VCG Auction-based Bandwidth Allocation with Network Coding in Wireless Networks

Transcription

1 VCG Auction-based Bandwidth Allocation with Network Coding in Wireless Networks PIRIYA CHAIKIJWATANA TAKUJI TACHIBANA Nara Institute of Science and Technology Graduate School of Information Science Takayama, Ikoma, Nara JAPAN {piriya-c, Abstract: Recently, it has been proven that network coding has a great potential to improve the throughput and robustness in wireless networks by exploit the broadcast nature of the wireless medium. In this paper, we propose Generalized VCG auction mechanism with network coding. Our proposed mechanism is effective for allocating bandwidth to each users in wireless networks so as to maximize the social surplus. In the auction, we allow users to specify theirs willing price per bandwidth unit and amount of request bandwidth. We evaluate the performance of our proposed auction mechanism with simulation. In numerical examples, we show the effectiveness of bandwidth allocation with auction mechanism and effectiveness of network coding. Key Words: VCG auction, wireless networks, bandwidth allocation, network coding, social surplus, optimization problem 1 Introduction In self-organizing multi-hop wireless networks [1], all nodes have to forward packet as relay node. The wireless network bandwidth is shared among each packet forwarding [2]. Because the number of users has been increasing, the demand of bandwidth has been growing exponentially [3]. Therefore, bandwidth allocation is one of the important issues. Several auction mechanisms have been proposed [4, 5, 6] for allocating bandwidth among users. In such mechanisms, social surplus (social welfare) that is defined as the summation of all users utilities in the network, is expected to be maximized. In [4], the downlink bandwidth can be allocated based on auction mechanism in a wireless future Internet network. In [5], a two-level bandwidth is shared among multiple users based on auction in integrated WiMax, UMTS and WLAN networks. [6] has introduced a double-sided auction for allocating resources such as bandwidth and energy among multiple users in wireless and P2P networks. Recently, network coding has been proposed as an excellent technique for improving potential network throughput and robustness in wireless networks [7]. It exploits the broadcast nature of the wireless medium to reduce the number of wireless transmissions, improving the overall network throughput. Therefore, it is expected that network coding is utilized for the auction mechanisms so as to improve the social surplus in the wireless networks. However, as far as the authors know, the utilization of network coding in bandwidth auction has not been considered yet. In this paper, in order to maximize the social surplus for efficient bandwidth allocation in wireless networks, we propose a Generalized Vickrey-Clarke- Groves (VCG) auction mechanism with network coding. In our proposed auction, wireless network bandwidth corresponds to divisible goods. Each user bids the amount of require bandwidth, price per bandwidth unit, and source-destination pair to a service provider (auctioneer). The service provider derives the optimization problem and determines the bandwidth allocation for each user based on the auction mechanism. Moreover, network coding is utilized in order to improve social surplus. We evaluate the performance of our proposed auction mechanism with simulation. Especially, we compare its performance with the existing auction mechanism without network coding. In numerical examples, we investigate the effectiveness of bandwidth allocation with auction mechanism and effectiveness of network coding. The rest of this paper is organized as follows. Section 2 explains some related works in terms of network coding and auction mechanism. Section 3 presents our proposed bandwidth auction mechanism. Section 4 shows some numerical examples and Section 5 denotes conclusion. ISBN:

