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Where Multiple Connectivity Brings Offloading Performance Boost? An Analytical Study

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Abstract

Recently multiple radio access technology (M-RAT) has been suggested as a solution to provide the currently congested wireless consumers with higher throughput and therefore improved performance mainly through offloading cellular traffic to alternative networks such as WiFi and femtocell. In this paper, we investigate analytically where M-RAT can accomplish this task. Specifically, a throughput comparison will be made between those users, who can benefit from M-RAT capability and those who cannot. More specifically, we consider three important scenarios: (1) simultaneous connection to WiFi and macrocell, (2) simultaneous connections to two macrocells, and (3) simultaneous connection to WiFi and femtocell. The main results of the analysis reveal that multiple connection capability improves the performance by increasing multiplexing gain, mostly observable when cells with access point of comparable size overlap completely.

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Notes

  1. We assume that media access mechanism is the same for both macrocell and femtocell.

  2. By saturation condition, we mean that users always have traffic for transmission; thus, contention for channel access exists all the times.

  3. Please note that superscript MC and SC denotes the scenario under consideration, thus, in both MC and SC scenarios, cellular-only users use the only available connection which is cellular, while WiFi users have two connections in MC scenario (WiFi and Cellular) and one connection in SC scenario (only WiFi).

  4. We differ between operators and cellular APs, since cellular APs might belong to different operators while covering the same area at the same time. For one operator, the whole bandwidth available to the operator might be shared between all the APs with spatial frequency reuse factor 1, as it is in LTE [26].

  5. In this case, we assume that a cooperation mechanism exists between APs of different operators such that the two APs belonging to two different operators and existing in the same place can use the whole bandwidth together.

  6. Number of users in a WiFi AP is usually in the order of a hundredth of that of a cellular AP due to the limited coverage area, while their bandwidth resources are of comparable sizes.

  7. This can be speculated from Fig. 4 as the distance between cell centers reduces.

  8. Here, we ignore the result for \(P_{A}\) less than 0.5, as it is not relevant for comparison of MC vs SC.

References

  1. White Paper, Cisco Visual Networking Index: Global Mobile Data Traffic Forecast Update, 2016–2021. http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html.

  2. Aijaz, A., Aghvami, H., & Amani, M. (2013). A survey on mobile data offloading: technical and business perspectives. IEEE Wireless Communications, 20(April), 104–112.

    Article  Google Scholar 

  3. Amani, M., Mahmoodi, T., Tatipamula, M., & Aghvami, H. (2014). SDN-based data offloading for 5G mobile networks. ZTE Communications, 2, 009.

    Google Scholar 

  4. Lee, K., Lee, J., Yi, Y., Rhee, I., & Chong, S. (2013). Mobile data offloading: How much can WiFi deliver? IEEE/ACM Transactions on Networking, 21(2), 536–550.

    Article  Google Scholar 

  5. He, Y., Chen, M., Ge, B., & Guizani, M. (2016). On WiFi offloading in heterogeneous networks: Various incentives and trade-off strategies. IEEE Communications Surveys and Tutorials, 18(4), 2345–2385.

    Google Scholar 

  6. Mehmeti, F., & Spyropoulos, T. (2016). Performance analysis of mobile data offloading in heterogeneous networks. IEEE Transactions on mobile computing, 16(2), 482–497.

    Article  Google Scholar 

  7. Rebecchi, F., Amorim, M. D., Conan, V., Passarella, A., Bruno, R., & Conti, M. (2015). Data offloading techniques in cellular networks: A survey. IEEE Communications Surveys and Tutorials, 17(2), 580–603.

    Article  Google Scholar 

  8. White paper. (2013). OUTLOOK, visions and research directions for the wireless world: Multi-rat network architecture. www.wwrf.ch/files/wwrf/content/files/publications/outlook/Outlook9.pdf.

  9. Sharafeddine, S., Jahed, K., Abbas, N., Yaacoub, E., & Dawy, Z. (2013). Exploiting multiple wireless interfaces in smartphones for traffic offloading. In Black Sea Conference on Communications and Networking (BlackSeaCom), pp. 142–146, July 2013.

