Price-based Congestion-Control in Wi-Fi Hot Spots

Transcription

1 Price-based Cogestio-Cotrol i Wi-Fi Hot Spots Roberto Battiti, Marco Coti, Erico Gregori, Mikalai Sabel To cite this versio: Roberto Battiti, Marco Coti, Erico Gregori, Mikalai Sabel. Price-based Cogestio-Cotrol i Wi-Fi Hot Spots. WiOpt 03: Modelig ad Optimizatio i Mobile, Ad Hoc ad Wireless Networks, Mar 2003, Sophia Atipolis, Frace. 10 p., HAL Id: iria Submitted o 23 Mar 2010 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Price-based Cogestio-Cotrol i Wi-Fi Hot Spots Roberto Battiti(*), Marco Coti(**), Erico Gregori(**), Mikalai Sabel(*) (*) Uiversity of Treto, Dip. Iformatica e Telecomuicazioi Treto, Italy { battiti, msabel}@dit.uit.it (**) Cosiglio Nazioale delle Ricerche IIT Istitute Pisa, Italy { marco.coti, erico.gregori}@iit.cr.it ABSTRACT. Wireless etworks are ow proliferatig due to the success of the IEEE b protocol, also kow as Wi-Fi (Wireless Fidelity). A Wi-Fi etwork is characterized by a set of base statios (also called access poits) placed throughout the eviromet ad coected to the traditioal wired LANs. This techology allows omadic users a broadbad access to the Iteret if they are i the trasmissio rage of a access poit. A ew busiess model, amed Wi-Fi Hot Spots, is ow emergig to exploit the potetialities of this techology. A hot spot is a critical busiess area, e.g., airports, statios, hotels, where users ca have wireless access by subscribig a cotract with the hot spot operator, or with a wireless Iteret service provider (WISP). Due to the radom access ature of the Wi-Fi techology, if the umber of users coected to the same access poit icreases, the QoS experieced may quickly degrade. This geerates complais from the users that, as a cosequece, may chage their WISP. I order to be competitive, a Wi-Fi hot spot operator eeds simple ad effective mechaisms to cotrol the cogestio therefore guarateeig the QoS, ad (at the same time) maximizig his/her reveues. I this paper we preset ad evaluate a price-based policy for the access cotrol i a Wi-Fi hot spot. Our policy, amed Price-based Cogestio Cotrol (PCC), cotrols the hot spot traffic by dyamically determiig the access cost as a fuctio of the curret load i the hot spot. We develop a theoretical framework to compute for ay load coditio the access cost to maitai the hot spot i its optimal operatig poit, for ay load coditio. The effectiveess ad robustess of the PCC policy has bee evaluated by simulatig a Wi-Fi hot spot. Both i saturated ad otsaturated coditios the PCC policy provides a better chael utilizatio tha the legacy Wi-Fi policy. 1. Itroductio Wi-Fi broadbad Iteret access is mushroomig ad competig with legacy wireless (cellular) etworks to support omadic ad data-cetric applicatios. Telecom operators ad service providers are gradually chagig their marketig strategies ad complemetig their traditioal offer with Wi-Fi. To reach a efficiet use of the scarce badwidth resources, market mechaisms appropriate for the Wi-Fi techology have bee proposed ([IEEE802], [IEEE11b]). A well-desiged market should ecourage spectrum efficiecy ad iovatio. The two drivig forces characterizig the Wi-Fi evolutio are: the low cost-barrier for a service provider to eter i the market (o expesive ifrastructure is required to start with), ad the emergig tedecy (maily i USA) to deregulate the spectrum eviromet to create a secodary wireless market [Cro02], secodary with respect to the primary cellular market. I this work we propose a price-based policy for access cotrol i a Wi-Fi hot spot. We cosider a WLAN Operator ad we ivestigate policies to cotrol the traffic geerated by its hot-spot users i order to maximize the reveues. The idea is to cotrol the hot spot traffic by implemetig a cogestio-cotrol pricig policy. At ay give time istat, the access cost depeds o the curret load i the hot spot. Whe cogestio icreases, the WLAN Operator icreases the access cost util some users give up trasmittig. The objective is to maitai the quality of service, ad to maximize the reveues. I this paper, we assume a per-packet cost model, i.e., the user will be charged for the umber of successfully trasmitted ad received packets. By icreasig or decreasig the per-packet cost the WLAN Operator will cotrol the umber of active users ad hece the cogestio i the hot-spot. The aim is to drive the hot spot to operate i a status that maximizes the operator reveues ad maitais the system i the most efficiet state, i.e., the maximum aggregated throughput. Specifically, we idetify as the optimal operatig poit of the system, the status correspodig to the maximum aggregated throughput. The ratioal for this choice is based o the followig observatios: - whe the offered load is below this poit, the WLAN Operator ca stimulate additioal traffic, thus icreasig the reveues without cogestig the system; - whe the offered load is higher tha this poit, to maitai the same reveues of the optimal operatig poit, the WLAN Operator must icrease the costs ad thus the users have to pay more for a worse quality of service. This policy, of course, caot be acceptable by the users that ca hece migrate to aother WLAN Operator implemetig a more fair policy ad therefore providig a better quality at a lower cost. This research is partially supported by the Provice of Treto (Italy) i the framework of the WILMA (Wireless Iteret ad Locatio Maagemet) project (

3 For example, a simple cost model which drives the system towards the optimal operatig poit ca be the followig. The cost model has oly two values: a low ad costat cost, say C 0, wheever the load is below the maximum aggregated throughput of the system, ad a very high value, say C ifiite wheever the load exceeds the maximum aggregated throughput. The flat access cost stimulates the users to access the system util it is ot overloaded, while the ifiite cost C ifiite forces the users to stop trasmittig whe the load exceeds the offered load. Obviously, this policy eeds to be refied to avoid sharp fluctuatios i the offered load as all users will give up whe the C ifiite is aouced. Simple mechaisms to smooth these fluctuatios ca be itroduced. For example gradually icreasig the costs, or the cost icrease ca be aouced to subsets of radomly selected users. I this paper we discuss a Price-based Cogestio Cotrol (PCC) policy that ca be implemeted i a based WLAN hot-spot. To defie the PCC policy we exploit some results derived i [CG02]. The cited work presets a distributed mechaism for cotrollig the cogestio i a IEEE etwork to guaratee that the system operates below the optimal operatig poit. Furthermore, the proposed mechaism asymptotically (with respect to the umber of active statios) drives the system close to the optimal operatig poit. The mai drawback of that approach is the requiremet that all the etwork statios use a etwork iterface card obtaied by modifyig the Wi-Fi cards. Obviously, this costrait is ot acceptable i a WLAN hot spot i which the WLAN operator caot force the use of modified Wi-Fi cards. I this paper, we exploit some of the ideas preseted i [CG02] to costruct a cetralized cogestio cotrol policy that oly requires additioal hardware/software i the Access Poit while the users adopt stadard Wi-Fi cards. The paper is orgaized as follows. I Sectio 2, we preset the mai Wi-Fi characteristics ad summarize previous results about Wi-Fi modelig that are relevat for this paper. I Sectio 3, we itroduce the theoretical framework of the Price-based Cogestio Cotrol policy. The performace of the PCC policy are ivestigated, via simulatio, i Sectio Wi-Fi Wi-Fi cards implemet the IEEE b stadard. The b stadard exteds the stadard by itroducig a higher-speed Direct Sequece Spread Spectrum (DSSS) Physical Layer i the 2.4 GHz frequecy bad. Specifically, b eables trasmissios at 5.5 Mbps ad 11 Mbps, i additio to 1 Mbps ad 2 Mbps supported by O top of the DSSS physical layer, the IEEE b stadard implemets a MAC layer that offers two differet types of service: a cotetio- based ad cotetio-free service. Hereafter, we focus o the cotetio- based service oly. This service is implemeted by the Distributed Coordiatio Fuctio (DCF). Accordig to the DCF, before trasmittig a data frame, a statio must sese the chael to determie whether aother statio is trasmittig. If the medium is foud to be idle for a iterval loger tha the Distributed IterFrame Space (DIFS), the statio cotiues with its trasmissio. If the medium is busy, the trasmissio is deferred util the ed of the ogoig trasmissio, ad the the backoff procedure is activated. A radom iterval, (the backoff time) is selected ad used to iitialize the backoff timer. The backoff timer is decreased for as log as the chael is sesed as idle, stopped whe a trasmissio is detected o the chael, ad reactivated whe the chael is sesed as idle agai for more tha a DIFS. The statio is eabled to trasmit its frame whe the backoff timer reaches zero. To guaratee fair access to the shared medium, a statio that has just trasmitted a packet ad has aother packet ready for trasmissio must perform the backoff procedure before iitiatig the secod trasmissio. The backoff time is slotted: its value is a iteger umber of slots uiformly chose i the iterval (0, CW-1). CW is defied as the Backoff Widow, also referred to as Cotetio Widow. At the first trasmissio attempt CW=CWmi, ad it is doubled at each retrasmissio up to CWmax. I the b stadard CWmi ad CWmax are 32 ad 1024, respectively. Obviously, it may happe that two or more statios start trasmittig simultaeously ad a collisio occurs. I the CSMA/CA scheme, statios caot detect a collisio by hearig their ow trasmissios (as i the CSMA/CD protocol used i wired LANs). Therefore, a immediate positive ackowledgemet scheme is employed to ascertai the successful receptio of a frame. I detail, upo receptio of a data frame, the destiatio statio iitiates the trasmissio of a ackowledgemet frame (ACK) after a time iterval called Short IterFrame Space (SIFS). The SIFS is shorter tha the DIFS i order to give priority to the receivig statio over other possible statios waitig for trasmissio. If the ACK is ot received by the source statio, the data frame is assumed to have bee lost, ad a retrasmissio is scheduled. The ACK is ot trasmitted if the received packet is corrupted. A Cyclic Redudacy Check (CRC) algorithm is used for error detectio. After a erroeous frame is detected (due to collisios or trasmissio errors), a statio must remai idle for at least a Exteded IterFrame Space (EIFS) iterval before it reactivates the backoff algorithm. Specifically, the EIFS shall be used by the DCF wheever the physical layer has idicated to the MAC that a frame trasmissio was begu that did ot result i the correct receptio of a complete MAC frame with a correct CRC value. Receptio of a error-free frame durig the EIFS re-sychroizes the statio to the actual busy/idle state of the medium, so the EIFS is termiated ad ormal medium access (usig DIFS ad, if ecessary, backoff) cotiues followig receptio of that frame. A MAC data frame cotais the cotrol iformatio (hereafter deoted as MAC hdr ), a variable legth data payload, which cotais the upper layers data iformatio, ad the CRC field. The MAC hdr cotais the MAC addresses i additio to other cotrol iformatio. The values for the protocol parameters are summarized i Table 1, where τ is the air propagatio time

4 Slot _ Time Table 1. IEEE b parameter values. t slot τ MAC hdr CRC Bit Rate(Mbps) 20 µsec 1 µsec 240 bits (2.4 t slot ) 32 bits (0.32 t slot ) 1, 2, 5.5, 11 DIFS SIFS ACK CW MIN CW MAX 50 µsec 10 µsec 112 bits 32 t slot 1024 t slot 2.1 Wi-Fi Modelig The behavior of a IEEE etwork ca be closely approximated by a equivalet p-persistet IEEE protocol [CCG00]. A p-persistet IEEE protocol differs from the stadard protocol oly i the selectio of the backoff iterval. Istead of the biary expoetial backoff used i the stadard, the backoff iterval of the p-persistet IEEE protocol is sampled from a geometric distributio with parameter p. I that paper it is also show that the p- persistet IEEE protocol closely approximates the stadard protocol (at least from the protocol capacity stadpoit) if the average backoff iterval is the same. This is obtaied whe the followig equatio holds: p= 1 ( E[ B]+ 1 ) (1) where EB [ ] is the average backoff iterval. Due to its memory-less backoff algorithm, the p-persistet IEEE protocol is suitable for aalytical studies. By exploitig the similarity of this protocol with the stadard oe, the aalytical results derived from the p-persistet model ca be used to ifer the behavior of the stadard protocol. I this sectio, we assume a system with a variable umber, M, of active statios accessig the chael accordig to the p-persistet IEEE protocol. The statios trasmit messages whose legth (expressed as umber of time slots) is a radom variable L with average l. The behavior of the chael access scheme ca be described as a sequece of virtual trasmissio times. A virtual trasmissio time is the iterval betwee two successful trasmissios ad, as show i Figure 1, it icludes a successful trasmissio ad may iclude several collisio itervals ad idle periods. 1 I the followig we use the followig otatio: Idle _ p is the umber of cosecutive empty slots; Coll is the time the chael is busy due to a collisio give that a trasmissio attempt occurs, also referred to as collisio iterval (see Figure 1). Obviously, Coll is equal to zero if the trasmissio attempt is successful, otherwise it is equal to the collisio legth; t slot is the time duratio of a time slot; E[ ] deotes the average operator, i.e., give a radom variable X, EX [ ] is its average; collisio collisio DIFS DIFS # # legth ##### legth... # MAC frame SIFS + ACK DIFS # : empty slot collisio iterval Virtual trasmissio time successful trasmissio iterval Figure 1: Structure of a virtual trasmissio time By exploitig the aalytical tractability of the p-persistet IEEE the followig property, relevat for defiig our cogestio pricig policy, was proved ([CG02], [BCG02]): Lemma 1. For M >> 1 ad l > 1 the optimal operatig poit, correspods to the system status that satisfies the followig equatio: E Idle _ p tslot E Coll [ ] = [ ] (2) 1 It is worth otig that, hereafter, the chael idle period that follows a collisio (at least a DIFS), ad the SIFS, ACK ad DIFS itervals that follow a successful trasmissio are couted as overheads associated to a collisio or to a successful trasmissio.

5 [ ] ad E[ Coll] i a p-persistet IEEE with M active statios are derived Closed form expressios for the E Idle _ p (by exploitig classical probabilistic argumets) i [CCG00a]. The formulas idicate that the optimal operatig poit is a fuctio of the umber of active statios (M), the legth of trasmitted packets, ad the average cogestio widow (i.e., the p value). Therefore, by fixig the p value, ad the legth of trasmitted packets, it is possible to determie the umber of statios that must be active (i a hot spot) i order to maitai the system i the optimal operatig poit. Therefore, Equatio (2) provides a effective tool that ca be used by the Access Poit to cotrol the cogestio status of the etwork. It is worth otig that the perspective of the PCC policy is complemetary to the policies developed i the previous works ([CG02], [BCG02]). I those papers p is a variable whose value is dyamically set, while M caot be modified. O the cotrary, i this paper the p value (i.e., the average backoff widow size) caot be cotrolled, while the Access Poit ca try to modify the value of M (by icreasig or decreasig the trasmissio cost). I detail, for each etwork coditio, the Access Poit tries to drive the M value to the optimal operatig poit, say M opt, that guaratees the satisfactio of Equatio (2). 3. The Price-based Cogestio Cotrol (PCC) policy To simplify the presetatio, we assume that the PCC policy takes its decisios (i.e., the ew price to be commuicated to the hot spot users) at the ed of each virtual trasmissio time. However, it is also possible to implemet the same policy takig as a referece other evets occurrig o the chael, or assumig a periodic re-computatio (ad aoucemet) of the optimal price. As show i the Sectio 4, the periodic aoucemet is the most robust solutio for PCC. The PCC policy operates i two steps. I the first step, it idetifies the percetage icrease/decrease i the umber of active statios i the cotrolled hot spot to drive the system to the optimal operatig poit. I the secod step, the policy idetifies the price level to achieve the desired icrease/decrease i the umber of active statios. Before presetig the PCC policy i detail, we itroduce some assumptios. i) After each successful trasmissio the AP decides the trasmissio-cost c to be applied to the future trasmissios. The time it takes to the AP to compute the ew cost ad to commuicate it to the cliets is assumed to be egligible. ii) To model the users behavior, we itroduce a probability fuctio Pgive_ up (). c Specifically, Pgive_ up () c represets the probability that a user does ot accept the c trasmissio-cost aouced by the AP, ad leaves the etwork for a radom time. It is reasoable to assume that Pgive_ up () c is mootoe ot decreasig, i.e., whe the cost icreases the probability that a user gives up icreases or remai the same. The AP estimates the Pgive_ up () c from the system history, by observig the users reactio to chages i the price level. 2 iii) We model the arrival of ew customers (ew cliets plus the cliets that make a ew attempt after a give-up period) i the hot spot through a Poisso process, with rate λ. λ is estimated by applyig a movig widow estimator to the recet history of the system. ed of the (-1)-th virtual trasmissio iterval M 1 -th virtual trasmissio iterval ed of the -th virtual trasmissio iterval M c 1 c collisio ad idle periods -th successful trasmissio E[ Coll] ad E[ Idle _ p] updatig E[ Coll] E[ Idle _ p] Figure 2: PCC policy behavior Step 1. Percetage icrease/decrease i the umber of active statios The behavior of the PCC policy is show i Figure 2. As metioed before, PCC operates at the ed of each virtual trasmissio time. As show i Figure 1, this time iterval (from the chael status stadpoit) cosists of oe or more 2 Pgive _ up ( c) ca be computed offlie by exploitig the statistics collected o a daily (or loger) basis.

6 idle periods, ad oe or more busy periods. A idle period is made up of cosecutive idle slots, while the busy period is a iterval i which the chael is busy due to either a collisio or a successful trasmissio. To explai the behavior of the PCC policy, we focus o the ed of the -th virtual trasmissio iterval. I additio, let s assume that, after computig E[ Idle _ p] ad E[ Coll], Equatio (2) does ot hold. Hece, the aim of PCC is to estimate the desirable umber of statios that should be active i the future i order to equate the collisio ad idle costs. Specifically, we deote with M the umber of active statios that will guaratee (i the future) the balace betwee the idle-period legth, ad the collisio cost, thus satisfyig Equatio (2). By assumig that M statios will be active i the future, by applyig the Taylor formula to the closed formulas, we ca express E[ Idle _ p] +1 ad E [ Coll ] +1 as follows3 ([BCG02], [CG02]): 1 p M p M E[ Idle _ p] E Coll l +, [ ] + (3) 1 1 p M 2 where l is the average collisio legth (i time uits) give that two statios collide. By exploitig (3), PCC icreases or decreases the M value to have E[ Idle _ p] + 1 = E[ Coll] + 1. The followig lemma directly defies M. Lemma 2. I a etwork adoptig the p-persistet access protocol, i which the message legths {Li}, ormalized to t slot, are a sequece of i.i.d. radom variables, the M value correspodig to the optimal operatig poit is: l M opt 1+ 2 l p Proof. Equatio (4) is obtaied by equatig the expressios of E Idle _ p [ ] +1 ad E [ Coll ] +1 defied by (3). Lemma 2 provides to the AP the expressio of the desirable M value. However, to exploit this expressio the AP must have a estimate of the average backoff widow size used i the WLAN (see Equatio (1)). 4 This may be difficult to determie. To avoid this problem, i the followig we preset a method that computes the percetage icrease/decrease of M without requirig ay kowledge of p. To this ed, we express M as a fuctio of a ukow quatity x, such that M = M 1( 1 + x) (5) where x deotes the percetage icrease/decrease i the M 1 value. By usig Equatios (5), (3) ad (2), we obtai Lemma 3. This Lemma provides a closed formula to estimate the x value startig from quatities that are easy to estimate. [ ] ad E[ Coll] the estimates of E[ Idle p] ad E[ Coll] that the Access Poit Lemma 3. By deotig with E Idle _ p _ has got at the ed of the -th virtual trasmissio iterval, the percetage icrease/decrease i the M value to achieve the optimal operatig poit is: (4) ( )+ ( ) + ( + ) tslot E Coll tslot E Coll tslot E Idle p x = [ ] 4 [ ] 1 [ _ ] 2E[ Coll] (6) [ ] +1 ad E [ Coll ] +1 Proof. By substitutig (5) i the expressios of E Idle _ p defied by (3), ad by assumig E[ Idle _ p] + 1 = E[ Coll] + 1, after some algebraic maipulatios, we obtai: tslot E Coll tslot E Coll tslot E Idle p x = ( + 2 )± ( ) 2 [ ] + 4 [ ] ( 1+ [ _ ] ) 2E[ Coll] (7) Equatio (7) provides two possible solutios to the problem, but oly the solutio give by (6) is correct for the PCC policy. This ca be verified by focusig o the case i which E[ Idle _ p] is equal to E[ Coll]. I this case oly by applyig Equatio (6) we obtai the correct value x=0, as the system is already i the optimal operatig poit. Oce the desirable percetage icrease/decrease i the M value, the PCC computes the cost level that must be applied i the future to stimulate/discourage the users to access the system i order to approach the desired M value. 3 The ew M value will of course ifluece the future average backoff widow (ad hece the p value), hereafter we assume that the impact of the M chage o p is egligible. The accuracy of this assumptio icreases with the icrease of the M value. 4 The AP ca approximate this value with the estimate of the average backoff widow size it has adopted i the past.

7 It is worth otig that Equatio (6) completely defies the percetage icrease/decrease i the M value i terms of quatities that the Access Poit estimates by measuremets o the etwork. These measuremets may itroduce fluctuatios. To avoid harmful fluctuatios, smoothig factors are ormally itroduced. Specifically, E[ Idle _ p], ad E[ Coll] ca be computed as: E[ Idle _ p] = α E Idle _ p E Idle _ p [ ] + ( 1 α 1 ) [ ( )] (8) [ ] = [ ] + ( ) [ ( )] E Coll α E Coll 1 α E Coll 1 where α is a smoothig factor ad E[ Idle _ p( ) ]ad E[ Coll( ) ] are the average legth of the idle periods ad collisio costs durig the -th virtual trasmissio time, respectively. The use of a smoothig factor is widespread i the etwork protocols to obtai reliable estimates from the etwork estimates by avoidig harmful fluctuatios, e.g. RTT estimatio i TCP. Previous work has show that α = 09. is a good compromise betwee accuracy ad promptess [CCG00b]. For this reaso we assume α = 09. as the default value. A similar approach ca be used to avoid cost fluctuatio. I this case the smoothig strategy must be modified to avoid frequet (small) modificatio i the trasmissio cost. Step 2. The cost level that stimulate/discourage ew users I the Step 1, the AP has idetified the percetage icrease/decrease of M (i.e., x). I this secod step, the x value is used to determie the trasmissio cost to apply for the future trasmissios i order to have M = M 1( 1 + x) active statios. The basis of this aalysis is give by: λ Et [ v] ( 1 P c)+ M ( P c)= M give_ up () 1 1 give_ up () (9) where Et [ v ] deotes the average legth of the virtual trasmissio time ad it is estimated by applyig a smoothig factor, α, see also Equatio (8). The ratioale behid Formula (9) is the followig. The first term i the left had side (l.h.s.) of (9) represets the expected umber of users that accept to start a ew sessio if a trasmissio cost equal to c is applied. The secod term i the l.h.s of (9) represets the umber of active users that accept to cotiue their trasmissios give that they will pay a cost per-packet equal to c for future trasmissios. With some algebraic maipulatio of Equatio (9) we fially obtai: λ Et [ v] M x Pgive up c 1 _ ( ) = max 0,. (10) λ Et [ v] + M 1 Equatio (10) idetifies the ew c value, say cew, to achieve the optimal operatig poit. To use Equatio (10) the AP requires the kowledge of M 1, i.e., the curret umber of active statios. I priciple the AP ca estimate this umber by observig the MAC addresses of the statios with a active sessio. This is a coservative estimate because it may happe that some statios are coected to the Access Poit but are ot curretly trasmittig ay data. However, we ca avoid this problem, ad at the same time further simplifyig Equatio (10), by otig that the itervals betwee successive updatig of the price are quite small (i the order of a secod). Hece, the umber of ew statios 5 that coects to the access poit durig a measuremet iterval is expected to be low. Therefore, we ca avoid the complexity i Equatio (10) by eglectig i Equatio (9) the impact of statios that will wake up i the future, ad simply cocetratig o optimizig the system for the curret umber of active statios. The impact of statios that will wake up i the ext future will be take ito accout i the successive price computatios phases. Accordig to this simplifyig assumptio, Equatio (9) reduces to: M 1 ( 1 Pgive_ up () c )= M (11) ad by substitutig (5) i (9) we obtai: Pgive_ up ()= c x (12) Equatio (12) provides a simple way to idetify the ew price without requirig ay explicit kowledge of the umber of the active statios. Ufortuately, (big) errors ca be itroduced if Pgive_ up () c is ot a cotiuous fuctio i the rage [0,1]. For example i the case i which all users have the same threshold, say c, Pgive_ up () c is: 0 c < c Pgive_ up ()= c (13) 1 c c I this case, give a value x (0<x<1), ay choice of the c value itroduces a big approximatio with egative effects o the overall system behavior. If the AP aouces a price c, such that c < c, the system remais cogested (as o statio gives 5 This umber icludes also the statios that recoect after a give-up period.

8 up) ad the QoS remais low. O the other had, by aoucig a price c, such that c c, all statios give up ad the system utilizatio, ad hece the provider reveue, drops to zero. The problem poited out i the previous example occurs wheever Pgive_ up () c is defied oly for a (small) subset of the price values. Hece give a value x it may ot exist a c value that satisfies Pgive_ up ()= c x but we have two possible approximatios, c low ad c up, such that: Pgive_ up ( clow )= xlow, Pgive_ up ( cup)= xup ad xlow < x < xup. To elimiate (or at least to reduce this error) we propose to use selective aoucemets to a radomly selected umber of users. Specifically, the algorithm operates as follows: - the AP selects the price c up ; - for each statio that is curretly coected, the AP aouces the ew price accordig the followig probability distributio: x x Pr { c = clow}= 1, Pr{ c = cup}= xup xup For example, i the 0-1 threshold case (see Equatio ()), x up = 1, ad if we wish to obtai a x = 040. reductio i the umber of active statios, the AP aouces the high cost, cup = c, to x x up = 040. of the statios. This policy based o selective aoucemets to radomly selected statios may be questioable i a real Wi-Fi hot spot. However, the focus of this work is to provide some theoretical foudatios to defie price-based cogestio cotrol policies i a Wi-Fi eviromet. The discussio of how such a policy ca be implemeted i a real eviromet is for further studies. I the followig sectios we show the effectiveess of PCC policy by simulatig the performace of a Wi-Fi etwork with or without the PCC policy. 4. PCC Performace Aalysis I this sectio we will extesively validate the capacity of the PCC policy to drive a Wi-Fi etwork close to the optimal operatig poit. We ivestigate the performace of a Wi-Fi hot spot i which there are M statios. A statio ca be either active, or i the sleep state. A active statio accesses the chael wheever it has a frame to trasmit. A statio eters the sleep state whe the AP aouces a trasmissio price that is too high for the statio to accept. Specifically, i our experimets we radomly assiged to each statio a threshold for the maximum price it accepts to pay. Whe the price exceeds its threshold a statio eters i the sleep state, ad all packets i its queue are discarded. A statio remais i the sleep state for a expoetially distributed radom time. The average sleep time is the same for all statios ad it is equal to 1 sec. Whe the statio wakes up it starts to fill its queue agai. I our study, the maximum price accepted by a statio is sampled from a trucated (to zero) ormal distributio with average 50 ad a stadard deviatio equal to 10, if ot metioed otherwise. We perform two types of experimets: saturated ad ot-saturated. I saturated mode each statio always has a oempty queue, ad therefore it is always willig to trasmit. I ot-saturated mode, each statio geerates ew packets to trasmit accordig to a Poisso distributio. The rate of the Poisso distributio is selected such that a statio's badwidth requiremet is oly a small fractio of the chael capacity. A large umber of statios is therefore required to saturate the chael. Hece, i the ot-saturated model, each statio alterates betwee ON (i.e., with packets to trasmit), ad OFF (i.e., o packets to trasmit) states. The performace idices computed via simulatio, have bee estimated with the idepedet-replicatio techique. The deviatio amog the estimates achieved i the differet rus are geerally very small, ad hece the error-bars are ofte ot visible. As said at the begiig of Sectio 3, the PCC policy may operate at differet time istats. Hereafter, we aalyze two differet possibilities: updatig at the virtual trasmissio time boudaries (deoted as Tv), ad periodic updatig (deoted as Tp). The periodic updates occur every msec, i.e., a typical value for Wi-Fi beaco sigal frequecy. 4.1 Log packets The mai target of this performace study is to ivestigate the relatioship betwee the chael utilizatio ad the etwork cogestio. I our experimets, several cogestio levels are simulated by ruig a set of experimets with differet M values. As explaied i Sectio 2, the MAC data frame cotais i additio to the MAC hdr, a variable legth data payload, hereafter referred to as packet. The results preseted i this sectio are obtaied with packets sizes sampled from a ormal distributio with a average packet size of 1500 bytes (value typical for wired etworks). The miimum packet size is 0, while the maximum size of 2130 bytes is below the limit of the Wi-Fi stadard, i.e., 2312 bytes. The effectiveess of the proposed PCC policy is show i Figures 3 ad 4. These figures show the chael utilizatio levels achieved by adoptig the PCC system, ad compare this idex with the utilizatio levels achieved with the legacy Wi-Fi approach (o pricig). The results show that the PCC policy sigificatly icreases the Wi-Fi chael utilizatio as the etwork cogestio icreases. Simulative rus performed i our ot-saturated sceario show that for etworks with less tha M=15 statios the price mechaism is ever active, ad the performace with or without the PCC

9 policy are similar. After this cogestio level, the PCC mechaism becomes more ad more effective. I this case there are ot sigificat differeces betwee the Tv ad the Tp policies. O the other had, results uder saturated coditios show iterestig ad relevat differeces betwee the Tv ad the Tp policies. Specifically, we observe that i few ulucky rus the PCC policy with the Tv updatig eters i a deadlock state. I detail, for small M values, it may happe that durig a ru all the statios give up due the price aouced by the access poit. I this case, a virtual trasmissio iterval ever eds (there are o successful trasmissios), ad hece o ew price aoucemet will occur. This is the reaso of the very wide error-bars that appear i Figure 4 associated to Tv plots. The deadlock problem caot occur with the Tp updatig policy, ad hece i this case the error-bars are always very small Chael Utilizatio "No-Pricig" "Tp-Pricig" "Tv-Pricig" Number of statios Figure 3: ot saturated (log packets) "No-Pricig" "Tp-Pricig" "Tv-Pricig" 0.8 Chael Utilizatio Number of statios Figure 4: saturated (log packets)

10 4.2 Mixed-packets I the previous sectio we assumed a traffic made of log messages oly. As show by TCP-traffic studies [Ste 94], o a byte cout basis, 90% of the traffic is made up of maximum size packets while the remaiig 10% cosists of very short packets. This meas that, o a packet-cout basis about 50% of the packets are log messages (up to 1500 bytes), ad the remaiig 50% are short packets 40-byte log. Hece, i a more realistic eviromet the message legth distributio should be bimodal. For this reaso, hereafter we evaluate our system by assumig a mixed distributio of the packet legths. I detail, we assume 50% log messages ad 50% short messages. Short messages have a 2-slot costat legth (about 40 bytes of payload). The legth of log messages is sampled from the same ormal distributio used i Sectio Chael Utilizatio "No-Pricig" "Tp-Pricig" "Tv-Pricig" Number of statios Figure 5: ot-saturated sceario (mixed traffic) 0.8 "No-Pricig" "Tp-Pricig" "Tv-Pricig" Chael Utilizatio Number of statios Figure 6: Saturated sceario (mixed traffic) From a qualitative stadpoit, Figures 5 ad 6 cofirm the behavior observed with log packets, oly:

11 i) the PCC policy is effective whe the etwork cogestio icreases; ii) the Tv ad Tp updatig policies have the same asymptotic (with respect to M) behavior; iii) the Tp policy is always deadlock free, while the Tv policy may produce deadlocks both i the saturated, ad i the ot-saturated coditios. These evets are highlighted by wide error-bars o the Tv plots ad correspod to deadlocks occurred i some simulative rus. 5. Coclusios I this paper we proposed ad evaluated a Price-based Cogestio Cotrol (PCC) policy that ca be applied to dyamically cotrol the etwork cotetio level i a Wi-Fi hot spot. The basic idea is to cotrol the traffic iside the hot spot by determiig the access cost as a fuctio of the curret load i the hot spot. By icreasig/decreasig the perpacket trasmissio cost, the hot spot operator ecourages/discourages her/his customers. I the paper we developed a theoretical framework to compute the access cost to maitai the hot spot i its optimal operatig poit, for ay load coditio. Via simulatio, we evaluated the performace of a Wi-Fi hot spot with or without the PCC policy. The performace aalysis was carried out both i saturated ad ot-saturated coditios. Results obtaied show that the PCC policy is effective for all the etwork ad traffic cofiguratios aalyzed. I detail, we cosidered two implemetatios of PCC depedig o the evet that triggers the otificatio of the ew price to the users: periodic ad virtual-trasmissiotime updatig. The results show that the periodic updatig policy is robust while the virtual-trasmissio-time updatig may geerate deadlock coditios. Several assumptios used i our implemetatio of the PCC policy may be cosidered urealistic (e.g., detailed kowledge of the chael status i terms of idle ad collisio periods) or too complex (e.g., the per-packet trasmissio cost) for a real settig. It is worth poitig out that the mai scope of this paper is to demostrate the feasibility of a price-based policy for cogestio cotrol i a Wi-Fi hot spot. I this sese, our results costitute a theoretical foudatio for price-based cogestio cotrol i Wi-Fi etworks. The defiitio of a realistic implemetatio of a PCC policy is beyod the scope of this paper ad it represets the ext step of this research activity. Refereces [BCG00b] R. Bruo, Coti M., E. Gregori, "Optimal Capacity of p-persistet CSMA Protocols", IEEE Comm. Letters (to appear). [CCG00a] F. Cali', Coti M., E. Gregori, "Dyamic Tuig of the IEEE Protocol to Achieve a Theoretical Throughput Limit", IEEE/ACM Trasactios o Networkig, Volume 8, No. 6 (Dec. 2000), pp [CCG00b] F. Cali', Coti M., E. Gregori, "Dyamic IEEE : desig, modelig ad performace evaluatio", IEEE Joural o Selected Areas i Commuicatios, 18(9), September pp [CG02] M.Coti, E. Gregori, Optimizatio of badwidth ad eergy cosumptio i Wireless Local Area Networks i Performace Evaluatio of Complex Systems: Techiques ad Tools (Performace 2002 Tutorial Lectures), LNCS 2459, M. Calzarossa, S. Tucci (Editors), pp [Cro02] J. Crowcroft, R. Gibbes, S. Hailes, BOURSE - Broadbad Orgaisatio of Uregulated Radio Systems through Ecoomics", [IEEE802] Official Homepage of the IEEE Workig Group, [IEEE11b] IEEE stadard b-1999, Wireless LAN Medium Access Cotrol (MAC) ad Physical Layer (PHY) Specificatios: Higher-Speed Physical Layer Extesio, i the 2GHz Bad. [Ste94] W.R.Steves. TCP/IP Illustrated, Volume 1: The Protocols, Addiso-Wesley, Readig, MA, 1994.