Game-theory-based lifetime maximization of multi-channel cooperative spectrum sensing in wireless sensor networks

Abstract

Accurate and efficient detection of the radio-frequency spectrum is a challenging issue in wireless sensor networks (WSNs), which are used for multi-channel cooperative spectrum sensing (MCSS). Due to the limited battery power of sensors, lifetime maximization of a WSN is an important issue further sensing quality requirements. The issue is more complex if the low-cost sensors cannot sense more than one channel simultaneously, because they do not have high-speed Analogue-to-Digital-Convertors which need high-power batteries. This paper proposes a novel game-theoretic sensor selection algorithm for MCSS that extends the network lifetime assuming the quality of sensing and the limited ability of sensors. To this end, an optimization problem is formulated using the “max–min” method, in which the minimum remaining energy of sensors is maximized to keep energy balancing in the WSN. This paper proposes a coalition game to solve the problem, in which sensors act as game players and decide to make disjoint coalitions for MCSS. Each coalition senses one of the channels. Other nodes, that decide to sense none of the channels, turn off their sensing module to reserve energy. First, a novel utility function for the coalitions is proposed based on the remaining energy and consumption energy of sensors besides their detection quality. Then, an algorithm is designed to reach a Nash-Equilibrium (NE) coalition structure. The existence of at least one NE, converging toward one of the NEs, and the computational complexity of the proposed algorithm are discussed. Finally, simulations are presented to demonstrate the ability of the proposed algorithm, assuming the systems using IEEE802.15.4/Zigbee and IEEE802.11af.

Appendices

Appendix 1

We want to define a suitable value function for coalitions in the proposed game such that it conducts the game to a proper NE, which maximizes the $E_{th}$, while it must capture the tradeoff between the GDP, the GFP, the $E_{th}$ and energy consumption. Based on the [31], for a finite OPG, the rank of a strategy is a proper candidate for the value of the strategy. This rank of a strategy is measured by counting the number of other strategies that have lower benefits than the strategy.

Now we use this idea to define a coalition utility for the proposed game. We define zero value for the coalition number of zero, i.e. the coalition of sensors which sense none of the channels. The result of selecting this zero value is that if sensors spend no cost, they receive no gain. Also, if a coalition does not satisfy at least one of the constraints ($P_{{d_{m} }} \left( {J_{m} } \right) < \beta$ or $P_{{f_{m} }} \left( {J_{m} } \right) > \alpha$ or both), its value is defined negatively because forming the coalition causes to consume energy but it cannot provide adequate detection quality. If a coalition satisfies all the constraints ($P_{{d_{m} }} \left( {J_{m} } \right) \ge \beta$ and $P_{{f_{m} }} \left( {J_{m} } \right) \le \alpha$), its value is defined positively.

Based on the lifetime maximization goal of the problem, the value of a coalition must be an increasing function of $E_{th}$ and a decreasing function of the energy consumption of sensors in the coalition. Between coalitions with positive values, the lower $E\left( {J_{m} } \right)$ and higher $E_{th}$, the more utility. So when a player wants to make a decision, it first determines the effect of its decision on the $E_{th}$. If its decision increases the $E_{th}$ in respect to the previous round of the game (the minimum of remaining energy of sensors after the previous sensor has played is denoted by $E_{th,0}$), selecting the decision causes more utility. If its decision does not change the ${\text{E}}_{\text{th}}$ level of sensors, the utility of selecting the decision only depends on the $E\left( {J_{m} } \right)$; hence, the decision with the lower $E\left( {J_{m} } \right)$ has more utility. Of course, these decisions have a lower utility than the decision which increases the $E_{th}$. Therefore, we have defined the utility of the coalition when the sensor decision provides $E_{th} > E_{th,0}$ as $1 + E_{th}$, which has a measure greater than one. Then, the utility of the coalition when the sensor decision provides $E_{th} = E_{th,0}$ as $\frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)}$, which is:

$$\frac{{min} \left( E \right)}{{max} \left( E \right) - {min} \left( E \right)} < \frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)} < 1$$

