Passenger Flow Management in Front of Ticket Booths in Urban Railway Stations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

I have read through the submitted manuscript and although the results of the study may seem interesting, the manuscript contains a number of methodological and other errors that raise doubts about the quality of the presented results.

First of all, I would like to state that one of the basic models of queueing systems is used for the analysis. The model is well-known and it is described in almost every book devoted to the queuing theory. There is no need to describe the model in such detail - there is no theoretical contribution to this model provided by the authors. Moreover, there are flaws and errors in its mathematical description. Let us mention some of the errors I identified in this section:

·         In line 380 the notation for performance measures L_q and W_q is used, whilst in the following text the same performance measures are denoted by E(L) and E(W). Does it mean the performance measures are different?

·         In formula (1) λ should be greater than 0 and x should be non-negative integers.

·         In formula (2) the meaning of S is not described.

·         In the sentence in lines 400 – 403 it should be written that the differential equations describe the probabilities that the system is in state j in time t+Δt.

·         There is a typo in formula (4) – the upper bound should be infinity in this system.

·         In formula (7) I do not understand what does p_m+1-∞ mean. The same is used in formula (16).

·         In formula (16) the symbols A(t) and B(t) are not explained.

In table 2 the authors present the data set containing the service times obtained in the modelled system. I think it is not suitable to present the rough data but its statistical processing should be presented instead. In the article only the average and the standard deviation is calculated. There is no other statistical analysis of the data set - does the data follow the exponential distribution of probability to be used in the model?

In addition, when looking at the average service time which is 87.47 seconds and the corresponding standard deviation 133.33 seconds (should this value be square rooted in formula in line 465?), the coefficient of variation, which is defined as the standard deviation divided by the average, is 133.33/87.47 = 1.52. For the exponential distribution the coefficient of variation should be about 1 (the mean value and the standard deviation are the same for the exponential distribution of probability). Are you really sure that this distribution is suitable for modelling the service times? I suggest to test the hypothesis about the exponential distribution of probability.

The main methodological problems are the following. The authors use the steady-state analysis for modelling short time periods. See for example table 5. I think the results are obtained by applying the same model in steady-state for the individual time periods. Are you sure that the system can be stabilized after two hours? Because the first time period takes 2 hours. This is the main reason why I think the results are not valid. In such cases using simulation methods could bring more accurate results.

The authors mistakes the service rate μ for the mean service time 1/μ (or at least their units) often in the manuscript. See for example the sentence in lines 500 – 502 – „The total average service time resulting from table 3 is 1/μ = 87.47 customers/hour… - is the service time really expressed in customers per an hour? This is a total nonsense. The same problem repeats many times thorough the following text – see for example tables 5 – 9 or the sentence in line 536.

After presenting the results for the current configuration of the system (section 4.2) the authors try to improve the system performance. But in the sentence in lines 618 – 622 it is stated that “As part of improving the quality of services provided to customers, it is also necessary to introduce a First in - First out queue mode in the station” Does it imply that the current system does not apply this queueing discipline? The model which was used in the previous text considers the FIFO queue.

Let us continue with some other minor problems I was able to find:

·         In line 206 it is stated that “The table shows that a total of 478,749 passengers purchased….” I do not know where the number can be seen in the table.

·         In table 10 – is the sum of p_k  really 0?

·         Is the sentence in line 511 valid – “At the intensity of operation ρ > 1, we can declare this system as stable…“? Really?

·         I think there is an error (or multiple errors) in results in table 14 in the first column (6 AM – 7 AM). Considering a single server queueing system with the arrival rate 85.5 customers per an hour and the service rate 88.25 customers per an hour the value of E(S) is not equal to 0.57.  

To be honest, I am not an expert in English language. Despite this fact, I would like to point out that the term queueing system instead of mass service system is used in English literature. Some sentences are confusing for me, for example the sentence in lines 339 – 343 or the sentence in lines 360 – 361.

Based on my comments provided above in the text I think the article is not suitable to be published and I recommend to reject it.

Author Response

Dear reviewer,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewers comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewer comments and they have all been incorporated into the article.

Comments from the Reviewer:

 

Reviewer 1

I have read through the submitted manuscript and although the results of the study may seem interesting, the manuscript contains a number of methodological and other errors that raise doubts about the quality of the presented results.

First of all, I would like to state that one of the basic models of queueing systems is used for the analysis. The model is well-known and it is described in almost every book devoted to the queuing theory. There is no need to describe the model in such detail - there is no theoretical contribution to this model provided by the authors. Moreover, there are flaws and errors in its mathematical description. Let us mention some of the errors I identified in this section:

In line 380 the notation for performance measures L_qand W_q is used, whilst in the following text the same performance measures are denoted by E(L) and E(W). Does it mean the performance measures are different?

- Thanks for the reminder. This has been edited.

In formula (1) λ should be greater than 0 and x should be non-negative integers.

- Thank you very much for your valuable comment. This has been edited. We provide an explanation below:
- parameter λ represents the expected number of events per unit of time and should be a positive number. In mathematical terms, λ > 0 should apply. Nevertheless, in reality, λ can be any non-negative number (λ≥0). At λ = 0, the Poisson distribution would mean that no event will occur with certainty (X = 0 with probability 1).
- In the Poisson distribution, x represents the number of events, so it must be an integer and cannot be negative. In mathematical terms, x≥0. The number of events cannot be a fractional number or a negative number; x must be a non-negative integer (0, 1, 2,...).

In formula (2) the meaning of S is not described.

- Thanks for the comment. This has been supplemented, and S represents the set of all possible states in which the system can be.

In the sentence in lines 400 – 403 it should be written that the differential equations describe the probabilities that the system is in state j in time t+Δt.

- Thank you. This has been edited to the correct format and description.

There is a typo in formula (4) – the upper bound should be infinity in this system.

- Thanks for the comment. This has been edited.

In formula (7) I do not understand what does p_m+1-∞ mean. The same is used in formula (16).

• Thank you. This was followed by an explanation, and p_m+1−∞ is the probability that the system is in a state that is greater than or equal to a certain threshold point m+1, after considering all possible states above that threshold. This threshold point can be interpreted in the context of an infinite number of states as "all states above a certain point,"  and the formula expresses the cumulative probability for all these states.

In formula (16) the symbols A(t) and B(t) are not explained.

• Thanks for the reminder. A(t) and B(t) are not found in formula (16), but we have modified them in formula (19).

In table 2 the authors present the data set containing the service times obtained in the modelled system. I think it is not suitable to present the rough data but its statistical processing should be presented instead. In the article only the average and the standard deviation is calculated. There is no other statistical analysis of the data set - does the data follow the exponential distribution of probability to be used in the model?

• Thank you for your valuable comment. Yes, we verified this with the Kolmogorov-Smirnov test, where the maximum difference between the empirical and theoretical cumulative distribution function of CDF was 0.12415. Based on this, we can conclude that the system is stable and the data follows an exponential probability distribution.

In addition, when looking at the average service time which is 87.47 seconds and the corresponding standard deviation 133.33 seconds (should this value be square rooted in formula in line 465?), the coefficient of variation, which is defined as the standard deviation divided by the average, is 133.33/87.47 = 1.52. For the exponential distribution the coefficient of variation should be about 1 (the mean value and the standard deviation are the same for the exponential distribution of probability). Are you really sure that this distribution is suitable for modelling the service times? I suggest to test the hypothesis about the exponential distribution of probability.

• Thanks for your comment and pointing out the importance of validating the fit of the exponential distribution for modelling service times. As you correctly noted, the coefficient of variation, which is defined as the ratio of the standard deviation to the mean, is equal to 1 for the exponential distribution because the mean and standard deviation are the same in this distribution.
• In our case, we calculated the coefficient of variation as 1.52 (133.33/87.47), which indicates that the standard deviation is greater than the mean. This would indicate that the data may be distributed differently from the exponential distribution, which would have a coefficient of variation around 1.
• We performed the Kolmogorov-Smirnov test to find the maximum difference between the empirical CDF and the theoretical CDF. The critical value of the Kolmogorov-Smirnov test was 0.12415. In this case, this means that our data can be modelled reasonably well by the assumed distribution.

