1. Introduction

The most widespread notion of a wireless ad hoc network is a network formed dynamically by autonomous systems of heterogeneous wireless devices interfaced through wireless links without an existing network infrastructure and without any central coordinator. The decentralized nature and quick deployment properties of these networks make them suitable for emergency and military application as well as any other communication related application. In future, it is possible to operate these networks in the unlicensed band without strict rules and regulation. The increasing demand for CDMA based wireless ad hoc networks has generated a number of issues. One such important issue is transmission power control that not only saves the battery power but also increases the system capacity by influencing the proper utilization of radio frequency. Since radio frequency is a scarce resource, it is advised to reuse the same frequency band at some distance to increase the system capacity. In order to reduce the reuse distance, it is needed to reduce the interference which, in turn, requires implementing transmission power control in a system [1].

The wireless devices, called nodes, take part in these networks are battery power operated and having an intention of using maximum transmission power for gaining higher signal to interference plus noise ratio (SINR) for better services. This tendency of a node makes its neighboring nodes to face higher interference. As a result, neighboring nodes also raise their transmission power to combat the posed interferences. All nodes, thereby, drain their battery power soon. There may be a solution to reduce the power level, but on the cost of compromising QoS [2]. Therefore, there is a need of minimizing interference by optimizing transmission power to increase channel capacity as well as to extend battery life.We found an optimum SINR in game theoretical framework and investigate how power management reduce the interference and increase energy efficiency, in turn, QoS in wireless ad hoc networks.

Many different approaches have been suggested on the power control issue in wireless networks, in general, and wireless ad hoc networks, in particular [3]. Recently an alternative approach in game theoretic framework drags the attention of researchers to study and analyze the power control problem in wireless ad hoc and sensor networks. Game theory is an art of decision making tool that helps several players, each having its own individual objectives, to increase their profit in a competitive situation. In some cases, like a wireless network, there may be a common goal, such as network performance, that has to be maximized by all players. In this case, some players may compromise with their personal objective and contend with each other under a set of rules. In a non cooperative situation the game finds an equilibrium solution at which each player is at a local maximum and it cannot do better by a unilateral deviation [4].

In this approach, the network operation is considered as a non cooperative game where the nodes of the networks are players who play the game to maximize their profit by applying some action plan in response to other nodes’ actions. In ad hoc networks scenario, the action plan could be the utilization of system resource, such as increasing transmission power, to get higher SINR or higher throughputs as their utility function. Since all the nodes want to maximize their own utility, the total utility of networks is typically difficult to achieve due to the rational behavior of the nodes. The purpose of applying game theory is to maximize total networks’ utility rather than individual node’s utility. To mitigate this problem, therefore, the game enforces price that causes the marginal loss in utility due to a marginal increase in interference. Now nodes change their behavior by knowing their profit and loss thus eventually the game reaches an equilibrium solution referred to as Nash Equilibrium, where all players have to use optimum transmission power to obtain networks’ goal.

Energy efficiency issue is well addressed in cellular networks and various power control algorithms are developed for cellular networks. It is of its first kind, an embodied energy is used in modeling the energy efficient cellular networks in [5]. The fast growth of data communication in cellular networks leads to a huge operating cost which can be reduced by designing energy efficient cellular networks. In [6] distribution of traffic load and power utilization is considered to model a Poisson-Voronoi tessellation (PVT) cellular networks to reduce the load of working outlay. The effect of interference on spectrum and energy efficiency is a critical issue in all form of wireless networks such as cellular networks and femtocell networks [7]. A Markov chain is used to determine the channel access in two tier femtocell networks and energy efficiency is analysed [8].Wireless body area network is a new wireless network with constrained energy. A new energy efficient relay mac protocol is proposed in [9] to save energy using dynamic power control mac methodology.

A Substantial amount of research on the power control issue in wireless networks has been done using game theoretic approach. Chao Liang and Kapil R. Dandekar [10] considered the power control problem in stationary MIMO ad hoc networks as a game theoretic problem and they devised a link shut down mechanism, if an inefficient link consuming more power. The link shut-down mechanism obviously increases the total networks’ performance, but it is a contentious approach as many people don’t want to get disconnected even if they have extremely low data rates. Joint beam forming and power allocation problem in MIMO cognetive radio is studied in [11] using game theory to avoide the excessive interference for primerary user.

