A probabilistic scheme of independent random elements distributed over a finite lattice. I. Exact probability distributions of functionals of union of random elements

A probabilistic scheme of independent random elements with values in a finite lattice is introduced. For the scheme, exact expressions of probability distributions of a functional of general form and union of random corank elements are obtained. Various probabilistic combinatorial problems (on probability distributions of the number of uncovered points in a generalized scheme of grouped distribution of particles, on the number of connectivity components of a random hypergraph, on the number of solutions of a system of random linear equations over a finite ring with unity, etc.) are demonstrated to be naturally formulated in terms of the scheme constructed.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 3–15, September–October 2004.     

Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 106–115, November–December 2008.     

A Model of Inventory Control. II

A model of inventory control is considered. It is described by a semi-Markovian random walk with a negative drift at an angle of (0° < < 90°), with positive random jumps, delays, and absorbing screens at zero and at a #gt; 0. The Laplace transformation is found for the distribution of the first moment of the stoking up of a warehouse, and its first and second moments are explicitly obtained.