The main methodological problems are the following. The authors use the steady-state analysis for modelling short time periods. See for example table 5. I think the results are obtained by applying the same model in steady-state for the individual time periods. Are you sure that the system can be stabilized after two hours? Because the first time period takes 2 hours. This is the main reason why I think the results are not valid. In such cases using simulation methods could bring more accurate results.

• Thank you for your comment and attention to methodological issues. We understand your concerns about using steady-state analysis to model short periods of time. We would like to emphasise that our approach was chosen considering the specific properties of the system and its ability to reach a steady state during the first time period of two hours. We have carefully considered the dynamics of the system, and based on our observations and calculations, we have come to the conclusion that stabilisation during this period is possible.

The authors mistakes the service rate μ for the mean service time 1/μ (or at least their units) often in the manuscript. See for example the sentence in lines 500 – 502 – „The total average service time resulting from table 3 is 1/μ = 87.47 customers/hour… - is the service time really expressed in customers per an hour? This is a total nonsense. The same problem repeats many times thorough the following text – see for example tables 5 – 9 or the sentence in line 536.

• Thank you. This has been edited for correct expression.

After presenting the results for the current configuration of the system (section 4.2) the authors try to improve the system performance. But in the sentence in lines 618 – 622 it is stated that “As part of improving the quality of services provided to customers, it is also necessary to introduce a First in - First out queue mode in the station” Does it imply that the current system does not apply this queueing discipline? The model which was used in the previous text considers the FIFO queue.

• Thanks for the comment. Yes, the FI-FO system was introduced originally, but there was a typo. This system is in place, and the proposal will continue to use this system. This has been edited.

Let us continue with some other minor problems I was able to find:

In line 206 it is stated that “The table shows that a total of 478,749 passengers purchased….” I do not know where the number can be seen in the table.

- Thank you. This has been edited and explained in the text.

In table 10 – is the sum of p_k really 0?

- Thanks for the reminder. This is a typo when writing the table. This has been edited and sum p_k=1.

Is the sentence in line 511 valid – “At the intensity of operation ρ > 1, we can declare this system as stable…“? Really?

- Thanks for the comment. This has been edited, and we provide the following explanation:
- For the system to be stable, ρ must be less than 1 (ρ<1).
- If ρ > 1, it means that customers are arriving faster than the system can handle them, leading to unlimited queue growth. Such a system is unstable.
- In our case, it is 0.98. So the system is stable.

I think there is an error (or multiple errors) in results in table 14 in the first column (6 AM – 7 AM). Considering a single server queueing system with the arrival rate 85.5 customers per an hour and the service rate 88.25 customers per an hour the value of E(S) is not equal to 0.57.  

- Thanks for the reminder. This was a typo and has been edited. This value should be 0.97.

To be honest, I am not an expert in English language. Despite this fact, I would like to point out that the term queueing system instead of mass service system is used in English literature. Some sentences are confusing for me, for example the sentence in lines 339 – 343 or the sentence in lines 360 – 361.

- Thanks for the comment. We reformulated the sentence in the original lines 339–343. It is about the use of mathematical methods and the ability to model the behaviour of the system when conditions change. The sentence in the original lines 360–361 has been reformulated. It means that the customer is in a situation where he has to wait until one of the service units (line or cash register) becomes available to provide the service.

Based on my comments provided above in the text I think the article is not suitable to be published and I recommend to reject it.

- Thank you for all your valuable comments. All comments were incorporated into the article, and errors were corrected and explained. We believe that we have properly corrected and explained all inconsistencies. Since in most cases there were typos or missing explanations, we are sending the edited manuscript together with the supplementary material.

Best regards

Authors

Reviewer 2 Report

Comments and Suggestions for Authors

The paper appears to focus on analyzing and improving the queuing system at a major train station in Slovakia. The authors attempt to develop an analytical model to optimize the ticketing process, particularly by assessing the introduction of a common queue system. However, the paper struggles with several issues:

• Writing and Clarity: The paper has significant issues with clarity and word choice, making it difficult to follow. There is also a lot of unnecessary detail that distracts a reader from the main points. The writing needs significant improvement. The choice of words is not consistent with the literature; for instance, "ticket office" can be replaced with "ticket booths," and "equipment" with "service." There are many long, confusing sentences, some repetition, and unnecessary context that should be removed. I believe the entire paper needs to be rewritten. For example, the detailed description of the queuing system is unnecessary and could be replaced with a short overview.
• Contribution: The paper lacks a clear contribution to the existing body of knowledge. While it presents an analytical approach to studying the ticket queues, it does not seem to add anything new or significant to the field.
• Abstract and Introduction: These sections are poorly written, failing to clearly present the research's core focus, the gaps it addresses, or its motivation. The introduction also fails to properly set up the paper, missing key elements like the importance of shifts and employee assignments in passenger flow management while it has a lot of texts in this regard later in the paper.
• Literature Review: The literature review is disorganized and lacks a clear focus. It jumps between topics without adequately explaining their relevance to the study and the current gaps.
• Methodology and Analysis: The methodology section is confusingly labeled and includes excessive detail that should be moved to an appendix. The mathematical analysis appears to rely on standard queuing theory without offering new insights, and the use of Markov chains is not clearly explained or justified in terms of a contribution.
• Results and Discussion: The results are unclear, and terms like "optimizing" and "equity" are used inaccurately. The final configuration is shown to be an improvement but lacks validation or proof that it is the best possible solution.
• Conclusion: The conclusion contains questionable claims, such as the assertion of equal waiting times for all passengers, which is not substantiated.

In my opinion, the paper attempts to address a practical problem of optimizing ticket queues in a train station but is hindered by poor writing, unclear contributions, and a lack of rigorous analysis. So, unfortunately, I have to reject this paper. The detailed comments are as follow:

Abstract: The abstract is poorly written; the grammar and clarity need improvement. The core of the research is not well presented, and it is unclear whether the focus is on improving queue locations/configurations or service elements, such as the number of service providers. The final results are also not well described. This section needs to be rewritten. The gaps the paper aims to address are not mentioned, and the motivation is not compelling.

Introduction: The gaps in the literature, the contribution of the paper, and its methodology are not well described. The general discussion about queuing and how it is perceived by customers and agencies can be shortened.

• Lines 70-72: Why is there no need to create complex simulations? Why are analytical models considered sufficient as a contribution to this field? What specific analytical models are being referred to?
• Lines 63-65: The data is only for one week—was this an average week? How was the data collected? Manually, or with detectors or counters? Passenger flow management can also be done through station configuration, which is mentioned later in the paper. However, there is no mention of pedestrian simulation tools like MassMotion, which are widely used for this purpose—why was this not considered? The paper also refers to shifts and employee assignments, but these are not mentioned in the introduction. Why are these important in passenger flow management, and how do they contribute to the overall system design?
• Moreover, the introduction does not fully describe what the paper will cover, so it needs improvement.

Literature Review: This section abruptly jumps into the routing problem without any introduction, raising the question of why routing is discussed here and how it relates to passenger flow management. The concept of centralized routing in parallel queues needs more explanation to help the reader understand the point of this paragraogh. The literature review lacks a focus on core flow management studies at stations and provides unnecessary summaries of papers without discussing their actual contributions and methods. The flow of studies discussed is not coherent. For instance, what is an "optimal algorithm" (Line 144)? The word "deal" is used repeatedly and sometimes inappropriately. The section lacks a final discussion to highlight the gaps in the literature and why it is important to address these issues. Up to this point, there are no clear statements about the paper's goals, methods, and contribution to the field.

Materials and Methods: First, this section should not be called "Materials and Methods" but rather "Data Description" or "Current System Configuration." The current title is confusing, especially since there is another section called "Research Methodology." The information regarding the station is excessive and unnecessary—many details could be moved to the appendix. For example, why is it important for the reader to know the number of passengers per month? An average number for a year would suffice to convey that the station is busy. The paper also does not specify when the one-week data collection took place, which month, and why; these details are more relevant. Most of the tables should be moved to the appendix. The details regarding the current service line practices are overwhelming and unnecessary, especially when the paper's focus is on flow management, and employee assignment is secondary—or at least, it is presented that way in the abstract and introduction. In Line 332, the reference to "15,159 employees or personnel*time" is confusing.