  10. Weihua, W., Yang, Q., Gong, P., & Kwak, K. S. (2017). Energy-efficient traffic splitting for time-varying multi-RAT wireless networks. IEEE Transactions on Vehicular Technology. https://doi.org/10.1109/tvt.2016.2643008.

    Google Scholar 

  11. Galinina, O., Pyattaev, A., Andreevy, S., Dohler, M., & Koucheryavy, Y. (2015). 5G multi-RAT LTE-WiFi ultra-dense small cells: Performance dynamics, architecture, and trends. IEEE Journal on Selected Areas in Communications, 33(6), 1224–1240.

    Article  Google Scholar 

  12. Yu, G., Jiang, Y., Xu, L., & Li, G. Y. (2015). Multi-objective energy-efficient resource allocation for multi-RAT heterogeneous networks. IEEE Journal on Selected Areas in Communications, 33(10), 21182127.

    Article  Google Scholar 

  13. Zhang, H., Chu, X., Guo, W., & Wang, S. (2015). Coexistence of Wi-Fi and heterogeneous small cell networks sharing unlicensed spectrum. IEEE Communications Magazine, 53(3), 158–164.

    Article  Google Scholar 

  14. Ibrahim, A. S., Darwish, A. M., Elbassiouny, S. O., & Elgebaly, H. (2013). QoS-aware traffic offloading in 4G/WiFi multi-RAT heterogeneous networks: Opnet-based simulations and real prototyping implementation, ICWMC, pp.82–89, July 2013.

  15. Gustafsson, E., & Jonsson, A. (2003). Always best connected. IEEE Wireless. Communication, 10(1), 49–55.

    Article  Google Scholar 

  16. Singh, S., Geraseminko, M., Yeh, S., Himayat, N., & Talwar, S. Proportional fair traffic splitting and aggregation in heterogeneous wireless networks. arxiv.org/abs/1508.05542.

  17. Singh, S., Yeh, S., Himayat, N., & Talwar, S. (2016). Optimal traffic aggregation in multi-RAT heterogeneous wireless networks. In Proceedings of the IEEE International Conference on Communications Workshops (ICC’16), pp. 626–631, Kuala Lumpur, Malaysia, May 2016.

  18. Elkourdi, T.,Chincholi, A., Le T., & Demir, A. (2015). Cross-layer optimization for opportunistic multi-MAC aggregation.In Proceedings IEEE Vehicular Technology (VTC Spring), pp. 1–5, June 2013.

  19. Goyal, S., Le, T., Chincholi, A., Elkourdi, T., Demir, A. (2015) On the packet allocation of multi-band aggregation wireless networks. Submitted. http://arxiv.org/abs/1508.05017.

  20. Seong, H. C., Hong, J. P., & Choi, W. (2016). Optimal access in OFDMA Multi-RAT cellular networks with stochastic geometry: Can a single RAT be better? IEEE Transactions on Wireless Communications, 15(7), 4778–4789.

    Google Scholar 

  21. Krishnamoorthy, K. (2006). Handbook of statistical distributions with applications. LLC: Taylor and Francis Group.

    Book  MATH  Google Scholar 

  22. Bianchi, G. (1998). IEEE 802.11—saturation throughput analysis. IEEE Communications Letters, 2, 318–320.

    Article  Google Scholar 

  23. Bianchi, G. (2000). Performance analysis of the IEEE 802.11 distributed coordinated function. IEEE Journal on selected areas in communications (JSAC), 18(3), 535–547.

    Article  Google Scholar 

  24. P. Kirby (2014), Properties of Functions, lecture notes on Discrete Mathematics. http://www.math.fsu.edu/~pkirby.

  25. Tesler, G. (2013). The Maximum of n random variables, hypothesis testing, long repeats of the same nucleotide, lecture notes on statistical methods in bioinformatics. http://www.math.ucsd.edu/~gptesler/283.

  26. Pauli, Volker, & Seidel, Eiko. (2010). ‘Inter-Cell Interference Coordination for LTE-A’. Munich: Nomor Research GmbH.