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The strategies that a sensor decision reduces the $E_{th}$ and causes to $E_{th} < E_{th,0}$ has the lower utility, hence we define its value as $\frac{0.5 {min} \left( E \right)}{{max} \left( E \right) - min\left( E \right)}\frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)}$, because:

$$0 < \frac{0.5 {min} \left( E \right)}{{max} \left( E \right) - {min} \left( E \right)}\frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)} < \frac{0.5 {min} \left( E \right)}{{max} \left( E \right) - {min} \left( E \right)} < \frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)} < 1$$

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A similar ordering of coalition values is done for the strategies that have a negative value. If a sensor decision increases the $E_{th}$ (i.e. $E_{th} > E_{th,0}$), selecting the decision has more utility. Therefore, we have defined the utility of the coalition as $\frac{ - {min} \left( E \right)}{{max} \left( E \right)}\frac{{E_{0} - E_{th} }}{{E_{0} }}$, which has a measure lower than zero, in which ${\text{E}}_{0}$ denotes the initial energy of sensors. The upper and lower bounds of the utility value are:

$$- 1 < \frac{ - {min} \left( E \right)}{{max} \left( E \right)} < \frac{ - {min} \left( E \right)}{{max} \left( E \right)}\frac{{E_{0} - E_{th} }}{{E_{0} }} < 0$$

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Then, the utility when the sensor decision provides $E_{th} = E_{th,0}$ as $\frac{{ - E\left( {J_{m} } \right)}}{{max} \left( E \right)}$, which is:

$$- 1 < \frac{{ - E\left( {J_{m} } \right)}}{{max} \left( E \right)} < \frac{ - {min} \left( E \right)}{{max} \left( E \right)}$$

(19)

The strategies that a sensor decision reduces the $E_{th}$ and causes to $E_{th} < E_{th,0}$ has the lower utility, hence we define its value as $- 1 - \frac{{E_{0} - E_{th} }}{{E_{0} }} - E\left( {J_{m} } \right)$, because:

$$- 3 < - 1 - \frac{{E_{0} - E_{th} }}{{E_{0} }} - \frac{{E\left( {J_{m} } \right)}}{{max} \left( E \right)} < - 1$$

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Finally, the utility value of every coalition is defined as:

$$u\left( {J_{m} } \right) = \left\{ {\begin{array}{*{20}l} {if\;P_{{d_{m} }} \left( {J_{m} } \right) \ge \beta ,\;P_{{f_{m} }} \left( {J_{m} } \right) \le \alpha \;\left\{ {\begin{array}{*{20}l} {1 + E_{th} ;} \hfill & {if\;E_{th} > E_{th,0} } \hfill \\ {\frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)};} \hfill & { if\;E_{th} = E_{th,0} } \hfill \\ {\frac{0.5 {min} \left( E \right)}{{max} \left( E \right) - {min} \left( E \right)}\frac{{{max} \left( E \right) - E\left( {J_{m} } \right)}}{{max} \left( E \right) - {min} \left( E \right)}} \hfill & {if\;E_{th} < E_{th,0} } \hfill \\ \end{array} } \right.} \hfill \\ {if\;P_{{d_{m} }} \left( {J_{m} } \right) < \beta \;or\;P_{{f_{m} }} \left( {J_{m} } \right) > \alpha \; \left\{ {\begin{array}{*{20}l} {\frac{ - {min} \left( E \right)}{{max} \left( E \right)}\frac{{E_{0} - E_{th} }}{{E_{0} }};} \hfill & {if\;E_{th} > E_{th,0} } \hfill \\ {\frac{{ - E\left( {J_{m} } \right)}}{{max} \left( E \right)} ;} \hfill & { if\;E_{th} = E_{th,0} } \hfill \\ { - 1 - \frac{{E_{0} - E_{th} }}{{E_{0} }} - E\left( {J_{m} } \right);} \hfill & {if\;E_{th} < E_{th,0} } \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right.$$

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Appendix 2

We define the following potential function for the proposed game:

$$\varPhi \left( {J_{0} , \ldots ,J_{M} } \right) = \mathop \sum \limits_{l = 1}^{M} u\left( {J_{l} } \right)$$

