Optimal Pricing in a Free Market Wireless Network

Transcription

1 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 1 Optimal Pricig i a Free Market Wireless Network Michael J. Neely Uiversity of Souther Califoria mjeely Astract We cosider a ad-hoc wireless etwork operatig withi a free market ecoomic model. Users sed data over a choice of paths, ad schedulig ad routig decisios are updated dyamically ased o time varyig chael coditios, user moility, ad curret etwork prices charged y itermediate odes. Each ode sets its ow price for relayig services, with the goal of earig reveue that exceeds its time average receptio ad trasmissio expeses. We first develop a greedy pricig strategy that maximizes social welfare while esurig all participats make o-egative profit. We the costruct a (o-greedy) policy that alaces profits more evely y optimizig a profit fairess metric. Both algorithms operate i a distriuted maer ad do ot require kowledge of traffic rates or chael statistics. This work demostrates that idividuals ca eefit from carryig wireless devices eve if they are ot iterested i their ow persoal commuicatio. Idex Terms Reveue Maximizatio, Ad-Hoc Moile, Queueig Aalysis, Stochastic Optimizatio, Cotrol y Pricig I. INTRODUCTION This paper presets a free market ecoomic model for ad-hoc wireless etworks. Multiple users desire to sed packet-ased traffic to their destiatios, potetially usig multi-hop paths. However, idividual wireless odes icur oth receptio ad trasmissio costs, ad hece will ot agree to act as itermediate relays for this traffic uless they are adequately compesated. Thus, each ode sets its ow price for hadlig ew data, ad ca dyamically adjust this price i reactio to time varyig etwork coditios. Additioally, odes dyamically choose ext-hop eighors for their data ased o chael coditios ad advertised prices. The goal of each user is to maximize its et utility, ad the goal of idividual wireless odes is to facilitate commuicatio while attemptig to make a profit. We desig distriuted pricig ad cotrol mechaisms for this system. The mechaisms yield fruitful markets, i the sese that the etwork takes maximum advatage of its multi-hop capacity while esurig that cooperatio is profitale for all participats. Specifically, we propose two differet market algorithms, oe that admits a greedy iterpretatio ad oe that does ot. I our first algorithm, each ode charges a per-uit price that is proportioal to its curret level of queue acklog. Neighorig trasmitters pay a hadlig charge accordig to this per-uit price, together with a receptio fee that is equal to the cost icurred y receivig a ew trasmissio. Every timeslot, idividual odes oserve the curret chael coditios o their outgoig liks ad the curret prices of the correspodig eighorig odes, ad determie which data to trasmit, how much to trasmit, ad which eighor to sed to. This choice is determied greedily every slot. Specifically, each ode compares the past reveue eared y acceptig data to the costs ad service charges associated with trasmittig o the curret timeslot, ad makes a greedy trasmissio decisio that maximizes istataeous profit. Note that a ode might decide to remai idle o a give timeslot i order to wait for etter chael coditios ad/or lower prices. However, the ode must evetually trasmit the data, as it is oligated to remai stale (so that the log term output rate is equal to the log term iput rate). We show that the algorithm yields a aggregate etwork utility that ca e pushed aritrarily close to optimal, with a correspodig tradeoff i ed-to-ed average delay. Further, the algorithm esures that This work was preseted i part at the IEEE INFOCOM coferece, Achorage, May 2007 [1]. This work is supported i part y oe or oth of the followig: The DARPA IT-MANET program grat W911NF , the Natioal Sciece Foudatio grat OCE

2 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 2 everyoe makes a o-egative profit. However, the resultig profits are ot ecessarily alaced evely across memers of the etwork. To yield a more fair profit distriutio, we propose a secod algorithm that seeks to maximize a geeral cocave profit metric. This algorithm uses a iterestig techique of ag-ag pricig, alteratig etwee periods of allowig free service (price = 0) ad periods where price is set to a pre-specified maximum value. The algorithm optimizes the target performace metric, although it relies o user cooperatio ad does ot ecessarily admit a greedy iterpretatio. A. Motivatios This work treats a fudametal prolem of etwork ecoomics, where multiple self-iterested users must coordiate to perform a routig task. Our results thus have road applicatios eyod wireless scearios. However, we focus o the ad-hoc wireless cotext. I this cotext, etwork ifrastructure is limited ad wireless users might eed to relay their data through other users efore eterig rage of a desired destiatio or access poit. Such scearios ca occur whe ifrastructure ever existed or was destroyed or overloaded. I such cases, privately owed wireless devices might iteract with each other ad with a orgaized team of other wireless odes to facilitate commuicatio. The profit eared y such iteractio ca ecourage others to eter the market, which i tur icreases etwork capailities. B. Related Work Prior work i the area of etwork pricig is foud i [2]-[27]. The prolem of allocatig flow rates to multiple users sharig a fixed capacity trasmissio lik is cosidered from a ecoomic perspective i [2]. Flow allocatio ad pricig i a multi-hop etwork is cosidered i [3]. Both [2] ad [3] cast the prolem as a static covex program, where Lagrage multipliers are iterpreted as prices charged y the lik to each user. It is show that there exist prices that yield the optimal flow rates if users greedily maximize their utility mius cost. Cotrol mechaisms that use price updates to coverge to the utility optimal flow rates are cosidered i [4] [5]. Pricig solutios applied to static wireless dowliks are cosidered i [6]. More recet work i [7] [8] [30] uses ack-pressure techiques for utility optimizatio i stochastic wireless etworks, ad relates queue acklog to prices charged to users at each etwork access poit. Related work is cosidered i [9]. Worst-case throughput utility results for a wireless lik with o-statioary chaels are preseted i [10]. I all of the works aove, pricig is itroduced oly to otai a fair sharig of resources over all users, so that idividual profit ojectives are ot directly cosidered. Prolems of pricig to maximize reveue are cosidered i [11] [12] [13] for static wireless dowliks, where structural properties of the resultig (o-covex) prolem are examied. Work i [14] [15] [16] cosiders game theory approaches to related prolems. Work i [17] cosiders admissio pricig to maximize reveue i a data lik with multiple traffic classes, ad develops a optimal algorithm ased o dyamic programmig. Simplificatios for large etworks are cosidered i [18]. Market mechaisms to stimulate cooperatio i ad-hoc wireless etworks are cosidered i [19]-[22]. The mechaisms i [19] provide moetary credits to each ode that forwards traffic, ut does ot cosider utility optimizatio ad does ot accout for heterogeeous etwork coditios. Work i [20] presets a simulatio study of more geeral pricig strategies, ased o pricig priciples of [4]. Related work i [21] cosiders su-cotractig strategies for distriutig a computatioal task over a moile etwork. Aalytical properties of pricig mechaisms for commuicatio i static etworks with fixed routes are cosidered i [22]. Our approach i this paper is quite differet tha the previous work, particularly that of [19]-[22], i that it provides aalytical guaratees for market mechaisms, ad is directly desiged for stochastic etworks. We treat the prolems of social welfare ad profit alacig, ad esure that all odes make a o-egative profit y itelligetly reactig to lik ad price iformatio. Differet from most work i this area, we cosider a packet ased model that fully icludes queueig. Our aalysis ad routig strategies are ispired y the ack-pressure cocepts developed i [28] [29] ad y the techiques for stochastic etwork optimizatio developed i [7] [8] [30].

3 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 3 C. Additioal Simplifyig Assumptios While our aalysis ca e applied to wireless etworks with geeral iterferece properties, for simplicity of expositio we cosider a simplified model where each ode trasmits usig sigals that are orthogoal to those of eighorig odes. This highlights the ecoomic issues ivolved i makig trasmissio decisios ased o advertised prices ad oserved chael coditios, without requirig additioal distriuted multiple access protocols to implemet these decisios. However, the multiple access prolem is aother importat issue for wireless etworks, ad we riefly descrie how radom ad scheduled access strategies ca e icorporated. Specifically, suppose there are oe or more owers of differet etwork regios, ad these owers schedule trasmissios ased o requests from odes withi their regios. We ca show that the aalytical results of the greedy algorithm preseted i this paper are preserved if odes pay owers a fixed fractio of their profits, so that each ower has a icetive to schedule to maximize the sum of istataeous profit withi its regio. This is cosidered i more detail i Sectio V. Alterative multiple access strategies ased o localized auctios withi differet etwork regios (usig recet etwork auctio results such as [23]-[27]), may also provide efficiet mechaisms ad suggest possile directios for future work. I the ext sectio we descrie our etwork model. Sectios III ad IV develop the greedy ad profit-alaced algorithms, respectively. Simulatios for fixed-topology etworks ad moile etworks are provided i Sectio V. II. NETWORK MODEL Cosider a ad-hoc wireless etwork with N odes. The etwork operates i slotted time with slots t {0, 1, 2,...}. Chael coditios o each lik are assumed to e costat over the duratio of a timeslot, ut ca vary from slot to slot (due, for example, to wireless fadig ad/or user moility). Specifically, let (, ) represet the wireless lik from ode to ode, ad let S (t) represet the curret chael state of the lik. The value of S (t) ca represet a quatized estimate of oe or more physical lik parameters (such as atteuatio), or ca represet a astract characterizatio of the chael (such as Good, Medium, Bad, or 0 ). We assume that there are a fiite (ut aritrarily large) umer of chael states, ad that each ode kows the state of its ow outgoig liks at the egiig of each timeslot. Let S(t) = (S (t)) represet the matrix of chael states over all etwork liks. For simplicitly of expositio, we assume throughout that chael state matrices S(t) are idepedet ad idetically distriuted (i.i.d) over timeslots. 1 For each matrix S we defie chael proailties π S =P r[s(t) = S]. The chael proailities are ot ecessarily kow to the etwork odes. A. Resource Allocatio Costraits ad Cost Exteralities Let S (t) = (S 1 (t), S 2 (t),..., S N (t)) represet the vector of chael states for outgoig liks of ode. We say that S (t) = 0 if ode caot trasmit to ode durig slot t. I most etworks of iterest, odes ca oly directly commuicate with a small suset of curret eighors, ad so each S (t) vector typically cotais oly a few o-zero chael states. Defie µ (t) as the trasmissio rate chose y ode for the (, ) data lik durig slot t (i uits of its/slot). Let µ (t) = (µ 1 (t), µ 2 (t),..., µ N (t)) represet the correspodig vector of trasmissio rates o outgoig liks of ode. The trasmissio rate vector for ode {1,..., N} is chose every timeslot i reactio to the curret chael states S (t), suject to the costrait: µ (t) Ω () S (t) (1) 1 This i.i.d. assumptio simplifies aalysis ut is ot essetial, ad our results ca e exteded to geeral ergodic chael processes with steady state proailities π S, usig the T -slot Lyapuov argumets of [29][30].

4 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 4 where Ω () S represets the compact set of all trasmissio rate optios for ode whe S (t) = S. We assume these sets are such that if µ Ω () S, the we also have µ Ω () S, where µ is ay vector formed from µ y settig oe or more etries to zero. That is, it is always possile to choose to trasmit othig over a particular lik, ad this choice does ot reduce the rate optios o other liks. Note that the costraits (1) are desiged for distriuted implemetatio, as they imply that the trasmissio rate optios availale to ode are ot affected y the trasmissio rates chose y other odes m. This assumptio is valid if all odes use orthogoal sigals, so that trasmissio rate choices at a particular ode do ot ifluece the optios of other odes. Alteratively, this assumptio holds if there is a implicit time divisio multiple access structure i the etwork, where sets of o-iterferig odes are scheduled either periodically or pseudo-radomly, ad this schedule is emedded ito the the chael state process S(t) y artificially settig lik states to zero at appropriate times. Extesios to iterferece etworks ca e treated y defiig costrait sets Ω S(t) specifyig the set of all optios for the joit rate vector (µ 1 (t),..., µ N (t)), as discussed i more detail i [30]. Defie C tra (µ (t), S (t)) as the exteral trasmissio cost icurred y ode due to choosig trasmissio rate µ (t) whe the chael state vector is S (t). This cost fuctio provides a moetary measure of the persoal resources (such as power) expeded y ode for this trasmissio decisio. A example trasmissio cost fuctio is give y: C tra (µ, S ) = { S >0} e µ 1 S (2) which correspods to idepedet outgoig liks, logarithmic rate-power curves µ = log(1 + S P ) for each lik (, ) (where S represets a atteuatio coefficiet for chael (, )), ad costs that are directly proportioal to power expediture. A example costrait set Ω () S is the set of all rates that ca e achieved y allocatig o-zero power P to at most oe outgoig lik (, ), where 0 P P max. Defie C rec(µ (t)) as the receptio cost icurred y ode due to receivig a icomig trasmissio from ode over lik (, ). This represets the exteral cost expeded whe demodulatig ad processig the received sigal. A example receptio cost fuctio is give y: { C rec σ if µ (µ ) = > 0 (3) 0 otherwise where σ is a value proportioal to the power expeded y ode whe receivig a trasmissio. The structure of the C rec(µ ) fuctio ca also e exteded to iclude depedece o the chael coditio S (t). We assume throughout that trasmissio ad receptio costs are zero wheever the correspodig trasmissio rates are zero. All costs are assumed to e o-decreasig i the trasmissio rate vector, ad are upper ouded y fiite costats. B. Network Queueig ad Routig Costraits Data might take multi-hop paths through the etwork, ad hece each etwork ode maitais a iteral set of queues to store data accordig to its fial destiatio. Ay data that is oud for a particular destiatio ode c is laeled as commodity c data. Let U (c) (t) represet the amout of commodity c data curretly queued i etwork ode (i uits of its). Node has accepted this data ad hece has resposiility for either deliverig this data to its destiatio or deliverig it to aother ode that accepts these resposiilities. Let µ (c) (t) represet the trasmissio rate offered to commodity c its over lik (, ) durig slot t. Node chooses µ (c) (t) suject to the followig routig costraits: N µ (c) (t) µ (t) for all liks (, ) ad all slots t (4) c=1

