A Coupon Collector based approximation for LRU cache hits under Zipf requests

The urgency of the problem of testing storage devices of modern computer systems is shown. The mathematical models of their faults and the methods used for testing the most complex cases by classical march tests are investigated. Passive pattern sensitive faults (PNPSFk) are allocated, in which arbitrary k from N memory cells participate, where k << N, and N is the memory capacity in bits. For these faults, analytical expressions are given for the minimum and maximum fault coverage that is achievable within the march tests. The concept of a primitive is defined, which describes in terms of march test elements the conditions for activation and fault detection of PNPSFk of storage devices. Examples of march tests with maximum fault coverage, as well as march tests with a minimum time complexity equal to 18N are given. The efficiency of a single application of tests such as MATS ++, March C− and March PS is investigated for different number of k ≤ 9 memory cells involved in PNPSFk fault. The applicability of multiple testing with variable address sequences is substantiated, when the use of random sequences of addresses is proposed. Analytical expressions are given for the fault coverage of complex PNPSFk faults depending on the multiplicity of the test. In addition, the estimates of the mean value of the multiplicity of the MATS++, March C− and March PS tests, obtained on the basis of a mathematical model describing the problem of the coupon collector, and ensuring the detection of all k2k PNPSFk faults are given. The validity of analytical estimates is experimentally shown and the high efficiency of PNPSFk fault detection is confirmed by tests of the March PS type.

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Antiduality and Möbius monotonicity: generalized coupon collector problem

ESAIM Probability and Statistics ◽

10.1051/ps/2019004 ◽

2019 ◽

Vol 23 ◽

pp. 739-769

Author(s):

Paweł Lorek

Keyword(s):

Markov Chain ◽

State Space ◽

Partial Ordering ◽

Absorption Time ◽

Absorbing Markov Chain ◽

Coupon Collector ◽

Strong Stationary Duality ◽

Finite State ◽

A Chain ◽

Finite State Space

For a given absorbing Markov chain X* on a finite state space, a chain X is a sharp antidual of X* if the fastest strong stationary time (FSST) of X is equal, in distribution, to the absorption time of X*. In this paper, we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence – utilizing known results on the limiting distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its FSST is distributed as a prescribed mixture of sums of geometric random variables.

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Some upper and lower bounds on the coupon collector problem

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2005.12.011 ◽

2007 ◽

Vol 200 (1) ◽

pp. 154-167 ◽

Cited By ~ 8

Author(s):

S. Shioda

Keyword(s):

Lower Bounds ◽

Upper And Lower Bounds ◽

Coupon Collector

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The Generalised Coupon Collector Problem

Journal of Applied Probability ◽

10.1017/s0021900200004599 ◽

2008 ◽

Vol 45 (03) ◽

pp. 621-629 ◽

Cited By ~ 5

Author(s):

Peter Neal

Keyword(s):

Probability Distribution ◽

Asymptotic Distribution ◽

Birthday Problem ◽

Coupon Collector ◽

Poisson Limit ◽

Special Case ◽

Independent And Identically Distributed

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

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Optimization results for a generalized coupon collector problem

Journal of Applied Probability ◽

10.1017/jpr.2016.27 ◽

2016 ◽

Vol 53 (2) ◽

pp. 622-629 ◽

Cited By ~ 2

Author(s):

Emmanuelle Anceaume ◽

Yann Busnel ◽

Ernst Schulte-Geers ◽

Bruno Sericola

Keyword(s):

Probability Distribution ◽

Uniform Distribution ◽

Closed Subset ◽

Probability Distributions ◽

Fixed Number ◽

Coupon Collector ◽

The One

Abstract In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

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Information Transmission under Random Emission Constraints

Combinatorics Probability Computing ◽

10.1017/s096354831400039x ◽

2014 ◽

Vol 23 (6) ◽

pp. 973-1009 ◽

Cited By ~ 3

Author(s):

FRANCIS COMETS ◽

FRANÇOIS DELARUE ◽

RENÉ SCHOTT

Keyword(s):

Limit Theorems ◽

Law Of Large Numbers ◽

Large Deviation Principle ◽

Large Deviation ◽

Total Duration ◽

Limited Resources ◽

Large Numbers ◽

Coupon Collector ◽

Selection Of ◽

Quantities Of Interest

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n → ∞, for the proportion of informed vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton–Watson tree with respect to the success epochs of the coupon collector problem.

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