Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or ppopularities). The induced Markov chain is known to be ergodic [4]. One main problem is the study of the distribution of the search cost defined as the position of the required item. Here we first establish the link between two recent papers [3, 8] that both extend results proved by Kingman [7] on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to infinity.normalized random measure; random discrete distribution; stable subordinator; problem of heaps

Antonio Lijoi

Christian Paroissin

Fabrizio Leisen

English

Research Papers in Economics

 
 
Quaderni di Dipartimento 
 
 
 
 
 
Limiting behavior of the search cost distribution 
for the move-to-front rule in the stable case 
 
 
 
Fabrizio Leisen 
(Universidad Carlo III de Madrid) 
 
Antonio Lijoi 
(Università di Pavia) 
 
Christian Paroissin 
(Université de Pau et des Pays de l'Adour) 
 
 
 
 
 
 
 
 
# 113 (04-10) 
 
 
 
 
 
 
 
Dipartimento di economia politica 
e metodi quantitativi 
Università degli studi di Pavia 
Via San Felice, 5 
I-27100 Pavia 
 
Aprile 2010 Limiting behavior of the search cost distribution
for the move-to-front rule in the stable case
F. Leisen, A. Lijoi1 and C. Paroissin
Universidad Carlo III de Madrid, Universit` a di Pavia and
Universit´ e de Pau et des Pays de l’Adour
Abstract
Move-to-front rule is a heuristic updating a list of n items accord-
ing to requests. Items are required with unknown probabilities (or
ppopularities). The induced Markov chain is known to be ergodic [4].
One main problem is the study of the distribution of the search cost
deﬁned as the position of the required item. Here we ﬁrst establish the
link between two recent papers [3, 8] that both extend results proved
by Kingman [7] on the expected stationary search cost. Combining
results contained in these papers, we obtain the limiting behavior for
any moments of the stationary seach cost as n tends to inﬁnity.
Keywords: normalized random measure; random discrete distribution; stable subordi-
nator; problem of heaps
AMS classiﬁcation: 60G57, 60G51, 60G52
1 Introduction
The heaps problem was ﬁrst considered, in independent works, by Tsetlin
[11] and McCabe [9]. Its basic description can be given as follows. Consider
a collection of n items stored into a list or heap and each of them is identiﬁed
by a label. Hence, the objects can be described by the set I = {1,...,n}.
The probability that the i–th item is requested by a user is denoted by pi,
for i = 1,...,n. Hence pi > 0, for any i, and
 n
i=1 pi = 1. At each unit of
time, an item is requested and it is searched for through the heap, starting
at the top. Once it is found, it is moved to the top of the heap. The search
cost is the position of the requested item in the heap or, equivalently, the
number of items to be removed from the heap in order to ﬁnd the requested
one. In this setting, it might be of interest to determine the distribution of
the search cost when the underlying Markov chain is at equilibrium.
Kingman [7] ﬁrst studied the case of random request probabilities, or ran-
dom popularities. His paper develops two important cases where request
1Also aﬃliated to Collegio Carlo Alberto, Moncalieri (TO) and CNR–IMATI, Milano
1probabilities are deﬁned in terms of: (a) the normalized increments of a
γ-stable subordinator; (b) the Dirichlet distribution on the simplex. The re-
sults contained therein provide an exact analytic evaluation of the expected
search cost either for any ﬁnite n or in the limit, as the number of items n
tends to inﬁnity. In particular, in the case of normalized γ-stable request
probabilities, it is found that the limiting expected search cost is ﬁnite if
and only if γ < 1/2.
These results have been recently extended in two independent papers. In [8]
Lijoi and Pr¨ unster studied the case of request probabilities derived from a
normalized random measures with independent increments, which general-
izes the result obtained by Kingman [7]. In [3] Barrera and Paroissin studied
the case of request probabilities based on exchangeable random partitions.
It is to be emphasized that all previous contributions on the subject is con-
ﬁned to the determination of the ﬁrst moment of the stationary search cost.
Here we wish to extend earlier work and determine the expression of the
limiting moments of any order in the γ–stable case. In particular, it will be
shown that the k–th moment exists if and only if γ < 1/(k + 1) which re-
duces to the condition provided by [7] when k = 1. See also [8]. The outline
of the paper is as follows. In Section 2 we provide a concise introduction
to some basic tools and notions that will be relevant for achieving the main
result in Section 3.
2 The γ-stable model
Before stating and proving our result, it might be worth recalling the main
ingredients that deﬁne the model we are going to use. As mentioned in the
previous section, the request probabilities pi, for i = 1,...,n, are going to
be random. Indeed, if (wi)i≥1 is a sequence of positive independent random
variables and Wn =
 n
i=1 wi, one can deﬁne
pi =
wi
Wn
i = 1,...,n
Hence, (p1,...,pn) is an exchangeable random partition of the unit interval.
A possible choice is wi := ξti − ξti−1 where 0 = t0 < t1 < ··· < tn = 1 and
ξ = {ξt : t ∈ [0,1]} is a subordinator that is a process with almost surely
increasing paths and with independents increments. In this case, one can
express the the Laplace transform of wi in terms of the L´ evy intensity ν of
2ξ. In other words
φi(s) := E
 
