Spectrum sharing via hybrid cognitive players evaluated by an M/D/1 queuing model
© The Author(s) 2017
Received: 3 November 2016
Accepted: 26 April 2017
Published: 10 May 2017
We consider a cognitive wireless network in which users adopt a spectrum sharing strategy based on cooperation constraints. The majority of cognitive radio schemes bifurcate the role of players as either cooperative or non-cooperative. In this work, however, we modify this strategy to one in which players are hybrid, i.e., both cooperative and non-cooperative. Using a Stackelberg game strategy, we evaluate the improvement in performance of a cognitive radio network with these hybrid cognitive players using an M/D/1 queuing model. We use a novel game strategy (which we call altruism) to “police” a wireless network by monitoring the network and finding the non-cooperative players. Upon introduction of this new player, we present and test a series of predictive algorithms that shows improvements in wireless channel utilization over traditional collision-detection algorithms. Our results demonstrate the viability of using this strategy to inform and create more efficient cognitive radio networks. Next, we study a Stackelberg competition with the primary license holder as the leader and investigate the impact of multiple leaders by modeling the wireless channel as an M/D/1 queue. We find that in the Stackelberg game, the leader can improve its utility by influencing followers’ decisions using its advertised cost function and the number of followers accepted in the network. The gain in utility monotonically increases until the network is saturated. The Stackelberg game formulation shows the existence of a unique Nash equilibrium using an appropriate cost function. The equilibrium maximizes the total utility of the network and allows spectrum sharing between primary and secondary cognitive users.
Demand is growing rapidly for wireless communication technologies, such as wireless data links, mobile telephones, and wireless medical technologies. This increasing demand places a significant burden on the limited wireless spectrum. Although the dominant spectrum allocation method (i.e., fixed allocations) is easy to implement, it does not maximize channel efficiency since the license holders (primary users) generally do not utilize their allocated spectrum at all times. A primary approach for increasing the efficiency of spectrum allocation is to allow a second group of unlicensed users to use it when the spectrum is idle. The users who wish to use the spectrum but do not have the primary license are called the secondary users, and they can opportunistically access the channel when the primary user is idle . To facilitate this, we introduce a self-organizing mechanism and assess it by modeling the network as a queue that allows both classes of user to wait in a queue to access the channel modeled as a server.
Game theory has played an important role in developing efficient algorithms for sharing a common spectrum between secondary users . Game theory is the study of cooperation and conflict between cognitive decision-makers, which, in this context, are represented by cognitive radios (a radio that changes its transmitter parameters based on feedback from the environment) in a wireless network . Spectrum sharing via game theory occurs in both licensed and unlicensed bands  and . Cognitive radio networks can be used for spectrum sharing both in unlicensed and licensed bands by using methods that can combine unused frequency bands and share them dynamically [6, 7] and . Heterogeneous wireless systems are an example of unlicensed-band devices that rely on games for spectrum sharing . Cellular operators that use WAN-WiFi are prime candidates for using games to share spectrum in licensed bands. Here, we focus on spectrum sharing in licensed frequency bands with primary users as license holders.
Game theory also plays an important role in deciding how a user must react to an event played by other users in order to maximize its utility (a measure of preferences over some set of strategies) [10, 11]. This decision is made by measuring the user’s throughput (packets successfully sent over some specified time frame) and waiting time as metrics for each player’s measured cost and gain.
Secondary users can be classified into cooperative and greedy players . Greedy players are not cooperative in the sense that their only objective is to maximize their throughput. In , we proposed an “altruistic” user that is cooperative until it senses the presence of a greedy player via observation (for instance, channel usage) similar to . In this situation, the altruistic player will turn into a non-cooperative player to punish the greedy players by jamming the wireless channel. This new altruistic player would subsequently back off when the greedy players act cooperatively with the other players. Adaptive greedy and altruistic players in spectrum-sharing games require an iterative method to study and predict their response. Here, we propose a new equilibrium concept, beyond that of Nash theory, that includes the strategy of a dynamically changing greedy player.
In the literature, spectrum allocation has been modeled with various pricing schemes as a non-cooperative game, with each cognitive radio acting as a player. References  and  propose a price-based spectrum-management system using a water-filling algorithm. Their algorithm employs a distributed pricing procedure that leads to an improved Nash equilibrium solution compared to iterative water-filling . However, our proposed pricing scheme to be used in the utility function, based on primary users only, is intuitively more realistic since the primary users are the license holders. A game-theoretic model is presented in  that achieves the optimal pricing for spectrum sharing based on competition between multiple primary users to give spectrum access to secondary users. However, here, we assume a generalized distributed system that uses a single pricing model for each primary user. Yet, to address the secondary users’ competition to maximize their spectrum access, we offer different pricing functions based on the traffic on the network and other variables such as available spectrum.
