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On the Network Sharing of Mixed Network Coding and Routing Data Flows in Congestion Networks Gang Wang, Member, IEEE, Xia Dai, and Yonghui Li, Senior Member, IEEE
Abstract—In this paper, we study the congestion game for a network where mutliple network coding (NC) and routing users sharing a single common congestion link to transmit their information. The data flows using NC and routing will compete network resources, and we need to determine the optimal allocation of network resources between NC and routing data flows to maximize the network payoff. To facilitate the design, we formulate this process using a cost-sharing game model. A novel average-costsharing (ACS) pricing mechanism is developed to maximize the overall network payoff. We analyze the performance of ACS in terms of price of anarchy (PoA). We formulate an analytical expression to compute PoA under the ACS mechanism. In contrast to the previous affine marginal cost (AMC) mechanism, where the overall network payoff decreases when NC is applied, the proposed ACS mechanism can considerably improve the overall network payoff by optimizing the number and the spectral resource allocation of NC and routing data flows sharing the network link. Index Terms—Affine marginal cost (AMC), average cost sharing (ACS), network coding (NC), price of anarchy (PoA).
I. I NTRODUCTION
W
ITH the ever-increasing bandwidth-hungry wireless applications with diverse quality-of-service (QoS) requirements, wireless network operators face a great challenge to meet such requirements with the limited available bandwidth. This is because the transmission rate of a user in the communication network is limited by the available transmission bandwidth. Users are generally selfish and pursue their own interests. As the data traffic transmitted through the network exceeds its transmission capacity, network congestion will occur. It is, thus, important to efficiently utilize limited resources in the congestion network to maximize the network throughput by designing novel network resource-allocation strategies.
Manuscript received October 30, 2012; accepted September 20, 2013. Date of publication November 20, 2013; date of current version June 12, 2014. This work was supported in part by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61221061; by the National Natural Science Foundation of China under Grant 61271194, Grant 60972007, Grant 61250001, and Grant 61231013; and by the China National 973 Program under Grant 2011CB707000. The work of Y. Li was supported in part by the Australian Research Council under Grant DPI20100190, Grant LP0991663, and Grant FT120100487. The review of this paper was coordinated by Prof. C. Assi. G. Wang and X. Dai are with the School of Electronics and Information Engineering, Beihang University, Beijing 100191, China (e-mail: gwang@ buaa.edu.cn;
[email protected]). Y. Li is with the School of Electrical and Information Engineering, The University of Sydney, Sydney, N.S.W. 2006, Australia (e-mail: yonghui.li@ sydney.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2291859
Pricing is an effective tool to control congestion and achieve QoS provisioning. It is essential to have an effective incentive mechanism of cost-sharing pricing to manage the resource allocation in a congestion network. In the incentive mechanism, each user is associated with a utility function, each link is entitled with a cost function, and the payoff of one user transmitting its information over a link is equal to its utility minus the cost. The design is to maximize the overall network payoff, which is defined as the aggregate payoff of all users in the network, while maintaining each user’s minimum data rate requirement. The cost-sharing pricing has been widely used as an incentive mechanism for allocating resources in communication networks [1]–[6] (see [7] for a comprehensive review). The commonly utilized cost-sharing pricing mechanisms include average cost pricing [8], affine marginal cost (AMC) pricing [9], and Aumann–Shapley pricing [10]. In [9], Johari and Tsitsiklis adopted the AMC pricing mechanism to investigate the loss of overall payoff in network resource allocation, where each user bids for the resource to maximize its payoff. Such a scheme is efficient when all users do not anticipate the effects of their choices on the cost-sharing prices; otherwise, there is a loss of efficiency. Network coding (NC) has been emerged as an effective technique to substantially increase network transmission efficiency [12]–[15]. In [16] and [17], the AMC pricing was investigated for the scenario where both NC and routing are applied in a congestion network. It is shown in [16] that intersession NC is very sensitive to the behavior of the strategies. For example, for the case of only two NC flows sharing a single bottleneck link, the worst-case efficiency bounds, i.e., the pricing of anarchy (PoA), can be as low as 48%. It is shown in [17] that the PoA can be as low as 20% in the butterfly network. In [18], it is shown that network payoff can be improve significantly by playing the same game repeatedly. In [19], an n-unicast problem in a wireless network that employs a restricted form of NC called reverse carpooling was studied. The impact of cost-sharing mechanisms on the global system performance was analyzed, when each unicast independently and selfishly chooses its route to minimize its individual cost. However, in all existing pricing mechanisms, the network payoff is reduced when NC is employed. To increase the network payoff, a more effective pricing mechanism is required for a congestion network using intersession NC. Generally, the AMC pricing mechanism only considers users’ strategies or actions in optimizing network resource without considering the number of users sharing the congestion link. To improve the network transmission efficiency through resource allocation,
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WANG et al.: ON NETWORK SHARING OF MIXED NC AND ROUTING DATA FLOWS IN CONGESTION NETWORKS
Fig. 1.
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Single link shared by K flows containing N NC flows and R routing flows.
the network resource should be allocated dynamically and adaptively as the number of users varies. When the capacity of a communication link is fixed, as the number of sharing users increases, the amount of spectral resource that each user can acquire becomes smaller. Therefore, the sharing price should be determined by the transmission rates demanded by users and by the number of sharing users. To achieve this, we can use a simple market mechanism for link resource allocation, where users adaptively choose their desired transmission rates and optimize their payoff only based on the individual average price (rather than the aggregate price). We propose a novel pricing mechanism referred to as average cost sharing (ACS) to take into account both users’ strategies and the number of sharing users in the network resource optimization. More specifically, in ACS, each user’s cost function is proportional to its action (or transmission rate) and inversely proportional to the number of users sharing the same link. In this paper, we consider a system where NC and routing data flows jointly share one communication network, and they compete network resources in their transmissions. To maximize the overall network payoff, the optimal allocation of network resources between NC and routing data flows needs to be determined. We formulate this optimization problem using a cost-sharing game model, where users determine the network resources based on their own payoff to maximize the overall network payoff. In this game model, the players are the users who choose the transmission protocols and the rate to transmit independently their own data through the single link. The payoff of each player is the profit for information transmission subtracting the cost for the energy consumed in the network. The players try to minimize the cost of the transmission and maximize their own payoff based on the ACS pricing mechanism. We focus on the solution concept of Nash equilibrium (NE), where each player has no incentive to unilaterally change his current choice of resource. However, a NE does not achieve social optimality in general. We will evaluate this inefficiency by studying the PoA [20]–[22]. We analyze the performance of ACS in terms of PoA, which is defined as the ratio between the “worst equilibrium” network payoff of a game and the optimal network payoff in the same
network system. Thus, the PoA implies the loss of network payoff in resource allocation compared with the optimal case, i.e., the higher the PoA, the smaller the loss of network payoff. We formulate an analytical expression to compute PoA under the ACS mechanism. As a result, the PoA increases with the growing number of NC data flows transmitted through the network, leading to a more efficient utilization of limited network resource. This is in contrast to previous work based on AMC [16], where the overall network payoff decreases when NC is applied. The remainder of this paper is organized as follows. In Section II, we present the system model, formulate the congestion game for the ACS pricing mechanism, and analyze its NE. The PoA is analyzed in Section III, and simulation results are provided in Section IV. Conclusions are drawn in Section V.
II. N ETWORK S HARING OF M ULTIPROTOCOL DATA F LOWS A. Problem Formulation Congestion networks usually have a complex topological structure. To gain more insights, we consider an onerous network topology and assume that all user choose either NC or routing to transmit their data by a common single congestion link, as shown in Fig. 1. Here, the common single congestion link between two intermediate nodes w and z is denoted by (w, z). We assume that K(K ≥ 2) independent users share the link to transmit their messages from w to z, where some users use NC, whereas others use routing protocol in their transmissions. Let N = {1, 2, . . . , N } and R = {N + 1, . . . , K} represent the sets of NC and routing users, respectively. Let N = N (N ≥ 2) and R = R = K − N denote the number of elements in the set N and R, respectively. Then, we have K = N + R. NC usually requires that all packets participating in NC must have equal packet length. In this paper, we have assumed that all NC packets are of the same packet length. Let Di denote the packet transmitted from sender si to destination ti . The intermediate node w performs linear NC on packets of N NC
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users. The node w will generate N NC packets using different combinations of N NC users’ messages and send them toward node z that further forwards them to the corresponding destinations t1 , . . . , tN . For each routing user i ∈ R, intermediate node w directly relays its packet to node z, which further forwards it to corresponding destination tN +1 , . . . , tK . For each user i ∈ N ∪ R, let ri represent the desired transmission data rate from its sender si to its corresponding receiver ti . For simplicity, we assume that all NC users transmit at the same data rate, i.e., r1 = r2 = · · · = rN = rNC . The obtained results can be also extended to the scenarios where the different end-to-end flows have different rate ratios. For example, we can assume that the N NC users have the different transmission rates of r1 ≥ r2 ≥ · · · ≥ rN . Then, we set the lowest transmission rate of NC users rN as the encoding rate of all NC users. According to this encoding rate rN , we divide the ith NC user’ traffic into two data flows, with respective rates of rN and ri − rN , which are treated as the NC data flow and the routing data flow and are transmitted through NC and routing protocols at the intermediate nodes. Let q = K i=N +1 ri + rNC denote the overall transmission rate of K users sharing link (w, z). The link will be overloaded when the transmitted rate q exceeds link capacity C. In this paper, we assume that the capacity C of link (w, z) is equal to 1. Hence, the total rate of all users transmitted throughthe link should satisfy the following condition: q = rNC + K i=N +1 ri ≤ C = 1. For simplicity of description, we arrange the rate of routing users in a decreasing order, i.e., rN +1 ≥ · · · ≥ rK ≥ 0. B. ACS Pricing Mechanism To improve the network transmission efficiency through resource allocation, the network resource should be allocated dynamically and adaptively as the number of users varies. Since the capacity of a communication link is limited, as the number of sharing users increases, the amount of spectral resource that each user can acquire decreases. Given the rate demanded by the sharing users, the network sets a price per unit of capacity to optimize its revenue from selling capacity by strategically choosing its price per unit of capacity. The price per unit of capacity affects the cost of the users, and the cost increases with the price. Therefore, the sharing price strongly depends upon the transmission rates demanded by users and the number of sharing users. We propose a novel pricing mechanism referred to as the ACS to take into account both users’ strategies and the number of sharing users in the network resource optimization. As shown in [9], [17], and [23], the price function for costsharing systems must satisfy the four well-known axioms of rescaling consistency, additivity, and positivity, and the linear price functions are the only price functions that satisfy this requirement. Thus, in this paper, we assume a linear price function p(x) = ax, a > 0. Parameter a is a constant value and can take any positive real number. We define the ACS pricing mechanism as follows: i ri i ri =a (1) p n n
where p(x) is the price function, which should be convex and nondecreasing, as detailed in [23]; n denotes the number of users using the same transmission protocol (NC or routing in this paper) on a link; and i ri is the total rates of n users. Let us consider the congestion network setting in Fig. 1. Following the ACS pricing mechanism defined in (1), we establish two discriminatory price functions μ(r) and δ(r) for routing and NC users, respectively, as follows: K i=N +1 ri μ(r) = a (2) R where K i=N +1 ri is the sum rate of all routing users. Obviously, the routing price is affected by the sum rate of all routing users and the number of routing users R. For each routing user i ∈ R, the cost Pi (ri ) imposed by transmission rate ri is equal to ri · μ(r). Similarly, the price function for NC users is r NC (3) δ(r) = a N where rNC is the rate of NC users transmitted through the single link, and N is the number of NC users. The cost of each NC user i ∈ N imposed by transmission rate rNC , which is denoted by Pi (ri ), is given by Pi (ri ) = rNC · δ(r). We define η as the ratio of two discriminatory price functions, i.e., η=
δ(r) . μ(r)
(4)
According to (2) and (3), we can obtain that η=
N
R rNC K
i=N +1 ri
.
Obviously, η depends on the number of users and the transmission rates allocated to NC and routing users. η will increase as the number of users. This is because, when the capacity of a communication link is fixed, as the number of sharing users increases, the amount of spectral resources that each user can acquire becomes smaller. Therefore, the sharing price should be determined by the transmission rates demanded by users and the number of sharing users. C. Payoff Function In the incentive mechanism, each user is associated with a utility function, each link is entitled with a cost function, and the payoff of one user transmitting its information over a link is equal to its utility minus the cost. The design is to maximize the overall network payoff, which is defined as the aggregate payoff of all users in the network, while maintaining each user’s minimum data rate requirement as the payoff of a network is generally the objective function to be optimized via resource allocation. To formulate this optimization problem, let us first define a utility function as the user’s profit for the information transmission. Then, the payoff of each user is its utility minus the cost.
WANG et al.: ON NETWORK SHARING OF MIXED NC AND ROUTING DATA FLOWS IN CONGESTION NETWORKS
The utility function Ui (ri ) for each user i ∈ N ∪ R is usually associated with the user’s strategy, i.e., its transmission rate ri . Thus, utility function Ui (ri ) can be expressed as Ui (ri ) = βi ri
(5)
where βi > 0, i = 1, 2, . . . , K is the utility parameter, and we assume maxi {βi } = b, i = 1, 2, . . . , K. In general, Ui (ri ) should be a nonnegative, increasing, and concave function of ri because each user is interested in taking up more network resource until it reaches its maximum available level [23]. Given the utility and cost functions, we define the payoff function of user i as the difference between the utility and cost, i.e., Si (ri , r −i ) = Ui (ri ) − Pi (ri ). The notation r −i denotes the vector of rates allocated to all users excluding user i. Thus, we have βi ri − ri μ(r), for i ∈ R Si (ri , r −i ) = (6) βi rNC − rNC δ(r), for i ∈ N . Our objective is to maximize the payoff of the whole network subject to the link capacity constraint. This can be formulated as follows:
(Ui (ri ) − Pi (ri )) Maximize i
subject to rNC +
K
ri ≤ C = 1, ri ≥ 0, i = 1, . . . , K.
i=N +1
(7) Objective function (7) is defined as the aggregate payoff. To optimize the network aggregate payoff, the optimal payoff of each user i ∈ N ∪ R should be considered first. For a given r −i , each user i will choose a transmission rate ri ≥ 0 to maximize its payoff, i.e., max Si (ri , r −i ) = max [Ui (ri ) − Pi (ri )] ri ≥0
ri ≥0
i = 1, . . . , K.
(8)
D. Game Model To formally study the interactions between NC and routing users in a congestion link, we formulate a noncooperative game where users interact with each other to acquire a share of the limited resource to maximize their own payoff. For the network setting in Fig. 1, we formulate a game by using the payoff functions defined in Section II-C as follows: G = N ∪ R, {Bi }i∈N ∪R , {Si (ri , r −i )}i∈N ∪R
(9)
where N ∪ R = {1, 2, . . . , K} is the set of players (all the sharing users), and the strategy of each player i ∈ N ∪ R corresponds to the demanded rate ri ∈ Bi . In the formulated game, each user i strategically selects its rate ri ≥ 0 to maximize its payoff Si (ri , r −i ). A system is said
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to achieve a NE when no individual user i ∈ N ∪ R can increase its own payoff by changing its strategy ri ultimately. Let ri∗ and r ∗−i denote the final rate selected by user i and all other users in the game; then, we have Si (ri∗ , r ∗−i ) ≥ Si (ri , r ∗−i ). E. Existence and Nonuniqueness of NE Here, let us first analyze the existence of the NE of the proposed game model. For simplicity of description, we use p(r) to represent both μ(r) and δ(r). Then, we can come to the following conclusion about the NE of the proposed game model. Proposition 1: The NE of the proposed game model (9) exists and is nonunique. Proof: We prove the existence of the NE in two steps. First, given price function p(r), for each user i ∈ N ∪ R, there exists one and only one transmission rate ri that maximizes its own payoff. For an individual payoff function, Si (ri , r −i ) is maximized when ∂p(r) ∂Si (ri , r −i ) = βi − p(r) − ri = 0. ∂ri ∂ri
(10)
∂p(r) when ri > 0. Furthermore, for ∀i ∈ {1, 2, . . . , K}, ∂r i ≥0 Since Ui (ri ) is a concave function of ri , the second-order derivative of Ui (ri ) should satisfy ∂βi /∂ri < 0 for ri > 0. Then, we have
∂ 2 Si (ri , r −i ) ∂βi ∂p(r) = −2 < 0 when ∂ 2 ri ∂ri ∂ri
ri > 0. (11)
Thus, ∂Si (ri , r −i )/∂ri is a monotonically decreasing function of ri . According to the properties of the utility function and the price function, ∀i ∈ {1, 2, . . . , K}, we can easily show that Si (ri , r −i ) is strictly concave. As shown in (10), ∂Si (ri , r −i )/∂ri < 0 as ri → ∞. Thus, we can prove that, given price function p(r), there exists one and only one transmission rate ri for each user i ∈ N ∪ R, which maximizes its own payoff. Next let us find the solution to NE. Since the payoff Si (ri , r −i ) is concave for fixed r −i , vector r ∗−i is a NE if and only if the following first-order conditions are satisfied for each i [9]: ∂Si (ri , r −i ) ∂Ui (ri ) ∂Pi (ri ) = − = 0, ∂ri ∂ri ∂ri βi (0) =
∂Ui (ri ) ∂Pi (ri ) ≤ , ∂ri ∂ri
if
ri > 0 (12)
if ri = 0.
(13)
In a network, transmission rate ri ≥ 0, and ∂Si (ri , r −i )/ ∗ ) is a NE if ∂ri |ri =0 > 0. Thus, vector r ∗ = (r1∗ , r2∗ , . . . , rK ∗ ∗ and only if ∂Si (ri , r −i )/∂ri = 0, i = 1, 2, . . . , K, and βi = p(r) + ri∗ (∂p(r)/∂ri∗ ) for ri∗ > 0 and βi ≤ p(r) for ri∗ = 0 are satisfiedfor each i. Moreover, the solution to the NE should ∗ ∗ satisfy K i=N +1 ri + rNC ≤ C = 1. Based on the given analysis, we can conclude that a NE vector r ∗ exists. We only consider the two blocks of resource allocated to two sets of NC and routing users, respectively, instead of individual resource allocation to each user. As a
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result, given the block of resource allocated to the routing users, an individual routing user’s transmission rate depends on other routing users. Thus, NE is nonunique for this game model under the ACS pricing mechanism in a congested link. This completes the proof of Proposition 1. III. P RICE - OF -A NARCHY A NALYSIS OF THE P ROPOSED AVERAGE -C OST-S HARING P RICING M ECHANISM A. Definition of PoA
K
We have shown that NE exists for this game G under the ACS pricing mechanism. Let us denote by B = B1 × · · · × BK the set of outcomes and by W (ri ) = K i=1 Si (ri , r −i ) the sum of all users’ payoff. Define subset E ⊆ B as a set of strategies in NE. Then, we can define the PoA as the ratio of all users’ payoff in the worst NE and that in the optimal network payoff case as follows: PoA =
minri∗ ∈E W (ri∗ ) . maxri ∈B W (ri )
(14)
Thus, PoA represents the loss of network payoff in resource allocation compared with the optimal case, i.e., the higher the PoA, the smaller the loss of network payoff. Assume r s is the optimal solution for the optimization problem (7) and r ∗ is a feasible NE for the proposedgame. Then, by using the fact that ∂Ui (ri )/∂ri = βi and q = K i=N +1 ri + rNC ≤ 1, we can prove that K i=1 K i=1
Ui (ri∗ ) − Ui (ris ) − ≥
K i=1 K i=1
For the optimal solution, all the utility parameters βi should be equal to the maximumvalue b. The rate of routing users is equal to 1 − rNC , i.e., K i=N +1 ri = 1 − rNC . For the rate allocation of routing users rN +1 ≥ · · · ≥ rK ≥ 0, it has been shown in [9] that the optimal solution that maximizes the routing aggregate payoff is to allocate all the routing rate to the optimal router, router N + 1, i.e., rN +1 = 1 − rNC , rN +2 = · · · = rK = 0, and βN +1 = b. Hence, the aggregate payoff can be rewritten as
Pi (ri∗ ) Pi (ris ) K
min1≥ri∗ ≥0
∗ i=1 βi ri −
K
∗ i=1 Pi ri
. s max1≥q≥0 (maxi βi )q − K P r i=1 i i
(15)
To calculate this lower bound, let us then analyze the optimal case and the worst case of NE in the proposed game.
2 Si (ri , r −i ) = N brNC − arNC
i=1
+ b · (1 − rNC ) −
K
i=1
Si (ri , r −i ) = rNC
N
According to the utility function definition in (5) and payoff condition Si (ri , r −i ) ≥ 0, we can prove that the utility parameter βi ∀i ∈ N ∪ R satisfies βi ≥ max (δ(r), μ(r)) .
i=1
+
K
i=N +1
βi ri − μ(r)
K
C. Worst Case of Aggregate Payoff in NE To determine the network payoff in the worst case, we need to optimize the aggregate payoff function at any NE of game G. The worst case of game G at any NE can be formulated to optimize the following problem:
(16)
i=N +1
using the assumption that rN +1 ≥ · · · ≥ rK ≥ 0 and q = By K i=N +1 ri + rNC ≤ 1, the optimal solution to the optimization problem (7) exists when the total rates transmitted through the single link is allocated with the total capacity 1.
K
(Ui (ri∗ ) − Pi (ri∗ ))
i=1
subject to ri∗ ≥ 0,
i = 1, . . . , K
(20)
∗ ) is a NE for game G, which can be where r ∗ = (r1∗ , r2∗ , . . . , rK obtained by (12) and (13). The aggregate payoff under the NE (20) can be calculated as follows. Proposition 2: The total aggregate payoff is given by K
ri .
(19)
s = Thus, we have b ≥ max(a/N, a/R), which implies rNC (R(N − 1)b + 2a)/(2aR + 2a) ≤ 1. When N ≥ R, we further have (N + 1)/(2(R + 1)) ≥ (R(N − 1) + 2N )/(2(R + 1)N ), and max(a/N, a/R) ≤ b ≤ 2a/(N − 1).
Minimize
2 βi − arNC
(17)
s From (17), the optimal solution can be obtained as rNC = (R(N − 1)b + 2a)/(2aR + 2a). As a result, the maximal aggregate payoff can be expressed as (N −1)2 b2 R+4abK−4a2 K
s , if rNC ∈ [0, 1) 4a(R+1) Si (ri , r −i ) = s N b − a, if rNC = 1. i=1 (18)
B. Optimal NE of Network Aggregate Payoff Let us first analyze the optimal NE for the proposed game. , r −i ) ≥ 0, ∀ri ≥ Since the payoff function should satisfy Si (ri 0, we can easily prove that i=1 Ui (ri ) ≥ i=1 Pi (ri ). The aggregate payoff can be calculated as follows:
a · (1 − rNC )2 . R
∗
)2 a · (q ∗ − rNC ∗ Si ri∗ , r ∗−i = a · (rNC )2 + . R2 i=1
(21)
Proof: We can observe that, for each NC user i ∈ N , vec∗ ∗ /N when rNC > 0, tor r ∗NC is a NE if and only if βi = 2arNC ∗ and βi ≤ δ(r) when rNC = 0. Similarly, for each routing user i ∈ R, ri∗ is a NE of game G when βi = μ(r) + ri∗ · ∂μ(r)/∂ri∗ for ri∗ > 0, and βi ≤ μ(r) for ri∗ = 0.
WANG et al.: ON NETWORK SHARING OF MIXED NC AND ROUTING DATA FLOWS IN CONGESTION NETWORKS
The aggregate payoff under the NE for NC users can be calculated as N
Si (ri∗ , r −i ) =
i=1
N
Ui (ri∗ ) −
i=1
N
(N βi )2 . (22) 4a
ri∗ δ(r) =
i=1
∗ Since there always exists a ri∗ = rNC for each NC user i ∈ N ∗ ∗ , where rNC ≤ N , we can get βi = βNC and i=1 βi = 2arNC bN/2a. Similarly, the aggregate payoff of a NE for the routing users equals to K
K K
Si ri∗ , r ∗−i = Ui (ri∗ ) − ri∗ μ(r)
i=N +1
i=N +1
(23)
K
Ui (ri∗ ) =
i=N +1
ri∗ βi
i=N +1
1 ≥ R =
K
ri∗
i=N +1
K
βi
i=N +1
∗ )2 a(R + 1) (q ∗ − rNC . R2
(24)
As the routing users use the same price function μ(r), we can obtain K
ri∗ μ(r) =
i=N +1
∗ )2 a (q ∗ − rNC . R
(25)
By using (24) and (25), we can prove that the total aggregate payoff satisfies the following inequality: K
∗
)2 a(R + 1) (q ∗ − rNC ∗ Si ri∗ , r ∗−i ≥ a (rNC )2 + R2 i=1
−
∗ )2 a (q ∗ − rNC R
∗ = a (rNC )2 +
∗ )2 a (q ∗ − rNC . R2
(26)
This completes the proof of Proposition 2. According to the given results, we can calculate the lower bound of the PoA, as shown in the following proposition. Proposition 3: The PoA under the ACS pricing mechanism can be lower bounded by 1 ≥ PoA ≥
Proof: According to the definition of PoA in (14), (17), and (21), this result is obvious. This completes the proof of Proposition 3. D. Implementation of the Proposed Algorithm Based on the given analysis, in the following, we describe in details how the proposed algorithm works in networks. We assume that there are already some number of users transmitting their packets in the network and that they have reached the NE in resource allocation. Next, let us consider the case when a new node joins the network. Algorithm 1 describes how the proposed algorithms dynamically allocate the resources to the new user to transmit its packets.
i=N +1
∗ ∗ q∗ = K i=N +1 ri + rNC , and the sum of the routing utility K ∗ ∗ ∗ ∗ parameters is i=N+1 βi = a · (q −rNC )+(a · (q −rNC )/R. From the conditions of NE r ∗ and the given condition rN +1 ≥ rN +2 ≥ · · · ≥ rK ≥ 0, we can infer that βN +1 ≥ βN +2 ≥ · · · ≥ βK > 0. Therefore, according to the Chebyshev’s sum inequality, we have K
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s − a (r s )2 + N brNC NC
1: Initialization: Set rNC , G = N ∪ R, {Bi }i∈N ∪R , {Si (ri , r −i )}i∈N ∪R 2: when a new node u would like transmit its packets via a common single congestion link (w, z), at the transmission rate ru 3: if u chooses to participate NC then 4: the intermediate node w calculates the new NC users’ transmission rate according to rNC = min{rNC , ru } 5: the intermediate node w broadcasts rNC and N + 1 to all users in the set N 6: each user i ∈ N calculates the NC price function according to (3) 7: else 8: u chooses to participate routing 9: the intermediate node w broadcasts the new routing users’ transmission rate ru and R + 1 to all users in the set R 10: each user i ∈ R calculates the routing price function according to (2) 11: end if 12: each user i ∈ N ∪ R calculates its own payoff function according to (6) and broadcast it to the others user in the set of N ∪ R 13: all of the user in the set of N ∪ R jointly calculate the optimal NE of the network aggregate payoff according to (17) iteratively. Then, each user i ∈ N ∪ R obtains a new transmission rate ri 14: each user i ∈ N ∪ R transmits {Si (ri , r −i )}i∈N ∪R and ri to the intermediate node w 15: the intermediate node w updates the game model G 16: for each user i ∈ N , the intermediate node w allocates a random NC coefficient, encodes the NC user’s packets, and transmits the encoded packet to node z at the transmission rate of rNC 17: for each user i ∈ R, the intermediate node w transmits its packets by direct routing at the transmission rate of ri
2
∗ a(q ∗ −rNC ) R2 s ) b (1 − rNC
∗ )2 + a (rNC
Algorithm 1 Dynamic Resource-Allocation Algorithm when a new node joins the network transmission.
−
s )2 a(1−rNC R
.
(27)
When some NC users or routing users leave the network, the price function would be also varied, and the intermediate node
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w has to reallocate the channel resources to the remaining users. This process is decrypted in Algorithm 2 as follows. Algorithm 2 Dynamic Resource-Allocation Algorithm when one node leaves. 1: Initialization: Set rNC , G = N ∪ R, {Bi }i∈N ∪R , {Si (ri , r −i )}i∈N ∪R 2: when a node v with the transmission rate of rv leaves the network 3: if node v is a NC user then 4: the intermediate node w updates the NC users set N = N − {v} 5: the intermediate node w calculates the NC users’ transmission rate according to rNC = min{ri }, i ∈ N 6: the intermediate node w broadcasts rNC and N − 1 to all users in the set N 7: each user i ∈ N calculates the NC price function according to (3) 8: else 9: node v is a routing user 10: the intermediate node w updates the routing users set R = R − {v} 11: the intermediate node w broadcasts rv and R − 1 to all users in the set R 12: each user i ∈ R calculates the routing price function according to (2) 13: end if 14: each user i ∈ N ∪ R calculates its own payoff function according to (6), and broadcasts it to the others user in the set of N ∪ R 15: all of the user in the set of N ∪ R jointly calculate the optimal NE of network aggregate payoff according to (17) iteratively. Then, each user i ∈ N ∪ R obtains a new transmission rate ri 16: each user i ∈ N ∪ R transmits {Si (ri , r −i )}i∈N ∪R and ri to the intermediate node w 17: the intermediate node w updates the game model G 18: for each user i ∈ N , the intermediate node w allocates a random NC coefficient, encodes the NC users’ packets, and transmits the encoded packets to node z at the transmission rate of rNC 19: for each user i ∈ R, the intermediate node w transmits its packets by direct routing at the transmission rate of ri From the given analysis, to achieve the NE, each user i ∈ N ∪ R needs to interact with the intermediate node w to determine its transmission rate. Specifically, the intermediate node w needs to calculate and transmit the new number of users and transmission rate to each user when nodes join or leave the network. Based on this information, every user i ∈ N ∪ R jointly calculates the optimal NE of the network aggregate payoff according to (17) iteratively and sends it to the intermediate node w. This is the only communication overheads and cost required to implement the proposed distributed resourceallocation algorithms in practice.
Fig. 2.
PoA under AMC and ACS when N = 2, R = 0, and 0 < η ≤ 1.
IV. S IMULATIONS To gain more insights into our theoretical results and understand the competition and selfish behaviors of sharing users on the limited resources in the congestion network, here, we present some simulation results to validate our analysis. We assume that all assumptions and definitions in Section II hold, and the capacity of single link (w, z) is equal to 1. We first consider the same setup as [17] where the network consists of only two NC users, i.e., N = 2. When N = 2, R = 0, and 0 < η ≤ 1, based on the analysis in Sections II and III, we can have the following calculations: 1) under the single pricing scheme, i.e., η = 1, PoA = 8/9; 2) under the discriminatory pricing scheme, i.e., η = 1/2, PoA = 16/25. Proof: See Appendix A. From these results under ACS pricing mechanism and from [17, Th. 7] under AMC pricing mechanism, we can see that, when η = 1, PoA = 8/9 < 1/3. When η = 1/2, PoA = 16/25 > 12/25. Moreover, when N = 2, R = 0 and 0 < η ≤ 1. Fig. 2 shows the simulation results and compares the PoA achieved by the AMS and the proposed ACS pricing mechanisms. Based on (4) and the simulation results in Fig. 2, we can obtain the following conclusions. 1) When 0 < η ≤ 1, the PoA strictly increases for the proposed ACS mechanism but decreases for the AMS mechanism. This indicates that, by using the proposed ACS mechanism, the NC can improve the PoA. This is in contrary to the conventional AMC mechanism where the PoA decreases as the NC applies. 2) When η ≤ 0.387, i.e., the pricing of NC users δ(r) ≤ 0.387μ(r), the PoA of the ACS mechanism is smaller than that of the AMC mechanism, meaning that the AMC is better than the ACS in this region. 3) When 0.387 < η ≤ 1, i.e., the pricing of NC users 0.387μ(r) < δ(r) ≤ μ(r), the PoA of ACS is larger than that of AMC, meaning that ACS performs better in this region. Then, let us evaluate the performance of the network with mixed NC and routing users. We assume that the total number of users is K = 100. Fig. 3 shows the PoA of the proposed ACS mechanism versus the ratios of NC and routing users under the different utility parameter η.
WANG et al.: ON NETWORK SHARING OF MIXED NC AND ROUTING DATA FLOWS IN CONGESTION NETWORKS
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Fig. 4. PoA with the number of NC users and routing users varies under η = 0.5. Fig. 3. PoA under ACS with the different ratios of NC users and routing users under the different η. TABLE I N UMBER OF NC U SERS N EED TO S ATISFY THE P OA U NDER ACS W ITH THE D IFFERENT η
In Fig. 3, we can obtain some conclusions as follows. 1) Given η, the PoA is a strictly increasing function of the ratio of NC users and routing users under ACS mechanism. This means that PoA will increase as the ratio of NC users increases and will achieve its maximum value when all the users in the network use NC protocols. 2) when η ≥ 0.5, i.e., the pricing of NC users δ(r) ≥ 0.5μ(r), the PoA increases rapidly even when the ratios of NC users and routing users are very little. To show this result clearly, the least number of NC users required to satisfy a specifc PoA under the different utility parameter η is shown in Table I as follows. Finally, let us investigate the effect of user dynamics on the PoA, i.e., how PoA will change with the number of nodes in the network increases or decreases. Fig. 4 shows a 3-D figure showing the PoA versus the number of NC and routing users. From this figure, we can see that PoA is only dependent on the ratio of NC and routing users, rather than the number of NC users in the network. Meanwhile, the PoA is still a strictly increasing function about the ratio of NC users and routing users under ACS mechanism. V. C ONCLUSION In this paper, a game-theoretic approach has been developed for allocating limited network resource between data streams using two different transmission protocols, i.e., routing and NC, to maximize the overall network payoff of a congestion network. We proposed a novel ACS pricing mechanism where
the price was designed based on the average network resource allocated to each user. The PoA of the proposed game under the ACS mechanism has been analyzed. The results are validated through simulations. It is shown that, in contrast to previous AMC pricing mechanism, the network efficiency under the ACS mechanism considerably improves when NC is employed in the network. Specifically, the PoA increases as the ratio of NC and routing users and/or the resource allocated to NC data flows increases, leading to a more efficient utilization of limited network resource. A PPENDIX P ROOF OF P RICE OF A NARCHY W HEN N = 2 According to the definition of (3) and (6), for user 1, the payoff function can be calculated as S1 (r1 , r 1 ) = U1 (r1 ) − P1 (r1 ) = β1 x1 − a ·
x21 . 2
(28)
To obtain the opimization value, let ∂S1 (r1 , r 1 ) = β1 − ax1 = 0. ∂r1
(29)
Then, x∗1 = β1 /a; for user 2, we have x∗2 = β2 /a. For the optimal NE of network aggregate payoff, according to (16), we have 2
Si (ri , r −i ) = (β1 + β2 ) · xNC − a · x2NC .
(30)
i=1
From (30), the optimal solution can be calculated as 2
i=1
SiS (ri , r −i ) =
(β1 + β2 )2 4a
(31)
when x∗NC = (β1 + β2 )/2a. Then, let us compute the worst case of aggregate payoff in NE. Without loss of generality, let β1 ≥ β2 ≥ a/2; the node w set the NC rate as min(x∗1 , x∗2 ) = β2 /a for the worst case. Then, the solution is given by 2 2
β2 β2 β1 β2 . (32) Si∗ (ri , r −i ) = (β1 + β2 ) − a = a a a i=1
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Based on (31) and (32), we can obtain PoA =
4β1 β2 . (β1 + β2 )2
(33)
In (33), it is shown that the PoA depends on parameters β1 and β2 . Let β1 /β2 = k; then, (33) can be rewritten as PoA =
4k . (k + 1)2
(34)
In (34), we can see that PoA is obviously a decreasing function of k. According to [17, Th. 6], parameter k is inversely proportional to parameter η, for 0 < η ≤ 1; thus, we can easily deduce the following conclusions. 1) Under the single pricing scheme, i.e., η = 1, PoA = 8/9 when k = 2. 2) Under the discriminatory pricing scheme, i.e., η = 1/2, if 1 < k ≤ (1+1/η), then PoA ≥ 4(1+1/η)/(1+1/η+1)2 = 3/4, and if 1 + 1/η < k ≤ 2/η, then PoA ≥ (4 · 2/η)/ (1+2/η)2 = 16/25. Therefore, we have PoA = min(3/4, 16/25) = 16/25. This completes the proof.
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Gang Wang (M’13) received the Ph.D. degree from Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, China, in 2003. From 2003 to 2006, he was a Postdoctoral Research Associate with Beihang University, where he is currently an Associate Professor with the School of Electrical and Information Engineering. His current research interests include wireless networks, with a particular focus on network coding, information dissemination, and resource allocation.
Xia Dai received the Master’s degree from Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, China, in 2013. Her current research interests include information network economics, with particular focus on game theory, congestion networks, network coding, and resource allocation optimization.
Yonghui Li (M’04–SM’09) received the Ph.D. degree from Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, China, in 2002. From 1999 to 2003, he was with Linkair Communication Inc., Santa Clara, CA, USA, where he was a Project Manager responsible for the design of physical-layer solutions for large-area-synchronous code-division multiple-access systems. Since 2003, he has been with the Center of Excellence in Telecommunications, The University of Sydney, Sydney, Australia. He is currently an Associate Professor with the School of Electrical and Information Engineering, The University of Sydney. He is a holder of a number of patents both granted and pending. His current research interests include wireless communications, with a particular focus on multiple-input–multiple-output systems, cooperative communications, coding techniques, and wireless sensor networks. Dr. Li was the Australian Queen Elizabeth II Fellow and is currently the Australian Future Fellow. He serves as an Executive Editor for the E UROPEAN T RANSACTIONS ON T ELECOMMUNICATIONS. He has also been involved with the technical committees of several international conferences, such as the IEEE International Conference on Communications and the IEEE Global Communications Conference.
