Price-based resource allocation for wireless ad hoc networks with multi-rate capability and energy constraints

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Price-based resource allocation for wireless ad hoc networks with multi-rate

capability and energy constraints

Yu-Fen Kao

a,b,*

, Jen-Hung Huang

a

Department of Management Science, National Chiao Tung University, Hsin-Chu 300, Taiwan

b_{Department of Information Management, Chung Hua University, Hsin-Chu 300, Taiwan}

a r t i c l e

i n f o

Article history: Received 1 July 2007

Received in revised form 6 June 2008 Accepted 15 June 2008

Available online 24 June 2008 Keywords: Ad hoc network Nonlinear programming Pricing Resource allocation Wireless communication

a b s t r a c t

Wireless ad hoc networks have attracted a lot of attentions recently. Resource allocation in such networks needs to address both fairness and overall network performance. Pricing is a prospective direction to reg-ulate behaviors of individual nodes while providing incentives for cooperation. In this work, we develop some pricing strategies for resource allocation by taking account of factors like multiple transmission rates and energy consumption of nodes, which have not been well studied in former works. Multi-rate transmission capability is commonly seen in most wireless products nowadays, while energy is one of the most important resources in portable devices. We propose a clique-based model which allows us to achieve optimal resource utilization and fairness among network flows when multi-rate transmission is considered. We also show how to extend the model to dynamically adjust prices based on energy con-sumptions of flows. In particular, our model takes into account energy concon-sumptions in the transmitters’ side, the receivers’ side, and those that are non-transmitters and non-receivers but are interfered by these activities. So our model can more accurately reflect the real energy constraint in a wireless network. Sim-ulation results are presented to show the convergence and other properties of these strategies.

1. Introduction

In recent years, we have seen rising demand for mobile comput-ing and communication services. The tremendous advancement in wireless network technologies has made the dream of ‘‘communi-cation anytime and anywhere” realizable. Users can experience full mobility, while at the same time maintaining the ability to connect with others as well as the Internet. Wireless networks provide peo-ple a more durable and ﬂexible way of communications. Successful wireless communication systems include GSM, PHS, 3G WCDMA, and WLAN (WiFi) systems.

One wireless network conﬁguration that has become a popular subject of research is the mobile ad hoc network (MANET) [2,5– 7,17,21–23]. A MANET is comprised of a collection of wireless nodes without a pre-existing infrastructure. Any device with a microprocessor and a wireless interface, whether highly mobile or static, may serve as a potential node in a MANET. Each node in the network acts as a router to relay data packets for others. Each ﬂow may travel over multiple hops of wireless links from its origin to its destination. In a MANET, multi-hop routing can achieve high degree of network connectivity, but this requires

the willingness of each node to forward packets for others. How-ever, constrained by limited power and communication resources, a selfish node may be reluctant to relay packets of others, but ex-pect others to relay its packets. Compared to wired networks, mul-ti-hop MANETs have several special characteristics as opposed to wireline networks. For example, nodes may suffer from a higher degree of interference and energy resources are more constrained. Also, since competition is related to the geographic distribution of nodes, some flows may unfairly consume more resources (such as bandwidths and energies) than others. This raises the problem of designing proper resource allocation mechanisms to encourage cooperation among nodes in such a way that competing multi-hop flows can share scarce channel as well as battery resources in a fair way, while the whole utility of all flows is maximized.

The aim of this paper is to explore the possibility of using price as incentives in multi-hop MANETs to encourage nodes to acquire resources in a reasonable way to maximize the aggregated utility (i.e., social welfare) of flows with fairness in mind. The use of pric-ing as a tool for allocatpric-ing resources in communication networks has drawn a lot of attention recently. Both utility and pricing are not new concepts and have been studied in economics for a long time. Utility is to reflect the level of users’ satisfaction from con-suming a resource and price is the cost per unit of resource charged to users. The intention is to influence users’ behaviors through pricing to achieve certain desired results, such as improving the overall system utilization and maintaining fairness among users.

* Corresponding author. Address: Department of Information Management, Chung Hua University, No. 707, WuFu Road, Sec. 2, Hsin-Chu 300, Taiwan.

E-mail addresses: yfkao@chu.edu.tw (Y.-F. Kao), jhh@ms1.hinet.net (J.-H. Huang).

