In this section, we validate the mathematical frameworks and findings derived in the previous sections with the aid of Monte Carlo simulations, as well as compare the IAM scheme against IAFPC and IUFPC schemes. The following setup compliant with LTE specifications is considered. The bandwidth is equal to 10 MHz, which implies b_{w}=9 MHz by excluding the guard bands. The noise power spectral density is n_{thermal}=− 174 dBm/Hz, and the noise figure of the receiver is n_{F} = 9 dB. Both GCA and SPLA criteria are studied, and the association weights are, unless otherwise stated, t^{(1)}/t^{(2)}=9 dB and t^{(1)}/t^{(2)} = 0 dB, respectively. The case study t^{(1)}/t^{(2)} = 9 dB is related to a cell association based on the average DL received power criterion, where the first tier of BSs (macro) has transmit power equal to 46 dBm and the second tier of BSs (smallcell) has transmit power equal to 37 dBm, which agrees with ([18], Annex A: Simulation Model). Other simulation parameters are provided in Table 3. As far as Monte Carlo simulations are concerned, they are obtained by considering 10^{4} realizations of channels and network topologies. In all the figures, analytical and Monte Carlo simulation results are represented with solid lines and markers, respectively.
Average transmit power, probability of being active, mean, and variance of the interference
In this section, we analyze the average transmit power of the MTs, the probability that the typical MT is active, which provides information on the system fairness, and the mean and variance of the interference.
Figures 1, 2, 3, and 4 confirm the conclusions drawn in Remark 1, i.e., the mathematical frameworks of average transmit power and probability of being active are exact while those of mean and variance of the interference are approximations that exploit Assumption 1. Such an assumption considers that the position of interfering MTs can be modeled as a conditionally thinned (i.e., nonhomogeneous) PPP. The difference between such a nonhomogeneous PPP, and the actual point process, which is on the other hand not tractable, explains also the difference between simulation and analytical results in all the metrics that depend on the interference (SINR, SE, BR). The conclusions drawn in Remark 2 are confirmed as well: the mean and variance of the interference decrease by decreasing i_{0}, which provide important advantages for implementing AMC schemes.
In the figures, IAM and IAFPC are compared as well. We observe that IAM reduces the average transmit power and the mean and variance of the interference.
Consider the SPLA criterion, which is illustrated with dashed lines in the figures. We observe that the findings in Remark 4 are confirmed: the system is interferenceaware and interferenceunaware if i_{0}<p_{0} and i_{0}>p_{0}, respectively. As expected, the crossing point occurs at p_{0} = − 70 dBm based on the simulation parameters used. In addition, the scaling laws of average transmit power and average interference are in agreement with the findings in Remark 7 and Remark 8.
All in all, the numerical illustrations reported in Figs. 1, 2, 3, and 4 confirm all the conclusions and performance trends discussed in the previous sections and highlight the advantages of IAM.
Complementary cumulative distribution function of the SINR
In this section, we analyze the coverage probability (CCDF of the SINR) of the active MTs. The results are illustrated in Figs. 5 and 6 for ε=1 and ε=0.75, respectively, and by assuming p_{max}→∞.
In both figures, we observe a good agreement between mathematical frameworks and Monte Carlo simulations. In particular, the figures confirm, once again, that the coverage probability of IAM increases as i_{0} decreases. In Fig. 5, for example, almost all the active MTs have a SINR greater than 20 dB if i_{0}=− 120 dBm. This good SINR is obtained because IAM keeps under control the interference by muting the MTs that create more interference. Based on Fig. 2, in fact, we note that only a small fraction of the MTs are allowed to be active for i_{0}=− 120 dBm. The active MTs, however, better exploit the available bandwidth. Similar conclusions can be drawn for ε=0.75 shown in Fig. 6. The main difference is that, in this latter figure, IAM provides almost the same coverage probability for i_{0} = − 60 dBm and i_{0} = − 90 dBm. The reason is that the MTs transmit with less power if ε = 0.75 and, thus, there is almost no difference between the two interference constraints. This brings to our attention that the design of the UL of HCNs requires to jointly optimize i_{0}, p_{0}, p_{max}, and ε, in order to identify the desired operating regime that fulfills the requirements in terms of system fairness and interference mitigation. The proposed mathematical frameworks can be used to this end.
Spectral efficiency and binary rate
In this section, the average SE and average BR are analyzed, as well as the IAFPC and IAM schemes are compared against each other for several system setups.
In Fig. 7, the average SE of IAFPC and IAM schemes is analyzed and three conclusions can be drawn. By comparing the average SE of the IAPFC scheme (i.e., for AMC schemes) and on the Shannon formula, we note, as expected, that the latter formula provides optimistic estimates of the average SE. By comparing the average SE of the IAM scheme for typical (active and muted) MTs and active (only) MTs, we note a different performance trend as a function of i_{0}. As for the active MTs, the average SE increases as i_{0} decreases. As for the typical MTs, on the other hand, the average SE decreases as i_{0} decreases. This is because the lower i_{0} is the more MTs are turned off, which on average, contributes to reduce the SE of the typical MT. By comparing the average SE of IAPFC and IAM schemes, we evince that IAFPC outperforms IAM for all relevant values of the maximum interference constraint i_{0}, since all the MTs are active under the IAFPC scheme. The average SE of the active MTs under the IAM scheme is, however, much better than that of the IAFPC scheme, since the othercell interference is reduced.
