Functional equations with multiple recursive terms

Ivo J.B.F. Adan; Onno J. Boxma; Jacques A.C. Resing

doi:10.1007/s11134-022-09861-9

Functional equations with multiple recursive terms

Ivo J.B.F. Adan (Corresponding author), Onno J. Boxma, Jacques A.C. Resing

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Abstract

In this paper, we study a functional equation for generating functions of the form
f(z) = g(z) ∑_i=1,...,M p_if(α_i(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the α_i(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

Original language	English
Pages (from-to)	7-23
Number of pages	17
Journal	Queueing Systems
Volume	102
Issue number	1-2
Early online date	4 Sept 2022
DOIs	https://doi.org/10.1007/s11134-022-09861-9
Publication status	Published - Oct 2022

Keywords

Generating function
Laplace–Stieltjes transform
Queueing model
Recursion
Stochastic process

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abstract = "In this paper, we study a functional equation for generating functions of the form f(z) = g(z) ∑i=1,...,M pi f(αi(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the αi(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.",

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AU - Adan, Ivo J.B.F.

AU - Boxma, Onno J.

AU - Resing, Jacques A.C.

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N2 - In this paper, we study a functional equation for generating functions of the form f(z) = g(z) ∑i=1,...,M pi f(αi(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the αi(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

AB - In this paper, we study a functional equation for generating functions of the form f(z) = g(z) ∑i=1,...,M pi f(αi(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the αi(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

KW - Generating function

KW - Laplace–Stieltjes transform

KW - Queueing model

KW - Recursion

KW - Stochastic process

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Functional equations with multiple recursive terms

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