Contents lists available atScienceDirect

Computer Communications

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In wireline networks, pricing mechanisms have been studied in

[3,8–11,13]. In wireless networks, a number of works[16,19,25]

have introduced pricing mechanisms to improve resource manage-ment. In the context of wireless LANs, price-based resource alloca-tion strategies have found applicaalloca-tion in power control[19] and call admission control[4]. However, these models only concentrate on single-hop infrastructure wireless networks. Price-based ap-proaches to bandwidth allocation in multi-hop MANETs are pro-posed in [18,24]. In [18], an iterative price and rate adaptation algorithm is proposed assuming that users set prices for forwarding packets to maximize their own net benefits. The result shows that using pricing to stimulate cooperation will generate a socially opti-mal bandwidth allocation, i.e., maximization of the total utility of all users. Ref.[24]introduces the concept of clique into the resource allocation problem to accommodate the unique characteristics of contention among wireless nodes. Based on this new model, they present a new pricing policy for end-to-end multi-hop flows. Their simulation results demonstrate that pricing can indeed lead to the maximization of aggregated utility of flows as well as fairness among flows.

In this work, we are interested in IEEE 802.11-based MANETs. IEEE 802.11[12]is one of the most widely used broadband wire-less access systems nowadays. In this particular domain, we ob-serve that there are important characteristics of MANETs that have not been carefully studied in existing works. First, the trans-mission rate of a wireless link is in fact environment-sensitive. Most of today’s wireless interfaces can support multiple modula-tions and thus can transmit at a wide range of rates. Second, trans-mitting a packet in IEEE 802.11 incurs energy consumptions not only at the transmitter and the receiver sides, but also at neighbor-ing stations of the transmitter and the receiver. We name the latter the idle-listening energy cost. It follows, interestingly, that the en-ergy cost incurred by a transmission also depends on the number of neighboring nodes. Without taking these factors into account, existing models can not accurately capture prices that should be charged to trafﬁc ﬂows in a MANET. Based on these observations, we then propose new pricing strategies for resource allocation in a MANET. Our contributions are twofold. First, by including the fac-tors of multiple transmission rates and prices of idle-listening en-ergy consumptions, our model and thus the derived results are more realistic. Second, we demonstrate that it is still feasible to use prices to control behaviors of nodes in a MANET to achieve maximal system utilization with proper fairness among nodes.

The rest of this paper is organized as follows. Some backgrounds are given in Section2. Section3presents our clique-based resource allocation strategy with multi-rate constraint. Section4further ex-tends our resource allocation strategy with both multi-rate and en-ergy constraints. Section 5 reports our experimental results. Finally, Section6concludes the paper.

2. Backgrounds and related works

We are interested in pricing mechanisms in IEEE 802.11-based MANETs. In this particular domain, we observe that there are two important characteristics of MANETs that have been ignored in existing works. First, the transmission rate of a wireless link is environment-sensitive. Most of today’s wireless interfaces can sup-port multiple modulations and thus can transmit at a wide range of rates. For example, IEEE 802.11b can operate at rates of 1, 2, 5.5, and 11 Mbps, while with OFDM (orthogonal frequency division multiplexing), IEEE 802.11a can support a wide range of rates of 6, 9, 12, 18, 24, 36, 48, and 54 Mbps. Second, transmitting a packet in IEEE 802.11 incurs energy costs not only at the transmitter and the receiver sides, but also at the neighboring stations of the trans-mitter and the receiver. For example, an evaluation shows that an

IEEE 802.11b card at transmit, receive, monitor, and sleep modes would cost around 280, 180, 70, and 10 mW, respectively[20]. When two nodes are communicating, a node that is within the transmitter’s transmission range will overhear the wireless signal, decode the packet, and eventually drop it because it is not the in-tended receiver. These receiving activities do not benefit the over-hearing node but would still cause significant energy consumption to the overhearing node. We name this the idle-listening energy cost. Experiences show that idle-listening energy cost is not much less than real receiving energy cost. It follows, interestingly, that the energy cost incurred by a transmission also depends on the number of neighbors of the transmitter. Further, because the IEEE 802.11 MAC protocol also requires extra control packets being sent by the receiver, there is also extra energy cost incurred to neigh-boring nodes of the receiver. This leads to an observation that the total energy consumption incurred by a multi-hop traffic flow in a MANET also depends on the number of neighboring nodes of the routing path. Based on these observations, we will propose our pricing strategies in a MANET.

Utilizing pricing as a means for fostering cooperation in a MAN-ET has been studied in[18]. However, it assumes a simpliﬁed mod-el, where each node k has a transmission capacity of Ck, which is disassociated with other nodes. This model ignores the unique characteristic of inter-node interference in wireless communica-tions. In[24], it is shown that cliques (to be deﬁned later) can bet-ter characbet-terize the inbet-terference nature. However, it is assumed that the channel capacity for each wireless link is equal. Thus, the multi-rate nature of wireless communications is ignored. Fur-ther, in both works, the factor of energy consumptions is ignored. A comparative study of two price-based algorithms is in [15], where it is shown that the gradient projection method has a better convergence property, but at the cost of performance.

Our work will model the prices by nonlinear programming techniques[1]. We will adopt the Lagrangian Primal–Dual solution, which is summarized as follows. Consider the following nonlinear problem P, which is called the primal problem.

maximize f ðxÞ

subject to giðxÞ 6 0 for i ¼ 1; . . . ; m ð1Þ

Several problems, closely associated with the above primal prob-lem, have been proposed and are called dual problems. Among the various dual functions, the Lagrangian dual function has perhaps drawn the most attention. The Lagrangian form of the optimization problem P is deﬁned as follows:

Lðx; kÞ ¼ f ðxÞ X

m

i¼1

kigiðxÞ: ð2Þ

where kiP0 is the Lagrange multiplier associated with the inequal-ity constraint giðxÞ 6 0. The Lagrange dual function hðkÞ is deﬁned as the maximized Lðx; kÞ over x, i.e.,

hðkÞ ¼ supx2XLðx; kÞ; ð3Þ

where sup stands for the least upper bound, or the supremum. The La-grange dual problem D is presented below.

minimize hðkÞ

subject to k P 0: ð4Þ

The optimal primal and dual objectives are equal. Any algorithms that ﬁnd a pair of primal–dual variables ðx; kÞ that satisfy the KKT optimality condition would solve the primal and its dual problem. One possible approach is to use the gradient projection method

[1], which updates the dual variables k to solve the dual problem D:

kðt þ 1Þ ¼ kðtÞ

a

ohðkðtÞÞ

ok

þ

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where t is the iteration number and

a

>0 is the step size. Certain choice of step sizes guarantee that the sequence of dual variables kðtÞ will converge to the dual optimal kas t ! 1. The primal var-iable xðkðtÞÞ will also converge to the primal optimal varvar-iable x_. 3. Resource allocation with transmission rate constraint 3.1. Network and contention models

We are given a multi-hop MANET. Each node has a maximum transmission distance of dtx. Two nodes are able to communicate with each other if their distance is no larger than dtx. Wireless channels are considered as resources. When a node is transmitting a packet, any node that is within the interference distance of dint can detect the carrier from its radio interface, where dintPdtx, and thus is prohibited from transmitting and receiving. We assume that each radio interface can support multiple modulations, and thus can transmit at multiple rates of r1;r2; . . . ;rm. Without loss of generality, let r1>r2> . . . >rm. The rate that a node can trans-mit depends on its distance to the receiver. Let d1;d2; . . . ;dmbe m distances such that d1<d2< . . . <dm¼ dtx. We assume that a transmitter can successfully transmit to a receiver at the rate of ri if the distance between them is no larger than di. The concept is illustrated inFig. 1. We assume that a node can determine, from past experience, the transmission rates that it can use with each neighboring node and will always choose the best (highest) rate for use.

We are interested in solving the resource allocation problem in a MANET by modeling the power consumption incurred by a rout-ing path by takrout-ing into account the energy cost for transmission, reception, and inter-node interference along the path. The network is modeled by a graph G ¼ ðV; EÞ, where VðGÞ is the set of mobile nodes and EðGÞ is the set of wireless links. For any two nodes u; v 2 VðGÞ, a link ðu; vÞ is included in EðGÞ iff their distance dðu; vÞ 6 dtx. For each link e ¼ ðu; vÞ 2 EðGÞ, depending on the dis-tance dðu; vÞ, we denote by rðeÞ the best transmission rate for e. We are also given a set of n traffic flows F in G. Each flow fi2 F; i ¼ 1 . . . n, goes from one source node to a destination node via a predefined routing path (typically a shortest path). The set of wireless links that are traversed by fiis denoted by EðfiÞ EðGÞ. The goal is to calculate a rate allocation vector A ¼ ðrðf1Þ; rðf2Þ; . . . ; rðfnÞÞ

such that each ﬂow fican transmit at the rate of rðfiÞ; i ¼ 1 . . . n. We will formulate the objectives and constraints later on.