    Google Scholar 

  27. Abramowitz, M., & Stegun, I. (1970). Handbook of mathematical functions. New York: Dover.

    MATH  Google Scholar 

  28. Cover, T. M., & Thomas, J. A. (1991). Elements of Information theory. New York: Wiley.

    Book  MATH  Google Scholar 

  29. Richards, D. (2002). Advanced mathematical methods with maple. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  30. Maple release 18. MapleSoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

  31. Andrews, J. G., Singh, S., Ye, Q., Lin, X., & Dhillon, H. S. (2014). An overview of load balancing in HetNets: Old myths and open problems. IEEE Wireless Communications magazine, 21(2), 18–25.

    Article  Google Scholar 

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Correspondence to Mehri Mehrjoo.

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Appendices

Appendix A

In this appendix, the asymptotic behavior of \(E\left[ {R_{W} } \right]\) and \(E\left[ {R_{C} } \right]\) will be derived as the number of users tends to its limit point, i.e., \(\left( {N_{U} ,N_{C} } \right) = \left( {1,1} \right)\) and \(\left( {N_{U} ,N_{C} } \right) \to \left( {\infty ,\infty } \right)\), as well as for the case when channel quality \(\left( {\left( {\lambda_{W} ,\lambda_{C} } \right)} \right)\) tends to zero and infinity, i.e., \(\left( {\lambda_{W} ,\lambda_{C} } \right) \to \left( {\infty ,\infty } \right)\) and. \(\left( {\lambda_{W} ,\lambda_{C} } \right) \to \left( {0,0} \right)\)

If we have \(N_{U} = 1\) and \(\lambda_{W} \to \infty\), then, \(E\left[ {R_{W} } \right]\) becomes:

$$\begin{aligned} {\mathop{\mathop{\lim }\limits_{\lambda_{W} \to \infty }}\limits_{N_{U} = 1}} E\left[ {R_{W} } \right] &={\mathop{\mathop {\lim }\limits_{\lambda_{W} \to \infty }}\limits_{N_{U} = 1}} \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)\lambda_{W} e^{{ - \lambda_{W} x}} {\text{d}}x \\ &{\mathop = \limits^{1}} \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop {\lim }\limits_{{\lambda_{W} \to \infty }} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + \frac{u}{{\lambda_{W} }}} \right)e^{ - u} {\text{d}}u \\ &\le \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop {\lim }\limits_{{\lambda_{W} \to \infty }} \mathop \int \limits_{0}^{\infty } \frac{u}{{\lambda_{W} }}e^{ - u} {\text{d}}x \\ &= \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop {\lim }\limits_{{\lambda_{W} \to \infty }} \frac{1}{{\lambda_{W} }} = 0. \\ \end{aligned}$$
(A-1)

where equality (1) comes from the variable change \(\lambda_{W} x = u\), and inequality is the result of logarithm inequality \({ \ln }(1 + {\text{u}}) \le {\text{u}}\) [27]. From (A-1), we can also conclude that when \(N_{U} \to \infty\) and \(\lambda_{W} \to \infty\), \(E\left[ {R_{W} } \right] \to 0\), since in this case, the average rate will be divided by the number of users \(N_{U}\). For the case of \(N_{U} = 1\) and \(\lambda_{W} \to 0\), we show that a lower bound of \(E\left[ {R_{W} } \right]\) will tends to infinity, thus \(E\left[ {R_{W} } \right]\) must go to infinity, too. We have:

$$\begin{aligned} \mathop {\lim }\limits_{{\lambda_{W} \to 0}} E\left[ {R_{W} } \right] &= \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop {\lim }\limits_{{\lambda_{W} \to 0}} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)\lambda_{W} e^{{ - \lambda_{W} x}} {\text{d}}x \\ &> \frac{{A_{C} B_{W} }}{{A_{W} }}\mathop {\lim }\limits_{{\lambda_{W} \to 0}} \mathop \int \limits_{0}^{1} \ln \left( {1 + \frac{u}{{\lambda_{W} }}} \right)e^{ - u} {\text{d}}u \\& > \frac{{A_{C} B_{W} }}{{A_{W} }}e^{ - 1} \mathop {\lim }\limits_{{\lambda_{W} \to 0}} \mathop \int \limits_{0}^{1} \ln \left( {1 + \frac{u}{{\lambda_{W} }}} \right){\text{d}}x \\ &= \frac{{A_{C} B_{W} }}{{A_{W} }}e^{ - 1} \mathop {\lim }\limits_{{\lambda_{W} \to 0}} \left( {\left( {\lambda_{W} + 1} \right)\ln \left( {1 + \frac{1}{{\lambda_{W} }}} \right) - 1} \right) \to \infty . \\ \end{aligned}$$
(A-2)

where here, by infinity of the rate, we mean the upper limit of the rate determined by such decisive parameters as power. When \(N_{U} \to \infty\) and \(\lambda_{W} \to 0\), we have:

$$\begin{aligned} {\mathop{\mathop{\lim}\limits_{\lambda_{W} \to 0}}\limits_{N_{U} \to \infty }} E\left[ {R_{W} } \right] &= {\mathop{\mathop{\lim }\limits_{\lambda_{W} \to 0}}\limits_{N_{U} \to \infty }} \frac{{A_{C} B_{W} }}{{A_{W} N_{U} }}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)\lambda_{W} e^{{ - \lambda_{W} x}} {\text{d}}x \\ & \mathop \le \limits^{1} \frac{{A_{C} B_{W} }}{{A_{W} }}{\mathop{\mathop{\lim }\limits_{\lambda_{W} \to 0}}\limits_{N_{U} \to \infty }} \frac{1}{{N_{U} }}\ln \left( {\mathop \int \limits_{0}^{\infty } \left( {1 + \frac{u}{{\lambda_{W} }}} \right)e^{ - u} {\text{d}}u} \right) \\ &= \frac{{A_{C} B_{W} }}{{A_{W} }}{\mathop{\mathop{\lim }\limits_{\lambda_{W} \to 0}}\limits_{N_{U} \to \infty }} \frac{1}{{N_{U} }}\ln \left( {1 + \frac{1}{{\lambda_{W} }}} \right) \\ & \approx \frac{{A_{C} B_{W} }}{{A_{W} }}{\mathop{\mathop{\lim }\limits_{\lambda_{W} \to 0}}\limits_{N_{U} \to \infty }} \frac{1}{{N_{U} }}\ln \left( {\frac{1}{{\lambda_{W} }}} \right) \to 0. \\ \end{aligned}$$
(A-3)

where the inequality comes from the combination of concavity of the logarithm function and Jensen’s inequality [28]. Thus we have \(E\left[ {R_{W} } \right] \to 0\) as \(N_{U} \to \infty\) and \(\lambda_{W} \to 0\).

For the case of average cellular data rate \(E\left[ {R_{C} } \right]\), when \(N_{C} = 1\), similar to (A-1) and (A-2), the average rate tends to zero and infinity, respectively, as \(\lambda_{W} \to \infty\) and \(\lambda_{W} \to 0\). On the other hand, for \(N_{C} \to \infty\), we must consider two cases, one for \(\lambda_{C} \to \infty\) (that is, when number of users is high and channel qualities are worst) and the other for \(\lambda_{C} \to 0\) (that is, when number of users is high and channel qualities are best). In order to facilitate the analysis, we assume that the rate at which \(N_{C} \to \infty\) is the same as the rate at which \(\lambda_{C} \to \infty\). Using the notation \(\alpha = N_{C} = \lambda_{C}\), thus we have:

$$\begin{aligned} {\mathop{\mathop{\lim}\limits_{\lambda_{C} \to \infty}}\limits_{N_{C} \to \infty }} E\left[ {R_{C} } \right] &= {\mathop{\mathop{\lim}\limits_{\lambda \to \infty}}\limits_{N_{C} \to \infty }} \lambda_{C} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{{ - \lambda_{C} x}} \left( {1 - e^{{ - \lambda_{C} x}} } \right)^{{N_{C} - 1}} {\text{d}}x \\ &{\mathop = \limits^{1}} B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \mathop \int \limits_{0}^{\infty } \ln \left( {1 + \frac{u}{\alpha }} \right)e^{ - u} \left( {1 - e^{ - u} } \right)^{\alpha - 1} {\text{d}}x \\& \mathop \le \limits^{2} B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \frac{1}{\alpha }\ln \left( {\mathop \int \limits_{0}^{\infty } \alpha \left( {1 + \frac{u}{\alpha }} \right)e^{ - u} \left( {1 - e^{ - u} } \right)^{\alpha - 1} {\text{d}}x} \right) \\ &{\mathop = \limits^{3}} B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \frac{1}{\alpha }\ln \left( {1 + \underbrace {{\mathop \int \limits_{0}^{\infty } ue^{ - u} \left( {1 - e^{ - u} } \right)^{\alpha - 1} {\text{d}}x}}_{f\left( \alpha \right)}} \right) \\ &\mathop \approx \limits^{4} B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \frac{1}{\alpha }\ln \left( {1 + \frac{\ln \left( \alpha \right) + \gamma }{\alpha } + O\left( {\frac{1}{{\alpha^{2} }}} \right)} \right) \\ & \approx B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \frac{1}{\alpha }\ln \left( {1 + \frac{\ln \left( \alpha \right) + \gamma }{\alpha } + O\left( {\frac{1}{{\alpha^{2} }}} \right)} \right) \approx B_{C} \mathop {\lim }\limits_{\alpha \to \infty } \frac{1}{\alpha }\ln \left( 1 \right) \approx 0. \\ \end{aligned}$$
(A-4)

where the equality (1) comes from the variable change \(\lambda_{C} x = u\), the inequality (2) comes from the concavity of logarithm function and Jensen’s inequality, (3) comes from the fact that the area under a probability density function is unity and the approximation (4) follows from the asymptotic expansion of the function \(f\left( \alpha \right)\) for \(\alpha \to \infty\) [29, 30]. In (A-4), \(\gamma\) is the Euler–Mascheroni constant [27].

When we have \(N_{C} \to \infty\) and \(\lambda_{C} \to 0\), using the notation \(\beta = N_{C} = 1/\lambda_{C}\), we have:

$$\begin{aligned} {\mathop{\mathop{\lim}\limits_{\lambda_{C} \to 0}}\limits_{N_{C} \to \infty }} E\left[ {R_{C} } \right] &= {\mathop{\mathop{\lim}\limits_{\lambda \to \infty}}\limits_{N_{C} \to \infty }} \lambda_{C} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{{ - \lambda_{C} x}} \left( {1 - e^{{ - \lambda_{C} x}} } \right)^{{N_{C} - 1}} {\text{d}}x \\ &{\mathop = \limits^{1}} B_{C} \mathop {\lim }\limits_{\beta \to \infty } \mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta u} \right)e^{ - u} \left( {1 - e^{ - u} } \right)^{\beta - 1} {\text{d}}x \\ & \mathop \le \limits^{2} B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {\mathop \int \limits_{0}^{\infty } \beta \left( {1 + \beta u} \right)e^{ - u} \left( {1 - e^{ - u} } \right)^{\beta - 1} {\text{d}}x} \right) \\ &= B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {1 + \beta^{2} \underbrace {{\mathop \int \limits_{0}^{\infty } ue^{ - u} \left( {1 - e^{ - u} } \right)^{\beta - 1} {\text{d}}x}}_{f\left( \beta \right)}} \right) \\ & \mathop \approx \limits^{3} B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {1 + \beta^{2} \left( {\frac{\ln \left( \beta \right) + \gamma }{\beta } + \frac{1}{{2\beta^{2} }} + O\left( {\frac{1}{{\alpha^{2} }}} \right)} \right)} \right) \\ & \approx B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {\beta \ln \left( \beta \right)} \right)\mathop \le \limits^{4} B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {\beta^{2} } \right) \approx B_{C} \mathop {\lim }\limits_{\beta \to \infty } \frac{2}{\beta }\ln \left( \beta \right) \approx 0. \\ \end{aligned}$$
(A-5)

where the inequality (4) is a result of \({ \ln }\left( x \right) < { \ln }\left( {1 + x} \right) \le x\). Taking into account the asymptotic results of average user rates for \(\left( {N_{U} ,N_{C} } \right) \to \left( {\infty ,\infty } \right)\), we conclude that the decisive factor for highly crowded APs is number of users, not channel quality.

