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A Game Theoretic Approach to CSMA/CA Networks Davood Shamsi

Abstract In today’s dynamic word, valuation of throughput is not similar for all users in a network; some users are willing to pay more for a given throughput than others. The network manager should grantee a higher rate for users with higher valuation to increase its revenue. In this project, we propose using a VCG mechanism in assigning backoff times (in 802.11 context) to achieve this goal optimally. In a case study, we show that revenue can be increased by 50% to 70% using VCG mechanism compared to traditional network with symmetric nodes.

I. I NTRODUCTION Consider N users that communicate which share a common communication media (e.g., a wireless scheme). There is one server that each user want to talk with. If users transmit simultaneously, the transmitted packets would collide and both packets are lost. Users should behave rationally to accommodate the transmission. A common practice to solve collision problem is the binary exponentially backoff. If a user sends a packet and transmission fails due to collision, it waits a random time W and retransmit again. If transmission was not successful again, the user increases the waiting time. If all users follow binary exponentially backoff, then a minimum rate is guaranteed for all users. This procedure in used in IEEE 802.11 protocol. However, there is no reason that make all users to follow the same backoff. If a user is willing to pay more, we should be able to provide him/her a higher rate. The higher rate can be achieved by letting the user to wait for a shorter time (a shorter contention windows). The main question here is how to set the price for different waiting times. If we setting too cheap, then every one would asks for a short waiting time and the system would collapse. The expensive price would keep users away from the short waiting times and then channel would be free for most of the time. The price should be set such that total social welfare is maximized. Cagalj et al. [7] study the cheating opportunities in the CSMA/CA networks. They model the cheater users in the presence of normal users. The cheating users can keep their waiting time (after March 14, 2010

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the collision) constant while normal users follow the binary exponentially backoff. They show that by cheating, a user can increase its rates by more than 100 times! They study the Nash equilibrium of the game and show that total throughput of the system is very low. Altman et al. [8] study retransmission in slotted ALOHA. They analyze equilibrium and optimal retransmission probabilities. A retransmission pricing mechanism is introduced to reach a better Nash equilibrium. In another work, WeiZhao et al. [6] address individual selfish behavior in the packet routing. They show that VCG is not an appropriate payment mechanism in this problem and they introduce a mechanism that incur an equilibrium with high social welfare. For more resources on application of game theory in wireless network see [4] [3] [1] [2] [4].

II. M ODEL We assume N users send data to a receiver. Time is slotted and they can send a new message at times ti , i = 1 . . . ∞ (Here assume ti = i). For simplicity we assume all users are symmetric and have the same channel and data rate. i.e., at start of each time slot ti , each user can start sending a message. It takes δ seconds to transmit the message block. If no one else sends any message between ti and ti + δ, then message can be received by base station (receiver) correctly; otherwise collision would happen and both messages would be lost. A waiting time Wi is assigned to each user. If collision happens, all users who participated in collision would back off for a random time with mean 2Wi . In the next consequent collision, the user back off for a random time with mean 4Wi and so on. After a successful transmission, the back off time is reset to Wi again. Users with small waiting time Wi with high probability have higher rates. Thus, if someone is willing to pay more, he/she should have a higher data rate. Thus, we assume a bidding mechanism for waiting time allocation. Traffic Model: We assume at beginning of each time slot, every user might start transmitting a new message with probability p. (if it is not already sending a message). Each user i specifies a waiting time Wi and the price pi which is willing to pay for the waiting time. The auctioneer decides to accept some users and deny others. Users should be accepted (to transmit) such that total welfare is maximized. The total welfare is defined as total throughput of the system. I am planning to use VCG mechanism for pricing and allocation. In this view, we allocate waiting time to users until we reach the limit of the system (adding one more user would decrease total throughput of the system). Then, we charge each user based on his/her effect on the network. Utility of each user can be defined as a function of its rate and the price pays to the auctioneer. The mechanism should be designed such that users do not violate the constraints. Note, as suggested in [5], the server (receiver) can force users to use specific waiting time(contention window). In this model, a user can bid for different waiting times. March 14, 2010

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To find optimal number of users in the system (considering their waiting times), I might use analytical results in this area or simulation. I have to read more about this problem and figure out if the analytical results already available can be applied in this scenario.

Fig. 1.

A MAC system with N users.

