Packet Fragmentation in Wi-Fi Ad Hoc Networks with Correlated Channel Failures

Transcription

1 Packet Fragmentation in Wi-Fi Ad Hoc Networks with Correated Channe Faiures Andrey Lyakhov Vadimir Vishnevsky Institute for Information Transmission Probems of RAS B. Karetny 19, Moscow, , Russia E-mai: {yakhov, Abstract In this paper, we present an anaytica method for estimating the saturation throughput of an ad hoc LAN caed aso the Wi-Fi (Wireess Fideity) LAN in the presence of noise distorting transmitted frames. This is the first method that aows studying anayticay the Wi- Fi network performance with consideration of correated channe faiures usuay inherent to wireess channes. With the study, we consider the possibe packet fragmentation that can be adopted to reduce the performance waste caused by noise-induced distortions. In addition to the throughput, our method aows estimating the probabiity of a packet rejection occurring when the number of packet transmission retries attains its imit. The obtained numerica resuts of investigating Wi-Fi LANs by the deveoped method are vaidated by simuation show high estimation accuracy as we as the method efficiency in determining the optima fragmentation threshod. I. Introduction IEEE [16] is one of the most popuar technoogies for wireess ad hoc mobie networking. The fundamenta access mechanism in the IEEE protoco is the Distributed Coordination Function (DCF), which impements the Carrier Sense Mutipe Access with Coision Avoidance (CSMA/CA) method. In eary studies, the DCF performance was evauated either by simuation (e.g., [1]) or by approximate anaytica modes [5], [8] based on assumptions simpifying consideraby the DCF access rues. The DCF was studied in depth in [2], [4], [12], where anaytica methods were deveoped for evauating the performance of wireess LANs in the saturation conditions, when there are aways queues for transmitting at every wireess LAN station. This performance index caed the saturation throughput in [2] was evauated in the assumption of idea channe conditions, i.e., in the absence of noise, causing the throughput overestimation. There may be different noise sources: other devices ocated in the LAN neighborhood operating on the same icense-free frequency b, mutipath fading, co- /adjacent channe interference, etc. (Detai arguing of noise sources can be found in [15], for exampe.) In [9], [10] [13], we have deveoped the methods of [2], [4], [12] to study the infuence of noise on the Wi-Fi LAN performance, assuming channe faiures (that is, noise-induced distortions) uncorreated, for instance, in case of a channe adding white gaussian noise. However, it is known (e.g., see [14] [17]) that the wireess-medium behavior is better characterized by the Gibert mode [6] representing a two-states Markov chain. There are Good Bad states, which differ in the Bit Error Rate (BER) being constant in each state. Obviousy, according to the mode, channe faiures caused by noise infuence are correated, this correation makes hard the Wi-Fi network performance anaysis, forcing previous investigators of the probem to adopt simuation (see [3], for instance). Nevertheess, in this paper, we succeed in studying anayticay the performance of the Wi-Fi network with correated channe faiures, assuming that stochastic sojourn times in Bad Good states are distributed exponentiay. Further, in Section II we briefy review the DCF operation in saturation noise. In Sections III V, we study a fragmented packet transmission process in a Wi-Fi ad hoc LAN with correated channe faiures deveop a nove anaytica method of estimating the saturation throughput the probabiity of a packet rejection occurring when the number of packet transmission retries attains its imit. In Section VI, we give some numerica research resuts of LAN performance evauation. These resuts obtained by both our anaytica method simuation aow us to vaidate the deveoped method to show /04/$20.00 '2004 IEEE 204

