Auction-Based Power Allocation for Many-to-One Cooperative Wireless Networks Mohammed W. Baidas and Allen B. MacKenzie Wireless @ Virginia Tech, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, USA (email:
[email protected],
[email protected])
Abstract—In this paper, a distributed efficient power allocation game-theoretic framework in wireless ad-hoc networks is proposed where multiple source nodes communicate with a single destination node via a relay node. The power allocation among the source nodes is formulated as an alternative ascending-clock auction (A-ACA) and achieved using a distributed algorithm that converges in a finite number of clocks and is proven to enforce truthful power demands at every clock and maximize the social welfare. Analytical and numerical results are presented to verify the efficient power allocation, truthtelling and social welfare maximization properties of the proposed AACA. It is concluded that the proposed A-ACA lends itself to practical implementation in wireless ad-hoc networks. Index Terms—Amplify-and-forward (AF), auction, cooperation, many-to-one, network coding, power allocation
I. I NTRODUCTION In decentralized and fully distributed ad-hoc wireless networks without a single authority and with network users acting as independent entities, rational users selfishly aim at maximizing their utility and use of resources. Moreover, selfish users might not have any incentive to share their resources if they cannot gain some reward in return. In other cases, several users could compete in order to consume transmission resources (e.g. bandwidth and power) from a particular user who might be welling to assist in relaying their data transmissions towards a destination, however, at a price. Since rational users are rather selfish, they may overstate or otherwise not truthfully report their resource demand if doing so can improve their utility. In other scenarios, revealing such proprietary information may have an adverse strategic long-term impact. Hence, a key problem in wireless ad-hoc networks is how distributively and efficiently allocate transmission resources among competing selfish users. Overall, the modeling of resource allocation and repeated interaction of network users is rooted in game and auction theories. An important thrust of recent research works deal with user selfishness and cooperation from a game-theoretic perspective. For instance, in [1], a Stackelberg game is proposed for multinode relay selection and power control when a single source node communicates with a single destination. In another line of work, Huang et al. propose two distributed auction mechanisms that achieve a unique Nash Equilibrium are proposed for relay selection and power allocation, namely, the SNR auction and the power auction [2]. It is shown that the SNR auction offers a flexible tradeoff between fairness and efficiency, while the power auction achieves efficient power allocation by maximizing the total rate increase. In [3], a second-price auction is proposed for fair allocation of a wireless fading channel for different channel state distributions. In this paper, the alternative ascending-clock auction (A-ACA) is proposed for efficient power allocation in many-to-one cooperative wireless networks, where multiple source nodes communicate with
978-1-4577-9538-2/11/$26.00 ©2011 IEEE
a single destination node via a relay node [4] [5]. In particular, a distributed algorithm is proposed to achieve the A-ACA based efficient power allocation algorithm which is proven to enforce truthful power demands among network source nodes and maximize the social welfare (i.e. the sum of utilities of the source nodes and the relay). It is envisioned that the presented work will pave the way for a practical distributed and efficient power allocation in manyto-one cooperative communications in wireless ad-hoc networks. In the remainder of this paper, the network model is presented in Section II. The multiuser game-theoretic framework and the proposed alternative ascending-clock auction are presented in Sections III and IV, respectively. Numerical results of the proposed ascending-clock auction based power allocation are presented in Section V. Finally, the conclusions are drawn in Section VI. II. N ETWORK M ODEL Consider a wireless network consisting of N source nodes (N ≥ 2), denoted as S1 , S2 , . . ., SN . The N nodes are assumed to have their own data symbols x1 , x2 , . . . , xN , respectively, and each node aims at communicating its data symbol to a common destination node D. In addition, assume that there is a relay node R that is welling to share its transmission power PR to forward source nodes data symbols to the destination. In this network (shown in Fig. 1 for N = 2), each node is equipped with a single antenna and the relay node’s cooperative transmission follows the amplify-andforward (AF) protocol [6]. The channels between any two nodes are modeled as narrow-band Rayleigh fading channels with additive white Gaussian noise (AWGN). Let hj,i denote a generic channel coefficient representing the channel between any two nodes j and 2 2 i, then hj,i ∼ CN (0, σj,i ), where σj,i = d−ν j,i is the channel gain with dj,i and ν being the distance between the two nodes and the path-loss exponent, respectively. The communication between the source nodes and the destination node is performed over a total of N + 1 time-slots and is split into two phases, namely the broadcasting phase (BP) (of N time-slots) and the cooperation phase (CP) (of a single time-slot). A. Broadcasting Phase In the broadcasting phase, each source node Sj for j ∈ {1, 2, . . . , N } is assigned a time-slot Tj in which it broadcasts its own data symbol xj to the rest of the network. The received signal yj,r (t) at the relay node R in time-slot Tj is expressed as yj,r =
PBj hj,r xj + nj,r ,
(1)
while the received signal at the destination node D is expressed as
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yj,d =
PBj hj,d xj + nj,d ,
(2)
n ¯ d (t) = nd (t) + hr,d
N
βm,r nm,r cm (t).
(7)
m=1
2) Multinode Signal Detection: Upon receiving signal Yr,d (t) from the relay node R, a multinode signal detection is performed by the destination D to extract each symbol xj , for j ∈ {1, 2, . . . , N }. This is achieved by passing the received signal Yr,d (t) through a matched filter bank (MFB) of N branches, matched to the corresponding nodes’ signature waveforms cj (t), yielding Fig. 1. Many-to-One Cooperative Network - Broadcasting and Cooperation Phases - N = 2 Nodes
Yj,r,d = Yr,d (t), cj (t) =
N
αm,r,d xm ρm,j + n ¯ j,d .
(8)
m=1
where PBj is the broadcasting transmit power at node Sj and nj,r and nj,d are the zero-mean additive white Gaussian noise (AWGN) processes with variance N0 , at the relay and destination nodes R and D, respectively. Upon completion of the broadcasting phase, relay and destination nodes R and D will have received a set of N N signals {yj,r }N j=1 and {yj,d }j=1 , respectively, comprising symbols {xj }N of the N source nodes. j=1 B. Cooperation Phase The cooperation phase involves two operations: 1) signal transmission via the relay node, and 2) multinode signal detection, which are discussed in the following subsections, respectively. 1) Signal Transmission: In the cooperation phase, relay node R in its assigned time-slot TN +1 forms a linear network code based on its received symbols {ym,r }N m=1 , during the broadcasting phase and transmits it to destination node D. For source separation of each transmitted symbol of the different source nodes at the destination, each received signal yj,r is spread using a signature waveform, cj (t). It is assumed that the relay and destination nodes know the signature waveforms of all the source nodes. The cross-correlation of cj (t) and ci (t) is ρj,i = cj (t), ci (t) (1/Ts ) 0Ts cj (t)c∗i (t)dt for j = i with ρj,j = 1, Ts being the symbol duration and (·)∗ denoting complex conjugation. The resulting signal transmitted by the relay, X (t) is written as X (t) =
N
βm,r ym,r cm (t),
(3)
m=1
where cm (t) is the signature waveform associated with source node Sm and βm,r is a scaling factor defined as [6] βm,r =
P Cm . PBm |hm,r |2 + N0
(4)
where PCm is the cooperative transmit power of symbol xm at the relay node R. The received signal at destination node D during time-slot TN +1 is given by Yr,d (t) = hr,d X (t) + nd (t),
(5)
where nd (t) is the AWGN at node D. Upon substitution of (1), (3) and (4) into (5), the received signal can be expressed as Yr,d (t) =
N m=1
αm,r,d xm cm (t) + n ¯ d (t),
(6)
where αm,r,d = βm,r PBm hm,r hr,d and n ¯ d (t) is the equivalent noise term, defined as
Recall that ρm,j is the correlation coefficient between cm (t) and cj (t). The output of the MFB can be put in a vector form as Y r,d = RAx + n ¯ d,
(9)
where Y r,d = [Y1,r,d , . . . , Yj,r,d , . . . , YN,r,d ]T , T
x = [x1 , . . . , xj , . . . , xN ] ,
(10) (11)
T n ¯ d = [¯ n1,d , . . . , n ¯ j,d , . . . , n ¯ N,d ] ∼ CN (0, N0B R ), and R , A , and B are N × N matrices with R being defined as ⎤ ⎡
1 ⎢ .. ⎢ . ⎢ R=⎢ ⎢ ρj,1 ⎢ . ⎣ ..
··· .. . ···
ρ1,j .. . 1 .. . ρN,j
···
ρ1,N .. ⎥ . ⎥ ⎥ ρj,N ⎥ ⎥, .. ⎥ . ⎦ 1
(12)
A = diag [α1,r,d , . . . , αj,r,d , . . . , αN,r,d ] ,
(13)
2 2 2 B = diag β˜1,r,d , . . . , β˜j,r,d , . . . , β˜N,r,d ,
(14)
ρN,1
··· ···
··· ··· .. . ···
while the diagonal matrices A and B are and
2 2 2 with β˜j,r,d being defined as β˜j,r = βj,r |hr,d |2 + 1. The received signal vector Y r,d can then be decorrelated (assuming matrix R is ˜ d , where n ˜ d = R −1n ¯d invertible) as Y˜ r,d = R −1Y r,d = Ax + n −1 and n ˜ d ∼ CN (0, N0R B ). Thus, the decorrelated received signal Y˜j,r,d corresponding to symbol xj is obtained as
Y˜j,r,d = βj,r
PBj hj,r hr,d xj + n ˜ j,d ,
(15)
2 where n ˜ j,d ∼ CN (0, N0 rj (βj,r |hr,d |2 + 1)) and rj is the j th −1 diagonal element of matrix R . Without loss of generality, it is assumed that ρj,i = ρ for all j = i and thus [7]
rj =
1 + (N − 2)ρ rN . 1 + (N − 2)ρ − (N − 1)ρ2
(16)
Upon the completion of the broadcasting and cooperation phases, destination node D will have received two signals of each symbol xj for j ∈ {1, 2, . . . , N }; a direct signal from the source node Sj in the broadcasting phase and a relayed signal in the cooperation phase. The detection of symbol xj , denoted as x ˜j , is achieved through maximal-ratio-combining (MRC) of the signals received in the broadcasting and cooperation phases. Thus, the instantaneous SNR at the output of the MRC of symbol xj is given by B C B γj = γj,d +γj,d , where γj,d is the SNR due to the broadcast “direct” transmission from the source to the destination and is defined as
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Rj,d =
PCj (ξ) = max 0, min
1 log2 N +1
1+
PBj PCj |hj,r |2 |hr,d |2 PBj |hj,d |2 + N0 N0 rN (PBj |hj,r |2 + PCj |hr,d |2 + N0 )
1 2(Ωj,r,d + 1)
Ω2j,r,d Υ2j,r,d
PBj PCj |hj,r |2 |hr,d |2 . N0 rN (PBj |hj,r |2 + PCj |hr,d |2 + N0 )
(17)
Since a single data symbols is exchanged between every source node Sj , for j ∈ {1, 2, . . . , N } and the destination once every N + 1 time-slots, the achievable rate in bits per time-slot can be determined using (18) (top of page). III. M ULTI -U SER G AME -T HEORETIC F RAMEWORK This section presents a game-theoretic framework for the cooperative relay power allocation among the N source nodes. A. Source Node Utility Function The utility function of each source node is based on the improvement in the transmission rate achieved by relay node’s cooperative transmission. Accordingly, the utility function of source node Sj for cooperative data transmission to the destination D via the relay node R is UjS (PCj ) = ΔRj,d (PCj ) − ξPCj ,
(19)
where ξ is the price per unit of power charged by the relay node to forward data symbols to the destination, and ΔRj,d (PCj ) is the improvement in the transmission rate due to the cooperative relaying which is given by ΔRj,d (PCj ) =
PCj Ωj,r,d 1 , log2 1 + N +1 PCj + Υj,r,d
(20)
where PBj |hj,r |2 , N0 rN (PBj |hj,d |2 + N0 )
(21)
Υj,r,d = (PBj |hj,r |2 + N0 )/|hr,d |2 .
(22)
Ωj,r,d =
and Thus, ΔRj,d (PCj ) defines the cooperation gain to node Sj in terms of an improvement in the transmission rate to the destination node D, when relay node R cooperates. Clearly, ΔRj,d (PCj ) is a monotonically increasing function of PCj . Each source node aims at maximizing its utility subject to the total transmit power PR available at the relay R. Thus, each source node’s cooperative power demand problem can be modeled as max PCj
s.t.
UjS (PCj ) = ΔRj,d (PCj ) − ξPCj , 0 ≤ P C j ≤ PR ,
∀j ∈ {1, 2, . . . , N }.
,
∀j ∈ {1, 2, . . . , N }.
4ηΩj,r,d Υj,r,d (Ωj,r,d + 1) + − (Ωj,r,d + 2)Υj,r,d ξ
B C γj,d = PBj |hj,d |2 /N0 , while γj,d is the SNR due to the cooperative transmission via relay node R and is defined as C γj,d =
(23)
Clearly, the utility function UjS (PCj ) is concave in PCj , and taking the derivative of UjS (PCj ) with respect to PCj , yields
(18)
, PR
, ∀j ∈ {1, 2, . . . , N }.
∂UjS (PCj ) ∂ΔRj,d (PCj ) = − ξ = 0. ∂PCj ∂PCj
(25)
(24)
1 By using the identity log2 (x) = ln(x)/ ln 2, defining η = (N +1) ln 2 and substituting ΔRj,d (PCj ) in (20) into (24), the utility function UjS (PCj ) is maximized at PCj (ξ) which is expressed in (25).
B. Relay Node Utility Function The relay node’s utility function is based on selling its cooperative transmit power PR to the source nodes to forward their data symbols to the destination. In this case, the relay’s utility is defined as the total payment it receives by selling its transmit power PR to the source nodes minus its own cost of cooperation ζ per unit power (i.e. for processing, transmitting and receiving). Thus, the relay’s utility function is given by N
UR (PR ) = ϑr (PR ) − ζPR ,
(26)
j=1 ϑj (PCj )
with ϑr (PR ) = being the total payment the relay receives from the N source nodes for transmitting their data symbols, ϑj (PCj ) is the payment source node Sj makes when it is assigned N cooperative transmit power PCj such that PR ≤ j=1 PCj (as will be formally established in the following section). It should be noted that ζPCj is the cooperation cost due to the transmission of source node Sj ’s data symbol. IV. A LTERNATIVE A SCENDING -C LOCK AUCTION The idea of pricing as a distributed control mechanism aims at encouraging autonomous and independent network users to make rational decisions that result in a social benefit for the entire network. Towards this end, a distributed algorithm based on the alternative ascending-clock auction (A-ACA) is proposed for efficient power allocation [4] [5]. In particular, the relay “auctioneer” announces a price, the N source nodes “bidders” report back their cooperative power demanded at that price and power is allocated to source nodes at the current price whenever they are “clinched”. The relay then raises the announced price and the process repeats until the total power demand meets the available power supply (in which case all the relay’s transmit power PR is allocated). In the A-ACA algorithm, at each time τ = 0, 1, . . ., the relay announces a price ξ τ to the N source nodes. In order for the relay to cover the cooperation cost per unit power ζ, it initially sets a reserve price of ξ 0 = ζ and announces it to the source nodes. Based on the announced price ξ τ , each source node Sj responds with a bid in the form of an optimal power demand PCj (ξ τ ) to the relay. After receiving all the demands at each clock, N the relay compares the total demanded power PTCotal (ξ τ ) = j=1 PCj (ξ τ ) and compares it with the total available power PR . If the total demand exceeds the supply (i.e. PTCotal (ξ τ ) > PR ), the auction
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1 Relay initializes clock index at τ = 0 and step size to μ > 0 and then announces initial price of ξ 0 = ζ . 2 Each source node Sj computes and submits its optimal power demand PCj (ξ 0 ). 0 3 Relay sums up all demands PTCotal (ξ 0 ) = N j=1 PCj (ξ ). C 0 4 IF PT otal (ξ ) ≤ PR , 5 the auction concludes and time is denoted T . 6 ELSE, 7 set ξ τ +1 = ξ τ + μ and τ = τ + 1. 8 Price ξ τ is announced to source nodes. 9 Each source node Sj computes and submits its optimal demand PCj (ξ τ ). τ 10 Relay sums up all demands PTCotal (ξ τ )= N j=1 PCj (ξ ). 11 IF PTCotal (ξ τ ) > PR , τ 12 compute Cj (ξ τ ) = max 0, PR − N i=1,i=j PCi (ξ ) and go to step 7. 13 ELSE, 14 the auction concludes and time is denoted T . 15 END. 16 END. 17 Compute Cj (ξ T ) and assign PC j = Cj (ξ T ) to source node Sj , which makes a payment of ϑj (PC j ) to the relay node.
proceeds to time τ + 1 and the associated price is increased to ξ τ +1 = ξ τ + μ = (ζ + μτ ) + μ, where μ is an appropriate step size. The relay then calculates the cumulative vector of quantities Cj (ξ τ ), clinched by source node Sj at prices up to ξ τ , defined as ⎛
Cj (ξ ) = max ⎝0, PR − τ
⎞
N
PCi (ξ )⎠ , τ
∀j ∈ {1, 2, . . . , N }.
i=1,i=j
(27)
On the other hand, if the supply meets the total demand (i.e. PTCotal (ξ τ ) ≤ PR ), the auction concludes and the current time is denoted T . However, as the price goes up, the demanded cooperative power by each source node decreases and it is possible, for a certain increase in price, that the supply is not covered at the final price ξ T (i.e. PTCotal (ξ T ) < PR ). In turn, a proportional rationing rule is applied and the power allocation is achieved according to [4][8] Cj (ξ T ) = PCj (ξ T )+ PCj (ξ T −1 ) − PCj (ξ T ) N N T −1 ) − T j=1 PCj (ξ j=1 PCj (ξ )
N
PR −
N
T
PCi (ξ ) ,
(28)
i=1
with j=1 Cj (ξ ) = PR . Hence, each source node Sj is assigned its demanded cooperative transmit power as PCj = Cj (ξ T ). Also, the payment from source node Sj to the relay is written as
TABLE I A LTERNATIVE A SCENDING -C LOCK AUCTION A LGORITHM
T
ϑj (PCj )
0
0
= Cj (ξ )ξ +
T
τ
τ
ξ (Cj (ξ ) − Cj (ξ
τ −1
)),
(29)
τ =1
where the total payment the relay node receives for allocating its N power PR is ϑr (PR ) = j=1 ϑj (PCj ). It is easily verified that PCj = Cj (ξ 0 ) +
T
(Cj (ξ τ ) − Cj (ξ τ −1 )).
(30)
τ =1
Also, by substituting ξ τ = ζ + μτ into (29), the payment of source node Sj can be rewritten as ϑj (PCj ) = ζPCj + B(PCj ),
where B(PCj ) =
T
μτ (Cj (ξ τ ) − Cj (ξ τ −1 ))
(31)
(32)
τ =1
is the surplus the relay obtains from node Sj upon allocating cooperative power PCj . The proposed distributed A-ACA algorithm is summarized in Table I. The main properties of the proposed A-ACA algorithm are briefly discussed in the following subsections. A. Convergence Theorem 1 (Convergence): The A-ACA algorithm concludes in a finite number of clocks. Proof: See Appendix 1.A. B. Truth-Telling Theorem 2 (Truth-Telling): Reporting optimal cooperative power demand truthfully at every clock in the alternative ascendingclock auction is a mutually best response for all source nodes. Proof: See Appendix 1.B. In the proposed distributed A-ACA algorithm, each participant has full incentive to truthfully reveal its true power demand and
this is because the price each source node pays depends solely on opposing nodes bids and thus need not report their private information (i.e. also preserves privacy). Thus, the proposed AACA algorithm enforces truth-telling and the best strategy of each source node is to truthfully report its power demand at every clock. C. Social Welfare Maximization Theorem 3 (Maximization of Social Welfare): The alternative ascending-clock auction based power allocation (PC1 , PC2 , . . . , PCN ) maximizes the social welfare. Proof: See Appendix 1.C. The proposed A-ACA algorithm maximizes the sum of the source nodes and the relay node utilities when the relay node fully sells out its cooperative transmit power PR . D. Simplicity and Applicability The key characteristic of the proposed A-ACA algorithm is the relative simplicity of the auction process during the iterative clock phase and the final power demand allocation. In order words, the computational complexity on both the N source nodes and the relay is low. Each source nodes needs to calculate its optimal power demand PCj (ξ τ ) based on the announced price ξ τ while ascertaining that it is within available power PR and then submit a bid. On the other hand, the assessment of excess demand and calculation of the cumulative clinch are relatively trivial. Another important key characteristic is the fact that the relay’s aim during the clock phase is to drive the excess demand out of the system by raising the price without ending up with excess power supply. This in turn implies the auction process achieves all gains from trade and is allocatively efficient [9]. Based on all the above discussed properties, it is concluded that the proposed A-ACA algorithm lends itself to practical ad-hoc network implementation.
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(a) Utilities Vs. Relay Power 1.5 Source Node S Utilities
1
Source Node S3
Allocated Cooperative Power (dBW)
Cooperative Relay Network - Simulation Scenario
V. N UMERICAL R ESULTS
Relay Node R Sum of Utilites
0.5
0
Fig. 2.
Source Node S2
1
0
5
10
15 20 Relay Power P (dBW)
25
30
35
25
30
35
25
30
35
R
(b) Allocated Cooperative Power Vs. Relay Power 25
Source Node S
1
20
Source Node S2
15
Source Node S3
10 5 0
0
5
10
15 20 Relay Power PR (dBW) (c) Payments Vs. Relay Power
0.2 Payments
Source Node S
1
0.15
Source Node S2 Source Node S
0.1
3
Sum of Payments 0.05 0
0
5
10
15 20 Relay Power P (dBW) R
Fig. 3.
Simulation Results: Utilities, Allocated Power and Payment Utilities Vs. Relay Power 0.5 Source Node S
1
Source Node S
0.4
2
Source Node S
3
Utilities
To validate the proposed A-ACA algorithm, a wireless network with three source nodes S1 , S2 and S3 and a relay R (see Fig. 2) is simulated. The network topology illustrates a scenario where the distance between any source node Sj for j = {1, 2, 3} and the destination is equal (i.e. d1,d = d2,d = d3,d ). Also, the relay is closer to the destination than to node S1 (i.e. d1,r > dr,d ); while it is closer to node S2 than to the destination (i.e. d2,r < dr,d ). Moreover, the relay is at an equal distance from the destination and node S3 (i.e. d3,r = dr,d ). The simulations assume a path-loss exponent of ν = 3, non-orthogonal signature waveforms with a cross-correlation of ρ = 0.5, reserve price of ζ = 10−5 , and source broadcasting transmit power PBj = 20 dBW, ∀j = {1, 2, 3}. It is evident from Fig. 3a that with the increase in the available relay power PR , the utility of each source node increases, with node S2 having the highest utility while node S1 having the lowest. This is attributed to the location of node S2 with the relay being closer to it than to the destination and hence the effect of path-loss and channel noise is less severe. Thus, using the relay is significantly beneficial to node S2 which is translated into a higher cooperative power demand and allocation (as evident from Fig. 3b). The lowest utility corresponding to node S1 is justified by a converse argument to that of node S2 . Interesting to observe in Fig. 3a is the utility of the relay which peaks at PR = 20 dBW and then starts to degrade. This is because when PR is high enough (i.e. PR > 20 dBW), there is abundant cooperative power for each of the source nodes (i.e. the total power demanded by the source nodes is less than the supply relay power PR ). Hence, the relay does not get to raise the price so much in the auction to extract higher revenue (i.e. higher payments) but instead sells most of the power early in the auction at a relatively lower price, which is seen in the form of lower payments in Fig. 3c. Overall, the sum of utilities keeps on increasing with the increase in available relay power PR . In Fig. 4, the utilities of the source nodes and the relay are evaluated at PR = 20 dBW when source node S1 falsely reports a demand of P˜C1 (ξ τ ) = max [0, min (κPC1 (ξ τ ), PR )], ∀τ = 0, 1, . . . , T with κ ≥ 0 being the demand factor. On the other hand, nodes S2 and S3 report their truthful demands PC2 (ξ τ ) and PC3 (ξ τ ), respectively, ∀τ = 0, 1, . . . , T . It is clear that as κ increases, the utility of S1 improves until it reaches its maximum value at κ = 1 (i.e. truthful demand), beyond which the utility degrades. Hence, the proposed A-ACA algorithm enforces truthtelling and each source node must report its true/optimal power demand to maximize their utility.
Relay Node R Sum of Utilites
0.3
0.2
0.1
0
Fig. 4.
0
0.5
1
1.5 Demand Factor κ
2
2.5
3
Simulation Results: Truthful Demand Verification - PR = 20 dBW
It is also interesting to notice that with the increase in the demand factor κ, the utilities of nodes S2 and S3 decrease and this is due to the increased overall demand on power PR which in turn improves the relay node’s utility (due to the higher revenue/payments). VI. C ONCLUSIONS In this paper, an alternative ascending-clock auction is proposed for efficient cooperative relay power allocation to geographically distributed source nodes in an ad-hoc wireless network. The proposed auction is applied via a distributed algorithm that is proven to enforce truth-telling and also maximize the sum of utilities. In conclusion, the proposed alternative ascending-clock auction lends itself to practical implementation in wireless ad-hoc networks. VII. A PPENDIX I A. Convergence From (25), it is clear that PCj (ξ τ ) is non-increasing in ξ τ and hence PCj (ξ τ ) ≥ PCj (ξ τ +1 ) with equality occurring when PCj (ξ τ +1 ) = PCj (ξ τ ) = 0 or PCj (ξ τ +1 ) = PCj (ξ τ ) = PR , ∀τ . Since for a sufficiently large τ , PCj (ξ τ +1 ) < PCj (ξ τ ) < PR , then there N exists a finite number T such that j=1 PCj (ξ T ) ≤ PR and thus the auction concludes at clock T .
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B. Truth-Telling Assuming all the other source nodes truthfully report their power demands, let Σj (T ) = {PCj (ξ 0 ), . . . , PCj (ξ T ); Cj (ξ 0 ), . . . , Cj (ξ T ); T }
which is convex, since ΔRj is concave in PCj and the constraint set is convex. Thus, the Lagrangian problem is formulated as [10] L(PCj , λ, ωj , υj ) = −
(33)
˜
˜j (ξ τ˜ ), . . . , C ˜j (ξ˜T ); T˜} Cj (ξ 0 ), . . . , C
(34)
+
j
j
−
j
UjS (Σj (T )) − UjS Σj (T˜) ≥ ΔRj,d (PCj (ξ T )) − ξ T PCj (ξ T ) ˜ ˜ + ξ T P˜Cj ξ˜T ≥ 0. − ΔRj,d P˜Cj ξ˜T (36)
•
≥ PCj (ξ T ), and thus ˜ − T˜ ≥ T . Similarly, it can be shown that ϑj P˜Cj ξ˜T T˜ T T ˜ ˜T˜ T T T ˜ ˜ − ϑj (PCj (ξ )) ≥ ξ Cj (ξ ) − ξ Cj (ξ ) = ξ PCj ξ ξ T PCj (ξ T ). Thus UjS (Σj (T )) − UjS Σj (T˜) ≥ ΔRj,d (PCj (ξ T )) − ξ T PCj (ξ T ) ˜ ˜ + ξ T P˜C ξ˜T ≥ 0. − ΔRj,d P˜C ξ˜T j
(37)
From (36) and (37), falsely demanding power at least once yields a lower utility than demanding power truthfully at every clock τ . Thus, the best strategy is to truthfully demand power at each clock τ for 0 ≤ τ ≤ T . C. Social Welfare Maximization The proof aims to show that the A-ACA based power allocation (PC1 , PC2 , . . . , PCN ) solves the following optimization problem: max PC j
s.t.
N
ΔRj,d (PCj )
j=1 N
υj PCj ,
j=1
PCj ≤ PR , (40)
1 (N +1) ln 2
where η = while λ ≥ 0, ωj ≥ 0, υj ≥ 0, ∀j ∈ {1, 2, . . . , N } are the dual variables associated with the power constraint and transmit power positivity. The solution to the optimization problem is known as water-filling [11], in the form of PCj (λ) in (25), ∀j ∈ {1, 2, . . . , N }, where λ satisfies N j=1 PCj (λ) = PR . Thus, the outcome of the A-ACA algorithm, (PC1 , PC2 , . . . , PCN ) is the solution to the optimization problem in (38) that maximizes the sum of of the source nodes and the relay node utilities when the relay node fully sells out its cooperative transmit power PR . R EFERENCES [1] B. Wang, Z. Han, and K. J. R. Liu, “Distributed relay selection and power control for multiuser cooperative communication networks using stackelberg game,” IEEE Trans. on Mobile Computing, vol. 8, pp. 975–990, Jul. 2009. [2] J. Huang, Z. Han, M. Chiang, and H. V. Poor, “Auction-based resource allocation for cooperative communications,” IEEE JSAC, Special Issue on Game Theory, vol. 26, pp. 1226–1238, 2008. [3] J. Sun, E. Modiano, and L. Zheng, “Wireless channel allocation using an auction algorithm,” IEEE Journal on Selected Areas in Communications, vol. 24, pp. 1085–1096, May 2006. [4] L. M. Ausubel, “An efficient ascending-bid auction for multiple objects,” Amer. Econ. Rev., vol. 94, pp. 1452–1475, 2004. [5] V. Krishna, Auction Theory. Academic Press, 2002. [6] K. J. R. Liu, A. K. Sadek, W. Su, and A. Kwasinski, Cooperative Communications and Networking. Cambridge University Press, 2008. [7] M. W. Baidas, H.-Q. Lai, and K. J. R. Liu, “Many-to-many communications via space-time network coding,” Proc. of IEEE Wireless Communcations and Networking Conference, pp. 1 – 6, April 2010. [8] Y. Saez, D. Quintana, P. Isasi, and A. Mochon, “Effects of a rationing rule on the ausubel auction: A genetic algorithm implementation,” Computational Intelligence, vol. 23, pp. 221–235, 2007. [9] T. K. Forde and L. E. Doyle, “Combinatorial clock auction for OFDMA-based cognitive wireless networks,” Proc. of 3rd Int. Symp. on Wireless Pervasive Computing (ISWPC), pp. 329–333, May 2008. [10] S. Boyd and L. Vandenberghe, “Convex optimization,” Cambridge University Press 2003. [11] T. Cover and J. Thomas, Elements of Information Theory. John Wiley Inc., 1991.
j=1
0 ≤ P C j ≤ PR ,
(39)
j=1
(38)
PCj ≤ PR
N
ωj (PCj − PR ) = 0, ∀j ∈ {1, 2, . . . , N }, υj PCj = 0, ∀j ∈ {1, 2, . . . , N }, 0 ≤ PCj ≤ PR , ∀j ∈ {1, 2, . . . , N },
˜ If P˜Cj (ξ τ˜ ) ≥ PCj (ξ τ ), then P˜Cj ξ˜T
j
ωj (PCj − PR ) −
N
˜ ≤ It can be shown that ϑj (PCj (ξ T )) − ϑj P˜Cj ξ˜T ˜ T T T T ˜ ˜T˜ T T T ˜ ξ Cj (ξ ) − ξ Cj (ξ ) = ξ PC (ξ ) − ξ PC ξ˜ . Hence,
P Cj − P R
j=1
j=1
j
UjS (Σj (T )) − UjS Σj (T˜ ) = ΔRj,d (PCj (ξ T )) ˜ ˜ − ΔRj,d P˜Cj ξ˜T − ϑj (PCj (ξ T )) + ϑj P˜Cj ξ˜T . (35)
ηΩj,r,d Υj,r,d + λ + ωj − υj = 0 ((Ωj,r,d + 1)PCj + Υj,r,d )(PCj + Υj,r,d ) N PCj − PR = 0, λ
j
N
N
and the Karush-Kuhn-Tucker (KKT) conditions are given by:
implies that T˜ ≤ T . From (19) and (29), it is clear that
ΔRj,d (PCj ) + λ
j=1
be the auction profile of source node Sj when it falsely demands power on clock τ = τ˜ for 0 ≤ τ˜ ≤ T˜ while nodes Si , ∀i ∈ {1, 2, . . . , N } andi = j truthfully report their power demands. ˜ Also, let P˜Cj ξ˜T be the power allocated to node Sj at the end of the auction when it falsely demands power. τ˜ τ ˜C ξ˜T˜ ≤ PC (ξ T ), which • If PC (ξ ) ≤ PC (ξ ), then P j
j=1
be the auction profile when source node Sj when it reports its power demand truthfully on every clock τ . Moreover, let ˜ Σj (T˜) = {PCj (ξ 0 ), . . . , P˜Cj (ξ τ˜ ), . . . , P˜Cj (ξ˜T );
N
∀j = 1, 2, . . . , N,
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