IEEE Wireless LAN: Saturation Throughput. Analysis with Seizing Eect Consideration. V.M.Vishnevsky and A.I.Lyakhov

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1 Cluster Computing 0 (2002)?{? 1 IEEE Wireless LAN: Saturation Throughput Analysis with Seizing Eect Consideration V.M.Vishnevsky and A.I.Lyakhov a Institute for Information Transmission Problems of RAS B. Karetny 19, Moscow, , Russia fvishn, lyakhovg@iitp.ru The IEEE network technology is the emerging standard for wireless LANs and mobile networking. The fundamental access mechanism in the IEEE MAC protocol is the Distributed Coordination Function. In this paper, we present an analytical method of estimating the saturation throughput of wireless LAN in the assumption of ideal channel conditions. The proposed method generalizes the existing LAN models and advances them in order to take the Seizing Eect into consideration. This real-life eect consists in the following: the station that has just completed successfully its transmission has a better chance of winning in the competition and therefore of seizing the channel than other LAN stations. The saturation throughput of wireless LANs is investigated by the developed method. The obtained numerical results are validated by simulation and lead to the change of the existing idea of the optimal access strategy in the saturation conditions. Keywords: IEEE LAN, Seizing Eect, saturation throughput, rejection probability, analytical method 1. Introduction In recent years, wireless data communications networks have become one of the major trends of the network industry development. According to market research rms data [8], by 2002from30to60million people will be connected to the world Web by wireless lines of communications. Wireless local area networks (WLANs) can be considered as an extension of the wired network with a wireless \last mile" link to attach the large number of mobile terminals. Obvious advantages of WLANs are both their installation simplicity with saving the cost of cabling and the possibility of dynamic change of their topologies (when mobile users become connected or disconnected as well as move inside the WLAN) without any signicant loss of time. The success of wireless networks is connected in many respects to the development This work was partially supported by NATO Science Programme in the Collaborative Linkage Grant PST.CLG \Wireless Access to Internet exploiting the IEEE technology"

2 2 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN of networking products providing multiple access to the wireless environment,and to the availability of appropriate network standards. The IEEE protocol [7] is the emerging standard, which provides detailed MAC and PHY layer specications for WLANs. Several large-scale telecommunication industries (such as CISCO) haverecently launched into the market their software and hardware products regulated by this standard. The fundamental access mechanism in the IEEE MAC protocol is the Distributed Coordination Function (DCF), which is based on a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme. According to this scheme, successive transmission attempts of each WLAN station are separated by a backo interval. Number b of slots constituting the backo interval is random and determined by a binary exponential rule described in section 2. On top of the DCF mechanism, the IEEE standard denes an optional Point Coordination Function (PCF). With the PCF, a coordinating station polls other stations and simultaneously transmits them data. The PCF performance has been studied in [1]. In this paper we concentrate on the DCF performance evaluation. In previous works, the throughput of WLANs controlled by the DCF mechanism has been studied either by simulation (for instance, see [9]) or by approximate models [5,6] based on assumptions simplifying signicantly the real-life backo rule. Most detailed analysis of the DCF scheme has been made in [2,4], where a throughput has been estimated in the saturation conditions when the transmission queue of each station is always non-empty. This performance index, called the saturation throughput in [2], has been evaluated in the assumption of ideal channel conditions, i.e. there is no noise and no hidden stations. As follows from assumptions adopted in [2,4], all WLAN stations work statistically identically and independently from each other. Therefore, at the beginning of each slot all stations have equal chances to start their transmissions. In reality, the station that has just completed successfully its transmission has a better chance of winning in the competition and therefore of seizing the channel than other LAN stations. We call this phenomenon as the Seizing Eect and describe it in the next section. The aim of this paper is to advance the analytical methods [2,4] in order to take the Seizing Eect into account. Further, in section 2 we briey review the DCF operation in the saturation and ideal channel conditions. In section 3, we generalize the saturation throughput evaluation method developed in [4], using some results obtained in [2]. In section 4, we develop a new analytical method for estimating the saturation throughput with taking the Seizing Eect into consideration. In section 5, we give some numerical research results of the saturation throughput of WLANs. These results obtained by both our analytical methods and simulation allow us to validate the developed method and to arrive at a conclusion about the optimal access strategy under the saturation conditions. Finally,

3 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 3 the obtained results are summarized in section DCF in saturation and ideal channel conditions In this section, we briey review the DCF mechanism and describe only those DCF features, which are displayed in the saturation and ideal channel conditions. In particular, we do not consider the optional packet fragmentation, because it is not eective inthe ideal channel conditions with zero bit error rate. For more details of the DCF, see [7]. With DCF, information packets are transmitted in general by the following two techniques. The Basic Access mechanism is used to transmit a short packet whose length does not exceed some limit P. With this mechanism shown in Figure 1, a station which receives a DATA frame containing an information packet replies by a positive acknowledgement ACK just after a short interframe interval SIFS. The Request-To- Send/Clear-To-Send (RTS/CTS) mechanism is applied if the packet length exceeds P. In this case shown in Figure 2, sending of the DATA frame is preceded by the requesting RTS (Ready-To-Send) frame directed to the destination station, which replies by the permitting CTS (Clear-To-Send) frame just after SIFS. Thus, the value of P is chosen as a result of a reasonable trade-o between the RTS/CTS mechanism overhead consisting in transmitting two additional control frames (RTS and CTS) and a gain in the collision duration. Comparing Figures 1 and 2, we see that the collision duration is determined by the length of the longest packet involved in collision for the Basic Access mechanism and only by the length of the short RTS frame for the RTS/CTS mechanism. Each station counts the number n c of attempts performed for transmitting the current packet. If n c reaches some threshold R, the current packet is rejected and (in case of saturation) the next packet is chosen to transmit with resetting the value of n c to zero. Upon nishing a packet transmission attempt, a station transits to the backo state after DIFS interval if this attempt was successful or after EIFS interval in case of collision. With this transition, the backo counter is set equal to initial value b, called the backo time, measured in backo slots (whose size is ) and uniformly chosen from the set (0 ::: w;1). The value w, called the Contention Window, depends on n c : w = CW min 2 nc with n c <mand w = CW max with n c m where CW max = CW min 2 m is the maximum contention window. As shown in Figures 1 and 2, the backo time is counted as long as the channel is sensed idle. The station starts its transmission when its backo time reaches zero. Counting the backo slots stops when the channel becomes busy, and backo time counters of all stations can decrement next time only when the channel will be sensed idle for the duration of +DIFS or +EIFS if the last sensed transmission is successful or failed, respectively. Let us consider the slot immediately following the DIFS interval