Research Methodology: As mentioned earlier, there is no need to explain queuing theory in detail here.

• Line 345: Why do you think analytical models are more accurate, and compared to which other methods? One important element missing from this section is that, instead of fitting a distribution to the arrival rate of passengers, it is assumed to follow a Poisson process. Certain tests and rules must be met before determining that the system follows a Poisson process. So essentially, by substituting numbers into formulas, the authors present some results. In my opinion, this is not a contribution but merely a simple calculation. Therefore, this section does not deserve the space it occupies. The same applied to the mathematical analysis part.
• The tables presented in this section merely explain the system's current state and do not represent the research methodology or the results of the main model of a scientific paper.
• Additionally, Markov chains are mentioned in the abstract, but it is unclear how they were implemented. Although it is mentioned on page 13 that Markov's theorem is used, this is still classic queuing theory—so what is the contribution?
• Line 359: Why are service lines defined as those where customers arrive at certain intervals? What else would constitute a service line? The explanation for Figure 11 is also confusing. The figure shows five service lines, but Line 368 mentions three. Does "service line" refer to a single common queue rather than individual queues in front of each service booth? This needs clarification.

Mathematical Analysis:

• Lines 496-498: The authors state that the aim of this section is to capture the current behavior of the system. However, the previous section was also dedicated to describing the current behavior. What is different here? It is important to clearly describe what each section covers and maintain consistency.

Results and Discussion:

• "Lead to the highest possible of the system utilization of issuing tickets"—what does this mean exactly? Highest possible in terms of what? Why mention "possibility" here?
• I believe the word "equity" in Line 624 is used incorrectly. "Equity" has specific definitions and metrics that need to be evaluated, which have not been discussed. Using it without relation to the other material is inappropriate.
• From what I understand, the final result is shown in Figure 13. I can accept that the authors have demonstrated that this new configuration works better than the previous one. But how can you be sure this is the best solution? The authors use the word "optimizing" in Lines 793 and 831, which is misleading since this approach is not an optimization per se. What is the validation and contribution here? What led to this new configuration? It seems that the main point of the paper is obscured by unnecessary material. Rewriting the paper might/could make it a strong publication. Including a flowchart of the steps taken in the study and a comprehensive figure or table comparing the benefits and changes of the new system would help the reader more than long texts and scattered material across different pages.

Conclusion:

• Lines 870-871: What is meant by "equal waiting time for all passengers"? This does not seem correct—please provide evidence and proof. Even with common sense, this claim does not seem plausible.

 

 

Comments on the Quality of English Language

The paper has significant issues with clarity and word choice, making it difficult to follow. There is also a lot of unnecessary detail that distracts a reader from the main points. The writing needs significant improvement. The choice of words is not consistent with the literature; for instance, "ticket office" can be replaced with "ticket booths," and "equipment" with "service." There are many long, confusing sentences, some repetition, and unnecessary context that should be removed. I believe the entire paper needs to be rewritten.

Author Response

Vážený recenzent,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewer comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewer comments and they have all been incorporated into the article.

Comments from the Reviewers:

Reviewer 2

The paper appears to focus on analyzing and improving the queuing system at a major train station in Slovakia. The authors attempt to develop an analytical model to optimize the ticketing process, particularly by assessing the introduction of a common queue system. However, the paper struggles with several issues:

Writing and Clarity: The paper has significant issues with clarity and word choice, making it difficult to follow. There is also a lot of unnecessary detail that distracts a reader from the main points. The writing needs significant improvement. The choice of words is not consistent with the literature; for instance, "ticket office" can be replaced with "ticket booths," and "equipment" with "service." There are many long, confusing sentences, some repetition, and unnecessary context that should be removed. I believe the entire paper needs to be rewritten. For example, the detailed description of the queuing system is unnecessary and could be replaced with a short overview.

- Thanks for the comment. All errors have been corrected.

Contribution: The paper lacks a clear contribution to the existing body of knowledge. While it presents an analytical approach to studying the ticket queues, it does not seem to add anything new or significant to the field.

Abstract and Introduction: These sections are poorly written, failing to clearly present the research's core focus, the gaps it addresses, or its motivation. The introduction also fails to properly set up the paper, missing key elements like the importance of shifts and employee assignments in passenger flow management while it has a lot of texts in this regard later in the paper.

Literature Review: The literature review is disorganized and lacks a clear focus. It jumps between topics without adequately explaining their relevance to the study and the current gaps.

Methodology and Analysis: The methodology section is confusingly labeled and includes excessive detail that should be moved to an appendix. The mathematical analysis appears to rely on standard queuing theory without offering new insights, and the use of Markov chains is not clearly explained or justified in terms of a contribution.

Results and Discussion: The results are unclear, and terms like "optimizing" and "equity" are used inaccurately. The final configuration is shown to be an improvement but lacks validation or proof that it is the best possible solution.

Conclusion: The conclusion contains questionable claims, such as the assertion of equal waiting times for all passengers, which is not substantiated.

In my opinion, the paper attempts to address a practical problem of optimizing ticket queues in a train station but is hindered by poor writing, unclear contributions, and a lack of rigorous analysis. So, unfortunately, I have to reject this paper. The detailed comments are as follow:

Abstract: The abstract is poorly written; the grammar and clarity need improvement. The core of the research is not well presented, and it is unclear whether the focus is on improving queue locations/configurations or service elements, such as the number of service providers. The final results are also not well described. This section needs to be rewritten. The gaps the paper aims to address are not mentioned, and the motivation is not compelling.

- Thank you for your valuable comment. The abstract has been carefully revised.

Introduction: The gaps in the literature, the contribution of the paper, and its methodology are not well described. The general discussion about queuing and how it is perceived by customers and agencies can be shortened.

- Thanks for the reminder. The entire Introduction section has been revised. All comments have been added to the introduction section.

 

Lines 70-72: Why is there no need to create complex simulations? Why are analytical models considered sufficient as a contribution to this field? What specific analytical models are being referred to?

• Thanks for the comment. This has been added to the Introduction section, and we have attached a short explanation:
• As part of the research, we decided to use analytical models of calculations of mass service systems, which provide explicit mathematical results and allow immediate application of the acquired knowledge into practice. These analytical models, specifically those based on the theory of Markov chains and systems with an infinite queue, were chosen because they can effectively describe and predict the behaviour of the system under different configurations and intensities of customer flow.
• Simulations, although useful, were considered unnecessarily complex in this case given the nature of the problem, where analytical models can provide sufficiently accurate and verifiable results without the need for additional complications. The advantage of these models is their ability to provide quick and accurate solutions for the optimisation of service systems, which is crucial for the operation of the railway station. Moreover, their application to real data from the Poprad-Tatry station demonstrated that the results are reliable and practically usable without the need for additional verification steps.
• The chosen analytical models are therefore considered sufficient not only because of their accuracy but also because of their ability to provide immediate results, which represents a significant contribution to the field of mass service optimisation in transport.

Lines 63-65: The data is only for one week—was this an average week? How was the data collected? Manually, or with detectors or counters? Passenger flow management can also be done through station configuration, which is mentioned later in the paper. However, there is no mention of pedestrian simulation tools like MassMotion, which are widely used for this purpose—why was this not considered? The paper also refers to shifts and employee assignments, but these are not mentioned in the introduction. Why are these important in passenger flow management, and how do they contribute to the overall system design?

