An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
Round 1
Reviewer 1 Report
Authors must make the following corrections in the paper:
- Authors should explain better the academic contribution of the work developed. Highlighting what is innovative / original about the existing literature.
--- Authors should develop the conclusions of the work and refer in more detail to the next steps of the work
- Authors should improve the literature review using more recent references.
Author Response
Responses to Reviewer1
An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs(Manuscript applsci-1199318)
We are grateful for your sincere advice. Following are the specific responses to the comments.
- Authors should explain better the academic contribution of the work developed. Highlighting what is innovative / original about the existing literature.
For helping readers to understand our contribution, we inserted some sentences in abstraction, Sections introduction, and conclusion.
(p.1, lines 1-2) Designing a discrete event system converging to steady temporal patterns is an essential issue of a system with time window constraints.
(p.2, lines 59-61) In conclusion, we provide a direct way to calculate an asymptotic cyclicity of a given TEG and to determine whether the TEG is stable without any conversion.
(p.17, lines 693-695) The main contribution of this paper is to propose a method to calculate the cyclicity directly from a timed event graph.
- Authors should develop the conclusions of the work and refer in more detail to the next steps of the work
We attached the future work as a paragraph like the following:
(p.18, lines 703-707) The result of this paper can improve a time-constrained system design. For example, we expect to use the suggested method to analyze time-constrained discrete event systems such as CVD or a hoist system and to simulate the effect to validate our theory practically. Finally, we can extend the investigated relationship between the properties of a Petri net and (max,+) matrix to apply other analyses of (max,+) algebra to a Petri net.
- Authors should improve the literature review using more recent references.
We have reinforced the recent references by adding eight articles: 11, 14-19, 23.
- Komenda, J.; Lahaye, S.; Boimond, J.L.; van den Boom, T. Max-plus algebra in the history of discrete event systems. Annual Reviews in Control 2018, 45, 240–249.
- Akian, M.; Gaubert, S.;Walsh, C. Discrete max-plus spectral theory. Contemporary Mathematics 2005, 377, 53–78.
- Akian, M.; Gaubert, S.; Guterman, A. Linear independence over tropical semirings and beyond. Contemporary Mathematics 2009, 495, 1.
- Butkoviˇc, P. Max-linear systems: theory and algorithms; Springer Science & Business Media, 2010.
- Kennedy-Cochran-Patrick, A.; Merlet, G.; Nowak, T.; Sergeev, S. New bounds on the periodicity transient of the powers of a tropical matrix: using cyclicity and factor rank. Linear Algebra and its Applications 2021, 611, 279–309.
- Sergeev, S. Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes. Linear algebra and its applications 2009, 431, 1325–1339.
- Gavalec, M.; Ponce, D.; Zimmermann, K. Steady states in the scheduling of discrete-time systems. Information Sciences 2019, 481, 219–228.
- Butkovic, P.; Schneider, H.; Sergeev, S. Core of a matrix in max algebra and in nonnegative algebra: a survey. 2019, 24, 252–271.
The changed parts are highlighted with blue color in the revised paper.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The purpose of this paper was to provide a theoretical foundation for investigating the stability and cyclicality of timed event graphs. Without a max-plus conversion process, the authors proposed converting one timed event graph to another with dynamic behaviour equivalent to the original.
The introduction section summarizes the basic problem's intro. I recommend a deeper search and more precise basic information in the section devoted to (max, +) algebra. Although the mentioned literature is basic, current research in this area has shifted. Consider the works of M. Akian, P. Butkovič, S. Gaubert, M. Gavalec, S.Sergeev and K. Zimmermann.
Theorem 1 is a statement, not an exact theorem.
Minor formal remarks: the text contains minor typos, and the equations in the text should be formatted exactly according to the basic rules of writing mathematical equations. The reference should be associated with the last word and should not be carried over to the next line.
Overall, the article is interesting, and the research question is reasonable. The outputs will be relevant if the author updates his knowledge in the field of max-plus algebra.
Author Response
Responses to Reviewer 2
An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs(Manuscript applsci-1199318)
We thank you for your keen advice. We have modified so many things, so please check the attached file if you want what is changed in detail.
Point 1: The introduction section summarizes the basic problem's intro. I recommend a deeper search and more precise basic information in the section devoted to (max, +) algebra.
-> We introduced the concepts of max tropical semiring and Perron-Frobenius theory. Additionally, we added eight recent references.
Point 2. Although the mentioned literature is basic, current research in this area has shifted. Consider the works of M. Akian, P. Butkovič, S. Gaubert, M. Gavalec, S.Sergeev and K. Zimmermann.
-> We have reinforced the recent references by adding eight articles: 11, 14-19, 23.
Point 3. Theorem 1 is a statement, not an exact theorem.
->You can check that we removed the theorem on pp. 11-12 (lines 396-465).
Point 4. the text contains minor typos
->We correct three typos and promise that I will use the “English Editing Service” of MDPI after accepting this paper (https://www.mdpi.com/authors/english)
Point 5. the equations in the text should be formatted exactly according to the basic rules of writing mathematical equations.-> We tried to reformat as many equations in the text according to the basic writing mathematical equations referring to the document "Rules and tips for writing mathematics” (http://www2.gcc.edu/dept/math/faculty/bancrofted/teaching/handouts/math_writing_rules_tips_for_discrete.pdf).
Point 6. The reference should be associated with the last word and should not be carried over to the next line.
-> In previous papers, some citations were used as the subject or object of a sentence, but in the revised version, we associated them with their last word according to your advice.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors made the requested corrections and improved the paper. The paper must be published.
Reviewer 2 Report
Thank you for making the changes to the text and responding to my comments. The expansion of the literature has filled basic gaps in the text, and I can now recommend it for publication.
