Adaptive and intelligent energy efficient routing for transparent heterogeneous ad-hoc network by fusion of game theory and linear programming

Abstract

In the modern era, the use of transparent Heterogeneous Ad-Hoc NETwork (HANET) has increased due to the numerous two-tier applications. It provides cross platform services. In this network, the concept of energy efficient routing is a vital issue due to the limited energy capacity of batteries. In response to this, the proposed work introduces an adaptive and intelligent energy efficient routing technique based on the fusion of game theory and linear programming methods. The combination of these two help to analyze multi-person decision making in the network, where each decision maker tries to maximize its own utility. The proposed model is simulated using the LINGO optimization software. The performance of the proposed model is validated with the existing approaches in several passes to justify its effectiveness in term of performance metrics.

Appendices

Appendix A

Lemma 1

The Player A is the CH based on strategy hop-count with respect to parameter energy.

Proof

Let probability of CH and CM selection of Player A nodes are p and (1-p). The expected utility of Player A nodes whenselecting CH is $U_{M_{CH}}$and selecting CM is $U_{M_{CM}}$.These expected utilities are expressed with the help of (39) and (40) are given as:

$$ \begin{array}{lllllll} & U_{M_{CH}}=H_{a} + r_{1} \times \delta_{1} - q \times r_{1} \times \delta_{1} \end{array} $$

(39)

$$ \begin{array}{lllllll} & U_{M_{CM}}=q \times H_{a} - \delta_{1} \end{array} $$

(40)

The average profit of Player A is denoted as in (41).

$$ \begin{array}{lllllll} & \overline{U}_{M}=p \times H_{a} + p \times r_{1} \times \delta_{1} - p \times q \times r_{1} \times \delta_{1}\\ & \qquad \;\; + q \times H_{a} - \delta_{1} - p \times q \times H_{a} + p \times \delta_{1} \end{array} $$

(41)

The dynamic replicator equation of Player A is obtained from (42) given as:

$$ \begin{array}{lllllll} & \frac{dp}{dt}=\overline{\Delta}_{1}(p)=p \times (U_{M_{CH}} - \overline{U}_{M})\\ & \qquad=p (1-p)[H_{a} + r_{1} \times \delta_{1} + \delta_{1} -q(r_{1} \times \delta_{1} + H_{a})] \end{array} $$

(42)

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Appendix B

Lemma 2

The Player B is the CM based on strategy hop-count with respect to parameter energy.

Proof

Let probability of CH and CM selection of Player B nodes are q and (1-q). The expected utility of Player B nodes whenselecting CH is $U_{L_{CH}}$and selecting CM is $U_{L_{CM}}$.These expected utilities are expressed with the help of (43) and (44) are given as:

$$ \begin{array}{lllllll} & U_{L_{CH}}=H_{b} + \delta_{1} - p \times \delta_{1} \end{array} $$

(43)

$$ \begin{array}{lllllll} & U_{L_{CM}}=p \times H_{b} + p \times r_{1} \times \delta_{1} - \delta_{1} + p \times \delta_{1} \end{array} $$

(44)

The average profit of Player B is denoted as in (45).

$$ \begin{array}{lllllll} & \overline{U}_{L}=q \times H_{b} + 2 \times q \times \delta_{1} + p \times H_{b} + p \times r_{1} \times \delta_{1}\\ & \qquad \;\; - \delta_{1} \,-\, p \!\times\! q \times H_{b} \,-\, p \times q \times r_{1} \times \delta_{1} - p \times q \times \delta_{1} \end{array} $$

(45)

The dynamic replicator equation of Player B is obtained from (46) given as:

$$ \begin{array}{lllllll} & \frac{dq}{dt}=\overline{\Delta}_{2}(q)=q \times (U_{L_{CH}} - \overline{U}_{L})\\ & \qquad =q(1-q)[H_{b} + 2 \times \delta_{1} - p(\delta_{1} + H_{b} + r_{1} \times \delta_{1})] \end{array} $$

(46)

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