The wireless sensor networks, a part of an ad hoc network, also consists of battery operated powered device, are used for sensing the environmental changes, data processing and communicating wirelessly [12]. The data processing in sensor networks consumes more power, therefore, transmission power control is equally important in wireless sensor networks too. To solve the distributed power control problem in wireless sensor networks, Shamik Sengupta, et al. framed a utility function as a non cooperative game and suggested ‘not to transmit’ if the channel condition is not suitable for transmission [13]. They advised to fix a threshold power level with price mechanism to punish the nodes that transmit above threshold limit of power level. Chengnian Long et al. applied SFP theory, instead of pricing mechanism, in non cooperative power control algorithm to gain requisite quality of services by scheduling and controlling the power level of users in ad hoc network [14]. The authors recommended a self incentive scheme which makes the user decide based on its own belief without falling in a selfish behavior. Nash Equilibrium is an important criterion to get the optimum solution in a game. But sometimes it is possible to increase users’ payoff beyond Nash Equilibrium. In this case the Pareto optimal theory is being used to measure the efficiency of an outcome which is defined as “An outcome is a Pareto optimal if and only if, there exists, no other outcome that makes every player at least as well off while making at least one player better off”. Using the Pareto optimality theory, Mehdi Rasti et al. provided a trade-off solution between fairness and aggregate throughput using a SIR based linear pricing scheme in wireless networks [15]. Recently, Mehdi Monemi et al. proposed a Pareto efficient distributed multiple target SINRs tracking power control algorithms. According to this algorithm, if a user cannot achieve target SINR, it selects some lower target SINR to continue transmission instead of get disconnected [16]. Game theory has been successfully applied to solve power control, rate control and spectral efficiency issues in cellular networks. Madhusudhan et al. used non cooperative game to provide pliable transmission rate by utilizing radio resource efficiently using the power and rate control in uplink single cell CDMA system [17]. Eirini et al. further accosted a convex pricing to balance the throughput and power consumption to maximize utility of each user in wireless networks [18]. The power control problemis again investigated by Yuan hang Xiao et al. using the concept of intervention device to monitor and accordingly, adjust the power level of selfish users in single hop ad hoc networks to get the desired result [19].

There exist several literatures on power control issues in wireless ad hoc networks. Most of the proposed works are framed in non cooperative strategic form game with incomplete information. This work also comes under the non cooperative strategic form game with pricing since the characteristics of ad hoc networks are very much similar to a non cooperative game. But the pricing scheme proposed in this work is different from many previously proposed pricing schemes. Maximum proposed pricing schemes are linear pricing scheme which charges a common price to all the nodes irrespective of the channel condition and their distances from receivers whereas this adaptive pricing scheme can be tuned according to transmission feasibility.

The rest of the paper is organized as follows. The next section provides the system model that comprises network topology and SINR model. The utility function is developed in section 3 using non cooperative power control game and validated analytically by proving the existence of Nash equilibrium in section 4. A new proposed pricing scheme is introduced in section 5 to obtain the net utility. Finally, the proposed NCPCG is simulated and evaluated in section 6 followed by the conclusion in section 7.

2. System Model

For easier reference the remaining works begin with the major notations, used in this paper, are listed in Table 1.

Table 1.Major Notation

A single hop wireless ad hoc network is considered, shown in Fig. 1 (a), with several nodes located in a neighborhood of a node. The SINR model shown in Fig. 1 (b) reveals that the transmission from T1 interferes R2 and R3.

Fig. 1.Network Topology: (a) A node is surrounded by N−1 nodes, (b) SINR model where transmitter 1 interfere receiver 2 and receiver 3.