e−swi 
= exp
 
−(ti − ti−1)
  ∞
0
 
1 − e−sy 
ν(dy)
 
(1)
with ν such that
  ∞
0 min{1,y}ν(dy) < ∞. According to the terminology
set forth in [10], (p1,...,pn) deﬁnes a normalized random measure with
independent increments (NRMI).
Lijoi and Pr¨ unster in [8] considered this general construction to determine
an expression of the expected value of the search cost Sn. In the special case
where
ν(dy) =
γ
Γ(1 − γ)
y−1−γ dy γ ∈ (0,1) (2)
they recovered an expression of the limiting expected search cost, as n tends
to inﬁnity, thus recovering a result proved by [7]. Note that if ν is as in (2),
then φi(s) = exp{−(ti − ti−1)sγ} for any s ≥ 0.
Barrera and Paroissin [3] have been able to determine an integral represen-
tation for the Laplace transform φSn of the search cost Sn in terms of the
Laplace transforms φi of the single random weights wi. In doing so they rely
on results proved by Fill and Holst [5]. The expression they obtain is, then,
used to derive a formula for the ﬁrst two moments. From these formulas,
they get an asymptotic equivalent for the Laplace transform of Sn and the
limit of the two ﬁrst moments. Only this last point needs the assumption
that the expectation of Sn is ﬁnite. Two examples are studied: the case of
deterministic weight and the case of gamma weight, which corresponds to
the Dirichlet partition. Notice that, for this case, some limiting results were
proved with an alternative way in [1]. The limiting distribution has been
also derived in the general iid case provided that the expectation µi of wi is
ﬁnite [2].
In the following section we will undertake the approach developed in [3] and
determine the k–th moment of Sn by working directly on φSn.
3 Moments of the stationary search cost
The main tool we are relying on for the evaluation of E[Sk
n] is the Laplace
transform of Sn as displayed in theorem 2.2 of [3] and recalled here below.
3Theorem 1 For a sequence (wi)i≥1 of independent random variables
φSn(s) =
n  
i=1
  ∞
0


  ∞
t
φ′′
i (r)
 
j =i
ht,s,j(r)dr

 dt , (3)
for all s > 0, where for all j ∈ {1,...,n},
ht,s,j(r) = φj(r) + e−s(φj(r − t) − φj(r)) , t > 0,r > 0 .
Using (3), we are able to compute moments of any order of the search
cost Sn. Before doing so we need to introduce the quantity
Mk,n(s) := e−ks  
i =i1 =··· =ik
  ∞
0
dt
  ∞
t
dr φ′′
i (r)
k  
l=1
(φil(r − t) − φil(r))
×
 
j ∈{i,i1,...ik}
[φj(r) + e−s(φj(r − t) − φj(r))] (4)
whose values, at s = 0, will determine the moments of Sn.
Proposition 1 If the (p1,...,pn) are determined by normalizing the in-
crements of a γ-stable subordinator with ti − ti−1 = 1/n in (1) for each
i ∈ {1,...,n}, then
lim
n→∞
Mk,n(0) =



(k!)2
( 1
γ−k−1)k if γ < 1
k+1
∞ otherwise
where (a)k = Γ(a + k)/Γ(a) is the k–th ascending factorial of a.
Proof. Note ﬁrst that φi(s) = exp{−sγ/n} for any s ≥ 0. Moreover
Mk,n(0) =
 
i =i1 =··· =ik
  ∞
0
  ∞
t
φ′′
i (r)
k  
l=1
(φil(r − t) − φil(r))
 
j ∈{i,i1,...ik}
φj(r − t) drdt
=
 
i =i1 =··· =ik
  ∞
0
  ∞
0
φ′′
i (r + t)
k  
l=1
(φil(r) − φil(r + t))
 