4 4 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN closing a successful packet transmission by station A. At the slot beginning, the value of the backo time counter of station A becomes equal to b, while backo time counters of other stations remain at the same values as at this transmission beginning. Thus this slot is non-contention: at this slot beginning, only station A may start its transmission if its backo time b is chosen equal to zero (we callsuch situation as the immediate repetition of transmission). Using the immediate repetition, station A can transmit the whole series of packets, all these transmissions being successful because other stations may not take part in competition. Similarly, at the beginning of the slot immediately following the EIFS interval closing a collision, only the stations involved in the collision may transmit by the immediate repetition of their attempts. These facts related to the Seizing Eect were reported in [3]. Upon successful transmission of a given station, the contention window of this station is minimal and equal to CW min, therefore just the considered station has a better chance of winning in the competition. This is another reason of the Seizing Eect. Further, we analyze the saturation throughput of WLAN consisting of n stations, controlled by the DCF mechanism and operating in the ideal channel conditions. For this aim, in the next section, we rstly generalize the saturation throughput evaluation methods developed in [2,4] where the Seizing Eect is neglected and advance them to take the retry number threshold R into account, estimating also the probability p rej of a packet rejection happening when the threshold is reached. Then, in section 4, we suggest a new method for estimating the saturation throughput, taking the Seizing Eect into consideration. 3. Generalization of existing methods Following Bianchi's approach [2], we adopt the following discrete and integer time scale: t and t + 1 correspond to the beginning of two consecutive virtual slots. Virtual slots are not uniform. Each of them can represent: i) \empty" backo slot when no station transmits, ii) \successful" slot when one and only one station transmits its packet, and iii) \collision" slot when two or more stations try to transmit their packets. According Bianchi's model, at the beginning of each slot, all stations are assumed to start their transmissions with equal probability. In the model of Cal et al. [4], a backo time b is assumed to be independent ofthenumber n c of current packet transmission retries and sampled from a geometric distribution with parameter, i.e. b =0 1 2 ::: with probabilities (1 ; ) (1 ; ) 2 :::.Obviously, these assumptions are equivalent with neglecting the Seizing Eect and therefore assuming that any station may start its transmission at the beginning of any slot. Adopting any of these assumptions, we obtain easily the probabilities of \empty" slot (p e ), \successful" slot (p s ), and \collision" slot

5 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 5 (p c ): p e =(1; ) n p s = n(1 ; ) n;1 p c =1; (1 ; ) n ; n(1 ; ) n;1 : (1) For a given transmission attempt, a collision probability is independent of the transmitted packet size. Hence, we can assume that the time sizes t s and t c of \successful" and \collision" slots are in turn independent of the retry number (or retry numbers, in case of collision) and determined only by a probability distribution function F () for the information packet size P 2 [0 P max ] normalized by thechannel bit rate V c. Proceeding from Figures 1 and 2, we obtain the following average values of time sizes t s and t c : T s =[1; F (P l )](t RT S + t CTS SIFS) + H + E[P ]+t ACK +2 + SIFS + DIFS (2) and T c = E[P ]+ + EIFS (3) where P l is the RTS/CTS threshold P normalized by V c t RT S t CTS t ACK and H are the times sizes required for transmitting RTS, CTS and ACK frames as well as the header of DATA frame, respectively is the propagation delay proposed the same for all pairs of stations E[P ] is the mean value of P nally, E[P ] is the mean collision duration, i.e. the average time size required for transmitting the longest frame involved in collision. The last value is obviously dened by formulae E[P ]=p ;1 c n k=2 n k! k (1 ; ) n;k E[P jk] (4) where E[P jk] is the mean collision duration in the case of k colliding stations. To nd E[P jk], we adopt the Bianchi's assumption [2]: i) the distribution function F () isthe same for all stations, ii) there exists a derivative f(p )=df=dp for all P 2 (0 P max ). Then E[P jk] =k (Z Pl 0 Z Pmax (x + H)f(x)F (x) k;1 dx + t RT S f(x)f (x) k;1 dx Z Pl = t RT S +(H + P l ; t RT S )F (P l ) k ; F (x) k dx: (5) 0 Substituting (5) in (4), we obtain after a simple transformation where E[P ]=t RT S + p ;1 c ( P l (H + P l ; t RT S )Z n (P l ) ; Z Pl 0 Z n (x)dx Z n (x) =[1; + F(x)] n ; (1 ; ) n ; nf (x)(1 ; ) n;1 : ) ) = (6)