An extensive survey presented in  reviews the state-of-the-art and advances in cognitive-radio medium access control protocols. A stochastic geometry framework that captures the performance of an asynchronous ALOHA network in which a subset of nodes operates in full-duplex mode is presented in . Compared to  and , in which an altruistic player can regain access to shared spectrum in an asynchronous ALOHA network,  only allows licensed primary users to access the network. In order to evaluate our game theory modeling approach, we used a queuing analysis that is used in . The opportunistic access used for the performance analysis in  does not consider different cost functions or pricing schemes, number of primary or secondary cognitive users, or congestion. An M/G/1 queuing system (a queue model in which arrivals are Markovian and service times have a general distribution with a single server) containing one primary and multiple secondary users is presented in . Here, we use an M/D/1 queuing system, merely to be used for analysis. Secondary users can gain access to the spectrum through an amplify-and-forward time-division multiple-access protocol. Our method is more generalized in that it supports multiple primary users as well as general cost functions that are not imposing any performance requirement for secondary users such as amplify and forward.
In this paper, we investigate a Stackelberg competition with the primary user as leader and find that in the Stackelberg game, the leader can improve its utility by influencing the follower’s decision using its advertised cost function and the number of followers accepted into the network. For a given stable system and for feasible transmission rate sets, based on the number of primary and secondary users, we find a Nash equilibrium for primary and secondary users. We study a network of cognitive radios competing to access the spectrum that are either cooperative or non-cooperative. We introduce a hybrid player, i.e., one which is both cooperative and non-cooperative. Using a Stackelberg game strategy, we evaluate the improvement in performance of the cognitive players using an M/D/1 queuing model. We use altruism to monitor the spectrum usage and find the non-cooperative players. We also study a Stackelberg competition with primary users as leaders and investigate the impact of multiple leaders by modeling the wireless channel as an M/D/1 queue.
The remainder of this paper is organized as follows. In Section 2, we describe the game with a greedy and normal player and demonstrate that a vigilante player mitigates the impact of a greedy player. We then describe the M/D/1 queuing modeling and the proposed cooperation scheme. In Section 3, we formulate and solve a Stackelberg game with the primary user as the leader and employ a Vickrey auction between secondary users. In Section 4, we provide the numerical results for several communication scenarios and observe the impact of the network parameters in each case. In Section 5, we discuss our results and conclusions.
2 Problem definition
Summary of variables used in paper
Total number of normal players in ith cell
Total number of players that are not vigilante in ith cell
The aggressiveness of the greedy player
The aggressiveness of the vigilante player
Total number of players in network
Total number of cells in network
Total number of players in ith cell
Number of primary players
Number of secondary players
Transmitting probability of greedy player
Transmitting probability of normal player
Transmitting probability of vigilante player
Throughput of greedy player
Throughput of normal player
Throughput of vigilante player
Greedy utility function
Primary utility function
Secondary utility function
Vigilante utility function
Primary waiting time
Secondary waiting time
Share of bandwidth used by primary
Share of bandwidth used by secondary
Packet rate for the primary
Packet rate for the secondary
Server rate or bandwidth
2.1 Vigilante player
where the vigilante player is aware of e g because of the nature of Stackelberg games, and N is total number of cognitive radios in the corresponding cell.
In our model, one must assume that a greedy player is able to move between the geographical cells; in which case, it can move from a cell with an active vigilante player to a cell in which the presence of a vigilante player is unknown. Once moved, then a cooperative player will turn into a vigilante player, and the same cyclic behavior occurs. If the greedy player is static, i.e., not able to move between cells, then it cannot achieve more than its share because of the presence of an active vigilante player. We investigate these behaviors and show that the same cyclic behavior happens in the new cell.
2.2 M/D/1 queueing model
3 System model
Below, we first study a game with three players who desire to maximize their utility functions, each using a unique strategy. Then, we formulate and solve a Stackelberg game for the communication scenario described in Section 2.2 with the primary license holder and secondary users as leader and followers of the Stackelberg game, respectively.
3.1 Game with three players
Without loss of generality, the utility functions defined in Section 2.1 are simplified versions of the utility functions defined in . Based on different pairs of (e v ,e g ), one can see either a cyclic behavior for the throughput of the players  or a Nash equilibrium . For the case of reaching an equilibrium, the vigilante player uses most of the shared bandwidth, which keeps the greedy player from increasing its transmission probability and, as a result, there is no fair resource sharing for cooperative players to use.
By moving from a cell that has an active vigilante player, the greedy player can minimize its utility function. In a distributed cognitive network, a predefined radio node in each cell can be considered/assigned as a vigilante player. For a dynamic greedy player, the measured throughput that is an indicator of e v is used to calculate the best e g and/or best time to move to a new cell.
By introducing a vigilante player and using non-traditional game strategy for decision-making, we hope to improve the performance of a cognitive radio network. To date, the application to cognitive radio networks of a hybrid player, which is both cooperative and non-cooperative, has not been studied significantly. We propose a play strategy (i.e., altruism) to police a wireless network. Using this new player, we will test a series of predictive algorithms to investigate a potential improvement in wireless channel utilization by punishing the non-cooperative players. Then, we will use this strategy to demonstrate the application of a vigilante player in an M/D/1 queue.