• Thanks for the reminder. This has been added in the Introduction section and discussed in the Discussion section.
• For research purposes, we collected data from the Poprad-Tatry railway station in the period from January 12 to 25. This timeframe included both weekdays and weekends in order to capture diversity in passenger arrivals and analyse a typical week in station operation. The data was obtained by personal measurement on the spot, while the measurements were carried out manually using a stopwatch at each sales window. This method of data collection, although difficult, provided a detailed view of the real state and dynamics of the system. Although we did not include advanced simulation tools such as MassMotion, we opted for an analytical approach based on empirical data and Markov chain modeling. The reason was mainly the pursuit of simplicity and the immediate applicability of the results. Advanced simulation tools were not considered necessary as the goal was to identify practical optimisation options based on directly acquired data.
• Passenger flow management, which also includes changes in staff allocation, is a key factor in the efficient management of a railway station. In our design, we took into account the configuration of the station and its possibilities for optimising the movement of passengers. Allocation of employees is important because the flexible redistribution of labour forces according to the current intensity of passenger arrivals enables more efficient processing of queues and shorter waiting times. This practice directly contributes to the optimisation of the overall system design and increases passenger satisfaction.

Moreover, the introduction does not fully describe what the paper will cover, so it needs improvement.

• Thank you for your valuable comment. The Introduction section has been carefully redesigned and supplemented with the required information.

Literature Review: This section abruptly jumps into the routing problem without any introduction, raising the question of why routing is discussed here and how it relates to passenger flow management. The concept of centralized routing in parallel queues needs more explanation to help the reader understand the point of this paragraogh. The literature review lacks a focus on core flow management studies at stations and provides unnecessary summaries of papers without discussing their actual contributions and methods. The flow of studies discussed is not coherent. For instance, what is an "optimal algorithm" (Line 144)? The word "deal" is used repeatedly and sometimes inappropriately. The section lacks a final discussion to highlight the gaps in the literature and why it is important to address these issues. Up to this point, there are no clear statements about the paper's goals, methods, and contribution to the field.

• Thank you very much. We appreciate your comments to improve our article. The literature review has been carefully revised and condensed. All errors have been correct.

Materials and Methods: First, this section should not be called "Materials and Methods" but rather "Data Description" or "Current System Configuration." The current title is confusing, especially since there is another section called "Research Methodology." The information regarding the station is excessive and unnecessary—many details could be moved to the appendix. For example, why is it important for the reader to know the number of passengers per month? An average number for a year would suffice to convey that the station is busy. The paper also does not specify when the one-week data collection took place, which month, and why; these details are more relevant. Most of the tables should be moved to the appendix. The details regarding the current service line practices are overwhelming and unnecessary, especially when the paper's focus is on flow management, and employee assignment is secondary—or at least, it is presented that way in the abstract and introduction. In Line 332, the reference to "15,159 employees or personnel*time" is confusing.

• Thanks for the comment. The overview of the number of passengers at the Poprad-Tatry station was presented to show that the railway station is really congested all year round and that only a small percentage of passengers buy their tickets at the Poprad-Tatry railway station in a form other than at the ticket office.
• The weekly data collection that was created for this research is presented in several places in the article, in almost every section.
• Calculations of the current state showed the number of employees as 19, but after recalculation, it was found that 15.159 employees are needed, so this number was reduced.

Research Methodology: As mentioned earlier, there is no need to explain queuing theory in detail here.

• Thank you. This has been edited and redundant text describing queuing theory has been removed.

Line 345: Why do you think analytical models are more accurate, and compared to which other methods? One important element missing from this section is that, instead of fitting a distribution to the arrival rate of passengers, it is assumed to follow a Poisson process. Certain tests and rules must be met before determining that the system follows a Poisson process. So essentially, by substituting numbers into formulas, the authors present some results. In my opinion, this is not a contribution but merely a simple calculation. Therefore, this section does not deserve the space it occupies. The same applied to the mathematical analysis part.

• Thank you for your comment, which leads us to think more deeply about the methodology of our research and the way of presenting the results.
• Our preference for analytical models is based on their ability to provide accurate and fast results when modelling complex systems. These models make it possible to analyse in detail the relationships between variables and predict the behaviour of the system under different conditions. However, we acknowledge that this claim has not been sufficiently substantiated by comparison with other methods, such as simulation models. We have added a short justification to the article about why we think that analytical models are more suitable for this study than, for example, simulation methods.
• We agree that analytical models are usually less computationally demanding because they do not require repeated simulations or iterations. Nevertheless, it allows for faster results, which is especially advantageous when analysing large systems or when quick decisions are needed. Our results can be applied to a wider range of situations because they are based on fundamental principles and relationships. Unlike simulations, which can be very specific to particular conditions, analytical models can be applied to similar systems with minimal modifications.
• The analytical models used in our research provide accuracy, reliability, and speed, as they are often unsurpassed in solving specific types of problems, especially where mathematical rigour and unequivocal results are required.

The tables presented in this section merely explain the system's current state and do not represent the research methodology or the results of the main model of a scientific paper.

• Thank you. Tables have been moved to Supplementary Materials. In the article, tables of mathematical analysis were left for each state, always for Monday.

Additionally, Markov chains are mentioned in the abstract, but it is unclear how they were implemented. Although it is mentioned on page 13 that Markov's theorem is used, this is still classic queuing theory—so what is the contribution?

• Thanks for the comment. The benefit is in the applicability of simple research directly into practice, taking into account the conditions of Slovak railway stations.

Line 359: Why are service lines defined as those where customers arrive at certain intervals? What else would constitute a service line? The explanation for Figure 11 is also confusing. The figure shows five service lines, but Line 368 mentions three. Does "service line" refer to a single common queue rather than individual queues in front of each service booth? This needs clarification.

• Thanks for the reminder. The description has been modified and errors have been removed. There are 5 service lines.
• Customers are often modelled as arriving at certain intervals, as this allows the creation of mathematical models that describe their behaviour within the service line. Interarrival intervals can be random and follow different distribution models. Alternatively, arrivals could be defined as fully deterministic (regular arrivals) or based on empirical data collected directly from the field.
• It is important to distinguish between the "service line" as an overall system and the "queues" in front of individual service points (e.g., checkouts). In this case, "service line" does not necessarily mean one common row but may include several parallel queues in front of different service boxes. Each of these fronts can be analysed separately, or they can all be included in one common model, depending on the goals of the analysis.

Mathematical Analysis:

Lines 496-498: The authors state that the aim of this section is to capture the current behavior of the system. However, the previous section was also dedicated to describing the current behavior. What is different here? It is important to clearly describe what each section covers and maintain consistency.

• Thanks for the comment. The article evaluates the current situation and suggests ways to improve and streamline the work of service lines.
• The obtained results provide valuable information for railway station managers, which will allow them to better plan and adapt to the future needs of passengers, thereby increasing efficiency and customer satisfaction.
• At the same time, this predictive ability of mathematical analysis allows us to simulate different scenarios and identify the best procedures that can minimise waiting times, improve the flow of passengers, and optimise the use of available resources. Thanks to this, it is possible to make decisions based on thorough analysis and not on assumptions, which contributes to more efficient management of the entire system.

Results and Discussion:

"Lead to the highest possible of the system utilization of issuing tickets"—what does this mean exactly? Highest possible in terms of what? Why mention "possibility" here?

• Thanks for the comment. The possibility was intended in the sense of maximising the effective use of the system when issuing tickets. This has been corrected to make the idea more precise and understandable.

I believe the word "equity" in Line 624 is used incorrectly. "Equity" has specific definitions and metrics that need to be evaluated, which have not been discussed. Using it without relation to the other material is inappropriate.

• Thank you. This has been edited.

Z toho, čo som pochopil, je konečný výsledok znázornený na obrázku 13. Môžem akceptovať, že autori preukázali, že táto nová konfigurácia funguje lepšie ako predchádzajúca. Ako si však môžete byť istý, že je to najlepšie riešenie? Autori používajú slovo „optimalizácia“ v riadkoch 793 a 831, čo je zavádzajúce, pretože tento prístup nie je optimalizáciou ako takým. Aké je tu potvrdenie a prínos? Čo viedlo k tejto novej konfigurácii? Zdá sa, že hlavná pointa papiera je zakrytá nepotrebným materiálom. Prepísanie papiera by z neho mohlo/mohlo urobiť silnú publikáciu. Zahrnutie vývojového diagramu krokov vykonaných v štúdii a komplexného obrázku alebo tabuľky porovnávajúcej výhody a zmeny nového systému by čitateľovi pomohlo viac ako dlhé texty a roztrúsený materiál na rôznych stranách.