6 6 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN Thus, we have determined the mean slot time sizes for all kinds of virtual slots. Similarly to [4], let us consider the time interval t v between two consecutive successful transmissions. More precisely, this interval referred to as a virtual transmission time is between the time instances when two consecutive DIFS intervals nish. Then the saturation throughput S is dened as follows: S = V ce[p ] E[t v ] (7) where E[t v ] is the mean value of t v. In general, a virtual transmission time t v can consist of l =1 2 :::::virtual slots, where the last slot is \successful", k =0 ::: l; 1 slots are \collision" ones and l ; 1 ; k slots are \empty", i.e. t v = T s + kt c +(l ; 1 ; k). Then E[t v ]=p s 1 l;1 l=1 k=0 Calculating this sum, we obtain [T s + kt c +(l ; 1 ; k)] l ; 1 k! p k c pl;1;k e : E[t v ]=T s + p c p s T c + p e p s : (8) An equivalent expression was derived in [4]. As follows from formulas (1), (7) and (8), to nd the saturation throughput S, it remains to determine the transmission probability. Tosolve this problem, we modify the algorithm proposed in [4] in order to generalize the values of backo rule parameters and to take into account of the retry number threshold R. Assuming a geometric distribution for the backo time b, wehave =1=(E[b] + 1) where E[b] is the average backo time. Since the values of b are sampled uniformly from set f0 ::: w; 1g, the value of E[b] is determined by theaverage contention window E[w] : E[b] =(E[w] ; 1)=2: In detail, the last equality was proved in Lemma 2 in [4]. Hence, =2=(E[w]+1) (9) and we needtondtheaverage contention window E[w]. A station can be at one of R stages where the number i (i =0 ::: R; 1) of stage is equal to the number of current packet retries suered. From stage i where contention window is equal to W i = CW min 2 i if i<mand W i = CW max = CW min 2 m if i m (10) a station transits to stage 0 with its successful transmission and to stage (i +1) mod R when its transmission attempt is failed due to collision. Note that a given transmission attempt is failed with probability p =1; (1 ; ) n;1 (11)

7 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 7 and hence the rejection probability is equal to p rej = p R =[1; (1 ; ) n;1 ] R : Lemma 1. Let the number of attempts n c performed for transmitting the current packet be restricted by threshold R. Then the average contention window is determined by the following formulas: in case of R m and E[w]= otherwise. ( m i=1 E[w]= ( R i=1 p i;1 W i + pr W R ; W 0 1 ; p ) (1 ; p) 2 1 ; p R (12) ) p i;1 W i + W m (i ; m +2)p i + (R ; m +1)W mp R ; W 0 (1 ; p) 2 1 ; p 1 ; p R i=m (13) Proof. Let us consider the process of transmitting a packet from the transmission beginning to either its successful completion or rejecting the packet due to reaching the maximal number R of attempts. During this process, i = 0 1 ::: R collisions may occur. The probability of the case i when exactly i collisions occur, is P coll i =(1; p)p i with i =0 1 ::: R; 1 and P coll R = pr with i = R: Let Wi be the sum of contention windows, which iscounted for the case i during the considered process in a whole. Then the average contention window (per one attempt of transmitting) is equal to E[w] = ( P coll i W i + P coll R W ) = ( Taking into account of (10), we nd the values of Wi : W i = i j=0 (i +1)P coll i + RP coll R ) : (14) W i = W i+1 ; W 0 i < m (15) m;1 W i =(i ; m +1)W m + W i =(i ; m +2)W m ; W 0 i m: (16) j=0 Since denominator of (14) is equal to (1 ; p R )=(1 ; p), we obtain the sought formulas by substituting (15) and (16) into numerator of (14). Solving numerically the system of equations (9) and (11) as well as (12) or (13), we nd the transmission probability and, hence, the saturation throughput S.

8 8 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 4. Seizing Eect consideration To evaluate the saturation throughput, we use the same formula (7) as in section 3. Thus, the problem is to estimate the average size E[t v ] of the virtual transmission time t v between two consecutive successful transmissions, taking the Seizing Eect into consideration. This eect causes the dierence in behavior of the station just completing successfully its transmission (let it be station 1 for deniteness) and other stations. As explained in section 2, station 1 has a better chance of winning in the competition for the channel than other stations. So we call the station 1 as the Privileged Station (or the station being in the privileged state) to distinguish it from other ordinary stations (stations being in the ordinary state). The virtual transmission time estimation. The virtual transmission time interval can be instantly closed by a successful transmission of the Privileged Station (PS) with probability W ;1 0. It happens in the case when the chosen backo time of the PS is found equal to 0. Otherwise, upon proceeding the \empty" backo slot, the PS begins to compete for the channel with other stations. We denote this remaining part of the virtual transmission time interval by t c v. Then E[t v ]=W ;1 0 T s +(1; W ;1 0 )( + E[tc v]) (17) where E[t c v ] is the mean size of interval tc v to be under analysis. With analyzing, we take into account of the possible immediate repetition (see section 2) of a transmission attempt after collision and assume the collision absence with this immediate repetition, i.e. an attempt immediately repeated is successful. Such assumption is based on the following: i) only stations involved in the initial collision can participate in the repeated collision, and ii) in most of cases, the contention windows of these stations are doubled after the initial collision. Therefore the repeated collision probability is less at least in several times than the initial collision probability. We distinguish the following kinds of collisions: O-collision when packets of only ordinary stations collide, and P-collision when the PS is also involved in collision. Similarly to section 3, we divide the interval t c v into virtual slots by following so principle that all ordinary stations may try to transmit at the beginning of each virtual slot. Then we have the following kinds of virtual slots. \Empty" backo slot when no station tries to transmit. Slot of an Ordinary station Success (OS-slot) when either one and only one ordinary station transmits or some station involved in O-collision succeeds by repeating its attempt. Slot of the Privileged or Ordinary station Success (POS-slot) when either only the PS transmits or P-collision is followed by a successful immediate attempt repetition.

9 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 9 Slots of O-collision (OC-slot) and P-collision (POC-slot) not followed by a successful immediate attempt repetition, each of these slots including an \empty" backo slot closing the collision. Let us consider the WLAN state evolution shown in Figure 3. We say the WLAN is at stage j (j =0 1 ::: R; 1) during a current virtual slot if the PS was also at stage j at the beginning of this slot. The start of the competition interval t c v means that the PS being at stage 0 has chosen the backo time b 0 > 0. So the WLAN transits to stage 0 and stays there during k =1 ::: b 0 virtual slots (see Figure 4), the PS transmitting only during the b 0 th slot and only in the case when no ordinary station succeeds during the previous b 0 ; 1 slots. If no one of b 0 virtual slots is found successful, the WLAN transits to stage 1 etc. If the WLAN reaches the stage R ; 1andallb slots of this stage are found unsuccessful, the WLAN returns to stage 0. To describe the behavior of ordinary stations, we assume: i) at the beginning of each virtual slot an ordinary station transmits with probability ii) after collision, an ordinary station repeats immediately (in course of the same virtual slot) its attempt with probability p rep. Let us consider in detail the WLAN evolution at some stage j: see Figure 4. Since the transition to this stage means the lack of immediate repetition of the PS attempt (when b j = 0), the PS backo time b j is sampled uniformly from the set f1 ::: W j ; 1g. Only ordinary stations may transmit during the rst b j ; 1 virtual slots, so each of these slots can be OS-slot, \empty" slot or OC-slot with probabilities n;1 p o s =^p o s + k=2! n ; 1 k (1 ; ) n;1;k kp rep (1 ; p rep ) k;1 = k =(n ; 1)[p rep (1 ; p rep ) n;2 +(1; p rep )(1 ; ) n;2 ] (18) p o e =(1; ) n;1 p o c =1; p o e ; p o s: (19) where ^p o s =(n ; 1)(1 ; )n;2 is the probability that one and only one ordinary station transmits. Let no one of the rst b j ; 1 virtual slots be successful. In this case, whose conditional probability is W j;1 j =(W j ; 1) ;1 (p o e + p o c) i;1 = 1 ; (po e + p o c) Wj;1 (W j ; 1)p o s i=1 the PS transmits in the b j th slot. Therefore, the b j th slot can be POS-slot or POC-slot

10 10 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN with conditional probabilities p s j and p c j =1; p s j, the rst one being dened as follows:! " # n;1 ;1 p p n ; 1 (1 s j =^pp s + k (1 ; ) n;1;k (1 ; p rep ) k ; W W h ;1 h k j + j )kp rep = 1 ; p rep k=1 =(1; W ;1 h j )(1 ; ) n;1 + W ;1 h j +(1; W h ;1 j ) (n ; 1)p rep (1 ; p rep ) n;1 (20) 1 ; p rep where ^p p s =(1;) n;1 is the probability that only the PS transmits, and h j =(j+1) mod R is the number of the backo stage next to j. To obtain the expressions (18){(20) (as well as with further analysis), we neglect the probability of repeated collisions (i.e. collisions of attempts repeated immediately after an initial collision), because a ratio r=i of the repeated collision probability to the initial collision one is quite small: r=i p 2 rep in case of O-collision and r=i W h ;1 j p rep +(1; W h ;1 j )(n ; 2)p 2 rep (these approximations are valid with n 1). Thus, the WLAN transits from the stage j to the next stage (see Figure 3) with probability j = j p p c j : (21) Let L j and L j (j =0 1 ::: R; 1) be the average time sizes of successful stage j closed by OS-slot or POS-slot and unsuccessful stage j closed by transition to the next stage. Then, analyzing the WLAN evolution shown in Figure 3, we obtain: where Q j = 1 E[t c v]= ; j )Q j (L j + Q j=0(1 j + rq R )Q r R r=0 j;1 = j=0 L i Q j = (1 ; j )Q j L j + Q 1 ; Q j + Q R Q R R 1 ; Q R j;1 Y i j =1 ::: R Q0 =0 Q 0 =1: We nd similarly the average number of the PS packet rejections counted during the virtual transmission time: N p rej =(1; W ;1 0 ) 1 r=0 Q r R 8 < 9 = (1 ; j )Q j +(p p ; s ^pp s) Q : r j=0 (22) = 1 ; W ;1 0 1 ; Q R [Q R +(p p s ; ^pp s) Q ]: (23)