The framework presented here is based on SINR calculation. To obtain SINR, many sets of power vectors P1,P2,…,PN for nodes 1,2,…,N are assumed where each set consists of power levels ranging from pmin to pmax. Let the node 1 selects power level p1 from set P1, node 2 selects power level p2 from set P2, and so on, then the power profile can be define as p={p1, p2,…,pN} ∈ known as space of power profile and Pi is the set of power level of node i. If all the nodes are considered to be identical, then the power profile of all nodes will also be identical i.e., . Suppose that nodes transmit packet at the rate of R bits/second over a bandwidth of W Hz, then the signal to interference plus noise ratio of link (i, j) is defined as:

Where

and ηj is the background noise at the receiver. Assuming the background noise is additive white Gaussian noise (AWGN) then ηj= σ2. Thus, the SINR at receiver is given by:

In dB, SINR can be obtained as:

The term hi,j is the path gain between the transmitter (say node i) and the receiver (say node j) and can be calculated using simple path loss model where K is a constant and ∝ is the path loss exponent [20]. The distance, di,j, between node i and node j can be found as:

Where x and y indicate coordinate positions of the nodes. The numerator term in equation (2) indicates the received power at the jth node, whereas denominator is the total interferences caused by the other nodes’ received power at the jth receiver plus noise power at the receiver.

4. Utility Function

Utility function in the proposed model is adopted from the data transmission model of D.J Goodman and N.B. Mandayam [21]. Data are transmitted in packets, each containing L bits of information plus overhead bits, at the rate of R bits/sec. A strong error detection code is assumed to be used for detecting errors and retransmitting packets, if error found, until the receiver receives error free data. Including error detecting code, the size of packet becomes M bits, where M > L and (M – L) bits are used for error detection code. The number of retransmission required to receive a packet correctly is a function of geometric random variable X. Let all retransmissions are statistically independent then the mean value of X is equal to E[X]= 1/q(γi,j) where γi,j is the SINR at node j (explain in previous section) and q(γi,j) is the probability of correct reception of data transmitted by node i. If the duration of each transmission is calculated as M/R seconds then the expected total transmission time for correct reception is E[X]M/R seconds. With transmitted power pi watts the expected energy consumed during transmission is E[X]piM/R joules. But the actual data transmitted is only L bits; therefore, the utility function is modeled as:

Where

and Pe is the bit error rate that depends on many factors such as channel condition, interference from other nodes and various modulation schemes.

Modulation schemes used in wireless networks are either coherent or non-coherent. Coherent techniques, such as, FSK, BPSK, QPSK and MSK, exhibit excellent performance, but they require prior knowledge of the phase information of the carrier signal which is impractical in a wireless environment. On the other hand, the performance of non-coherent technique, DPSK and FSK, are almost equal to coherent PSK and coherent FSK [22]. Moreover, they are robust with respect to amplitude variation hence they are widely accepted schemes in practices. Both the modulation schemes (DPSK and NCFSK) are suitable for the proposed utility model, albeit, NCFSK is more suitable for the pricing scheme proposed in this paper.

Hence, by selecting non-coherent FSK modulation scheme and substituting its BER Pe=0.5e−γi,j/2 in equation (5), the packet success probability is expressed as:

The utility function defined in equation (4) has a mathematical anomaly. If Pi=0, the utility becomes infinity. To avoid this problem, q(γi,j) can be modified in such a way that it could give a close approximation of the packet success rate given in equation (6) and also it should yield f(0) = 0 and f(∞) = 1. Thus, the efficiency function is now defined as:

This function gives the desirable properties, e.g., for γi,j = 0; f(0) = 0 and for Also f(γi,j)/Pi = 0 for Pi = 0. The close behavior of PSR and efficiency function is depicted in Fig. 2. Finally the modified utility function is formulated as [23]:

Fig. 2.Close approximation of packet success rate and efficiency function with respect to SINR.

Equation (8) represents the total number of information bits that are transmitted to the receiver without an error per joule of energy consumed and well balances between throughput and battery life. The similar shapes of q(γi,j) and f(γi,j) ensures that a power profile that maximizes utility given in equation (8) will be close to the power profile that maximizes the utility given in equation (4). Also, it is observed that the utility function obtained in equation (8) by each node is a function of its own power level plus the power level selected by other nodes.

