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1 A hydrodynamc lmt of move-to-front rules and ts applcaton to web rankngs START
2 Move-to-front cf. Move-to-front N N 1( )
3 (move-to-front ) N =1,,N t 0 t X (N) (t) X (N) (0) = x (N) X (N) (t) = x (N) + ν (N) + t 0 N k=1 t 0 1 X (N) k (s 0)>X (N) (s 0) ν(n) k (ds) (1 X (N) (s 0)) ν (N) (ds), =1,,N, t 0 : (Ω, B(R + )) Z + ρ (N) =E[ν (N) ] X (N) =(X (N) 1,,X (N) N ) 2( )
4 = E[ ν (N) ((0,t]) = w (N) ρ (N) ρ (N) ] R + (ρ (N) ({t}) = 0) t ν (N) : (Ω, B(R + )) Z + ν (N) (A) ρ (N) (A) A B = ν (N) ω Ω ν (N) (A) ν (N) (B) ν (N) (ω)(ds) = (ω)((a, b]) t (a, b] j=1 1 {τ (N),j, j} F (s)ν(ds) = F (τ j ) 1 A A j t X (N) (t) = x (N) N t t + (ds)+ k=1 0 1 X (N) k (s 0)>X (N) (s 0) ν(n) k 0 δ (N) τ,j (ω) (ds) (1 X (N) (s 0)) ν (N) (ds) 3( )
5 τ (N),j {τ (N),j, j} 0 <τ (N),1 <τ(n),2 1) X (N) (τ (N),j )=1(, j) 2) X (N) (τ (N),j )=X (N) < (τ (N),j 0) + 1 (,j ) t=τ 1,1 t=τ 2,1 t=τ 1,2 t=τ 3,1 x (N) 3,0 =1,x(N) 2,0 =2, τ 1,1 <τ 2,1 <τ 1,2 <τ 3,1 < 1 4( )
6 (M.L.Tsetln, 1963) (Least-Recently-Used cachng) v.s. (?) top-to-random shufflng w 5( )
7 Move-to-front Amazon 2ch.net 6( )
8 Y (N) C (t) = 1 N λ (N) t = 1 N N =1 δ ρ (N) Y (N) C (t) y C(t) =1 N =1 ((0,t]) λ t (N ) 0 e s λ t (ds) (N ) Y (N) (N) C (t) E[ Y C (t) ] 0 P[ τ 1 >t]=p[ν((0,t]) = 0 ] = e ρ((0,t]] 1 τ (N),1 t 7( )
9 x (N) = =1,0 t<τ (N) N t k=1 0 1 X (N) k X (N) (t) 1 (s 0)>X (N) (s 0) ν(n) k (ds) = N =1 t =0 1 N N =1 1 τ (N),1 δ (N) ρ ((0,t]) λ t Y (N) C (t) y C(t) =1 (N) = NY t C (t) 0 e s λ t (ds) ( w ) y C (t) λ 8( )
10 ρ Y (N) µ (N) t = 1 N N =1 δ (ρ (N),Y (N) (t)) = 1 N (X(N) 1) µ (N) 1 N N =1 0 µ 0 (N ) 0 s<t N δ (N) ρ ((s,t]) δ ρ((s,t]) Λ(dρ); Λ(dρ) =µ 0 (dρ [0, 1)) M(R + ) t >0 µ (N) t µ t (N µ t 9( )
11 µ (N) t µ t µ t U(dρ, y, t) :=µ t (dρ [y, 1)) { e ρ((t t0(y,t),t]) Λ(dρ), 0 y y = C (t), e ρ((0,t]) U(dρ, ŷ(y, t), 0), y C (t) y<1. t 0 y A (t 0,t)=1 e ρ((t t 0,t]) Λ(dρ) ŷ y B (y, t) =1 M(R + ) M(R + ) y C (t) =y A (t, t) =y B (0,t) e ρ((0,t]) µ 0 (dρ [y, 1)) 10( )
12 Λ= r β δ ρβ, ρ β (A) = β A w β (u) du, w β,r β > 0 β r β =1, β r β w β (t) <, u α : [0, 1) R + : u β (y) =1 y U α (y, t) =U({ρ α },y,t) β U α t (y, t)+ w β (t) U β (y, t) U α y (y, t) = w α(t)u α (y, t), β U α (0,t)=r α U α (, 0) = u α ( ), (t 0,ŷ) (y A,y B,y C ) 1 t y α 11( )
13 (cf. Amazon.co.jp N = O(10 6 )) jump tal (y y C (t)) (µ 0 = U(,, 0)) randomness cf. TASEP 1 nvscd Burgers PDE 12( )
14 (t = ) ρ((0,t]) = wt w λ(dw) =µ (dw [0, 1)) µ (dw [y, 1)) = e wt 0(y) λ(dw); 1 y = R + e wt 0(y) λ(dw) t I(t) C N (t) =X (N) I (N) (t) (t) [y, 1) tal P[ 1 N C N(t) y ] = P[ Y (N) (t) y ]P[ I (N) R w µ (dw [y, 1)) (t) = ] + R wλ(dw) + LRU-cachng cache fault Amazon.co.jp Pareto a, b > 0, λ([0,w]) = 1 (a/w) b (w a) (w = a(n/) 1/b ) y>0 w µ (dw [y, 1)) R + b>1 wλ(dw) = ab,0<b<1 = y =+0 R + b 1 b>1 13( )
15 1 Ω sampleω f f(ρ, y)µ (N) t (dρ dy) = f(ρ, y)µ t (dρ dy) lm N M(R + ) [0,1) M(R + ) [0,1) M(R + ) [0, 1) f = f n Ω n Ω = n Ω n y χ[y, 1) 1 N lm g(ρ (N) N N ) 1 (N) Y (ω)(t) y = g(ρ)µ t (dρ [y, 1)) =1 M(R + ) y y C (t) 1 N N =1 1 Y (N) (t)<y 1 (N) 0, a.s. ν ((t t 0 (y,t),t])>0 14( )