11 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 11 It remains to nd the values L j and Lj for all j =0 1 ::: R; 1. We begin with determining the mean slot time sizes for all kinds of virtual slots. For any stage, the \empty" slot size is, and the mean time sizes of OC-slot, OS-slot and POC-slot are equal to t OC = T O c + t OS = T s + T O c p o s ; ^po s p o s and t POC = T P c + respectively. The mean POS-slot time size depends on the stage number j: t (j) POS = T s + T P c p p s j ; ^pp s : In these formulas, Tc O and Tc P are the average time sizes of O-collision and P-collision. Tc O and Tc P are calculated similarly to T c in section 3, i.e. by formulas (3) and (6) providing that (6) is modied as follows: p c and Z n (x) are replaced by 1; p o e ; ^p o s and Z n;1 (x) with Tc O calculation or by 1; ^p p s and ^Z n;1 (x) =[1; + F(x)] n;1 ; ^p p s with Tc P calculation. Using the obtained mean time sizes of virtual slots, we analyze the Figure 4 showing the WLAN evolution at stage j and nd the sought values L j and L j : P Wj;1 L j = T s + po s i=3 g j (i)+p p s j g j(w j )+(p o s ; ^p o s)(1 ; j )Tc O =p o s +(p p s j ; ^pp s) j Tc P 1 ; j where g j (i) = p p s j (24) L j = g j (W j )= j + T P c + (25) t i;1 ce (h ; 1)(p o e + p o W j ; 1 c) h;1 t ce = + po c p o e + p o c h=2 Thus, substituting, rstly, the obtained values L j and L j in (22), and secondly, the value E[t c v] in (17), we nd the mean virtual transmission time E[t v ] and hence (by (7)) the saturation throughput S. However, to apply these formulas, we should know the values of parameters and p rep dening the behavior of ordinary stations. Estimating the probabilities of transmission and immediate repetition for ordinary stations. An evolution of a station being in the ordinary state is shown in Figure 5. Upon transition from the privileged state, the evolution starts from the backo stage j with probability j. A station transits to the next backo stage h j if the current stage j closed by a collision not followed by an immediate repetition of this station attempt. Such collision probability is equal to T O c : q j =(1; W ;1 h j )[1 ; (1 ; ~q)(1 ; ) n;2 ] (26)

12 12 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN where ~q is the probability that the current transmission of the given ordinary station collides with the PS transmission. Thus, we should estimate the probabilities ~q j j = 0 ::: R; 1 before starting to search thevalues and p rep. As ~q estimation, we can use the fraction of virtual slots when the PS transmits, i.e. ~q = N p slot =N slot (27) where N p slot and N slot are the average quantities of such slots and all virtual slots, respectively, which are counted during the virtual transmission time t v. Similarly to (22), we have: N p slot = N slot = j=0 j=0 ( ) (1 ; j )Q j j p p s j + j + RQ R (28) 1 ; Q R 1 ; j 1 ; Q R ( (1 ; j )Q j 1 ; Q R j;1 n j + Q R n i + 1 ; Q R where n j and n j are the average quantities of slots in successful and unsuccessful stages j =0 ::: R; 1, and these quantities are found similarly to L j and Lj : n j = < W j;1 (1 ; j )t ce : po s i=3 9 = n j ) g j (i)+p p s j g j(w j ) n j = g j (W j )=( j t ce )+1: So ~q has been determined. Let us consider the case when the given station is in the privileged state at the stage j before another station transmission, which causes the given station transition to the ordinary state. This transition can happen in the following two cases. Case A. Another station succeeds in that slot when the given station does not transmit. Case B. P-collision happens with an immediate attempt repetition of another station participating in the collision and lack of an immediate attempt repetition of the given station. The probabilities of cases A and B determined under the condition of the successful stage j are equal to A j =(1 ; j)and B j =(1 ; j) where A j =1; j and (see (20)) (29) B j = j (1 ; W ;1 h j )(n ; 1)p rep (1 ; p rep ) n;2 : (30) Upon transitting to the ordinary state, the considered station is at the same stage j in case A and at the next stage h j in case B. Hence, an ordinary station evolution begins from the backo stage j with probability o j = nq j j A + Q j B j = Q i (i A + i B ) (31)

13 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 13 where j is the number of stage previous to j, i.e. j = j ; 1 with j>0 and 0 = R ; 1. Obviously, =1=(E[b ]+1) (32) where E[b ] is the average number of virtual slots (counted per a given ordinary station attempt being not an immediate repetition) when an ordinary station defers its transmission. The values of E[b ] and immediate repetition probability p rep as well as the rejection probability are found by the following Lemma. Lemma 2. Let a station be in the ordinary state. Then the values of E[b ], immediate repetition probability p rep and the average number Nrej o of this station packet rejections (counted from the instance of this station transition to the ordinary state till the station success) are obtained by the following formulas: ( ) ( E[b ]= j v ij a ij + AV ) = v ij i +1+ RV (33) 1 ; V 1 ; V where b 0j = p rep = A = j=0 h=0 j=0 ( j v ij (j+i)mod R ) = N o rej = W h ; 2 V = 2 W j ; 1 ; 2 p o s v ij =(1; q i ) Y h=0 j=0 ( j v ij 1 ; V ) v ij i + (j+i)mod R + RV (34) 1 ; V V 1 ; V + ij q h v 0j =1; q j a 0j = A j Q jb 0j + Wj;2 2 Q j B j Q j A j + Q j B j Wj ; ; (1 ; p o s ) Wj;1 2 p o s p o s Y i+j;1 v ij =(1; q i ) q h a ij = a 0 j + 0 h=j h=j q h 1 Y A i+j; h=0 q h! i+j h=j+1 W h ; 2 2 a ij = a 0 j + (35) ; 1 =[(W j ;1)(1; j )] (36) h=j with i<r; j W h ; 2 2 i+j;r + h=0 W h ; 2 2 with R ; j i<r probability q j was dened in (26) for all j =0 ::: R; 1 j = q j =[(W hj ; 1)(1 ; q j )] is the conditional probability that the given station succeeds at stage j just as a result of an immediate repetition after collision at last, ij = 1 with i R ; j ij = with i + j = R ; 1 and ij = 0 otherwise.

14 14 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN Proof. Let us consider the evolution shown in Figure 5 from the instance when the given station transits to the ordinary state to the instance of the given station successful transmission. In the course of this evolution, we'll count the number B of virtual slots when a given ordinary station defers transmitting, the number of the considered station collisions N c, and the number of its packet rejections N r. Denote by C(j i r) the case when the evolution begins at stage j and i + rr this station collisions not repeated immediately occur during the evolution. (Here j i =0 ::: R; 1 and r =0 1 :::.) The probability of case C(j i r) is ^P [C(j i r)] = j v ij V r : In this case, the last evolution stage with non-zero backo time is stage (j + i) mod R. Just after transition to the ordinary state at stage j accordingly to Case A, it remains to defer transmitting till the next attempt during b 0j = p o W j;1 s (W j ; 1)(1 ; j ) i=2 i k=2 (i ; k)(1 ; p o s) k;2 virtual slots (calculating the sum, we obtain easily (36)). If the station appears at stage j as a result of a collision or transiting to the ordinary state accordingly to Case B, then it defers transmitting in average during b j =(W j ; 2)=2 virtual slots, because, rstly, the backo time is sampled uniformly from the set f1 ::: W j ;1g with lack of an immediate repetition and, secondly, the rst backo slot is not accounted (it is included in the same virtual slot as the last attempt). Therefore, in case C(j i r) wehave B = a ij + ra N c = (j+i)mod R + i + rr and N r = r + ij. Hence, E[b ]= 1 j=0 r=0 (a ij + ra) ^P[C(j i r)]= 1 j=0 r=0 (i +1+rR) ^P[C(j i r)] the probability that the given ordinary station collision is followed by its immediate repetition is equal to p rep = 1 j=0 r=0 (j+i)mod R ^P[C(j i r)]= 1 j=0 r=0 (i + (j+i)mod R + rr) ^P[C(j i r)] and the average number of this station packet rejections is determined by N o rej = j=0 1 r=0 (r + ij ) ^P[C(j i r)]: Calculating the sums for r, we obtain the sought formulas (33) { (35).

15 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 15 To nd the packet rejection probability, we use the following expression: p rej = N rej =(N rej +1) (37) where N rej is the average number of rejections counted between the successive instances of the same station success. Obviously, N rej = N p rej +(1; W ;1 0 ) " Q i ( A i + B i ) # N o rej (38) where N p rej and N rej o were dened by (23) and Lemma 2. Solving the system of equations (26) and (32) { (34) with known ~q and j j = 0 :::, we nd the values of and p rep.thus, to estimate the saturation throughput S, we can adopt the following iterative technique: Step 0. Choose some initial values = 0 and p rep = p 0 rep. Step 1. Calculate the values of ~q and j j =0 ::: R; 1byformulas (27) { (31). Step 2. Find the modied values 1 and p 1 rep byformulas (32) { (34). Step 3. Compare 1 with 0 and p 1 rep with p0 rep. If the dierence of these values exceeds some pre-dened threshold, return to Step 1, choosing ( )=2 and (p 0 rep + p1 rep )=2 as initial values. Step 4. Calculate, rstly, the packet rejection probability by formulas (23), (37) and (38), secondly, the mean stage time sizes L j and Lj byformulas (24) and (25), thirdly, the average virtual transmission time E[t v ]byformulas (17) and (22), and nally,the saturation throughput S by equality (7). We don't prove exactly the convergence of this iterative technique due to its complexity and lack of space. It is clear intuitively that equation (31) has a unique solution, because a growth of transmission probability leads to increasing the collision probability and, hence, to increasing the average number E[b ] of slots anticipating an ordinary station attempt. In practice, numerous examples of adopting the suggested technique with various values of WLAN parameters have shown that this technique provides very fast convergence to the solution and high speed of calculating the values of estimated performance indices. It takes less than a second to calculate S and p rej with running this technique program implementation at Intel Celeron 400 MHz. We shall call further the analytical models developed in sections 3 and 4 as the Initial Model (I-model) and the Seizing Eect Model (SE-model), respectively.

16 16 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 5. Numerical results To validate the Initial Model and the Seizing Eect Model, we have compared their results with that obtained by GPSS 1 simulation [10]. The object of our numerical investigations was a WLAN controlled by the DCF scheme of the IEEE protocol with the FHSS (Frequency Hopping Spread Spectrum) PHY layer. The values of protocol parameters used to obtain numerical results for both analytical models and simulation were the default values [7] for the FHSS PHY layer and summarized in Table 1. In Figure 6, we present some results of this investigation for the following case: the number of WLAN stations n =20, only Basic Access Mechanism is adopted, maximum Contention Window CW max = 1024, which isthedefaultvalue [7], the information packet size (in bytes) is sampled uniformly from the set f0 1 ::: 1250g. In Figure 6, the results are presented as broken lines S(CW min )showing the dependence of the saturation throughput on the minimum Contention Window CW min given in the logarithmic scale. To plot these broken lines, we have connected the adjacent points corresponding to possible discrete values of CW min by direct lines. These broken lines are solid, dashed and dotted for results obtained by simulation, Seizing Eect Model and Initial Model, respectively. Comparing the broken lines in Figure 6, we see a high accuracy of both analytical models with large values of CW min : the errors of Initial Model and Seizing Eect Model do not exceed 4% 1%, respectively, with CW min 16. With decreasing CW min below CW min = 16, the Seizing Eect causes a sharp rise of the Initial Model error and makes it inecient withcw min = 2, while the Seizing Eect Model error remains small and does not exceed 7%. Some results of estimating the rejection probability p rej with n =50andR = 7 are given in Table 2. As we see, the Initial Model works well only with CW min 16 (where its error does not exceed 3%) and becomes inecient with small values of CW min. As opposed to the Initial Model, the error of the Seizing Eect Model does not exceed 3% for all values of CW min except of CW min = 2 where the error reaches to 20% due to quite large probability of repeated collisions. In particular, with the value CW min =16 recommended in [7] for WLANs with the FHSS PHY layer, the errors of Initial Model and Seizing Eect Model are equal to 6.9% and 0.1%, respectively. Let us return to Figure 6. As the Seizing Eect Model and simulation show, each curve S(CW min )hastwo local maximums with CW min = 2 and CW min = CW opt min 1 GPSS (General Purpose Simulation System) was developed by Gordon at IBM. Today there are several of dierent GPSS implementations with partly dierent syntax and semantics.

17 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 17 = 256 in Figure 6). The behavior of these curves is not changed radically with varying the number n of stations and with adapting the RTS/STS mechanism to transmit all packets or some fraction of packets. An increase of n leads to the rise of CWmin opt, while adopting the RTS/STS mechanism makes the curves more smooth. In the case when the threshold R restricting the number of attempts performed for transmitting a packet is equal to its default value [7], i.. R = 7, the saturation (CW opt min throughput maximum is reached at CW opt min = 256. This optimal value of CW min is signicantly larger than the recommended value CW min = 16. The studies conducted in [2,4] have arrived at a similar conclusion, so some mechanisms have been suggested in [3,4] to be on top of the DCF and to increase in fact the standard value CW min =16. In case of R = 15, the saturation throughput is maximalwithcw min =2: S(2) = 0:801, while S(256) = 0:744. Moreover, if we increase CW max together with R, thevalue S(2) becomes very close to the maximal possible throughput value equal to E[P ]V c =T s. This fact is due to the Seizing Eect becomes the most clear with CW min = 2: one of stations seizes the channel for a long time, while other stations are at backo stages with large sizes of their contention windows and so not participate really in the competition. Thus, if we take into account only the throughput value, then, in the saturation condition, the following access strategy is optimal: CW min = 2 and maximal values of R and CW max. However, seizing the channel for a long time by a station seems unfair with respect to other stations. As an index of such unfairness, we suggest the Seizing Probability P Seize dened as follows. Let some station A completes successfully a packet transmission. Then the next station, which will transmit successfully its packet, will be just station A with the Seizing Probability P Seize. Studying the WLAN state evolution shown in Figures 3 and 4, we obtain similarly to (28): P Seize = W ;1 0 +(1;W ;1) j Q j 0 [(1;W h ;1 1 ; Q j )(1;) n;1 +W h ;1 j (1;p rep ) n;1 ]: (39) R j=0 Curves P Seize (CW min ) presented in Figure 7 correspond to the curves S(CW min )in Figure 6. As in Figure 6, the results obtained by simulation and formula (39) are shown by solid and dashed broken lines, respectively. Comparing the curves, we see an excellent accuracy of approximation (39) for all values of CW min : solid and dashed curves are almost coincident. With R =15andCW min = 2, the unfairness index is nearly one: P Seize =0:991, while the unfairness is small with CW min = CWmin opt : P Seize =0:033. Thus, we can now determine the optimal values of protocol parameters providing that the saturation throughput is maximal with restricting the unfairness, i.e. the unfairness index should not exceed some pre-dened threshold P seize : P seize P seize. As numerical investigations show, CWmin opt is the optimal value of CW min with P seize < 0:98.

18 18 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 6. Conclusions In this paper, we have developed two simple analytical models to estimate the saturation throughput of a wireless LAN controlled by thedcfmechanism of IEEE protocol in the ideal channel conditions. The rst model, called the Initial Model, is based on generalizing the existing saturation throughput evaluation methods [2,4] and advancing them in order to take the retry number threshold R into account. The aim of developing the second model, called the Seizing Eect Model, was to take the Seizing Eect into consideration. This real-life phenomenon not considered yet in the known studies of the DCF mechanism consists in the following: the station that has just completed successfully its transmission has a better chance of winning in the competition and therefore of seizing the channel than other LAN stations. In addition to the saturation throughput, our models allow to estimate the probability ofapacket rejection happening when the threshold R is reached. Using the developed analytical models together with simulation, we have investigated numerically the LAN throughput. As comparing the obtained results shown, the Seizing Eect Model provides a high accuracy of estimating the saturation throughput and the rejection probability forallvalues of investigated protocol parameters, while the Initial Model happens to be inecient with small sizes of the minimum Contention Window where the Seizing Eect becomes the most clear. Investigating the saturation throughput, we have arrived at the following conclusion: an access strategy is optimal if CW min = 2 and the values of R and CW max are maximal. However, with such strategy, one of stations seizes the channel for a long time that is unfair with respect to other stations. Therefore, for the cases when it is required to restrict this unfairness, we extend our Seizing Eect Model by anapproximation to estimate the unfairness index. Extensions of the developed models to take into account ofchannel unreliability (because of noises) and a possible presence of hidden stations as well as to consider the real-life situations when trac generated by wireless LAN stations is non-uniform and non-saturating, seem possible and are proposed as a future research activity. In order to tackle new research issues generated by the use of wireless LANs as Internet access networks, we plan also to apply the results of studying the MAC layer for investigating the interaction between this protocol and the TCP/IP protocol stack (i.e. the protocols of Internet). References [1] A.S. Bakanov, A.I. Lyakhov and V.M. Vishnevsky, A Method for Evaluating Performance of Wireless Communication Networks with Centralized Control, Automation and Remote Control 61(4) (April 2000),

