12,110 research outputs found
Limiting search cost distribution for the move-to-front rule with random request probabilities
Consider a list of files whose popularities are random. These files are
updated according to the move-to-front rule and we consider the induced Markov
chain at equilibrium. We give the exact limiting distribution of the
search-cost per item as tends to infinity. Some examples are supplied.Comment: move-to-front, search cost, random discrete distribution, limiting
distribution, size biased permutatio
Limiting behavior of the search cost distribution for the move-to-front rule in the stable case
Move-to-front rule is a heuristic updating a list of n items according to
requests. Items are required with unknown probabilities (or popularities). The
induced Markov chain is known to be ergodic. One main problem is the study of
the distribution of the search cost dened as the position of the required item.
Here we first establish the link between two recent papers that both extend
results proved by Kingman 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 innity
Limiting behavior of the search cost distribution for the move-to-front rule in the stable case
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
Limiting behavior of the search cost distribution for the move-to-front rule in the stable case
Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or popularities). The induced Markov chain is known to be ergodic. 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 of Barrera and Paroissin and Lijoi and Pruenster that both extend results proved by Kingman 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
The limiting move-to-front search-cost in law of large numbers asymptotic regimes
We explicitly compute the limiting transient distribution of the search-cost
in the move-to-front Markov chain when the number of objects tends to infinity,
for general families of deterministic or random request rates. Our techniques
are based on a "law of large numbers for random partitions," a scaling limit
that allows us to exactly compute limiting expectation of empirical functionals
of the request probabilities of objects. In particular, we show that the
limiting search-cost can be split at an explicit deterministic threshold into
one random variable in equilibrium, and a second one related to the initial
ordering of the list. Our results ensure the stability of the limiting
search-cost under general perturbations of the request probabilities. We
provide the description of the limiting transient behavior in several examples
where only the stationary regime is known, and discuss the range of validity of
our scaling limit.Comment: Published in at http://dx.doi.org/10.1214/09-AAP635 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Limiting behaviour of the stationary search cost distribution driven by a generalized gamma process
Consider a list of labeled objects that are organized in a heap. At each time, object j is selected with probability pj and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities pj's are defined as the normalization of the increments of a subordinator. In this setting, we provide an exact formula for the moments of any order of the stationary search cost distribution. We illustrate the new findings in the case of a generalized gamma subordinator and deal with an extension to the two-parameter Poisson-Dirichlet process, also known as Pitman-Yor process
The Employed, the Unemployed, and the Unemployable: Directed Search with Worker Heterogeneity
We examine the implications of worker heterogeneity on the equilibrium matching process, using a directed search model. Worker abilities are selected from a general distribution, subject to some weak regularity requirements, and the firms direct their job offers to workers. We identify conditions under which some fraction of the workforce will be "unemployable": no firm will approach them even though they offer positive surplus. For large markets we derive a simple closed form expression for the equilibrum matching function. This function has constant returns to scale and two new terms, which are functions of the underlying distribution of worker productivities: the percentage of unemployable workers, and a measure of heterogeneity (?).The equilibrium unemployment rate is increasing in ? and, under certain circumstances, is increasing in the productivity of highly skilled workers, despite endogenous entry. A key empirical prediction of the theory is that ? ? 1. We examine this prediction, using data from several countries.Directed search; worker heterogeneity; unemployment
Stochastic ranking process with space-time dependent intensities
We consider the stochastic ranking process with space-time dependent jump
rates for the particles. The process is a simplified model of the time
evolution of the rankings such as sales ranks at online bookstores. We prove
that the joint empirical distribution of jump rate and scaled position
converges almost surely to a deterministic distribution, and also the tagged
particle processes converge almost surely, in the infinite particle limit. The
limit distribution is characterized by a system of inviscid Burgers-like
integral-partial differential equations with evaporation terms, and the limit
process of a tagged particle is a motion along a characteristic curve of the
differential equations except at its Poisson times of jumps to the origin
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