3.2. Clique-based rate allocation strategy

Below, we will derive our node interference model. Then we will present our rate allocation problem, followed by an iterative scheme to solve this problem. Our results are based on[14,24]with some extensions.

First, we will formulate the constraints of inter-node interfer-ence by modifying the model in[24]. Since flows in G will contend with each other for transmission, we first convert G into a link con-tention graph Gc¼ ðVc;EcÞ [14]. Each link in EðGÞ of the original graph G is converted into a vertex in Vc. Each pair of links e1and e2in EðGÞ with a contention relation is converted to a link ðe1;e2Þ in Ec, where a contention relation is established if the distance be-tween any endpoint of e1and any endpoint of e2is 6 dint. The rea-son for such a definition is to model the behavior of the IEEE 802.11 MAC protocol, as shown inFig. 2. For each data packet being trans-mitted on a wireless link, RTS/CTS/ACK control packets need to be sent. This calls for two-way communications, so we can model the contention relation without regarding the directions of flows.

With graph Gc, we define our clique-based rate allocation prob-lem as follows. In a graph, a complete subgraph is called a clique. A maximal clique is a clique such that no other clique is its superset. The set of all maximal cliques, or simply cliques, in Gcis denoted by Q.Fig. 3shows a network G and its corresponding Gc. Two example maximal cliques (marked by dotted circles) are identified inFig. 3. For each q 2 Q, the set of vertices of q (i.e., the set of wireless links in EðGÞ which forms clique q) is denoted by VðqÞ. Maximal cliques (or simply called cliques below) in Q will be the units of resource allocation in our scheme. For any feasible rate allocation vector A and for each link e that is traversed by fi, the air time ratio rðfiÞ=rðeÞ is the amount of air time occupied by fiper time unit. Be-cause no two members in a maximal clique are allowed to transmit at the same time (otherwise, collision will happen), this enforces that the sum of air time ratios seen by all links belonging to the same clique be no more than 100%. More specifically, for each cli-que q 2 Q, the total of air time ratios occupied by all links of all flows that go through q at any time unit must be no more than 100%, i.e.,

Fig. 1. Relationship of transmission distances and rates.

Fig. 2. IEEE 802.11 MAC protocol.

7 8 6 5 4 3 2 1 10 9 (8,3) (9,10) (7,8) (5,6) (1,2) (3,4) (4,5) (2,3) (4,9)

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(b)

Fig. 3. (a) network G and (b) link contention graph Gcand two example maximal

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8

q 2 Q : X 8e2VðqÞ X 8fi2F:e2EðfiÞ rðfiÞ rðeÞ 0 @ 1 A 6 1: ð6Þ

For example, the total of air time ratios of members in each of the dotted circles inFig. 3should be bounded by 100%. We say that a

rate allocation vector A is feasible if all inequalities in Eq.(6)are satisﬁed.

We now present our price-based resource allocation scheme with the above air time constraints. Our derivation will be based on a social welfare model to calculate a rate allocation vector A such that the total utility of all flows is maximized and fairness among flows is maintained. We will associate with the rate rðfiÞ of each fia utility function UðrðfiÞÞ, which represents the degree of satisfaction of figiven rate rðfiÞ. Following typical definitions of utility, we as-sume that the function UðÞ is strictly increasing, concave, and twice continuously differentiable. The primal problem P can be for-mulated by a nonlinear optimization problem as follows:

maximize X 8fi2F UðrðfiÞÞ subject to

8

q 2 Q : X 8e2VðqÞ X 8fi2F:e2EðfiÞ rðfiÞ rðeÞ 0 @ 1 A 6 1: ð7Þ

The goal is to maximize the total of all flows’ utilities. However, be-cause of the way that utility functions are defined, it also has a sense of fairness behind. Since traffic flows have to compete with each other, they have to share the resources provided by cliques.

Fig. 4. Power consumption model. For each node, the corresponding Px=Pymeans

the energy consumption incurred by transmissions of u=v, respectively.

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The way utility functions are defined will enforce a flow’s utility to gradually saturate as more and more resources are taken by it. Intu-itively, when approaching the saturation point, it would be better to reduce its traffic rate and give the saved resource to other traffic flows, which may generate higher utility margins. This is what we mean by social welfare. Also, utility functions are based on users’ psychological feelings to prices and can be defined differently. Sev-eral examples of utility functions can be found in our simulations.

In order to solve problem P, we turn our attention to the dual problem D of P deﬁned as follows. For each q 2 Q, let

l

qbe the cost of the usage of one air time unit charged by clique q. Problem D is deﬁned as the following min–max problem:

min 8l1;l2;...;ljQj max 8rðf1Þ;rðf2Þ;...;rðfnÞ fDðrðf1Þ; rðf2Þ; . . . ; rðfnÞ;

l

1;

l

2; . . . ;

l

jQ jÞg ; where Dðrðf1Þ; rðf2Þ; . . . ; rðfnÞ;

l

1;

l

2; . . . ;

l

jQ jÞ ¼X 8fi2F UðrðfiÞÞ X 8e2EðfiÞ X 8q2Q:e2VðqÞ rðfiÞ rðeÞ

l

q ! 0 @ 1 A þX 8q2Q

l

q; ð8Þ under the same constraints as in P, where the expression inside the first summation can be considered as the net benefit of flow fiand the second term can be considered as the total value of the potential capacities of all cliques that can be offered to flows. Eq.(8)can be rewritten as Dðrðf1Þ; rðf2Þ; . . . ; rðfnÞ;

l

1;

l

2; . . . ;

l

jQ jÞ ¼X 8fi2F UðrðfiÞÞ rðfiÞ X 8q2Q

l

q X 8e2EðfiÞ:e2VðqÞ 1 rðeÞ 0 @ 1 A 0 @ 1 A þX 8q2Q

l

q; ð9Þ which satisﬁes the Lagrangian form of the optimization problem P, where ð

l

1;

l

2; . . . ;

l

jQ jÞ is a vector of Lagrange multipliers. In Eq.(9), the term X 8q2Q

l

q X 8e2EðfiÞ:e2VðqÞ 1 rðeÞ 0 @ 1 A ð10Þ

can be regarded as the unit path cost charged to flow fi. From Eq.(10), we see that the difference between our formulation and that of[24]is that we take into account the actual air time occupied for a flow in each clique, while[24]only counts the number of links appearing in each clique. This does matter when two links belong to the same clique, one transmitting at a higher speed and the other transmitting at a lower speed; although they may transmit the same amount of information, the occupied air time ratios should be differentiated. Thus, our formulation can more accurately model the cost charged to each flow.

Next, we develop an iterative algorithm to determine the rate allocation vector A. Intuitively, each clique can be regarded as a pro-vider and each ﬂow can be regarded as a buyer. Clique q may gradu-ally adjust its unit price

l

qdepending on the demands of buyers. On the other hand, each buyer fimay gradually adjust its ﬂow rate rðfiÞ depending on its current utility value and the accumulated price charged by all cliques that it will go through. More speciﬁcally, the algorithm goes in a sequence of steps. At step t, the unit cost of each clique q is denoted by

l

qðtÞ, and the rate of each flow fiis denoted by rðfi;tÞ. In each iteration, the clique costs will be updated first, fol-lowed by updates of flow rates. The algorithm is a distributed one executed by individual cliques and sources of flows.

A1. For each clique q, one node Lqis pre-elected as the leader of that clique. Lqthen collects the rate rðfi;tÞ of each fisuch that EðfiÞ \ VðqÞ 6¼ ;. (How to elect a leader is trivial, so we omit the details.)

A2. Lqwill determine the price of q in the next step t þ 1 based on its current price at step t using the gradient projection method[1]as follows:

l

_qðt þ 1Þ ¼

l

_qðtÞ

c

oDðÞ

o

l

q

" #þ

; ð11Þ

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where

c

is a small step size and ½þ_{will return 0 when the} va-lue inside the brackets is negative. Since the utility function is strictly concave, DðÞ is continuously differentiable. From Eq.

