Hedonic Coalition Formation for Distributed Task Allocation among Wireless Agents
Walid Saad, Zhu Han, Tamer Basar, Merouane Debbah, Are Hjørungnes
aa r X i v : . [ c s . I T ] O c t Hedonic Coalition Formation for Distributed TaskAllocation among Wireless Agents
Walid Saad, Zhu Han, Tamer Bas¸ar, M´erouane Debbah, and Are Hjørungnes
Abstract —Autonomous wireless agents such as unmannedaerial vehicles, mobile base stations, or self-operating wirelessnodes present a great potential for deployment in next-generationwireless networks. While current literature has been mainly fo-cused on the use of agents within robotics or software engineeringapplications, we propose a novel usage model for self-organizingagents suited to wireless networks. In the proposed model, anumber of agents are required to collect data from severalarbitrarily located tasks. Each task represents a queue of packetsthat require collection and subsequent wireless transmission bythe agents to a central receiver. The problem is modeled as a hedonic coalition formation game between the agents and thetasks that interact in order to form disjoint coalitions. Eachformed coalition is modeled as a polling system consisting ofa number of agents which move between the different taskspresent in the coalition, collect and transmit the packets. Withineach coalition, some agents can also take the role of a relayfor improving the packet success rate of the transmission. Theproposed algorithm allows the tasks and the agents to takedistributed decisions to join or leave a coalition, based on theachieved benefit in terms of effective throughput, and the costin terms of delay. As a result of these decisions, the agents andtasks structure themselves into independent disjoint coalitionswhich constitute a Nash-stable network partition. Moreover, theproposed algorithm allows the agents and tasks to adapt thetopology to environmental changes such as the arrival/removal oftasks or the mobility of the tasks. Simulation results show how theproposed algorithm allows the agents and tasks to self-organizeinto independent coalitions, while improving the performance, interms of average player (agent or task) payoff, of at least . (for a network of agents with up to tasks) relatively to ascheme that allocates nearby tasks equally among agents.Keywords: wireless networks, multiagent systems, game theory,hedonic coalitions, task allocation, ad hoc networks. I. I
NTRODUCTION
Next generation wireless networks will present a highlycomplex and dynamic environment characterized by a largenumber of heterogeneous information sources, and a varietyof distributed network nodes. This is mainly due to the re-cent emergence of large-scale, distributed, and heterogeneouscommunication systems which are continuously increasingin size, traffic, applications, services, etc. For maintaining asatisfactory operation of such networks, there is a constantneed for dynamically optimizing their performance, moni-toring their operation and reconfiguring their topology. Due
W. Saad and A. Hjørungnes are with the UNIK Graduate University Center,University of Oslo, Oslo, Norway, e-mails: { saad,arehj } @unik.no .Z. Han is with Electrical and Computer Engineering Department, Uni-versity of Houston, Houston, Tx, USA, email: [email protected] .T. Bas¸ar is with the Coordinated Science Laboratory, University of Illi-nois at Urbana Champaign, IL, USA, email: [email protected] .M. Debbah is the Alcatel-Lucent chair, SUPELEC, Paris, France e-mail: [email protected] . This research is supported by theResearch Council of Norway through the projects 183311/S10, 176773/S10and 18778/V11, and by a grant from AFOSR. A preliminary version of thiswork [35] appeared in the proceedings of the International Conference onGame Theory for Networks (GameNets), May, 2009. to the ubiquitous nature of such wireless networks, it isinherent to have self-organizing autonomous nodes (agents),that can service these networks at different levels such asdata collection, monitoring, optimization, maintenance, andothers [1]–[6]. These nodes can be owned by the authoritymaintaining the network, and must be able to survey largescale networks, and perform very specific tasks at differentpoints in time, in an autonomous manner, with little relianceon any centralized authority [1]–[6].While the use of autonomous agents has been well investi-gated in robotics, computer systems or software engineering,research models tackling the use of such agents in wireless andcommunication networks are few. However, recently, the needfor such agents in wireless networks has become of notice-able importance as many next-generation networks encompassseveral types of wireless devices, such as cognitive devicesor unmanned aerial vehicles (UAVs), that are autonomousand self-adapting [1]–[6]. A key challenge in this area is theproblem of task allocation among a group of agents that needto execute a number of tasks. This problem has been alreadytackled in areas such as robotics control [7]–[9] or softwaresystems [10], [11]. However, most of the existing models areunsuitable for task allocation in wireless networks due to manyreasons: (i)- The task allocation problems studied in existingwork are tailored for military operations, computer systems,or software engineering and, thus, cannot be readily appliedin models pertaining to wireless networks, (ii)- the tasks aregenerally considered as static abstract entities with very simplecharacteristics and no intelligence (e.g. the tasks are just pointsin a plane) which is a major limitation, and (iii)- the existingmodels do not consider any aspects of wireless networkssuch as the characteristics of the wireless channel, the datatraffic, the need for wireless transmission, or other wireless-specific specifications. In this context, numerous applicationsin next-generation wireless networks require a number ofagent-nodes to perform specific wireless-related tasks thatemerge over time and are not pre-assigned. One example isthe case where a number of wireless nodes are required tomonitor the operation of the network or perform relayingat different times and locations [1], [2], [4]–[6]. In suchapplications, the objective is to develop algorithms enablingthe agents to autonomously share the tasks among each other.The main existing contributions within wireless networking inthis area [12]–[16], are focused on deploying UAVs which canact as self-deploying autonomous agents that can efficientlyperform pre-assigned tasks in applications such as connectivityimprovement in ad hoc network [13], routing [14], [15], andmedium access control [16]. However, these contributionsfocus on centralized solutions for specific problems such asfinding the optimal locations for the deployment of UAVsor devising efficient routing algorithms in ad hoc networks in the presence of UAVs. In these papers, the tasks that theagents must accomplish are pre-assigned and pre-determined .In contrast, many applications in wireless networks require theagents to autonomously assign the tasks among themselves.Hence, it is inherent to devise algorithms, in the context ofwireless networks, that allow an autonomous and distributedtask allocation process among a number of wireless agents with little reliance on centralized entities.The main contribution of this paper is to propose a novelwireless-oriented model for the problem of task allocationamong a number of autonomous agents. The proposed modelconsiders a number of wireless agents that are required tocollect data from arbitrarily located tasks. Each task representsa source of data, i.e., a queue with a Poisson arrival of packets,that the agents must collect and transmit via a wireless link to acentral receiver. This formulation is deemed suitable to modelseveral problems in next-generation networks such as videosurveillance in wireless networks, self-deployment of mobilerelays in IEEE 802.16j networks [2], data collection in ad hocand sensor networks [6], operation of mobile base stations invehicular ad hoc networks [4] and mobile ad hoc networks [5](the so called message ferry operation), wireless monitoringof randomly located sites, autonomous deployment of UAVsin military ad hoc networks, and many other applications. Forallocating the tasks, we introduce a novel framework fromcoalitional game theory, known as hedonic coalition formation .Albeit hedonic games have been widely used in game theory,to the best of our knowledge, no existing work utilized thisframework in a communication or wireless environment. Thus,we model the task allocation problem as a hedonic coalitionformation game between the agents and the tasks, and weintroduce an algorithm for forming coalitions. Each formedcoalition is modeled as a polling system consisting of a numberof agents, designated as collectors , which act as a singleserver that moves continuously between the different tasks(queues) present in the coalition, gathering and transmittingthe collected packets to a common receiver. Further, withineach coalition, some agents can act as relays for improvingthe packet success rate during the wireless transmission. Forforming coalitions, the agents and tasks can autonomouslymake a decision to join or leave a coalition based on welldefined individual preference relations. These preferences arebased on a coalitional value function that takes into account thebenefits received from servicing a task, in terms of effectivethroughput (data collected), as well as the cost in terms ofthe polling system delay incurred from the time needed forservicing all the tasks in a coalition. We study the propertiesof the proposed algorithm, and show that it always convergesto a Nash-stable network partition. Further, we investigatehow the network topology can self-adapt to environmentalchanges such as the deployment of new tasks, the removalof existing tasks, and low mobility of the tasks. Simulationresults show how the proposed algorithm allows the networkto self-organize, while ensuring a performance improvement, The term wireless agent refers to any node that can act autonomously andcan perform wireless transmission. Examples of wireless agents are UAVs[13], mobile base stations [4]–[6], cognitive wireless devices [3], or self-deploying mobile relay stations [2]. in terms of average player (task or agent) payoff, compared toa scheme that assigns nearby tasks equally among the agents.The remainder of this paper is organized as follows: Sec-tion II presents and motivates the proposed system model. InSection III, we model the task allocation problem problem asa transferable utility coalitional game and propose a suitedutility function. In Section IV, we classify the task allocationcoalitional game as a hedonic coalition formation game, wediscuss its key properties and we introduce the algorithmfor coalition formation. Simulation results are presented, dis-cussed, and analyzed in Section V. Finally, conclusions aredrawn in Section VI.II. S
YSTEM M ODEL
Consider a network having M wireless agents that belongto a single network operator and that are controlled by acentral command center (e.g., a central controller node ora satellite system). These agents are required to service T tasks arbitrarily located in a geographic area that has anassociated central wireless receiver connected to the commandcenter. In general, the tasks are entities belonging to one ormore independent owners . The owners of the tasks can be,for example, service providers or third party operators. Wedenote the set of agents and tasks by M = { , . . . , M } ,and T = { , . . . , T } , respectively. We consider only the casewhere the number of tasks is larger than the number of agents,hence, T > M . The main motivation behind this considerationis that, for most networks, the number of agents assigned to aspecific area is generally small, e.g., due to cost factors. Eachtask i ∈ T represents an M/D/1 queueing system , wherebypackets of constant size B are generated using a Poissonarrival with an average arrival rate of λ i . Hence, we considerdifferent classes of tasks each having its corresponding λ i .The tasks we consider are sources of data that cannot sendtheir information to the central receiver (and, subsequently,to the command center) without the help of an agent. Thesetasks can represent a group of mobile devices, such as sensors,video surveillance devices, or any other static or dynamicwireless nodes that have limited power and are unable toprovide long-distance transmission. Such devices (tasks) needto buffer their data locally and wait to be serviced by an agentthat can collect the data. For example, an agent such as amobile station or a UAV can provide a line-of-sight link tofacilitate the transmission from the tasks to the receiver. Thetasks can also be mapped to any other source of packet datathat require collection by an agent for transmission . To servicea task, each agent is required to move to the task location,collect the data, and transmit it using a wireless link to thecentral receiver. The command center periodically downloadsthis data from the receiver, e.g., through a backbone network.Each agent i ∈ M offers a link transmission capacity of µ i , inpackets/second, using which the agent can service the tasks’data. The quantity µ i thus represents the well-known servicetime for a single packet that is being serviced by agent i . The The scenario where all tasks and agents are owned by the same entity isa particular case of this generic model. Other queue types, e.g., M/M/1, can be considered without loss ofgenerality. The tasks can also be moving with a periodic low mobility. agent which is collecting the data from a task is referred to as collector . In addition, each agent i ∈ M can transmit the datato the receiver with a maximum transmit power of P i = ˜ P ,assumed the same for all agents with no loss of generality.The proposed model allows each task to be serviced bymultiple agents, and also, each agent (or group of agents) toservice multiple tasks. Whenever a task is serviced by multipleagents, each agent can act as either a collector or a relay . Anygroup of agents that act together for data collection from thesame task, can be seen as a single collector with improvedlink transmission capacity. In this paper, we consider that thelink transmission capacity depends solely on the capabilitiesof the agents and not on the nature of the tasks. In this context,given a group of agents G ⊆ M that are acting as collectorsfor any task, the total link transmission capacity with whichtasks can be serviced with by G can be given by µ G = X j ∈ G µ j . (1)For forming a single collector, multiple agents can easilycoordinate the data extraction and transmission from everytask, so as to allow a larger link transmission capacity for theserviced task as in (1). The transmission of the packets by theagents from a task i ∈ T to the central receiver is subject topacket loss due to the fading on the wireless channel. Thus,in addition to acting as collectors, some agents may act as relays for a task. These relay-agents would locate themselvesat equal distances from the task (given that the task is alreadybeing served by at least one collector), and, hence, thecollectors transmit the data to the receiver through multi-hopagents, improving the probability of successful transmission.In Rayleigh fading, the probability of successful transmissionof a packet of size B bits from the collectors present at a task i ∈ T through a path of m agents, Q i = { i , . . . , i m } , where i = i is the task being serviced, i m is the central receiver,and any other i h ∈ Q i is a relay -agent, is given byPr i, CR = m − Y h =1 Pr Bi h ,i h +1 , (2)where Pr i h ,i h +1 is the the probability of successful transmis-sion of a single bit from agent i h to agent (or the centralreceiver) i h +1 . This probability can be given by the probabilityof maintaining the SNR at the receiver above a target level ν as follows [17]Pr i,i h +1 = exp (cid:18) − σ ν ( D i h ,i h +1 ) α κ ˜ P (cid:19) , (3)where σ is the variance of the Gaussian noise, κ is a path lossconstant, α is the path loss exponent, D i h ,i h +1 is the distancebetween nodes i h and i h +1 , and ˜ P is the maximum transmitpower of agent i h .For servicing a number of tasks C ⊆ T , a group of agents G ⊆ M (collectors and relays) can sequentially move fromone task to the other in C with a constant velocity η . The group G of agents, servicing the tasks in C , stop at each task, withthe collectors collecting and transmitting the packets using therelays (if any). The collectors would move from one task to Fig. 1. An illustrative example of the proposed model for task allocation inwireless networks (the agents are dynamic, i.e., they move from one task tothe other continuously). the other, only if all the packets in the queue at the currenttask have been transmitted to the receiver (the process throughwhich the agents move from one task to the other for datacollection is cyclic). Simultaneously with the collectors, therelays also move, positioning themselves at equal distanceson the line connecting the task being currently served by thecollectors, and the central receiver. With this proposed model,the final network will consist of groups of tasks serviced bygroups of agents, continuously. An illustration of this model isshown in Fig. 1. Given this proposed model, the main objectiveis to provide an algorithm for distributing the tasks between theagents, given the operation of the agents previously describedand shown in Fig. 1.III. C
OALITIONAL G AME F ORMULATION
A. Game Formulation
By inspecting Fig. 1, one can clearly see that the taskallocation problem among the agents can be mapped intothe problem of the formation of coalitions. In this regard,coalitional game theory [18, Ch. 9] provides a suitable an-alytical tool for studying the formation of cooperative groups,i.e., coalitions, among a number of players. For the proposedmodel, the coalitional game is played between the agentsand the tasks. Hence, the players set for the proposed taskallocation coalitional game is denoted by N , and contains bothagents and tasks, i.e., N = M ∪ T . Hereinafter, we use theterm player to indicate either a task or an agent.For any coalition S ⊆ N containing a number of agentsand tasks, the agents belonging to this coalition can struc-ture themselves into collectors and relays. Subsequently, asexplained in the previous section, within each coalition, thecollector-agents will continuously move from one task to theother, stopping at each task, and transmitting all the packetsavailable in the queue to the central receiver, through therelay-agents (if any). This proposed task servicing schemecan be mapped to a well-known concept that is ubiquitousin computer systems, which is the concept of a polling system [19]. In a polling system, a single server moves betweenmultiple queues in order to extract the packets from eachqueue, in a sequential and cyclic manner. Models pertaining to polling systems have been widely developed in variousdisciplines ranging from computer systems to communicationnetworks, and different strategies for servicing the queuesexist [19]–[21]. In the proposed model, the collectors of everycoalition are considered as a single server that is servicingthe tasks (queues) sequentially, in a cyclic manner, i.e., afterservicing the last task in a coalition S ⊆ N , the collectors of S return to the first task in S that they previously visited hencerepeating their route continuously. Whenever the collectorsstop at any task i ∈ S , they collect and transmit the datapresent at this task until the queue is empty. This method ofallowing the server to service a queue until emptying the queueis known as the exhaustive strategy for a polling system, whichis applied at the level of every coalition S ⊆ N in our model.Moreover, the time for the server to move from one queue tothe other is known as the switchover time. Consequently, wehighlight the following property: Property 1:
In the proposed task allocation model, everycoalition S ⊆ N is a polling system with an exhaustive pollingstrategy and deterministic non-zero switchover times. In eachsuch polling system S , the collector-agents are seen as thepolling system server, and the tasks are the queues that thecollector-agents must service.For any coalition S , once the queue at a task i ∈ S isemptied, the collectors and relays in a coalition move from task i to the next task j ∈ S with a constant velocity η , incurringa switchover time θ i,j . The switchover time in our modelcorresponds to the time it takes for all the agents (collectorsand relays) to move from one task to the next. Assuming allagents start their mobility at the same time, this switchovertime maps to the time needed for the farthest agent to movefrom one task to the next. Since we consider only straight linetrajectories for collectors and relays, and due to the fact thatthe relays always position themselves at equal distances on theline connecting the tasks in a coalition to the receiver, we havethe following property (clearly seen through the geometry ofFig. 1). Property 2:
Within any given coalition S , the switchovertime between two tasks corresponds to the constant time ittakes for one of the collectors to move from one of the tasksto the next.Having modeled every coalition S ⊆ N as a polling system,we investigate the average delay incurred per coalition. Forpolling systems, finding exact expressions for the delay atevery queue is a highly complicated task and no generalclosed-form expressions for the delay at every queue in apolling system can be found [19], [20] . A key criterion usedfor the analysis of the delay incurred by a polling system isthe pseudo-conservation law that provides closed-form expres-sions for weighted sum of the means of the waiting times atthe queues [19], [20]. For providing the pseudo-conservationlaw for a coalition S ⊆ N composed of a number of agentsand a number of tasks, we make the following definitions.First, within coalition S , a group of agents G S ⊆ S ∩ M are designated as collectors. Second, for each task i ∈ S ∩ T Note that some approximations [20] exist for polling systems under heavytraffic or large switchover times, but in our problem, they are not suitable aswe require a more general delay expression. with an average arrival rate of λ i , and served by a numberof collectors | G S | with a link transmission capacity of µ G S (as given by (1)), we define the utilization factor of task iρ i = λ i µ GS . Further, we define ρ S , P i ∈ S ∩T ρ i . Given thesedefinitions, for a coalition S , the weighted sum of the meansof the waiting times by the agents at all the tasks in thecoalition are given by the pseudo-conservation law as follows[20, Section. VI-B] (taking into account that our switchoverand service times are deterministic) X i ∈ S ∩T ρ i ¯ W i = ρ S P i ∈ S ∩T ρ i µ GS − ρ S ) + ρ S θ S (4) + θ S − ρ S ) " ρ S − X i ∈ S ∩T ρ i , where ¯ W i is the mean waiting time at task i and θ S = P | S ∩T | h =1 θ i h ,i h +1 is the sum of the switchover times given apath of tasks { i , . . . , i | S ∩T | } followed by the agents, with i h ∈ S ∩ T , ∀ h ∈ { , . . . , | S ∩ T |} and i | S ∩T | +1 = i .The first term in the right hand side of (4) is the well knownexpression for the average queueing delay for M/D/1 queues,weighed by ρ S . The second and third terms in the right handside of (4) represent the average delay increase incurred by thetravel time required for the collectors to move from one task tothe other, i.e., the delay resulting from the switchover period.Further, for any coalition S that must form in the system, thefollowing condition must hold: ρ S < . (5)This condition is a requirement for the stability of any pollingsystem [19]–[21] and, thus, must be satisfied for any coalitionthat will form in the proposed model. In the event where thiscondition is violated, the system is considered unstable andthe delay is considered as infinite. In this regard, the analysispresented in the remainder of this paper will take into accountthis condition and its impact on the coalition formation process(as seen later, a coalition where ρ S ≥ will never form). B. Utility Function
In the proposed game, for every coalition S ⊆ N , the agentsmust determine the order in which the tasks in S are visited,i.e., the path { i , . . . , i | S ∩T | } which is an ordering over theset of tasks in S given by S ∩ T . Naturally, the agents mustselect the path that minimizes the total switchover time for oneround of data collection. This can be mapped to the followingwell-known problem: Property 3:
The problem of finding the path that mini-mizes the total switchover time for one round of data collectionwithin a coalition S ⊆ N is mapped into the travelingsalesman problem [22], where a salesman, i.e., the agents S ∩ M , is required to minimize the time of visiting a seriesof cities, i.e., the tasks S ∩ T .It is widely known that the solution for the travelingsalesman problem is NP-complete [22], and, hence, there hasbeen numerous heuristic algorithms for finding an acceptablenear-optimal solution. One of the simplest of such algorithmsis the nearest neighbor algorithm (also known as the greedyalgorithm) [22]. In this algorithm, starting from a given citythe salesman chooses the closet city as his next visit. Using the nearest neighbor algorithm, the ordering of the cities whichminimizes the overall route is selected. The nearest neighboralgorithm is sub-optimal but it can quickly find a near-optimalsolution with a small computational complexity (linear in thenumber of cities) [22] which makes it suitable for problemssuch as the proposed task allocation problem. Therefore, in theproposed model, for every coalition S , the agents can easilywork out the nearest neighbor route for the tasks, and operateaccording to it.Having modeled each coalition as a polling system, thepseudo-conservation law in (4) allows to evaluate the cost,in terms of average waiting time (or delay), from forming aparticular coalition. However, for every coalition, there is abenefit, in terms of the average effective throughput that thecoalition is able to achieve. The average effective throughputfor a coalition S is given by L S = X i ∈ S ∩T λ i · Pr i, CR , (6)with Pr i, CR given by (2). By closely inspecting (1), one cansee that adding more collectors improves the transmission linkcapacity, and, thus, reduces the service time that a certain taskperceives. Based on this property and by using (4) one caneasily see that, adding more collectors, i.e., improving theservice time, reduces the overall delay in (4) [19]–[21]. Fur-ther, adding more relays would reduce the distance over whichtransmission is occurring, thus, improving the probability ofsuccessful transmission as per (2) [4], [17]. In consequence,using (6), one can see that this improvement in the probabilityof successful transmission is translated into an improvementin the effective throughput. Hence, each agent role (collectoror relay) possesses its own benefit for a coalition.A suitable criterion for characterizing the utility in networksthat exhibit a tradeoff between the throughput and the delayis the concept of system power which is defined as theratio of some power of the throughput and the delay (or apower of the delay) [23]. Hence, the concept of power isan attractive notion that allows to capture the fundamentaltradeoff between throughput and delay in the proposed taskallocation model. Power has been used thoroughly in theliterature in applications that are sensitive to throughput as wellas delay [24]–[26]. Mainly, for the proposed game, the utilityof every coalition S is evaluated using a coalitional valuefunction based on the power concept from [26] as follows v ( S ) = δ L βS ( P i ∈ S ∩T ρ i ¯ W i ) (1 − β ) , if ρ S < and | S | > , , otherwise , (7)where β ∈ (0 , is a throughput-delay tradeoff parameter. In(7), the term δ represents the price per unit power that thenetwork offers to coalition S . Hence, δ represents a genericcontrol parameter that allows the network operator to somehowmonitor the behavior of the players. The use of such controlparameters is prevalent in game theory [27]–[31]. In certainscenarios, δ would represent physical monetary values paidby the operator to the different entities (agents and tasks). Insuch a case, on one hand, for the tasks, the operator simplywould pay the tasks’ owners for the amount of data (and its corresponding quality as per (7)) each one of their tasks hadgenerated. On the other hand, for the agents, the paymentwould, for example, represent either a reward for the behaviorof the agents or the proportion of maintenance or servicingthat each agent would receiver from its operator. In this sense,the utility function in (7) would, thus, represents the totalrevenue achieved by a coalition S , given the network powerthat coalition S obtains. For coalitions that consist of a singleagent or a single task, i.e., coalitions of size , the utilityassigned is due to the fact that such coalitions generate nobenefit for their member (a single agent can collect no dataunless it moves to task, while a single task cannot transmit anyof its generated data without an agent collecting this data).Further, any coalition where condition (5) is not satisfied isalso given a zero utility, since, in this case, the polling systemthat the coalition represents is unstable, and hence has aninfinite delay.Given the set of players N , and the value function in(7), we define a coalitional game ( N , v ) with transferableutility (TU). The utility in (7) represents the amount of moneyor revenue received by a coalition, and, hence, this amountcan be arbitrarily apportioned between the coalition members,which justifies the TU nature of the game. To divide this utilitybetween the players, we adopt the equal fair allocation rule,where the payoff of any player i ∈ S , denoted by x Si is x Si = v ( S ) | S | . (8)The payoff x Si represents the amount of revenue that player i ∈ S receives from the total revenue v ( S ) that coalition S generates. The main motivation behind adopting the equal fairallocation rule is in order to highlight the fact that the agentsand the tasks value each others equally. As seen in (7), thepresence of an agent in a coalition is crucial in order for thetasks to obtain any payoff, and, vice versa, the presence ofa task in a coalition is required for the agent to be able toobtain any kind of utility. Nonetheless, the proposed modeland algorithm can accommodate any other type of payoffallocation rule. Although in traditional coalitional games, theallocation rule may have a strong impact on the game’ssolution, for the proposed game, other allocation rules can beused with little impact on the analysis that is presented in therest of the paper. This is due to the class of the proposed gamewhich is quite different from traditional coalitional games. Asseen from (4) and (7), whenever the number of tasks in acoalition increases, the total delay increases, hence, reducingthe utility from forming a coalition. Further, in a coalitionwhere the number of tasks is large, the condition of stabilityfor the polling system, as given by (5), can be violated due toheavy traffic incoming from a large number of tasks yieldinga zero utility as per (7). Hence, forming coalitions betweenthe tasks and the agents entails a cost that can limit the sizeof a coalition. In this regard, traditional solution concepts forTU games, such as the core [18], may not be applicable. Infact, in order for the core to exist, a TU coalitional game mustensure that the grand coalition, i.e., the coalition of all playerswill form. However, as seen in Fig. 1 and corroborated by theutility in (7), in general, due to the cost for coalition formation, the grand coalition will not form. Instead, independent anddisjoint coalitions appear in the network as a result of thetask allocation process. In this regard, the proposed gameis classified as a coalition formation game [27]–[31], andthe objective is to find an algorithm that allows to form thecoalition structure, instead of finding only a solution concept,such as the core, which aims mainly at stabilizing a grandcoalition of all players.IV. T ASK A LLOCATION AS A H EDONIC C OALITION F ORMATION G AME
A. Hedonic Coalition Formation: Concepts and Model
Coalition formation games have been a topic of high interestin game theory [27]–[31]. Notably, in [29]–[31], a classof coalition formation games known as hedonic coalitionformation games is investigated. This class of games entailsseveral interesting properties that can be applied, not only ineconomics such as in [29]–[31], but also in wireless networksas we will demonstrate in this paper. The two key requirementsfor classifying a coalitional game as a hedonic game are [29]:
Condition 1: - (Hedonic Conditions) - A coalition forma-tion game is classified as hedonic if1) The payoff of any player depends solely on the membersof the coalition to which the player belongs.2) The coalitions form as a result of the preferences of theplayers over their possible coalitions’ set.These two conditions characterize the framework of hedonicgames. Mainly, the term hedonic pertains to the first conditionabove, whereby the payoff of any player i , in a hedonic game,must depend only on the identity of the players in the coalitionto which player i belongs, with no dependence on the otherplayers. For the second condition, in the remainder of thissection, we will formally define how the preferences of theplayers over the coalitions can be used for the formationprocess. To use hedonic games in the proposed model, wefirst introduce some definitions, taken from [29]. Definition 1: A coalition structure or a coalition partition is defined as the set Π = { S , . . . , S l } which partitions theplayers set N , i.e., ∀ k , S k ⊆ N are disjoint coalitions suchthat ∪ lk =1 S k = N (an example partition Π is shown in Fig. 1). Definition 2:
Given a partition Π of N , for every player i ∈ N we denote by S Π ( i ) , the coalition to which player i belongs, i.e., coalition S Π ( i ) = S k ∈ Π , such that i ∈ S k .In a hedonic game, each player must build preferences overits own set of possible coalitions. Hence, each player mustbe able to compare the coalitions and order them based onwhich coalition prefers being a member of. To evaluate theseplayers’ preferences over the coalitions, we define the conceptof a preference relation or order as follows [29]: Definition 3:
For any player i ∈ N , a preference relation or order (cid:23) i is defined as a complete, reflexive, and transitivebinary relation over the set of all coalitions that player i canpossibly form, i.e., the set { S k ⊆ N : i ∈ S k } .Thus, for a player i ∈ N , given two coalitions S ⊆ N and, S ⊆ N such that i ∈ S and i ∈ S , S (cid:23) i S indicatesthat player i prefers to be part of coalition S , over being partof coalition S , or at least, i prefers both coalitions equally. Further, using the asymmetric counterpart of (cid:23) i , denoted by ≻ i , then S ≻ i S , indicates that player i strictly prefersbeing a member of S over being a member of S . Forevery application, an adequate preference relation (cid:23) i can bedefined to allow the players to quantify their preferences. Thepreference relation can be a function of many parameters, suchas the payoffs that the players receive from each coalition, theweight each player gives to other players, and so on. Giventhe set of players N , and a preference relation (cid:23) i for everyplayer i ∈ N , a hedonic coalition formation game is formallydefined as follows [29]: Definition 4:
A hedonic coalition formation game is acoalitional game that satisfies the two hedonic conditionspreviously prescribed, and is defined by the pair ( N , ≻ ) where N is the set of players ( |N | = N ), and ≻ is a profile ofpreferences , i.e., preference relations, ( (cid:23) , . . . , (cid:23) N ) definedfor every player in N .Having defined the main components of hedonic coalitionformation games, we utilize this framework in order to providea suitable solution for the task allocation problem proposed inSection II. The proposed task allocation problem is modeled asa ( N , ≻ ) hedonic game, where N is the set of agents and tasksand ≻ is a profile of preferences that we will shortly define.For the proposed game model, given a network partition Π of N , the payoff of any player i , depends only on the identityof the members of the coalition to which i belongs. In otherwords, the payoff of any player i depends solely on the playersin the coalition in which player i belongs S Π ( i ) (easily seenthrough the formulation of Section III). Hence, our gameverifies the first hedonic condition.Moreover, to model the task allocation problem as a hedoniccoalition formation game, the preference relations of theplayers must be clearly defined. In this regard, we define twotypes of preference relations, a first type suited for indicatingthe preferences of the agents, and a second type suited forthe tasks. Subsequently, for evaluating the preferences ofany agent i ∈ M , we define the following operation (thispreference relation is common for all agents, hence we denoteit by (cid:23) i = (cid:23) M , ∀ i ∈ M ) S (cid:23) M S ⇔ u i ( S ) ≥ u i ( S ) , (9)where S ⊆ N and S ⊆ N are any two coalitions thatcontain agent i , i.e., i ∈ S and i ∈ S and u i : 2 N → R is apreference function defined for any agent i as follows u i ( S ) = ∞ , if S = S Π ( i ) & S \ { i } ⊆ T , , if S ∈ h ( i ) ,x Si . otherwise , (10)where Π is the current coalition partition which is in placein the game, x Si is the payoff received by player i from anydivision of the value function among the players in coalition S such as the equal fair division given in (8), and h ( i ) is thehistory set of player i . At any point in time, the history set h ( i ) is a set that contains the coalitions that player i was a partof in the past, prior to the formation of the current partition Π .Note that, by using the defined preference relation, the players can compare any two coalitions S and S independently ofwhether these two coalitions belong to Π or not.The main rationale behind the preference function u i in(10) is as follows. In this model, the agents, being entitiesowned by the operator, seek out to achieve two conflictingobjectives: (i)- Service all tasks in the network for the benefitof the operator, and (ii)- Maximize the quality of service, interms of power as per (7), for extracting the data from thetasks. The preference function u i must be able to allow theagents to make coalition formation decisions that can capturethis tradeoff between servicing all tasks (at the benefit ofthe operator) and achieving a good quality of service (at thebenefit of both agents and operator). For this purpose, as per(10), any agent i that is the sole agent servicing tasks in itscurrent coalition S = S Π ( i ) such that S Π ( i ) ∩ M = { i } ,assigns an infinite preference value to S . Hence, in orderto service all tasks, the agent always assigns a maximumpreference to its current coalition, if this current coalition iscomposed of only tasks and does not contain other agents.This case of the preference function u i forbids the agent fromleaving a group of tasks, already assigned to it, unattendedby other agents. In this context, this condition pertains tothe fist objective (objective (i) previously mentioned) of theagents and it implies that, whenever there is a risk of leavingtasks without service, the agent do not act selfishly, in contrast,they act in the benefit of the operator and remain with thesetasks, independent of the payoff generated by these tasks. Sucha decision allows an agent to avoid making a decision thatcan incur a risk of ultimately having tasks with no servicein the network, in which case, the network operator wouldlose revenue from these unattended tasks and it may, forexample, decide to replace the agent that led to the presence ofsuch a group of tasks with no service. Otherwise, the agents’preference relation u i would highlight the second objective ofthe agent, i.e., maximize its own payoff which maps into therevenue generated from the quality of service, i.e., the poweras per (7). with which the tasks are being serviced. In thiscase, the preference relation is easily generated by the agentsby comparing the value of the payoffs they receive from thetwo coalitions S and S . Further, we note that no agent hasany incentive to revisit a coalition that it had left previously,and hence, the agents assign a preference value of for anycoalition in their history (this can be seen as a basic learningprocess). In summary, taking into account the conflicting goalsof the agents, between two coalitions S and S , an agent i prefers the coalition that gives the better payoff, given thatthe agent is not alone in its current coalition, and the coalitionwith a better payoff is not in the history of agent i .To evaluate the preferences of any task j ∈ T , we definethe following operation (this preference relation is commonfor all tasks, hence, we denote it by (cid:23) j = (cid:23) T , ∀ j ∈ T ) S (cid:23) T S ⇔ w j ( S ) ≥ w j ( S ) , (11)with the tasks’ preference function w j defined as follows. w j ( S ) = ( , if S ∈ h ( j ) ,x Sj , otherwise . (12) The preferences of the tasks are easily captured using thefunction w j . The preference function of the tasks is differ-ent from that of the agents since the tasks are, in general,independent entities that act solely in their own interest. Thus,based on (12), each task prefers the coalition that providesthe larger payoff x Sj unless this coalition was already visitedpreviously and left. In that case, the preference function ofthe tasks assigns a preference value of for any coalitionthat the task had already visited in the past (and left to joinanother coalition). Using this preference relation, every taskcan evaluate its preferences over the possible coalitions thatthe task can form. Consequently, the proposed task allocationmodel verifies both hedonic conditions, and, hence, we have: Property 4:
The proposed task allocation problem amongthe agents is modeled as a ( N , ≻ ) hedonic coalition formationgame, with the preference relations given by (9) and (11).Note that the preference relations in (9) and (11) are alsodependent on the underlying TU coalitional game described inSection III. Having formulated the problem as a hedonic game,the final task is to provide a distributed algorithm, based on thedefined preferences, for forming the coalitions. However, priorto deriving the algorithm for coalition formation, we highlightthe following result: Proposition 1:
For the proposed hedonic coalition forma-tion model for task allocation, assuming that all collector-agents have an equal link transmission capacity µ i = µ, ∀ i ∈M , any coalition S ⊆ N with | S ∩ M| agents, must have atleast | G S | min collector-agents ( G S ⊆ S ∩ M ) as follows | G S | > | G S | min = P i ∈ S ∩T λ i µ . (13)Further, when all the tasks in S belong to the same class, wehave | G S | min = | S ∩ T | · λµ , (14)which constitutes an upper bound on the number of collector-agents as a function of the number of tasks | S ∩T | for a givencoalition S . Proof:
As per the defined preference relations in (10) and(12), any coalition that will form in the proposed model mustbe stable since no agent or task has an incentive to join anunstable coalition, hence, we have, for every coalition S ⊆ N having | G S | collectors with G S ⊆ S ∩ M , we have from (5) ρ S < , and thus X i ∈ S ∩T λ i µ G S < , which, given the assumption that µ i = µ, ∀ i ∈ M yields | G S | · µ · X i ∈ S ∩T λ i < , which yields | G S | > | G S | min = P i ∈ S ∩T λ i µ . Further, if weassume that all the tasks belong to the same class, hence, λ i = λ, ∀ i ∈ S ∩ T , we immediately get | G S | min = | S ∩ T | · λµ . (15) Consequently, for any proposed coalition formation algorithm,the bounds on the number of collector-agents in any coalition S as given by Proposition 1 will always be satisfied. B. Hedonic Coalition Formation: Algorithm
In the previous subsection, we modeled the task allocationproblem as a hedonic coalition formation game and, thus, theremaining objective is to devise an algorithm for forming thecoalitions. While literature that studies the characteristics ofexisting partitions in hedonic games, such as in [29]–[31],is abundant, the problem of forming the coalitions both inthe hedonic and non-hedonic setting is a challenging problem[27]. In this paper, we introduce an algorithm for coalitionformation that allows the players to make selfish decisions asto which coalitions they decide to join at any point in time. Inthis regard, for forming coalitions between the tasks and theagents, we propose the following rule for coalition formation:
Definition 5: Switch Rule -
Given a partition
Π = { S , . . . , S l } of the set of players (agents and tasks) N , aPlayer i decides to leave its current coalition S Π ( i ) = S m , for some m ∈ { , . . . , l } and join another coalition S k ∈ Π ∪ {∅} , S k = S Π ( i ) , if and only if S k ∪ { i } ≻ i S Π ( i ) .Hence, { S m , S k } → { S m \ { i } , S k ∪ { i }} .Through a single switch rule made by any player i , anycurrent partition Π of N is transformed into Π ′ = (Π \{ S m , S k } ) ∪ { S m \ { i } , S k ∪ { i }} . In simple terms, for everypartition Π , the switch rule provides a mechanism throughwhich any task or agent, can leave its current coalition S Π ( i ) ,and join another coalition S k ∈ Π , given that the newcoalition S k ∪ { i } is strictly preferred over S Π ( i ) throughany preference relation that i is using (in particular usingthe preference relations defined in (9) and (11)). Independentof the preference relations selected, the switch rule can beseen as a selfish decision made by a player, to move from itscurrent coalition to a new coalition, regardless of the effectof this move on the other players. Furthermore, we considerthat, whenever a player decides to switch from one coalitionto another, the player updates its history set h ( i ) . Hence,given a partition Π , whenever a player i decides to leavecoalition S m ∈ Π to join a different coalition, coalition S m isautomatically stored by player i in its history set h ( i ) .Consequently, we propose a coalition formation algorithmcomposed of three main phases: Task discovery, hedoniccoalition formation, and data collection. In the first phase,the central command receives information about the existenceof tasks that require servicing and informs the agents of thelocations and characteristics of the tasks (e.g., the arrivalrates). Hence, the agents start by having full knowledge ofthe initial partition Π initial . Once the agents are aware ofthe tasks, they broadcast their own presence to the tasks.Consequently, the players can interact with each other, forperforming coalition formation. Hence, the second phase ofthe algorithm is the hedonic coalition formation phase. Inthis phase, all the players (tasks and agents) investigate thepossibility of performing a switch operation. For identifyingpotential switch operations, given complete knowledge aboutthe network (which can be gathered in different methods as will be discussed in Subsection IV-C)), each agent investigatesits top preference, and decides to perform a switch operation,if possible through (9). As one can easily see in (7), for theproposed model, no coalition composed of tasks-only wouldever form since such a coalition would always generate a utility. Therefore, the tasks are only interested in switching tocoalitions that contain at least a single agent. From a tasks’perspective, for determining its preferred switch operation,each task needs only to negotiate with existing agents in orderto enquire on the amount of utility it can obtain by joiningwith this agent. By doing so, each task can determine theswitch operation it is interested in making at a given time. Weconsider that, the players make sequential switch decisions(the order of switch operations is referred to as the order ofplay hereinafter). For any agent, a switch operation is easilyperformed as the agent can leave its current coalition and jointhe new coalition, if (9) is satisfied. For the tasks, any task thatfinds out a possibility to switch, can autonomously request,over a control channel with the concerned agent, to be addedto the coalition of interest (which would always contain atleast one agent with whom the task previously negotiated).The convergence of the proposed algorithm during the hedoniccoalition formation phase is guaranteed as follows: Theorem 1:
Starting from any initial network partition Π initial , the proposed hedonic coalition formation phase ofthe proposed algorithm always converges to a final networkpartition Π f composed of a number of disjoint coalitions. Proof:
For the purpose of this proof, we denote Π kn k as thepartition formed during the time k when player i ∈ N decidesto act after n k switch operations have previously occurred (theindex n k denotes the number of switch operations performedby one or more players up to time k ). Given any initialstarting partition Π initial = Π , the hedonic coalition formationphase of the proposed algorithm consists of a sequence ofswitch operations. As per Definition 5, every switch operationtransforms the current partition Π into another partition Π ′ ,hence, hedonic coalition formation consists of a sequence ofswitch rules, yielding, e.g., the following transformations Π = Π → Π → . . . → Π Ln L . . . → . . . → Π Tn T , (16)where the operator → indicate the occurrence of a switchoperation. In other words, Π kn k → Π k +1 n k +1 , implies that duringturn k , a certain player i made a single switch operation whichyielded a new partition Π k +1 n k +1 at the turn k + 1 . By inspectingthe preference relations defined in (9) and (11), it can be seenthat every single switch operation leads to a partition thathas not yet been visited (new partition). Hence, for any twopartitions Π kn k and Π ln l in the transformations of (16), suchthat n k = n l , i.e., Π ln l is a result of the transformation of Π kn k (or vice versa) after a number of switch operations | n l − n k | ,we have that Π kn k = Π ln l for any two turns k and l .Given this property and the well known fact that the numberof partitions of a set is finite and given by the Bell number [27],the number of transformations in (16) is finite, and, hence, thesequence in (16) will always terminate and converge to a finalpartition Π f = Π Tn T which completes the proof.The stability of the final partition Π f resulting from theconvergence of the proposed algorithm can be studied using TABLE IT
HE PROPOSED HEDONIC COALITION FORMATION ALGORITHM FOR TASKALLOCATION IN WIRELESS NETWORKS . Initial State
The network is partitioned by Π initial = { S , . . . , S k } . At the beginningof all time Π initial = N = M ∪ T with no tasks being serviced.