III. VCG M ECHANISM First, I explain settings of the Vickrey Clarke Groves (VCG) mechanism. Consider a set W = {w1 , w2 . . . wn }. Each wi is an allocation of backoff times. i.e., wi = (ti1 , ti2 , . . . tiM ) where tij is backoff time of j-th user when backoff assignment wi is chosen. Each agent j has a private valuation for each out come. We show valuation functions by Vj : W → +

R . Valuation function is private and each user just know its own valuation function. This valuation function is also unknown for auctioneer (The agent who assign backoff times). In order to assign backoff times, each agent provides a value function Vˆj . Note, they are free to report whatever value function they prefer; not necessary their own private value function. Then, the auctioneer decide which outcome w ∈ W should be chosen to maximize social welfare. Social welfare is defined by: S(w) =

M ∑

Vˆj (w)

j=1

. For example value function of j-th user, Vj (w) can be throughput of the user times the price it is willing to pay for unit throughput (say kbps). Note that throughput of the j-th user is a function of backoff assignment to it wj and backoff time assigned to other users wk , k ̸= j. Utility of a user for a specific assignment w ∈ W is its value function in w plus price it has to pay for it, pj . i.e., Vj (w) + pj . March 14, 2010

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The auctioneer should charge each user such that they reveal their private valuation function. i.e., pricing pj should such that the best policy of each user is to reports its true valuation function. Such mechanism are called truthful. Payment of users is determined by following statement. pj =

∑

Vk (w(−j) ) −

k̸=j

∑

Vk (w)

k̸=j

where w(−j) is same assignment as w expect assigning infinity large backoff time to user j. In particular, we charge each user based on its effect on the market.

IV. VCG

FOR

CSMA/CA N ETWORKS

Now, we apply VCG framework to CSMA/CA Networks. First, we should determine value functions. If w ∈ W is a backoff time assignment then value function of the j-th user would be its throughput times its unit throughput valuation. Unit throughput valuation might be different for different users. Some user are willing to pay more for a fixed throughput than other users. For example, if you are making an important transaction over the internet you are willing to pay more than a person who just brows the web to kill time! Efficiency of CSMA/CA networks is widely studied [10]–[14]. However, by best of my knowledge, there is not a complete analysis of the network that addresses networks with multiple nodes, nodes with different backoff time, and when nodes are randomly deployed in the network. Also, hidden terminal is not addressed well. Thus, we can not determine efficiency of the network theoretically. Instead, we use simulation to evaluate performance of the network in different setting. The simulation is based on a network with 5 nodes. As you can see in Figure 2, nodes are uniformly distributed in the unit disk. We have following settings in our simulation: 1) Time is slotted. We run simulation for T = 10000 time steps. 2) There are 5 nodes and one base station. All nodes send packets to the base station. 3) At each node, packets are generated in the beginning of the time slot with probability p (which is the same for all nodes). Having the same packet generation probability is not an essential assumption and can be removed if it is necessary. 4) Each customer has a different valuation for transmission rate. In our simulation these number are randomly drawn from uniform distribution. Note, in the last section, we mention that users have different valuations for different sets of backoff time. While we can continue having the same assumption, for simplicity, we slightly changed the assumption. Now, valuation of each set of backoff times is just function of the user’s throughput. 5) All packets have length equal to 5. 6) In case of collision, back of time is doubled.

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1 Nodes Base Station

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

Fig. 2.

−0.8

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Configuration of the nodes and the base station in our simulation

7) Maximum backoff time is wM AX = 500 8) To reduce complexity of simulation, we assume each user can bid on backoff time 1 and 2. 9) If two nodes transmit simultaneously, they interfere and collision happens. If distance between two nodes is less than 1, they can hear each other and avoid collision but if they are far they can not hear each others signal (hidden terminal). Using above simulation, we can find transmission rate of each node for a given set of backoff times. Thus, we can find throughput of each user for all possible backoff time assignment. Base on the VCG mechanism, payments would be pj = F(w−(j) ) − F(w) where w is set of selected backoff times (such that maximizes NPV) and w−(j) is set of selected backoff times setting backoff time of j-th user to infinity. F(.) is given by above simulation. Figure 3 compares performance of different schemes. VCG mechanism chooses nodes optimally. ”All user” mechanism forces all users transmit with large backoff time and ”random users” mechanism randomly chooses users with small backoff time. The horizontal axes is traffic rate of nodes (p) and the vertical axes is total NPV (throughput times price). When p is small, almost there is no collision and everyone can transmit. Thus, all three mechanism are similar. As traffic increases, we should allocate more transmission rates to users who are willing to pay more. Thus, VCG starts to gain compared to others. When p is increased from zero to 0.1, first all schemes are similar. For a short interval, ”all users” March 14, 2010

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performs better than ”random user” scheme since it can increase total throughput. Finally, ”random user” scheme outperforms ”all user” scheme since ”all user” scheme has large backoff times.

