In this note we study a class of 2× 2 Pólya-Eggenberger urn models, with ball transition matrix given by M = ( 0 −b c −d ) , for arbitrary parameters b, c ∈ N, assuming the balancing condition d = c+ b. This urn model serves as a stochastic model in biology describing cannibalistic behavior of populations. The special case b = c = 1 and d = 2 was studied by Pittel [16] using asymptotic approxim...

We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelia...

In this work we analyze a class of 2× 2 Pólya-Eggenberger urn models with ball replacement matrix M = (−a 0 c −d ) , a, d ∈ N, and c = p · a with p ∈ N0. We obtain limiting distributions for this 2 × 2 urn model by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain ...

Suppose an urn contains M balls, of different types, which are removed from the urn in a uniform random manner. In the hypergeometric coupon collection problem, we are interested in the set of balls that have been removed at the moment when at least one ball of each type has been removed. In its dual, we are interested in the set of removed balls at the first moment that this set contains all o...

We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for t...

The User Requirements Notation (URN), standardized by the International Telecommunication Union in 2008, is used to model and analyze requirements with goals and scenarios. This paper describes the first ten years of development of URN, and discusses ongoing efforts targeting the next ten years. We did a study inspired by the systematic literature review approach, querying five major search eng...

The User Requirements Notation (URN), standardized by the International Telecommunication Union in 2008, is used to model and analyze requirements with goals and scenarios. This paper describes the first ten years of development of URN, and discusses ongoing efforts targeting the next ten years. We did a study inspired by the systematic literature review approach, querying five major search eng...

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonali...

The User Requirements Notation (URN), standardized by the International Telecommunication Union in 2008, is used to model and analyze requirements with goals and scenarios. This paper describes the first ten years of development of URN, and discusses ongoing efforts targeting the next ten years. We did a study inspired by the systematic literature review approach, querying five major search eng...