The SE, however, does not provide information on the amount of bandwidth that the scheduler allocates to each active MT.
This tradeoff is captured by the average BR, which is shown Fig. 8. As far as the average BR is concerned, in particular, we note that IAFPC and IAM schemes provide opposite trends compared to those evinced from the analysis of the average SE of the typical MT. More precisely, IAM provides a better average BR than IAFPC, and there exists an optimal value of i_{0} that maximizes it. This optimal value of i_{0} emerges if the typical MT is considered, i.e., the MT may be either active or inactive. The figure, however, shows the average BR achieved only by the active MTs. In this case, we note that the MTs that satisfy both power and interference constraints achieve a very high throughput due to the reduced level of interference that is generated in this case. In a nutshell, IAM outperforms IAFPC in terms of average BR because the available bandwidth is shared among fewer MTs (only those active), which results in a higher throughput for each of them. Even though some MTs may be turned off in IAM, this may not necessarily be considered as a downside from the user’s perspective: in highmobility scenarios, for example, some MTs may prefer to be muted for some periods of time if their reward is achieving a higher throughput once they are allowed to transmit. In Fig. 9, we study the impact of p_{max} for a given maximum interference constraint i_{0}. We observe that p_{max} plays a critical role as well and highly affects the average BR. This figure confirms, once again, that both p_{max} and i_{0} constraints need to be appropriately optimized in order for IAM to outperform IAFPC.
In Fig. 10, we illustrate the potential of IAM of reducing the variance of the interference compared with IUM, while still guaranteeing the same average BR. As discussed in the previous sections, this is beneficial for implementing AMC schemes. The figure shows a fourorder magnitude reduction of the variance of the interference for the considered setup of parameters.
Impact of the association weights: on ULDL decoupling
As shown in [2] and [5], optimizing the performance of HCNs for DL transmission does not necessarily results in optimizing their performance in the UL. Based on the GCA criterion, this implies that different cell association weights (i.e., a different ratio t^{(1)}/t^{(2)} for twotier HCNs) may be needed in the DL and in the UL. However, this approach, which is referred to as ULDL decoupling, introduces additional implementation challenges, which require the modification of the existing network architecture and control plane.
In this section, motivated by these considerations, we analyze and compare IAM, IAFPC, and IUFPC schemes as a function of t^{(1)}/t^{(2)}. The setup t^{(1)}/t^{(2)} = 0 dB corresponds to the SPLA criterion. Some numerical illustrations are provided in Figs. 11 and 12, where the probability that the typical MT is active and the average BR are shown, respectively.
In Fig. 12, in particular, we compare the average BR of IUFPC and IAM schemes. The figure highlights important differences between these two interference management schemes for improving the performance of the UL of HCNs. First of all, we note that the average BR of the IUFPC scheme decreases as the ratio t^{(1)}/t^{(2)} increases. More specifically, the best average BR is obtained if the SPLA criterion is used, which is in agreement with previously published papers [16]. This originates from the fact that the larger t^{(1)}/t^{(2)} is, the more MTs are associated with more distance BSs, which, due to the use of power control, results in increasing the interference in the UL. The performance trend is, on the other hand, different if the IAM scheme is used. In this case, there are several values of i_{0} that provide a better average BR compared with IUFPC. In addition, the average BR increases as t^{(1)}/t^{(2)} increases, since the excess interference that is generated under the IUFPC scheme is now kept under control by imposing the maximum interference constraint i_{0}. As observed in previous figures, Fig. 11 confirms that this gain is obtained since more MTs are turned off.
Figures 11 and 12 confirm the findings in Remark 3 and, in particular, the existence of an operating regime where the performance of IAM is independent of the association weights. Let us consider, for example, the setup for i_{0} = − 60 dBm. In this case, i_{0}>p_{0} and hence, according to Remark 3, the system is interferenceunaware if t^{(1)}/t^{(2)}∈[− 10,+ 10] dB. Figure 12, more specifically, confirms that IAM is interferenceunaware since it provides the same average BR as IUFPC for t^{(1)}/t^{(2)}∈[− 10,+ 10] dB^{Footnote 6}. Similar conclusions can be drawn for other values of i_{0}, where different operating regimes can be identified as predicted in Remark 3. If i_{0} = − 90 dBm, in particular, then i_{0}<p_{0} and the system is independent of the cell association criterion for t^{(1)}/t^{(2)}∈[− 20,+ 20], which is confirmed in Figs. 11 and 12. It is worth mentioning that the values of t^{(1)}/t^{(2)} for which the considered system model is cell association independent are usually adopted in practical engineering applications. In particular, the authors of [2, 19] have shown that the optimal cell association ratio that optimizes the DL is usually less than 20 dB. This is in agreement and compatible with the findings in Figs. 11 and 12.
In view of the numerical results and theoretical insights derived in this work, it is possible to state the following arguments in favor of IAM:

1
Taking into account the periods where the typical MT is active and those where it is muted, the average BR is increased with IAM compared to IAFPC and IUFPC.

2
Thanks to mobility and shadowing, MTs are only muted for a given period of time.

3
Since the muted MTs do not transmit, their average transmitted power is reduced compared to IAFPC and IUFPC. This has been studied with Fig. 1.

4
With IAM, there is a regime where the UL performance is independent of cell association, which eases the joint design of UL and DL transmissions as it have been discussed above.

5
The IAM scheme can be further enhanced as discussed in Section 7. Some numerical illustrations are provided in the next section.
Hybrid scheme—complementary cumulative distribution function of the SINR
In this section, we analyze the coverage probability (CCDF of the SINR) of the hybrid scheme introduced in Section 7.
The results are illustrated in Figs. 13, 14, and 15 for i_{0} = − 70, i_{0} = − 90, and i_{0} = − 120, respectively, and by assuming ε = 1 and p_{max}→∞.
In these figures, we observe a good agreement between analytical frameworks and Monte Carlo simulations.
In Fig. 13, it can be observed that the coverage probability of the hybrid scheme increases as t_{1} increases. In Fig. 14, it can be observed that the coverage probability of the hybrid scheme decreases first and then increases as t_{1} increases. In Fig. 15, it can be observed that coverage probability of the hybrid scheme decreases as t_{1} decreases. As for high i_{0}, these trends are obtained because the coverage probability in time slot 1 is better than in time slot 2 for high i_{0} since more MTs are active. Then, increasing t_{1} improves the coverage probability. As for low i_{0}, the trend is opposite for similar reasons. The comparison of these figures also shows that the coverage probability of the hybrid scheme decreases as i_{0} increases for small value of t_{1} because time slot 2 dominates the performance. On the contrary, the coverage probability of the hybrid scheme increases as i_{0} increases for large value of t_{1}.
Hybrid scheme—spectral efficiency and binary rate
In this section, the average SE and average BR are analyzed.
In Fig. 16, the average SE of the hybrid system is reported. By comparing the average SE of the typical MTs, we note a different performance trend as a function of i_{0} for different values of t_{1}. If t_{1} = 1, which is the IAM scheme, the average SE increases as i_{0} increases as discussed in the previous sections. If t_{1} = 0, only the muted MTs transmit signals and the average SE decreases as i_{0} increases. If, e.g., t_{1} = 0.5, the average SE increases first and then decreases as i_{0} increases.
This result is reasonable because the lower i_{0} is the fewer MTs transmit in time slot 1 and more MTs transmit in time slot 2, which on average, contributes to reduce the SE if t_{1} = 1 and to increase the SE if t_{1} = 0. As for the hybrid scheme, the SE is somehow in between.
The average BR for different values of t_{1} is illustrated in Fig. 17. We note that the average BRs are quite similar if t_{1} = 0, while the gap increases if t_{1} = 1. We observe a quite large gap as a function of t_{1}. This highlights that t_{1} should be carefully allocated in order to fulfill system fairness and interference mitigation. The proposed mathematical frameworks can be used to optimize these competing performance metrics.