• dakujem za vase komentare. Slovo optimalizácia sme zvolili trochu nešťastne. Toto bolo upravené.
• V časti Úvod bol pridaný vývojový diagram, ktorý popisuje postupné kroky nášho výskumu, ktoré sú uvedené v článku.
• Tabuľka porovnávajúca výhody a zmeny nového systému oproti starému je uvedená v sekcii Diskusia.

Záver:

Linky 870-871: Čo znamená „rovnaká čakacia doba pre všetkých cestujúcich“? Zdá sa, že to nie je správne – poskytnite dôkazy a dôkazy. Aj so zdravým rozumom sa toto tvrdenie nezdá byť pravdepodobné.

• dakujem. Táto myšlienka bola trochu zle prezentovaná a opísaná. Toto bolo upravené.

Príspevok má značné problémy s jasnosťou a výberom slov, čo sťažuje sledovanie. Je tu tiež veľa zbytočných detailov, ktoré odvádzajú pozornosť čitateľa od hlavných bodov. Písanie potrebuje výrazné zlepšenie. Výber slov nie je v súlade s literatúrou; napríklad „predaj lístkov“ môže byť nahradený „predajňami lístkov“ a „vybavenie“ výrazom „servis“. Existuje veľa dlhých, mätúcich viet, niektoré opakovania a zbytočné súvislosti, ktoré by sa mali odstrániť. Myslím si, že je potrebné prepísať celý dokument.

• Ďakujeme za každý cenný komentár, ktorý ste nám poskytli. Napriek vášmu odporúčaniu odmietnuť náš článok sme celý článok starostlivo upravili a niektoré časti úplne prepracovali. Opravili sme chybné výrazy a presunuli sme väčšinu tabuliek do doplnkových materiálov. Všetky požadované informácie boli vyplnené. Dúfame, že to pomohlo zlepšiť náš článok.

S pozdravom

Autori

Reviewer 3 Report

Comments and Suggestions for Authors

The key idea of this paper is to use Markov chains of mass service to optimize the process of issuing travel tickets. This is done by adjusting the number of service points based on the frequency of customer arrivals, line types, and the speed of service. The goal is to minimize waiting times in queues and improve efficiency by accurately estimating the number of service points needed to handle customer demand. The object of the study is the Poprad-Tatry railway station in Slovakia. The authors use analytical models. The article makes a good impression, but I have questions and comments about its design and content.

 1. Literature Review. The authors should explain clearly the difference between the approach proposed and known studies.

2. Line 206. The authors point out “The table shows that a total of 478,749 passengers purchased a ticket in 2023 in a form other than purchasing a travel ticket at the railway station”. However, Table 1 does not present these data. Perhaps a column was skipped.  

3. Sections 3 needs to be corrected. In particular, the justification of the object choice (lines 217 and 218) and a description of its structure (lines 225-227) should be moved to the beginning of the section. Lines 272-275 repeat the information presented at the beginning of Section 3.

4. Line 368. “Figure 11 represents a mass service system with THREE service lines into which customers who request service enter in random order.” At the same time, Figure 11 shows 5 channels.

5. Lines 373-375. As far as we understand, the authors use analytical methods to calculate performance indicators of the resulting QS. In this regard, the mention of simulation modeling requires further clarification. Please explain in detail what for you apply simulation.

6. Line 465. The formula should be corrected, in particular, the “=” symbol should be removed from under the root sign.

7. Lines 471, 478, and 481. The authors use the same intensity notations “λ” for different request flows. Also it is unclear whether “Intensity of customers entering with a request to issue an international travel ticket between 9.00 AM and 3.00 PM” (line 481) is a part of  “The average total intensity of customer access to the system from 6.00 AM – 6.00 PM” (line 471).

8. Line 500. “The total average service time resulting from table 3 is 1/μ = 87.47 customers/hour, AFTER ADJUSTMENT μ = 41.16 customers/hour”. It is necessary to clarify what exactly “ADJUSTMENT” consists of.

9. Sections 4 and 5 are a bit long, it is recommended to simplify. 

10. The article also does not discuss verification of the model constructed. Please, add some details.

11. I recommend including possible directions for further research in the conclusion.

Author Response

Dear reviewer,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewer comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewer comments and they have all been incorporated into the article.

Comments from the Reviewer:

Reviewer 3

The key idea of this paper is to use Markov chains of mass service to optimize the process of issuing travel tickets. This is done by adjusting the number of service points based on the frequency of customer arrivals, line types, and the speed of service. The goal is to minimize waiting times in queues and improve efficiency by accurately estimating the number of service points needed to handle customer demand. The object of the study is the Poprad-Tatry railway station in Slovakia. The authors use analytical models. The article makes a good impression, but I have questions and comments about its design and content.

1. Literature Review. The authors should explain clearly the difference between the approach proposed and known studies.

- Thanks for the comment. This has been added to the Literature Review section.

2. Line 206. The authors point out “The table shows that a total of 478,749 passengers purchased a ticket in 2023 in a form other than purchasing a travel ticket at the railway station”. However, Table 1 does not present these data. Perhaps a column was skipped.  

- Thank you for the valuable reminder. This has been edited. The data indicates the purchase of tickets via the application Ideme vlakom or via www.slovakrail.sk. Table 1 shows only the number of issued tickets at the cash desks in the Poprad-Tatry railway station. This is described more in Section 3.

3. Sections 3 needs to be corrected. In particular, the justification of the object choice (lines 217 and 218) and a description of its structure (lines 225-227) should be moved to the beginning of the section. Lines 272-275 repeat the information presented at the beginning of Section 3.

- Thank you. This has been moved as requested, explained and edited.

4. Line 368. “Figure 11 represents a mass service system with THREE service lines into which customers who request service enter in random order.” At the same time, Figure 11 shows 5 channels.

- Thank you for the comment. This was a typo and we have corrected it to 5 service lines.

5. Lines 373-375. As far as we understand, the authors use analytical methods to calculate performance indicators of the resulting QS. In this regard, the mention of simulation modeling requires further clarification. Please explain in detail what for you apply simulation.

- Thank you. This has been supplemented with further explanation.

6. Line 465. The formula should be corrected, in particular, the “=” symbol should be removed from under the root sign.

- Thanks for the comment. This has been edited.

7. Lines 471, 478, and 481. The authors use the same intensity notations “λ” for different request flows. Also it is unclear whether “Intensity of customers entering with a request to issue an international travel ticket between 9.00 AM and 3.00 PM” (line 481) is a part of  “The average total intensity of customer access to the system from 6.00 AM – 6.00 PM” (line 471).

- Thank you. Original lines 471, 478 and 481 have been modified.

- Yes, it is included as it is stated that λ is calculated for all ticket types.

8. Line 500. “The total average service time resulting from table 3 is 1/μ= 87.47 customers/hour, AFTER ADJUSTMENT μ= 41.16 customers/hour”. It is necessary to clarify what exactly “ADJUSTMENT” consists of.

- Thanks for the reminder. It was a typo. This is table 2, the adjustment results from the calculated average service time and those from formula (20).

9. Sections 4 and 5 are a bit long, it is recommended to simplify. 

- Thanks for the reminder. These sections have been shortened and the calculation tables of the mathematical analysis have been moved to the supplementary materials.

10. The article also does not discuss verification of the model constructed. Please, add some details.

- Thanks for the comment. This was added in the Discussion section.

11. I recommend including possible directions for further research in the conclusion.

• Thanks for the comment. This has been added to the Conclusion section.

Best regards

Authors

Reviewer 4 Report

Comments and Suggestions for Authors

A thorough literature review taking into account recent publications has been conducted, but a summary showing the research gap is missing. Reference should be made to the method developed.

In Chapter 3 I would add a description comparing the Poprad-Tatry railway station with other stations. What special/different conditions are there at this station compared to other stations?