5. Existence of Nash Equilibrium

Non-cooperative Game tries to predict stable outcomes of a competition that is defined by Nash equilibrium. It is defined based on the best action plan chosen by a player to maximize its utility for a given action profile of other players [24][25]. Specifically, in wireless ad hoc network, is the best response by node i to p−i if

Definition: A power profile is a Nash Equilibrium for non cooperative power control game G = if, for every i∈N.

At Nash equilibrium, for the given power levels of other nodes, no node can improve its utility by changing the power level of its own. However, such a point is not guaranteed to exist in a game. The sufficient conditions at which a unique Nash equilibrium is guaranteed to exist are delineated in following theorem.

Theorem: A Nash equilibrium exists in game G= for ∀i∈N if:

(i) The power profile Pi is a nonempty, convex and compact subset of some Euclidean space Rn.

(ii) ui(P) is continuous in P and quasi-concave in pi.

Proof: To satisfy the above conditions it is required to calculate the power that maximizes the utility of node i by equating the first order derivative of ui to zero [25]. The first order derivative of ui from equation (8) is as follows:

Where

Equation (11) yields (∂ui)/(∂pi ) = 0 for pi=0. Therefore, pi=0 is a stationary point at which the utility becomes zero. With some small positive value of pi, say ε, the utility becomes positive, that means, the utility is increasing at pi = 0. But zero cannot be a local maximum. To find a local maximum for a non-zero value of power, further study is required to examine the value of SINR. Equating second part of the equation (11) to zero:

and substituting (12) and (7) in (13) it gives:

The LHS of the equation (15) is monotonically increasing in γi,j, and the RHS is convex in γi,j. Moreover, it is satisfied at γi,j = 0. That leads to the existence of a single value γT which satisfies the equation (15) for γi,j>0, where γT (12.4205 = 10.94 dB for M=80) is numerically derived from the above equation. Substituting γT in equation (11) and taking its second order partial derivative, it reveals that ∂2ui/∂pi2 < 0 and this point is a local maximum and therefore the function is global maximum. Hence, the utility is quasi-concave in pi for ∀i and Pi is nonempty. This completes the proof of existence of a unique Nash equilibrium [25].

6. Net Utility

In the absence of any punishment, the selfish nodes may try to maximize their throughput by consuming high power and cause interferences to other nodes. Thereby, it may result a cascading effect where all nodes lose their energy. In order to restrict a node using a high power level to achieve maximum throughput, the proposed algorithm NCPCG inflicts a pricing function defined as:

where the parameters β and τ are used as control parameters in order that SINR does not deviate from threshold and maintained greater than or equal to 1. Now the net utility of node i can be redefined as:

With this pricing scheme, the node that produces more interference to other nodes gets punished in terms of poor SINR and thereby forced to use less transmission power.

Based on the discussion presented in section 2 through section 6, algorithm is listed in Table 2.

Table 2.Algorithm

7. Numerical Analysis

Existence of Nash Equilibrium for the utility function is proven in section 5. In order to show the effect of the proposed adaptive pricing scheme on net utility, an ad hoc network is set up with 10 nodes (for avoiding the run time delay), deployed randomly in 100 x 100 meters of area. The values of control parameters β and τ and other parameters are summarized in Table 3. For emphasizing on the pricing effect of the proposed work, hidden nodes and exposed nodes are ignored.

Table 3.Major Notation

7.1 Effect of β while keeping τ constant (τ = 10)

With the above network setup, the resulting utility and net utility curves with respect to SINR are shown in Fig. 3 to Fig. 5. It is observed that both the utility and net utility of node i increase as SINR increases.

Fig. 3.Less value of β has negligible effect on utility and net utility.

Fig. 4.The effect of β is visible when increased. Difference between utility and net utility is also increased.

Fig. 5.Both utility and net utility are improved at the higher value of β, however, the difference between these two curves also increased (709.93) when weight is factor increased.

Once SINR obtains its equilibrium state, the utility starts decreasing as expected from NCPCG. Also, it can be noticed that all the upper set of utility curve are convex that illustrates the property of quasi-concavity of utility function. Furthermore the effect of weight factor in pricing scheme is evident from these figures.

