Distributed Spectrum Allocation via Local Bargaining

Transcription

1 Distibuted Spectum Allocation via Local Bagaining Lili Cao Shanghai Jiaotong Univesity, Shanghai, P.R. China cao li Haitao Zheng Micosoft Reseach Asia, Beijing, P.R. China Abstact In this pape, we pesent an adaptive and distibuted appoach to spectum allocation in mobile ad-hoc netwoks. We popose a local bagaining appoach whee uses affected by the mobility event self-oganize into bagaining goups and adapt thei spectum assignment to appoximate a new optimal assignment. The numbe of computations equied to adapt to topology changes can be significantly educed compaed to that of the conventional topology-based optimizations that ignoe the pio assignment. In paticula, we popose a Fainess Bagaining with Feed Povety to impove fainess in spectum assignment and deive a theoetical lowe bound on the minimum assignment each use can get fom bagaining fo cetain netwok configuations. Such bound can be utilized to guide the bagaining pocess. We also show that the diffeence between the poposed bagaining appoach and the tue optimal appoach is uppe-bounded. Expeimental esults demonstate that the poposed bagaining appoach povides simila pefomance as the topology-based optimization but with moe than 5% of eduction in complexity. I. INTRODUCTION Wieless devices ae becoming ubiquitous, placing additional stess on the fixed adio spectum available to all access technologies. To eliminate intefeence between diffeent wieless technologies, cuent policies allocate a fixed spectum slice to each wieless technology. This static assignment pevents devices fom efficiently utilizing allocated spectum, esulting in spectum holes (no devices in aea) and vey poo utilization [6]. These esults futhe motivate the Open Spectum [3] appoach to spectum access. Open Spectum allows unlicensed (seconday) uses to coexist with legacy (pimay) spectum holdes, theeby ceating new capacity and commecial value fom existing spectum anges. Seconday uses oppotunistically utilize unused licensed spectum on a non-intefeing o leasing basis based on ageements and constaints imposed by pimay uses. Open spectum focuses on contolling the behavio of seconday uses while keeping the system tanspaent to pimaies. While maximizing spectum utilization is the pimay goal of dynamic spectum systems, a good allocation scheme is also needed to povide fainess acoss uses. We heeby use use to epesent seconday use. A use seizing spectum without coodinating with othes can cause hamful intefeence with its suounding neighbos, and thus educing available spectum. Given a fixed topology, existing appoaches can efficiently allocate spectum to uses by educing the poblem to a vaiant of the gaph coloing poblem [23], [8]. A conflict fee spectum assignment is obtained fo the given topology. In geneal, a topology-optimized allocation algoithm begins with no pio infomation, and assigns each use an optimal assignment. In a mobile netwok, howeve, uses ae constantly moving and the netwok topology changes. Using this global optimization appoach, the netwok needs to completely ecompute spectum assignments fo all uses afte each change, esulting in high computational and communication ovehead. This costly opeation needs to be epeated fequently to maintain spectum utilization and fainess. In this wok, we conside a distibuted appoach to spectum allocation that stats fom the pevious spectum assignment, and pefoms a limited numbe of computations to adapt to ecent topology changes. We popose a local bagaining appoach whee uses affected by the mobility event self-oganize into bagaining goups and adapt thei spectum assignment to appoximate a new optimal conflict fee assignment. The key contibutions of this pape ae thee-fold: Local Bagaining Stategy. We popose a local bagaining famewok, and two bagaining stategies: one-to-one fainess bagaining and feed povety bagaining, to impove fainess based system utility. Bound of Local Bagaining Pefomance. We deive a theoetical lowe bound on the numbe of channels each use can get fom bagaining, efeed to as Povety Line. The bound eflects the level of fainess enfoced by the poposed bagaining stategies, and can seve as a guidance fo bagaining. We also deive a uppe bound on the pefomance diffeence between the local bagaining and the global optimal solution, efeed to as pice of anachy. Simulation of Efficiency and Complexity. We conduct extensive simulations to quantify the pefomance of local bagaining. Results indicate the poposed bagaining pefoms similaly to the gaph-coloing solution[23], [8] but with significantly educed algoithm complexity. We also validate the coectness of the povety line and the effectiveness of povety guided bagaining. The est of the pape is oganized as follows. We begin in Section II by defining the spectum allocation poblem and the system utility functions. Next, we popose a local bagaining famewok in Section III and develop specific stategies to impove system fainess in Section IV. Next in Section V, we conduct expeiments to evaluate the pefomance of bagaining stategy and validate the theoetical lowe bound. We then deive analytical esults in tems of pice of anachy in

2 Section VI. Finally, we summaize elated wok in Section VII, discuss implications and futue diections and conclude in Section VIII. II. BACKGROUND ON SPECTRUM ALLOCATION The two key goals of a spectum allocation algoithm ae spectum utilization and fainess. Combinations of these two goals fom specific utility functions that can be customized fo diffeent types of netwok applications. As backgound, we descibe in this section ou pevious wok on efficient and globally optimized spectum allocation. We stat with the theoetical model used to epesent the geneal allocation poblem, and two utility functions that maximize spectum utilization and fainess. We descibe how to educe the optimal allocation poblem to a vaiant of a gaph multi-coloing poblem and descibe the pevious solution that optimize the spectum allocation fo a given topology. A. Poblem Model and Utility Functions We conside the case whee the collection of available spectum anges foms a spectum pool, divided into nonovelapping othogonal channels. We