A Markovian Perspective on the Classical Occupancy Problem with a Generalization to Pure Birth Processes

TL;DR

This paper offers a Markovian framework for the classical occupancy problem, deriving new formulas and recursive systems, and extends these results to pure birth processes, enhancing analytical tools for stochastic modeling.

Contribution

It introduces a novel Markovian approach, new recursive relations, and extends occupancy problem analysis to pure birth processes.

Findings

Derived new probability mass and distribution functions

Developed a sparse bidiagonal recursive system

Extended results to pure birth processes

Abstract

We study the classical occupancy problem from the viewpoint of its embedding Markov chain. We derive new expressions for the probability mass function and (complementary) distribution function in generalized form. Furthermore, we derive a completely novel sparse bidiagonal system of recursion relations for the (complementary) distribution function and provide its efficient matrix implementation. Importantly, we generalize these results to the entire class of discrete-time pure birth processes.