Sequential Filling of a Line by Intervals Placed at Random and Its Application to Linear Adsorption

Mackenzie, J. K.

doi:10.1063/1.1733154

Cited by 109 publications

(64 citation statements)

References 11 publications

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“…Similarly, the combined results of [5] and [6] showed that the variance ofĨ x has the estimate in (2), once the limit t → ∞ is taken; in this limit,…”

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confidence: 82%

“…In [3], it is observed that one can adapt the technique to the interval packing problem so as to obtain a central limit theorem for any fixed t. The details of a rigorous proof can be found in [2]. Bankovi [1] investigated the distribution of vacant intervals for the parking problem, as did Mackenzie [6] for a discretized version of the problem. This paper generalizes their results to the interval packing problem.…”

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confidence: 99%

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Interval packing: the vacant interval distribution

Coffman¹,

Flatto²,

'c}³

2000

Ann. Appl. Probab.

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Starting at time 0, unit-length intervals arrive and are placed on the positive real line by a unit-intensity Poisson process in two dimensions; the left endpoints of intervals appear at the rate of 1 per unit time per unit distance. An arrival is accepted if and only if, for some given x, the interval is contained in 0 x and overlaps no interval already accepted. This stochastic, on-line interval packing problem generalizes the classical parking problem, the latter corresponding only to the absorbing states of the interval packing process, where successive packed intervals are separated by gaps less than or equal to 1 in length.In earlier work, the authors studied the distribution of the number of intervals accepted during 0 t . This paper is concerned with the vacant intervals (gaps) between consecutive packed intervals. Let p t y be the limit x → ∞ of the fraction of the gaps at time t which are at most y in length. We prove that 1 − e −v /v dv . We briefly discuss the recent importance acquired by interval packing models in connection with resource-reservation systems. In these applications, our vacant intervals correspond to times between consecutive reservation intervals. The results of this paper improve our understanding of the fragmentation of time that occurs in reservation systems.1. Introduction. Unit intervals arrive at random times and at random locations in R + . The arrival times and left endpoints comprise a unit-intensity Poisson process in two dimensions. Thus, the probability of an arrival in the time interval t t + dt with left endpoint in y y + dy is dt dy + o dt dy . The number of intervals packed entirely within 0 x during 0 t is denoted by I t x and has a mean denoted by K t x . In [3], it is shown that, for any given T > 0,

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“…Similarly, the combined results of [5] and [6] showed that the variance ofĨ x has the estimate in (2), once the limit t → ∞ is taken; in this limit,…”

mentioning

confidence: 82%

mentioning

confidence: 99%

Interval packing: the vacant interval distribution

Coffman¹,

Flatto²,

'c}³

2000

Ann. Appl. Probab.

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“…What is the expected number of persons to sit down? Solutions to this problem were provided by Friedman, Rothman and MacKenzie [3,7] who show that as n tends to infinity the expected fraction of the seats that are occupied goes to 1 2 − 1 2e 2 . (For a nice exposition of this and related problems see [1].)…”

Section: Related Workmentioning

confidence: 99%

Maintaining Privacy on a Line

Kranakis

Kriz̧anc

2011

Theory Comput Syst

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Consider a situation where people have to choose among a sequence of n linearly ordered positions to perform some task requiring a certain amount of privacy. Which position should one choose so as to maximize one's privacy, i.e., minimize the chances that one of your neighboring positions becomes occupied by a later arrival? In this paper, we attempt to answer this question under a variety of models for the behavior of the later arrivals. Our results suggest that for the most part one should probably choose one of the extreme positions (with some interesting exceptions). We also suggest a number of variations on the problem that lead to many open problems.

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“…The gap distribution for various versions of the random sequential adsorption model has been studied in detail in [8,9] since it has direct consequences among others for the distribution of cracks in brittle materials [10]. In the most simple continuous case (random car parking model) the spacing distribution P (D) behaves like [11]. But the results from London demonstrated clearly that in reality P (D) → 0 as D → 0.…”

Section: Theorymentioning

confidence: 99%