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

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 ≥ 3), only prospective comments are made, because then no concrete theory exists.
Type de document :
Communication dans un congrès
ACMPT 2017 - Analytical and Computational Methods in Probability Theory and its Applications, Oct 2017, Moscou, Russia. pp.1-25, 2017, Analytic and Computational Methods in Probability Theory and its Applications
Liste complète des métadonnées

https://hal.inria.fr/hal-01657154
Contributeur : Guy Fayolle <>
Soumis le : mercredi 6 décembre 2017 - 20:11:11
Dernière modification le : jeudi 26 avril 2018 - 10:27:54

Fichiers

Fayolle-ACPMT-2017.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01657154, version 1
  • ARXIV : 1712.02271

Collections

Citation

Guy Fayolle. Functional equations as an important analytic method in stochastic modelling and in combinatorics. ACMPT 2017 - Analytical and Computational Methods in Probability Theory and its Applications, Oct 2017, Moscou, Russia. pp.1-25, 2017, Analytic and Computational Methods in Probability Theory and its Applications. 〈hal-01657154〉

Partager

Métriques

Consultations de la notice

153

Téléchargements de fichiers

30