On vector-valued functional equations with multiple recursive terms

Ioannis Dimitriou; Ivo J.B.F. Adan

arxiv: 2507.01871 · v3 · submitted 2025-07-02 · 🧮 math.PR

On vector-valued functional equations with multiple recursive terms

Ioannis Dimitriou , Ivo J.B.F. Adan This is my paper

Pith reviewed 2026-05-19 06:26 UTC · model grok-4.3

classification 🧮 math.PR

keywords vector-valued functional equationsmultiple recursive termsmultiplicative Lindley recursionssemi-Markovian queueingvector autoregressive processesstochastic recursions

0 comments

The pith

A detailed framework solves vector-valued functional equations with multiple recursive terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a systematic framework for solving vector-valued functional equations containing multiple recursive terms. These equations arise from vector-valued multiplicative Lindley-type recursions. The approach matters because it applies directly to models of waiting times and state evolution in semi-Markovian queueing systems as well as vector autoregressive processes. By supplying a solution method the framework makes previously intractable equations amenable to analysis. This enables derivation of performance measures and distributional results in multi-dimensional stochastic models.

Core claim

We study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. We provide a detailed framework for the solution of such equations. Our theoretical results are applied in a wide range of semi-Markovian queueing, and vector-valued autoregressive processes.

What carries the argument

the detailed framework for solving vector-valued functional equations with multiple recursive terms from multiplicative Lindley recursions

Load-bearing premise

The functional equations arise naturally from vector-valued multiplicative Lindley-type recursions in the context of semi-Markovian queueing and autoregressive processes.

What would settle it

Constructing an explicit two-dimensional multiplicative Lindley recursion, deriving its functional equation, and verifying that the framework produces an incorrect or inconsistent solution would disprove the central claim.

read the original abstract

In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. We provide a detailed framework for the solution of such equations. Our theoretical results are applied in a wide range of semi-Markovian queueing, and vector-valued autoregressive processes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a workable framework for vector-valued functional equations with multiple recursive terms from Lindley-type recursions, but the advance looks like a targeted extension rather than a major shift.

read the letter

Hi, the main takeaway is that the authors lay out a solution method for vector-valued functional equations that include several recursive terms, equations that show up when you look at vector multiplicative Lindley recursions in queueing and autoregressive models. They start from the recursions, write down the corresponding functional equations, and then give a way to solve them, with the results fed back into semi-Markovian queues and vector AR processes. That connection between the abstract equations and the applied models is the part that feels most useful on a first read. The derivations appear systematic and the applications are stated clearly enough that a specialist could check them against known scalar cases. What they do well is keep the focus on the vector setting and multiple terms without overclaiming generality. The framework seems to handle the added complexity in a direct way, and the queueing and time-series examples are the natural places where this kind of tool would be tried. On the soft spots, the novelty feels incremental; it is not obvious from the setup whether the vector case requires genuinely new techniques or mostly reduces to solving a coupled system that could have been handled with existing scalar methods plus linear algebra. A reader would want to see explicit comparisons to prior work on functional equations in stochastic processes and at least one fully worked numerical example to judge how practical the method is. Minor gaps like that are common in this style of paper and do not sink the central claim. This is written for people already working on exact analysis of multi-dimensional recursive distributions in applied probability. A queueing theorist or someone doing vector time-series modeling would get the most out of it, while a general probabilist might find the scope narrow. It deserves a serious referee because the claims are concrete, the math is checkable, and the applications are in an active area. I would send it out for review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript studies vector-valued functional equations with multiple recursive terms arising from vector-valued multiplicative Lindley-type recursions. It claims to provide a detailed framework for solving these equations and applies the theoretical results to semi-Markovian queueing systems and vector-valued autoregressive processes.

Significance. If the framework is rigorously derived with explicit solution methods, existence/uniqueness results, and validated applications, the work could contribute useful tools for analyzing recursive stochastic systems in probability theory. The asserted connections to queueing and autoregressive models indicate potential relevance in applied probability, though the precise novelty and generality remain to be confirmed from the derivations.

minor comments (2)

Clarify the precise form of the vector-valued multiplicative Lindley-type recursion at the outset, including any assumptions on the vector operations or commutativity of terms.
Provide at least one fully worked example of the framework applied to a semi-Markovian queue in the applications section to illustrate the solution procedure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful summary and for recognizing the potential utility of our framework for analyzing recursive stochastic systems in probability theory. We address the concerns regarding novelty and generality below.

read point-by-point responses

Referee: the precise novelty and generality remain to be confirmed from the derivations.

Authors: The manuscript develops a general solution framework for vector-valued functional equations featuring multiple recursive terms, extending scalar multiplicative Lindley recursions via matrix-valued iterations and fixed-point arguments in suitable Banach spaces. Sections 3 and 4 derive an explicit recursive solution procedure together with existence and uniqueness results under Lipschitz-type contraction conditions. This vector setting with simultaneous multiple recursions has not appeared in the literature with the same level of explicitness. The applications to semi-Markovian queues and vector autoregressive processes illustrate the framework's reach across different dependence structures. We can add a short comparison paragraph in the introduction highlighting these distinctions if the referee finds it helpful. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a framework for solving vector-valued functional equations with multiple recursive terms from Lindley-type recursions in queueing and autoregressive contexts. The abstract and structure indicate a self-contained mathematical development that specifies equations, derives solution methods, and applies results without reducing any central claim to a self-citation chain, fitted parameter renamed as prediction, or definitional equivalence. No load-bearing step collapses to its own inputs by construction; external applicability to semi-Markovian processes is asserted independently of the core derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5567 in / 962 out tokens · 54992 ms · 2026-05-19T06:26:14.249795+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

vector-valued functional equations ... ˜Z(r, s, η) = G(r, s, η) ∑ ˜P(i) ˜Z(r, αi(s), η) + K ... commutative contraction mappings
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

solution via Liouville’s theorem and Wiener-Hopf boundary value theory; iteration of contractions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

I. Adan, O. Boxma, and J. Resing. Functional equations with multiple recursive terms. Queueing Systems, 102(1-2):7–23, 2022

work page 2022
[2]

I. Adan, B. Hathaway, and V. G. Kulkarni. On first-come, first-served queues with two classes of impatient customers. Queueing Systems, 91(1-2):113–142, 2019

work page 2019
[3]

Adan and V

I. Adan and V. Kulkarni. Single-server queue with Markov-dependent inter-arrival and service times. Queueing Systems, 45:113–134, 2003

work page 2003
[4]

Albrecher and O

H. Albrecher and O. J. Boxma. A ruin model with dependence between claim sizes and claim intervals. Insurance: Mathematics and Economics , 35(2):245–254, 2004

work page 2004
[5]

Allan, G

G. Allan, G. R. Allan, and H. G. Dales. Introduction to Banach spaces and algebras. Number 20 in Oxford Graduate Texts in Math. Oxford University Press, 2011

work page 2011
[6]

Asmussen and O

S. Asmussen and O. Kella. A multi-dimensional martingale for Markov additive processes and its applica- tions. Advances in Applied Probability, 32(2):376–393, 2000

work page 2000
[7]

Borst, O

S. Borst, O. Boxma, and M. Comb´ e. An M/G/1 queue with customer collection. Stochastic Models, 9(3):341–371, 1993

work page 1993
[8]

Boxma, A

O. Boxma, A. L¨ opker, and M. Mandjes. On two classes of reflected autoregressive processes. Journal of Applied Probability, 57(2):657–678, 2020

work page 2020
[9]

Boxma, A

O. Boxma, A. L¨ opker, M. Mandjes, and Z. Palmowski. A multiplicative version of the Lindley recursion. Queueing Systems, 98:225–245, 2021

work page 2021
[10]

Boxma and M

O. Boxma and M. Mandjes. Shot-noise queueing models. Queueing Systems, 99(1):121–159, 2021

work page 2021
[11]

Boxma and M

O. Boxma and M. Mandjes. Queueing and risk models with dependencies. Queueing Systems, 102(1-2):69– 86, 2022

work page 2022
[12]

Boxma, M

O. Boxma, M. Mandjes, and J. Reed. On a class of reflected AR(1) processes.Journal of Applied Probability, 53(3):818–832, 2016

work page 2016
[13]

J. W. Cohen. The Wiener-Hopf technique in applied probability. Journal of Applied Probability, 12(S1):145– 156, 1975

work page 1975
[14]

J. H. De Smit. The queue GI/M/s with customers of different types or the queue GI/H m/s. Advances in Applied Probability, 15(2):392–419, 1983

work page 1983
[15]

Dieudonne

J. Dieudonne. Foundations of Modern Analysis . Academic Press, New York, 1969

work page 1969
[16]

Dimitriou

I. Dimitriou. On Markov-dependent reflected autoregressive processes and related models. arXiv preprint, arXiv:2404.10361v2, 2024

work page arXiv 2024
[17]

Dimitriou and D

I. Dimitriou and D. Fiems. Some reflected autoregressive processes with dependencies. Queueing Systems, 106:67–127, 2024

work page 2024
[18]

D. Huang. On a modified version of the Lindley recursion. Queueing Systems, 105(3):271–289, 2023. 18

work page 2023
[19]

Kumar and N

V. Kumar and N. S. Upadhye. On first-come, first-served queues with three classes of impatient customers. International Journal of Advances in Engineering Sciences and Applied Mathematics , 13(4):368–382, 2021

work page 2021
[20]

Lancaster and M

P. Lancaster and M. Tismenetsky. The theory of matrices: with applications . Elsevier, 1985

work page 1985
[21]

Ramankutty

P. Ramankutty. Extensions of Liouville theorems. Journal of Mathematical Analysis and Applications , 90(1):58–63, 1982

work page 1982
[22]

G. J. K. Regterschot and J. H. A. de Smit. The queue M/G/1 with Markov modulated arrivals and services. Mathematics of Operations Research, 11(3):465–483, 1986

work page 1986
[23]

W. Rudin. Functional Analysis. Tata McGraw-Hill, New York, 1974

work page 1974
[24]

P. Tin. A queueing system with Markov-dependent arrivals. Journal of Applied Probability, 22(3):668–677, 1985

work page 1985
[25]

E. C. Titchmarsh. The theory of functions . Oxford University Press, USA, 1939

work page 1939
[26]

M. Vlasiou. Lindley-type recursions. PhD thesis, Technische Universiteit Eindhoven, 2006

work page 2006
[27]

Vlasiou, I

M. Vlasiou, I. Adan, and O. Boxma. A two-station queue with dependent preparation and service times. European Journal of Operational Research, 195(1):104–116, 2009. 19

work page 2009

[1] [1]

I. Adan, O. Boxma, and J. Resing. Functional equations with multiple recursive terms. Queueing Systems, 102(1-2):7–23, 2022

work page 2022

[2] [2]

I. Adan, B. Hathaway, and V. G. Kulkarni. On first-come, first-served queues with two classes of impatient customers. Queueing Systems, 91(1-2):113–142, 2019

work page 2019

[3] [3]

Adan and V

I. Adan and V. Kulkarni. Single-server queue with Markov-dependent inter-arrival and service times. Queueing Systems, 45:113–134, 2003

work page 2003

[4] [4]

Albrecher and O

H. Albrecher and O. J. Boxma. A ruin model with dependence between claim sizes and claim intervals. Insurance: Mathematics and Economics , 35(2):245–254, 2004

work page 2004

[5] [5]

Allan, G

G. Allan, G. R. Allan, and H. G. Dales. Introduction to Banach spaces and algebras. Number 20 in Oxford Graduate Texts in Math. Oxford University Press, 2011

work page 2011

[6] [6]

Asmussen and O

S. Asmussen and O. Kella. A multi-dimensional martingale for Markov additive processes and its applica- tions. Advances in Applied Probability, 32(2):376–393, 2000

work page 2000

[7] [7]

Borst, O

S. Borst, O. Boxma, and M. Comb´ e. An M/G/1 queue with customer collection. Stochastic Models, 9(3):341–371, 1993

work page 1993

[8] [8]

Boxma, A

O. Boxma, A. L¨ opker, and M. Mandjes. On two classes of reflected autoregressive processes. Journal of Applied Probability, 57(2):657–678, 2020

work page 2020

[9] [9]

Boxma, A

O. Boxma, A. L¨ opker, M. Mandjes, and Z. Palmowski. A multiplicative version of the Lindley recursion. Queueing Systems, 98:225–245, 2021

work page 2021

[10] [10]

Boxma and M

O. Boxma and M. Mandjes. Shot-noise queueing models. Queueing Systems, 99(1):121–159, 2021

work page 2021

[11] [11]

Boxma and M

O. Boxma and M. Mandjes. Queueing and risk models with dependencies. Queueing Systems, 102(1-2):69– 86, 2022

work page 2022

[12] [12]

Boxma, M

O. Boxma, M. Mandjes, and J. Reed. On a class of reflected AR(1) processes.Journal of Applied Probability, 53(3):818–832, 2016

work page 2016

[13] [13]

J. W. Cohen. The Wiener-Hopf technique in applied probability. Journal of Applied Probability, 12(S1):145– 156, 1975

work page 1975

[14] [14]

J. H. De Smit. The queue GI/M/s with customers of different types or the queue GI/H m/s. Advances in Applied Probability, 15(2):392–419, 1983

work page 1983

[15] [15]

Dieudonne

J. Dieudonne. Foundations of Modern Analysis . Academic Press, New York, 1969

work page 1969

[16] [16]

Dimitriou

I. Dimitriou. On Markov-dependent reflected autoregressive processes and related models. arXiv preprint, arXiv:2404.10361v2, 2024

work page arXiv 2024

[17] [17]

Dimitriou and D

I. Dimitriou and D. Fiems. Some reflected autoregressive processes with dependencies. Queueing Systems, 106:67–127, 2024

work page 2024

[18] [18]

D. Huang. On a modified version of the Lindley recursion. Queueing Systems, 105(3):271–289, 2023. 18

work page 2023

[19] [19]

Kumar and N

V. Kumar and N. S. Upadhye. On first-come, first-served queues with three classes of impatient customers. International Journal of Advances in Engineering Sciences and Applied Mathematics , 13(4):368–382, 2021

work page 2021

[20] [20]

Lancaster and M

P. Lancaster and M. Tismenetsky. The theory of matrices: with applications . Elsevier, 1985

work page 1985

[21] [21]

Ramankutty

P. Ramankutty. Extensions of Liouville theorems. Journal of Mathematical Analysis and Applications , 90(1):58–63, 1982

work page 1982

[22] [22]

G. J. K. Regterschot and J. H. A. de Smit. The queue M/G/1 with Markov modulated arrivals and services. Mathematics of Operations Research, 11(3):465–483, 1986

work page 1986

[23] [23]

W. Rudin. Functional Analysis. Tata McGraw-Hill, New York, 1974

work page 1974

[24] [24]

P. Tin. A queueing system with Markov-dependent arrivals. Journal of Applied Probability, 22(3):668–677, 1985

work page 1985

[25] [25]

E. C. Titchmarsh. The theory of functions . Oxford University Press, USA, 1939

work page 1939

[26] [26]

M. Vlasiou. Lindley-type recursions. PhD thesis, Technische Universiteit Eindhoven, 2006

work page 2006

[27] [27]

Vlasiou, I

M. Vlasiou, I. Adan, and O. Boxma. A two-station queue with dependent preparation and service times. European Journal of Operational Research, 195(1):104–116, 2009. 19

work page 2009