Abstract
To mitigate the interference in co-located wireless body area networks (WBANs), this paper proposes an inter-WBAN priority-based capacity allocation scheme based on the Nash bargaining game, and an intra-WBAN priority-based power control scheme based on the Stackelberg game. Moreover, under the network capacity imposed by the Nash bargaining solution, two pricing mechanisms: non-uniform pricing and uniform pricing, are introduced in the Stackelberg game and the Stackelberg equilibrium under each mechanism is achieved analytically. Additionally, owing to the special features of WBANs, the players priorities indicated by exigency of the sensed data and the energy consumption of sensors are considered in the design of utility functions. Extensive simulations show that the proposed schemes are energy efficient and can improve the network quality of service in terms of real time and reliability of critical data transmission.
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This research was supported by National Natural Science Foundation of China (No. 61372118).
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Appendix
Appendix
1.1 A. Proof of Proposition 4.1
Obviously, Problem 4.2 is a convex optimization problem, which can be solved using the Lagrange Multiplier approach. The Lagrange function is given by
Then, the KKT conditions can be written as follows:
From formulas (31) and (32), we have
Thus, the following solution can be obtained
Lemma 1
\({\alpha _i} = 0, \quad \forall i\)
Proof
Suppose that \({\alpha _i} \ne 0\) for some i. it can be deduced from formula (33) that \({S_i} = S_i^{\min }\), which contradicts the meaning of cooperative game and formula (37). Therefore, the assumption that \({\alpha _i} \ne 0\) for some i doesn’t hold, and we have \({\alpha _i} = 0, \ \forall i\). \(\square \)
Lemma 2
\(\sum \limits _{i = 1}^N {{S_i}} = C\)
Proof
Suppose that \(\sum _{i = 1}^N {{S_i}} \ne C\), according to formula (34), we have \(\beta = 0\). Further, it has been proved in Lemma 1 that \({\alpha _i} = 0, \ \forall i\). Then, it can be seen that the equation in formula (37) isn’t true in this case. Thus, the assumption doesn’t hold and we have \(\sum _{i = 1}^N {{S_i}} = C\).
According to Lemmas 1 and 2, it can be computed that \(\beta = {\left( {\sum _{i = 1}^N {{w_i}} } \right) } \bigg /{\left( {C - \sum _{i = 1}^N {S_i^{\min }} } \right) }\). Substituting it into formula (38), we have
Proposition 4.1 is thus proved. \(\square \)
1.2 B. Proof of Proposition 5.1
The first- and second-order partial derivatives of \({U_k^i}\) with respect to \({p_k^i}\) are given as
As \(\lambda _k^i > 0\) and \(\vartheta _k^i \ge 0\), it can be obtained that \(\frac{{{\partial ^2}U_k^i}}{{\partial p{{_k^i}^2}}} \le 0\) Thus, \({U_k^i}\) is a concave function of \({p_k^i}\). We can get the optimal solution of Problem 5.1 by setting \(\frac{{\partial U_k^i}}{{\partial p_k^i}} = 0\), i.e., \(p_k^i = \frac{{\lambda _k^i \cdot \vartheta _k^i}}{{\ln 2 \cdot c_k^i}} - \frac{{{\sigma ^2}}}{{h_k^i}}\). Due to the limited energy, sensor nodes active under the constraint of maximum feasible transmission power \({P_{\max }}\), i.e., \(p_k^i = \left( {\frac{{\lambda _k^i \cdot \vartheta _k^i}}{{\ln 2 \cdot c_k^i}} - \frac{{{\sigma ^2}}}{{h_k^i}}} \right) _0^{{P_{\max }}}\). Therefore, we can get
The solution can also be obtained using the Lagrange Multiplier approach, which is similar to part A of appendix, and it is omitted here. Thus, Proposition 5.1 is proved.
1.3 C. Proof of Proposition 5.2
Problem 5.3 is a convex optimization problem and can be solved using the Lagrange Multiplier method. The Lagrange function is given by
Then, the KKT conditions can be written as follows:
From formulas (44)–(47), it can be seen that \({\beta _i} = {\gamma _i} = 0,\forall i\). Since \(\frac{{\partial L}}{{\partial c_k^i}} = - \frac{{{\sigma ^2}}}{{h_k^i}} + {\beta _i} - {\gamma _i} + \frac{{\alpha \cdot B}}{{\ln 2 \cdot c_k^i}} = 0\) , it can be obtained that
From formulas (48) and (49), it can be deduced that
Substituting (49) into (50), we can get that
Substituting (51) into (49), the price strategy can be obtained as follows:
Proposition 5.2 is proved.
1.4 D. Proof of Proposition 5.3.
It is observed that the solution in (19) is the optimal solution of Problem 5.3 if the following condition is satisfied:
Substituting (19) into these inequalities, it can be deduced that
and
These inequalities can be transformed as formulas (20) and (21) compactly, respectively. Thus, Proposition 5.3 is proved.
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Wang, J., Sun, Y. & Ji, Y. Priority-based capacity and power allocation in co-located WBANs using Stackelberg and bargaining games. J Supercomput 74, 3114–3147 (2018). https://doi.org/10.1007/s11227-018-2364-z
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DOI: https://doi.org/10.1007/s11227-018-2364-z