5 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 5 where µ (t) is the rate selected for lik (, ) y the resource allocatio decisio at ode. This model allows for dyamic routig of data, as oth µ (c) (t) ad µ (t) ca e chaged from slot to slot. Let R (c) (t) represet the amout of ew exogeous commodity c data that the user at ode admits ito the etwork durig slot t. The oe-step queueig dyamics for each ode ad each commodity c thus satisfies: U (c) (t + 1) max[u (c) (t) µ (c) (t), 0] + a µ (c) a(t) + R (c) (t) (5) This is expressed as a iequality ecause idividual odes may ot have eough commodity c data to sed to ode at the full offered trasmissio rate µ a(t). (c) We assume that U () (t) = 0 for all t, as data that reaches its destiatio is immediately removed from the etwork. C. Data Admissio ad Relay Pricig We assume that each etwork layer ode has either zero or oe user at its trasport layer. Nodes with users are source odes. Whe the user at ode admits a amout of data R (c) (t) to the etwork layer, it pays this ode a amout α (c) (t) (i uits of dollars). This amout is to e determied y the dyamic pricig rule estalished y ode. This distictio etwee the user at ode ad ode shall e coveiet, eve i cases whe the user i fact ows ode ad hece pays itself for acceptace of ew data. Whe a give ode trasmits data at rate µ (t) to aother ode durig slot t, it pays oth a receptio fee ad a hadlig charge. The receptio fee is exactly equal to the receptio cost C rec(µ (t)) icurred y ode upo receivig the trasmissio. The hadlig charge is give y c β(c) (t), where β (c) (t) is the charge for acceptig resposiility of ew commodity c data, ad is determied every slot y ode. The total paymet from ode to ode is thus: C rec (µ (t)) + c β (c) (t) We shall cosider hadlig charges of the form β (c) (t) = q(c) (t)µ (c) (t), where q(c) (t) is a per-uit price for acceptig commodity c data at ode. We assume that the curret price q (c) (t) is set y ode ad is advertised at the egiig of the timeslot, as is the receptio cost fuctio C rec(µ (t)). I this way, a trasmittig ode ca assess the paymets required for makig a trasmissio decisio. Note that if the receptio cost fuctios are give y (3), the each receiver ode ca commuicate its fixed receptio fee σ at time 0, ad every slot it eeds oly to advertise its curret price q (c) (t) for each commodity c. D. Time Average Profits ad the Social Welfare Ojective The user at each source ode has a utility fuctio g (c) (r) that represets a moetary measure of the satisfactio it receives y sedig commodity c data to its destiatio at a log term average rate r its/slot. Utility fuctios g (c) (r) are assumed to e o-egative ad cocave, with ouded right derivatives. We assume each user has elastic traffic, i the sese that it always has a ifiite reservoir of data to sed, ad the log term sed rate ca e adapted to whatever rate the etwork allows. 2 I the case whe the user at ode does ot desire to sed ay data of a particular commodity c, we set g (c) (r) =0. Without loss of geerality, we treat o-source odes as if they are sources of users with utility fuctios that are idetically zero for every commodity c. 2 Ielastic traffic ca e treated via techiques of [8] [30].

6 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 6 For each ode {1,..., N}, defie the ode profit variale φ (t) as follows: φ (t) = α (c) (t) + β a (c) (t) β (c) (t) c a c c C tra (µ (t), S (t)) C rec (µ (t)) (6) The value φ (t) represets the istataeous profit (total reveue mius total cost) associated with trasmissio decisios i the curret timeslot. The total cost icludes the iteral paymets to other odes as well as the exteral trasmissio ad receptio costs. The fial term i the right had side of (6) represets the sum of all receptio fees paid y ode. The exteral receptio costs icurred y ode ad the correspodig receptio fees paid to ode do ot appear, as these terms exactly cacel each other. Defie the expected time average profit of ode over t slots as follows: φ (t) = 1 t 1 E {φ (τ)} (7) t Likewise, for each user {1,..., N} we defie the expected time average user profit ψ (t) as follows: ψ (t) = g (c) (r (c) (t)) 1 t 1 E { α (c) (t) } (8) t where: c r (c) (t) = 1 t t 1 c E { R (c) (τ) } (9) That is, the user profit ψ (t) represets the differece etwee the throughput utility ad the time average paymets associated with user over the course of t slots. The idividual goal of each user ad each ode is to maximize its ow time average profit. Our overall etwork ojective is to maximize the sum of profits over all users ad all odes. However, there is a additioal costrait that all queues of the etwork must e stale. 3 This esures that the log term iput rate to the etwork is exactly the same as the log term output rate. The followig simple lemma relates this sum profit ojective to maximizig social welfare. Lemma 1: (Social Welfare) Ay etwork cotrol ad pricig algorithm that stailizes the etwork yields time average profits ψ (t) ad φ (t) that satisfy: lim sup t lim sup t N [ ψ (t) + φ (t) ] = =1 [,c g (c) (r (c) (t)) 1 t where C (t) represets the exteral cost of ode : ] t 1 E {C (t)} C (t) =C tra (µ (τ), S (t)) + C rec (µ (t)) We call the right had side of (10) the social welfare of the etwork, ad ote that it ivolves oly exteral utilities ad costs. The proof of the lemma is trivial, ad follows y oticig that the sum of iteral paymets over all odes is exactly equal to the sum of iteral reveues eared from takig these paymets. Therefore, the iteral moetary costs ad reveues cacel each other out i the sum profit 3 P 1 We say a queue U(t) is stale if lim sup t 1 t t E {U(τ)} <. This type of staility is usually referred to as strog staility. (10)

7 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 7 metric. The lemma implies that the ojective of desigig a etwork cotrol ad pricig algorithm to stailize the etwork while maximizig sum profit is equivalet to the ojective of desigig a etwork cotrol algorithm to maximize the social welfare metric, without regard to etwork prices. The followig theorem estalishes that ay achievale social welfare value (ad hece ay achievale sum profit value) ca e achieved aritrarily closely via a statioary radomized policy that ases decisios oly o the curret chael state, ad that sets all moetary charges α (c) (t) ad β (c) (t) to zero. Theorem 1: Suppose there exists a cotrol strategy that stailizes the etwork ad yields a positive lim sup social welfare value g : [ ] g = lim sup g (c) (r (c) (t)) 1 t 1 E {C (t)} t t,c The for ay ρ such that 0 < ρ < 1, there exists a statioary radomized cotrol algorithm that stailizes the etwork, sets all α (c) (t), β (c) (t) to zero, sets all admissios R (c) (t) to particular costat values R (c) for all time, ad that chooses trasmissio rates µ (t) accordig to a statioary ad radom fuctio of the oserved chael state matrix S(t). Further, this statioary radomized policy yields for all t ad all (, c): R (c) + a E { µ (c) a (t) } { } E µ (c) (t) ad yields the followig social welfare result for all slots t: g (c) (R (c) ) E {C (t)} ρg (12),c Proof: The proof is similar to the ecessary coditios for etwork staility ad miimum average eergy expediture prove i [29] ad [31], ad is omitted for revity. The proailities ad trasmissio rate modes required of the statioary policy i Theorem 1 could i priciple e computed y a offlie algorithm with cetralized kowledge of all chael proailities, cost fuctios, ad user utilities. However, the resultig algorithm might cause some odes to receive egative profit, ad hece these odes would have o icetive to cotiue participatig. The desig of a olie cotrol algorithm that maximizes social welfare i this cotext ad esures all users ad odes receive o-egative profit is a ope questio. We resolve this questio i the ext sectio y a simple ad direct olie algorithm that makes use of ackpressure [28] [29]. The algorithm has the additioal desirale feature that idividual cotrol actios ca e iterpreted as greedily maximizig istataeous profit. E. Discussio of Alterative Approaches It is possile to use a modified Lagrage multiplier argumet, similar to [3], to prove existece of fixed flows ad prices that yield the desired o-egative profit result i a static etwork with o chael variatio ad with more assumptios imposed o the structure of the cost fuctios. However, the resultig multipliers (prices) are ot kow a-priori, ad it would require a extesive offlie computatio to estimate them to withi a adequate degree of accuracy. Olie techiques related to dual sugradiet algorithms, as i [29] [8] [9], ca also e cosidered whe prices are suitaly defied for this free market cotext. However, we ca show that these algorithms do ot always lead to o-egative profits, particularly whe there are cetrally located odes with varyig chaels ad o-egligile costs. III. A GREEDY PRICING STRATEGY The followig algorithm makes distriuted ad greedy decisios at each ode usig local lik coditios ad prices of eighorig odes. It uses a positive costat V that determies a tradeoff i ed-to-ed (11)

8 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 8 etwork delay. We shall also require the followig fiite ouds o the maximum trasmissio rate ito ad out of a give ode : = sup µ µ max,out µ max,i h S,µ Ω () S = sup» S, i o µ i Ω (i) N S i i=1 Stochastic Greedy Pricig Algorithm (SGP): Pricig: Every timeslot t, each ode sets the per-uit price q (c) (t) for hadlig ew commodity c data as follows: (t) = U (c) (t)/v The correspodig charge for acceptig R (c) (t) uits of exogeous data is give y: q (c) α (c) (t) = R (c) (t)q (c) The charge for acceptig edogeous commodity c data at rate µ (c) a(t) is give y: (t) β a (c) (t) = µ a(t)q (c) (c) (t) Admissio Cotrol: Every timeslot t, each user oserves the curret prices q (c) (for all commodities c such that the utility fuctio g (c) values solve: r (c), where the r (c) where R max Maximize: Suject to: is a costat such that R max c g(c) (r (c) c r(c) µ max,out. a µ a (t) i its source ode (r) is ot idetically zero), ad chooses R (c) (t) = ) c r(c) R max 0 r (c) for all c q (c) (t) (13) Resource Allocatio: Every timeslot t, each ode oserves the curret prices q (c) eighorig odes. It the computes the differetial price W (c) (t) as follows: W (c) (t) advertised y (t) =q (c) (t) q (c) (t) δ max /V (14) where δ max = max {µ max,out, µ max,i + R max }, ad represets the largest chage i ay queue acklog durig a slot. The etwork parameter δ max is assumed to e kow y all odes at the egiig of operatio. The optimal differetial price ad the correspodig optimal commodity is computed: W (t) c (t) = max c = arg max c W (c) (t) W (c) (t) (15) The curret chael states S (t) of all outgoig liks are oserved, ad the trasmissio rate vector µ (t) is allocated as the solutio of the followig optimizatio prolem: Maximize: Suject to: µ W (t) Crec (µ ) C tra (µ, S (t)) (16) µ Ω () S (t) Routig/Schedulig: Wheever W (t) > 0, data of commodity c (t) is trasmitted over lik (, ) at a rate µ (t), where µ (t) is determied y the resource allocatio algorithm aove.

9 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 9 A. Greedy Iterpretatio of SGP The admissio cotrol strategy ca e viewed as a greedy optimizatio every timeslot, where ode maximizes its utility associated with admittig ew traffic to its source ode (as measured y the g (c) (r) fuctios) mius the total cost of admittig this traffic, suject to a costrait R max o the total sum of admitted data. Note that i the special case whe user has oly a sigle active commodity (that is, it has a sigle o-zero g (c) (r) fuctio), the the algorithm reduces to settig R (c) (t) = r, where r maximizes g (c) (r) rq (c) (t) suject to 0 r R max. The resource allocatio computes the differetial prices W (c) (t). Note from (14) that: W (c) (t)µ(c) (t) = [q(c) (t) q (c) (t)]µ (c) (t) µ(c) (t)δ max/v The first term o the right had side of the aove equality represets the differece etwee the charge required for trasmittig commodity c traffic to ode ad the reveue it would ear y acceptig this same amout of traffic from other odes, chargig these odes the curret price q (c) (t). From a greedy perspective, it makes sese to trasmit commodity c data to ode oly whe the price differetial q (c) (t) q (c) (t) is positive. The value µ (c) (t)δ max/v ca e viewed as a fudge factor that decreases the price differetial to accout for the fact that ode may ot receive ew data at its curretly advertised price. The value W (c) (t)µ(c) (t) ca thus e viewed as ode s estimate of its istataeous profit associated with relayig µ (c) (t) uits of commodity c data (ot icludig trasmissio costs or receptio fees). Hece, commodity c (t) defied i (15) is the most valuale commodity to trasfer over lik (, ). The resource allocatio (16) ca thus e viewed as a greedy attempt y ode to allocate resources to maximize its total istataeous profit. Note also that the SGP algorithm trasmits commodity c data from ode oly if W (c) (t) > 0 for some receiver ode. It follows from (14) that such a trasmissio ca oly take place if U (c) (t) > δ max (recall that SGP uses price q (c) (t) = U (c) (t)/v ). Therefore, there is always eough data availale to fill the offered trasmissio rates. B. Algorithm Performace Assume all queues of the etwork are iitially empty, ad that the SGP algorithm is implemeted with a fixed cotrol parameter V > 0. Assume all utility fuctios g (c) (r) have fiite right derivatives at r = 0. Because utilities are cocave, it follows that right derivatives exist ad are o-icreasig over the iterval r 0. Defie η as the maximum right derivative of ay utility fuctio. Note that 0 η <. Additioally assume that utility is zero whe rate is zero, i.e., g (c) (0) = 0 for all ad c. Theorem 2: (SGP Performace) For aritrary S(t) processes ad for ay fixed parameter V > 0, the SGP algorithm esures: (a) U (c) (t) V η + δ max for all slots t ad all (, c). () All odes ad users receive o-egative profit. Specifically, for all slots t ad odes {1,..., N}, we have: 1 t 1 φ (τ) 0 (17) t Likewise, for all users {1,..., N}, all commodities c, ad all slots t, we have: ( ) t 1 g (c) 1 R (c) (τ) 1 t 1 α (c) (τ) 0 (18) t t

10 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 10 (c) If chael state matrices S(t) are i.i.d. over timeslots, the the achieved social welfare of the algorithm satisfies: [ ] lim if g (c) (r (c) (t)) 1 t 1 E {C (t)} t t,c g O(1/V ) where g is the correspodig social welfare value achieved y ay other stailizig cotrol algorithm. 4 The parameter V thus determies a explicit tradeoff etwee welfare utility ad queue cogestio (ad hece, y Little s Theorem [32], ed-to-ed average delay). It is iterestig to ote that the o-egative profit result of part () holds determiistically o every timeslot t ad for ay ode that is implemetig the SGP algorithm, regardless of whether or ot the other odes are implemetig SGP. Proof: (Theorem 2 part (a)) Fix ay (, c) pair, ad cosider the admissio variale R (c) (t) chose y the SGP algorithm accordig to (13) at a particular time t. The right derivative of g (c) (r) evaluated at ay poit r 0 is less tha or equal to η. Hece, if q (c) (t) > η, the g (c) (r) rq (c) (t) 0 for all r 0, with equality holdig oly at r = 0 (recall that g (c) (0) = 0). It follows that if q (c) (t) > η, the R (c) (t) = 0 (otherwise, the solutio to (13) could e improved y settig R (c) (t) = 0). Notig that q (c) (t) = U (c) (t)/v, we have estalished the followig importat property (Property P1) of SGP: (P1) For ay (, c), t, if U (c) (t) > V η, the R (c) (t) = 0. Now suppose that for a particular timeslot t, we have U (c) (t) V η + δ max for all (, c) (this certaily holds for t = 0, as all queues are iitially empty). We prove that the same holds for time t + 1. Cosider ay particular (, c). If U (c) (t + 1) U (c) (t), the clearly U (c) (t + 1) V η + δ max. Else, queue (, c) must have received ew commodity c arrivals durig slot t (either edogeous, exogeous, or oth). If it received a positive amout of exogeous commodity c arrivals from the source user, the R (c) (t) > 0 (refer to the oe-step queueig dyamics (5)). By Property P1, this implies that U (c) (t) V η. As δ max represets the largest chage i queue acklog durig ay sigle timeslot, it follows that U (c) (t + 1) V η + δ max. Fially, i the case that this queue did ot receive ay exogeous arrivals ut did receive a positive amout of edogeous data trasmitted from other odes, the y the SGP routig policy we kow that W a (c) (t) > 0 for at least oe other ode a (where a ). It follows from (14) ad the price defiitios (t) = U (c) (t)/v that: q (c) Therefore: U (c) U a (c) (t) U (c) (t) δ max > 0 (19) (t + 1) U (c) (t) + δ max < U a (c) (t) δ max + δ max (20) V η + δ max (21) where (20) follows from (19), ad (21) follows ecause all queues are ouded y V η + δ max o slot t. Hece, i all cases we have U (c) (t + 1) V η + δ max. This holds for all queues (, c), ad y iductio it holds for all timeslots t {0, 1, 2,...}, provig the result. Proof: (Theorem 2 part ()) For each user, the SGP admissio decisios R (c) (t) are chose to optimally solve (13). Hece, for ay (, c) ad ay slot τ we have: g (c) (R (c) (τ)) q (c) (τ)r (c) (τ) 0 (22) Ideed, the left had side eig egative would create a cotradictio, as the the solutio to (13) could e strictly improved y chagig R (c) (τ) to 0. Takig a time average of (22) over τ {0,..., t 1} ad usig cocavity of the utility fuctio together with the fact that α (c) (τ) = q (c) (τ)r (c) (τ) yields (18). 4 A oud o the O(1/V ) term i the theorem ca e computed explicitly, ut we omit this computatio for revity.

11 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 11 To prove (17), assume that a give ode implemets SGP. For each slot τ, defie h (τ) as follows: h (τ) = µ (c) (c) (τ)w (τ) C rec (µ (τ)) c C tra (µ (τ), S (τ)) (23) For ay lik (, ), SGP trasmits oly a sigle commodity c (τ), ad this commodity receives the full trasmissio rate µ (τ). It follows that µ (τ)w (τ) = c µ(c) (c) (τ)w (τ). Thus, h (τ) is the same as the maximizatio metric (16) used i the resource allocatio algorithm of SGP. Further ote that µ (τ) = 0 is always a optio i the resource allocatio optimizatio (16), ad hece this optimizatio metric is always o-egative. That is, h (τ) 0 for all τ. Usig simple arithmetic together with the defiitios of h (τ) i (23) ad φ (τ) i (6), we have for all τ: φ (τ) = h (τ) + c,c µ (c) = h (τ) + c,c α (c) (τ) + c a β (c) a (τ) c (τ)[q(c) (τ) q (c) (t) δ max /V ] α (c) (τ) + c µ (c) (τ)[q(c) (τ) δ max /V ] a β (c) a (τ) β (c) (τ) where the fial equality holds ecause, y defiitio, β (c) (τ) = µ(c) (τ)q(c) (τ). Because h (τ) 0, we have: [ t 1 t 1 φ (τ) α (c) (τ) + ] β a (c) (τ) c c a t 1 µ (c) (τ)[q(c) (τ) δ max /V ] (24),c It suffices to prove that the right had side of the aove iequality is o-egative. That is, we desire to prove: t 1 Reveue (t) µ (c) (τ)[q(c) (τ) δ max /V ] (25),c where Reveue (t) represets the first term o the right had side of (24), ad is equal to the total reveue eared y ode from hadlig charges paid to it durig the course of the first t slots. To show this, oserve that all sample paths ad queueig values are preserved if the actual data chose to e trasmitted from each queue U (c) (τ) takes place accordig to the Last I First Out (LIFO) strategy. Usig this iterpretatio, we ote that every it of data that arrives to ode is charged a particular price y ode. Thus, for a particular slot τ, the data associated with the µ (c) (τ) trasmissios out of ode is composed of its that may have arrived at differet times ad may have ee charged differet prices. However, uder the LIFO trasmissio rule, all of this data was trasmitted ito ode durig slots whe the queue acklog was greater tha or equal to U (c) (τ) δ max. This is ecause µ (c) (τ) δ max, ad so the LIFO trasmissio of µ (c) (c) (τ) data leaves at least U (τ) δ max its of data ehid, all of which must have ee there whe the trasmitted data arrived. It follows that the price charged to each it of this trasmitted data whe it arrived to ode was at least (U (c) (τ) δ max )/V. This is ecause ode uses the SGP pricig rule that sets curret price equal to curret acklog divided y V, ad holds regardless

12 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 12 of the pricig ad decisio strategies of the other odes. Therefore, the total reveue eared y acceptig this data is at least as large as the right had side of (25). Thus, (25) holds, ad the result follows. Part (c) of Theorem 2 is derived via the followig Lyapuov drift lemma from [8][30]. Let U(t) = (t)) represet the matrix of queue acklogs, ad let L(U) e a o-egative fuctio of the etwork queue state, called a Lyapuov fuctio. Formally defie the Lyapuov drift as follows: (U (c) (U(t)) =E {L(U(t + 1)) L(U(t)) U(t)} Let f(t) represet some real valued stochastic reward process related to the system, ad assume f(t) f max for all t, for some fiite upper oud f max. Lemma 2: (Lyapuov drift [8][30]) If there exist costats B > 0, ɛ > 0, V > 0 such that for all slots t ad all queue states U(t), the Lyapuov drift satisfies: (U(t)) V E {f(t) U(t)} B ɛ,c U (c) (t) V f for some target utility value f, the:,c lim sup t 1 t t 1 lim if t,c 1 t E { U (c) (τ) } B + V (f max f ) ɛ t 1 E {f(τ)} f B/V (c),c (U Proof: (Theorem 2 part (c)) Defie L(U) = 1 2 ) 2. The queueig dyamics (5) ca e used to oud the Lyapuov drift (U(t)) accordig to a stadard computatio [29][30]: (U(t)) V { } E g (c) (R (c) (t)) C (t) U(t) c B V { } E g (c) (R (c) (t)) C (t) U(t) c { U (c) (t)e µ (c) (t) } µ (c) a(t) R (c) (t) U(t) a where B is a costat that depeds o N ad δ max. 5 The key oservatio is that, give U(t) ad give the pricig rule q (c) (t) = U (c) (t)/v, the SGP algorithm comes withi a additive costat B =δ max µmax,out of miimizig the right had side of the aove iequality over all possile cotrol decisios for {R (c) (t)}, {µ (t)}, ad {µ (c) (t)}. The additive costat is ecause we use the modified differetial acklog i (14) (with a δ max /V fudge factor ), rather tha the pure differetial acklog. The detailed demostratio of this is similar to related demostratios i [8][30], ad is omitted for revity. However, this immediately implies that pluggig the alterative statioary radomized decisios {R (c) }, {µ (t)}, ad {µ (c) (t)} from Theorem 1 ito the right had side ad addig B preserves the iequality. The statioary radomized algorithm makes decisios idepedet of U(t), ad yields a sigificat simplificatio. Specifically, pluggig (11) ad (12) directly ito the right had side of the aove iequality ad addig B yields: (U(t)) V { } E g (c) (R (c) (t)) C (t) U(t) c (B + B) V ρg 5 Specifically, it ca e show that B 1 2 P (µmax,out ) P (µmax,i + R max ) 2, as i [29].

13 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 13 where g is the social welfare of ay particular stailizig strategy, ad ρ is ay value such that 0 < ρ < 1 (from Theorem 1). Usig Lemma 2 yields: { } 1 t 1 lim if E g (c) (R (c) (τ)) C (τ) t t c ρg (B + B)/V Takig a limit as ρ 1 ad usig cocavity of the utility fuctios g (c) (r) proves part (c) of Theorem 2. IV. PRICING FOR BALANCED PROFITS While the SGP algorithm makes greedy decisios ad esures o-egative profit for all participats, the profits might ot e distriuted evely. To provide more alaced profits, we defie profit metrics accordig to geeral o-decreasig cocave fuctios Φ (φ ) ad Ψ (ψ ) for all odes ad users. Suppose that φ ad ψ respectively represet the time average profit of each ode ad user {1,..., N}. The goal is to desig a cotrol ad pricig algorithm that stailizes the etwork, esures o-egative profits, ad that optimizes: [ Φ (φ ) + Ψ (ψ ) ] (26) N =1 We impose a additioal ouded price assumptio o the prolem: all per-uit prices are ouded y a maximum price Q max. To solve this prolem, we use our stochastic etwork optimizatio framework from [30] [31] [8]. Let γ (t) ad ν (t) represet auxiliary variales, ad cosider the equivalet prolem: Maximize: N [Φ (γ ) + Ψ (ν )] =1 Suject to: 1) φ γ, ψ ν for all 2) Network Staility That this ew prolem is equivalet to the prolem of maximizig (26) suject to etwork staility follows immediately from the fact that the Φ (φ) ad Ψ (ψ) fuctios are o-decreasig. The ituitio ehid why it is importat to modify the prolem y itroducig auxiliary variales i this way is the followig: The auxiliary variales γ (t) ad ν (t) are chose i software ad ca e set to ay desired value at ay time, regardless of the etwork state. Thus, it is easier to use such auxiliary variales i the maximizatio of the cocave (ad hece potetially o-liear) fuctios Φ () ad Ψ (). I cotrast, the actual user ad ode profit variales φ (t) ad ψ (t) caot e chose as desired every slot, as these istataeous profits deped highly o the curret etwork state. Thus, it would e difficult (or impossile) to use φ (t) ad ψ (t) directly i a o-lie oliear optimizatio every slot, ut it is possile to esure that their time averages coform to the liear time average iequality costraits φ γ ad ψ ν (further details o this importat stochastic optimizatio techique are discussed i [30]). This issue did ot arise i our previous aalysis of SGP, as the variales that appeared i the oliear optimizatio metrics were admissio decisios R (c) (t), which could e chose as desired each slot ecause of the elastic traffic assumptio that users always have data to sed. 6 Defie φ pos (t) ad φ eg (t) respectively as the sum of positive terms i the φ (t) fuctio (6) ad the asolute value of the sum of the egative terms. Thus, φ (t) = φ pos (t) φ eg (t). The iequality costraits 6 Ideed, without assumig users have ifiite acklog for admissio decisios (as, for example, whe ew data is ot always availale ut arrives radomly to the trasport layer at each source), the SGP algorithm would eed to e modified y itroducig auxiliary variales or flow state variales, a techique developed i a related flow cotrol cotext (without profit metrics) i [8] [30].

14 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 14 i the aove prolem are treated via stailizatio of a ew set of virtual queues X (t) ad Y (t), defied accordig to the followig dyamics: X (t + 1) = max[x (t) φ pos (t), 0] + γ (t) + φ eg (t) Y (t + 1) = max[y (t) g (c) (R (c) (t)), 0] c +ν (t) + c α (c) (t) (27) Stailizig the X (t) ad Y (t) queues implies that the time average of the queue iput variales is less tha or equal to the time average of the queue service variales. Specifically, stailizig X (t) esures γ + φ eg φ pos, which is equivalet to the costrait φ γ. Likewise, stailizig Y (t) esures that ν c [g(c (r (c) ) α (c) ]. We ext defie the pricig variales: α (c) (t) = p (c) (t)r (c) (t), β (c) (t) = q(c) (t)µ(c) (t) where p (c) (t) is the per-uit price for exogeous commodity c arrivals at ode, ad q (c) (t) is the per-uit price for commodity c data trasmitted over lik (, ). Recall that i SGP we foud it was sufficiet to use a sigle price q (c) (t) that is charged to ay user or icomig lik that seds ew commodity c data to ode o slot t (so that p (c) (t) = q a (c) (t) = q (c) (t) for each lik (a, )). However, here we fid it is importat to allow the possiility of differetiated pricig for each lik ad user sedig data to ode. Let Z(t) =(U(t); X(t); Y (t)) represet the comied queue state. Defie the Lyapuov fuctio L(Z(t)) as follows: L(Z(t)) = 1 (U (c) (t)) X (t) Y (t) ,c The followig oud ivolvig Lyapuov drift (Z(t)) ca e computed via the queueig dyamics (5) ad (27): (Z(t)) V E {Φ (γ (t)) + Ψ (ν (t)) Z(t)} D V E {Φ (γ (t)) + Ψ (ν (t)) Z(t)} {,c U (c) (t)e µ(c) (t) } a µ(c) a(t) R (c) (t) Z(t) X (t)e {φ (t) γ (t) Z(t)} { Y (t)e c g(c) (R (c) (t)) } c α(c) (t) ν (t) Z(t) where D is a costat. The followig algorithm is otaied y makig cotrol ad pricig decisios that miimize the right had side of the aove drift oud o every slot t. Bag-Bag Pricig Algorithm for Stochastic Networks: Pricig: Every slot t, each ode oserves its virtual queues X (t), Y (t) ad chooses p (c) (t) as follows: { p (c) Qmax if Y (t) = (t) < X (t) 0 otherwise Each ode also oserves the virtual queues X a (t) of its eighors, ad chooses q a (c) (t) as follows: { q a (c) Qmax if X (t) = a (t) < X (t) 0 otherwise of: Admissio Cotrol: Every slot t, each ode chooses R (c) (t) (for each commodity c) as the maximum Y (t)g (c) (r) r[u (c) (t) p (c) (t)(x (t) Y (t))]

15 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 15 over the iterval 0 r R max. For each, the auxiliary variale γ (t) is chose as the maximum of V Φ (γ) X (t)γ suject to 0 γ Q max δ max. Likewise, the auxiliary variale ν (t) is chose as the maximum of V Ψ (ν) Y (t)ν suject to 0 ν ηr max. Resource Allocatio: Each ode oserves the chael states ad queue acklogs of its eighors, ad computes: Θ (c) (t) =U (c) (t) U (c) (t) q (c) (t)(x (t) X (t)) The optimal weight ad commodity is the chose as follows: Θ (t) = max Θ (c) c, c (t) = arg max Θ (c) c The trasmissio vector µ (t) is allocated as the solutio to: Maximize: Suject to: Routig/Schedulig: The µ (c) (t) rates are selected as: µ (c) µ Θ (t) X (t)c tra (µ, S (t)) X (t) Crec (µ ) µ Ω () S (t) { (t) = µ (t) if Θ (t) > 0 ad c = c (t) 0 otherwise The actual queues U(t) are the updated accordig to (5), ad the virtual queues X(t), Y (t) are updated accordig to (27). Ulike SGP, the pricig here does ot deped o the commodity c, ut is potetially lik depedet. Theorem 3: (Bag-Bag Pricig Performace) The Bag-Bag pricig algorithm stailizes all actual ad virtual queues of the system, esures all participats make o-egative time average profit, ad yields: lim if t lim sup t 1 t t 1,c E { U (c) (τ) } O(V ) N [ Φ (φ (t)) + Ψ (ψ (t)) ] Φ O(1/V ) =1 where φ (t) ad ψ (t) are defied i (7) ad (8), ad where Φ is the lim sup of the achieved profit metric (26) uder ay other stailizig cotrol algorithm. Proof: The result uses the Lyapuov drift lemma (Lemma 2) together with the fact that (i) The give cotrol decisio variales miimize the right had side of the drift oud, ad (ii) There exists a statioary radomized cotrol algorithm that stailizes the system ad achieves a profit metric of at least ρφ (for ay ρ such that 0 < ρ < 1). A complete derivatio is omitted for revity. A. A Fixed-Topology Network with Access Poits V. SIMULATION Cosider the etwork of Fig. 1, where there are seve wireless odes, four of which have sources that desire to sed data to ay of the three wirelie access poits. This is a sigle commodity prolem, as the three access poits ca e viewed collectively as a sigle ode. Assume traffic is i uits of packets, ad suppose each wireless ode ca trasmit over at most oe outgoig lik per timeslot. The dashed liks idicate time varyig ON/OFF chaels with i.i.d. ON proailities of 1/2 (so that a sigle packet ca e trasmitted whe ON, ad zero whe OFF). Trasmissio costs over these liks are 1 cet/packet, ad

16 5 7 6 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 16 S 1 S 3 S 4 S 2 C 2 C 5 C 7 Fig. 1. A sigle commodity etwork where four differet wireless sources desire to sed data to ay of the three wirelie access poits. SGP Profit BB Profit SGP U BB U User User User User Node Node Node Node Node Node Node Total Fig. 2. A tale of simulatio results for the sigle commodity etwork of Fig. 1, showig time average profit (i cets/slot) ad queue acklog (i packets) for the SGP ad Bag-Bag algorithms. V = 50 for SGP, V = 500 for BB. R max = 1, Q max = g(1 + R max), C 2 = C 5 = C 7 = 1. receptio costs of all liks are equal to 0.5 cet/packet. The solid liks to the access poits are always ON, ad ca trasmit oe packet per slot with trasmissio costs C2 = C5 = C7 = 1. Suppose utility fuctios for each of the four users are give y g(r) = 10 log(1 + r). We simulate the SGP ad Bag- Bag pricig algorithms over 10 millio timeslots. We use Φ(x) = Ψ(x) = log(1 + x) for the Bag-Bag implemetatio. Profit results are preseted i Fig. 2. Note that odes 5, 6, ad 7 make positive profits y actig as pure relays. This shows that odes ca eefit from participatig i the free market eve if they do ot desire their ow persoal commuicatio. Decreasig the V parameter i the Bag-Bag algorithm leads to less precisely alaced profits ut also decreases cogestio. Uder SGP, odes 6 ad 7 support roughly 1/3 of the traffic from S4, ut receive oly 0.02 profit. I Fig. 3 we illustrate results for the case whe C2 ad C5 are icreased to 3. I this case, it is apparet that the sum profit decreases, ad the idividual profits of odes 2 ad 5 sigificatly decrease uder oth SGP ad BB, while the profit of odes 6 ad 7 either stay the same or icrease. B. A Ad-Hoc Moile Network Cosider ow a ad-hoc moile etwork with 7 odes that move aout a etwork regio that is partitioed ito a 5 5 grid (see Fig. 4). Nodes 1 ad 4 are statioary ad stay i cell locatios (4, 4) ad (2, 2), respectively, for all time. Nodes 3, 5, 6, ad 7 are fully moile ad take idepedet ad uiform Markov radom walks throughout the etwork. Specifically, every slot each fully moile ode idepedetly decides with proaility α to stay i its same cell, ad else (with proaility 1 α) it decides to move to a adjacet cell either to the North, West, South, or East, equally likely i all 4 directios. If it decides to move i a ifeasile directio, it stays i its same cell. Node 2 is partially moile, i that it takes a Markov radom walk that is restricted to oly the two shaded cells i the ottom corer of Fig. 4. For simplicity, we assume that odes ca oly commuicate whe they are i the same