The main importance of introducing β is to improve the net utility even if the transmitting node is far away from the receiving node but inside the transmission range. A node far away from the receiving node normally has a small path gain, so it is more likely to get a lower SINR. In this case either transmission may fail or delayed. So by increasing β, it is possible to improve the utility, consequently, net utility, as depicted in Figures for β = 100, 500 and 1000.

7.2 Effect of τ while keeping β constant (β = 500 and β = 5000)

Fig. 6 to Fig. 8 depicts the effect of τ on utility as well net utility function with respect to transmission power. The effect of pricing is evident when τ is set to minimum value (τ = 1) but there could be a further improvement in utility as τ is increased. It is apparent that when τ is nearly set equal to γT at the Nash equilibrium, the utility and hence net utility is raised to the highest value while taking lowest value transmission power. It is not necessary that the net utility will increase always with increasing τ. Even though τ is increased up to 100 (Fig. 8), the utility (also net utility) does not increase while consuming more power compared to τ = 10.

Fig. 6.Utility and net utility with respect to transmission power.

Fig. 7.At τ = 10, the nodes achieve the highest level of maximum utility and net utility at the lowest transmission power.

Fig. 8.When τ is deviated from Nash equilibrium, both utility and net utility decrease and transmission power increases.

Fig. 9 to Fig. 10 further demonstrate the effect of τ on net utility with respect to both SINR and transmission power. As shown in the figures the net utility is maximum when the value of τ is nearer to optimum SINR (γT = 10.94 dB). When a node tries to deviate from optimum SINR unilaterally, it has to pay in terms of reducing its net utility. As a result, the nodes who increase power to get high SINR in order to maximize their net utility, are charged at high prices, in turn, the nodes will rationally reduce their transmit power.

Fig. 9.Net utilities of nodes with respect to transmission power at different level of τ.

Fig. 10.Net utilities of nodes with respect to SINR at different level of τ.

7.3 Comparison with Existing Pricing Schemes

The other pricing schemes, found in various literatures [13], are given in equation (18) along with their respective curves in Fig. 11.

Fig. 11.Curves of various pricing schemes.

Where λ is a positive scalar quantity having the same unit as bits/joule. Fig. 11 exemplified that all these pricing schemes except the proposed pricing scheme (for β = 100 and τ = 10) are increasing function of SINR. The existing pricing schemes shown in equation (18) charges a common price to all nodes, whereas the proposed pricing scheme can be tuned by using adaptive parameter β and τ depend on their attainable SINR. The nodes that use more power to gain higher SINR have to compromise with net utility. As a result, they reduce power, in turn, reduce interference and the purpose of this scheme is fulfilled.

Plot of net utilities at various pricing functions are shown in Fig. 12. The utility curve obtained from the proposed pricing scheme shows nearly 10 % improvement in net utility and nearly 17 % reduction in transmission power as compared to a nearest exponential pricing scheme.

Fig. 12.Comparision curves of net utilities at various pricing schemes.

Finally the net utility verses τ and transmission power verses τ is shown in Fig. 13 and Fig. 14 respectively. It is seen in these figures that at the equilibrium point (τ = 10 or SINR = 0.94 dB) nodes achieves maximum net utility while using minimum transmission power as the objective of this paper.

Fig. 13.Plot for net utility with respect to τ at different level of β.

Fig. 14.Plot for transmission power with respect to τ at different level of β.

8. Conclusion

The control of transmitting power by controlling SINR using game theory offers a synergistic way of reducing interference while improving connectivity. In this approach, the power control problem in wireless ad hoc networks is framed as a non cooperative power control game. A new SINR based pricing scheme with two control parameters is proposed for non cooperative power control game which is validated with the proof of the existence of Nash equilibrium. The improvements of nodes’ utility at minimum transmission power by using control parameters of the proposed pricing scheme are demonstrated using simulation while SINR was kept fixed. It is investigated that with this pricing scheme the nodes are encouraged to use a minimum power level to save their battery power as well as improving channel capacity, in turn, net utility without compromising QoS.