16 Amazon Amazon.co.jp 2ch.net (ameblo.jp) 15( )
17 Amazon.co.jp Amazon 2006 ML 16( )
18 rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date (FAQ) 17( )
19 Pareto x (N) =1,0 t<τ (N) Y (N) (t) =Y (N) C (t)+ 1 N t =0 NY C (t) x C (t) N N 0 e s λ t (ds) λ t Pareto (Zpf ) w (N) = a ( ) N 1/b ; a: b: x C (t) Ny C (t) N(1 b(at) b Γ( b, at)); Γ(z,p) = p e x x z 1 dx N, a, b 18( )
20 Amazon.co.jp (N, a, b )= (8 10 5, , 0.81) 500,000 rankng 1,000 2,000 hrs b<1 o(n) 19( )
21 2ch.net rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date rankng 500,000 Jan 09 May 09 Sep 09 date Amazon.co.jp 1 20( )
22 w (N) 21( )
23 λ t : w (N) (t) =w (N) a(t), w (N) > 0 λ (N) t = 1 N t δ (N) N w A(t) λ t; A(t) = a(u)du =1 0 a(t) 24 A(t) =t+ 22( )
24 2ch.net (2ch.net) web 1 move-to-front sage λ t 23( )
25 :00 18:00 00:00 06:00 12: N = /10/ jump y C (t) 2008 M2 24( )
26 A(t) = S (N) (t) n =1 t y C (t) 1 e ws(n) (t)/z(n) λ(dw); Z(N) = R + 0 a(u)du w (N) (t) =w (N) a(t) > 0 ρ (N) ((0,t]) = A(t) Z(N) N =1 w (N) M2 12:00 18:00 00:00 06:00 12:00 25( )
27 b y C (A 1 t) Pareto b =0.872 Amazon.co.jp 2ch.net b <1 26( )
28 Amazon.co.jp Chevaler, Goolsbee b =1.2 (Onlne bookstore ) Brynjolfsson, Hu, Smth b =1.148 (consumer welfare ) Onlne retal C. Anderson, The Long tal Amazon 27( )
29 2ch.net :00 18:00 00:00 06:00 12: Amazon.co.jp O(10 2 ) /1 (10 ) 9 /1 (30 ) (cf =20 /, 10 2 =1 / ) 28( )
30 Amazon.co.jp aa(t) 1 x C (t) NΓ(1 b)a b A(t) b + O(A(t)) 24 N, b, a ( 1.5 ) 2ch.net 29( )
31 Amazon.co.jp ( ) 0 MTF ( )
32 move-to-front (stochastc rankng process) nvscd Burgers PDE Amazon.co.jp 2ch.net Amazon.co.jp 2ch.net Amazon.co.jp 2ch.net 31( )
33 Stochastc rankng process long tal 32( )
34 (b <1) b<1 2ch b>1 b<1 33( )
35 End of sldes. Clck [END] to fnsh the presentaton. K. Hattor, T. Hattor, Stochastc Processes and ther Applcatons 119 (2009) K. Hattor, T. Hattor, Funkcalaj Ekvacoj 52 (2009) K. Hattor, T. Hattor, preprnt (2009). Y. Harya, K. Hattor, T. Hattor, Y. Nagahata, Y. Takeshma, T. Kobayash, preprnt (2010). Google twtter tetshattor END Bye
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2012.08.10 START 3 B(3, 0.5) N =8 3 2 1 0 1 3 3 1 w, =1, 2,...,N w 3 2 1 1 N 0 1 N 1( ) w, =1, 2,...,N w > 0 w = c log N, =1, 2,...,N, c>0 P[ [x, ) ]= N 1 { w x} = 1 N { Ne x/c } e x/c X w x 1 N 2( ) Zpf
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Amazon.co.jp 2008.09.02 START Amazon.co.jp Amazon.co.jp Amazon.co.jp Amazon Internet retailers are extremely hesitant about releasing specific sales data 1( ) ranking 500,000 100,000 Jan.1 Mar.1 Jun.1
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4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
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72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
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