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Throughput Analysis and Optimization Based on Mobility Analysis and Markov Process for Heterogeneous Wireless Networks

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Abstract

One of the hot issues in heterogeneous wireless networks (HWNs) is radio resource utilization. The objective of this paper is to solve this problem. By considering the diversity of radio access technologies (RATs) in HWNs, the differences between them should be studied and exploited, e.g., coverage radius of each network, service arrival rate of each region, data transmission rate. The paper proposes a novel method for HWNs throughput analysis and optimization on the basis of these differences. Users engaging in calls in overlapping regions need to conduct network selection. To enhance the throughput of HWNs, users in these regions should be reasonably allocated to each network. Hence, the users’ proportion accessing each network is an important factor in the HWNs utilization. The mean total throughput of HWNs can be formulated by the Markov Model, which is determined by the distribution of service arrival rate and the analysis of handoff rate. Users’ mobility, furthermore, is important in network analysis because of its effect upon the handoff rate, which is one of the parameters to decide the throughput of HWNs. The service access proportion should be optimized to maximize the throughput of HWNs. By considering the convexity of the objective function, the subgradient method is employed in the solution of the optimization problem. Meanwhile, quadratic programming is used to reduce the computational complexity. Finally, a throughput optimization algorithm is proposed for HWNs on the basis of the architecture of common radio resource management, which can jointly manage diverse RATs. Then the validity of the proposed algorithm is illustrated through the simulation results, from which the paper simultaneously draw some important conclusions.

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References

  1. ETSI. (2001). Requirements and Architectures for Interworking between HIPERLAN/3 and 3rd Generation Cellular Systems. TR, 101, 957.

  2. 3GPP. (2001). Improvement of RRM across RNS and RNS/BSS (release 5). TR 25.881 v5.0.0.

  3. 3GPP. (2003). Improvement of RRM across RNS and RNS/BSS (release 6). TR 25.891 v0.3.0.

  4. Feng, Z., Yu, K., Ji, Y., Zhang, P., Li, V. O. K., & Zhang, Y. (2006). Multi-access radio resource management using multi-agent system. In IEEE Wireless Communications and Networking Conference (pp. 63–68).

  5. Serrador, A., & Correia, L. M. (2011). A Cost Function Model for CRRM over Heterogeneous Wireless Networks. Wireless Personal Communications, 59(2), 313–329.

    Article  Google Scholar 

  6. Gozalvez, J., Lucas-Estan, M. C., & Sanchez-Soriano, J. (2012). Joint radio resource management for heterogeneous wireless systems. Wireless Networks, 18(4), 443–455.

    Article  Google Scholar 

  7. Ding, Z., & Xu, Y. (2011). A novel joint radio resource management with radio resource reallocation in the composite 3G scenario. Information Technology Journal, 10(6), 1228–1233.

    Article  Google Scholar 

  8. Skehill, R., Barry, M., Kent, W., Ocallaghan, M., Gawley, N., & McGrath, S. (2007). The common RRM approach to admission control for converged heterogeneous wireless networks. IEEE Wireless Communications, 14(2), 48–56.

    Article  Google Scholar 

  9. Buljore, S., Harada, H., Filin, S., Houze, P., Tsagkaris, K., Holland, O., et al. (2009). Architecture and enablers for optimized radio resource usage in heterogeneous wireless access networks: The IEEE 1900.4 Working Group. IEEE Communications Magazine, 47(1), 122–129.

    Article  Google Scholar 

  10. Gavrilovska, L. M., & Atanasovski, V. M. (2009). Resource management in Wireless Heterogeneous Networks (WHNs). In 9th International Conference on Telecommunications in Modern Satellite, Cable, and Broadcasting Services (pp. 97–106).

  11. Wang, J., Xu, Y., Ma, L., & Chen, L. (2012). Call admission control in CDMA2000/WLAN network based on user position information. Journal of Networks, 7(8), 1220–1226.

    Article  Google Scholar 

  12. Yao, J., Guo, J., & Bhuyan, L. N. (2008). Ordered Round-Robin: An efficient sequence preserving packet scheduler. IEEE Transactions on Computers, 57(12), 1690–1703.

    Article  MathSciNet  Google Scholar 

  13. Cao, G., Yang, D.-C., Zhang, X., & Zhu, X.-Y. (2011). A joint resource allocation and power control algorithm for heterogeneous network. Journal of China Universities of Posts and Telecommunications, 18(2), 17–22.

    Article  Google Scholar 

  14. Hasib, A., & Fapojuwo, A. O. (2008). Analysis of common radio resource management scheme for end-to-end QoS support in multiservice heterogeneous wireless networks. IEEE Transactions on Vehicular Technology, 57(4), 2426–2439.

    Article  Google Scholar 

  15. Sabbagh, A. A. L. (2011). A Markov chain model for load-balancing based and service based RAT selection algorithms in heterogeneous networks. World Academy of Science, Engineering and Technology, 73, 146–152.

    Google Scholar 

  16. Leijia, W., Sandrasegaran, K., & Elkashlan, M. (2010). A Markov Model for Performance Evaluation of CRRM Algorithms in a Co-Located GERAN/UTRAN/WLAN Scenario. In IEEE Wireless Communications and Networking Conference Workshops (pp. 1–6).

  17. Ma, F., Xu, G.-X., & Yang, F.-X. (2011). Capability adaptation algorithm based on joint network and terminal selection in heterogeneous networks. Journal of China Universities of Posts and Telecommunications, 18(1), 76–82.

    Article  Google Scholar 

  18. Chamodrakas, I., & Martakos, D. (2012). A utility-based fuzzy TOPSIS method for energy efficient network selection in heterogeneous wireless networks. Applied Soft Computing Journal, 12(7), 1929–1938.

    Google Scholar 

  19. Gelabert, X., Perez-Romero, J., Sallent, O., & Agusti, R. (2008). A markovian approach to radio access technology selection in heterogeneous multiaccess/multiservice wireless networks. IEEE Transactions on Mobile Computing, 7(10), 1257–1270.

    Article  Google Scholar 

  20. Azhari, V. S., Smadi, M., & Todd, T. D. (2008). Fast client-based connection recovery for soft WLAN-to-cellular vertical handoff. IEEE Transactions on Vehicular Technology, 57(2), 1089–1102.

    Article  Google Scholar 

  21. Pengbo, S., Yu, F. R., Hong, J., & Leung, V. C. M. (2009). Optimal Network Selection in Heterogeneous Wireless Multimedia Networks. In IEEE International Conference on Communications (pp. 1–5).

  22. Shi, Z., & Zhu, Q. (2012). Performance analysis and optimization based on markov process for heterogeneous wireless networks. Journal of Electronics & Information Technology, 34(9), 2224–2229.

    Article  Google Scholar 

  23. Perez-Romero, J., & Salient, O. (2006). Loose and Tight Interworking between Vertical and Horizontal Handovers in Multi-RAT Scenarios. In IEEE Mediterranean Electrotechnical Conference (pp. 579–582).

  24. Leijia, W., & Sandrasegaran, K. (2007). A Survey on Common Radio Resource Management. In The 2nd International Conference on Wireless Broadband and Ultra Wideband Communications (pp. 66–66).

  25. Wei, S., & Qing-An, Z. (2008). Cost-function-based network selection strategy in integrated wireless and mobile networks. IEEE Transactions on Vehicular Technology, 57(6), 3778–3788.

    Article  Google Scholar 

  26. Lusheng, W., & Binet, D. (2009). Mobility-Based Network Selection Scheme in Heterogeneous Wireless Networks. In IEEE 69th Vehicular Technology Conference (pp. 1–5).

  27. Shi, Z., Zhu, Q., & Zhao, S. (2012). A vertical handoff rate analysis based on angle mobility model in heterogeneous networks. Signal Processing, 28(7), 1029–1036.

    Google Scholar 

  28. Rappaport, T. (2002). Wireless communications: principles and practice. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  29. Lopez-Benitez, M., & Gozalvez, J. (2011). Common radio resource management algorithms for multimedia heterogeneous wireless networks. IEEE Transactions on Mobile Computing, 10(9), 1201–1213.

    Article  Google Scholar 

  30. Hao, W., Lianghui, D., Ping, W., Zhiwen, P., Nan, L., & Xiaohu, Y. (2011). QoS-Aware Load Balancing in 3GPP Long Term Evolution Multi-Cell Networks. In ICC (pp. 1–5).

  31. Palomar, D. P., & Mung, C. (2006). A tutorial on decomposition methods for network utility maximization. IEEE Journal on Selected Areas in Communications, 24(8), 1439–1451.

    Article  Google Scholar 

  32. Wei, Y., & Lui, R. (2006). Dual methods for nonconvex spectrum optimization of multicarrier systems. IEEE Transactions on Communications, 54(7), 1310–1322.

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by National Natural Science Foundation of China (61171094), National Basic Research Program of China (973 program: 2013CB329005), National Science & Technology Key Project(2011ZX03001-006-02, 2011ZX03005-004-03)) and Key Project of Jiangsu Provincial Natural Science Foundation (BK20110270).

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Correspondence to Qi Zhu.

Appendices

Appendix A

To illustrate the optimization problem is a convex optimization problem, we can prove the Hessian matrix \(\frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}\) is negative definite matrix. \(\frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}\) can be expressed as

$$\begin{aligned} \frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}=\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {R_1 \frac{\partial ^{2}M\left( {\xi _1 ,C_1 } \right) }{\partial \overrightarrow{p_{O_i }^1 }^{2}}}&{} 0&{} \cdots &{} 0 \\ 0&{} {R_2 \frac{\partial ^{2}M\left( {\xi _2 ,C_2 } \right) }{\partial \overrightarrow{p_{O_i }^2 }^{2}}}&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {R_{N} \frac{\partial ^{2}M\left( {\xi _N ,C_N } \right) }{\partial {\overrightarrow{p_{O_i }^N}}^{2}}} \\ \end{array} }} \right] \end{aligned}$$
(42)

where \(\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial {\overrightarrow{p_{O_i }^{j}}}^{2}}\) can be expressed as

$$\begin{aligned} \frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial {\overrightarrow{p_{O_i }^{j}}}^{2}}=\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial {p_{O_i }^{j}}^{2}}}&{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_i }^j \partial p_{O_k }^j }}&{} \cdots &{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_i }^j \partial p_{O_t }^j}} \\ {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_k}^j\partial p_{O_i }^j }}&{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j} \right) }{\partial {p_{O_k }^{j}}^{2}}}&{} \cdots &{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_k }^j \partial p_{O_t }^j}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_t }^j \partial p_{O_i }^j }}&{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_t }^j \partial p_{O_k }^j }}&{} \cdots &{} {\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial {p_{O_t }^{j}}^{2}}} \\ \end{array} }} \right] _{m\times m} \end{aligned}$$
(43)

where \(m=N\left( {O^{\prime }\left( {A_j } \right) } \right) \) and \(i,k,t\in O^{\prime }\left( {A_j } \right) \). \(\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_i}^j \partial p_{O_k }^j }\) is given by

$$\begin{aligned} \frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_i }^j \partial p_{O_k }^j }=\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial p_{O_k }^j \partial p_{O_i }^j }=\left\{ {\begin{array}{l@{\quad }l} \lambda _{o_i } \lambda _{o_k } W^{j}&{}<0;i,k\in O^{\prime }\left( {A_j } \right) \\ 0;&{} else \\ \end{array}} \right. \end{aligned}$$
(44)

where

$$\begin{aligned} W^{j}&= \left( {Q^{j}} \right) ^{2}\left( {1+\frac{{2v}/{\left( {\pi d_j } \right) }}{\mu +{2v}/{\left( {\pi d_j } \right) }}\frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j }\cdot } \right. \nonumber \\&\left. {\left( {1-\frac{{2v}/{\left( {\pi d_j } \right) }}{\mu +{2v}/{\left( {\pi d_j } \right) }}\frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j }} \right) ^{-1}} \right) \frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \xi _j^{2}}<0\end{aligned}$$
(45)
$$\begin{aligned} Q^{j}&= \frac{1}{\mu +{2v}/{\left( {\pi d_j } \right) }}\left( {1+\frac{2vP_j^{nb} }{\left( {\pi d_j \mu +2vP_j^b } \right) }} \right) \cdot \nonumber \\&\left( {1+\frac{2\pi P_j^b d_j v\left( {\lambda _j +\sum _{o_i \in O^{\prime }\left( {A_j } \right) } {p_{o_i }^j \lambda _{o_i } } } \right) \left( {C_j -\xi _j P_j^{nb} } \right) }{\left( {\pi d_j \mu +2vP_j^b } \right) ^{2}\xi _j }} \right) ^{-1}\quad \quad \end{aligned}$$
(46)
$$\begin{aligned} 0&< \frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j }=1-P_j^b \left( {C_j +1 -\xi _j P_j^{nb} } \right) <1\end{aligned}$$
(47)
$$\begin{aligned} \frac{\partial ^{2}M\left( {\xi _i ,C_i } \right) }{\partial \xi _i ^{2}}&= -\frac{P_i^b }{\xi _i }\left( {\left( {C_i +1 -\xi _i P_i^{nb} } \right) \left( {C_i -\xi _i P_i^{nb} +\xi _i P_i^b } \right) -\xi _i } \right) <0 \end{aligned}$$
(48)

Hence, (42) can be rewritten as

$$\begin{aligned} \frac{\partial ^{2}T}{\partial \overrightarrow{p_{O_i }^j }^{2}}=\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {R_1 \frac{\partial ^{2}M\left( {\xi _1 ,C_1 } \right) }{\partial {\overrightarrow{p_{O_i }^{1}}}^{2}}}&{} 0&{} \cdots &{} 0 \\ 0&{} {R_2 \frac{\partial ^{2}M\left( {\xi _2 ,C_2 } \right) }{\partial \overrightarrow{p_{O_i }^2 }^{2}}}&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {R_N \frac{\partial ^{2}M\left( {\xi _N ,C_N } \right) }{\partial {\overrightarrow{p_{O_i}^N}}^{2}}} \\ \end{array} }} \right] \end{aligned}$$
(49)

By substituting (44) into (43), we obtain

$$\begin{aligned} \frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \overrightarrow{p_{O_i }^j }^{2}}=W^{j}\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\left( {\lambda _{o_i } } \right) ^{2}}&{} {\lambda _{o_i } \lambda _{o_k } }&{} \cdots &{} {\lambda _{o_i } \lambda _{o_m } } \\ {\lambda _{o_k } \lambda _{o_i } }&{} {\left( {\lambda _{o_k } } \right) ^{2}}&{} \cdots &{} {\lambda _{o_l } \lambda _{o_m } } \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {\lambda _{o_m } \lambda _{o_i } }&{} {\lambda _{o_m } \lambda _{o_l } }&{} \cdots &{} {\left( {\lambda _{o_m } } \right) ^{2}} \\ \end{array} }} \right] \end{aligned}$$
(50)

Clearly, the principal minor sequence of matrix \(\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \overrightarrow{p_{O_i }^j }^{2}}\) is less than or equal to zero. Because \(\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \overrightarrow{p_{O_i }^j }^{2}}\) is a Hermitian matrix, and its characteristic roots are less than or equal to zero, \(\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \overrightarrow{p_{O_i }^j }^{2}}\) can be by using eigendecomposition written as

$$\begin{aligned} \frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial \overrightarrow{p_{O_i }^j }^{2}}=P_j \Lambda _j P_j^{T} \end{aligned}$$
(51)

By substituting (50) into (49), we can further obtain

$$\begin{aligned} \frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}&= \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {R_1 P_1 \Lambda _1 P_1^{T}}&{} 0&{} \cdots &{} 0 \\ 0&{} {R_2 P_2 \Lambda _2 P_2^{T}}&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {R_N P_N \Lambda _N P_N^{T}} \\ \end{array} }} \right] \nonumber \\&= \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {P_1 }&{} 0&{} \cdots &{} 0 \\ 0&{} {P_2 }&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {P_N } \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {R_1 \Lambda _1 }&{} 0&{} \cdots &{} 0 \\ 0&{} {R_2 \Lambda _2 }&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {R_N \Lambda _N } \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {P_1 }&{} 0&{} \cdots &{} 0 \\ 0&{} {P_2 }&{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0&{} \cdots &{} {P_N } \\ \end{array} }} \right] ^{T}\nonumber \\ \end{aligned}$$
(52)

From (52), the characteristic roots of matrix \(\frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}\) are less than or equal to zero, hence \(\frac{\partial ^{2}T}{\partial \overrightarrow{p}^{2}}\) is a negative definite matrix. So the convexity of objective function is proved.

Appendix B

On the basis of (16), the partial derivative of \(M_j \left( {\xi _j ,C_j } \right) \) with respect to \(v\) can be written as

$$\begin{aligned} \frac{\partial M_j \left( {\xi _j ,C_j } \right) }{\partial v}=\frac{\partial \xi _j }{\partial v}\frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j } \end{aligned}$$
(53)

From (15) to (18), the partial derivative of \(\xi _j \) with respect to \(v\) can be derived as

$$\begin{aligned} \frac{\partial \xi _j }{\partial v}=2\frac{v\left( {\mu +{2v}/{\left( {\pi d_j } \right) }} \right) \frac{\partial M_j \left( {\xi _j ,C_j } \right) }{\partial v}+M_j \left( {\xi _j ,C_j } \right) \mu -\gamma _j }{\pi d_j \left( {\mu +{2v}/{\left( {\pi d_j } \right) }} \right) ^{2}} \end{aligned}$$
(54)

By substituting (54) into (53), we have

$$\begin{aligned} \frac{\partial M_j \left( {\xi _j ,C_j } \right) }{\partial v}&= 2\left( {1-\frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j }\frac{{2v}/{\left( {\pi d_j } \right) }}{\mu +{2v}/{\left( {\pi d_j } \right) }}} \right) ^{-1}\nonumber \\&\times \left( {\frac{M_j \left( {\xi _j ,C_j } \right) -\xi _j }{\pi d_j \left( {\mu +{2v}/{\left( {\pi d_j } \right) }} \right) }} \right) \frac{\partial M\left( {\xi _j ,C_j } \right) }{\partial \xi _j } \end{aligned}$$
(55)

On the basis of \(M_j \left( {\xi _j ,C_j } \right) =\xi _j P_j^{nb} <\xi _j \) and (47), \(\frac{\partial M_j \left( {\xi _j ,C_j } \right) }{\partial v}<0\) is proved, so the average number of users \(M\left( {\xi _j ,C_j } \right) \) in each RAN is a decreasing function of \(v\).

Appendix C

Actually, the average throughput of each HWN is a function of \(\gamma _j \), so the average throughput of each RAN can be expressed as

$$\begin{aligned} T_j \left( {\gamma _j } \right) =R_j M_j \left( {\xi _j ,C_j } \right) \end{aligned}$$
(56)

On the basis of (56), the average throughput of HWNs can be written as

$$\begin{aligned} T\left( {\overrightarrow{\gamma }} \right) =\sum _{j=1}^N {T_j \left( {\gamma _j} \right) } \end{aligned}$$
(57)

where \(\overrightarrow{\gamma }=\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\gamma _1 }&{} {\gamma _2 }&{} \cdots &{} {\gamma _N } \\ \end{array} }} \right] \). By considering \(\frac{\partial T}{\partial p_{O_i }^j }=\frac{\partial T_j \left( {\gamma _j } \right) }{\partial \gamma _j }\frac{\partial \gamma _j }{\partial p_{O_i }^j }=\lambda _{O_i } G^{j}\), the partial derivative of \(T_j \left( {\gamma _j } \right) \) with respect to \(\gamma _j \) can be derived as

$$\begin{aligned} T_j {\prime }\left( {\gamma _j } \right) =\frac{\partial T_j \left( {\gamma _j } \right) }{\partial \gamma _j }=G^{j} \end{aligned}$$
(58)

And the second-order partial derivative of \(T_j \left( {\gamma _j } \right) \) with respect to \(\gamma _j \) can be derived as

$$\begin{aligned} T_j ^{\prime \prime }\left( {\gamma _j } \right) =\frac{\partial ^{2}T_j \left( {\gamma _j } \right) }{\partial \gamma _j ^{2}}=\frac{\partial G^{j}}{\partial \gamma _j }=\frac{R_j }{\left( {\lambda _{O_i } } \right) ^{2}}\frac{\partial ^{2}M\left( {\xi _j ,C_j } \right) }{\partial {p_{O_{i}}^{j}}^{2}}=R_j W^{j}<0 \end{aligned}$$
(59)

Let \({\overrightarrow{\gamma ^*}}\) denote an optimization result of (33), where \(\overrightarrow{\gamma }^{*}=\left[ {\gamma _1^{*}}\; {\gamma _2 ^{*}}\; \ldots \; {\gamma _N^{*}} \right] \) and \(T\left( {\overrightarrow{\gamma }^{*}} \right) =\sum _{j=1}^N {T_j \left( {\gamma _j ^{*}} \right) } \). On the basis of (39), we obtain

$$\begin{aligned} T_{j}^{\prime }\left( {\gamma _j^{*}} \right) =\left. {G^{j}} \right| _{\gamma _j =\gamma _j^{*}} =\kappa ^{*} \end{aligned}$$
(60)

Here, let \(\overrightarrow{\gamma }^{\prime *}\) denote another optimization result of (33), where \(\overrightarrow{\gamma _j^2 }^{*}=\small \left[ {\gamma _1^{*}+\Delta \gamma _1 }\; \gamma _2^{*}+\Delta \gamma _2 \; \ldots \;{\gamma _N^{*}+\Delta \gamma _N } \right] \), \(\Delta \gamma =\sum _{j=1}^N {\Delta \gamma _j } \ge 0\). Let \(\delta =\sum _{j=1}^N {\left( {\Delta \gamma _j } \right) ^{2}} \), where \(\delta \) is an infinitesimal. Hence, by using Taylor formula, \(T\small \left( {\overrightarrow{\gamma _j^2 }^{*}} \right) \) can be written as

$$\begin{aligned} T\left( {\overrightarrow{\gamma ^{\prime }}^{*}} \right)&= \sum _{j=1}^N {T_j \left( {\gamma _j^{*}+\Delta \gamma _j } \right) }\nonumber \\&= \sum _{j=1}^N {T_j \left( {\gamma _j^{*}} \right) } +\sum _{j=1}^N {T_j ^{\prime }\left( {\gamma _j^{*}} \right) \Delta \gamma _j }\nonumber \\&+\frac{1}{2!}\sum _{j=1}^N {T_j^{{\prime }{\prime }}\left( {\gamma _j^{*}} \right) \left( {\Delta \gamma _j } \right) ^{2}} +o\left( {\max \left( {\left( {\Delta \gamma _j } \right) ^{2}} \right) } \right) \nonumber \\&= T\left( {\overrightarrow{\gamma }^{*}} \right) +\kappa ^{*}\Delta \gamma +\frac{1}{2!}\sum _{j=1}^N {\left. {R_j W^{j}} \right| _{\gamma _j =\gamma _j^{*}} \left( {\Delta \gamma _j } \right) ^{2}} +o\left( {\max \left( {\left( {\Delta \gamma _j } \right) ^{2}} \right) } \right) \nonumber \\ \end{aligned}$$
(61)

Thus,

$$\begin{aligned} T\left( {\overrightarrow{\gamma ^{\prime }}^{*}} \right)&= \sum _{j=1}^N {T_j \left( {\gamma _j^{*}} \right) } +\kappa ^{*}\Delta \gamma +o\left( {\max \left( {\left| {\Delta \gamma _j } \right| } \right) } \right) \ge \sum _{j=1}^N {T_j \left( {\gamma _j^{*}} \right) }\end{aligned}$$
(62)
$$\begin{aligned} T\left( {\overrightarrow{\gamma ^{\prime }}^*} \right)&= T\left( {\overrightarrow{\gamma }^{*}} \right) +\kappa ^{*}\Delta \gamma +\frac{1}{2!}\sum _{j=1}^N {\left. {R_j W^{j}} \right| _{\gamma _j =\gamma _j^{*}} \left( {\Delta \gamma _j } \right) ^{2}} +o\left( {\max \left( {\left( {\Delta \gamma _j } \right) ^{2}} \right) } \right) \nonumber \\ \end{aligned}$$
(63)

To maximize the throughput of HWNs, the \(\Delta \gamma _j \) should satisfy

$$\begin{aligned} \begin{array}{l} \max \sum _{j=1}^N {\left. {R_j W^{j}} \right| _{\gamma _j =\gamma _j^{*}} \left( {\Delta \gamma _j } \right) ^{2}} \\ s.t.\quad \Delta \gamma =\sum _{j=1}^N {\Delta \gamma _j } \\ \end{array} \end{aligned}$$
(64)

By using Langrage multiplier method, \(\Delta \gamma _j \) can be resolved

$$\begin{aligned} \Delta \gamma _j =\frac{{\Delta \gamma }/{\left( {\left. {R_j W^{j}} \right| _{\gamma _j =\gamma _j^{*}} } \right) }}{\sum _{j=1}^N {1/{\left( {\left. {R_j W^{j}} \right| _{\gamma _j =\gamma _j^{*}} } \right) }} }\ge 0 \end{aligned}$$
(65)

Thus, all of the \(\Delta \gamma _j\) are greater than or equal to zero if \(\Delta \gamma \ge 0\). According to this conclusion, we can get that there is at least one optimal result which can make sure that the service arrival rate of each RAN won’t decrease under the OA scheme if the service arrival rate of any region in the HWNs increases. Specifically, if the optimization problem (40) has the unique solution, i.e., \(N\left( {O^{\prime }} \right) +N=\sum _{O_i \in O^{\prime }} {N\left( {O_i } \right) } +1\), the service arrival rate of all RANs will increase with an increase in service arrival rate of any region.

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Shi, Z., Zhu, Q. Throughput Analysis and Optimization Based on Mobility Analysis and Markov Process for Heterogeneous Wireless Networks. Wireless Pers Commun 77, 1091–1116 (2014). https://doi.org/10.1007/s11277-013-1556-3

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