j ∈{i,i1,...ik}
φj(r) drdt
=
 
i =i1 =··· =ik
  ∞
0
 
j ∈{i,i1,...ik}
φj(r)
  ∞
0
φ′
i(r + t)
k  
l=1
φ′
il(r + t)
×
k  
m=1
m =l
(φim(r) − φim(r + t))dtdr
4Taking into account the form of φi in the γ–stable case, one has
k  
m=1
m =l
(φim(r) − φim(r + t)) =
 
al∈{0,1}k−1
k  
m=1
m =l
(−1)amφam
im (r + t)φ1−am
im (r)
=
 
al∈{0,1}k−1
(−1)|al|e−
(r+t)γ
n |al| e− rγ
n (k−1−|al|)
where al = (a1,...,al−1,al+1,...,ak) and |al| =
 
m =l am. Summing up,
in the γ–stable case one has
Mk,n(0) =
γ2
n2
 
i =i1 =··· =ik
k  
l=1
 
al∈{0,1}k−1
(−1)|al|
  ∞
0
e−rγ(1− 1
n− k
n)
×
  ∞
0
(r + t)2γ−2 e−
(r+t)γ
n (2+|al|) e− rγ
n (k−1−|al|) dtdr
=
γ2
n2
 
i =i1 =··· =ik
k  
l=1
 
al∈{0,1}k−1
(−1)|al|
  ∞
0
e−rγ(1− 2
n−
|al|
n )
×
  ∞
0
(r + t)2γ−2e−
(r+t)γ
n (|al|+2) dtdr
The change of variable (x,y) = ((r + t)γ,rγ) yields
Mk,n(0) =
1
n2
 
i =i1 =··· =ik
k  
l=1
 
al∈{0,1}k−1
(−1)|al|
  ∞
0
y
1
γ−1e−y(1− 2
n− 1
n|al|)
×
  ∞
y
x
1− 1
γe− x
n(2+|al|) dxdy
5Using formulae (3.381.6) and (7.621.3) in [6], one ﬁnds out that
Mk,n(0) =
1
n2
 
i =i1 =··· =ik
k  
l=1
 
al∈{0,1}k−1
(−1)|al|
  ∞
0
y
1
2γ− 1
2 (n−1(|al| + 2))
− 3
2+ 1
2γ
× e−y(1− 1
n− 1
2n|al|) W 1
2− 1
2γ,1− 1
2γ
 
y
|al| + 2
n
 
dy
=
γ
n2
 
i =i1 =··· =ik
k  
l=1
 
al∈{0,1}k−1
(−1)|al|
2F1
 
2,1;
1
γ
+ 1;1 −
2 + |al|
n
 
=
γ
n2
 
i =i1 =··· =ik
k  
l=1
k−1  
r=0
(−1)r
 
k − 1
r
 
2F1
 
2,1;
1
γ
+ 1;1 −
2 + r
n
 
=
γk(n − 1)(n − 2)···(n − k)
n
k−1  
r=0
(−1)r
 
k − 1
r
 
2F1
 
2,1;
1
γ
+ 1;1 −
2 + r
n
 
Since the Gauss hypergeometric function 2F1 can be rewritten as
2F1
 
2,1;
1
γ
+ 1;1 −
2 + r
n
 
=
∞  
l=0
(2)l(1)l
l!(1 + 1
γ)l
l  
j=0
(−1)j
 
l
j
  
2 + r
n
 j
the expression of Mk,n(0) can be further simpliﬁed as follows
Mk,n(0) =
γk(n − 1)(n − 2)···(n − k)
n
k−1  
r=0
(−1)r
 
k − 1
r
 
×
∞  
j=0
∞  
l=j
(2)l(1)l
l!(1 + 1
γ)l
(−1)j
 
l
j
  
2 + r
n
 j
=
γk(n − 1)(n − 2)···(n − k)
n
k−1  
r=0
(−1)r
 
k − 1
r
  ∞  
j=0
(−1)jaj
 
2 + r
n
 j
where
aj =
∞  
l=j
(2)l(1)l
l!(1 + 1
γ)l
 
l
j
 
6A simple change of variable m = l − j leads to write aj as
aj =
∞  
m=0
(2)m+j(1)m+j
(m + j)!(1 + 1
γ)m+j
 
m + j
j
 
=
∞  
m=0
(m + j + 1)!(m + j)!
j!(1 + 1
γ)m+j
1
m!
=
(j + 1)!
(1 + 1
γ)j
∞  
m=0
(j + 2)m(j + 1)m
(j + 1 + 1
γ)m
(1)m
m!
=
(j + 1)!
(1 + 1
γ)j
2F1(j + 2,j + 1,j + 1 +
1
γ
,1)
and, consequently,
Mk,n(0) =
γk(n − 1)(n − 2)···(n − k)
n
k−1  
r=0
(−1)r
 
k − 1
r
  ∞  
j=0
(−1)j (j + 1)!
(1 + 1
γ)j
× 2F1(j + 2,j + 1,j + 1 +
1
γ
,1)
 
