-------- SUMMARY --------

This paper is about heterogeneous networks (HetNets) in which we can find three kinds of cells : macrocells to cover large areas, and picocells and femtocells to cover smaller areas.

Usually, two parameters are studied : the coverage performance and the energy efficiency. The former is about accessing all the users of the networks with every cells while achieving a specified quality of service. It depends on the density of the cells. The latter is the ratio between the throughput of the network and the power consumption.

Previous studies rarely focus on the two parameters in an three-tier HetNets. Usually, either the picocells of the femtocells are considered.

This paper fills this lack by studied the relationship between those two parameters in a three-tier HetNets.

The paper is organized into two main parts: - an analytic study where the model is explained - simulations curves of the model

-------- GENERAL COMMENTS --------

From my point of view, the subject is well introduced. I must admit I am not an expert in cellular networks and the associated physical aspects. However I have the feeling that I understood the important parts and I could follow the calculations of the article.

Considering the english, it seems good.

The analytical parts and the proofs seem correct. However some (major and minor) mistakes appears in the equations. Particularly, some parts are missing, leaving blank spaces in the middle of the computations... In Lemma 1, there is at least one mistake in each equation. In addition, I think the last part with the linear program could go further. (See major comments)

On the contrary, the simulation part is, in my mind, not really useful. It seems to be no more than a duplicate of the analytical part and does not provide any new result. See the major comments for details. However, as I am not an expert, I could have miss something...

Thus, I suggest Major revisions.

-------- MAJOR COMMENTS --------

Some parts of equations are missing, particularly in the appendix : - line 147 - line 186, equation (10), between the middle dot and p (twice) - line 187 - lines 420, equation (A.4), between the middle dot and lambda_M - line 421 (twice)

- line 179, equation 7 ; and line 186, equation 10, the integral should be bounded from 0 to Rp, as r is no more than Rp
- line 193, equation 12 ; and line 197, equation 13, the integral should be bounded from 0 to Rf, as r is no more than Rf

Page 8, once we see Figure 2, solving the linear system is trivial. The optimal solution is one of the two non null vertices of the feasible triangle. This vertex depends only on the slope of the energy consumption function. It is then surprising not to see the analytic relation between the optimal solution and the parameters.

In Section 6, the two parameters l and epsilon are never explicitly set.

Section 6 seems only a duplicate of the previous sections.

What is not clear is that I do not understand how the authors did the simulations. - did they only compute, for each value of the parameters Tf, Tp, lambda_f and lambda_p, the theoretical optimal value n_EE and print it? In that case, each curve is drawn using numeric simulations of the formulas of previous sections and the discussion about the results explains how the curves evolves with the different parameters. However we could have done the same explanations using directly the formulas without drawing any curve. If I have y = 2x, I know that y increases with x and I do not need the curve to see that. In my mind, such a simulation is necessary if we cannot exactly compute the solutions. As there is an analytical function of n_EE depending on the other parameters, I do not see the point. As a consequence, the simulation part could be removed. - did they place the cells and the users on a map, and then simulate the performances of each cell independently? In that case, how the physical parameters are simulated. How do we decide the quality of service of a user? Is the source code accessible somewhere? ... In addition, why are the previous sections ignored? It could have been a good idea to validate the analytical model by measuring of the curves fit with the equations. However, why would this not happen? Except if there is a mistake in the equations, as long as the simulations are simulations (and not an experiment in real life with real cells), it necessarily follows the calculations of the model. As a consequence, the simulation part could be, again, removed.

In the two cases, the simulations are just a duplicate and are not necessary. I suggest to remove the simulations from section 6, except for some figures in order to give an idea of how n_EE evolves with the other parameters.

Maybe I missed something. In that case, I suggest the authors write a complete transition between section 5 and section 6 to explain why they do the simulations.

-------- MINOR COMMENTS --------

- line 38, "between 2010 and 2020, 2025" -> "between 2010 and 2025"
- line 75, Is this a good thing or a bad thing? In what this is an interesting result?
- line 90, "the majority of previous works", which ones?
- line 134, equation 1, why the notation f_{r_p}, why not f_p or just f?
- line 136-138, "Differing from picocells users ... uniform distribution" Is there anything that can justify those distributions? In real cases, how are the picocells and the femtocells deployed?
- line 140, "based on probability theory": the result should be either cited from somewhere or proved in the lemma or the appendix.
- line 156, "R_i and R_j" I do not know why, but the two "R" does not seem written the same way. The second seems larger than the first one.
- line 171,172, "and can be also found in other literatures" Where?
- line 183, equation 8, could the author add a note on what is the random variable of that equation? Is that r?
- line 184, equation 9, put parenthesis or square brackets instead of brackets. Brackets are used for mathematical sets. The same remarks occurs every time there are brackets in equations.
- line 186, equation 10, a "+ delta^2" is added to equation 10, and then it vanishes.
- line 186, equation 10, there is a lambda_S that appears from nowhere
- line 187, equation 11, Tp became T
- line 187, equation 11, E_Ir(exp[...]|r) should be E_Ir(p(exp[...]|r))
- lines 195 eq 13, 197 eq 14, 199, 200 eq 15 : each probability symbol is P instead of p
- lines 198-199, the sentence is weird. Why not just recall the femto tiers follows a uniform distribution?
- Lemma 1 and 2 are never explicitly used.
- Line 217, "Accordingly" I do not see why this word is used. Equation 17 is not a consequence of equation 16.
- Line 224 "no matter in the" -> "no matter the"?
- Line 226-227, "channel ressources channel"
- Line 242, equation 23, this equation is redundant with equations 24,25,26
- line 244, a space is missing before "The"
- line 245, a dot appears between "Figure" and "2".
- line 253 Table 1, why do not describe the parameters we are interested in? What is alpha? What is the link between the System carrier, the system bandwidthn the path losses and the paremeters of the analytical study?
- line 259-260 "the key parameters are set up according to [35,36]" why are those particular papers chosen?
- Figure 3, 4, if it is possible, rotate qp and qf
- Line 283 "There results" -> "These result"
- Figure 5,6,7,8,9,10, Why is the abscissa equal to lambda_f/lambda_p and not only lambda_f as lambda_p is fixed? If lambda_f and lambda_p are multiplied by 2, do the same curves occur?
- Figure 5,6,7 : the highest ordinate is 180 in the twi first images, and 200 in the last one. I suggest to put 200 every where.
- Line 341 "for or each choice" -> "for each choice"