Appendix B

For the limiting behavior of \(T_{W}\), from Eqs. (6 and 7) and using limiting behavior in appendix A, and further assuming \(\lambda_{W} = \lambda_{C} = \lambda\) and \(B_{W} = B_{C}\), we have:

$${\mathop{\mathop {\lim }\limits_{ \lambda_{C} \to \infty }}\limits_ {N_{U} = 1} } T_{W} = 1 + \frac{{A_{W} }}{{A_{C} }},$$
(B-1)
$$\begin{aligned} {\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_ {N_{U} \to \infty }} T_{W} &= 1 + {\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_{N_{C} \to \infty }} \frac{{A_{W} N_{U} }}{{A_{C} }}\frac{{\int_{0}^{\infty } {\ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{N_{U} - 1}} {\text{d}}x} }}{{\int_{0}^{\infty } {\ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} dx} }} \\ &= 1 + \frac{{A_{W} }}{{A_{C} }}\mathop {\lim }\limits_{\beta \to \infty } \beta \frac{{\overbrace {{\int_{0}^{\infty } {\ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{\beta - 1} {\text{d}}x} }}^{{(.) < \frac{2\ln \left( \beta \right)}{\beta }}}}}{{\underbrace {{\int_{0}^{\infty } {\ln \left( {1 + \beta x} \right)e^{ - x} dx} }}_{(.) > \ln \left( \beta \right)}}} < 1 + \frac{{2A_{W} }}{{A_{C} }}. \\ \end{aligned}$$
(B-2)

Since in real scenarios we have \(A_{C} \gg A_{W}\), thus:

$${\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_ {N_{U} \to \infty }}T_{W} \approx 1$$
(B-3)

(see Fig. 4). Similarly, for \(T_{C}\) and from Eqs. (8 and 9), we have:

$$\begin{aligned} {\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_ {N_{U} \to \infty }} T_{C} = {\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_ {N_{U} \to \infty }} \frac{{B_{C} \lambda \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - \lambda x} \left( {1 - e^{ - \lambda x} } \right)^{{N_{U} - 1}} {\text{d}}x}}{{B_{C} \lambda \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - \lambda x} \left( {1 - e^{ - \lambda x} } \right)^{{N_{U} \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right) - 1}} {\text{d}}x}} \\ = \mathop {\lim }\limits_{\beta \to \infty } \frac{{\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{\beta - 1} {\text{d}}x}}{{\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right) - 1}} {\text{d}}x}} \\ = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \frac{{\mathop \int \limits_{0}^{M} \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{\beta - 1} {\text{d}}x}}{{\mathop \int \limits_{0}^{M} \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right) - 1}} {\text{d}}x}} \\ = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \frac{{\left. {\frac{1}{\beta }\ln \left( {1 + \beta x} \right)\left( {1 - e^{ - x} } \right)^{\beta } } \right|_{0}^{M} - \overbrace {{\mathop \int \limits_{0}^{M} \frac{{\left( {1 - e^{ - x} } \right)^{\beta } }}{1 + \beta x}{\text{d}}x}}^{{f_{1} }}}}{{\left. {{1 \mathord{\left/ {\vphantom {1 {\left( {\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)} \right)}}} \right. \kern-0pt} {\left( {\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)} \right)}}\ln \left( {1 + \beta x} \right)\left( {1 - e^{ - x} } \right)^{{^{{\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right) - 1}} }} } \right|_{0}^{M} - {1 \mathord{\left/ {\vphantom {1 {\left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)}}} \right. \kern-0pt} {\left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)}}\underbrace {{\mathop \int \limits_{0}^{M} \frac{{\left( {1 - e^{ - x} } \right)^{{\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)}} }}{1 + \beta x}{\text{d}}x}}_{{f_{2} }}}}, \\ \end{aligned}$$
(B-4)