2 (a) Transmission without network coding. (b) Transmission with network coding. Figure 1: Number of wireless transmissions. 2 Related Work 2.1 Network coding Network coding has been proposed in order to improve the usage of network resources [8, 9, 10]. Figure 1 shows how the network coding works in wireless networks called Alice-Bob structure. Now, Alice wants to send packet A to Bob, while Bob wants to send packet B to Alice. Alice and Bob cannot communicate directly due to a short transmission range, and hence they have to send each packet to Sam. In the traditional packet forwarding, four wireless transmissions are needed for the packet transmission because Sam has to send packets A and B individually (see Fig. 1(a)). When the network coding (throughout this paper, we use XOR [9]) is used, on the other hand, Sam makes a new packet by encoding packet A and packet B. Then Sam broadcasts the new packet to Alice and Bob (see Fig. 1(b)). After Alice (Bob) receives the new packet, Alice (Bob) recovers packet B (A) from the received packet by using original packet A (B). Thus, the number of wireless transmissions is reduced to three, improving the overall network throughput. Now, we consider the broadcast rate of a broadcast transmission for the network coding. Let e denote a wireless unicast transmission link, and R e denote the transmission rate of link e. Bj i denote a wireless broadcast transmission at node i on the jth subset (j = 1, 2, ) of its outgoing links. When the broadcast transmission consists of multiple links ( Bj i 2), the transmission rate is R(Bi j ). If the broadcast transmission consists of a single unicast link e (e = 1, 2, ), the transmission rate R(Bj i) = R e. Here, the success of a wireless transmission is affected by the amount of interference. This interference can be modeled based on a conflict graph [11] by extending a connectivity graph shown in Fig. 1. In the connectively graph G = (N, E), vertices (N) of G correspond to wireless nodes and edges (E) correspond to the wireless links between the nodes. Note that when d ij is the distance between node i and node j, l i denotes the transmission range at node i, there is a directed link (e(i, j)) from node i to node j, if d ij l i and i j. On the other hand, in a conflict graph C, vertices correspond to edges in the network connectivity graph G. Two vertices are connected by an (undirected) edge if the corresponding links in G cannot be scheduled simultaneously. Moreover, broadcast conflict graph F has been considered for broadcast transmission as the extension of the conflict graph C [12]. Each vertex in F represents a broadcast transmission Bj i instead of unicast transmission. Here, let y(bj i ) denote the amount of broadcast traffic on Bj i. Let d(bi j ) denote the set of destination nodes for the links in broadcast set j at node i. When two broadcasts B j i and Bl m interfere with each other, there is an edge between them in the broadcast conflict graph. Note that B j i and Bl m interfere with each other if either some node k d(b j i ) is within the interfere range of Bm l or some node k d(bl m ) is within the interfere range of B j i. In the broadcast conflict graph F, there are one clique C(Bj i) for each broadcast transmission Bi j [13]. In each clique, only one wireless transmission can be active at any time. Here, the fraction of time that Bj i is active is given by y(bi j )/R(Bi j ). Hence, y(bj i)/r(bi j ) is restricted as follows. y(bm n ) R(B m Bn m C(Bi j ) n ) 1, Bi j 2, i, j. 2.2 Auction mechanism Several types of auctions have been proposed and utilized in order to provide goods with users [14, 15]. Among such auction mechanisms, it is well known that VCG auction mechanism satisfies incentive compatibility where the weakly dominant strategy for bidders is to bid truthfully [15]. In VCG auction mechanism, the auctioneer selects winners among all bidders so that social surplus can be maximized. VCG auction mechanism is extended to generalized VCG auction mechanism in order to consider multiple divisible units of goods. Recently, some auction mechanisms have been utilized for the bandwidth allocation in order to maximize the social surplus. In [2], VCG-like auction mechanism has been proposed for allocating bandwidth in a communication network. In [16], a combinatorial double auction mechanism called c-sebida ISBN:

3 that maximizes the social surplus has been proposed. In this combinatorial double auctions, network operators provide discrete bandwidth on only one link with users. On the other hand, users bid the amount of bandwidth that they want to use and the price that they are willing to pay. Users can win the desired amount of bandwidth or cannot bandwidth. [17] has proposed a real-time spectrum auction mechanism to provide spectrum with a large number of users. [4] has proposed an auction-based mechanism for allocating the downlink bandwidth of a wireless future Internet network. Thus, in wireless networks, bandwidth auction is expected to be utilized widely to provide bandwidth for potential users. However, as far as the authors know, the utilization of network coding has not been considered yet. 3 The Proposed Bandwidth Auction Mechanism In this section, we explain our proposed Generalized VCG mechanism that can provide bandwidth for users by using network coding in wireless networks. Figure 2 shows our proposed bandwidth auction mechanism. In this mechanism, generalized VCG auction is performed to provide bandwidth for users. When user h (h = 1, 2,, H) requires for an available bandwidth, he joins the auction by sending his bid (b h, r h, s(h), d(h)) to a network service provider (auctioneer). Here, b h (b h > 0) is his desired price per unit bandwidth and r h (r h > 0) is the maximum amount of bandwidth that he is willing to utilize. In the following, we assume that the total require bandwidth is always greater than the total available bandwidth. Moreover, s(h) and d(h) are source node and destination one of user h, respectively. The service provider decides the bandwidth allocation for each user with the auction. Then, the service provider returns the information about (P h, x h, p h ) to user h. Here, P h denotes a path between s(h) and d(h), and x h denotes the amount of bandwidth that is provided for user h. Moreover, p h is the amount of money that user h has to pay. Note that x h and p h are zero if user h becomes a loser in the auction. The user h who won the auction can transmit the data according to provided path P h with bandwidth x h after completes the payment p h to the service provider. 3.1 Utility Function and Social Surplus In this subsection, we explain how x h and P h are determined with generalized VCG auction. Figure 2: Proposed bandwidth auction mechanism. When the bid of user h is (b h, r h, s(h), d(h)), the declared utility function for user h is given by F (h) = min (b h r h, b h x h ). (1) From the declared utility of each user, the social surplus S for this auction is equal to S = H F (h). (2) Here, we assume that K flows have been existed in this wireless network. Let P k (k = 1,, K) be the routing path for existing flow k. In addition, let denote the traffic for the flow k. On the other hand, the possible path set for user h is denoted as R h [2]. If a flow of user h is routed on path P R h, f(p h ) = 1. Otherwise, f(p h ) is equal to zero. Moreover, when a link between node i and node j is denote as e, ē denote a link between node j and node i. Input links and output links for node i are represented as E i and E i. In this case, the maximal social surplus S max is calculated from the following optimization problem. Moreover, x h and P h are also derived from the following problem. subject to S max = max x h S = max x h H F (h), (3) P h R h f(p h ) = 1, h H, (4) K H ē 1 e 2 P k P h R h,ē 1 e 2 P h f(p h )x h, e 1, e 2 E i, (5) ISBN:

4 y(e) = H K H H ē 2 e 1 P k P h R h,ē 2 e 1 P h f(p h )x h, P h R h,e P h,s(h)=i K e 1 E i P h R h,e 1 e P y(bm n ) R(B m Bn m C(Bj i) n ) e 1, e 2 E i, (6) e P k,s(k)=i K f(p h )x h e 1 e P k f(p h )x h y(e, ē 1 ), e E i, (7) 1, B i j 2, i, j, (8) x h 0, h H. (9) Constraints (4) denote that only one route is provided for user h. Constraints (5) and (6) denote the maximum amount of coded traffic that can be broadcast on outgoing links {e 1, e 2 } at node i after network coding. Moreover, constraints (7) denote the amount of traffic y(e) that is unicast traffic on outgoing link e at node i. This unicast traffic is divided into two parts. The first part is the unicast traffic that originally traverses at node i and is sent on link e. The amount of this