − 2)(n − 3) Qx/n n−4 Qx/n αe αQe = n3 2 6(n − 1)(n − 2) Qx/n n−3 Qx/n + αQ2 eQx/n αe αQe n3 3(n − 1) Qx/n n−2 2 Qx/n 2 αe αQ e + n3 4(n − 1) Qx/n n−2 αe + αQeQx/n αQ3 eQx/n n3 n−1 1 + 3 αeQx/n αQ4 eQx/n , n =. . .

(136) $&.

(137) . - x = 0 {Θn > 1} = {Θn > 2} = {Θn > 3} = {Θn > 4} =. αQ, n−1 1 2 (αQ) + αQ2 , n n (n − 1)(n − 2) 3(n − 1) 1 (αQ)3 + αQαQ2 + 2 αQ3 , n2 n2 n (n − 1)(n − 2)(n − 3) 6(n − 1)(n − 2) 4 2 (αQ) + (αQ) αQ2 3 n n3 3(n − 1) 2 2 4(n − 1) 1 + αQαQ3 + 3 αQ4 . αQ + 3 3 n n n. inria-00485667, version 1 - 25 May 2010. !

(138)

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

(140) . . m k k () . g (x) dk f (j) (g(x)) j! f (g(x)) = k! , dxk j! m1 ! · · · mk ! ! j=1 m∈Tj,k. =1.

(141) Tj,k . Tj,k =. k . m = (m1 , . . . , mk ) m = j. . =1. k .

(142) m = k. .. A$*B. =1. .+ g(x) = αeQx/n f (x) = xn 0 + f (j) (x) =. n! xn−j 1{j≤n} (n − j)!. g() (x) = n1 αQ eQx/n ,. 0 - n ≥ 1 k ≥ 1 Fn (x) = f (g(x)) Fn(k) (x) =. m k∧n k . αQ eQx/n j! k! n Q nx n−j αe . nk j=1 j m1 ! · · · mk ! ! m∈Tj,k. =1. .+ x = 0 0 {Θn > k} =. m k∧n k . αQ k! n j! . nk j=1 j m1 ! · · · mk ! ! m∈Tj,k. A$$B. =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. + . {Θn > k}. + -0+ /. A$)B.

(143) $D.

(144)

(145) . " p,n = 1/n

(146)

(147) = 1, . . . , n

(148)

(149) k ≥ 0 . {Θn > k} = (αQ). k. lim. n−→∞. . lim E(Θn ) =. n−→∞. ⎧ ⎨. 1 1 − αQ ⎩ ∞. αQ < 1. . αQ = 1.. ,

(150) 2 - - Fn 0

(151) . / > z ∈ 0 . ∞ zk Fn (z) = {Θn > k}, k!. inria-00485667, version 1 - 25 May 2010. 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 =. Fn (eiη )e−ipη dη {Θn > k} k!. 2π. iei(k−p)η dη. 0. {Θn > p} , p! . -. 0 2iπ. 0. , - p ≥ 0 {Θn > p} =. . 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. C. |dz| = e. 2π. dη = 2eπ.. 0. - + 0 + lim. n−→∞. {Θn > p}. = =. p! 2iπ. . C. 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 ) =. ∞ . {Θn > k}.. k=0. , 2 0 αQ = 1/ > - - 0 - k ≥ 0 lim. n−→∞. {Θn > k} = 1.. C + > 9 - 0 + lim inf. inria-00485667, version 1 - 25 May 2010. n−→∞. ∞ . {Θn > k} ≥. k=0. ∞ k=0. lim. n−→∞. {Θn > k} = ∞,. limn−→∞ E(Θn ) = ∞ 0 - - 0 αQ = 1/ , 0 0 αQ < 1/ C + A$$B A$)B 0 {Θn > k}. ≤. k∧n k m. k! n j! 1/ ) (αQ nk j=1 j m1 !(1!)m1 m2 !(2!)m2 · · · mk !(k!)mk m∈Tj,k =1 k max (αQ )1/. ≤. k sup(αQ )1/ .. =. =1,...,k. ≥1. #- 0 0 sup≥1 (αQ )1/ < 1 -0 + + - / 4 0 sup≥1 (αQ )1/ < 1/ > - αQ < 1 0 αQ < 1 - ≥ 1 1/ αQ <1. - ≥ 1. A$'B , 2 - 2 -

(154) M ||M || = sup i. |Mi,j |.. j. , ||α|| = 1 |||| = 1 - ≥ 1 1/ (αQ )1/ ≤ ||α||||||||Q || = ||Q ||1/ .. A$?B # 0.0 - ;$$< ρ(Q) -

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

(156) $E.

(157)

(158) . > A$'B A$DB 0 + sup≥1 (αQ )1/ < 1/ G + - 0 lim E(Θn ) =. n−→∞. ∞ k=0. (αQ)k =. 1 . 1 − αQ. -/ # 0 - A$$B/ / 0 0 - + - Θn - + - n - k - . X / > 1 ≤ k ≤ n A$$B 0 . inria-00485667, version 1 - 25 May 2010. {Θn > k} =. k k! n uj,k (Q), nk j=1 j. A$@B. 0 uj,k (Q) =. m∈Tj,k. k. m j! . αQ m1 !(1!)m1 m2 !(2!)m2 · · · mk !(k!)mk =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) = (αQ) , k−1 k−2 (αQ) uk−1,k (Q) = αQ2 1{k≥2} , 2 2 (k − 2)(k − 3) (αQ)k−4 αQ2 1{k≥4} uk−2,k (Q) = 8 k−2 k−3 (αQ) + αQ3 1{k≥3} , 6 3 (k − 3)(k − 4)(k − 5) k−6 (αQ) αQ2 1{k≥6} uk−3,k (Q) = 48 (k − 3)(k − 4) (αQ)k−5 αQ2 αQ3 1{k≥5} + 12 k−3 (αQ)k−4 αQ4 1{k≥4} , + 24 4 (k − 4)(k − 5)(k − 6)(k − 7) k−8 (αQ) αQ2 1{k≥8} uk−4,k (Q) = 384 2 (k − 4)(k − 5)(k − 6) k−7 (αQ) + αQ2 αQ3 1{k≥7} 48 (k − 4)(k − 5) k−6 (αQ) + αQ2 αQ4 1{k≥6} 48 2 (k − 4)(k − 5) k−6 (αQ) + αQ3 1{k≥6} 72 k−4 k−5 (αQ) + αQ5 1{k≥5} . 120. inria-00485667, version 1 - 25 May 2010. k. 5 -+ Q 0