The mean value of a received signal in a certain frequency range is an indicator of the presence of a primary user. Since malicious users are more effective in acting in a cooperative manner with other malicious users to change the mean and make a false pretense that a primary user is active, one can suggest finding these users in an iterative manner . With this method, one can find their intention for changing the mean by averaging their advertised signal power and treating them as a separate group inside each cell, which is plausible since one can argue that by introducing a fusion center, the algorithm will be capable of disregarding the malicious users as a group. If a user is falsely accused of being malicious due to multipath fading and/or shadowing, it can be reclassified as a normal user if the weight assignment method is implemented .
3.2 Stackelberg game in M/D/1 queue
where C is the cost function for that pair, for example, the simple cost function defined in Eq. 11. If there is no answer for α p , then the primary user has no motive to share the spectrum because it makes the network unstable.
The Stackelberg game using three types of secondary users introduced in Section 3.1, and a primary user as follower will result in a cyclic behavior. The leader cannot stop a greedy player, instead it will not share the spectrum when the network is saturated, according to Eq. 21. In this scenario, the vigilante player will force the greedy player to move to another cell.
4 Numerical analysis
Here, we present the simulation results of cyclic behavior of the three players’ utility functions introduced in Section 2.1. Then, we introduce the numerical analysis of a Stackelberg game introduced in Section 2.2 with parameters inside the feasible set defined in Section 3.2.
4.1 Cyclic behavior for vigilante player
4.2 Spectrum sharing performance analysis of a Stackelberg game using an M/D/1 queueing model
For the Stackelberg game’s Nash equilibrium analysis, we first present the simulation results analyzed via an M/D/1 queue, with one primary user as the leader, and then extend the results with multiple leaders. We evaluate the utilities of the leader and follower at the equilibria found in Sections 2.2 and 3.2. We omit the equilibria found in the feasible set defined by Eqs. 15, 16, and 17 when the utility function for both leaders and followers yields zero. As shown later in this section, this happens when the network is close to saturation. The available bandwidth is between 40 to 160 kbps (we use actual numbers to compare the results for different scenarios). We vary the remaining parameters, such as the number of primary and secondary users (n p , n s ), cost function, and accessible spectrum μ, in order to assess their impact on the utilities.
Figure 7 shows the relationship between the cost function and varying normalized utility of both players versus the number of primary users. Here, the parameters for our game are n s =3, λ p =10 kbps, λ s = 1 kbps, and μ = 100 kbps. It can be concluded that via a Vickrey auction, we can have different saturation points for the number of secondary users. For Eqs. 11, 22, and 23, we have a saturated network for n p = 5, 6, and 6, respectively.
Figure 8 provides a comparison of the utility functions of both players versus the number of secondary users, where n p = 2, λ p = 10 kbps, λ s = 1 kbps, and μ = 100 kbps. Here, for Eqs. 11, 22, and 23, we have saturation for n s = 8, 9, and 9, respectively. As can be seen in Fig. 8, there will be a cutoff point for the number of secondary users. This means that, no matter what cost function we use, there is a point beyond which the queue will be saturated. By choosing an appropriate cost function, one can modify the maximum number of secondary users admitted in to the network.
Traditional game strategy for cognitive radio networks generally only includes static non-cooperative players. More efficient cognitive radio networks can be constructed by modeling more realistic dynamic players with various goals that lead to different strategies. In this paper, an altruistic cognitive player is introduced to monitor and police the network. A dynamic greedy player and vigilante player in each cell are used to study the cyclic behavior of a game to maximize the throughput of greedy and cooperative (non-vigilante) players, respectively. In our simulations, without loss of generality, we assumed that the network is divided into cells containing the same number of nodes. We assumed a static vigilante player because any cooperative player can sense its throughput and follow an altruistic strategy. We studied the correlation between the number of players in a cell and the aggression factor of a vigilante player with the greedy player’s throughput. The result is used in the study of a Stackelberg game and assessed in an M/D/1 queue.
We studied the spectrum sharing cooperation by modeling the spectrum and users as an M/D/1 queue, with the goal of encouraging the cognitive players to cooperate. We have focused on the system model that, despite the desire to maximize their individual utilities, the cognitive players find it beneficial to cooperate. We have formulated a Stackelberg game in which the primary license holder and secondary user are leader and follower, respectively, and studied how the leader can influence the follower’s decision of participating in the game by varying the cost function. Additionally, we observed that a pricing scheme can be employed to improve all utilities to the social optimality of an M/D/1 queue. In this scenario, cognitive users can employ the cost function to decide how much of the spectrum is used by primary users and secondary users.
A future direction is to study the impact on performance of full and partial knowledge of the game strategies for all players. The partial knowledge is a more realistic study of cognitive radio to be used for wireless transmission. The throughput used by a vigilante player to make the greedy player migrate or cooperate needs to be studied to assess the performance accurately. The complexity of our network can be investigated by modeling it with an embedded Markov chain using an approach similar to that in , which investigated consecutive loss in a simple queue. By introducing cells into their scheme, one can use an approach similar to the one presented in that work to study large networks. Naturally, computational complexity will increase significantly if cognitive radios act in a strategy that is between greedy and hybrid. Investigating these tradeoffs is left as future work.
KK and SGB conceived and designed the study. Both authors read and approved the manuscript.
The authors declare that they have no competing interests.
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