Modelling and Analysis of the Distributed Coordination Function of IEEE 802.11 with Multirate Capability

1 Modelling and Analysis of the Distrib uted Coordination Function of IEEE 802.11 with Multirat e Capability F . Danesh garan, M. Laddomada, F . Mesiti, an d M. Mondin Abstract — The aim of this paper is two fold. On one hand, it presents a multi-dimensional Markovian state transition model characterizing the behavior at the Medium Access Control (MA C) layer by includ ing transmission states that account for packet transmission failures due to errors caused by p ropagation through the channel, along with a state characterizing the system when there a re no packets to be transmitted in the queue of a station (to model non-saturated traffic condit ions). On the other hand, it pro vid es a thro u ghput analysis of the IEEE 802.11 protocol at the data link layer in both saturated and non-saturated traffic conditions taking into account the impact of both transmission ch annel and mult irate transmission in Rayleigh fadi ng en vironment. Simulation results closely match the theoretical derivations confirming the effectiveness of the proposed model. I . I N T RO D U C T I O N The IEEE80 2.11 MA C [1] presents a mand atory o p- tion, namely the Distributed Coo rdination Function (DCF), a medium access mechanism based on th e CSMA/CA access method, that h as receiv ed considerably attention in the past years [2]-[14]. A number o f p apers [3]-[5], after the semin al work by Bianchi, have addre ssed th e problem of m odelling the DCF in a variety of traffic load and channel transmission con ditions. Most of them fo cuses on a scenario p resenting N saturated stations that transmit toward a common access po int (AP) under the h ypothe ses that the packet rates along with th e probab ility of transmission in a r andom ly chosen slot tim e is common to all the in volved stations, while error e vents on the tran smitted packets are m ainly due to c ollisions be tween packets belon ging to dif fer ent stations. Modeling of the DCF of I EEE 8 02.1 1 WLANs in unsat- urated tra ffic conditions has been analyzed in a n umber of papers [6]-[10]. In [ 6] the auth ors extend ed the under lying model in or der to consider non- saturated traffic condition s by introdu cing a n ew state, not present in the origin al Bianchi’ s model, accoun ting for the case in which the station queue is empty after succ essful completion of a packet transmission. Paper [7] propo ses an extension of the Bianchi’ s model considerin g a new state f or each backoff stage accou nting fo r the absence of ne w packets to be transmitted, i.e ., in unloaded traffic cond itions. This work was supported through funds provided by PRIN-ICON A project. F . Daneshgaran is wit h ECE Dept., California State Uni versi ty , Los Ange- les, USA. M. Laddomada, F . Mesiti, and M. Mondin are with DELE N, Politecn ico di T orino, Italy . In [11], the authors look at the impact of channel in duced errors an d the received SNR on the achievable thro ughp ut in a system with rate adap tation whereby the tr ansmission rate of the terminal is adap ted based on either direct or indirect measure ments of the link quality . In [12], authors observed th at in multirate networks the aggregate throughpu t is strongly in fluenced b y the b it rate of the slowest c ontendin g station: such a phen omeno n is termed perfor mance anomaly of the DCF of the IEEE 80 2.11 protoc ol. In [14], author s provide DCF mod els for finite lo ad sources with mu ltirate capabilities, wh ile in [13] authors pro pose a DCF mo del for multirate networks an d d erive the saturatio n th rough put. In both previous works, packet errors are o nly due to collisions between dif feren t contend ing station s. In this paper , we substantially extend a previous work propo sed in th e co mpanio n pap ers [8]- [9] co nsidering real channel con ditions, both saturated and n on-satur ated tr affic, and multirate capabilities. As a re ference stan dard, we use network para meters belongin g to the IEEE8 02.1 1b pro tocol, ev en thou gh the proposed mathematical model hold s for any flav or of the IEEE802.11 family o r oth er wireless protocols with si m ilar MAC layer fu nctionality . This pape r is o rganized as f ollows. After a b rief revie w of th e fun ctionalities of the contention window proced ure at MAC layer, section II substantially extends the Markov model initially p roposed by Bianchi, presen ting mod ifications that acco unt fo r transmission errors. Section III provides an expression for the ag gregate thr oughp ut of the link, while Section IV derives the time slot du ration needed f or throughp ut ev aluation. The ad opted traffic mo del is discussed in Sec- tion V. Section VI briefly addr esses th e modellin g o f the physical layer of IEEE 802.1 1b in a variety of c hannel fadin g condition s. I n section VII we present simulation results where typical MA C layer parameters for I EEE802 .11b are u sed to obtain the through put as a function of v ario us system level parameters, an d the SNR un der typical traf fic conditions. I I . M A R K O V I A N M O D E L In a p revious pape r [8], we pr oposed a bi-d imensional Markov model for characterizin g th e behavior of the DCF under a v ariety o f real traf fic conditions, b oth no n-saturated and saturated tr affic load, with packet queues of small sizes, and c onsidered the IEEE 8 02.11 b protocol with the basic 2- way ha ndshakin g mechanism. Many of the basic h ypothe ses are the same as the ones adopted by Bianch i in the seminal paper [2 ]. 2 As a starting po int fo r the deriv ation s which f ollow , we adopt the bi-dimension al model propo sed in the compan ion paper [ 8], app ropria tely modified in order to accoun t for a scenario of N con tending station s each one employing a specific bit rate and a different transmission pac ket rate. For conciseness, we invite the interested reader to refer to [8]- [9] for many details on the considered bi-dimension al Mar kov model. Consider th e fo llowing scenario: N station s transmit to ward a commo n AP , whereby each station, characterized b y an o wn traffic load, can access the ch annel using a da ta rate in the set { 1 , 2 , 5 . 5 , 11 } Mbps dep ending on ch annel cond itions. Any bit rate is associated with a different modulation form at, whereas the basic rate is 1 Mb ps with DBPSK modulatio n (2Mbps with DQPSK if short preamble is used) [1]. W e iden tify a generic station with the index s ∈ S = { 1 , 2 , · · · , N } , where N is the number of stations in the network, an d S is th e set o f station indexes. As far as the transmission data ra te is co ncerned , we define four r ate-classes id entified b y a rate-class identifier r taking values in the set R = { 1 , 2 , 3 , 4 } order ed by increasing data rates R D = { 1 , 2 , 5 . 5 , 11 } Mb ps (as an e x ample, ra te- class r = 3 is r elated to the bit rate 5.5 Mbps). Con cerning control packets and PLCP header transmissions, the basic rate is identified b y R C . The traffic load of the s-th station is iden tified by a packet arriv al rate (P AR) λ ( s ) ev aluated in packets p er seco nd. Up on defining both r ate-classes and traf fic, we can associate a generic station s with a rate-class r ∈ R and a pro per traf fic load λ ( s ) . Therefore , we need to specify , with respect to the model proposed in [8], specific probabilities along with different Markov chains fo r each contend ing station in the network. The two sources o f errors on the transmitted p ackets are both co llisions between p ackets and ch annel ind uced erro rs. In relation to the s -th station in the network , co llisions can o ccur with pr obability P ( s ) col , while transmission errors due to im per- fect chann el c an occur with prob ability P ( s ) e . Notice that P ( s ) e depend s upo n the station rate-class r , which is in turn related to the received Sig nal-T o-No ise ( SNR) (appro priate expr essions will b e p rovided in Sectio n VI f or eac h rate-class). W e assume that collisions and tr ansmission error e vents are statistically indepen dent. In this scen ario, a packet from the s - th station is successfu lly transmitted if there is no co llision (this e vent has pr obability 1 − P ( s ) col ) a nd the packet encounters n o chan nel errors durin g transmission (this ev en t has probability 1 − P ( s ) e ). The probab ility of su ccessful transmission is th erefore eq ual to (1 − P ( s ) e )(1 − P ( s ) col ) , while th e equ iv alent probability of failed tra nsmission is defined as P ( s ) eq = P ( s ) col + P ( s ) e − P ( s ) e · P ( s ) col (1) T o simplify the an alysis, we make the assump tion that the impact of chan nel in duced errors on the p acket head ers ar e negligible becau se of their short len gth with respect to the data payload size [8]. The modified Markov model r elated to th e s -th conten ding station is depicted in Fig. 1. W e consider ( m + 1) different backoff stage s in cluding th e zero-th stage. The max imum ... m,1 m,W -1 m 1 1 1 ... 0,1 1 1 1 ... ... ... ... m-1,1 m-1, W -1 m-1 1 1 1 ... ... P (s) eq 1 /W P (s) eq m /W ... ... P (s) eq m-1 /W P (s) eq m /W ... (1-P ) I,0 (s) (1-q )(1- ) eq (s) (s) P P (s) I,0 0 /W P (s) I,0 q P (s) (s) (1- )/W eq 0 m,0 m-1,0 0,W -1 0 ... I 0,0 Fig. 1. Ma rkov c hain for the contention m odel of a generi c station s in general traf fic condit ions, based on the 2-w ay handsha king technique , consideri ng the effe cts of channe l induced errors. contention windows (CW) size is W max = 2 m W 0 , and th e notation W i = 2 i W 0 is used to define the i th contention window size ( W 0 is th e minimum co ntention window size). A packet transmission is attempted o nly in the ( i, 0) s tates, ∀ i = 0 , . . . , m . In case of collision , or due to the fact that transmission is unsuc cessful b ecause of chann el err ors, the backoff stage is incr emented, so th at the new state can b e ( i + 1 , k ) with unifor m probability P ( s ) eq /W i +1 . Th e co ntention window is suppo sed to b e com mon to all the stations in the network; for this r eason th e station index s is drop ped from the contention m odel d epicted in Fig. 1. In o rder to a ccount fo r n on-saturate d tr affic co nditions, we introdu ced a ne w state labelled I , fo r the following two situations: • Immediately after a successful tr ansmission, the queue of the transmitting station is empty . This e vent o ccurs with prob ability (1 − q ( s ) )(1 − P ( s ) eq ) , whereby q ( s ) is the probab ility that there is at least one packet in the queue after a successful tran smission, • The statio n is in the idle state with an empty queue until a new packet a rriv al in the queue. Probab ility P ( s ) I , 0 represents the probability that wh ile the station resides in the idle state I th ere is at least on e packet arri val, an d a new b ackoff pro cedure is scheduled. W e notice that th e probability P I , 0 of residing in the idle state is strictly related to the ado pted traffic model. I I I . M A R K O V I A N P RO C E S S A N A LY S I S This section focuses on th e evaluation of the station ary state distribution of the Markov m odel pr oposed in the p revious section. The objective is to find the probability that a station occupies a given state at any discrete tim e slot along with the stationary probab ility b ( s ) I of being in the idle state. This mathematical deri vation is at the basis for the deri vation of the p robab ility τ ( s ) that a station will attempt transmission in a random ly cho sen slot time . For the sake of simplifying 3 the no tation, in what f ollows we will omit th e apex ( s ) sinc e the mathematical der iv ations are valid for any c ontendin g station s = 1 , . . . , N . F or future de velop ments, fro m the model depicted in Fig . 1 we note the following relations: b i, 0 = P eq · b i − 1 , 0 = P i eq · b 0 , 0 , ∀ i ∈ [1 , m − 1] b m, 0 = P m eq 1 − P eq · b 0 , 0 , i = m (2) whereby b i,j is the station ary proba bility to be in the state labelled i , j of the Markov chain in Fig. 1 . Let us focu s on the m eaning of the idle state I noted in Fig. 1 to which the stationa ry prob ability b I is associated. It co nsiders both th e situation in whic h a fter a successful transmission there a re no packets to be transmitted in the station queue, and the situation in which the packet q ueue is empty and the station is waiting for ne w packet arriv als. The stationary pr obability o f being in state b I can be ev alua ted as b I = (1 − q )(1 − P eq ) P m i =0 b i, 0 + (1 − P I , 0 ) b I = (1 − q )(1 − P eq ) P I , 0 · P m i =0 b i, 0 (3) Upon employing the probabilities b i, 0 noted in (2), it is straightfor ward to obtain: m X i =0 b i, 0 = b 0 , 0 " m − 1 X i =0 P i eq + P m eq 1 − P eq # = b 0 , 0 1 − P eq (4) By u sing the previous result, (3) simplifies to b I = 1 − q P I , 0 · b 0 , 0 (5) The other stationary probabilities for any k ∈ [1 , W i − 1] follow by r esorting to th e state transition d iagram shown in Fig. 1: b i,k = W i − k W i        q (1 − P eq ) · P m i =0 b i, 0 + + P I , 0 · b I , i = 0 P eq · b i − 1 , 0 , i ∈ [1 , m − 1] P eq ( b m − 1 , 0 + b m, 0 ) , i = m (6) Employing the n ormalization co ndition , af ter som e m athe- matical manipulations, and remember ing (4), it is possible to obtain: 1 = m X i =0 W i − 1 X k =0 b i,k + b I = α · b 0 , 0 + b I (7) whereby α = 1 2  W 0  1 − (2 P eq ) m 1 − 2 P eq + (2 P eq ) m 1 − P eq  + 1 1 − P eq  (8) From (7), the following equation for computatio n of b 0 , 0 easily follows: b 0 , 0 = 1 − b I α (9) Equ. (9) is u sed to com pute τ ( s ) , the prob ability th at the s -th station starts a tran smission in a rando mly chosen time slot. In fact, takin g into accou nt that a packet transmission occurs when th e backof f counter reaches zero, we have: τ ( s ) = m X i =0 b ( s ) i, 0 = b ( s ) 0 , 0 1 − P ( s ) eq = 1 − b ( s ) I α ( s ) (1 − P ( s ) eq ) = (10) = 2(1 − b ( s ) I )(1 − 2 P ( s ) eq ) ( W 0 + 1)(1 − 2 P ( s ) eq ) + W 0 P ( s ) eq (1 − (2 P ( s ) eq ) m ) whereby we r e-introd uced the apex ( s ) since this expression will be used in the following. The collision proba bility P ( s ) col needed to compute τ ( s ) can be found co nsidering that using a 2 -way han d-shakin g mechanism, a p acket fro m a transmitting station en counte rs a collision if in a g iv en tim e s lo t, at least one of the rem aining ( N − 1) stations tran smits simu ltaneously o ne packet. Since each station has its o wn τ ( s ) , the collision probability for the s - th contendin g station depends on the transmission prob abilities of the remaining stations as follows: P ( s ) col = 1 − N Y j =1 j 6 = s (1 − τ ( j ) ) (11) Giv en th e set of N eq uations (1) and ( 10), a non-lin ear system of 2 N equ ations can be solved in order to determin e th e values of τ ( s ) and P ( s ) col for any s = 1 , . . . , N : this is the operating point corresponding to the N station s in t h e network, needed in order to d etermine the aggregate thro ughp ut of the network, defined as the fraction of time the c hannel is used to successfully tr ansmit pa yload bits: S = N X s =1 1 T av P ( s ) s · (1 − P ( s ) e ) · P L (12) whereby the summation is ov er the throughp ut relate d to the N conten ding stations, P L is the average payloa d size, an d T av is the exp ected time per slot defined in the fo llowing. Probabilities inv olved in (1 2) are as fo llows: P ( s ) e is th e PER (or FER) of the s -th station du e to imperfect channel transmissions, a nd P ( s ) s is the probability that a p acket trans- mission from the s -th station is succ essful. I n the next section , we derive the mathematical relations defining both T av and the probab ilities inv o lved in (12). I V . E S T I M A T I N G T H E A V E R AG E T I M E S L OT D U R AT I O N In order to proceed further , we need to ev aluate the a verage time T av spent by a station in any possible state. The average duration T av of a time slot can be evaluated by weigh ting the times spent by a station in a particu lar state with th e probab ility of being in that state. It is possible to note f our kind of time slots. The av er age idle slot duration, identified by T I , in wh ich no st atio n is tran smitting over the channel. The a verage collision slo t duration, id entified by T C , in wh ich more than one station is attempting to acce ss the ch annel. The av era ge duration of the slot du e to erron eous tr ansmissions because o f imper fect chan nel conditio ns, identified b y T E . The average slot duration of a succe ssful transmission, identified by T S . The average idle slot duration. Th e average idle slot duration can be ev alu ated as the prob ability (1 − P t ) that n o station is attempting to g ain the access to the chann el times the duration σ of an em pty slot time. Let P t be the pr obability that the chan nel is busy in a slot because at least one station is transmitting. Th en, it is P t = 1 − Q N s =1 (1 − τ ( s ) ) . The average idle slot duratio n can be defined as T I = P I · σ = (1 − P t ) · σ , 4 where ea ch idle slot is assumed to ha ve duration σ . The average slot duratio n of a successful transmission. Consider a tagg ed station between the N stations in th e un- derlined network, and let s be its index in the set { 1 , . . . , N } . The proba bility that only the s -th tagged station is suc cessfully transmitting ov er th e chan nel can be defined as P ( s ) s = τ ( s ) N Y j =1 j 6 = s (1 − τ ( j ) ) (13) Then, the av er age slot du ration of a successful transmission, which depend s on the rate-class ( r ) of the tagged station, can be e valuated as fo llows: T ( s ) s = H P H Y R ( r ) C + H M AC + P L R ( s ) D + δ + (14) + S I F S + H P H Y + AC K R C + δ + D I F S whereby P L is the a verage payload length, H P H Y and H M AC are, respecti vely , the phy sical and MA C header sizes, τ p is the propag ation delay , DI FS is the dur ation of the Distributed In- terFrame Space, R C is the b asic data rate used f or transmitting protoco l data, an d R ( s ) D is the d ata rate o f the s -th station . W ith this setu p, the average slot duratio n of a successful transmission can b e evaluated as T S = P N i =1 P ( i ) s  1 − P ( i ) e  · T ( i ) s . The average collision slot duration. In a n etwork of stations transmitting e qual length packets with d ifferent da ta rates, the a verage duration T C of a collision is largely dominated by the slo west transmitting stations. Th is p henom enon is called performan ce anomaly of 8 02.11 b, a nd it has been firstly obser ved in [12 ]. As an example, suppo se that a frame transmitted by a station using the rate 1 Mbps (class 1) collides with the packet of a station transmitting at th e bit rate 11 Mbps (class 4). Of cou rse, both f rames get lost wh ile the channel ap pears as busy to th e rem aining sensing stations for the whole dur ation o f the frame transm itted by the low r ate station. Therefor e, fast st atio ns (higher class es) ar e penalized by th e slow stations (low classes), causing a decrease of the throug hput. In or der to evaluate the collision probab ility , we define the class ( r ) collision duration as T ( r ) c = H P H Y R C + H M AC + P L R ( r ) D + AC K timeout , which ta kes in to account the b asic rate R C along with the data rate R ( r ) D of the class ( r ) . For the deriv ations which follow , we co nsider a set of indexes which identify the station s transmitting with the r -th data rate: n ( r ) = { identifiers of stations belon ging to rate- class (r ) } ∀ r ∈ R = { 1 , . . . , N R } such that P N R r =1 | n ( r ) | = N ( | · | is the cardinality o f the embr aced set). W ith this setup, we no tice two different collision s: • intra-class collisions b etween at least two frames belong- ing to the same class r ate ( r ) ; • inter -class collisions between a t least o ne fram e of class ( r ) and at least one frame belongin g to a class ( j ) > ( r ) As far as intra-class ( r ) co llisions are concerne d, th e collision probab ility P ( r ) c 1 can be ev aluated as follows: 8 > > > > < > > > > : 1 − 2 6 6 6 6 4 Y s ∈ n ( r ) (1 − τ ( s ) ) + X s ∈ n ( r ) τ ( s ) Y j ∈ n ( r ) j 6 = s (1 − τ ( j ) ) 3 7 7 7 7 5 9 > > > > = > > > > ; · Y s ∈{ S − n ( r ) } (1 − τ ( s ) ) (15) Notice that the latter is the pr obability th at the stations not belongin g to the same data rate set n ( r ) , do not transmit, times the pr obability th at there are at least two s tatio ns in the same rate class n ( r ) tr ansmitting over the channel. Notice that the first pro duct w ithin brace br ackets acco unts for the scen ario in which th e statio ns with r ate in the set n ( r ) are silent, or there is only a station tran smitting with rate in the set n ( r ) . As a note aside, notice that P ( r ) c 1 = 0 if th ere are no collisions between s tatio ns b elongin g to the same rate class. Follo wing a similar reasoning , the inter-class ( r ) co llision probab ility P ( r ) c 2 can be ev aluated as: 2 6 4 1 − Y s ∈ n ( r ) (1 − τ ( s ) ) 3 7 5 · 2 6 4 1 − N R Y j = r +1 Y s ∈ n ( j ) (1 − τ ( s ) ) 3 7 5 · 2 6 4 r − 1 Y j =1 Y s ∈ n ( j ) (1 − τ ( s ) ) 3 7 5 (16) which considers the scenario in wh ich at least one st atio n of class ( r ) and at least one station be longing to a higher rate class (i.e., ( j ) > ( r ) ) transmit in the same slot time, while all th e other s tatio ns b elongin g to lo wer indexed classes (i.e., ( j ) < ( r ) ) are silent. As a note asid e, notice that P ( r ) c 2 = 0 if there a re n o collisions between stations belo nging to d ifferent rate classes . The total class ( r ) collision pro bability is the su m o f the previous two pro babilities: P ( r ) c = P ( r ) c 1 + P ( r ) c 2 (17) while the average collision slot duratio n can be comp uted considerin g the whole set o f classes r ∈ R along with their collision p robab ilities weigh ted b y the respecti ve d urations: T C = N R X r =1 P ( r ) c · T ( r ) c (18) The av era ge duration of the slot due to erroneous trans- missions. The average du ration of the slot due to err oneou s transmissions can be ev aluated in a way similar to th e one used fo r evaluating T S and T C : T E = N X i =1 P ( i ) s · P ( i ) e · T ( i ) e (19) whereby P ( i ) s is defined in (13), and T ( s ) e is assum ed to be equal to T ( s ) c since the tran smitting station does not re ceiv e the acknowledgment before the en d of th e A CK timeout in the presence o f ch annel erro rs. A verage time slo t duration. Giv en the average slot durations derived in the previous sectio ns, the average du ration of a slot time can b e evaluated as follows: T av = T I + T C + T S + T E (20) 5 T ABLE I P H Y S E T U P O F T H E I E E E 8 0 2 . 1 1 B S TA N D A R D Frequenc y [GHz] 2.4 2.4 2.4 2.4 Bit rate [Mbps] 1 2 5.5 11 Modulat ion DBSPK DQPSK CCK CCK Chips per symbol, C s 11 11 8 8 Bits per symbol, B s 1 2 4 8 Channel band, B w [MHz] 22 22 22 22 Recei ver Sensit ivit y -85 -82 -80 -76 A WGN-[dBm] V . T R A FFI C M O D E L This section p resents the traffic model employed in ou r setup alo ng with the derivation of the key probab ilities q ( s ) and P ( s ) I , 0 shown in Fig . 1. The of fer ed load r elated to each station is characterized by the p arameter λ ( s ) representin g the rate at which p ackets arrive at the s -th station buffer from the upp er layer s, a nd measu red in pkt/s . Th e time between two p acket arriv als is d efined as interarrival time , and its mean value is e valuated as 1 λ ( s ) . One of the most comm only used traffic mo dels assumes that the packet arriv al process follows a Poisson d istribution. The resulting interarrival times are e xp onentially distributed. In the proposed model shown in Fig. 1, we n eed a prob- ability q ( s ) that indicate s if the re is at least one packet to be transmitted in the queue. Prob ability q ( s ) can b e well approx imated in a situation with sma ll buffer size [7], [8], [9] th rough the following r elation: q ( s ) = 1 − e − λ ( s ) · T av (21) where T av is the e xp ected time per slot , useful to relate the state o f th e Markov chain with the actual time spent by a station in each state. Such a time h as b een der iv ed in (20). Under th e h ypothesis of systems e mploying small queues, probab ilities q ( s ) and P ( s ) I , 0 can be appr oximated co nsidering that th e prob ability that at least one pa cket arrives in the queue at the end of a succ essful packet transmission is the same as having at least one packet arriv al in an average time slot du ration. As a result of this simp le appro ximation, it is q ( s ) = P ( s ) I , 0 . Up on remembering (5) and (9), τ ( s ) in (10) can be e valuated as fo llows: τ ( s ) = q ( s )  1 − P ( s ) eq  − 1 q ( s ) ( α ( s ) − 1) + 1 (22) Thoug h simple, this appro ximation prov ed to be quite ef fective for p redicting th e agg regate thr oughp ut th rough simulation . V I . P H Y S I C A L L A Y E R M O D E L L I N G In a scen ario with N contend ing stations ran domly d is- tributed around a comm on AP , throu ghput performan ce d e- pends on the channel conditio ns experienced by any station in the network. Consider a contending station at a distance d from the AP . Gi ven the o ne-sided no ise power spectral density , N o = − 174 dBm | T =273 K , th e r eceived SNR can be evaluated as [1 6]: S N R dB = P ( d ) | dBm − N o − B w | dB − N F (23) T ABLE II T Y P I C A L N E T W O R K PA R A M E T E R S MA C heade r 28 byt es Prop ag. delay τ p 1 µs PLCP Preamble 144 bit PLCP Header 48 bit PHY header 24 byt es Slot time 20 µs basic rate 1Mbps W 0 32 No. back-off stages, m 5 W max 1024 Payl oad size 1028 bytes SIFS 10 µs A CK 14 byt es DIFS 50 µs A CK timeout 364 µs E IFS 364 µs whereby N F is the receiver noise figure (10dB), while P ( d ) | dBm , the power received at a d istance d , corresp onds to P ( d ) | dBm = P tx | dBm − P L dB (24) Based o n FCC regulation s, in the 2 .4GHz ISM band the tran s- mitted power P tx | dBm amount to 20 dBm, or , equ iv alently , 100 mW , wh ile P L dB is the so-called path- loss [16]: P L dB = P L o | dB + 10 · n p log 10  d d 0  whereby P L o | dB = − 10 log 10  G t G r λ 2 (4 π ) 2 d n p 0  . T he path-loss ex- ponen t, n p , depend s on the specific pro pagation environment, and it ranges fr om 2 (free space prop agation) to 3.5-4 for non- line-of-sig ht propagation, o r multi-pa th f ast fading conditions, in indo or en v ironmen ts [16]. Th e SNR per transmitted bit, γ , is defin ed as: γ | dB = S N R dB + 10 lo g 10  C s B s  (25) whereby C s stands f or chips per sy mbol, while B s is the number of bits per transmitted symbol. Bo th C s and B s are summ arized in T able I. BER perfor mance of th e various transmitting modes o f I EEE802 .11b are shown in (26) for Rayleigh fadin g cond itions [15], [ 17]: DBPSK 1 2(1+ γ ) DQPSK 1 2 » 1 − r γ √ 2 2 1+ γ √ 2 2 – CCK-5.5/11 Mbps 2 ( α − 1) 2 α − 1 P α − 1 i =1 ( − 1) i +1 C α − 1 i 1+ i + i · γ (26) whereby α = 4 for 5.5 Mb ps, and 8 for 11 M bps, and C α − 1 i = ( α − 1)! i ! · ( α − 1 − i )! . V I I . S I M U L A T I O N R E S U LT S A N D M O D E L V A L I D A T I O N This section focuses on simulation results for v alid ating the theoretical m odels an d derivations presen ted in the previous sections. W e have developed a C++ simulator mode lling the DCF pro tocol de tails in 8 02.11 b for a specific nu mber of indepen dent transmitting stations. The simulator consider s an Infrastru cture BSS (Basic Serv ice Set) with an AP and a certain nu mber of fixed stations which commun icates only with the AP . For the sake of simplicity , in side each station there are only three fundam ental work ing le vels: traffic model generato r , M A C and PHY laye rs. Traf fic is gener ated fo l- lowing the expo nential d istribution fo r the packet interar riv al times. More over , the M A C layer is managed by a state 6 machine wh ich follows the main directiv es spe cified in the standard [1], namely waiting times (DIFS, SIFS, E IFS), p ost- backoff, b ackoff, basic and R TS/CTS access mode. The typical MA C layer par ameters for IEEE 802.1 1b noted in T a ble II [1] have been used for performan ce validation. For co nciseness, in this p aper we presen t a set of results related to fo llowing scenarios A n umber o f 9 con tending stations are rand omly placed along a cir cle of rad ius R , while the AP is placed at the center of the ar ea. Upon employing Equ.s (23)- (25) with n p = 4 (typical of heavy faded Rayleigh channel condition s), we have chosen a distance R = 20 m in such a way that the SNR per tran smitted b it is above the minimum sensiti vity , specified in T able I, relativ e to the bit rate 11 Mb ps. Suc h stations are in saturated cond itions and have P AR λ = 8 kpkt/s. The p ayload size, assumed to be common to all the transmitting stations, is equal to 1028 bytes. In this scenario, another s tatio n, in the following identified as the s low station, is placed at 4 dif f erent distances fro m the AP in such a way that tran smission occurs with the four bit rates envisaged within the 802.1 1b p rotoco l. The th eoretical agg regate th rough put in this scenario is depicted in Fig. 2 as a func tion of the P A R of the slow station. Curves in b oth subplots have been par ameterized with respect to the bit r ate of the slow station. Simulated po ints are no ted with cross-po ints over th e respec ti ve theo retical curves. The u pper cu rves refer to ideal ch annel co nditions, i.e., P E R = 0 , while the lo wer subplot rep resents a scenario in which the p ackets transmitted by all th e stations are affected by a P E R = 8 · 1 0 − 2 , whic h is the worst-case situa tion related to the minimum sensitivity [1]. Som e considera tions are in order . Both subplots sh ow that the aggregate throug hput is significantly lower than 11 Mb ps even tho ugh all the statio ns transmit at the high est b it rate (con tinuou s cu rve). Mo reover , such a throug hput r educes as a far as the P AR λ slow increases reaching saturation values stro ngly influenced by the rate of the slowest station. A comparative an alysis of the set of cur ves depicted in b oth su bplots reveal the thr oughp ut reduction due to the p resence o f ch annel indu ced erro rs. V I I I . C O N C L U S I O N S In this paper, we ha ve presented a multi-dim ensional Marko- vian state transition m odel characterizing th e DCF b ehavior at the MA C layer of the IEEE8 02.1 1 series of standards b y ac- counting for ch annel induced erro rs and multirate tran smission typical of fading en viron ments, under both non -saturated an d saturated traf fic cond itions. Th e modelling allows taking into consideratio n the impact of chan nel contention in throug hput analysis which is o ften not co nsidered o r it is considered in a static mode by u sing a mean con tention per iod. Theoretical d eriv ations were supported by simulations. R E F E R E N C E S [1] IEEE Standard for W ir eless LAN Medium A ccess Contr ol (MAC ) and Physical Layer (PHY) Speci fications , Novembe r 1997, P802.11 [2] G. Bianchi, ”Performance analysis of the IEEE 802.11 distribut ed coordina tion functio n”, IEEE JSAC , V ol.18, No. 3, March 2000. [3] Ha Cheol Lee, ”Impact of bit errors on the DCF throughput in wireless LAN o ver ric ean fading ch annels”, In Pr oc. of IEEE ICDT ’06 , 200 6. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 4 5 6 x 10 6 λ slow [pkt/s] S [bps] 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 4 5 6 x 10 6 λ slow [pkt/s] S [bps] +1 @ 11 Mbps slow @ 5.5 Mbps slow @ 2 Mbps slow @ 1 Mbps +1 @ 11 Mbps slow @ 5.5 Mbps slow @ 2 Mbps slow @ 1 Mbps PER=0 PER=8 × 10 −2 Fig. 2. Theoret ical an d simulated throughput for the 2-way mechanism as a func tion of the packe t rate λ slow of the slo w statio n, for four differe nt bit rates, shown in the le gends. Simulated points are identi fied by cr oss-m arke rs ov er the respe ctiv e theoret ical curves. [4] Q. Ni, T . Li, T . Tu rletti, and Y . Xiao, ”Saturati on throughput analysis of error -prone 802.11 wireless networks”, W ile y J ournal of W ire less Communicat ions and Mobile Computin g , V ol. 5, No. 8, pp. 945-9 56, Dec. 200 5. 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