(8), Lqcan derive that

oDðÞ o

l

q ¼ 1 X 8fi2F X 8e2EðfiÞ:e2VðqÞ rðfiÞ rðeÞ 0 @ 1 A: ð12Þ

Plugging Eq.(12)into Eq.(11), Lqdetermines its unit price in step t þ 1 as

l

qðt þ 1Þ ¼

l

qðtÞ

c

1 X 8fi2F X 8e2EðfiÞ:e2VðqÞ rðfiÞ rðeÞ 0 @ 1 A 0 @ 1 A 2 4 3 5 þ : ð13Þ

Then Lqsends the updated price

l

qðt þ 1Þ to all members in VðqÞ.

A3. On receiving

l

qðt þ 1Þ, each e 2 VðqÞ notifies the updated price to each flow that goes through it. Each flow should for-ward the new price to its source node.

A4. When the source of ficollects all updated prices at step t þ 1, it derives its updated net beneﬁt function as

BðrðfiÞÞ ¼ UðrðfiÞÞ X 8e2EðfiÞ X 8q2Q:e2VðqÞ rðfiÞ rðeÞ

l

qðt þ 1Þ ! ð14Þ

and takes the ﬁrst derivative of BðrðfiÞÞ by setting it to 0

oBðrðfiÞÞ orðfiÞ ¼ U0_ðrðf iÞÞ X 8e2EðfiÞ X 8q2Q:e2VðqÞ 1 rðeÞ

l

qðt þ 1Þ ! ¼ 0: ð15Þ

The next injection rate that would maximize its net beneﬁt is

rðfi;t þ 1Þ ¼ argrðfiÞ oBðrðfiÞÞ orðfiÞ ¼ 0 : ð16Þ

A5. The source of fi then communicates its updated rate rðfi;t þ 1Þ to all cliques ﬂowed by it by piggybacking the value with its data packets. The above procedure then loops back to step A2 and repeats in each time step.

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4. Resource allocation with both transmission rate and energy constraints

A radio channel is a kind of replenishable resource in the sense that in every time unit, the same amount of resource can be pro-vided again. On the contrary, it is not so for battery energy in a mo-bile node because after each usage, the remaining energy decreases until the battery is exhausted. Below, we will develop an extension to our model to include energy price.

We ﬁrst develop the energy consumption model in IEEE 802.11 MAC, where each transmission of a data packet is accompanied by RTS/CTS/ACK control packets, as illustrated in Fig. 2. Let the amounts of energy consumption per time unit for transmission, reception, and idle-listening be Ptx;Prx, and Pidle, respectively. For each directional wireless link ~e ¼ ðu; vÞ 2 EðGÞ, the amount of en-ergy required to transmit one data bit from u to v can be written as

Pð~eÞ ¼ ð1 þ dtxÞ 1

rðeÞ ðPtxþ Prxþ ðjNðuÞj 1ÞPidleÞ þ drx

1

rðeÞ ðPtxþ Prxþ ðjNðvÞj 1ÞPidleÞ; ð17Þ

where the ﬁrst term is the cost incurred by the transmission activ-ities at u and the second term is the cost incurred by the transmis-sion activities at v. NðuÞ and NðvÞ are the sets of neighbors of u and v in G, respectively. The terms dtxand drxare to account for the ratios of extra control overheads per data bit incurred for u and v, respec-tively. Note that since ~e is directional, Pððu; vÞÞ may not be equal to Pððv; uÞÞ.Fig. 4shows an example.

We utilize energy price Pð~eÞ in two ways. First, Pð~eÞ will be sent to each clique leader Lqto differentiate the unit price of q charged to each ﬂow. More speciﬁcally, the unit cost

l

qwill be extended to

l

q;~e to account for the energy cost of link ~e. Second, the energy price will also be sent to each source node to be included in its net beneﬁt function. The detail procedure is shown below.

B1. Each directional link ~e will calculate its energy cost Pð~eÞ. At step t, the leader Lq of each clique q will collect the rate rðfi;tÞ of each fisuch that EðfiÞ \ VðqÞ 6¼ ; and the energy cost Pð~eÞ of each link ~e 2 VðqÞ.

