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1

Leader election : A Markov chain approach

Grübel R., Hagemann K.

Mathematica Applicanda

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2016

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Vol. 44, No. 1

113--134

EN

A well-studied randomized election algorithm proceeds as follows: In each round the remaining candidates each toss a coin and leave the competition if they obtain heads. Of interest is the number of rounds required and the number of winners, both related to maxima of geometric random samples, as well as the number of remaining participants as a function of the number of rounds. We introduce two related Markov chains and use ideas and methods from discrete potential theory to analyse the respective asymptotic behaviour as the initial number of participants grows. One of the tools used is the approach via the Rényi-Sukhatme representation of exponential order statistics, which was first used in the leader election context by Bruss and Grübel in [BG03].

PL

W artykule przywołany jest dobrze znany i szczegółowo zbadany następujący algorytm losowego wyboru lidera. W kolejnych krokach każdy kandydat rzuca monetą. Jeśli wyrzuci orła, to kończy eliminację (nie przechodzi do następnej tury). Interesuje nas liczba rund do wyłonienia lidera bądź liczba pozostałych kandydatów w powiązaniu z maksimum ciągu zmiennych losowych o rozkładzie geometrycznym. Również wyznaczamy rozkład liczby pozostałych kandydatów jako funkcji liczby tur. W celu odpowiedzi na postawione pytania konstruowane są dwa powiązane ze sobą łańcuchy Markowa. Wykorzystując metody teorii potencjału badana jest asymptotyka przy rosnącej początkowej liczbie kandydatów. Jednym z wykorzystywanych narzędzi jest reprezentacja Rényi-Sukhatme dla statystyk porządkowych rozkładu wykładniczego, która została po raz pierwszy użyta do zagadnienia wyborów lidera przez Brussa i Grübela w [BG03].

2

Stochastic analysis of dis-similar standby system with discrete failure, inspection and replacement policy

Bhatti J., Chitkara A. K., Kakkar M. K.

Demonstratio Mathematica

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2016

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Vol. 49, nr 2

224--235

EN

The device named autoclave used in pharma industries has been studied in this paper. This study provides the discussion how to obtain the reliability testing strategy of two dis-similar parallel units. The system is considered to be in operative condition if at least one out of two is in operative state. A single repair facility is available for repairing both kinds of failed units. In addition to repair mechanism, inspection policy has also been introduced for failed automatic unit. But the manual one does not require any such supervision facilty. Various essential measures of system effectiveness such as mean time to system failure (MTSF), steady state availability, busy period of supervisor and repairman are examined probabilistically by using geometric distribution and regenerative point techniques. A graph has been plotted to represent the behaviour of profit function and MTSF with respect to different failure and repair rate.

3

On Flooding in the Presence of Random Faults

Cichoń J., Klonowski M.

Fundamenta Informaticae

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2013

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Vol. 123, nr 3

273--287

EN

In this paper we study the efficiency of information flooding protocols in various communication networks, and in the presence of random faults. We show big differences between the flooding performance of networks with a seemingly similar structure. Since real-life systems usually consist of a moderate number of devices, the analysis presented in this paper is not limited to the asymptotic behavior of the flooding protocol. Instead, exact formulas are provided whenever possible. The presented results can be useful building blocks for the analysis of other, more sophisticated protocols. In particular, they may be used for planning and analyzing sensors network deployed in an environment subject to communication failures.

4

Distributional properties of the negative binomial Lévy process

Kozubowski T. J., Podgórski K.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 1

43--71

EN

The geometric distribution leads to a Lévy process parameterized by the probability of success. The resulting negative binomial process (NBP) is a purely jump and non-decreasing process with general negative binomial marginal distributions. We review various stochastic mechanisms leading to this process, and study its distributional structure. These results enable us to establish strong convergence of the NBP in the supremum norm to the gamma process, and lead to a straightforward algorithm for simulating sample paths. We also include a brief discussion of estimation of the NPB parameters, and present an example from hydrology illustrating possible applications of this model.

5

The construction of the random variable with the geometric distribution

Habaj R.

Journal of Mathematics and Applications

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2006

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Vol. 28

71-74

EN

In probability theory the random variable with the geometric distribution is used very often. But unfortunateIy the literature shows us readymade shape of the probability function. However its origin is neglected. This paper shows the full educe of the probability function of the random variable with the geometric distribution.

6

Linearity of regression for adjacent order statistics-discrete case

Dembińska A.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

711-721

EN

Let X1X2 be a random sample from a discrete distribution on the integers with the corresponding order statistics X1:2 m) = ajXj:2+bj where m is a nonnegative integer, a, b -some constants and i,j należy {1,2}, i = j. However some of the main results of that paper are not complete or incorrect. In this note, developing Nagaraja's ideas, the family of distributions with the linearity of regression property is completely characterized.

7

Geometric approximation of mixture of discrete life time distributions

Popowska B.

Fasciculi Mathematici

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1998

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Nr 28

135-140

EN

We consider a mixture of a distribution from the class (D)NBUE and a distribution, which describes the life time of smal1 part of lower reliability elements. Given the first two moments of this mixture, the distribution can be approximated by a geometric distribution with some parametr Ds. mixture of discrete distributions, geometric distribution, error of approximation is also estimated.