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in which $J_{l}$ is the current coalition of sensors that cooperatively sense channel l. Given any coalition $J_{m}$, an improvement step of player n is a change of its strategy from $a_{n} = m$ to $a_{n} = m^{\prime}$, such that the reward utility of player n increases. This move is performed in two possible ways. The first way is adding to the coalition $m^{\prime}$ when other nodes in the coalition remain. In this ways, the player n does the move when he receives more reward, i.e.:

$$Re_{n} \left( {J_{{m^{\prime}}} + \left\{ n \right\}} \right) > Re_{n} \left( {J_{m} + \left\{ n \right\}} \right) \to u\left( {J_{{m^{\prime}}} + \left\{ n \right\}} \right) - u\left( {J_{{m^{\prime}}} } \right) > u\left( {J_{m} + \left\{ n \right\}} \right) - u\left( {J_{m} } \right)$$

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Now the potential function of the game before and after the move of player n are called as $\varPhi_{1}$ and $\varPhi_{2}$, respectively. The measures of potential function are calculated as:

$$\varPhi_{1} = \mathop \sum \limits_{l = 1}^{M} u\left( {J_{l} } \right) = u\left( {{\text{J}}_{{m^{\prime}}} } \right) + u\left( {J_{m} + \left\{ {\text{n}} \right\}} \right) + \mathop \sum \limits_{{\begin{array}{*{20}c} {l = 1} \\ {l \ne m,m^{\prime}} \\ \end{array} }}^{M} u\left( {J_{l} } \right)$$

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$$\varPhi_{2} = \mathop \sum \limits_{l = 1}^{M} u\left( {J_{l} } \right) = u\left( {{\text{J}}_{{m^{\prime}}} + \left\{ {\text{n}} \right\}} \right) + u\left( {J_{m} } \right) + \mathop \sum \limits_{{\begin{array}{*{20}c} {l = 1} \\ {l \ne m,m^{\prime}} \\ \end{array} }}^{M} u\left( {J_{l} } \right)$$

(25)

Therefore, the change in the potential function of the game is calculated as:

$$\Delta \varPhi = \varPhi_{2} - \varPhi_{1} = u\left( {{\text{J}}_{{m^{\prime}}} + \left\{ {\text{n}} \right\}} \right) + u\left( {J_{m} } \right) - u\left( {{\text{J}}_{{m^{\prime}}} } \right) - u\left( {J_{m} + \left\{ {\text{n}} \right\}} \right) > 0$$

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The second way for the player n is adding to the coalition $m^{\prime}$ when a node $n_{0}$ in the coalition removes. This move is done if:

$$\begin{aligned} & Re_{n} \left( {{\text{J}}_{{m^{\prime}}} + \left\{ n \right\} - \left\{ {n_{0} } \right\}} \right) \ge Re_{n} \left( {J_{m} + \left\{ {\text{n}} \right\}} \right) \\ & \quad \to u\left( {J_{{m^{\prime}}} + \left\{ n \right\} - \left\{ {n_{0} } \right\}} \right) - u\left( {J_{{m^{\prime}}} } \right) > u\left( {J_{m} + \left\{ n \right\}} \right) - u\left( {J_{m} } \right) \\ \end{aligned}$$

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Now the measure of $\varPhi_{2}$ is calculated as:

$$\varPhi_{2} = \mathop \sum \limits_{l = 1}^{M} u\left( {J_{l} } \right) = u\left( {J_{{m^{\prime}}} + \left\{ n \right\} - \left\{ {n_{0} } \right\}} \right) + u\left( {J_{m} } \right) + \mathop \sum \limits_{{\begin{array}{*{20}c} {l = 1} \\ {l \ne m,m^{\prime}} \\ \end{array} }}^{M} u\left( {J_{l} } \right)$$

(28)

Therefore, the change in the potential function of the game is calculated as:

$$\Delta \varPhi = \varPhi_{2} - \varPhi_{1} = u\left( {J_{{m^{\prime}}} + \left\{ n \right\} - \left\{ {n_{0} } \right\}} \right) + u\left( {J_{m} } \right) - u\left( {{\text{J}}_{{m^{\prime}}} } \right) - u\left( {J_{m} + \left\{ {\text{n}} \right\}} \right) > 0$$

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Hence; an improvement step of an individual player increases also the potential function, in both possible ways. This is concluded that the proposed game is an OPG.