assume a netwok of N uses indexed fom to N competing fo M spectum channels indexed to M. Each use can be a tansmission link o a boadcast access point. Uses select communication channels and adjust tansmit powe accodingly to avoid intefeing with pimaies. The channel availability and thoughput fo each use can be calculated based on the location and channel usage of neaby pimaies. The spectum access poblem becomes a channel allocation poblem, i.e. to obtain a conflict fee channel assignment fo each use that maximizes system utility. The key components of ou model ae: - Channel availability L(n). Γ = {l m,n l m,n {, }} M N is a M by N binay matix epesenting the channel availability: l m,n = if and only if channel m is available at use n. In geneal, l m,n = when channel m is occupied by a pimay use who conflicts with use n, so that the tansmissions of n on this channel will intefee with the pimay s activity if they use channel m concuently. Let L(n) = { m M l m,n = } be the set of channels available at n. - Intefeence constaint C. Let C = {c n,k c n,k {, }} N N, a N by N matix, epesents the intefeence constaints among uses. If c n,k =, uses n and k would intefee with each othe if they use the same channel. The intefeence constaint depends on the signal stength of tansmissions and the distance between uses. A simple model of intefeence constaint is the binay geomety metic, i.e. two tansmissions conflict if they ae within π distance fom each othe. This povides an appoximation to the effects of intefeence in eal wieless systems. It should be noted that the intefeence constaint could also depend on the fequency location of the channel (i.e. m), since powe and tansmission egulations vay significantly acoss fequencies. The wok in [23], [8] consides the channel-dependency and uses a M by N by N intefeence matix C. In this pape, fo simplicity, we conside a channelindependent intefeence constaint, assuming channels have simila powe and tansmission egulations. It is staightfowad to extend the poposed appoaches to account fo channel-dependent o othe intefeence conditions [4]. - Use dependent channel thoughput B. Let B = {b m,n > } M N descibe the ewad that a use gets by successfully acquiing a spectum band, i.e. b m,n epesents the maximum bandwidth/thoughput that use n can acquie though using spectum band m (assuming no intefeence fom othe neighbos). Let b m,n = if l m,n =. So that B epesents the bandwidth weighted use available spectum. - Conflict fee assignment A. A = {a m,n a m,n {, }} M N whee a m,n = denotes that spectum band m is assigned to use n, othewise. A satisfies all the constaints defined by C, that is, a m,n + a m,k, if c n,k =, n, k < N, m < M. Let Λ N,M denote the set of conflict fee spectum assignments fo a given set of N uses and M spectum bands. - Use thoughput of a conflict fee assignment. Let T P A (n) epesent the thoughput that use n gets unde assignment A, i.e. T P A (n) = M m= a m,n b m,n. Given this model, the goal of spectum allocation is to maximize netwok utilization, defined by U. We can define the spectum assignment poblem by the following optimization function: A = max agmax U(A), A Λ N,M We can obtain utility functions fo specific application types using sophisticated subjective suveys. An altenative is to design utility functions based on taffic pattens and fainess inside the netwok. In this pape, we conside and addess fainess based system utility. Consistent with pio wok[7], [2], [2], we addess fainess fo single-hop flows since they ae the simplest fomat in wieless tansmissions. We postpone the discussion of outing elated utility functions to a futue pape. Simila to [7], we define fainess in tems of maximizing total logaithmic use thoughput, efeeed to as popotional fainess. The utility can be expessed as U(A) = N n= log T P A (n) = N n= a m,n b m,n. () M log m= As a efeence, anothe utility function is the total spectum utilization in tems of total use thoughput, U(A) = N n= T P A(n). Maximizing utilization does not conside fainess, and the esulting channel assignment is in geneal unbalanced. B. Colo-Sensitive Gaph Coloing In [23], [8], it is shown that by mapping each channel into a colo, the channel assignment poblem can be educed to a gaph multi-coloing (GMC) poblem.

3 Definition : Given the channel assignment poblem in above, the system can be epesented by a Conflict Gaph G = (V, E, B) whee V is a set of vetices denoting the uses that shae the spectum, B epesents the bandwidth weighted available spectum, mapping to the colo list at each vetex, and E is a set of undiected edges between vetices epesenting intefeence constaint between two vetices defined by C. Fo any two distinct vetices u,v V, an edge between u and v, is in E if and only if c u,v =. Fig. illustates an example of GMC gaph. Thee ae 5 colos available. The numbes outside the backets attached to each node denote the colos assigned to that node, while the numbes inside the backets denote the available colo list of each node. Fig.. An example of GMC gaph A GMS poblem is to colo each vetex using a numbe of colos fom its colo list, and find the colo assignment that maximizes system utility. The coloing is constained by that if an edge exists between any two distinct vetices, they can t be coloed with the same colo. Most impotantly, the objective of coloing is to maximize system utility. This is diffeent fom taditional gaph colo solutions that assign one colo pe vetex. Notice that the solution to this gaph coloing poblem is to maximize system utility fo a given gaph, i.e. a given topology and channel availability. This chaacteizes the optimal solution fo a static envionment. The optimal coloing poblem is known to be NP-had [7]. Efficient algoithms to optimize spectum allocation fo a given netwok topology exist. In [23], the authos pesented a set of sequential heuistic based appoaches that poduce good coloing solutions. The algoithm stats fom empty colo assignment and iteatively assign colos to vetices to appoximate the optimal assignment. In each stage, the algoithm labels all the vetices with a non-empty colo list accoding to some policy-defined labeling. The algoithm picks the vetex with the highest valued