traffic is equal to H P h R h,e P h,s(h)=i f(p h)x h K e P k,s(k)=i f(pk ). The second part is the transit traffic at node i with the next-hop e that could not be coded with the other flows. The amount of this traffic is equal to [ K e 1 E e i 1 e P k f(pk ) ] H P h R h,e 1 e P f(p h)x h y(e, ē 1 ). Constraints (8) denote the broadcast transmission scheduling constraints corresponding to cliques in broadcast conflict graph F. The number of wireless transmission in clique C(Bj i ) is restricted to at most two due to Alice-Bob structure. Moreover, constraints (9) denote the bandwidth providing for a user is always positive. Among those constraints, integer variable (0 or 1) is utilized in constraints (4). Therefore, the above problem is Mixed Integer Linear Programming (MILP). In order to reduce the computation time, the problem can be reduced to a class of linear programming by modifying the objective function to max xh H b h x h and adding a new constraints x h r h. 3.2 Payment In this subsection, we determine the payments of users. Let δ (δ H \ h) denote the other user excluding user h and Smax h denote the maximal social surplus in a case where user h does not join the auction. Smax h is calculated from the following optimization problem by excluding user h. Smax h = max S h = max F (δ) (10) x δ x δ subject to θ H\h P δ R δ f(p δ ) = 1, δ H \ h, (11) K ē 1 e 2 P k K y(e) = δ H\h δ H\h P δ R δ,ē 1 e 2 P δ f(p δ )x δ, ē 2 e 1 P k δ H\h e 1, e 2 E i, (12) P δ R δ,ē 2 e 1 P δ f(p δ )x δ, δ H\h P δ R δ,e P δ,s(δ)=i K y(bm n ) R(B m Bn m C(Bi j ) n ) e 1, e 2 E i, (13) f(p δ )x δ e P k,s(k)=i K e 1 E e i 1 e P k f(p δ )x δ y(e, ē 1 ), P δ R δ,e 1 e P δ e E i, (14) 1, B i j 2, i, j, (15) x δ 0, δ H \ h. (16) From (3) and S h max, payment p h of user h is given by p h = S h max (S max F (h)). (17) ISBN:

5 Figure 3: Sample network topology. After user h pays p h to the auctioneer, user h can transmit data from s(h) to d(h) along P h. 4 Numerical Examples In this section, we evaluate the performance of our proposed auction with simulation. In our simulation, six nodes are randomly placed in a square whose sides are 400 [unit] shown in Fig. 3. For each node, the transmission range is set to 100 units and the interfere range is 200 units. There is a directed link between the two nodes when the distance between them is smaller that the transmission range. The data transmission rate for each link is given by 12 Mbps. In this network, bandwidth is allocated to users by using our proposed auction. Here, the number of users is from 2 to 8. For each user, b h is randomly chosen from [15, 20] and r h is randomly chosen among (0, 20]. In addition, s(h) and d(h) are randomly selected among the six nodes. There is only one existing flow (K = 1) and its source and destination nodes are selected at random. We also evaluate the existing auction mechanism for the performance comparison. In this auction, network coding cannot be utilized at every node. For both our propose auction and the existing auction, only one candidate route is pre-determined for each pair of s(h) and d(h). Moreover, we evaluate the performance of bandwidth allocation method without auction mechanism. In this method, bandwidth is allocated to each user so as to maximize the overall network throughput. In the following, we solve all optimization problems by using CPLEX [18]. 4.1 Effectiveness of bandwidth allocation with auction mechanism In this section, we investigate the effectiveness of bandwidth allocation with auction mechanism. Figure 4 shows the maximum social surpluses S max for our proposed auction and bandwidth allocation without auction against the number of users. In this graph, for both the two bandwidth allocations, the social sur- Figure 4: Performance comparison between our proposed mechanism with the bandwidth allocation without auction. Figure 5: Performance comparison between our proposed auction with the existing bandwidth auction without network coding. plus increases as the number of users increases. This is because a larger number of auctions are performed when a larger number of users want bandwidth. By comparing both the methods, we can find that the maximum social surplus of our proposed mechanism is larger than that of the existing bandwidth allocation without auction regardless of the number of users. Therefore, our proposed method is effective to increase the social surplus. 4.2 Effectiveness of network coding We investigate the effectiveness of network coding by comparing with the existing bandwidth auction without network coding. Figure 5 shows the maximum ISBN:

6 social surpluses for our proposed auction and the existing bandwidth auction against the number of users. From this figure, we can find that the social surplus for our proposed mechanism becomes large as the number of users increases. On the other hand, the social surplus for the existing auction does not increase so much when the number of users is larger than five. Moreover, the maximum social surplus of our proposed mechanism is larger than that of the existing bandwidth allocation without auction regardless of the number of users. This denotes that network coding is effective to improve the social surplus for the bandwidth auction when it is applied to our proposed mechanism effectively. 5 Conclusion In this paper, we proposed a generalized VCG auction mechanism for efficiently allocating bandwidths in wireless networks. In the proposed mechanism, the network coding is utilized in order to maximize the social surplus. We evaluated the performance of our proposed mechanism with simulation. We compared its performance with the performance of the existing bandwidth auction without auction and the performance of the existing bandwidth auction without network coding. From numerical examples, we found that the proposed mechanism can significantly increase the social surplus regardless of the number of users. References: [1] N. Abramson, THE ALOHA SYSTEM Another Alternative for Computer Communications, in Proc. AFIPS Fall Joint Computer Conference, pp , April [2] R. Jain and J. Walrand, An Efficient Mechanism for Network Bandwidth Auction, in Proc. IEEE INFOCOM 2004, March [3] P. Maille and B. Tuffin, Multi-Bid Auctions for Bandwidth Allocation in Communication Networks, in Proc. IEEE/IFIP Network Operations and Management NOMS 2008, April [4] M. Dramitinos and I. G. Lassous, Auctionbased Bandwidth Allocation Mechanisms for Wireless Future Internet, in Proc. INRIA 2010, January [5] L. Ming, J. Hong and L. Xi, VCG Auction Based Hierarchical Bandwidth Sharing Scheme for Integrated Heterogenous Networks, IEEE Wireless Communications, Networking and Mobile Computing 2009 (WiCom 09), September [6] G. Iosifidis and I. Koutsopoulos, Double Auction Mechanisms for Resource Allocation in Autonomous Networks, in IEEE Journal on Selected Areas in Communications, vol. 28, no. 1, January [7] C. Fragouli, JY. Le Boudec, and J. Widmer, Network Coding: An Instant Primer, Proc. ACM Sigcomm 2006, January [8] R. Ahlswede, N. Cai, S.Y. Li, and R. Yeung, Network Information Flow, IEEE Transactions on Information Theory, vol. 46, pp , July [9] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft, XORs in the air: Practical Wireless Network Coding, ACM Sigcomm 2006, September [10] S. Sengupta, S. Rayanchu and S. Banerjee, An Analysis of Wireless Network Coding for Unicast Sessions: The Case for Coding-Aware Routing, in Proc. IEEE INFOCOM, May [11] K. Jain and J. Padhye, V.N. Padmanabhan, and L. Qiu, Impact of Interference on Multi-hop Wireless Network Performance, ACM MOBI- COM 2003, September [12] J. Zhang and Q. Zhang, Cooperative Network Coding-Aware Routing for Multi-Rate Wireless Networks, in Proc. IEEE INFOCOM 2009, April [13] J.W. Moon and L. Moser, On cliques in graphs, Israel Journal of Mathematics, vol. 3, pp , [14] P. Cramton, Y. Shoham, and R. Steinberg, Combinatorial Auctions, Cambridge, Mass. : MIT Press, [15] V. Krishna, Auction Theory, 2nd ed., Department of Economics, Pennsylvania State University, University Park, [16] R. Jain and P. P. Varaiya, Combinatorial Exchange Mechanisms for Efficient Bandwidth Allocation, Communications in Information and Systems, vol. 3, no. 4, pp , September [17] S. Gandhi, C. Buragohain, L. Cao, H. Zheng and S. Suri, A General Framework for Wireless Spectrum Auctions, in Proc. IEEE DySpan 2007, April [18] ILOG CPLEX, ISBN: