Abstract
In heterogeneous wireless networks (HWNs), paths constituting multipath routing are characterized by selfish rationality. Each path’s intentions of pursuing individual profits may cause unreasonable competition for limited wireless resources, which leads to the unreliable data transport problem. Therefore, multipath routing optimization is a challenge issue in HWNs. This paper provides a novel approach to study this issue by employing game theory. By taking the utility maximization as the design goal, limited bandwidth resources as the constraint and path reliability as the key metric, a noncooperative stochastic differential game model is constructed for HWNs. With the feedback Nash equilibrium solution, an optimal multipath routing strategy is obtained. Theoretical derivations and simulation results verify the validity of the method present in this paper.
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Acknowledgments
The authors are grateful to the anonymous reviewer for constructive suggestions that have improved the quality of this paper. This work has been supported by the National Key Technology R&D Program of P. R. China (Grant No. 2011BAH10B03-1) and the National High-Tech Research and Development Program of P. R. China (Grant No. 2012AA121604).
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Appendices
Appendix 1: Proof of Corollary 1
Proof
Maximizing the right-hand-side of (10) for path-\(i\), and arranging the equation we get
For \(i = 1,2,\ldots ,n\), there is an system of linear equations
Summing over the right-hand-side and the left-hand-side respectively in (18), we get
There, we have two methods to obtain the solution of \(\phi _i^*\left( {t,x} \right) \). \(\square \)
Solution 1
Substituting (20) into (18), we get
Arranging (21), then
Solution 2
Letting (18) minus (20), we directly get
This completes the proof of Corollary 1.
Appendix 2: Proof of Corollary 2
Proof
From (14), we get
Substituting (22) into (11) produces
Meanwhile, substituting (22) into (13) produces
Thus
Let \(\Theta (t)=\left( {\lambda {-}\delta q_i } \right) \left( {f_i {+}\Gamma (t)\sum _{i=1}^n {g_i } } \right) -\left[ {\xi \left( {f_i {+}\Gamma (t)\sum _{i=1}^n {g_i } } \right) {+}\Gamma (t)} \right] \sum _{i=1}^n \Big ( f_i {+}\Gamma (t)\sum _{i=1}^n {g_i } \Big ),\)
Then
This completes the proof of Corollary 2. \(\square \)
Appendix 3: Derivation of (16)
Derivation Equation (15) is constituted by
and
It is noted that (15-1) is a homogeneous linear differential equation, while (15-3) is a nonhomogeneous linear differential equation, and both of them have their general solutions. The derivation of (16) is related to the solution process of linear differential equation. The derivation process is detailed as following:
Separate the variable of (15-1), we get
Integrate both sides of (26), we get
Then,
When \(t=T\),
Combining (28) and (15-2), we get
Let
be the solution of the nonhomogeneous linear differential equation (15-3).
Substitute \(\dot{\Lambda }(t)\) and \(\Lambda (t)\) into (15-3), then
Arrange (30), we get
Integrate both sides of (31), we get
Substitute (32) into (29), we get
When \(t=T\),
Combining (33) and (15-4), we get
Equations (16-1), (16-2), (16-3) and (16-4) constitute (16).
This completes the derivation of (16).
Appendix 4: Proof of Corollary 3
Proof
Solving the linear differential equation (9), we get
Substituting the optimal load-control strategy (24) into (34) produces
This completes the proof of Corollary 3. \(\square \)
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Hu, J., Xie, Y. A Stochastic Differential Game Theoretic Study of Multipath Routing in Heterogeneous Wireless Networks. Wireless Pers Commun 80, 971–991 (2015). https://doi.org/10.1007/s11277-014-2065-8
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DOI: https://doi.org/10.1007/s11277-014-2065-8