label and assigns the colo associated with the label to the vetex. The algoithm then deletes the colo fom the vetex s colo list, and fom the colo lists of the constained neighbos. The colo list and the intefeence constaint of a vetex keep on changing as othe vetices ae pocessed, and the labels of the coloed vetex and its neighbo vetices ae modified accoding to the new gaph. The algoithm can be implemented using a centalized contolle who obseves global topology and makes decisions, o though a distibuted algoithm whee each vetex pefoms a distibuted voting pocess. Results in [8], [23] show that the heuistic based algoithms pefom similaly to the global optimum (deived off-line fo simple topologies), and the centalized and distibuted algoithms pefom similaly. III. LOCAL FRAMEWORK The appoach descibed in Section II globally optimizes spectum allocation fo a given topology. In a mobile netwok model, node movements lead to constant changes in netwok topology. Using the existing appoach, we can eapply the spectum allocation algoithm afte each change in the conflict gaph. This appoach assumes no pio allocation infomation, and incus high computation and communication oveheads. To educe these oveheads, we popose the use of an adaptive and obust distibuted algoithm that takes pio allocation into account in new spectum assignments. An efficient dynamic allocation algoithm can un evey time use movement causes a change in the coesponding netwok conflict gaph. Theefoe, an adaptive algoithm needs to only compensate fo small changes affecting a local netwok egion. The algoithm stats fom a non-optimal spectum allocation, which can be constucted fom the allocation pio to the topology change. Conside a conflict gaph with N nodes (indexed fom to N ) and M channels (indexed fom to M ), whee the optimized assignment is A M N. When a new node (indexed as N) joins the netwok, the assignment afte intoduction of the node becomes A M (N+) whee { A Am,n : n N, m < M m,n = : n = N, m < M O if a pimay use i entes the netwok and wants to use channel m, the nodes within impact of pimay i (denoted by Nb(i)) need to stop using channel m within a given time. Hence, the assignment becomes A M N such that { A : m = m m,n = and n Nb(i) A m,n : othewise Assuming the spectum allocation was nea optimal befoe the topology change, local bagaining between affected vetices can quickly optimize allocations fo utilization and fainess. Duing local bagaining, sets of neighboing vetices, each of which fom a connected component of the conflict gaph, self-oganize into bagaining goups. Each goup modifies spectum assignment within the goup to impove system utility while ensuing that the change in spectum assignment does not equie any change at othe nodes outside the goup (due to intefeence constaints). Note that a node can epesent a tansmission link o an access point. Bagaining elated to a tansmission link is caied out by the tansmitte o eceive while bagaining elated to an access point is caied out by the access point. A. Bagaining constaints To pefom bagaining, we must fist detemine the size and membeship of local bagain goups. Lage goups incease

4 the complexity of bagaining due to high synchonization and communication costs. In addition, inteactions might occu between bagain goups if they shae neighboing uses. To facilitate bagaining, we popose two constaints to egulate the pocedue and simplify the pocess. Constaint : Limited Neighbo Bagaining. While pai-wise bagaining is aleady a had convex optimization poblem, bagaining within a lage goup implies even highe computation and communication oveheads. Coodinating aound a cental leade pe goup can geatly simplify spectum assignment. In this pape, we popose a simple goup fomation whee a node who wants to impove its spectum assignment boadcasts a bagaining equest to its k-hop neighbos, whee k is the atio of intefeence ange and tansmission ange. These neighbos ae connected to the node in the coesponding conflict gaph. Those neighbos whose ae willing to paticipate eply to the sende and fom a bagaining goup. Note that it is possible that two connected neighbos in a conflict gaph might not be able to communicate diectly with each othe, i.e. when k >. The bagaining infomation can be elayed by the nodes in between. These elay nodes do not necessay paticipate in the bagaining goup. In this following, we will use node to pesent any vetex in a conflict gaph. Fo each bagaining goup, the equeste becomes the goup coodinato and pefoms the bagaining computation. The bagaining stategies can be divided into the following fomats:.a) One-to-one bagaining- The node n who initiates the bagaining can choose to bagain with only one neighboing node n 2 at a time. They exchange some channels to impove system utility while complying with the conflict constaints fom the othe neighbos. This is the simplest bagaining pocess and the equeste only needs appoval fom one of his neighbos to pefom the bagaining. When multiple neighbos e.g. n 2 and n 3 acknowledge the bagaining equest, n can sequentially compute assignment assuming bagaining with n 2 fist, and then with n 3. n boadcasts the assignment to both n 2 and n 3. This expands the bagaining goup to (n, n 2, n 3 ) without adding exta signalling ovehead. Howeve, this also equies that n chooses a sequential bagaining ode and gets appoval fom all the goup membes on the ode befoe conducting bagaining. If one of neighbos disappoves the equest, n needs to pefom anothe equest. Hence, fo simplicity, we estict this fomat to only one-to-one bagaining. Fig. 2 illustates an example of one-to-one bagaining..b) One-buye-multi-selle bagaining- A buye node n puchases a set of channels M, fom its neighbos who ae cuently using any channel in M, such that to impove system utility. In this case, the bagaining equies concuent appoval fom multiple neighbos. As we will show late, this type of bagaining is necessay to eliminate use stavation. Fig. 2 illustates an example with one buye and fou selles. Constaint 2: Self-contained Goup Bagaining Once the bagaining goups ae oganized, the bagaining Fig. 2. An example of Bagaining Goups. inside each goup should not distub the spectum assignment at nodes outside the goup. That is, afte the bagaining, the modified channel assignment should not lead to any conflict with nodes outside the goup. This helps to maintain system stability, so that a bagaining may not invoke a seies of eactions due to violations in intefeence constaints. Moe impotantly, this guaantees that if a bagaining impoves the utility in a local aea, it also impoves the system utility. O in othe wods, a local impovement will lead to a system impovement. This constaint has two components. 2.a) Resticted Bagainable Channels- This esticts the set of channels that ae exchangeable between nodes inside each bagaining goup, such that when a node gets one channel fom its neighbo, the assignment does not conflict with its neighbos outside the bagaining goup. 2.b) Isolated Bagaining Goup- This not only esticts each node to paticipate in at most one bagaining goup at any time, but also equies that the membes of any two bagaining goups can not be diectly connected. Having nodes between goups egulate spectum adjustment and pevents conflict between goups. The necessity of this equiement can be explained by the following example. Assume thee ae two neighboing nodes A and B (two nodes in the conflict gaph connected with an edge) who ae the membes of two diffeent bagaining goups. Befoe assignment, A and B ae not using channel. Afte the bagaining, both A and B ae ganted with channel fom thei bagaining neighbos. Howeve, as A and B conflict with each othe if using the same channel, the bagaining poduces intefeence conflict among nodes. The detailed pocedue to fom isolated bagaining goups will be intoduced in Section III-B. An example of isolation between bagaining goups is shown in Fig. 2. B. Bagaining steps in detail We design a local bagaining pocedue based on the above constaints, assuming a distibuted achitectue. We popose a distibuted, iteative gouping and bagaining pocess. We assume that nodes peiodically boadcast thei cuent channel

5 assignment and intefeence constaints to thei neighbos. Each node has thee states: bagaining, disabled and enabled (see Fig. 3). Only enabled nodes can pefom bagaining. The actual bagaining involves the following 4 steps, and epeats until no futhe bagaining can impove system utility. Hee nodes efe to the vetices in the conflict gaph, and two connected nodes might be physically k (k > hops away. Infomation exchange between them is done though elay..) Initialize Bagaining Request In geneal, nodes affected by mobility events initialize bagaining. Based on boadcasts of channel assignments and intefeence constaints fom neighbos, an enabled node detemines if bagaining with a neighbo will lead to an impovement in system utility. If such neighbos exist, the node boadcasts a bagain equest to the neighbos along with its cuent channel assignment and intefeence constaints. Such boadcasts educes communication ovehead. As we will show in Section IV-C, additional citeion exists to guide nodes on geneating bagaining equests. 2.) Acknowledge Bagaining Request Neighbos who ae enabled and willing to bagain eply an ACK message with its cuent channel assignment and intefeence constaints. We assume that nodes ae willing to collaboate to impove system utility, and accept equests that impove system utility even if it might degade thei individual channel assignments. Incentive systems to encouage such collaboation will be investigated in a futue pape. If a node eceives multiple concuent equests fom its neighbos, it acknowledges the equest that leads to the highest bagaining gain as calculated based on infomation embedded in the equest. 3.) Bagain Goup Fomation When the equeste eceives the eplies, it selects the membes of the bagaining goup, and boadcasts this infomation along with the poposed modification of the channel assignment to neighbos. Once the bagaining goup is set, its membes ente bagaining state. They boadcast a DISABLE message with a time equal to the estimated duation of the bagain pocess to neighbos not in the bagaining goup. Note that the DISABLE message can be embedded in the ACK messages to educe ovehead. Nodes eceiving the message ente disabled state fo the duation of the time. This pocedue pevents nodes who ae neighbos of existing bagaining goup to paticipate in any futue bagaining befoe the time expies. Following this, all bagaining goups ae isolated. 4.) Bagaining Once all membes acknowledge the changes to the channel assignment, each membe updates its local channel assignment. This is staightfowad fo one-to-one bagaining. Fo onebuye-multiple-selle bagaining, inteactions among membes can be coodinated by the bagain equesto. Afte bagaining, each membe entes enabled state. Fig. 3 and 4 illustate the node state tansition and messages duing bagaining. Once the local bagaining pocedue is set, the specific Fig. 4.! Fig. 3. Node State and Tansitions Messages exchanged duing bagaining. bagaining stategy may be customized fo diffeent utility functions. It is easy to show that fo utilization based utility (total use thoughput), the optimization can be educed to solving M optimization poblems fo each colo espectively. On each colo, the coesponding optimization poblem is exactly a Weighted Independent Set (WIS) poblem []. WIS poblem is a special case of Weighted Set Packing poblem, and can be appoximated by a local impovement heuistic algoithm, geneally called t impovement [8], []. It is staightfowad to convet this algoithm to local bagaining among neighbos. We omit the bagaining pocedue due to space constaints, and next focus on the local bagaining stategy fo the fainess-based utility. IV. LOCAL TO IMPROVE FAIRNESS In this section, we focus on the local bagaining stategy optimizing fo fainess. Based on its definition in (), the optimization aims to maximize the total logaithmic use thoughput, i.e. the poduct of use thoughput. Theefoe, the global fainess utility inceases if nodes with many assigned channels give some channels to nodes with few assigned channels. In this section, we stat by descibing basic oneto-one bagaining whee two unbalanced nodes exchange channels to impove the local thoughput poduct. We show that such bagaining is limited by the numbe of bagainable channels and thus not effective against the node stavation poblem. We then develop a special case of one-buye-multiselle bagaining, efeed to as Feed Povety to eliminate node stavation. We also deive a theoetical lowe bound of use thoughput using local bagaining unde a simplified netwok