B2. To reﬂect the difference in energy cost of each link, we mod-ify Eq.(13) such that Lq assigns a different step size

c

~eto each link ~e 2 VðqÞ. We intentionally let links with higher

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energy costs get larger step sizes, and vice versa. The intui-tion is to let links with higher energy costs adjust prices more quickly. So ﬂows passing high energy consumption areas will be more sensitive to price changes. Speciﬁcally, Lqsets the unit price of link ~e in step t þ 1 as

l

q;~eðt þ 1Þ ¼

l

q;~eðtÞ

c~

e 1 X 8fi2F X 8e2EðfiÞ:e2VðqÞ rðfiÞ rðeÞ 0 @ 1 A 0 @ 1 A 2 4 3 5 þ ; ð18Þ

where

l

q;~eðtÞ is the unit price charged by each link ~e in step t. Then Lqsends the updated price to all members in VðqÞ. The value of

c

~eis deﬁned as follows. Let step size variance b be a positive constant such that b <

c

(for example, if

c

¼ 0:01, then b can be 0.005). Let Pavg¼2jVðqÞj1

P

8~e2VðqÞPð~eÞ. For link ~e, we let

c~

e¼

c

þ b h Pð~eÞ Pavg Pavg ; ð19Þ where hðyÞ ¼ y if 1 6 y 6 1 1 if y < 1 1 if y > 1 8 > < > : : ð20Þ

Function hðyÞ is to constrain the returned value within ½1; 1 when y is outside that range.

B3. On receiving

l

q;~eðt þ 1Þ, each ~e 2 VðqÞ notifies the updated price to each flow that goes through it. Each flow should carry the new price to its source node.

B4. When the source of ficollects all updated prices at step t þ 1, it derives its updated net beneﬁt function as

BðrðfiÞÞ ¼ UðrðfiÞÞ X 8e2EðfiÞ X 8q2Q:e2VðqÞ rðfiÞ

rðeÞ

l

q;~eðt þ 1Þ þ weng rðfiÞ Pð~eÞ

!

; ð21Þ where wengis a constant representing the weight of the price of energy, considering that one may give more or less

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sis on the cost of energy consumption. Taking the ﬁrst deriv-ative of BðrðfiÞÞ by setting it to 0, we have

oBðrðfiÞÞ orðfiÞ ¼ U0_ðrðf iÞÞ X 8e2EðfiÞ X 8q2Q:e2VðqÞ

l

q;~eðt þ 1Þ

rðeÞ þ weng Pð~eÞ

!

¼ 0: ð22Þ The next injection rate that would maximize its net beneﬁt is

rðfi;t þ 1Þ ¼ argrðfiÞ

oBðrðfiÞÞ

orðfiÞ ¼ 0

: ð23Þ

B5. The source of fithen communicates its updated rate to all cli-ques ﬂowed by it by piggybacking the value of rðfi;t þ 1Þ with its data packets. The above procedure then loops back to step B2 and repeats in each time step.

5. Experimental results

To understand the convergence property and performance of the proposed protocols, we have developed a simulator. We con-sider the effect of multi-rate transmission, without the effect of en-ergy price. A network area of size 1500 m 1500 m is simulated, on which 50 nodes are randomly generated. We assume that the IEEE 802.11b wireless interface cards are used, which support four transmission rates of r1¼ 11 Mbps, r2¼ 5:5 Mbps; r3¼ 2 Mbps, and r4¼ 1 Mbps, with transmission distances of d1¼ 30 m; d2¼ 50 m; d3¼ 80 m, and d4¼ 145 m, respectively. Therefore, dtx¼ 145 m. Unless stated otherwise, we set dint¼ 2 dtxand initial price

l

qð0Þ ¼ 1:00 for each q. For each ﬂow, the initial rate is set to 0. The step size

c

is set to 0.05. In the fol-lowing simulations, we ﬁrst assume weng¼ 0 (i.e., no energy price). At the end, we will evaluate the impact of weng.

(A) Convergence test: First, we inject different initial values to verify the convergence property of our scheme. We adopt the util-ity function UðxÞ ¼ x1=2_{. There are n ¼ 5 ﬂows each with an initial} ﬂow rate of 0 Mbps. The initial unit price for each clique is 1.0. We test different step sizes

c

¼ 0:08, 0.18, and 0.28. The results are in Fig. 5, which shows that in all step sizes, the clique unit prices and ﬂow rates will converge to the same values. A smaller step size will lead to slower convergence, which is reasonable. We also conduct simulations with different initial clique unit prices, under a ﬁxed

c

¼ 0:08. AsFig. 6shows, initial unit prices do affect the speed of convergence. However, all cases converge to the same ﬂow rates. A similar test of convergence using different initial ﬂow rates are shown inFig. 7.

