Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Stochastic geometry modeling and energy efficiency analysis of millimeter wave cellular networks

Abstract

Current wireless networks face unprecedented challenges because of the exponentially increasing demand for mobile data and the rapid growth in infrastructure and power consumption. This study investigates the optimal energy efficiency of millimeter wave (mmWave) cellular networks, given that these networks are some of the most promising 5G-enabling technologies. Based on the stochastic geometry, a mathematical framework of coverage probability is proposed and the optimal energy efficiency is obtained with coverage performance constraints. Numerical results show that increasing the base station density damages coverage performance exceeding the threshold. This work demonstrates an essential understanding of the deployment and dynamic control of energy-efficient mmWave cellular networks.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Akdeniz, M., Liu, Y., Samimi, M., Sun, S., Rangan, S., Rappaport, T., et al. (2014). Millimeter wave channel modeling and cellular capacity evaluation. IEEE Journal on Selected Areas in Communications, 32(6), 1164–1179.

  2. 2.

    Akoum, S., El Ayach, O., Heath, R., et al. (2012). Coverage and capacity in mmWave cellular systems. Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (pp.688–692).

  3. 3.

    Andrews, J., Baccelli, F., Ganti, R., et al. (2011). A tractable approach to coverage and rate in cellular networks. IEEE Transactions on Communications, 59(11), 3122–3134.

  4. 4.

    Bai, T., Desai, V., Heath, R., et al. (2014). Millimeter wave cellular channel models for system evaluation. International Conference on Computing, Networking and Communications (pp.178–182).

  5. 5.

    Bai, T., Heath, R., et al. (2013). Coverage analysis for millimeter wave cellular networks with blockage effects. IEEE Global Conference on Signal and Information Processing (pp.727–730).

  6. 6.

    Bai, T., Heath, R., et al. (2014). Coverage and rate analysis for millimeter wave cellular networks. IEEE Transactions on Wireless Communications, 14(2), 1100–1114.

  7. 7.

    Bai, T., Vaze, R., Heath, R., et al. (2012). Using random shape theory to model blockage in random cellular networks. International Conference on Signal Processing and Communications (pp.1–5).

  8. 8.

    Bai, T., Vaze, R., Heath, R., et al. (2014). Analysis of blockage effects on urban cellular networks. IEEE Transactions on Wireless Communications, 13(9), 5070–5083.

  9. 9.

    Baykas, T., Sum, C. S., Lan, Z., Wang, J., Rahman, M., Harada, H., et al. (2011). IEEE 802.15.3c: The first IEEE wireless standard for data rates over 1 Gb/s. IEEE Communications Magazine, 49(7), 114–121.

  10. 10.

    Blaszczyszyn, B., Karray, M., et al. (2013). Quality of service in wireless cellular networks subject to log-normal shadowing. IEEE Transactions on Communications, 61(2), 781–791.

  11. 11.

    Blaszczyszyn, B., Karray, M., Keeler, H., et al. (2013). Using poisson processes to model lattice cellular networks. Proceedings IEEE INFOCOM (pp.773–781).

  12. 12.

    Cao, D., Zhou, S., Niu, Z., et al. (2012). Optimal base station density for energy-efficient heterogeneous cellular networks. IEEE International Conference on Communications (pp. 4379–4383).

  13. 13.

    Cao, D., Zhou, S., Niu, Z., et al. (2013). Optimal combination of base station densities for energy-efficient two-tier heterogeneous cellular networks. IEEE Transactions on Wireless Communications, 12(9), 4350–4362.

  14. 14.

    Dhillon, H., Ganti, R., Baccelli, F., Andrews, J., et al. (2012). Modeling and analysis of k-tier downlink heterogeneous cellular networks. IEEE Journal on Selected Areas in Communications, 30(3), 550–560.

  15. 15.

    Di Renzo, M., Guan, P., et al. (2014). A mathematical framework to the computation of the error probability of downlink MIMO cellular networks by using stochastic geometry. IEEE Transactions on Communications, 62(8), 2860–2879.

  16. 16.

    Di Renzo, M., Guan, P., et al. (2014). Stochastic geometry modeling of coverage and rate of cellular networks using the Gil-Pelaez inversion theorem. IEEE Communications Letters, 18(9), 1575–1578.

  17. 17.

    Di Renzo, M., Guidotti, A., Corazza, G., et al. (2013). Average rate of downlink heterogeneous cellular networks over generalized fading channels: A stochastic geometry approach. IEEE Transactions on Communications, 61(7), 3050–3071.

  18. 18.

    Di Renzo, M., Lu, W., et al. (2014). The equivalent-in-distribution (EiD)-based approach: On the analysis of cellular networks using stochastic geometry. IEEE Communications Letters, 18(5), 761–764.

  19. 19.

    Fettweis, G., et al. (2008). ICT energy consumption trends and challenges. The 11th International Symposium on Wireless Personal Multimedia Communications.

  20. 20.

    Gao, Y., Wang, G., Jafar, S., et al. (2015). Topological interference management for hexagonal cellular networks. IEEE Transactions on Wireless Communications, 14(5), 2368–2376.