I suggest marking the paths described from line 189 with different colors in Figure 1 and including this in the description.

Please justify on what day and at what times were the measurements described in section 3.2 conducted? Doesn't the length of service time depend, for example, on the age of the traveler? Does this not affect the developed model?

In chapter 5 lacks information on how was the proposed method verified with reality?

In the discussion, a table comparing the required number of employees on lines 1-5 after the changes can be added - with reference to Table 3.

Author Response

Dear reviewer,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewer comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewer comments and they have all been incorporated into the article.

Comments from the Reviewer:

Reviewer 4

A thorough literature review taking into account recent publications has been conducted, but a summary showing the research gap is missing. Reference should be made to the method developed.

- Thanks for the comment. This has been added in the Literature Review section.

In Chapter 3 I would add a description comparing the Poprad-Tatry railway station with other stations. What special/different conditions are there at this station compared to other stations?

- Thanks for the reminder. This comparison has been added to section 3.

I suggest marking the paths described from line 189 with different colors in Figure 1 and including this in the description.

- Thank you for your valuable comment. This has been modified in both the text and Figure 1.

Please justify on what day and at what times were the measurements described in section 3.2 conducted? Doesn't the length of service time depend, for example, on the age of the traveler? Does this not affect the developed model?

- Thanks for the comment. This was added in section 3.2.

In chapter 5 lacks information on how was the proposed method verified with reality?

- Thank you. This was added in section 5.

In the discussion, a table comparing the required number of employees on lines 1-5 after the changes can be added - with reference to Table 3.

• Thanks for the comment. The table was added to the Discussion section.

Best regards
Authors

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I read through the revised manuscript and I must admit that many minor typos and errors have been corrected. The main methodological problems I provided in my previous review are briefly discussed in the author responses but there are no corrections in the article (only some basic pieces of information why simulation is not good for solving the problem). I suggested to test the hypothesis about the exponential distribution of probability – this is still missing in the article. And I still have doubts about using the steady-state analysis for short time periods.

However the authors still misinterpret the meaning of μ and 1/μ. The value of μ is the service rate expressed in customers per an hour and 1/μ is the mean service time expressed in hours per an customer. It is not true that the total average service time is μ = 87.47 customers/hour as stated in lines 626-627. In addition, in table 5 (and also other tables) it is stated that 1/μ (hours/customer) is 41.16 – does they really need 41.16 hours on average to serve a customer?

My opinion is still the same, I think the article is not suitable to be published and I recommend to reject it.

Author Response

Dear editor and reviewers,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewers comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewers' comments and they have all been incorporated into the article. We have highlighted the incorporated comments of the second review round in green throughout the document.

Comments from the Reviewers:

 

Reviewer 1

I read through the revised manuscript and I must admit that many minor typos and errors have been corrected. The main methodological problems I provided in my previous review are briefly discussed in the author responses but there are no corrections in the article (only some basic pieces of information why simulation is not good for solving the problem). I suggested to test the hypothesis about the exponential distribution of probability – this is still missing in the article. And I still have doubts about using the steady-state analysis for short time periods.

• Thank you for the constructive comment. We have supplemented the testing of the hypothesis about the exponential distribution. Testing was performed with the Kolmogorov-Smiring test. The chosen level of significance was 0.05. Test statistic was D = 0.0647; P = 0.6727. Since the p-value is quite large (greater than 0.05), we failed to reject the null hypothesis. This means that the data follows an exponential distribution.
• All required information was added to the article. We hope it will be okay.

However the authors still misinterpret the meaning of μ and 1/μ. The value of μ is the service rate expressed in customers per an hour and 1/μ is the mean service time expressed in hours per an customer. It is not true that the total average service time is μ = 87.47 customers/hour as stated in lines 626-627. In addition, in table 5 (and other tables) it is stated that 1/μ (hours/customer) is 41.16 – does they really need 41.16 hours on average to serve a customer?

• Thank you for your valuable comment. This has been edited. We checked all the procedures and calculations, nevertheless our results are checked and the value of μ = 87.47 customers/hours. In Table 5 and other tables, inconsistencies have been corrected. Value 1/μ = 41.16 minutes/customers.

My opinion is still the same, I think the article is not suitable to be published and I recommend to reject it.

• Thank you for your opinion on our research. Nevertheless, we believe that this practical orientation of the research is our main added value, and we hope that you will reconsider your position on this aspect of our article.

Best regards

Authors

Reviewer 2 Report

Comments and Suggestions for Authors

• Be consistent: In the abstract, “ticketing counter” should be changed to “ticket booth.”
• The abstract is much better now; however, the last sentence could be improved. Is the use of a common queue their main contribution? Is it one of the gaps in the literature?
• L39-41: This needs references: “Also, the literature often underestimates the influence of queue proposal on customers' perceptions of waiting time, which can significantly affect their satisfaction and tolerance for waiting.” Are you going to discuss or address users’ perceptions? If not, why mention it here?
• L43: What do you mean by “removal of the queue”? Also, which queue? Replace "The" with "A."
• L45: Again, it should be "A company’s," not "the." Which company?
• L76-85: It still does not clearly show why using this approach is a contribution. How is this different from classic queue theory?
• L76-85 and L99-110: These sections can be combined and made more specific to avoid repetition.
• L94: What exactly is the significant contribution?
• L86-98: You need to justify your claims with some references. Otherwise, they may not seem valid.
• In the future, for the introduction and any editing, I suggest only highlighting the modified parts to make it easier to detect changes.
• I don’t think Figure 1 is necessary. You could summarize the important sections of the paper in a couple of sentences. For instance, determining the research objective doesn’t need to be mentioned at all.
• I generally accept your explanation for not using the simulation model, but you still need to add some references.
• There is still no mention of staff allocation in the introduction or why it needed to be considered in their study.
• The main gaps and how their approach addresses these gaps are still missing from the introduction section.
• L132: How should I know what studies 10 and 11 have done by just saying “studies expand on this concept”?
• L152: What models?
• As mentioned in the introduction, many of the studies you cited used simulation. So, either you need to add and discuss more studies that did not use simulation or better justify why you are not using simulation. You need to elaborate more on studies 22-29 instead of those using the simulation modeling approach.
• What exactly is your contribution compared to study 30?
• L189-192: This could also be the case for your model, so it’s not very convincing.
• L192: You are correct that mathematical models can sometimes find optimal solutions with fewer iterations and less effort compared to simulations. However, you need to show that the solutions are indeed optimal. Optimizing a system to get an optimal solution can take as much time/iterations as simulation models. Be more specific and include references to verify your claim.
• The third section title is still not correct. What methods exactly?
• L420-428: Add some references. And what if we have a stochastic system, which a rail station ticket booth likely has? Why exaggerate deterministic results, then?
• L441: Use "he/she" or "they."
• One of the main concerns was whether the authors conducted any pre-analysis to ensure that the arrival patterns of their system follow the requirements of the Poisson process. This concern was not adequately addressed. The authors added some explanation in L476-482 regarding the Poisson process but did not specify whether the inter-arrival time of their data can be fitted to an exponential distribution.
• Another missing point is the need for more explanation on how the Markov Chain is used in their formulation and how it differs from classical queue theory formulations. In fact, comparing their results with those from classic queueing theory would be very beneficial, as well as clarifying why their output is considered optimized.

Comments on the Quality of English Language

Much better, still another round of proof-reading would be great. 

Author Response

Dear editor and reviewers,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewers comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewers' comments and they have all been incorporated into the article. We have highlighted the incorporated comments of the second review round in green throughout the document.

Comments from the Reviewers:

Reviewer 2

• Be consistent: In the abstract, “ticketing counter” should be changed to “ticket booth.”

- Thanks for the comment. This has been edited.

• The abstract is much better now; however, the last sentence could be improved. Is the use of a common queue their main contribution? Is it one of the gaps in the literature?

- Thanks for the comment. This has been edited.