(B) Impact of utility functions: Next, we test on different utility functions: UðxÞ ¼ x1=2_;_x1=4_{, and ln x. Five traffic flows are injected.} Then we observe the changes of unit prices of some cliques (Fig. 8(a), (c), (e)) and changes of rates of some flows (Fig. 8(b), (d), (f)). It can be seen that in all cases, flow rates will converge within short times. The convergence speed of UðxÞ ¼ ln x is rela-tively slower. Overall, we see that when UðxÞ ¼ x1=2_{or x}1=4_{, the flow} rates converge at faster speeds than the case UðxÞ ¼ ln x. This is be-cause the degree of satisfaction is less sensitive to rate change in the latter case. Interestingly, we also see that even after all flow rates have converged, some cliques’ unit prices will converge quickly, but some may keep on increasing or decreasing. Decreas-ing ones are due to the correspondDecreas-ing cliques are not 100% satu-rated yet. So their prices will keep on decreasing. However, flow rates may not be increased any more (observe that some cliques may be saturated already and become the bottlenecks of these flows). This causes such cliques drop their unit prices gradually to 0. This can also explain why some flows will keep on increasing their prices. As a flow sees a dropping path price, it will try to in-crease its rate. However, since no more inin-crease is possible, this only causes those already saturated cliques to become over-satu-rated and thus increase their unit prices.

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(C) Varying the network density: In the next set of simulations, we fix the utility function at UðxÞ ¼ x1=2_{and vary the network} den-sity. The network density can be changed by varying the interfer-ence range or the number of nodes. The results in Fig. 9 are obtained by setting dint¼ 2 dtx;3 dtx and 4 dtx. The conver-gence property remains true. However, since the definitions of cli-ques will change as the interference ranges change, the convergence speeds and the final flow rates are not necessarily the same. The results inFig. 10are obtained by setting the num-bers of nodes to 50 and 100. While the convergence is guaranteed, the speed of convergence is slower as there are more nodes, which is reasonable.

(D) Impact of number of ﬂows: Finally, we ﬁx the utility function at UðxÞ ¼ x1=2_{and the interference range at d}

int¼ 2:0 dtxand vary the number of flows among 5, 10, and 25. The results are inFig. 11. The convergence speeds are not sensitive to the number of flows, so the proposed protocol should be quite scalable to the number of flows.

(E) Impact of energy price: The above results assume no energy price (i.e., steps A1-A5 are adopted). In this simulation, we set

UðxÞ ¼ x1=2_;_d

int¼ 2:0 dtx, and vary the weight weng (i.e., steps B1-B5 are adopted). The results are inFig. 12. We see both the con-vergence property and the impact of energy cost. Flows 1 and 3 consume the most energy, so their stable rates decrease as weng in-creases. On the contrary, flows 2 and 4 consume relatively less en-ergy, so their stable rates, benefiting from the channel resources released by flows 1 and 3, increase as wengincreases.Fig. 13shows the impact of weng by varying it between 0.1 and 2.0. As can be seen, the cost of energy can suppress the rates of flows 1 and 3 effectively. As some channel resources are released by flows 1 and 3, flows 2 and 4 will first benefit from these new resources. However, as wengkeeps on increasing, flows 2 and 4 will eventually see higher overall prices, enforcing them to reduce their rates. This explains why we see increment followed by decrement in stable rates for them as wengkeeps on increasing.

6. Conclusions

We have addressed the resource allocation problem in MANETs by using pricing to regulate individual ﬂows’ behaviors. Two

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ing strategies have been proposed, which take the factors of multi-ple transmission rates and energy consumptions into account.

These two factors are critical ones for MANETs, but have not been well studied in former works. Therefore, our results can more clo-sely reflect realistic wireless network environments under current technologies. Our schemes do not rely on global network informa-tion. Each clique will run as an individual to adjust its unit price. Similarly, each flow will run as an individual to adjust its flow rate depending on its current utility value and the external charges. As shown by our simulations, the system will gradually reach a bal-ance point. Our simulation results have verified the convergence properties of the proposed clique-based and clique-plus-energy-based models. Various factors have been studied in our simulation experiments.

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Fig. 12. Impact of energy price. (Set A considers only channel cost, while set B considers both channel and energy costs.)

Fig. 13. Impact of weight wengwhen energy price is considered. (Set A considers

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