Three Phases for the Proposed Hedonic Coalition Formation Algorithm
Phase I - Task Discovery: a) The command center is informed by one or multiple owners about theexistence and characteristics of new tasks.b) The central command center conveys the information on the initial networkpartition Π initial using the methods of Subsection IV-C Phase II - Hedonic Coalition Formation: repeat
For every player i ∈ N , given a current partition Π current .a) Player i investigates possible switch using the preferencesgiven, respectively, by (9) and (11) for the agents and tasks.b) Player i performs the switch operation that maximizes its payoff:b.1) Player i updates its history h ( i ) by adding S Π current ( i ) .b.2) Player i leaves its current coalition S Π current ( i ) .b.3) Player i joins the new coalition that maximizes its payoff. until convergence to a final Nash-stable partition Π f . Phase III - Data Collection a) The network is partitioned using Π final .b) The agents in each coalition S k ∈ Π final continuously perform thefollowing operations, i.e., act as a polling system with exhaustive strategyand switchover times:b.1) Visit a first task in their respective coalitions.b.2) The collector-agents collect the data from the task being visited.b.3) The collector-agents transmit the data using wireless links to thecentral receiver either directly or using other relay-agents.b.4) Once the queue of the current is empty, visit the next task.The order in which the tasks are visited is determined by the nearestneighbor solution to the traveling salesman problem as in Property 3.This third phase is continuously repeated and performed by all the agentsin Π final for a fixed period of time Ψ (for static environments Ψ = ∞ ). Adaptation to environmental changes (periodic process) a) In the presence of environmental changes, such as the deployment ofnew tasks, the removal of existing tasks, or periodic low mobility ofthe tasks, the third phase of the algorithm is performed continuouslyonly for a fixed period of time Ψ .b) After Ψ elapses, the first two phases are repeated to allow the playersto self-organize and adapt the network to these environmental changes.c) This process is repeated periodically for networks where environmentalchanges may occur. the following stability concept from hedonic games [29]: Definition 6:
A partition
Π = { S , . . . , S l } is Nash-stable if ∀ i ∈ N , S Π ( i ) (cid:23) i S k ∪ { i } for all S k ∈ Π ∪ {∅} (for agents (cid:23) i = (cid:23) M , ∀ i ∈ N ∩ M and for tasks (cid:23) i = (cid:23) T , ∀ i ∈ N ∩ T ).In other words, a coalition partition Π is Nash-stable, if noplayer has an incentive to move from its current coalition toanother coalition in Π or to deviate and act alone. Furthermore,a Nash-stable partition Π implies that there does not exist anycoalition S k ∈ N such that a player i strictly prefers to bepart of S k over being part of its current coalitions, while allplayers of S k do not get hurt by forming S k ∪ { i } . This isthe concept of individual stability, which is formally definedas follows [29]: Definition 7:
A partition
Π = { S , . . . , S l } is individuallystable if there do not exist i ∈ N , and a coalition S k ∈ Π ∪{∅} such that S k ∪ { i } ≻ i S Π ( i ) and S k ∪ { i } (cid:23) j S k for all j ∈ S k (for agents (cid:23) i = (cid:23) M , ∀ i ∈ N ∩ M and for tasks (cid:23) i = (cid:23) T , ∀ i ∈ N ∩ T for tasks).As already noted, a Nash-stable partition is individually stable[29]. For the hedonic coalition formation phase of the proposedalgorithm, we have the following: Proposition 2:
Any partition Π f resulting from the hedo-nic coalition formation phase of the proposed algorithm isNash-stable, and, hence, individually stable. Proof:
For any partition Π , no player (agent or task) i ∈ N has an incentive to leave its current coalition, andact alone as per the utility function in (7). Assume that thepartition Π f resulting from the proposed algorithm is notNash-stable. Hence, there exists a player i ∈ N , and acoalition S k ∈ Π f such that S k ∪ { i } ≻ i S Π f ( i ) , hence,player i can perform a switch operation which contradictswith the fact that Π f is the result of the convergence ofthe proposed algorithm (Theorem 1). Thus, any partition Π f resulting from the hedonic coalition formation phase of theproposed algorithm is Nash-stable, and, hence, by [29], thisresulting partition is also individually stable.Following the formation of the coalitions and the convergenceof the hedonic coalition formation phase to a Nash-stablepartition, the last phase of the algorithm entails the actual datacollection by the agents. In this phase, the agents move fromone task to the other, in their respective coalitions, collectingthe data and transmitting it to the central receiver, similar to apolling system, as explained in Sections II and III. A summaryof the proposed algorithm is shown in Table I.The proposed algorithm, as highlighted in Table I, can adaptthe network topology to environmental changes such as thedeployment of new tasks, the removal of a number of existingtasks, or a periodic low mobility of the tasks (in the casewhere the tasks represent mobile sensor devices for example).For this purpose, the first two phases of the algorithm shownin Table I are repeated periodically over time, to adapt toany changes that occurred in the environment. With regardsto mobility, we only consider the cases where the tasks aremobile for a fixed period of time with a velocity that is smallerthan that of the agents η . In the presence of such a mobileenvironment, the central command center, through Phase Iof the algorithm in Table I informs the agents of the newtasks locations (periodically) and, thus, during Phase II of theproposed algorithm, both agents and tasks can react to theenvironment changes, and modify the existing topology. Asper Theorem 1 and Proposition 2, regardless of the startingposition, the players will always self-organize into a Nash-stable partition, even after mobility, the deployment of newtasks or the removal of existing tasks. In summary, in achanging environment, the first two phases of the algorithm inTable I are repeated periodically, after a certain fixed periodof time Ψ has elapsed during which the players were involvedin Phase III and the actual data collection and transmissionoccurred. Finally, whenever a changing environment is con-sidered, the players are also allowed to periodically clear theirhistory, so as to allow them to explore all the new possibilitiesthat the changes in the environment may have yielded. C. Distributed Implementation Possibilities
For implementation, as shown in Fig. 1, we clearly distin-guish between two inherently different entities: The commandcenter, which is the intelligence that has some control over theagents and the central receiver which is a node in the networkthat is connected to the command center and which wouldreceive the data transmitted by the agents (this distinction canbe, for example, analogous to the distinction between a radionetwork controller and a base station in cellular networks). Inpractice, the central command can be, for example, a node that owns a number of agents and controls a large area whichis divided into smaller areas with each area represented bythe illustration of Fig. 1. Hence, each such small area is aregion having its own central receiver and where a subset ofagents needs to operate and perform coalition formation usingour model. In other scenarios, the command center can alsobe a satellite system that controls groups of agents with eachgroup deployed in a different area (notably when the agents areUAVs for example). In contrast, the central receiver is simplya wireless node that receives the data from the agents and,subsequently, the command center can obtain this data fromall receivers in its controlled area (e.g., through a backbone) .For performing coalition formation, the agents and tasksare required to know different types of information. In orderto perform a switch operation, each agent is required to obtaindata on the location of the tasks (hence, consequently deducingthe hop distance D ij between any two tasks i and j ) as wellas on the arrival rates λ i , i ∈ T of these tasks. As a first step,whenever a tasks’ owner (e.g., a service provider or a thirdparty) requires that its tasks be serviced, it will give the detailsand characteristics of these tasks to the network operator(through service-level agreements for example) which wouldenter these details into the command center. Subsequently, thecommand center can insert this information into appropriatedatabases that the agents can access through, for example, anInternet connection. Such a transfer of information throughactive databases has been recently utilized in many communi-cation architectures, for example, in cognitive radio networkfor primary user information distribution [32], or in UAVsoperation [33]. In cases where the command center controlsonly a single set of agents and a single area, this informationcan be, instead, broadcast directly to the agents. Further, theagents are also required to know the capabilities of eachothers, notably, the link transmission capacity µ i , ∀ i ∈ M and the velocity (which can be used to deduce the switchovertimes). As the agents are all owned by a single operator, thisinformation can be easily fed to the agents at the beginning ofall time prior to their deployment, and, thus, does not requireany additional communication during coalition formation.From the tasks perspective, the amount of information thatneeds to be known is much less than that of the agents,notably since the tasks are, in general, resource-limited en-tities. For instance, as mentioned in the previous section,for performing coalition formation, the tasks do not needto know about the existence or the characteristics of eachothers. The main information that needs to be known bythe tasks is the actual presence of agents. The agents caninitially announce/broadcast their presence to the tasks as soonas they enter into the network. Subsequently, the tasks needonly to be able to enquire, over a control channel, about thepotential utility they would receive from joining the coalitionof a particular agent (which can contain other tasks or agentsbut this is transparent from the perspective of the tasks). Themain reason for this is that the tasks have no benefit in formingcoalitions that have no agents since such coalitions generate utility for the tasks. Hence, from the point of view of the tasks, Our model can accommodate the case in which the command center andthe central receiver coincide, e.g., in a small single-area network. they would see every agent as a black box which can providea certain payoff (communicated over a control channel duringnegotiation phase), and, based on this, they decide to join thecoalition one or another agent (if multiple agents are in thesame coalition then they would offer the same benefit fromthe tasks perspective). Note that, for