(163) - H - uj,k (Q) 0 + uj,k (I) 0 αQ = 1 0 Q = I / k k! n {Θn > k} = k uj,k (Q) + en,k , j n 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≤. {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 ∈ | bn,k ≤ ε}.. >+ $ + - k∗ - = - ε n/ # 0 - - k ≤ 53 0 0≤. {Θ1000 > k} −. k 1000 k! uj,k (Q) ≤ 10−2 . j 1000k. inria-00485667, version 1 - 25 May 2010. 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 + / αQ + + 0 - 1 0 0 u + + αQu = 1 /. αQ . u = sup{ ≥ 0 | αQ = 1}.. G - ρ(Q) - Q 2 ρ(Q) <. .+ ε ρ(Q) < ρ(Q) + < 1

(169) . Cε > 0 - ≥ 0 0 αQ ≤ Cε (ρ(Q) + ε) / - + + 0 + q ρ(Q) < q < 1 + 0 . αQ ≤ 1 0 0 ≤ ≤ 0 , αQ ≤ q 0 ≥ 0 + 1. 1. 4 0 {Θn > k} + A B / C + 3 ' 0 - k ≥ 0 n ≥ 1 n. k! 1 {Θn > k} = k αQkr ≤ An,k , n k1 ! · · · kn ! r=1 k∈Sk,n. 0 An,k. = =. n. k! 1 1{kr ≤0 } + q kr 1{kr ≥0 +1} nk k1 ! · · · kn ! r=1 k∈Sk,n n k . 1 k! q 1 + 1 {kr ≤0 } {kr ≥0 +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. k! 1 qk k n k1 ! · · · kn ! r=1 q 0 k∈Sk,n. =. q. nk. inria-00485667, version 1 - 25 May 2010. = q. . k−n0. k−n0. k∈Sk,n. k! k1 ! · · · kn !. .. 0 An,k 0 2 0 q k ≤ An,k ≤ q k−n0 .. > + - n // 0 n −→ ∞ 0 - 0 0 {Θn > k} −→ αQk .. 5 An,k / > + - k 0 . {Θn > k} ≤ q k−n0 ,. 0 {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 {Θn > k} =. k k s (1 − su,n )k− {Θu > } {Θ n−u > k − } u,n. A$EB. =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 0 pr,u = pr,n /su,n /. > k ≥ 0 0 - 3 '. {Θn > k} =. n. k! k (pr,n ) r αQkr k1 ! · · · kn ! r=1. k∈Sk,n. =. k . . =0 k∈S,u (ku+1 ,...,kn )∈Sk−,n−u. =. k k. . =0. u. ! (pr,n )kr αQkr k1 ! · · · ku ! r=1. . ×. inria-00485667, version 1 - 25 May 2010. k∈S,u. (ku+1 ,...,kn )∈Sk−,n−u. =. k =0. n. k! (pr,n )kr αQkr k1 ! · · · kn ! r=1. n. (k − )! k (pr,n ) r αQkr ku+1 ! · · · kn ! r=u+1. u . k ! k su,n (1 − su,n )k− (pr,u ) r αQkr k1 ! · · · ku ! r=1 k∈S,u. . ×. (ku+1 ,...,kn )∈Sk−,n−u. kr n. pr,n (k − )! αQkr ku+1 ! · · · kn ! r=u+1 1 − su,n. k k = s (1 − su,n )k− {Θu > } {Θ n−u > k − } u,n =0. 0 -/. " pr,n = 1/n

(185)

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

(187)

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

(189)

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

(191) - - -

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

(193)

(194) . )'. inria-00485667, version 1 - 25 May 2010. (

(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. Smax − 1 q. q. p a. inria-00485667, version 1 - 25 May 2010. >+ $ . - / - 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 ⎧ - X0 = Smin + 1 pk ⎨ - 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 + + lim. n−→∞. {Θn > k} =. 1 . 2k. # 0 0 + + . - - n/ .

(206) )&.

(207) 1. 0.8. 0.6. 0.4. inria-00485667, version 1 - 25 May 2010. 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) )D.

(213)

(214) 1. 0.8. 0.6. 0.4. inria-00485667, version 1 - 25 May 2010. 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. >+ ? > - n/. 10 E(Θn ). 15. 20. 25. - X0 = 5, 6, 7, 8, 9, 10 - . .

(215) )@.

(216) 1. 0.8. 0.6. 0.4. inria-00485667, version 1 - 25 May 2010. 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. 0. - X0 = 5, 6, 7, 8, 9, 10 - . 76. 32. 00. 30 0. 00. 25 0. 00. 20 0. 00. E(Θn ). 15 0. . 00. 10. 00. 50. 1. >+ D > - n - n = 1, 2, 4, 8, . . . , 215 /.

(217) )E.

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(234)

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

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

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(287) inria-00485667, version 1 - 25 May 2010. 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)