  21. 21.

    Ghosh, A., Thomas, T., Cudak, M., Ratasuk, R., Moorut, P., Vook, F., et al. (2014). Millimeter-wave enhanced local area systems: A high-data-rate approach for future wireless networks. IEEE Journal on Selected Areas in Communications, 32(6), 1152–1163.

  22. 22.

    Ghosh, S., Sinha, B., Das, N., et al. (2013). Channel assignment using genetic algorithm based on geometric symmetry. IEEE Transactions on Vehicular Technology, 52(4), 860–875.

  23. 23.

    Gilhousen, K., Jacobs, I., Padovani, R., Viterbi, A. J., & Weaver, L. A. (1991). On the capacity of a cellular CDMA system. IEEE Transactions on Vehicular Technology, 40(2), 303–312.

  24. 24.

    Heath, R., Kountouris, M., Bai, T., et al. (2013). Modeling heterogeneous network interference using poisson point processes. IEEE Transactions on Signal Processing, 61(16), 4114–4126.

  25. 25.

    Lee, D., Zhou, S., Zhong, X., Niu, Z., Zhou, X., Zhang, H., et al. (2014). Spatial modeling of the traffic density in cellular networks. IEEE Wireless Communications, 21(1), 80–88.

  26. 26.

    Peng, J., Hong, P., Xue, K., et al. (2015). Energy-aware cellular deployment strategy under coverage performance constraints. IEEE Transactions on Wireless Communications, 14(1), 69–80.

  27. 27.

    Rappaport, T., Gutierrez, F., Ben-Dor, E., Murdock, J., Qiao, Y., Tamir, J., et al. (2013). Broadband millimeter-wave propagation measurements and models using adaptive-beam antennas for outdoor urban cellular communications. IEEE Transactions on Antennas and Propagation, 61(4), 1850–1859.

  28. 28.

    Rappaport, T., Sun, S., Mayzus, R., Zhao, H., Azar, Y., Wang, K., et al. (2013). Millimeter wave mobile communications for 5g cellular: It will work!. IEEE Access, 1, 335–349.

  29. 29.

    Roh, W., Seol, J. Y., Park, J., Lee, B., Lee, J., Kim, Y., et al. (2014). Millimeter-wave beamforming as an enabling technology for 5g cellular communications: Theoretical feasibility and prototype results. IEEE Communications Magazine, 52(2), 106–113.

  30. 30.

    Singh, S., Kulkarni, M. N., Ghosh, A., Andrews, J. G., et al. (2014). Tractable model for rate in self-backhauled millimeter wave cellular networks. IEEE Journal on Selected Areas in Communications, 33(10), 2196–2211.

  31. 31.

    Soh, Y. S., Quek, T., Kountouris, M., Shin, H., et al. (2013). Energy efficient heterogeneous cellular networks. IEEE Journal on Selected Areas in Communications, 31(5), 840–850.

  32. 32.

    Stoyan, D. (2013). Stochastic geometry and its applications (3rd ed. ). The Atrium, Southern Gate, Chichester, West Sussex: Wiley series in probability and statistics, Wiley.

  33. 33.

    Tabassum, H., Siddique, U., Hossain, E., Hossain, M., et al. (2014). Downlink performance of cellular systems with base station sleeping, user association, and scheduling. IEEE Transactions on Wireless Communications, 13(10), 5752–5767.

  34. 34.

    Tsilimantos, D., Gorce, J.M., Altman, E., et al. (2013). Spectral and energy efficiency trade-off with joint power-bandwidth allocation in OFDMA networks. http://arxiv.org/abs/1311.7302.

  35. 35.

    Tsilimantos, D., Gorce, J.M., Altman, E., et al. (2013). Stochastic analysis of energy savings with sleep mode in OFDMA wireless networks. Proceedings IEEE INFOCOM (pp.1097–1105).

Download references

Author information

Correspondence to Song Cen.

Appendix

Appendix

Derivation of laplace transform of interference

The derivation process is shown in as follows:

$$\begin{aligned} {\mathcal {L}}_{I_L}(s) \mathop {=}\limits ^{\triangle }&{\mathbb {E}}\left[ {\mathrm{e}}^{-s(I_L+I_N) }\right] \\ \mathop {=}\limits ^{(a)}&{\mathbb {E}}\left[ {\mathrm{e}}^{-sI_L}\right] {\mathbb {E}}\left[ {\mathrm{e}}^{-sI_N}\right] \\ \mathop {=}\limits ^{(b)}&{\mathbb {E}}\left[ exp\left( -s\sum _{i\in \phi _{b_L} \backslash b_u} P_t g_i D_i \ell _L(r_i) \right) \right] \\&{\mathbb {E}}\left[ exp\left( -s\sum _{i\in \phi _{b_N} \backslash b_u} P_t g_i D_i \ell _N(r_i) \right) \right] \\ \mathop {=}\limits ^{(c)}&{\mathbb {E}}\left[ \prod _{i\in \phi _{b_L} \backslash b_u}{\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i D_i \ell _L(r_i)} \right] \right] \\&{\mathbb {E}}\left[ \prod _{i\in \phi _{b_N} \backslash b_u}{\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i D_i \ell _N(r_i)} \right] \right] \\ \mathop {=}\limits ^{(d)}&exp\left\{ \left( - \sum _{k=1}^{4}b_k \int _{r}^{\infty } \left( 1 - {\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i a_k \ell _L(t)} \right] \right) \right. \right. \\&tdt -\sum _{k=1}^{4}b_k \int _{\psi _L(r)}^{\infty } \left( 1 - {\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i a_k \ell _N(t)} \right] \right) \\&\left. \left. tdt \right) 2\pi \lambda _b \right\} \\ \mathop {=}\limits ^{(e)}&exp\left\{ -2\pi \lambda _b \sum _{k=1}^{4}b_k \left( \int _{r}^{\infty } \frac{t}{1+\frac{\tau }{sP_t \ell _L(r)a_k}}dt \right. \right. \\&\left. \left. + \int _{\psi _L(r)}^{\infty } \frac{t}{1+\frac{\tau }{sP_t \ell _N(t)a_k}}dt \right) \right\} \\ =&exp\left\{ -\pi \lambda _b \sum _{k=1}^{4}b_k \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \end{aligned}$$

where (a) is due to the independence of \(\varPhi _{b_L}\) and \(\varPhi _{b_N}\), (b) (c) follows the independence within the PPP of \(\varPhi _{b_L}\) or \(\varPhi _{b_N}\). (d) follows the Laplace transform of PPP [32], and (e) follows the integral transform.

Derivation of conditional coverage probability

For a LOS link with the radius of r, the conditional coverage probability \(P_{c,L}(\lambda _b | r)\) can be derived from Eq. (10) by adding the condition of radius of r, it is given as:

$$\begin{aligned} P_{c,L}(\lambda _b | r) =&{\mathrm{e}}^{-sN_u}{\mathbb {E}}\left[ {\mathrm{e}}^{-sI}\right] \\ \mathop {=}\limits ^{(a)}&exp\left\{ -\frac{\tau \rho N_u}{P_t Q_r Q_t}(\mu _L r)^{\alpha _L} -\pi \lambda _b \sum _{k=1}^{4}b_k \right. \\&\left. \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \\ =&exp\left\{ -\frac{\tau \rho N}{P_t Q_r Q_t} \frac{\lambda _b}{\lambda _u}(\mu _L r)^{\alpha _L} -\pi \lambda _b \sum _{k=1}^{4}b_k \right. \\&\left. \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \end{aligned}$$

where (a) follows the Laplace transform of PPP [32].

Derivation of the SINR coverage probability

When considering blockage effects, the SINR coverage probability should be the weighted summation of LOS and NLOS circumstances, which is given as:

$$\begin{aligned} P_c(\lambda _b) \mathop {=}\limits ^{(a)}&A_L P_{c,L}(\lambda _b) + A_N P_{c, N}(\lambda _b) \nonumber \\ \mathop {=}\limits ^{(b)}&A_L \int _0^\infty \hat{f_L} P_{c, L}(\lambda _b|r) dr + A_N \int _0^\infty \hat{f_N}P_{c, N}(\lambda _b|r)dr \end{aligned}$$

According to the Ref. [6], these equations can be introduced as follows:

$$\begin{aligned} A_L= & {} \int _{0}^{\infty }{\mathrm{e}}^{-2\pi \lambda \int _{0}^{\psi _{L}(r)}t(1-p(t))dt }f_L(r)dr\\ {\hat{f}}_L(r)= & {} \frac{B_L f_L(r)}{A_L} {\mathrm{e}}^{-2\pi \lambda _b \int _{0}^{\psi _L(r)} (1-p(t))tdt}\\ {\hat{f}}_N(r)= & {} \frac{B_N f_L(r)}{A_N} {\mathrm{e}}^{-2\pi \lambda _b \int _{0}^{\psi _N(r)} p(t)tdt} \end{aligned}$$

where \(A_L\) is the probability that a user is associated with an LOS BS. The probability density functions of the distance to a serving BS are denoted as \(\hat{f_L}\) and \(\hat{f_N}\) at the conditions that the typical user is associated with a LOS BS and a NOS BS respectively. (a) follows the weighted summation of LOS and NLOS circumstances, \(P_{c,L}(\lambda _b)\) and \(P_{c, N}(\lambda _b)\) are the SINR probabilities under LOS and NLOS circumstances, respectively. The probability that a user is associated with an NLOS BS is denoted as \(A_N\). Then \(A_N=1-A_L\). (b) follows the expansion with conditional probability density function.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cen, S., Zhang, X., Lei, M. et al. Stochastic geometry modeling and energy efficiency analysis of millimeter wave cellular networks. Wireless Netw 24, 2565–2578 (2018). https://doi.org/10.1007/s11276-016-1441-6

Download citation

Keywords

  • Millimeter wave
  • Energy efficiency
  • Stochastic geometry
  • Dynamic control