Analytic combinatorics aims to enable precise quantitative predictions of the properties of
large combinatorial structures. The theory has emerged over recent decades as essential …

This paper develops a unified enumerative and asymptotic theory of directed two-
dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specified …

Many important stochastic counting models can be written as general birth‐death processes
(BDPs). BDPs are continuous‐time Markov chains on the non‐negative integers in which …

We establish a fundamental isomorphism between discrete-time balanced urn processes
and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple …

A birth–death process is a continuous-time Markov chain that counts the number of particles
in a system over time. In the general process with n current particles, a new particle is born …

In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects
of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of …

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of
“particles” in a system over time. While widely used in population biology, genetics, and …

In this paper, we study birth/immigration-death processes under mild (binomial)
catastrophes. We obtain explicit expressions for both the time-dependent (transient) and the …

This report is part of a series whose aim is to present in a synthetic way the major methods
and models in analytic combinatorics. Here, we detail the case of rational and algebraic …

We show in this paper that the computation of the distribution of the sojourn time of an
arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS …