Abstract
A distributed queue system is presented for wireless mesh/relay networks with half-asleep and wakeup strategy and double-state channel. Each node could receive/transmit data packets from itself or from other nodes by acting as a relay node. We model the process at a mesh/relay node as a queue system with two types of data packets: the relayed packets and the node’s own packets, and the former have higher priority with non-preemption than the latter. Based on the efficient power saving strategy, each node is considered to have two modes of operation: half-asleep and active. Meanwhile, owing to the requirement of the quality of service (QoS), a dynamic channel bonding strategy is proposed which will lead to the different service rates. To obtain the steady-state distribution of the queue length, we build a four-dimensional discrete time Markov chain, and give the major performance expressions. Moreover, some numerical experiments are provided to analyze the influence of various parameters on the performance measures. Finally, we obtain the Nash equilibrium solution for individual benefit and the socially optimal solution. In addition, a cooperative bargaining game is formulated by balancing the arrivals of the data packets to reach a satisfied agreement under QoS constraints.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 61472342), Key Foundation of Higher Education Science and Technology Research of Hebei Province (No. ZD2017079), Youth Foundation of Higher Education Science and Technology Research of Hebei Province (No. QN2016016), and Innovation Foundation for Graduate Student of Yanshan University (No. 2017XJSS045).
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Appendix
Appendix
In order to simplify the matrices with complicated construction, we define some symbols as following:
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1.
The block \({\varvec{P}}_{00}\) is a (2 K + 3) dimensional matrix with its elements given by:
$${\varvec{P}}_{00} = \left[ {\begin{array}{*{20}c} {\bar{\lambda }_{1} \bar{\lambda }_{2} } & {\lambda_{1} \bar{\lambda }_{2} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \theta } & {} & {} & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} } & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & {\delta_{1} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } & {} & {} & {} & {} \\ {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \theta } & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } & {} & {} \\ {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & {\delta_{1} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } & {} & {} \\ {} & {} & {} & {} & \ddots & \ddots & \ddots & {} & {} \\ {} & {} & {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \theta } & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } \\ {} & {} & {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & {\delta_{1} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } \\ {} & {} & {} & {} & {} & {\delta_{2c} \bar{\theta }} & {\delta_{2c} \theta } & {\delta_{3c} \bar{\theta }} & {\delta_{3c} \theta } \\ {} & {} & {} & {} & {} & 0 & {\delta_{2} } & 0 & {\delta_{3} } \\ \end{array} } \right],$$ -
2.
The block \({\varvec{P}}_{01}\) is a (2 K + 3) × (4 K + 4) dimensional matrix with its elements given by:
$${\varvec{P}}_{01} = \left[ {\begin{array}{*{20}c} {\bar{\lambda }_{1} \lambda_{2} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\theta }} & {\lambda_{1} \lambda_{2} \theta } & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } & {} & {} & {} & {} & {} & {} \\ 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{4} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } & {} & {} \\ {} & {} & {} & {} & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{4} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu } & 0 & 0 & 0 & {\delta_{4} } & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {\delta_{5c} \bar{\theta }} & {\delta_{5c} \theta } & {\delta_{6c} \bar{\theta }} & {\delta_{6c} \theta } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & {\delta_{5} } & 0 & {\delta_{6} } \\ \end{array} } \right],$$ -
3.
The block \({\varvec{P}}_{10}\) is a (4 K + 4) × (2 K + 3) dimensional matrix with its elements given by:
$${\varvec{P}}_{10} = \left[ {\begin{array}{*{20}c} {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2c} } & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \theta } & {} & {} & {} & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \mu_{2} } & {} & {} & {} & {} & {} & {} & {} \\ {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \theta } & {} & {} & {} & {} & {} \\ {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \mu_{2} } & {} & {} & {} & {} & {} \\ {} & 0 & 0 & 0 & 0 & {} & {} & {} & {} & {} \\ {} & 0 & 0 & 0 & 0 & {} & {} & {} & {} & {} \\ {} & {} & \ddots & \ddots & \ddots & {} & {} & {} & {} & {} \\ {} & {} & {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \mu_{2c} \theta } & {} & {} & {} \\ {} & {} & {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \mu_{2} } & {} & {} & {} \\ {} & {} & {} & 0 & 0 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & 0 & 0 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & {} & {} & {\beta_{1c} \bar{\theta }} & {\beta_{1c} \theta } & 0 & {} & 0 \\ {} & {} & {} & {} & {} & 0 & {\beta_{1} } & 0 & {} & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ \end{array} } \right],$$ -
4.
The block \({\varvec{P}}_{0}\) is a (4 K + 4) dimensional matrix with its elements given by:
$${\varvec{P}}_{0} =\left[ {{\begin{array}{*{20}c} {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & 0 & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & 0 & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & 0 & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & \ddots & \ddots & \ddots & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\bar{\theta } & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2c}}\theta & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & {{{\bar{\lambda }}}_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & 0 & 0 & 0 & {{\lambda }_{1}}{{{\bar{\lambda }}}_{2}}{{\mu }_{2}} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {{\beta }_{1c}}\bar{\theta } & {{\beta }_{1c}}\theta & 0 & {} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & {{\beta }_{1}} & 0 & {} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {} & 0 \\ \end{array} }} \right],$$ -
5.
The block \({\varvec{P}}_{1}\) is a (4 K + 4) dimensional matrix with its elements given by:
$${\varvec{P}}_{1} = \left[ {\begin{array}{*{20}c} {\beta_{2c} \bar{\theta }} & {\beta_{2c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \theta } & {\lambda_{1} \lambda_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \mu_{2c} \theta } & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {\beta_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2} } & 0 & {\lambda_{1} \lambda_{2} \mu_{2} } & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \theta } & {\bar{\lambda }_{1} \lambda_{2} \mu_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{2c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \theta } & {\lambda_{1} \lambda_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \mu_{2c} \theta } & {} & {} & {} & {} \\ 0 & 0 & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2} } & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2} } & 0 & {\lambda_{1} \lambda_{2} \mu_{2} } & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & 0 & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } & {} & {} & {} & {} \\ 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{1} } & 0 & 0 & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } & {} & {} & {} & {} \\ {} & {} & \ddots & {} & \ddots & {} & \ddots & {} & \ddots & {} & {} & {} & {} & {} \\ {} & {} & 0 & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \theta } & {\bar{\lambda }_{1} \lambda_{2} \mu_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{2c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2c} \theta } & {\lambda_{1} \lambda_{2} \mu_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \mu_{2c} \theta } & {} & {} \\ {} & {} & 0 & 0 & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \bar{\mu }_{2} } & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{2} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{2} } & 0 & {\lambda_{1} \lambda_{2} \mu_{2} } & {} & {} \\ {} & {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & 0 & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } & {} & {} \\ {} & {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{1} } & 0 & 0 & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } & {} & {} \\ {} & {} & {} & {} & {} & {} & 0 & 0 & {\beta_{3c} \bar{\theta }} & {\beta_{3c} \theta } & {\beta_{4c} \bar{\theta }} & {\beta_{4c} \theta } & 0 & 0 \\ {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {\beta_{3} } & 0 & {\beta_{4} } & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{1c} \bar{\theta }} & {\delta_{1c} \theta } & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1c} \theta } \\ {} & {} & {} & {} & {} & {} & 0 & {\bar{\lambda }_{1} \bar{\lambda }_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{1} } & 0 & {\lambda_{1} \bar{\lambda }_{2} \bar{\mu }_{1} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {\delta_{2c} \bar{\theta }} & {\delta_{2c} \theta } & {\delta_{3c} \bar{\theta }} & {\delta_{3c} \theta } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & {\delta_{2} } & 0 & {\delta_{3} } \\ \end{array} } \right],$$ -
6.
The block \({\varvec{P}}_{2}\) is a (4 K + 4) dimensional matrix with its elements given by:
$${\varvec{P}}_{2} = \left[ {\begin{array}{*{20}c} {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2} } & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{2} } & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & 0 & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & {} & {} & {} & {} & {} & {} \\ 0 & 0 & 0 & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{2} } & {} & {} & {} & {} & {} & {} \\ {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } & {} & {} & {} & {} \\ 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{4} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } & {} & {} & {} & {} \\ {} & {} & \ddots & {} & \ddots & {} & \ddots & {} & \ddots & {} & {} & {} & {} & {} \\ {} & {} & 0 & 0 & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{2c} \theta } & {} & {} & {} & {} \\ {} & {} & 0 & 0 & 0 & {\bar{\lambda }_{1} \lambda_{2} \bar{\mu }_{2} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{2} } & 0 & {} & {} & {} \\ {} & {} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } & {} & {} \\ {} & {} & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{4} } & 0 & 0 & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } & {} & {} \\ {} & {} & {} & {} & {} & {} & 0 & 0 & {\beta_{5c} \bar{\theta }} & {\beta_{5c} \theta } & 0 & 0 & 0 & 0 \\ {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {\beta_{5} } & 0 & 0 & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \bar{\theta }} & {\bar{\lambda }_{1} \lambda_{2} \mu_{1c} \theta } & 0 & 0 & {\delta_{4c} \bar{\theta }} & {\delta_{4c} \theta } & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \bar{\theta }} & {\lambda_{1} \lambda_{2} \bar{\mu }_{1c} \theta } \\ {} & {} & {} & {} & {} & {} & 0 & {\bar{\lambda }_{1} \lambda_{2} \mu_{1} } & 0 & 0 & 0 & {\delta_{4} } & 0 & {\lambda_{1} \lambda_{2} \bar{\mu }_{1} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {\delta_{5c} \bar{\theta }} & {\delta_{5c} \theta } & {\delta_{6c} \bar{\theta }} & {\delta_{6c} \theta } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 & {\delta_{5} } & 0 & {\delta_{6} } \\ \end{array} } \right].$$.
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Wang, W., Ma, Z., Cao, J. et al. Analysis of Half-Asleep and Wakeup Strategy in Wireless Mesh/Relay Networks with Double-State Channel. Int J Wireless Inf Networks 25, 173–185 (2018). https://doi.org/10.1007/s10776-018-0390-7
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DOI: https://doi.org/10.1007/s10776-018-0390-7