coalitions that containmultiple agents, the task needs only to ask one agent abouttheir potential utility. In fact, this agent can append, alongwith the information on the utility, a signal to the tasks aboutother agents that belong to the same coalition. By doing so,the tasks would no longer need to assess whether to join acoalition by enquiring from other agents that belong to thesame coalition, i.e., having redundant information. Hence, bysending this additional information, the agent will enable thetasks to avoid doing multiple processing for the same enquiry.Given the information that needs to be known by eachplayer, the proposed algorithm can be implemented in adistributed way since the switch operations can be performedby the tasks or the agents independently of any centralizedentity. In this regard, given a partition Π , in order to determineits preferred switch operation, an agent would assess the payoffit would obtain by joining with any coalition in Π , exceptfor singleton coalitions composed of agents only. For thetasks, given Π , each task negotiates with only the agents(and the coalition to which they belong) in the network inorder to evaluate its payoff and decide on a switch operation.By adopting a distributed implementation, one would reducethe overhead and computational load on the command center,notably when this command center is controlling numerousareas with different groups of agents (each such area isrepresented by the model of Fig. 1). Further, the distributedapproach allows to decentralize the intelligence, and, thus,reduces the detrimental effects on the network and the tasks’owners that can be caused by failures or malicious behaviorat the command center level. It is also important to notethat the distributed approach complies better with the natureof both the agents and the tasks. On one hand, the agentsare inherently autonomous nodes (partially controlled by thecommand center) that need to operate on their own and, thus,make distributed decisions [1]–[6]. On the other hand, thetasks are independent entities that belong to different owners.Consequently, the tasks are apt to make their own decisionsregarding coalition formation and are, generally, unwilling toaccept a coalitional structure imposed by an external entitysuch as the command center. Nonetheless, we note that acentralized approach can also be adopted for the proposedalgorithm notably in small networks where, for example, thecommand center coincides with the central receiver and ownsall the tasks.Regarding complexity, the main complexity lies in theswitch operation, the solution to the traveling salesman prob-lem, i.e., determining the order in which the tasks are visitedwithin a coalition in order to evaluate the utility function,and the assignment of agents as either collectors or relays.For instance, given a present coalitional structure Π whereeach coalition in Π has at least one task , for every agent, thecomputational complexity of finding its next coalition, i.e.,performing a switch operation, is easily seen to be O ( | Π | ) , and the worst case scenario is when all the tasks act alone,in that case | Π | = T . In contrast, for the tasks, the worstcase complexity is O ( M ) since, in order to make a switchoperation, the tasks need only to negotiate with agents. Withregards to the traveling salesman solution, the complexity ofthe used nearest neighbor solution is well known to be linearin the number of cities, i.e., tasks [22], hence, for a coalition S k ∈ Π , the complexity of finding the traveling salesmansolution is simply O ( | S k ∩ T | ) , where S k ∩ T is the setof tasks inside coalition S k . During coalition formation, i.e.,Phase II of the algorithm, whenever a coalition S k is formed,this coalition needs to compute its own traveling salesmanproblem, which has a linear complexity O ( | S k ∩T | ) , as alreadymentioned. Certainly, the overall number of traveling salesmanproblems that should be solved also depends on the numberof new coalitions that were potentially evaluated for coalitionformation prior to convergence to the Nash-stable partition.Hence, the number of traveling salesman solutions that needto be computed is proportional to the number of coalitions (andthe identity and number of the tasks within) that negotiated apotential coalition formation prior to convergence. This cer-tainly depends on the number of iterations till convergence andthe number of switch operations that occurred. Nonetheless,for static environments, after coalition formation ends , i.e., inPhase III of the algorithm in Table I, the traveling salesmansolution needs to be computed only once for each coalition and, afterwards, the network can operate indefinitely (if theenvironment is static) without any need for the coalitions tore-compute the traveling salesman solution.Additionally, for determining whether a agent acts as acollector or relay within any coalition, we consider that theplayers would compute this configuration by inspecting allcombinations and selecting the one that maximizes the utilityin (7). This computation is done during coalition formationfor evaluating the potential utility, and, upon convergence,is maintained during network operation. As the number ofagents in a single coalition is generally small, this computationis straightforward, and has reasonable complexity. Finally, indynamic environments, as the algorithm is repeated period-ically and since we consider only periodic low mobility, thecomplexity of hedonic coalition formation is comparable to thestatic environment case, but with more runs of the algorithm.V. S IMULATION R ESULTS AND A NALYSIS
For simulations, the following network is set up: A centralreceiver is placed at the origin of a km × km square areawith the tasks appearing in the area around it. The path lossparameters are set to α = 3 and κ = 1 , the target SNR isset to ν = 10 dB, the pricing factor is set to δ = 1 , and thenoise variance σ = − dBm. All packets are consideredof size bits which is a typical IP packet size. The agentsare considered as having a constant velocity of η = 60 km/h,a transmit power of ˜ P = 100 mW, and a transmission linkcapacity of µ = 768 kbps (assumed the same for all agents).Further, we consider two classes of tasks in the network.A first class that can be mapped to voice services havingan arrival rate of kbps, and a second class that can bemapped to video services, such as the widely known Quarter −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 Task 7Task 2 Task 6Task 8Task 3Task 1 Task 5Task 9Task 4Task 10 Position in x (km) P o s i t i on i n y ( k m ) Agent 4 − CollectorCoalition S −Agent 4 visits tasks4 −> 9 −> 2 −> 4 Agent 1 − CollectorCoalition S −Agent 1 visits tasks6 −> 7 −> 5 −> 8 −> 6Agents 3 and 5 −Collectors Central ReceiverCoalition S − Agents 3 and 5 visit tasks1 −> 3 −> 10Agent 2 relays for these tasksin that order.Agent 2 − Relay Fig. 2. A snapshot of a final coalition structure resulting from the proposedalgorithm for a network of M = 5 agents and T = 10 tasks. In everycoalition, the agents (collectors and relays) visit the tasks continuously in theshown order. Common Intermediate Format (QCIF) [34], having an arrivalrate kbps. Tasks belonging to each class are generated withequal probability in the simulations. Unless stated otherwise,the throughput-delay tradeoff parameter β is set to . , toindicate services that are reasonably delay tolerant.In Fig. 2, we show a snapshot of the final network partitionreached through the proposed hedonic coalition formationalgorithm for a network consisting of M = 5 agents and T = 10 arbitrarily located tasks. In this figure, Tasks , ,and belong to the QCIF video class with an arrival rate of kbps, while the remaining tasks belong to the voice classwith an arrival rate of kbps. In Fig. 2, we can easily see howthe agents and tasks can agree on a partition whereby a numberof agents service a group of nearby tasks for data collectionand transmission. For the network of Fig. 2, the tasks aredistributed into three coalitions, two of which (coalitions S and S ) are served by a single collector-agent. In contrast,coalition S is served by two collectors and one relay. Theagents in coalition distributed their roles (relay or collector)depending on the achieved utility. For instance, for coalition S , having two collectors and one relay provides a utility of v ( S ) = 52 . while having three collectors yields a utilityof v ( S ) = 10 . , and having one collector and two relaysyields a utility of v ( S ) = 45 . . As a result, the case of twocollectors and one relay maximizes the utility and is agreedupon between the players. Further, the coalitions in Fig. 2 aredynamic, in the sense that, within each coalition, the agentsmove from one task to the other, collecting and transmittingdata to the receiver continuously. The order in which the agentsvisit the tasks, as indicated in Fig. 2, is generated using anearest neighbor solution for the traveling salesman problemas given by Property 3. For example, consider coalition S inFig. 2. In this coalition, agents an act as a single collectorand move from task , to task , to task , and then back totask repeating these visits in a cyclic manner. Concurrentlywith the collectors movement, agent of coalition S , movesand positions itself at the middle of the line connecting thetask being serviced by agents and to the central receiver.In other words, when the collectors are servicing task agent
10 15 20 25 30 35 40020406080100120140160 Number of tasks (T) P a y o ff (r e v enue ) pe r p l a y e r ( agen t o r t a sk ) Proposed hedonic coalition formation (maximum)Proposed hedonic coalition formation (average)Proposed hedonic coalition formation (minimum)Equal allocation of neighboring tasks
