Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate

Round 1

Reviewer 1 Report

Please see the attached referee report.

Comments for author File: Comments.pdf

Stated in the referee report.

Author Response

Thank you very much for important comments. All comments have been taken into account in the paper. Let us response to the first major comment: "On Page 3, Lines 71 - 74: Authors claim that the PDE given in equation (1) is hard, and
hence they consider the asymptotic diffusion analysis where the limit condition σ → 0 (long delay). I am not sure what is the practical significance of this regime. Doesn’t it mean that every member in orbit stays in there forever in this regime? If so, how is this of any practical use?"

The asymptotic limit condition sigma ->0 does mean that calls stay in the orbit for a long time. But it is a mathematical approach and let us note that sigma≠0 anywhere. Really, in examples, we can see that the asymptotic results are good enough for sigma<0.1. So practically, the results may be used in case when a delay rate is 10 times less than an arrival rate.

As for the second major comment, we have added more references and rewritten text.

Also, the paper has been edited according to other reviewer's comments.

Reviewer 2 Report

-page 3 The sigle PDE (partial differatial equation) has not be prealabely defined. Idem for FANET(Flying A-hoc  Network).

My recommandation concerns essentially english language although english is not my native language and some typos. For example:

-page 1 switching in place of swiching.

-page 2  is dedicated to some concluding remarks (in place dedicated for)

-page 3  The two dismentionnal process...

Also, the following notations.... (inplace of denotations)

page 4  ...is an infinisesimal element...

page 5... ...the asymptotic analysis alow us... (in palce of let us)

page 6  As it can be seen... (in place  of As you can see...)

page 9 In first phrase of Conclusion and Discussion.

-with dependent servicerete (delete the words "the state").

Please check also some sentences whic seem to me be , not incorrects, but not used in thi maeer.

Author Response

Thank you very much for your review, all typos have been corrected. 

Reviewer 3 Report

 

Manuscript Review Report

Title: Asymptotic Diffusion Method for Retrial Queues with State

Dependent Service Rate

 

In this manuscript, the author has made an effort to propose a retrial queue with state-dependent service rate as a mathematical model of a node of FANET communications. The author assumes that the orbit is limitless, there is multiple random accesses from orbit, the arrival process is Poisson and the delay time is exponentially distributed. The service time of each call is distributed exponentially with a variable parameter based on the number of calls in the orbit, and there is only one server. There are an infinite number of values for the service rate. The probability distribution of the number of calls in the orbit in the model under consideration is derived using the asymptotic diffusion approximation. Numerical analysis is also performed which demonstrates the validity of the queuing model in real-life congestion situations.

From my point of view, the outcomes are quite interesting. The exposition goes concise and accessible. There are some strength to this study including its novelty and its applications. Please take into consideration my observations and recommendations below.

1) There are so many grammatical errors in this manuscript. Commas are used multiple times which must be reduced. For example delete the extra commas in lines 124 and 139. Also in the Keywords in line-8, write the word “Retrial” in the lower-case alphabet (retrial).

2)  In lines 73 and 74 the sentence “Also, System (1) in a steady state can be solved numerically by a truncated method, but generally its dimension is infinite (i = 0, ∞).”  Please mention what is the truncated method, give references for it, and please write it properly it has no meaning or sense about the dimension.

3)Write the full form of FANET and write it in the bracket. There is also some spelling mistake such as in line 25. Also look into this throughout the article. Don’t use the same word multiple times such as in lines 34, 35, 36 and 44. Reframe those type of sentences.

4)  Please check equation 10. It has some ‘plus notation’ missing. Also, change the line “Then we need derive function Π (y, τ).”  It has no meaningful sense. Check equations 15 again as you are giving the same notation g0(x).

5) Write some lines about what Figure 2, 3 and table is representing in sentences and what it is demonstrating. Also emphasize on what will be the effect of parameters if we change other parameters and how is it affecting or influencing our queueing model?

6) Authors should take their own examples into account. Nowadays, nearly anybody can access queuing data from public sources, thus authors should attempt to resolve a real-world queueing problem and explain how this might benefit facility and queue managers in the communications industry. Otherwise, this paper adds nothing new or useful to the body of literature. Writers ought to use actual situations to illustrate which problems are taken into account when and why. It cannot contain hypothetical, arbitrary, or default data.

7)  Update the literature review by adding few recent references from 2020 onwards to create interests to the readers.

8)  The conclusion section should also address the following key questions: (a) what is the novelty of your work? (b) What is the contribution of your paper to society in resolving real life issues? (c) What advantages would queue managers experience if they choose to utilize your model?

Decision: After the necessary revisions, I recommend publishing this article in “MDPI JOURNAL” in accordance with the aforementioned suggestions and remarks.

 

 

Please correct the English at some places as mentioned in the review report.

Author Response

Thank you very much for important comments. All comments have been taken into account in the paper. Let us response on some of them.

2) In lines 73 and 74 the sentence “Also, System (1) in a steady state can be solved numerically by a truncated method, but generally its dimension is infinite (i = 0, ∞).” Please mention what is the truncated method, give references for it, and please write it properly it has no meaning or sense about the dimension.

In this method, some boundary I is chosen so that probability P(I) is close to zero, and the system of I equations is solved (i = 0, I). We have added some references and text in the paper.

6) Authors should take their own examples into account. Nowadays, nearly anybody can access queuing data from public sources, thus authors should attempt to resolve a realworld queueing problem and explain how this might benefit facility and queue managers in the communications industry. Otherwise, this paper adds nothing new or useful to the body of literature. Writers ought to use actual situations to illustrate which problems are taken into account when and why. It cannot contain hypothetical, arbitrary, or default data.

This paper focuses on the quite complex mathematical model of retrial queues with dependent parameters and proposes a novel analytical method of its study. The application of the results of the mathematical modelling for FANETs using real data will be in the future.  We have mentioned this in the new paper version.

Also, the paper have been edited according to other comments.