For functions \(f_{1}\) and \(f_{2}\), we have:

$$\begin{aligned} \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } f_{1} = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \mathop \int \limits_{0}^{M} \frac{{\left( {1 - e^{ - x} } \right)^{\beta } }}{1 + \beta x}{\text{d}}x \\ < \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \frac{1}{\beta }\mathop \int \limits_{0}^{M} \frac{{\left( {1 - e^{ - x} } \right)^{\beta } }}{x}{\text{d}}x \\ < \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \frac{1}{\beta }\mathop \int \limits_{0}^{M} \frac{1}{x}{\text{d}}x \\ = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \mathop {\lim }\limits_{\varepsilon \to 0} \frac{1}{\beta }\mathop \int \limits_{\varepsilon }^{M} \frac{1}{x}{\text{d}}x \\ = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \mathop {\lim }\limits_{\varepsilon \to 0} \frac{1}{\beta }\ln \left( {\frac{M}{\varepsilon }} \right), \\ \end{aligned}$$
(B-5)

Now, considering the same rate of tendency to the limiting points for all parameters \(M, \varepsilon\) and \(\beta\), we write:

$$\mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \mathop {\lim }\limits_{\varepsilon \to 0} \frac{1}{\beta }\ln \left( {\frac{M}{\varepsilon }} \right) \approx \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\ln \left( {\beta^{2} } \right) \approx \mathop {\lim }\limits_{\beta \to \infty } \frac{2}{\beta }\ln \left( \beta \right) = 0.$$
(B-6)

In a similar manner, the value of \(f_{2}\) will be zero, as the parameter \(\beta\) tends to infinity. Thus, the limiting behavior of \(T_{C}\) becomes:

$$\begin{aligned} {\mathop{\mathop {\lim }\limits_{\lambda \to 0}}\limits_ {N_{U} \to \infty }}T_{C} = \mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \frac{{\left. {\frac{1}{\beta }\ln \left( {1 + \beta x} \right)\left( {1 - e^{ - x} } \right)^{\beta } } \right|_{0}^{M} }}{{\left. {{1 \mathord{\left/ {\vphantom {1 {\left( {\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)} \right)}}} \right. \kern-0pt} {\left( {\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)} \right)}}\ln \left( {1 + \beta x} \right)\left( {1 - e^{ - x} } \right)^{{^{{\beta \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right) - 1}} }} } \right|_{0}^{M} }} \\ = \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)\mathop {\lim }\limits_{\beta \to \infty } \mathop {\lim }\limits_{M \to \infty } \left( {1 - e^{ - M} } \right)^{{^{{\beta \left( {\frac{{N_{AP} A_{W} }}{{A_{C} }}} \right)}} }} = \left( {1 - \frac{{N_{AP} A_{W} }}{{A_{C} }}} \right). \\ \end{aligned}$$
(B-7)

which is a linear decreasing function of the number of APs, \(N_{AP}\) (see Fig. 4).

Appendix C

In this appendix, in a similar manner as the two previous appendices, we derive the limiting behavior of the gain of MC scenario over SC scenario.

From (18), we have:

$$\begin{aligned} {\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}} T_{MC - SC}^{macro - macro} = \mathop {\lim }\limits_{{\begin{array}{*{20}c} {\begin{array}{*{20}c} {\lambda \to 0} \\ {N_{U} \to \infty } \\ \end{array} } \\ {P_{A} \to 1} \\ \end{array} }} \frac{{B_{C} P_{A} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x}}{{\frac{{B_{C} P_{A} }}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x + \frac{{B_{C} \left( {1 - P_{A} } \right)}}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\left( {1 - P_{A} } \right)N_{U} - 1}} {\text{d}}x}} \\ = {\mathop{\mathop {\lim }\limits_{\beta \to \infty} }\limits_ {P_{A} \to 1}} \frac{{P_{A} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} \beta - 1}} {\text{d}}x}}{{\frac{{P_{A} }}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} \beta - 1}} {\text{d}}x + \underbrace {{\frac{{\left( {1 - P_{A} } \right)}}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\left( {1 - P_{A} } \right)\beta - 1}} {\text{d}}x}}_{H}}}, \\ \end{aligned}$$
(C-1)