2 + r
n
 j
If one resorts to identity (0.154.6) in [6], it follows that
Mk,n(0) =
γk
 k
l=1(n − l)
n
∞  
j=k−1
(−1)j (j + 1)!
(1 + 1
γ)j
× 2F1(j + 2,j + 1,j + 1 +
1
γ
,1)
k−1  
r=0
(−1)r
 
k − 1
r
  
2 + r
n
 j
Finally, using formula (0.154.5) in [6] one has
Mk,n(0) =
γk
 k
l=1(n − l)
n
 
1
nk−1
k!(k − 1)!
(1 + 1
γ)k−1
2F1(k + 1,k,k +
1
γ
,1)
+o(
1
nk−1)
 
as n → ∞. If γ < 1
k+1 then
2F1(k + 1,k,k +
1
γ
,1) =
Γ(k + 1
γ)Γ(1
γ − k − 1)
Γ(1
γ − 1)Γ(1
γ)
.
7Otherwise the Gauss hypergeometric function diverges (see paragraph 9.102
in [6]). After some little algebra the results is proved. ￿
The study of the limiting behavior of Mk,n(0) is crucial for understanding
the limiting behavior of the moments. Indeed,
E(Sk
n) = (−1)k φ
(k)
Sn(s)
     
s=0
In particular, we have:
E(Sn) = M1,n(0)
E(S2
n) = M1,n(0) + M2,n(0)
E(S3
n) = M1,n(0) + 3M2,n(0) + M3,n(0)
E(S4
n) = M1,n(0) + 7M2,n(0) + 6M3,n(0) + M4,n(0)
E(S5
n) = M1,n(0) + 15M2,n(0) + 25M3,n(0) + 10M4,n(0) + M5,n(0)
···
In general
E(Sk
n) = a
(k)
1 M1,n(0) + ··· + a
(k)
k Mk,n(0) (5)
where
a
(k)
1 = 1
a
(k)
l = a
(k−1)
l−1 + la
(k−1)
l l = 2,...,k − 1
a
(k)
k = 1
(6)
The last recursion follows from the fact that
M′
k,n(s) = −kMk,n(s) − Mk+1,n(s)
From proposition 1 and equation (5), we have the following theorem.
Theorem 2 If the (p1,...,pn) are determined by normalizing the incre-
ments of a γ-stable subordinator with ti − ti−1 = 1/n in (1) for each i ∈
{1,...,n}, then
lim
n→∞E(Sk
n) =
   k
l=1
(l!)2
( 1
γ−l−1)la
(k)
l if γ < 1
k+1
∞ otherwise
8The previous theorem allows to calculate all the moments of the limiting
search cost distribution in the stable case. For example the second moment
is
lim
n→∞
E(S2
n) =
 
γ(1+γ)
(1−3γ)(1−2γ) if γ < 1
3
∞ otherwise
and the third moment
lim
n→∞E(S3
n) =
 
γ(1+5γ)
(1−4γ)(1−3γ)(1−2γ) if γ < 1
4
∞ otherwise
Acknowledgements. Antonio Lijoi is partially supported by MIUR grant
no. 2008MK3AFZ and by the Collegio Carlo Alberto project “New ap-
proaches to stochastic modeling in Economics via Bayesian Nonparamet-
rics”.
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10

A note on the problem of heaps.

Distributional results for means of normalized random measures of indepenent increments.

Finite automata and models of simple forms of behavior.

Limiting search cost distribution for the move-to-front rule with random request probabilities.

On the distribution of search cost for the move-to-front rule.

On the distribution of the stationary search cost for the move-to-front rule with random weights.

Random discrete distributions.

Size-biased permutation of Dirichlet partitions and search-cost distribution.

Tables of integrals, series and products

The heaps process, libraries, and size-biased permutations.

Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

http://economia.unipv.it/docs/dipeco/quad/ps/RePEc/pav/wpaper/q113.pdf

Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

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