scholarly journals On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Frosso S. Makri ◽  
Zaharias M. Psillakis
Keyword(s):  
Stochastic Processes ◽  
Numerical Results ◽  
Urn Models ◽  
Urn Model ◽  
Critical Events ◽  
Expected Number ◽  
Limited Length ◽  
Binary Strings

The expected number of 0-1 strings of a limited length is a potentially useful index of the behavior of stochastic processes describing the occurrence of critical events (e.g., records, extremes, and exceedances). Such model sequences might be derived by a Hoppe-Polya or a Polya-Eggenberger urn model interpreting the drawings of white balls as occurrences of critical events. Numerical results, concerning average numbers of constrained length interruptions of records as well as how on the average subsequent exceedances are separated, demonstrate further certain urn models.

scholarly journals The Number of Moves of the Largest Disc in Shortest Paths on Hanoi Graphs

10.37236/4252 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Simon Aumann ◽  
Katharina A.M. Götz ◽  
Andreas M. Hinz ◽  
Ciril Petr
Keyword(s):  
Shortest Paths ◽  
General Setting ◽  
Optimal Number ◽  
Numerical Results ◽  
Breadth First Search ◽  
Tower Of Hanoi ◽  
Optimal Paths ◽  
Widespread Interest ◽  
Binary Strings

In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs $H_p^n$ in a more general setting. Here $p$ stands for the number of pegs and $n$ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length $p-1$ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all $n\geqslant p(p-2)$, of optimal paths where $p-1$ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.

scholarly journals On Generalized Pólya Urn Models

2013 ◽  
Vol 50 (4) ◽  
pp. 1169-1186 ◽  
Author(s):  
May-Ru Chen ◽  
Markus Kuba
Keyword(s):  
Discrete Time ◽  
Higher Moments ◽  
Urn Models ◽  
Urn Model ◽  
Time Step ◽  
Model Case ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

OPTIMAL INSPECTION ORDER WHEN PROCESS’ FAILURE RATE IS CONSTANT

1996 ◽  
Vol 03 (01) ◽  
pp. 25-41 ◽  
Author(s):  
QI-MING HE ◽  
YIGAL GERCHAK ◽  
ABRAHAM GROSFELD-NIR
Keyword(s):  
Failure Rate ◽  
Optimal Policy ◽  
A Priori ◽  
Numerical Results ◽  
Expected Number ◽  
Process Failure ◽  
Inspection Policy

Suppose that a lot has been produced by a process with a constant failure rate. So either the entire lot is good, or all units up to some point are good and from that point on are all defective. We wish to determine the order in which units in such lot should be inspected so as to minimize the expected number of inspections needed to identify all defectives. Unlike previous work in this area, we do not a priori assume that the last unit in the lot is defective, and that key difference turns out to dramatically influence the nature of the optimal inspection policy and the expected number of units inspected. After analyzing the optimal policy, we suggest a very simple and intuitive heuristic, which turns out to perform extremely well. Numerical results are provided.

scholarly journals Model biological population by branching processes

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Antoanela Terzieva
Keyword(s):  
Population Dynamics ◽  
Cell Division ◽  
Stochastic Processes ◽  
End Of Life ◽  
Branching Processes ◽  
Expected Number ◽  
Biological Population ◽  
Different Types ◽  
Unicellular Organisms ◽  
Number Of Particles

Consider a population of two or more different types of cells that at the end of life create two new cells through cell division. We model the population dynamics using a multitype branching stochastic processes. Under consideration are processes of Bieneme-Galton-Watson and of Bellman-Harris for the Markovian case.   drawn Conclusions about the expected number of particles of each type after a random time are drawn. The proposed models could be applicable not only for populations of a unicellular organisms, but also for sets of objects which operate a certain period of time and then split into two new objects or change their type.

STATISTICAL APPROACH FOR OPEN BONUS MALUS

2013 ◽  
Vol 44 (1) ◽  
pp. 63-83 ◽  
Author(s):  
Gracinda Rita Guerreiro ◽  
João Tiago Mexia ◽  
Maria de Fátima Miguens
Keyword(s):  
Statistical Approach ◽  
Confidence Regions ◽  
Numerical Results ◽  
Asymptotic Distributions ◽  
Expected Number ◽  
System Evolution ◽  
Portfolio Approach ◽  
New Policies

AbstractIn this paper, following an open portfolio approach, we show how to estimate a Bonus-malus system evolution.Considering a model for the number of new annual policies, we obtain ML estimators, asymptotic distributions and confidence regions for the expected number of new policies entering the portfolio in each year, as well as for the expected number and proportion of insureds in each bonus class, by year of enrollment. Confidence regions for the distribution of policyholders result in confidence regions for optimal bonus scales.Our treatment is illustrated by an example with numerical results.

scholarly journals On Generalized Pólya Urn Models

2013 ◽  
Vol 50 (04) ◽  
pp. 1169-1186 ◽  
Author(s):  
May-Ru Chen ◽  
Markus Kuba
Keyword(s):  
Discrete Time ◽  
Higher Moments ◽  
Urn Models ◽  
Urn Model ◽  
Time Step ◽  
Model Case ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

scholarly journals Strong Laws for Balanced Triangular Urns

2009 ◽  
Vol 46 (2) ◽  
pp. 571-584 ◽  
Author(s):  
Arup Bose ◽  
Amites Dasgupta ◽  
Krishanu Maulik
Keyword(s):  
Urn Models ◽  
Urn Model ◽  
Strong Laws

Consider an urn model whose replacement matrix is triangular, has all nonnegative entries, and the row sums are all equal to 1. We obtain strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use these strong laws to study further behavior of certain three-color urn models.

scholarly journals Subgraphs in preferential attachment models

2019 ◽  
Vol 51 (03) ◽  
pp. 898-926
Author(s):  
Alessandro Garavaglia ◽  
Clara Stegehuis
Keyword(s):  
Power Law ◽  
Optimization Problem ◽  
Preferential Attachment ◽  
Urn Model ◽  
Expected Number ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Attachment Model ◽  
Law Degree

AbstractWe consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$ . For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

Modeling of repairable systems

1993 ◽  
Vol 25 (04) ◽  
pp. 926-938
Author(s):  
Nader Ebrahimi
Keyword(s):  
Stochastic Processes ◽  
Traditional Approach ◽  
Physical Characteristics ◽  
Time Dependent ◽  
Repairable System ◽  
Repairable Systems ◽  
Expected Number ◽  
Inference Procedure ◽  
Repair Models

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes. Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

scholarly journals Strong Laws for Balanced Triangular Urns

2009 ◽  
Vol 46 (02) ◽  
pp. 571-584
Author(s):  
Arup Bose ◽  
Amites Dasgupta ◽  
Krishanu Maulik
Keyword(s):  
Urn Models ◽  
Urn Model ◽  
Strong Laws

Consider an urn model whose replacement matrix is triangular, has all nonnegative entries, and the row sums are all equal to 1. We obtain strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use these strong laws to study further behavior of certain three-color urn models.

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