Assuming the same rate of tendencies for both \(\beta\) and \(P_{A}\) to the limit points, such that we have \({\mathop{\mathop {\lim }\limits_{\beta \to \infty} }\limits_ {P_{A} \to 1}} \beta \left( {1 - P_{A} } \right) = 1\), or equivalently, \(\mathop {\lim }\limits_{{P_{A} \to 1}} \left( {1 - P_{A} } \right) = \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\), we have:

$$\begin{aligned} {\mathop{\mathop {\lim }\limits_{\beta \to \infty} }\limits_ {P_{A} \to 1}} H = {\mathop{\mathop {\lim }\limits_{\beta \to \infty} }\limits_ {P_{A} \to 1}} \frac{{\left( {1 - P_{A} } \right)}}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\left( {1 - P_{A} } \right)\beta - 1}} {\text{d}}x \\ \le \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{2\beta }\mathop \int \limits_{0}^{\infty } \ln \left( {1 + \beta x} \right)e^{ - x} {\text{d}}x \\ = \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{2\beta }e^{1/\beta } Ei\left( {1,{1 \mathord{\left/ {\vphantom {1 \beta }} \right. \kern-0pt} \beta }} \right){\text{d}}x = \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{2\beta }\left( { - \gamma + \ln \left( \beta \right) + \frac{1 - \gamma + \ln \left( \beta \right)}{\beta } + O\left( {\beta^{ - 2} } \right)} \right) = 0. \\ \end{aligned}$$
(C-2)

where function \(Ei\left( {a,z} \right)\) is the exponential integral defined as \(\mathop \int \limits_{1}^{\infty } \frac{{e^{ - xz} }}{{x^{a} }}dx\) [27]. Thus we have:

$${\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}}T_{MC - SC}^{macro - macro} = {\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}} \frac{{B_{C} P_{A} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x}}{{\frac{{B_{C} P_{A} }}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + {x \mathord{\left/ {\vphantom {x \lambda }} \right. \kern-0pt} \lambda }} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x}} = 2.$$
(C-3)

(See Fig. 6).

Appendix D

The limiting behavior of (28), would be:

$${\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}} T_{MC - SC}^{WiFi - Femto} = {\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}} \frac{{\frac{{B_{A} }}{{N_{U} }}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - x} {\text{d}}x + P_{A} B_{A} \mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x}}{{\frac{{B_{A} }}{{N_{U} }}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - x} {\text{d}}x + \frac{{B_{A} P_{A} }}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{P_{A} N_{U} - 1}} {\text{d}}x + \frac{{B_{A} \left( {1 - P_{A} } \right)}}{2}\mathop \int \limits_{0}^{\infty } \ln \left( {1 + x} \right)e^{ - x} \left( {1 - e^{ - x} } \right)^{{\left( {1 - P_{A} } \right)N_{U} - 1}} {\text{d}}x}},$$
(D-1)

According to (A-3), the first integral term in the numerator and denominator becomes both zero in the limit. On the other hand, from (C-2), the last integral term in the denominator becomes zero, too. Taking these consideration into account, then, the numerator becomes equal to two times of the denominator, thus we have:

$${\mathop{\mathop{\mathop{\lim}\limits_{\lambda \to 0}}\limits_{N_{U} \to \infty}}\limits_{P_{A} \to 1}} T_{MC - SC}^{WiFi - Femto} = 2.$$
(D-2)

(See Fig. 7).

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Hosseini, S.M., Mehrjoo, M. & Amindavar, H. Where Multiple Connectivity Brings Offloading Performance Boost? An Analytical Study. Wireless Pers Commun 106, 1507–1530 (2019). https://doi.org/10.1007/s11277-019-06227-y

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