- Yes, this is one of the gaps in the literature. The queue and passenger flow management literature identifies several important gaps, which include decentralized control systems, the impact of new technologies and pandemics, the need for station-specific solutions, and the limitations of simulations. Other gaps are the lack of an integrated approach to queue management, such as the number of service providers or the configuration of queues, as well as the insufficient distribution of employees (respectively their working hours). Research also often underes-timates the impact that queue design has on customers' perception of waiting time, which can have a significant impact on their satisfaction and tolerance of waiting. Many queue management systems are not flexible enough to handle unexpected events, such as peaks during special events or system failures. The way the approach solves these shortcomings is by using Markov chains, which allows the queue management system to be dynami-cally adapted to different configurations and customer flow intensities.

• L39-41: This needs references: “Also, the literature often underestimates the influence of queue proposal on customers' perceptions of waiting time, which can significantly affect their satisfaction and tolerance for waiting.” Are you going to discuss or address users’ perceptions? If not, why mention it here?

- Thank you. This has been removed.

• L43: What do you mean by “removal of the queue”? Also, which queue? Replace "The" with "A."

- Thank you for your comment. I understand that the term "queue removal" can be confusing, so I will explain it in more detail. The term "queue elimination" in this context means minimizing or eliminating customer queues before entering the system. This means that customers should be served almost immediately after entering the system, reducing or eliminating the need to queue.

- As for the second part of your question - it concerns the waiting line of customers who are waiting to be served in the system. This queue is created as a result of customers entering the system randomly, but the attendant is unable to process them immediately, resulting in queues.

• L45: Again, it should be "A company’s," not "the." Which company?

- Thanks for the comment. This has been edited. The word company has been removed for better understanding of the meaning.

• L76-85: It still does not clearly show why using this approach is a contribution. How is this different from classic queue theory?

- Thank you. This has been supplemented with further clarification. The use of Markov chains is beneficial because it allows dynamic transitions between system states to be modeled and provides more accurate predictions of system behavior in real time. Unlike classical queuing theory, which focuses on simpler models, Markov chains can work flexibly with different intensities of customer arrivals and system configurations. This approach enables better optimization of service processes, especially in complex systems with multiple states and variable intensity of customer flow.

• L76-85 and L99-110: These sections can be combined and made more specific to avoid repetition.

- Thank you. The text has been carefully revised and combined for a better understanding of the meaning and to avoid repetition.

 

• L94:What exactly is the significant contribution?
• - Thank you. This was explained in the Introduction section.

• L86-98: You need to justify your claims with some references. Otherwise, they may not seem valid.

- Thanks for the comment. This has been justified and the references have been supplemented.

• In the future, for the introduction and any editing, I suggest only highlighting the modified parts to make it easier to detect changes.

- Thanks for the reminder. Since the Introduction section has been completely redesigned, we have highlighted this entire section.

• I don’t think Figure 1 is necessary. You could summarize the important sections of the paper in a couple of sentences. For instance, determining the research objective doesn’t need to be mentioned at all.

- Thank you for the reminder. Based on other reviewers, Figure 1 supplemented in the first review round. The reviewers requested a schematic representation of the methodological procedure of our research solutions. In the second review round, this image was modified as required. We hope it will be ok if we keep the image.

• I generally accept your explanation for not using the simulation model, but you still need to add some references.

- Thank you for your comment. Links have been added along with an explanation.

• There is still no mention of staff allocation in the introduction or why it needed to be considered in their study.

- Thank you for the reminder. This has been supplemented.

• The main gaps and how their approach addresses these gaps are still missing from the introduction section.

- Thank you. The main research gaps were filled.

• L132: How should I know what studies 10 and 11 have done by just saying “studies expand on this concept”?

- Thank you for your valuable comment. The study by Abdollahi and Khorasani (2008) focuses on the H∞ control strategy for the design of a robust dynamic routing algorithm in traffic networks, where the emphasis is placed on the stability and robustness of the system in handling unpredictable changes in traffic flows. Their approach uses centralized control, which is different from the decentralized approaches later developed by other studies.

- Manfredi's work (2014) introduces the concept of decentralized queue balancing and differentiated services within cooperative control, where the approach focuses on traffic queue management and service differentiation in decentralized systems. This model is more adaptable and does not require centralized control, creating a more flexible way to optimize in real time.

- Cogill, Rotkowitz and Van Roy (2006) in their work on decentralized control of stochastic systems apply approximate dynamic programming, which enables the solution of complex problems of decentralized control without the need for centralized information. This study is key in that it creates a framework for solving stochastic systems, which can also be applied to various traffic management models.

- These studies extend the concept of decentralized control and optimization by providing different approaches to improve the control of traffic flows in systems with centralized and decentralized control mechanisms. While Abdollahi and Khorasani focus on centralized robust control, Manfredi and Cogill contribute to the development of decentralized approaches emphasizing cooperative control and flexibility in systems with variable conditions.

• L152: What models?

- Thank you. This has been supplemented.

• As mentioned in the introduction, many of the studies you cited used simulation. So, either you need to add and discuss more studies that did not use simulation or better justify why you are not using simulation. You need to elaborate more on studies 22-29 instead of those using the simulation modeling approach.

- Thanks for the comment. These studies were discussed in the Literature review section. We also elaborated studies 22-29 (now 26-30, 19,20).

• What exactly is your contribution compared to study 30?

- Thank you. The benefit of our research lies in its specific focus on optimizing the passenger queuing system at railway stations, which brings practical solutions to shorten waiting times and improve the efficiency of customer service. Unlike the study of Vojtek et al. (2020), which focuses on the use of applied mathematics to improve the operational efficiency of a transport company, our research focuses on the practical application of mathematical analysis in comparison with the theoretical knowledge of the use of applied mathematics in the study of Vojtek et al. (2020).

• L189-192: This could also be the case for your model, so it’s not very convincing.

- Thank you. This was elaborated more in the Introduction section.

• L192: You are correct that mathematical models can sometimes find optimal solutions with fewer iterations and less effort compared to simulations. However, you need to show that the solutions are indeed optimal. Optimizing a system to get an optimal solution can take as much time/iterations as simulation models. Be more specific and include references to verify your claim.

- Thanks for the reminder. We described this in the section Literature review, references [32,33].

• The third section title is still not correct. What methods exactly?

- Thanks for the reminder. The title has been changed to Research Background.

• L420-428: Add some references. And what if we have a stochastic system, which a rail station ticket booth likely has? Why exaggerate deterministic results, then?

- Thank you. I understand your comment about the stochastic nature of a system such as ticket sales at a train station. I agree that stochastic models can better capture the random behavior of customers and real-time conditions. However, my emphasis on deterministic models is based on the fact that, despite the stochastic elements, deterministic models can provide accurate and fast solutions for stable and predictable aspects of the system, which can be supplemented by simulations for probabilistic analyses. This was described in the Research Methodology section, references [35,36].

• L441: Use "he/she" or "they."

- Thanks for the comment. This has been edited to the word customer.

• One of the main concerns was whether the authors conducted any pre-analysis to ensure that the arrival patterns of their system follow the requirements of the Poisson process. This concern was not adequately addressed. The authors added some explanation in L476-482 regarding the Poisson process but did not specify whether the inter-arrival time of their data can be fitted to an exponential distribution.

- Thank you. We performed a statistical test - the Kolmogorov-Smiring hypothesis test - which confirmed that the interarrival time in our data follows an exponential distribution. Thus, this result satisfies the condition of a Poisson process, which means that our arrival patterns can be modeled by a Poisson distribution. We thereby fulfilled the requirement for the validation of the arrival process and strengthened the reliability of our results.

• Another missing point is the need for more explanation on how the Markov Chain is used in their formulation and how it differs from classical queue theory formulations. In fact, comparing their results with those from classic queueing theory would be very beneficial, as well as clarifying why their output is considered optimized.

- Thanks for the comment. This is described in the Discussion section.