19 V.M.Vishnevsky and A.I.Lyakhov / IEEE Wireless LAN 19 [2] G. Bianchi, Performance Analysis of the IEEE Distributed Coordination Function, IEEE Journal on Selected Areas in Communications 18(3) (March 2000), [3] L. Bononi, M. Conti and L. Donatiello, Design and Performance Evaluation of Distributed Contention Control (DCC) Mechanism for IEEE Wireless Local Area Network, Journal of Parallel and Distributed Computing 60(4) (April 2000), [4] F. Cal, M. Conti and E. Gregory, IEEE Wireless LAN: Capacity Analysis and Protocol Enhancement, in: Proceedings of INFOCOM'98 (1998), pp [5] H.S. Chhaya and S. Gupta, Performance Modeling of Asynchronous Data Transfer Methods of IEEE MAC Protocol, Wireless Networks 3(3) (March 1997), [6] T.S. Ho and K.C. Chen, Performance Analysis of IEEE CSMA/CA Medium Access Control Protocol, in: Proceedings of PIMRC'96 (October 1996), pp [7] IEEE Standard , Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specications, IEEE Press (November 1997). [8] S. Saunders, P. Heywood, A. Dornan, L. Bruno and L. Allen, Wireless IP: Ready or Not, Here it Comes, Data Communications (September 1999), [9] J. Weinmiller, M. Schlager, A. Festag and A. Wolisz, Performance Study of Access Control in Wireless LANs - IEEE DFWMAC and ETSI RES 10 HIPERLAN, Mobile Networks and Applications 2(1) (1997), [10] T.J. Schriber, Simulation using GPSS, John Wiley & Sons, 1974.

20 20 V.M.Vishnevsky and A.I.Lyakhov / Figures FROOLVLRQ 6WDWLRQ $ EV EV EV '$7$ '$7$ ',)6 (,)6 6WDWLRQ % EV EV EV $&. '$7$ V ',)6 (,)6 Figure 1. Basic Access Mechanism (s - SIFS, b.s - backo slots) FROOLVLRQ 6WDWLRQ $ EV EV EV 576 '$7$ 576 V ',)6 (,)6 6WDWLRQ % EV EV EV &76 $&. 576 V V ',)6 (,)6 Figure 2. RTS/CTS Mechanism : ρ ρ 6WDJH 6WDJH «ρ # 6WDJH # σ : ρ ρ ρ # 6FFHVVIOWUDQVPLVVLRQ Figure 3. Network state evolution during a virtual transmission time

21 V.M.Vishnevsky and A.I.Lyakhov / Figures 21 6ORW 6ORW 6ORW E M 6ORW E M )URP WKH SUHYLRV VWDJH ZLWK S R H S R H S R H 1 1 «1 S S F M 32& 7R WKH QH[W VWDJH - - M S R F 2& 2& 2& S R F S R F S R V S R V S R V S S V M (QGRIDYLUWDOWUDQVPLVVLRQ Figure 4. Evolution at jth stage )URPSULYLOHJHGVWDWH # )URP T T T # 7R 6WDJH 6WDJH # 6WDJH 6WDJH «6WDJH # ±T ±T ±T # 6FFHVVIOWUDQVPLVVLRQ Figure 5. Evolution of an ordinary station

22 22 V.M.Vishnevsky and A.I.Lyakhov / Figures 0.9 R=7 0.9 R=15 Saturation Throughput, Mbit/s log CW 2 min log CW 2 min Figure 6. Saturation Throughput versus Minimum Contention Window 1.0 R=7 1.0 R= Seizing Probability log CW 2 min log CW 2 min Figure 7. Seizing Probability versus Minimum Contention Window

23 V.M.Vishnevsky and A.I.Lyakhov / Biographies 23 Vladimir M. Vishnevsky is a full professor, and a deputy director and a head of department of the Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia. Since 1971, his principal research interests have been in developing mathematical methods based on the queuing theory for performance analysis and structure optimization of computer and communication systems and networks. His current interests include topological design of large-scale communication networks and performance evaluation of wireless networks. Prof. Vishnevsky received a M.S. degree in computer science from Moscow Institute of Electronic and Mathematics in 1971, and Candidate and Doctoral Degrees in computer science from the Institute of Control Sciences of Russian Academy of Sciences (Moscow, Russia) in 1974 and 1989, respectively. He has written two textbooks in communication network study and design and published more than 100 papers in refereed journals and conferences. He is an associate member of the IEEE and an active member of the New York Academy of Sciences. In addition to serving as a program committee member of various conferences, Prof. Vishnevsky serves as a member of editorial boards of such journals as Automation and Remote Control and Electronica (in Russian).

24 24 V.M.Vishnevsky and A.I.Lyakhov / Biographies Andrey I. Lyakhov is a leading researcher of the Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia. Since 1982, his research interests have been in performance evaluation of parallel and distributed computer and communication systems, including multiprocessors and local area cable networks, using both queuing theory methods and asymptotic analysis of large scale queuing networks. His recent interest is in estimating the performance indices of local and metropolitan area wireless networks. Dr. Lyakhov received a M.S. degree in computer science from Moscow Engineering and Physics Institute in 1983, and Candidate and Doctoral Degrees in computer science from the Institute of Control Sciences of Russian Academy of Sciences (Moscow, Russia) in 1989 and 1996, respectively. He has written a textbookinmultiprocessor study and published about 50 papers in refereed journals and conferences.