Fig. 3. Performance statistics, in terms of maximum, average and minimum(over the order of play) player payoff, of the proposed algorithm comparedto an algorithm that allocates the neighboring tasks equally among the agentsas the number of tasks increases for M = 5 agents. is at the middle of the line connecting task to the centralreceiver, subsequently when the collectors are servicing task agent takes position at the middle of the line connectingtask to the central receiver and so on. Finally, note that, forall the coalitions in Fig. 2 one can verify that the minimumnumber of collectors, as per Proposition 1 is approximately ,(e.g., for coalition S , we have | G S | min = thus collectoris a minimum), and, hence, this condition is easily satisfiedby the coalition formation process.In Fig. 3, we assess the performance of the proposedalgorithm, in terms of the payoff (revenue) per player (agentor task) for a network having M = 5 agents, as the numberof tasks increases. The figure shows the statistics (averagedover the random positions of the tasks), in terms of maximum,average, and minimum over the random order of play. Wecompare the performance with an algorithm that assigns thetasks equally among the agents (i.e., an equal group ofneighboring tasks are assigned for every agent). Fig. 3 showsthat the performance of both algorithms is bound to decreaseas the number of tasks increases. This is mainly due to thefact that, for networks having a larger number of tasks, thedelay incurred per coalition, and, thus, per user increases.This increase in the delay is not only due to the increase inthe number of tasks, but also to the increase in the distancethat the agents need to travel within their correspondingcoalitions (increase in switchover times). In Fig. 3, we notethat the minimum payoff achieved by the proposed algorithmis comparable to that of the equal allocation. Hence, the per-formance of the proposed algorithm is clearly lower boundedby the equal allocation algorithm. However, Fig. 3 shows thatthe average and maximum payoff resulting by the proposedalgorithm is significantly better than the equal allocation at allnetwork sizes up to T = 25 tasks. Albeit this performanceimprovement decreases with the increase in the number oftasks, the performance, in terms of average payoff per player,yielded by the proposed algorithm is no less than . better than the equal allocation for up to T = 25 tasks. Beyond T = 25 tasks, Fig. 3 shows that the average and maximum P a y o ff (r e v enue ) pe r p l a y e r ( agen t o r t a sk ) Proposed hedonic coalition formation (maximum)Proposed hedonic coalition formation (average)Proposed hedonic coalition formation (minimum)Equal allocation of neighboring tasks
Fig. 4. Performance statistics, in terms of maximum, average and minimum(over the order of play) player payoff, of the proposed algorithm comparedto an algorithm that allocates the neighboring tasks equally among the agentsas the number of agents increases for T = 20 tasks. performance of the proposed algorithm is comparable to thatof the equal allocation, notably at T = 40 tasks. The reductionin the performance gap between the two algorithms for largenetworks stems from the fact that, as more tasks exist inthe network, for a fixed number of agents, the possibilityof forming large coalitions, using the proposed algorithm isreduced, and, hence, the structure becomes closer to equalallocation.In Fig. 4, we show the statistics (averaged over the randompositions of the tasks), in terms of maximum, average, andminimum (over the random order of play) payoff per playerfor a network with T = 20 tasks as the number of agents M increases. The performance is once again compared with analgorithm that assigns the tasks equally among the agents (i.e.,an equal group of neighboring tasks are assigned for everyagent). Fig. 4 shows that the performance of both algorithmsincreases as the number of agents increases. This is mainly dueto the fact that when more agents are deployed, the tasks canbe better serviced as the delay incurred per coalition decreasesand the probability of successful transmission improves. Forinstance, as more agents enter the network, they can actas either collectors (for improving the delay) or relays (forimproving the success probability). We note that, at M = 3 ,the performance statistics of the proposed algorithm convergeto the equal allocation algorithm since, for a small number ofagents, the flexibility of forming coalitions is quite restrictedand equal allocation is the most straightforward coalitionalstructure. Nonetheless, Fig. 4 shows that, as M increases, theperformance of the proposed algorithm, in terms of maximumand average payoff achieved, becomes significantly larger thanthat of the equal allocation algorithm, and this performanceadvantage increases as more agents are deployed. Finally,Fig. 4 also shows that the minimum performance of theproposed algorithm is comparable to the equal allocationalgorithm for network with a small number of agents, but asthe number of agents increases, the minimum performance ofhedonic coalition formation is better than equal allocationat M = 7 , and this advantage increases further with M .
10 12 15 17 20 25246810121416 Number of tasks (T) C oa li t i on s i z e Proposed hedonic coalition formation (maximum)Proposed hedonic coalition formation (average)Equal allocation of neighboring tasks
Fig. 5. Average and maximum (over order of play) coalition size yieldedby the proposed algorithm and an algorithm that allocates the neighboringtasks equally among the agents, as a function of the number of tasks T for anetwork of M = 5 agents. In Fig. 5, we show the average and maximum (over therandom order of play) coalition size resulting from the pro-posed algorithm as the number of tasks T increases, for anetwork of M = 5 agents and arbitrarily deployed tasks.These results are averaged over the random positions of thetasks and are compared with the equal allocation algorithm.Fig. 5 shows that, as the number of tasks increases, theaverage coalition size for both algorithms increases. For theproposed algorithm, the maximum coalition size also increaseswith the number of tasks. This is an immediate result of thefact that, as the number of tasks increases, the probability offorming larger coalitions is higher and, hence, our proposedalgorithm yields larger coalitions. Further, at all network sizes,the proposed algorithm yields coalitions that are relativelylarger than the equal allocation algorithm. This result impliesthat, by allowing the players (agents and tasks) to selfishlyselect their coalitions, through the proposed algorithm, theplayers have an incentive to structure themselves in coalitionswith average size lower bounded by the equal allocation. Ina nutshell, through hedonic coalition formation, the resultingtopology mainly consists of networks composed of a largenumber of small coalitions as demonstrated by the averagecoalition size. However, in a limited number of cases, thenetwork topology can also be composed of a small numberof large coalitions as highlighted by the maximum coalitionsize shown in Fig. 5.In Fig. 6, we show, over a period of minutes, thefrequency in terms of average switch operations per minuteper player (agent or task) achieved for various velocities ofthe tasks in a mobile wireless network with M = 5 agentsand different number of tasks. As the velocity of the tasksincreases, the frequency of the switch operations increasesfor both T = 10 and T = 20 due to the changes in thepositions of the various tasks incurred by mobility. Fig. 6shows that the case of T = 20 tasks yields a frequencyof switch operations significantly higher than the case of T = 10 tasks. This result is interpreted by the fact that,as the number of tasks increases, the possibility of findingnew partners as the tasks move increases significantly, hence
10 15 20 25 30 35 40 45 50024681012141618 Velocity of tasks (km/h) A v e r age nu m be r o f s w i t c h ope r a t i on s pe r m i nu t e pe r p l a y e r Network with T = 10 tasksNetwork with T = 20 tasks
Fig. 6. Frequency of switch operations per minute per player achievedover a period of minutes for different tasks’ velocities in a network having M = 5 agents and different number of mobile tasks. yielding an increase in the topology variation as reflected bythe number of switch operations. In summary, this figure showsthat hedonic coalition formation allows the agents and thetasks to self-organize and adapt their topology to mobility,through adequate switch operations.The network’s adaptation to mobility is further assessed inFig. 7, where we show, over a period of minutes, the averagecoalition lifespan (in seconds) achieved for various velocitiesof the tasks in a mobile wireless network with M = 5 agentsand different number of tasks. The coalition lifespan is definedas the time (in seconds) during which a coalition is presentin the mobile network prior to accepting new members orbreaking into smaller coalitions (due to switch operations).Fig. 7 shows that, as the velocity of the tasks increases,the average lifespan of a coalition decreases. This is dueto the fact that, as mobility increases, the possibility offorming new coalitions or splitting existing coalitions increasessignificantly. For example, for T = 20 , the coalition lifespandrops from around seconds for a tasks’ velocity of km/hto just under a minute as of km/h, and down to around seconds at km/h. Furthermore, Fig. 7 shows that asmore tasks are present in the network, the coalition lifespandecreases. For instance, for any given velocity, the lifespan ofa coalition for a network with T = 10 tasks is significantlylarger than that of a coalition in a network with T = 20 tasks.This is a direct result of the fact that, for a given tasks’ velocity,as more tasks are present in the network, the players are ableto find more partners to join with, and hence the lifespan ofthe coalitions becomes shorter. In brief, Fig. 7 provides aninteresting assessment of the topology adaptation aspect ofthe proposed algorithm through the process of forming newcoalitions or breaking existing coalitions.Moreover, for further analysis of the self-adapting aspectof the proposed algorithm, we study the variations of thecoalitional structure over time for a network where tasksare entering and leaving the network. For this purpose, inFig. 8, we show the variations of the average (over the randompositions of the tasks) number of players per coalition, i.e., theaverage coalition size, over a period of minutes, as new
10 15 20 25 30 35 40 45 50406080100120140160180200 Velocity of tasks (km/h) A v e r age c oa li t i on li f e s pan ( s e c ond s ) Network with T = 10 tasksNetwork with T = 20 tasks
Fig. 7. Average coalition lifespan (in seconds) achieved over a period of minutes for different tasks’ velocities in a network having M = 5 agents anddifferent number of mobile tasks. tasks join the network and/or existing tasks leave the network.The considered network in Fig. 8 possesses M = 5 agents andstarts with T = 15 tasks. The results are shown for differentrates of change which is defined as the number of tasks thathave either entered the network or left the network per minute.For example, a rate of change of tasks per minute indicatesthat either tasks enter the network every minute, tasksleave the network every minute, or tasks enters the networkand another tasks leaves the network every minute (these casesmay occur with equal probability). In this figure, we can seethat, as time evolves, the structure of the network is changing,with new coalitions forming and other breaking as reflected bythe change in coalitions size. Furthermore, we note that, as therate of change increases, the changes in the topology increase.For instance, it is seen in Fig. 8 that for a rate of change of tasks per minute, the variations in the coalition size aremuch larger than for the case of tasks per minute (which isalmost constant for many periods of time). In summary, Fig. 8shows the network topology variations as tasks enter or leavethe network. Note that, after the minutes have elapsed, thenetwork re-enters in the Phase III of the algorithm where datacollection and transmission occurs.In Fig. 9, we assess the performance of the proposedalgorithm, in terms of the payoff (revenue) per player (agent ortask) for a network having M = 5 agents and T = 20 tasks,as the throughput-delay tradeoff parameter β increases. Thefigure shows the statistics, in terms of maximum, average,and minimum over the random order of play between theplayers. Fig. 9 shows that, for small β , the performance ofthe proposed algorithm is comparable to the equal allocationalgorithm and the payoffs are generally small. This result isdue to the fact that, for small β , the tasks are highly delaysensitive, and the delay component of the utility governs theperformance. Hence, for such tasks, the proposed algorithmyields a performance similar to equal allocation. However, asthe tradeoff parameter β increases, the maximum and averageutility yielded by our proposed algorithm outperforms theequal allocation algorithm significantly. For instance, as of β = 0 . , hedonic coalition formation is highly desirable, A v e r age c oa li t i on s i z e Rate of change 2 tasks per minRate of change 5 tasks per min