• Markov chains in our formulations serve to model transitions between different states of the system (e.g. number of customers in a queue), assuming that the future behavior of the system depends only on the current state, not on the previous one. This model captures dynamic queue changes based on state transition probabilities, allowing us to better manage systems with variable arrival and service intensity. Unlike classical queuing theory, which often works with fixed parameters (e.g. constant service and arrival rates), the Markov model enables dynamic adaptation to changes in the system. Classical queuing theory models can be limited by the fact that they do not take into account random events that can affect the entire system (eg the sudden arrival of a large number of customers). The Markov chain is therefore more flexible in handling such changes. Classical models provide results based on average values, while our approach using Markov chains provides more dynamic and accurate predictions because it takes into account changes in the system in real time. The optimization comes from the fact that this model can more effectively redistribute resources (eg manpower) based on the current state of the system, thus minimizing waiting times and increasing efficiency.

Best regards

Authors

Reviewer 3 Report

Comments and Suggestions for Authors

The article presents significant improvements, although it still has some flaws that need improvement:

1) Figure 1. How does the methodological procedure relate to the rest of the article? It may be worth combining stages into blocks or introducing two-level numbering. The first level would correspond to the section. Circuit elements could also be placed in two columns for a better visual display.

2) Lines 177-197. The detailed description of the study is provided in [30]. It is followed by “Studies focusing on simulations of queuing in front of ticket booths can be specific and only for certain conditions or settings, which limits their applicability outside these conditions...". It seems that there are two distinct ideas in this paragraph. It would be better to divide this into two separate paragraphs, or simplify.

3) It would be helpful to know which parts of the statistical data were used to determine the model's parameters and which were used to evaluate its accuracy.

 

Best Regards

The reviewer

Author Response

Dear editor and reviewers,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewers comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewers' comments and they have all been incorporated into the article. We have highlighted the incorporated comments of the second review round in green throughout the document.

Comments from the Reviewers:

Reviewer 3

The article presents significant improvements, although it still has some flaws that need improvement:

• Figure 1. How does the methodological procedure relate to the rest of the article? It may be worth combining stages into blocks or introducing two-level numbering. The first level would correspond to the section. Circuit elements could also be placed in two columns for a better visual display.

- Thank you for your valuable comment. The figure has been edited.

2) Lines 177-197. The detailed description of the study is provided in [30]. It is followed by “Studies focusing on simulations of queuing in front of ticket booths can be specific and only for certain conditions or settings, which limits their applicability outside these conditions...". It seems that there are two distinct ideas in this paragraph. It would be better to divide this into two separate paragraphs, or simplify.

- Thank you for the comment. This has been edited in the Literature review section.

3) It would be helpful to know which parts of the statistical data were used to determine the model's parameters and which were used to evaluate its accuracy.

- Thank you. This was added in the Discussion section.

• Statistical data from railway stations were divided to determine model parameters and evaluate model accuracy. Statistical data were used to determine the parameters of the model, which include time intervals between passenger arrivals, queue lengths, service speed, and the total number of passengers during different time intervals. These data were used to calibrate the model and determine parameters such as arrival intensity (λ) and average service time. To evaluate the model's accuracy, it was used to compare the model's predictions with the real behavior of the system to verify its accuracy and robustness.

Best regards

Authors

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

It is difficult to me to accept the article when I see that the authors do not understand basic principles and terminology of queueing theory. I wrote two times that the authors misinterpreted the service rate μ and the mean service time 1/μ and each correction made by the authors is making it worse and worse. The service rate defines the mean number of customers that is a server able to service per a time unit. Its reciprocal value defines the mean service time.

In lines 756-757 the authors write: “The total average service time resulting from table 2 is μ = 87.47 customers/hours, after 1/μ = 41.16 minutes/customers (after recalculation 1/μ = 0.686 hours/customers).“ In table 2 we can see the observed service times (as I wrote in my first review I think it is better to present the statistical analysis of the data set instead of the raw data) and based on the data the authors calculated that the average (mean) service time is 87.47 seconds – in the notation of the queueing model the authors applied this is the mean service time 1/μ.

And in the sentence I referenced the authors state the μ = 87.47 customers/hours – I do not understand how it is possible to get this service rate from the data set – the value is the same as the mean service time calculated from the data set but the authors write that the value is the mean service rate μ which is a nonsense. Such value of μ implies that 1/ μ is about 0.011 hours/customer = 41.16 seconds/customer – this is the mean service time and the value does not correspond to the data set, moreover the authors use the wrong time unit – they say the value is in minutes per a customer which is another nonsense.

And in table 5 (and in other tables as well) the authors repeat the same error. The value of 1/μ is 41.16 minutes per a customer according to the authors. Is it true that to serve a customer we need about 40 minutes on average?

After seeing such errors in the manuscript twice revised I have serious doubts about other results – how can I trust the results obtained by the queueing model if the authors repeatedly misinterpret the basic notation of the model.     

Author Response

Dear reviewer,

We sincerely thank you to review team for the insightful and constructive comments on our article. The article has been carefully revised according to the reviewers comments.

We look forward to hearing from you on the revised article. In the remainder of this letter, we provide detailed answers to each of the comments. We remind you that we have tried to process all reviewers' comments and they have all been incorporated into the article.

Comments from the Reviewer:

 

Reviewer 1

It is difficult to me to accept the article when I see that the authors do not understand basic principles and terminology of queueing theory. I wrote two times that the authors misinterpreted the service rate μ and the mean service time 1/μ and each correction made by the authors is making it worse and worse. The service rate defines the mean number of customers that is a server able to service per a time unit. Its reciprocal value defines the mean service time.

In lines 756-757 the authors write: “The total average service time resulting from table 2 is μ = 87.47 customers/hours, after 1/μ = 41.16 minutes/customers (after recalculation 1/μ = 0.686 hours/customers).“ In table 2 we can see the observed service times (as I wrote in my first review I think it is better to present the statistical analysis of the data set instead of the raw data) and based on the data the authors calculated that the average (mean) service time is 87.47 seconds – in the notation of the queueing model the authors applied this is the mean service time 1/μ.

And in the sentence I referenced the authors state the μ = 87.47 customers/hours – I do not understand how it is possible to get this service rate from the data set – the value is the same as the mean service time calculated from the data set but the authors write that the value is the mean service rate μ which is a nonsense. Such value of μ implies that 1/ μ is about 0.011 hours/customer = 41.16 seconds/customer – this is the mean service time and the value does not correspond to the data set, moreover the authors use the wrong time unit – they say the value is in minutes per a customer which is another nonsense.

And in table 5 (and in other tables as well) the authors repeat the same error. The value of 1/μ is 41.16 minutes per a customer according to the authors. Is it true that to serve a customer we need about 40 minutes on average?

After seeing such errors in the manuscript twice revised I have serious doubts about other results – how can I trust the results obtained by the queueing model if the authors repeatedly misinterpret the basic notation of the model.     

 

• Thank you for the constructive comment. We think that there was a big mistake on our part and somehow we got involved in this issue. That is why this misunderstanding arose. To put things into perspective, the original interpretation of service rate μ and mean service time 1/μ was misinterpreted. After reviewing the data, that our calculations were correct, but there was an error in the presentation of the units and the interpretation of the given indicators and we found that the average service time for one customer is 87.47 seconds. This means that the service rate μ (number of customers served per hour) is approximately 41.16 customers/hour. Based on this value, 1/μ equals 87.47 seconds/customer, which matches our data.

- This means that we have become more and more entangled in this data with each review round. But here is a step-by-step explanation:

• μ must be expressed as the number of customers served per hour. That is, if the average service time is 87.47 seconds (which is the figure obtained from the data file), we calculate the service rate μ as: μ=3600/87.47  and we get 41.16 customer/hour
• This means that the system can serve 41.16 customers per hour.

Average service time is expressed as:

1/μ=1/41.16 hour/customer and we get 0.0243 customer/hour.

This means that the system can serve approximately 41.16 customers per hour.

After converting to seconds, we get 0.0243 x 3600=87.47 sec/customer, which corresponds to the original calculation of the average service time.

We have corrected errors and reformulated relevant sections to clearly reflect the correct units and interpretation of indicators. We apologize for the misunderstanding and believe that now the calculations and their interpretation are clear and accurate.

 

Best regards

Authors