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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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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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Authors and Affiliations

  1. Jiangsu Key Lab of Wireless Communications, Key Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education, Nanjing University of Posts and Telecommunications, Nanjing, 210003, Jiangsu, China

    Zheng Shi & Qi Zhu

Authors
  1. Zheng Shi
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  2. Qi Zhu
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Corresponding author

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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