6 configuation. We fist define the following notations. n: a node n in the conflict gaph ( n N ); Nb(n) = {v V (n, v) E}: neighbos of n; Nb(X) = n X Nb(n) \ X: neighbos of node set X; f A (n) = { m M a m,n = }: the set of channels assigned to node n unde cuent assignment A. A. One-to-One Fainess Bagaining As we descibed, one-to-one bagaining allows two neighboing nodes n and n 2 to exchange channels to impove system utility while complying with conflict constaints fom the othe neighbos. Fo n and n 2 to bagain, they need to fist obtain the channels that ae bagainable to avoid distubing othe neighbos, efeed to as C b (n, n 2 ): C b (n, n 2 ) = L(n ) L(n 2 ) {{..M }\ f A (n)}. n Nb(n,n 2 ) Given C b (n, n 2 ), we can define the one-to-one bagaining egading fainess as follows: Definition 2: Fo an assignment A M N, an One-to-One Fainess Bagaining finds nodes n and n 2, and thei bagaining channel set C b (n, n 2 ), and modifies A M N to A M N elated to n, n 2 and channels C b (n, n 2 ), such that T P A (n ) T P A (n 2 ) > T P A (n ) T P A (n 2 ). The One-to-One Fainess Bagaining inceases the poduct of the bagaining uses while othe nodes thoughput values emain unaffected. Hence, the system fainess inceases with each bagaining. The impovement between each pai of nodes (n, n 2 ) can be calculated as G(n, n 2 ) = T P A (n)t P A (n2) T P A (n )T P A (n 2 ). This is used in the bagaining pocess (in Section III) to detemine whethe a bagaining can impove system utility. Given (n, n 2 ), assigning channels to n and n 2 to maximize thei thoughput poduct is a difficult task. This is because node thoughput depends on all channels (including non-bagainable ones) assigned to a node, and the available bandwidth on a channel diffes between nodes. The poblem is shown to belong to the class of convex pogamming poblems [22]. When the numbe of bagainable channels ( C b (n, n 2 ) ) is small (e.g. C b (n, n 2 ) < ), exhaustive seach may be feasible. Othewise we need to use appoximations based on heuistics such as the one given in [22]: fist sot channels in C b (n, n 2 ) (by channel bandwidth), then use a two-band patition to detemine the allocation. The effectiveness of One-to-One bagaining is constained by the size of C b (n, n 2 ). In geneal, due to heavy intefeence constaints among neighboing nodes, C b (.) could be vey small. Figue 5 illustates an example whee the conflict gaph is a chain topology consisting of thee nodes A,B,C. Node B is not assigned with any channel and the system utility is zeo. We efe to this as use stavation. Node a and b cannot bagain due to the constaint fom c (i.e. C b (a, b) = ), while node b and c also cannot bagain due to the constaint fom a (i.e. C b (b, c) = ). Hence, the Fainess Bagaining is not effective to eliminate use stavation. A B C,2,3 Fig. 5. B. Feed Povety Bagaining,2,3 An example of Stavation We obseve that use stavation in most cases is a esult of the lack of flexibility in bagaining. As fo the example in Figue 5, by allowing A and C to give up channel at the same time and feed it to B, we can emove the stavation at B. This is an example of one-buye-multi-selle bagaining. In this pape, we popose a special one-buye-multi-selle bagaining, called Feed Povety whee if a node (buye) has vey poo channel assignment, the neighboing nodes can collaboate togethe to feed it with some channels. Definition 3: Fo an assignment A M N, a Feed Povety Bagaining is to find some node n and channel m, modify A M N to A M N, such that A m,n = : m = m and n = n : m = m and n Nb(n ) A m,n : othewise (intuitively, the assignment let some of n s neighbos give up channel m and feed it to n ) and G F P (n ) = (T P A (n )) (T P A (n)) (T P A (n )) n Nb(n ) A m,n= n Nb(n ) A m,n= (T P A (n)) >. This means the poduct-thoughput of the uses involved in the bagaining is locally inceasing, while the othe uses thoughput ae not affected. So geneally the bagaining impoves system utility, except that, in case of stavation of othe uses, the system utility emains. A special case of Feed Povety is when A m,n = fo all n Nb(n ). This means none of n s neighbos ae using channel m, and n simply seizes it. When thee is no feasible One-to-One Fainess Bagaining, i.e. C b =, the equesto initializes a Feed Povety Bagaining on all neighbos who acknowledge the equest. The equesto sequentially selects multiple channels to maximize goup utility. C. BF-Optimal Assignment and Bound on Use Thoughput We popose to combine one-to-one Fainess Bagaining and Feed Povety Bagaining into a Fainess Bagaining with Feed Povety (BF). Each node who wants to impove its spectum usage stats with negotiating one-to-one Fainess Bagaining with its neighbos to impove system utility. If thee is no bagainable channels between it and any of its neighbos, a staved node can boadcast a Feed-Povety equest to its neighbos to initialize Feed Povety Bagaining. Oveall, a channel assignment A is said to be BF-optimal if no futhe

7 Fainess Bagaining with Feed Povety can be pefomed on it. It is useful to deive a theoetical lowe bound on each use s thoughput fo a BF-optimal assignment. Howeve, it is difficult to analyze the system pefomance when the channel availability and bandwidth vay acoss channels and uses. In the following, we show that when channels ae of unifom bandwidth (W.L.O.G. b m,n = fo all m, n), we can deive the theoetical lowe bound on the total numbe of channels that each use can get, which is equivalent to the use thoughput. This also shows an intuition of fainess enfoced by Fainess Bagaining-with-Feed-Povety. Theoem : Unde a BF-optimal assignment A, fo each vetex n in the conflict gaph G, n N with degee d(n) and channel availability list L(n), its spectum usage T P (n) has a lowe bound, i.e. L(n) T P (n) = P L(n). d(n) + The poof is included in the appendix. The degee of a vetex d(n) is defined as the numbe of edges it is associated with, a measue of the numbe of channel shaes in the neighbohood it has to compete with. Theoem shows that the poposed Fainess Bagaining with Feed Povety guaantees a povety line P L(n) to each vetex n. The povety line of a vetex, i.e. the thoughput a vetex deseves, scales invesely with the numbe of shaes, which is also the spiit of some geedy allocation algoithms [23], [8]. The povety line also povides a guideline in bagaining in eal systems whee a vetex is entitled to equest bagaining if its cuent thoughput is below its povety line. We efe to this as the Povety guided bagaining. It is easy to show that if channels ae fully available at each vetex, i.e. L(n) = M, and the maximum degee in the gaph = max n<n d(n), a BF-optimal assignment can eliminate use stavation if the numbe of channels M +. This matches the well-known conclusion in gaph coloing that, the chomatic numbe of a gaph is at most + [5]. It can be shown that the deived povety bound is also tight, fo many typical gaph topologies, e.g. clique and ing topologies [4]. V. EXPERIMENTAL RESULTS We conduct expeimental simulations to quantify the pefomance of bagaining-based spectum allocation. We also validate the poposed local bagaining algoithms against the theoetical lowe bounds. Fo simulations, we assume a noiseless, mobile adio netwok. We simulate an ad-hoc netwok by andomly placing a set of nodes on a aea. We assume that each active node boadcasts data packets to some of its neighbos. We futhe abstact the netwok into a Conflict Gaph whee each vetex epesents a tansmitting node. Any two nodes intefee with each othe (i.e. connected in the conflict gaph) if they ae within distance of 2. The actual distance theshold depends on the choice of tansmission powe and adio hadwae. We simply use 2 as an illustative example. Fo simplicity, we assume that channels ae equally weighted and all the channels ae available fo each node, i.e. l n,m =, b n,m =. Ou simulations can be easily extended to the cases with patial channel availability and non-unifom channel bandwidth. In tems of taffic demands, all tansmitting nodes ae assumed backlogged. We focus on maximizing fainess, because bagaining unde utility based on spectum utilization can be educed to the classical local seach of weighted independent set poblem, and has been investigated extensively. Unde the simulation setting, fo one-to-one fainess bagaining, the optimal assignment of channels between two nodes (n, n 2 ) can be deived easily to maximize the poduct of the numbe of channels assigned to n and n 2. Fo Feed Povety Bagaining, we select channel m, i.e. feeding channel to be the one that geneates the minimum distubance to the neighbos, m = ag min m n Nb(n ) A m,n= T P A (n) T P A (n). Topology dynamics ae modeled by having nodes andomly moving to new locations. We divide time into slots, and in each time slot, p% of nodes move to a new andomly selected location. The model captues the way mobility is manifested in ad hoc netwoks without delving into complex potocols. A moving node takes the oiginal channel assignment but disables the channels that conflict with its new neighbos. In each time slot, afte the topology change, nodes adjust thei channel usage. We use two metics to evaluate the pefomance. System utility: We conside fainess defined in (). Note that if thee exists a use with no channel assigned, the utility becomes. Fo bette epesentation, we modify N the utility to U(A) = /N n= T P A(n) and U(A) = if thee is any T P A (n) =. Communication ovehead: We quantify algoithm complexity as the communication ovehead, i.e. total numbe of messages exchanged among nodes, since tansmission and handling of messages will likely dominate computations fo channel assignment. In both local bagaining and gaph coloing appoaches, each iteation of spectum assignment o bagaining involves a 4-way handshake between neighbos, i.e. (equest, acknowledgement, action, acknowledgement). We fist compae the pefomance of local bagaining to the gaph coloing appoaches that appoximate to the solution that maximizes system utility fo a given conflict gaph [23], [8]. We also validate the impact on system pefomance when nodes use the deived povety line to guide its bagaining decision. We then examine the effectiveness of using local bagaining to optimize spectum assignment fo fixed topologies. A. Compaison with Centalized Gaph Coloing Appoach We now compae the poposed local bagaining to the gaph-coloing appoach. We efe to these two as BARGAIN- ING and, espectively. We andomly deploy 4 links

8 Y BAR. w/o FEED POV Pobability of mean utility <Y 2 (a) System Utility Mean utility % 4% 6% 8% % Pecentage of mobility (a) System Utility 4 Y Mean ovehead Fig Pobability of Ovehead <Y (b) Algoithm Complexity Pefomance compaison of and 5 2% 4% 6% 8% % Pecentage of mobility (b) Algoithm Complexity Fig. 7. Pefomance compaison of and unde diffeent use mobility. with 3 channels in a given aea and poduce the coesponding conflict gaph. We use the gaph coloing appoach to deive an initial spectum assignment fo the given conflict gaph. We simulate mobility events in the next time slots, one event pe time slot whee up to 6 nodes move to new locations. Afte each event, we apply both local bagaining and gaph coloing appoaches to deive the new spectum allocation. Figue 6(a) illustates the soted fainess utility using both appoaches, and local bagaining pefoms nealy as good as the gaph coloing appoach. The gaph coloing appoach makes decisions with the knowledge of global topology, while using local bagaining, each use makes decisions based on only neighbo infomation. Figue 6(b) compaes the communication ovehead in each time slot. We obseve that local bagaining achieves simila pefomance while incuing much lowe complexity in tems of messages exchanged. This significant ovehead eduction allows quick adaptation to netwok dynamics. In Figue 6(a), we also examine the pefomance of local bagaining using only one-to-one fainess bagaining, without Feed Povety Bagaining. The esults confim that Feed Povety Bagaining is equied to effectively eliminate use stavation. Next, we extend the simulation to allow p% of vetices move to new andomly selected locations. In geneal, lage p implies moe distubance to the conflict gaph and thus moe vetices will pefom bagaining to adapt thei spectum usage to the new topology. Figue 7 illustates the system utility and algoithm ovehead fo local bagaining and gaph coloing fo inceasing values of p. The utility is geometically aveaged ove time slots, and ovehead is aveaged ove time slots. As befoe, local bagaining pefoms similaly to gaph coloing appoach in system utility. The ovehead of gaph coloing is not sensitive to the value of p as it mainly depends on the size of the gaph in numbe of vetices and channels. The ovehead complexity of local bagaining inceases with p as moe vetices need to pefom local bagaining. We obseve that even unde % mobility, local bagaining esults oughly /2 the ovehead in tems of messages exchanged compaed to the gaph coloing appoach. Theefoe, local bagaining appeas to be an attactive altenative to gaph coloing fo optimizing spectum allocation on a given netwok topology. In Figue 8, we fix the numbe of channels at 4 and examine the impact of the numbe of vetices fo p = 2% mobility. Inceasing the density of vetices in a fixed aea ceates additional intefeence constaints and thus inceases aveage vetex degee in the conflict gaph. Theefoe, system utility scales invesely with the numbe of vetices while the algoithm complexity inceases. As befoe, esults show that local bagaining compaes favoably with gaph coloing in quality of allocation while incuing significantly less ove-

9 2 Mean utility 5 POVERTY GUIDED BAR. Pecentage of instances 2% 5% % 5% Numbe of uses (a) System Utility Node utility / lowe bound Fig. 9. Histogam of the atio between the actual use thoughput and the povety bound. Mean ovehead POVERTY GUIDED BAR. Mean utility Numbe of uses (b) Algoithm Complexity Fig. 8. Pefomance compaison of and unde diffeent vetex density. head. B. Tightness of the Povety bound We now examine the appopiateness of the use povety bound deived in Theoem. Figue 9 illustates the histogam of the atio of the actual use thoughput and the povety bound assuming 4 vetices and time slots. Results show that the theoetical bound is valid and faily tight. As we descibed, nodes can use the povety line to decide whethe futhe local bagaining is necessay. A vetex with an assignment below the povety line should bagain with additional neighbos to acquie additional channels. In addition, if a vetex pedicts that a bagaining equest fom a neighbo will dop its assignment below the povety line, it can eject the equest. In Figue 8, we also compae the pefomance of Povety guided bagaining to the gaph coloing and bagaining appoaches. We show that Povety guided bagaining pefoms close to that of the bagaining but with % less ovehead, again demonstating the tightness of the povety line bound. C. Use local bagaining to optimize fo a given topology As stated befoe, it is possible to use local bagaining to appoximate the gaph coloing appoach and deive the spectum allocation fo a given topology. Local bagaining stats fom a andom allocation and gadually impove the system utility. Figue compaes the system utility and algoithm Ovehead 6... Geedy Bagaining Random topology index (a) System Utility Mean Mean topology Index (b) Algoithm Complexity Fig.. Pefomance compaison of gaph coloing, local bagaining and andom spectum assignment fo static envionments. complexity using gaph coloing (), local bagaining () and andom assignment (RANDOM). Use stavation is common when using andom assignment, esulting in many zeo values fo system utility. Local bagaining can effectively eliminate use stavation and pefoms only slightly wose compaed to gaph coloing appoach. Figue (b) shows that local bagaining can significantly educe communication ovehead. VI. THEORETICAL DISTANCE TO SOCIAL OPTIMAL In this section, we analyze the pefomance of the poposed bagaining stategy, by compaing it to that of the socially

10 optimal assignment. A socially optimal assignment is one maximizes global system utility. We ae inteested in the atio of the utility of social optimum to that of ou bagaining optimum, often efeed to as the pice of anachy (POA) as defined in [4]. The analysis equies a numbe of case study and illustations. We only give a sketch hee, and the details will be included in a futue pape [4]. It can be shown that the POA is unbounded in geneal, when thee is no estiction on channel bandwidth o numbe of channels [4]. In the following we assume that channels ae of unifom bandwidth (W.L.O.G. b m,n = fo all m, n), and the numbe of nodes ae bounded. In this case the POA can be bounded, based on the lowe bound deived in Section III. Theoem 2: Fo a topology with unifom channel availability and channel bandwidth, i.e. b m,n =, l m,n =, and M >, the pice of anachy fo a BF-optimal channel assignment is at most M N u,v ( d u+d v ) du N n= The poof is in the appendix. M d n + VII. RELATED WORK d u+d v ) ( du. (2) Optimal conflict-fee channel assignment satisfying a global optimal objective is often NP-had, even when global topology infomation is available [6]. Centalized appoximations ae widely used in single hop wieless netwoks such as cellula netwoks. This can be easily extended to multi-hop wieless netwoks by flooding connectivity and taffic equiements acoss the netwok, and equiing all uses to un a vaiant of the centalized algoithm. Howeve, this appoach clealy does not scale as netwoks become lage and moe dynamic. An altenative decentalized allocation, whee uses act based on locally available infomation is much moe attactive. Both analytical famewok and pactical stategies have been poposed. Analytical famewoks in [7], [2] addess fainess fo single-hop flows, and deive an estimate of the ate at each flow to achieve Max-Min fainess. Howeve, thee is no guaantee that a feasible scheme exists to achieve the ate. Pactical stategies have been poposed fo shaing a single channel. Contention based schemes invoke a andom access potocol like ALOHA and CSMA, whee uses contend in time to shae a common channel [5], [2], [7]. While this scheme povides fainess and utilization on a single channel system pobabilistically, its application to a multi-channel system equies each use to know how many and