MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback

Round 1

Reviewer 1 Report

The authors study a priority queueing system where the high-priority customer at service completion either leaves the system or - wp theta - rejoins the system as a low-priority customer.
An MMAP arrival process is considered and service times have class-dependent PH distributions.

Although the mathematical analysis is sound, I have two major remarks:
* The motivation is weak: a vague explanation is given on why feedback queues are studied, but no motivation is given for the particular models studied in this paper.
* The introduction should be split in introduction and a literature review section, to keep the introduction concise so that the main contribution of the paper is more clear in the introduction.

Some smaller remarks:
* In preemptive priority queueing systems, a distinction is made between preemptive repeat and preemptive resume. I presume the paper considers preemptive repeat, but this should be mentioned explicitly while explaining the model.
* general remark: 'et al.' should be used when citing a paper in the text in case of 3 or more authors.
* starting from p9 a lot of typos occur, for example:
    p9 l307: probabaility --> probability
    p9 l308: immedietly --> immediately
    p9 l310: vecror --> vector
    the authors should carefully proofread the paper

Author Response

The authors study a priority queueing system where the high-priority customer at service
completion either leaves the system or - wp theta - rejoins the system as a low-priority customer.
An MMAP arrival process is considered and service times have class-dependent PH
distributions.
Although the mathematical analysis is sound, I have two major remarks:
* The motivation is weak: a vague explanation is given on why feedback queues are studied, but
no motivation is given for the particular models studied in this paper.
Response: Explanation given in the revised version: see lines (171-184).
* The introduction should be split in introduction and a literature review section, to keep the
introduction concise so that the main contribution of the paper is more clear in the introduction.
Response: Corrected
Some smaller remarks:
* In preemptive priority queueing systems, a distinction is made between preemptive repeat and
preemptive resume. I presume the paper considers preemptive repeat, but this should be
mentioned explicitly while explaining the model.
Response: In this paper, we consider preemptive repeat (identical). It is now included in the
description of Model II.

* general remark: 'et al.' should be used when citing a paper in the text in case of 3 or more
authors.
Response: Corrected

* starting from p9 a lot of typos occur, for example:
    p9 l307: probabaility --> probability
    p9 l308: immedietly --> immediately
    p9 l310: vecror --> vector
    the authors should carefully proofread the paper
Response: Corrected

Reviewer 2 Report

Summary: This paper compares two single server queuing systems with customers of two distinct priorities. The service time distributions vary based on customer type and feedback. Overall this paper is well-written from the mathematical perspective and has a good background of prior work. 

Comments:

• Some assumptions made in the model are not clearly elaborated to connect with an actual scenario, this should be checked and more details should be added, for example, section 2, formulation of model I, the physical meaning or application of the model, the assumptions and their translation to a real scenario could be explained.
• In section 5, the utility of other performance measures is not clear. More details should be added.
• Section 10, numerical results, the 5 sets mentioned for customer arrival process, however, the scenario interpretation is unclear
• 10.1 - what is the reason for fixing theta as 0.6?
• 10.2 - Expected waiting time, what are the units of the quantities shown in Tables 5 and 6.
• Tables and Figures in the result sections are missing units. Also, the figure legends can add information regarding the values of theta, MNA, and MPA arrivals for each model.

Author Response

Summary: This paper compares two single server queuing systems with customers of two distinct priorities. The service time distributions vary based on customer type and feedback.
Overall this paper is well-written from the mathematical perspective and has a good background of prior work. 
Comments:
 Some assumptions made in the model are not clearly elaborated to connect with an actual scenario, this should be checked and more details should be added, for example, section 2, formulation of model I, the physical meaning or application of the model, the assumptions and their translation to a real scenario could be explained.
 Response: Details given in the revised version: please see lines 171-184.
 In section 5, the utility of other performance measures is not clear. More details should be added.
Response: We have given several performance measures to understand the system performance in sections 5 (for model I) and in section 9 (for model II). We illustrated the performance of the system based on some of these measures in section 10.
 Section 10, numerical results, the 5 sets mentioned for customer arrival process, however, the scenario interpretation is unclear
Response: We have chosen five sets of matrices for 10,DD and 2D for MMAP arrival process.
These processes are normalized to have an arrival rate of 0.1 and are qualitatively different in that they have different variance and correlation structure. The purpose of the illustration is to study how various system performance measures behave under different scenarios.

 10.1 - what is the reason for fixing theta as 0.6?
Response: We have selected just a value between 0 and 1 to study the effect of L on various performance measures. We have also illustrated the effect of theta on various performance measures when L is fixed.
 10.2 - Expected waiting time, what are the units of the quantities shown in Tables 5 and 6.
Response: The units used are time; the same time unit is used for the distribution functions also.
For example that could be seconds/minutes/hours or even day(s).

 Tables and Figures in the result sections are missing units.

Response: Regarding time unit, the response is the same as that for the immediately previous query.
 The figure legends can add information regarding the values of theta, MNA, and MPA arrivals for each model.
Response: Figure legends are brought in. However, these are given immediately below the corresponding figures.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors of this paper consider two single server queueing systems to which customers of two distinct priorities arrive following a Marked Markovian arrival process (MMAP). Service time of both types of customers are having two distinct phase-type distributions. The probability of a P1 customer to feedback is q on completion of his service. The feedback (P1)
customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. It is also assumed that P1 and P2 customers have a finite and an infinite waiting space, respectively.

The authors provide the steady-state analysis for both models, which is correct (to the best of the reviewer's understanding) and many numerical experiments in order to compare the different performance characteristics of both models.

As a general comment the reviewer believes that the paper deserves publication. Some grammatical/syntax errors can be corrected during the preparation of the camera ready version.

Author Response

The authors of this paper consider two single server queueing systems to which customers of two distinct priorities arrive following a Marked Markovian arrival process (MMAP). Service time of both types of customers are having two distinct phase-type distributions. The probability of a P1 customer to feedback is q on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch.
It is also assumed that P1 and P2 customers have a finite and an infinite waiting space, respectively.
The authors provide the steady-state analysis for both models, which is correct (to the best of the reviewer's understanding) and many numerical experiments in order to compare the different performance characteristics of both models.
As a general comment the reviewer believes that the paper deserves publication. Some grammatical/syntax errors can be corrected during the preparation of the camera ready version.
Response: grammatical/syntax errors corrected

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have split the introduction and the literature review now. The introduction is now more concise, but the main contribution of the paper is not yet clear in the introduction. The authors just briefly mention what model they will study. But before going to the literature review, the authors should briefly discuss what is new as compared to literature (the further details are in the literature review, but the introduction should contain a to the point concise summary of the contribution) and why it is an interesting problem/model to investigate/analyse.

Author Response

Explanation given in the revised version: see lines (40-71) and (218-221).

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

the authors have taken care of my remarks. Some minor remarks:

p1 l28-30 this text seems to be at an incorrect place, it should follow after the summary of the main contributions in my opinion.

p2 l 48 'is done away with' is incorrect English

 

Author Response

p1 l 28-30 this text seems to be at an incorrect place, it should follow after the summary of the main contributions in my opinion.

Response

Corrected

p2 l 48 'is done away with' is incorrect English

Response

We replaced part of line 47 and THE ABOVE INDICATED PART OF LINE 48 AS FOLLOWS: 

provided the direct admission of low priority customers to the second queue is done away with.--> provided low priority customers are not admitted to the system.

Author Response File: Author Response.pdf