Abstract
In this paper, we study the distributed DOS problem for wireless multiple relay networks. Formulating the problem as an extended three-level optimal stopping problem, an optimal strategy is proposed guiding distributed channel access for multiple source-to-destination communications under the help of multiple relays. The optimality of the strategy is rigorously proved, and abides by a tri-level structure of pure threshold. For network operation, easy implementation is presented of low complexity. The close-form expression of the maximal expected system throughput is also derived. Furthermore, numerical results are provided to demonstrate the correctness of our analytical expressions, and the effectiveness is verified.
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- 1.
Note the superscript \(\overline{\infty }\) means the summation includes term at \(n=\infty \).
- 2.
For finite n, the existence proof is similar to Theorem 6, while \(n=\infty \) means the main layer does not stop.
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A Proof of Lemma 1
A Proof of Lemma 1
For relay number \(k\in \{1,2,...,L\}\), we define \(r_n':=r_n\!-\!P_s|h(n)|^2\). For function \(Z_k\), the first-order derivative is calculated as that
By replacing \(1\!-\!e^{-\frac{r_n'}{\sigma _g^2}}\) with \(y(r_n')\), the derivative is rewritten in Eq. (15).
In the region \(r_n'\!\ge \!0\), the factor in Eq. (15) satisfies \((1-y(r_n'))y^{1-k}(r_n')>0\). Therefore, it suffices to compare \(\frac{\tau _d\log _2e}{1+r_n'+P_s|h(n)|^2} \sum \limits _{i=0}^{k-1}y^{-i}(r_n')\) and \(\frac{1}{\sigma _g^2} k \big (\log _2(1\!+\!r_n'\!+\!P_s|h(n)|^2)\tau _d\!-\!\lambda \tau _d\big )\) to determine the derivative.
It can be proved that \(\sum \limits _{i=0}^{k-1}y^{-i}(r_n')\frac{1}{1+r_n'+P_s|h(n)|^2}\) is decreasing and \(\log _2(1+r_n'+P_s|h(n)|^2)-\lambda \) is increasing in \(r_n'\!\ge \!0\), respectively. Meanwhile, as \(r_n'\) is large, \(\sum \limits _{i=0}^{k-1}y^{-i}(r_n')\) will approach to k and \(\frac{1}{1+r_n'+P_s|h(n)|^2}\) is small, while \(\{\log _2(1+r_n'+P_s|h(n)|^2)\tau _d-\lambda \tau _d'\}\) is large. Thus, the existence of stationary point, denoted by \(\zeta _k\!-\!P_s|h(n)|^2\) is guaranteed, which are unique such that \(\frac{\partial {Z}_k}{\partial r_n'}\!=\!0\).
As a result, function \(Z_k\) increases in \(r_n\!<\!\zeta _k\), and decreases in \(r_n\!\ge \!\zeta _k\).
Then, relation of these points \(\{\zeta _k\},k=1,2,...,L\) is further investigated.
For \(k\!>\!1\), by valuing \(r_n'\!=\!0\), we have the derivative satisfy
which means \({\zeta }_k\!>\!P_s|h(n)|^2\).
For \(k\!=\!1\), we have the derivative satisfy that
Suppose \(\zeta _1\) satisfies \(\frac{\partial {Z}_k}{\partial r_n'}=0\), we compare the points \(\{\zeta _1,\zeta _2,...,\zeta _L\}\). The situation where \(\zeta _1\!=\!P_s|h_{s(n)}(n)|^2\) is similar as \({\zeta }_k\!>\!P_s|h_{s(n)}(n)|^2\) for \(\forall k\ge 2\).
Since \(\zeta _k\) such that \(\frac{\partial {Z}_k}{\partial \bar{R}_n}\!=\!0\), by induction it suffices from (15) to compare \(\frac{1}{k}\sum \limits _{i=0}^{k-1}y^{-i}(r_n')\).
Based the relation shown in the following that
it proves that \(\zeta _k<\zeta _{k+1}\) for \(\forall k=1,2,...,L\!-\!1\). In other words, \(\zeta _1<\zeta _2<...<\zeta _L\).
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Zhang, Z., Yan, Y., Sang, W., Xu, Z. (2020). Distributed Scheduling in Wireless Multiple Decode-and-Forward Relay Networks. In: Li, B., Zheng, J., Fang, Y., Yang, M., Yan, Z. (eds) IoT as a Service. IoTaaS 2019. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 316. Springer, Cham. https://doi.org/10.1007/978-3-030-44751-9_24
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