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Fig. 3.

Performance of VCG vs ”all user” and ”random user” approach

NOTE: MATLAB codes are provided in the project homepage.

V. P OSSIBLE E XTENSIONS Here we considered a simple model for nodes and transmission. These assumptions are not exactly compatible with the practice. It would be worthy to consider a model with more practical assumptions or even implement the VCG mechanism in a real network to compare the performance. The complexity of VCG computation is very high. It is not practical to find VCG optimal mechanism for a network with large number of users. Providing heuristics and approximation algorithms is a key point to make the model implementable. Another issue to address is bidding mechanism. While auctions are well studied and applied in the market, it is not clear how they can be implemented in wireless ad hoc networks.

VI. C ONCLUSION We studied CSMA/CA networks using a game theoretic approach. Specifically, we used VCG mechanism to provide a truthful mechanism for backoff time assignment. In this context, performance of the network is measured by valuation of throughput for users. Our analysis shows that one can increase total efficiency of the network using VCG mechanism.

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R EFERENCES [1] M. Xiao, N. Shroff, E. Chong, Utility-Based Power Control in Cellular Wireless Systems, Proceedings IEEE INFOCOM’2001, pp. 412-421 [2] M. Raya, J. P. Hubaux, I. Aad, ”DOMINO: A System to Detect Greedy Behavior in IEEE 802.11Hotspots” in Proceedings of MobiSys 2004, June 2004 [3] T. Alpcan, T. Basar, R. Srikant, E Altman, CDMA uplink power control as a non-cooperative game, Proceedings of the 40th IEEE Conference on Decision and Control, pp. 197-202, December 2001 [4] M. Felegyhazi and J.-P. Hubaux, Game theory in wireless networks: A tutorial, EPFL Laboratory for Computer Communications and Applications, Lausanne, Switzerland, Tech. Rep. LCA-REPORT-2006-002, June 2006. [5] P. Kyasanur and N. Vaidya. Detection and handling of MAC layer misbehavior in wireless networks. In Dependable Systems and Networks, June 2003. [6] Wang, WeiZhao and Li, Xiang-Yang and Wang, Yu, ”Truthful multicast routing in selfish wireless networks”, obiCom ’04: Proceedings of the 10th annual international conference on Mobile computing and networking, 2004, 245–259 [7] M. Cagalj, S. Ganeriwal, I. Aad, J.-P. Hubaux, ”On Selfish Behavior in CSMA/CA Networks”, IEEE INFOCOM, 2005, VOL 4, pages 2513-2524 [8] E. Altman and R. El-Azouzi and T. Jimenez, ”Slotted Aloha as a stochastic game with partial information”, In Proceedings of WiOpt’03, Sophia-Antipolis, France, 3-5, March 2003. [9] M. Felegyhazi, J.-P. Hubaux and L. Buttyan, ”Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks” in IEEE Transactions on Mobile Computing (TMC), volume 5, number 5, May 2006 [10] Eustathia Ziouva and Theodore Antonakopoulos, ”CSMA/CA performance under high traffic conditions: throughput and delay analysis” Computer Communications, Volume 25, Issue 3, 15 February 2002, Pages 313-321 [11] F. Cali, M. Conti, and E. Gregori, ”IEEE 802.11 wireless LAN: Capacity analysis and protocol enhancement,” presented at the INFOCOM’98, San Francisco, CA, Mar. 1998. [12] G. Bianchi, L. Fratta, and M. Oliveri, Performance analysys of IEEE 802.11 CSMA/CA medium access control protocol, in Proc. IEEE PIMRC, Taipei, Taiwan, Oct. 1996, pp. 407411. [13] Y.C. Tay and K.C. Chua, ” A Capacity Analysis for the IEEE 802.11 MAC Protocol” Wireless Networks, Volume 7, Number 2 / March, 2001 [14] Frederico Cal and Marco Conti and Enrico Gregori, ”Dynamic tuning of the IEEE 802.11 protocol to achieve a theoretical throughput limit” IEEE/ACM Transactions on Networking (TON), Volume 8 , Issue 6, 2000

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