17 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT 17 SGP Profit BB Profit SGP U BB U User User User User Node Node Node Node Node Node Node Total Fig. 3. A tale of simulatio results for the same sigle commodity etwork as Figs. 1 ad 2, with the exceptio that C 2 ad C 5 are icreased to 3. cell, that each cell ca support at most oe packet trasmissio per slot (cosistig of 1.0 uits of data), ad that there is o iter-cell iterferece. The capacity of such a etwork whe all odes are fully moile is derived i [33]. We assume a costat receptio cost C r = 0.25 ad trasmissio cost of C t = 0.5 uits for each ode. The α moility proaility is set to α = 1/2. We assume there are 3 sources, odes 1, 2, ad 3, ad that each source ode i desires to sed a stream of data to a particular destiatio ode j. The source-destiatio pairigs are: 1 4, 2 5, ad 3 6. Thus, this is a 3-commodity etwork, with commodities correspodig to the destiatio odes 4, 5, ad 6. Note that the statioary ode 1 desires to sed data to the other statioary ode 4. Hece, the source-destiatio pair 1 4 requires moile relays to support its commuicatio. The other source-destiatio pairs ca i priciple exchage data without relays y waitig util they are i the same cell, ut ca cosideraly expad their throughput capailities y usig moile relays. We assume that all odes participate i the market ad ca act as relays of other data (icludig the statioary odes 1 ad 4). Note that ode 7 is either a source or a destiatio, ad hece it participates oly to make a profit. For revity, we simulate oly the case whe all odes use the SGP algorithm. It is importat to ote that this ad-hoc moile etwork satisfies either the i.i.d. chael assumptio or the orthogoal chael assumptio. Ideed, the chael process etwee ay ode pair is effectively a ON/OFF process that depeds o whether or ot the odes are i the same cell (which is ot i.i.d. over slots due to the Markovia moility). However, the SGP algorithm guaratees ouded queue acklog ad o-egative profits for ay stochastic chael process, icludig o-i.i.d. chaels (recall Theorem 2 parts (a) ad ()). Further, while the maximum sum-profit result of Theorem 2 part (c) was prove for i.i.d. chaels, the same result ca e show for chaels modulated y ay fiite state ergodic Markov chai. The oly differece is that the O(1/V ) term that ouds deviatio from optimality is icreased y a costat coefficiet that depeds o the mixig time of the Markov chai. This ca e prove usig a more detailed K-slot Lyapuov drift argumet (see [30] for details o the techique), although the aalysis is omitted for revity. Further, the requiremet that each cell ca support at most oe trasmissio per slot violates the chael orthogoality assumptio. This ecessitates trasmissio schedulig i cases whe there are multiple competig trasmissio possiilities i a give cell. However, it is simple to see that the proof of Theorem 2 holds exactly i this trasmissio schedulig sceario uder the assumptio that trasmissios are chose to maximize h (τ) every slot, where h (τ) is defied i (23). Ideed, the orthogoality assumptio was used oly ecause it implies that maximizig this sum is equivalet to idividually maximizig each h (τ) term, which ca e doe idepedetly y each ode. However, i this cell-partitioed etwork sceario, maximizig the sum amouts to havig each cell with two or more users select the trasmitter-

18 WIRELESS NETWORKS, VOL. 15, PP , 2009 (THE ORIGINAL PUBLICATION IS AVAILABLE AT Fig. 4. A ad-hoc moile etwork uder SGP. SGP Profit SGP U i SGP r i User User User Node Node Node Node Node Node Node Total Fig. 5. A tale of simulatio results for SGP applied to the ad-hoc moile etwork of Fig. 4, with V = 20. The average queue values U i correspod to the sum over all 3 queues withi each ode i. receiver pair (, ) with the largest differetial price parameter W (τ), ad trasmittig 1.0 uits of the optimal commodity c (τ) wheever W (τ) C r C t > 0. This trasmissio schedulig optio has the followig ecoomic iterpretatio: Suppose that there is a cell ower that receives a portio of profits from each trasactio i its regio, ad that aritrates schedulig opportuities. The cell ower thus greedily selects the schedulig opportuity that maximizes istataeous profit W (τ) C r C t withi the cell. This decisio ca alteratively e motivated with a auctio strategy, uder the assumptio that each user ids for a trasmissio opportuity i proportio to its istataeous profit W (τ) C r C t, ad the highest idder wis. We simulate the etwork over 1 millio timeslots, usig utility fuctios g(r) = 5 log(1 + r) for each user 1, 2, 3, ad R max = 1.0. Note that this implies each queue has a worst case cogestio of 5V + δ max, where δ max = 2.0 i this sceario. Figs. 6 ad 7 plot sum profit ad average etwork cogestio versus the V parameter, illustratig that sum profit coverges to its maximum value as V is icreased, with a correspodig liear tradeoff i average queue cogestio. The kee of the curve appears roughly whe V = 20, ad hece we provide more detailed profit ad queue acklog iformatio for the V = 20 simulatio i Fig. 5. Here we agai make a distictio etwee the user at source ode i ad source ode i itself (although the total profit of ode i ca also e cosidered as the sum of its user ad ode profits). Here we see that each of the three sources i achieves a throughput r i of at least 0.1 packet/slot, ad all odes ad users make a o-egative profit. 7 It is iterestig to ote that the destiatio odes 4, 5, 6 make the least profit, as these odes are maily 7 Note that the largest possile throughput etwee two moile odes would e oly 1/25 = 0.04 packets/slot if they did ot use ay relays (as this is the proaility that oth odes are i the same cell). Hece, as all three sources achieve throughput larger tha 0.1 packets/slot, we see that they are all actively utilizig relays.