Fig. 8. Topology variation over time as new tasks join the network andexisting tasks leave the network with different rates of tasks arrival/departurefor a network starting with T = 15 tasks and having M = 5 agents. β P a y o ff (r e v enue ) pe r p l a y e r ( agen t o r t a sk ) Proposed hedonic coalition formation (maximum)Proposed hedonic coalition formation (average)Proposed hedonic coalition formation (minimum)Equal allocation of neighboring tasks
Fig. 9. Performance statistics, in terms of maximum, average and minimum(over the order of play) player payoff, of the proposed algorithm compared toan algorithm that allocates the neighboring tasks equally among the agents asthe throughput-delay tradeoff parameter β increases for M = 5 agents and T = 20 tasks. and presents a performance improvement in terms of averagepayoff of around . relative to the equal allocationalgorithm (at β = 0 . , the proposed algorithm has anaverage payoff of . while equal allocation has an averagepayoff of . ). This advantage increases with β . Note that,for all tradeoff parameters, the performance of the proposedalgorithm, in terms of minimum (over order of play) payoffgained by a player is lower bounded by equal allocation and,in average, outperforms the equal allocation algorithm.VI. C ONCLUSIONS
In this paper, we introduced a novel model for task allo-cation among a number of autonomous agents in a wirelesscommunication network. In this model, a number of wirelessagents are required to service several tasks, arbitrarily locatedin a given area. Each task represents a queue of packets thatrequire collection and wireless transmission to a centralizedreceiver by the agents. The task allocation problem is modeledas a hedonic coalition formation game between the agents andthe tasks that interact in order to form disjoint coalitions. Eachformed coalition is mapped to a polling system which consists of a number of agents continuously collecting packets from anumber of tasks. Within a coalition, the agents can act either ascollectors that move between the different tasks present in thecoalition for collecting the packet data, or relays for improvingthe wireless transmission of the data packets. For forming thecoalitions, we introduce an algorithm that allows the players(tasks or agents) to join or leave the coalitions based on theirpreferences which capture the tradeoff between the effectivethroughput and the delay achieved by the coalition. We studythe properties and characteristics of the proposed model, weshow that the proposed hedonic coalition formation algorithmalways converges to a Nash-stable partition, and we studyhow the proposed algorithm allows the agents and tasks totake distributed decisions for adapting the network topologyto environmental changes such as the deployment of newtasks, the removal of existing tasks or the mobility of thetasks. Simulation results show how the proposed algorithmallows the agents and tasks to self-organize into independentcoalitions, while improving the performance, in terms ofaverage player (agent or task) payoff, of at least . (fora network of agents with up to tasks) relatively to ascheme that allocates nearby tasks equally among the agents.In a nutshell, by combining concepts from wireless networks,queueing theory and novel concepts from coalitional gametheory, we proposed a new model for task allocation amongautonomous agents in communication networks which is wellsuited for many practical applications such as data collection,data transmission, autonomous relaying, operation of messageferry (mobile base stations), surveillance, monitoring, or main-tenance in next-generation wireless networks.R EFERENCES[1] M. Debbah, “Mobile flexible networks: The challenges ahead,” in
Proc.International Conference on Advanced Technologies for Communica-tions , Hanoi, Vietnam, Oct. 2008.[2] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Bas¸ar, “A game-based self-organizing uplink tree for VoIP services in IEEE 802.16jnetworks,” in
Proc. Int. Conf. on Communications , Dresden, Germany,Jun. 2009.[3] D. Niyato, E. Hossein, and Z. Han,
Dynamic spectrum access incognitive radio networks . Cambridge, UK: Cambridge University Press,2009.[4] S. Yousefi, E. Altman, R. El-Azouzi, and M. Fathy, “Connectivity invehicular ad hoc networks in presence of wireless mobile base-stations,”in
Proc. 7th International Conference on ITS Telecommunications ,Sophia Antipolis, France, Jun. 2007.[5] M. M. B. Tariq, M. Ammar, and E. Zegura, “Message ferry routedesign for sparse ad hoc networks with mobile nodes,” in
Proc. of ACMMobiHoc , Florence, Italy, May 2006.[6] Y. Shi and Y. T. Hou, “Theoretical results on base station movementproblem for sensor network,” in
Proc. IEEE Conf. on Computer Com-munications (INFOCOM) , Phoenix, AZ, USA, Apr. 2008.[7] B. Gerkey and M. J. Mataric, “A formal framework for the study oftask allocation in multi-robot systems,”
International Journal of RoboticsResearch , vol. 23, no. 9, pp. 939–954, Sep. 2004.[8] M. Alighanbari and J. How, “Robust decentralized task assignment forcooperative UAVs,” in
Proc. of AIAA Guidance, Navigation, and ControlConference , Colorado, USA, Aug. 2006.[9] D. M. Stipanovic, P. F. Hokayem, M. W. Spong, and D. D. Siljak,“Cooperative avoidance control for multi-agent systems,”
ASME Journalof Dynamic Systems, Measurement, and Control , vol. 129, no. 5, pp.699–706, Sep. 2007.[10] Q. Chen, M. Hsu, U. Dayal, and M. Griss, “Multi-agent cooperation,dynamic workflow and XML for e-commerce automation,” in
Proc. ofInt. Conf. on Autonomous agents , Catalonia, Spain, Jun. 2000.[11] O. Shehory and S. Kraus, “Methods for task allocation via agentcoalition formation,”
Artifical Intelligence Journal , vol. 101, pp. 165–200, May 1998.[12] R. Beard, D. Kingston, M.Quigley, D. Snyder, R. Christiansen, W. John-son, T. Mclain, and M. Goodrich, “autonomous vehicle technologiesfor small fixed wing UAVs,”
AIAA Journal of Aerospace Computing,Information, and Communication , vol. 2, no. 1, pp. 92–108, January2005. [13] Z. Han, A. Swindlehurst, and K. J. Liu, “Optimization of MANETconnectivity via smart deployment/movement of unmanned air vehicles,”
IEEE Trans. Vehicular Technology , to appear, 2009.[14] D. L. Gu, G. Pei, H. Ly, M. Gerla, B. Zhang, and X. Hong, “UAV aidedintelligent routing for ad hoc wireless network in single area theater,” in
Proc. IEEE Wireless Communications and Networking Conf. , Chicago,IL, Sep. 2000, pp. 1220–1225.[15] K. Xu, X. Hong, M. Gerla, H. Ly, and D. L. Gu, “landmark routingin large wireless battlefield networks using UAVs,” in in Proc. of IEEEMilitary Communication Conference , Washington DC, Oct. 2001, pp.230–234.[16] D. L. Gu, H. Ly, X. Hong, M. Gerla, G. Pei, and Y. Lee, “C-ICAMA,a centralized intelligent channel assigned multiple access for multi-layer ad-hoc wireless networks with UAVs,” in
Proc. IEEE WirelessCommunications and Networking Conf. , Chicago, IL, Sep. 2000, pp.879–884.[17] J. Proakis,
Digital Communications . New York, USA: 4th ed., McGraw-Hill, 2001.[18] R. B. Myerson,
Game Theory, Analysis of Conflict . Cambridge, MA,USA: Harvard University Press, Sep. 1991.[19] H. Takagi,
Analysis of Polling Systems . Cambridge, MA, USA: TheMIT Press, Apr. 1986.[20] H. Levy and M. Sidi, “Polling systems: applications, modeling, andoptimization,”
IEEE Trans. Commun. , vol. 10, pp. 1750–1760, Oct.1990.[21] Y. Li, H. S. Panwar, and J. Shao, “Performance analysis of a dual roundrobin matching switch with exhaustive service,” in
Proc. IEEE GlobalCommunication Conference , Taipei, Taiwan, Nov. 2002.[22] D. Applegate, R. M. Bixby, V. Chvatal, and W. J. Cook,
The travelingsalesman problem: a computational study . Princeton, NJ, USA:Princeton University Press, 2006.[23] L. Kleinrock, “Power and deterministic rules of thumb for probabilisticproblems in computer communications,” in
Proc. Int. Conf. on Commu-nications , Boston, USA, Jun. 1979.[24] W. K. Ching, “A note on the convegence of asynchronous greedyalgorithm with relaxation in a multiclass queueing system,”
IEEECommun. Lett. , vol. 3, pp. 34–36, Feb. 1999.[25] E. Altman, T. Bas¸ar, and R. Srikant, “Nash equilibria for combined flowcontrol and routing in networks: asymptotic behaviour for a large numberof users,”
IEEE Trans. Automtatic Control , vol. 47, pp. 917–930, Jun.2002.[26] V. Vukadinovic and G. Karlsson, “Video streaming in 3.5G: Onthroughput-delay performance of proportional fair scheduling,” in
Proc.Int. Symp. on Modeling, Analysis and Simulation of Comp. and Telecom.Systems , California, USA, Sep. 2006.[27] D. Ray,
A Game-Theoretic Perspective on Coalition Formation . NewYork, USA: Oxford University Press, Jan. 2007.[28] G. Demange and M. Wooders,
Group Formation in Economics . NewYork, USA: Cambridge University Press, 2006.[29] A. Bogomonlaia and M. Jackson, “The stability of hedonic coalitionstructures,”
Games and Economic Behavior , vol. 38, pp. 201–230, Jan.2002.[30] E. Diamantoudi and L. Xue, “Farsighted stability in hedonic games,”
Social Choice Welfare , vol. 21, pp. 39–61, Jan. 2003.[31] J. Dr`eze and J. Greenberg, “Hedonic coalitions: Optimality and stability,”
Econometrica , vol. 48, pp. 987–1003, Jan. 1980.[32] S. Shellhammer, A. Sadek, and W. Zhang, “Technical challenges forcognitive radio in TV white space spectrum,” in
Proc. of InformationTheory and Applications Workshop (ITA) , San Diego, USA, Feb. 2009.[33] M. Brohede and S. F. Andler, “Distributed simulation communicationthrough an active real-time database,” in
Proc. of NASA Goddard/IEEESoftware Engineering Workshop , Greenbelt, MD, USA, Dec. 2002.[34] L. Wainfan,
Challenges in Virtual Collaboration: Videoconferencing Au-dioconferencing and Computer-Mediated Communications . California,USA: RAND Corporation, Jul. 2005.[35] W. Saad, Z. Han, T. Bas¸ar, M. Debbah, and A. Hjørungnes, “A selfishapproach to coalition formation among unmanned aerial vehicles inwireless networks,” in