which channel(s) to access. Anothe appoach, conflict fee time slot scheduling, povides guaanteed channel usage by eseving time slots fo each flow. Solutions in [2], [2], [9] assign exactly one time slot to each flow. This appoach can be used in multichannel systems if each use uses only one channel. Anothe solution [2] allows uses to use multiple slots/channels to achieve Max-Min-fai, but does not conside intefeence fom neighbo tansmissions. othewise thee may exist stavation and the atio may be meaningless. Multi-channel assignment stategies wee developed mostly fo cellula netwoks. The wok in [3] povides solutions to assign fequency bands among base stations to minimize call blocking pobability fo voice taffic. Thee is no notion of fainess as the taffic detemines the numbe of channels each base station should use. In [9], the authos poposed a gaphtheoetic model and discussed the pice of anachy unde vaious topology conditions such as diffeent channel numbes and bagaining stategies. The main diffeence between [9] and the poposed wok is that the poposed model allows multicoloing of a vetex, while in [9] each vetex can only be assigned with at most one colo. In [23], [8], the authos pesented a genealized spectum allocation poblem whee intefeence constaint C is channel dependent. The authos developed a set of geedy coloing appoach to optimize spectum allocation fo a given conflict gaph. We use the poposed appoaches in [23], [8] as the efeence algoithm, i.e. in this pape. Coopeative/non-coopeative bagaining is also used in pevious eseach to optimize channel allocation fo cellula netwoks. In [] and [], the authos poposed a set of bagaining stategies fo OFDMA based netwok, focusing on one-to-one bagaining. In these cases, nodes ae mutually intefeed, i.e. the coesponding conflict gaph is fully-connected. Foming bagaining goup is to find any two uses netwok and let them exchange cetain channels to impove system pefomance. The main diffeence between these wok and the poposed wok is that the poposed wok povides solutions fo geneal conflict gaphs whee goup setup needs to conside local topology (i.e. isolated goup and self-contained channel adjustment). We popose a feed-povety bagaining to eliminate use stavation which can not addessed by one-toone bagaining. In addition, the poposed wok deives a lowe bound fo each node s channel assignment that does not limit to fully-connected topology. VIII. CONCLUSION AND FUTURE WORK In this pape, we pesent an adaptive and distibuted appoach to spectum allocation in mobile ad-hoc netwoks. We popose a local bagaining appoach whee uses affected by the mobility event self-oganize into bagaining goups and adapt thei spectum assignment to appoximate a new optimal assignment. The bagaining stats fom the spectum assignment etained befoe the mobility event, and equies significantly less computation and communication ovehead. We popose a Fainess Bagaining with Feed Povety to impove fainess in spectum assignment. We also deive a lowe bound on the spectum assignment that each node can get fom bagaining, efeed to as the povety line. It eflects the level of fainess enfoced by the bagaining, and seves as a guideline to oganize bagaining. Expeimental esults show that the poposed bagaining appoach pefoms similaly as the topology-optimized appoach but with much less complexity. We also veify the coectness of the povety line and the effectiveness of the povety guided bagaining.

11 While we only poposed a specific bagaining stategy to maximize fainess based system utility, the poposed bagaining famewok can be extended towads othe utility functions o optimization goal. We intend to examine this in a futue study. One of the biggest attactions of local bagaining is the low complexity. We have conducted simulations to evaluate algoithm complexity in tems of the numbe of bagaining iteations, which is quite intuitive. We ae cuently woking on a theoetical analysis on the complexity of local bagaining. In addition, the poposed bagaining famewok assumes that netwok nodes collaboate to impove system utility while in eal systems, nodes can be selfish so that a picing based bagaining o a ule based bagaining would be moe pactical. We ae cuently investigating this issue. REFERENCES [] ARKIN, E. M., AND HASSIN, R. On local seach fo weighted k-set packing. In poceedings of ESA, LNCS 284 (997), pp [2] BAO, L., AND GARCIA-LUNA-ACEVES, J. J. 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FDMA capacity of gaussian multi-access channels with ISI. IEEE Tans. Commun. 5, (Jan 22), 2. [23] ZHENG, H., AND PENG, C. Collaboation and fainess in oppotunistic spectum access. In Poc. 4th annual IEEE Intenational Confeence on Communications (June 25). A. Poof of Theoem APPENDIX The poof fo Theoem is complex. We stat by poving the following coollay, which is a special case of the Theoem. We then povide a sketch of genealization to Theoem. Coollay : Unde a BF-optimal assignment A, fo each vetex in the conflict gaph G, with degee d(n) and channel availability list L(n) = {,,, M }, the following holds: M T P (n), n N. (3) d(n) + This is the case whee each vetex has full channel availability. A. Poof of Coollay We fist show the following lemma exists. Lemma : Let B P Q epesent a P Q binay matix. Define T P (n) = P m= Bm,n, fo n Q ; and Lc =. Thee exists c P, s.t. i Q,B c,i = L c, = P/Q. (4) Poof of Lemma. Poof: The poof is tivial when < < since the ight side is negative. Next, we assume and use an induction on Q to pove the lemma. When Q =, = P, B becomes a binay matix. If fo some c, B c, =, then c satisfies (4). Othewise, if B c, = c, then T P () =, and c, L c =. Now suppose the lemma holds fo all k < Q (Q 2). a) We fist assume that thee exists n Q, s.t. T P (n ), e.g. n =. Hence, j, j P, B j, =. Then, B ( :P ),(:Q ) is a binay matix constucted by deleting the fist ows and the fist column of B. Its size is (P ) (Q ). By induction hypothesis, thee exists c P, s.t. i Q,B c,i = whee = P Q L c = ( P m= B m,i) ( P m= Bm,i), (5) P =. Following this, we can deive Lc as Q = (B c, = ) = i Q,B c,i = i Q,B c,i = i Q,B c,i = i Q,B c,i = ( P m= B m,i) P m= Bm,i ( P m= B m,i) P m= Bm,i ( P m= Bm,i) P m= B m,i (by(5)). (6)