Functional equations as an important analytic method in stochastic modelling and in combinatorics

TL;DR

This paper highlights the significance of functional equations in analyzing stochastic systems and combinatorics, demonstrating methods to solve them and exploring their properties and applications across various models.

Contribution

It reviews the development of functional equation techniques in stochastic modeling and combinatorics, including boundary value problem reduction and classification of solutions.

Findings

Functional equations can be reduced to boundary value problems for solutions.

Solutions often exhibit rational, algebraic, or holonomic properties.

Asymptotic analysis provides insights into the behavior of solutions.

Abstract

Functional equations (FE) arise quite naturally in the analysis of stochastic systems of different kinds : queueing and telecommunication networks, random walks, enumeration of planar lattice walks, etc. Frequently, the object is to determine the probability generating function of some positive random vector in $Z\_n^+$. Although the situation n = 1 is more classical, we quote an interesting non local functional equation which appeared in modelling a divide and conquer protocol for a muti-access broadcast channel. As for n = 2, we outline the theory reducing these linear FEs to boundary value problems of Riemann-Hilbert-Carleman type, with closed form integral solutions. Typical queueing examples analyzed over the last 45 years are sketched. Furthermore, it is also sometimes possible to determine the nature of the functions (e.g., rational, algebraic, holonomic), as illustrated in a combinatorial context, where asymptotics are briefly tackled. For general situations (e.g., big jumps, or n $\ge$ 3), only prospective comments are made, because then no concrete theory exists.