The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof The Limiting Search-Cost of the Move-to-Front Strategy in a Law of Large Numbers Asymptotic Regime AofA, april 2008 Javiera BARRERA and Joaquín FONTBONA 1 Departamento de Matemática Universidad Tec. Federico Sta. Maria 2 Departamento de Ing. Matemática Universidad de Chile J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Outline The MtF strategy for deterministic popularities 1 Main Result: The limiting transient search-cost 2 Examples 3 Idea of the Proof 4 J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Request Process 1 2 3 Poisson ( ω i ) 4 1 4 2 3 2 ? �� � Poisson ω i ω i � ω i i with prob. p i = J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Request Process Consider a list of n objects labelled { 1 , . . . , n } ; Assume that at time t = 0 objects are arranged in a permutation π of { 1 , . . . , n } ; Let w = ( ω 1 , . . . , ω n ) be a deterministic nonnegative vector and consider a Poisson point process in I R + × { 1 , . . . , n } with intensity measure dt ⊗ w . The request instant for an item i is given by the restriction of the point measure to I R + × { i } and follows a Poisson process of rate ω i . For t > 0 each object is request independently. J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Request Process Denote by N t the total number of requests, which is also a standard Poisson process of rate � n i = 1 ω i . At each request the probability that object i is the one requested is ω i p i = � n , j = 1 ω j we call p i the “popularity” of object i . We denote by S ( n ) ( t ) the position of file i at time t and by I k i the k-th request file. By convention the list is update after t then I N t − + 1 is the label of the first object to be requested. J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof The Move-to-front rule 4 1 2 5 7 3 8 9 6 ↓ 4 1 2 5 7 3 8 9 6 ↓ 5 4 1 2 7 3 8 9 6 Figure: Ilustration of the MtF strategy J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof The MtF with independent requests Search-cost of the next requested item, define by: n � S ( n ) S ( n ) ( t ) := ( t ) 1 { I Nt − + 1 = i } . i i = 1 R t is the subset of objets which have been requested at least once in [ 0 , t [ . We decompose the search-cost into two r. v., S ( n ) ( t ) = S ( n ) eq ( t ) + S ( n ) oe ( t ) where S ( n ) S ( n ) ( t ) 1 { I Nt − + 1 ∈ R t } eq ( t ) := S ( n ) S ( n ) ( t ) 1 { I Nt − + 1 �∈ R t } . oe ( t ) := J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Proposition: the search-cost for the MtF rule We set 1 ij = 1 π ( i ) <π ( j ) . Let T i be the time that has past since the last request of i or t if it has never been request a) For all k , i ∈ { 1 , . . . , n } , n � P { S ( n ) I I ( t ) = k , i ∈ R t } = I E T i P { 1 T j < T i = k − 1 | T i } 1 T i < t i j = 1 j � = i b) For all k , i ∈ { 1 , . . . , n } , n � � � P { S ( n ) I ( t ) = k , i �∈ R t } = I P 1 ji + 1 T j < t 1 ij = k − 1 I P { T i = t } i j = 1 , j � = i J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Proposition: the search-cost for the MtF rule We set 1 ij = 1 π ( i ) <π ( j ) . Let B 1 ( q 1 ) . . . , B n ( q n ) independent Bernoulli r. v. with given parameters q 1 , . . . , q n . a) For all k , i ∈ { 1 , . . . , n } , � t P { S ( n ) p i e − p i u I P { J n I ( t ) = k , i ∈ R t } = eq ( u ) = k − 1 } du i 0 n � where J n eq ( u ) = d B j ( 1 − e − p j u ) . j = 1 , j � = i b) For all k , i ∈ { 1 , . . . , n } , P { S ( n ) P { J n oe ( t ) = k − 1 } e − p i t I ( t ) = k , i �∈ R t } = I i n � � � where J n oe ( t ) = d 1 − e − p j t 1 ij B j . j = 1 , j � = i J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof c) For all k , i ∈ { 1 , . . . , n } , � t n � P { S n p 2 i e − p i u I P { J n I eq ( t ) = k } = eq ( u ) = k } du . 0 i = 1 n � where J n eq ( u ) = d B j ( 1 − e − p j u ) . j = 1 , j � = i d) For all k , i ∈ { 1 , . . . , n } , n � P { S ( n ) P { J ( n ) oe ( t ) = k } e − p i t . I oe ( t ) = k } = p i I i = 1 n � � � where J ( n ) oe ( t ) = d 1 − e − p j t 1 ij B j . j = 1 , j � = i The proof is direct consequence of J. Fill and L. Holst results. J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Expected stationary search-cost (McCabe 65) � p i p j E ( S ( n ) ( ∞ )) = I p i + p j i � = j Laplace transform of the transient Search-Cost φ S ( n ) ( t ) ( z ) = A n ( t , z ) + B n ( t , z ) � t n n � � � e p j u − 1 ) e − z � p 2 i e − u du A n ( t , z ) = ( 0 i = 1 j = 1 j � = i n n � � � 1 i < j + ( e p j t − 1 i < j ) e − z � p i e − t B n ( t , z ) = i = 1 j = 1 j � = i Flajolet et al. ( t = ∞ ) 92, Bodell 97 and Fill and Holst 96. J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof The conditional MtF To do the asymptotic in the number of files we must do some assumption over w ( n ) . For each n we consider a random or deterministic vector of intensities w ( n ) = ( ω ( n ) 1 , . . . , ω ( n ) n ) . Consider: The Poisson process N t = N ( n ) and the search-cost t S ( n ) ( t ) , both defined conditionally on w ( n ) Let p ( n ) = ( p ( n ) 1 , . . . , p ( n ) n ) be the vector of popularities, / � n p ( n ) = ω ( n ) j = 1 ω ( n ) . i i j J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Previous Result, the limiting stationary search-cost Proposition JB., C. Paroissin, T. Huillet 2005 Let ω i be n i.i.d variable with law P. The limiting distribution of the stationary search-cost S ( n ) ( ∞ ) satisfies S ( n ) ( ∞ ) d → S ∞ n when n → ∞ , where S ∞ has the density function φ ′′ � � φ − 1 ( 1 − x ) f S ∞ ( x ) = − 1 φ ′ � � 1 [ 0 , 1 − p 0 ] , µ φ − 1 ( 1 − x ) E ( e − ω 1 t ) and µ = I with p 0 = P ( ω 1 = 0 ) , φ ( t ) = I E ( ω 1 ) . J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
The MtF strategy for deterministic popularities Main Result: The limiting transient search-cost Examples Idea of the Proof Definition: The LLN- P Condition We say that a sequence of (random or deterministic) vectors � � w ( n ) = ω ( n ) 1 , . . . , ω ( n ) N satisfies a law of large numbers n n ∈ I with limiting law P ( LLN- P ), if there exist a probability measure P ∈ P ( I R + ) with finite first moment µ � = 0 and positive random variables Z n , such that the empirical measures n � ν ( n ) := 1 → law P ˆ δ Z n ω ( n ) n i i = 1 and their empirical means n � 1 → law µ. Z n ω ( n ) i n i = 1 J. BARRERA & J. FONTBONA Search-Cost of the MtF Strategy in a LLN Asymptotic Regime
Recommend
More recommend