2 ' ' Fig. 1. asic Access Mechanism s - S FS, b.s - backoff sots how the correation of channe faiures affects the LAN performance packet fragmentation efficiency. Finay, the obtained resuts are summarized in Section VII. II. DCF in Saturation Now we briefy outine the DCF scheme, considering ony the aspects that are exhibited in saturation. This scheme is described in detai in [16]. Under the DCF, data packets are transferred in genera via two methods. With the Basic Access mechanism shown in Fig. 1, a station confirms the successfu reception of a DATA frame by a positive acknowedgment ACK after a short SIFS interva. The optiona Request-To-Send/Cear- To-Send (RTS/CTS) mechanism, where an inquiring RTS frame a granting CTS frame anticipate the DATA transmission, is not considered in the paper, since previous studies [1], [10], [13] have shown that it is efficient ony in rare cases, when the number of active stations is very arge (more than 20). After a packet transfer attempt, the station passes to the backoff state after a DIFS interva if the attempt was successfu (i.e., there was no coision, a frames of a packet were transferred without noise-induced distortions) or after an EIFS interva if the attempt faied. Further, we use the notation δ, δ d, δ e for SIFS, DIFS, EIFS intervas. The backoff counter is reset to the initia vaue, which is caed the backoff time, measured in units of backoff sots of duration σ, chosen uniformy from a set (,..., ). The vaue, caed the contention window, depends on the number r of attempts performed for transmitting the current packet: W ( r ) W ( r ), where { W0 2 nr for r r W max for r r, i.e., is equa to the minimum W 0 before the first attempt, then is doubed after every faied attempt of the current packet transmission, reaching the maximum W max W 0 2 r. Backoff interva is reckoned ony as ong as the channe is free: the backoff counter is decreased by one ony if the channe was free in the whoe previous sot. Counting the backoff sots stops when the channe (1) %()%+* # $ &% ' # $4 " #! " %(,%.- %(,%/-! " %576 %576 " # " Fig. 2. Fragmented packet transmission becomes busy, backoff time counters of a stations can decrement next time ony when the channe is sensed ide for the duration of σ δ d or σ δ e if the ast sensed transmission is successfu or faied, respectivey. When the backoff counter attains its zero vaue, the station starts transmission. In the course of transmission of a packet, the transmitting station counts the number s of retries that is imited by s, the current packet is rejected when s attains the imit. After the rejection or success of a packet transmission, the next packet is chosen (we consider the saturation case) with the vaues of r s equa to 0. For reducing the infuence of noise, the stard [16] recommends subdividing a packet onger than a fragmentation threshod L f into fragments of size L f (except for the ast fragment). Thus, a packet is transferred as a continuous chain of DATA frames, which contain sequentia fragments are interspaced with ACK frames SIFS intervas (see Fig. 2). If a fragment (fragment 2 in Fig. 2) or an ACK frame is distorted, the station passes to the backoff state, advancing the retry counters r s by one, thereafter the packet transmission is resumed precisey with this distorted fragment. Thus, the transmission of a packet can be considered as a transfer of one or severa continuous chains of frames (there are two chains in Fig. 2), these chains are separated by backoff intervas. Ony the DATA frame being the first in a chain can be invoved into coisions, whie subsequent DATA frames as we as a ACK frames are not affected by coisions, because a other stations hear the transmission of previous DATA ACK frames defer from their attempts. Notice that, in contrary to the r - counter referring to a whoe packet, the s-counter refers to a fragment is zeroed after the fragment transmission success. III. Throughput Evauation Let us consider a sma-size Wi-Fi ad hoc LAN of statisticay homogeneous stations working in saturation. In fact, we mean by not a number of a stations of the LAN, but a number of active stations, whose queues are not empty for a quite ong observation interva. By statistica homogeneity of stations, we mean the foowing: 205

3 (i) the engths of packets (in bytes) chosen by every station from the queue have an identica probabiity distribution {,,..., L}; (ii) a stations adopt the same fragmentation threshod L f ; (iii) since the distance between stations is sma, we negect the propagation deay assume that there are no hidden stations noise occurs concurrenty at a stations. The ast assumption impies that a stations sense the common wireess channe identicay. Aso, to describe the channe state change, we adopt the two-states Gibert mode [6] modified as foows: the channe stays in state (, ) during a time interva distributed exponentiay with parameter λ. The channe states differ in BER. More precisey, in state, BER is equa to µ 8 µ 8 with transmitting a PHY -byte header the other frame part, respectivey, an f- byte frame transmitted entirey with the channe state is distorted with probabiity exp{ µ µ f}. We have to adopt different BERs, since PHY headers are usuay transmitted with a ower channe rate, but more reiabe coding moduation scheme. The channe state change rates λ 0 λ are assumed to be not too high, so that no more than one state change can happen during a frame transmission or an interframe space. As in [2] [13], et us subdivide the time of the LAN operation into non-uniform virtua sots such that every station changes its backoff counter at the start of a virtua sot can begin transmission if the vaue of the counter becomes zero. Such a virtua sot is either (a) an empty sot in which no station transmits, or (b) a successfu sot in which one ony one station transmits, or (c) a coisiona sot in which two or more stations transmit. As in [2], [4] [13], we assume that the probabiity that a station starts transmitting a packet in a given sot does not depend neither on the previous history, nor on the behavior of other stations, is equa to τ, which is the same for a stations depends ony on the current channe state. Hence the probabiities that an arbitrariy chosen virtua sot starting, when the channe is in state, is empty ( e ), successfu ( s ), or coisiona ( c ) are e ( τ ), s τ ( τ ), c e s. (2) With every packet transmission attempt, a chain of data frames is tried to be transferred, the first frame of the chain containing the first fragment which has not been transferred correcty yet. So et us associate every packet transmission attempt starting, when the channe is in state, with a pair (, ), where is the ength (in bytes) of the packet which the chain is reated to is the number of the packet fragments remaining to be transferred, ca it the (,, )-attempt. Let be the probabiity that an arbitrariy chosen packet transmission attempt carried out when the channe is in state is an (,, )-attempt. The throughput S is defined as the average number of successfuy transferred payoad bits per second. Obviousy, S 0 ψ S, (3) where S is the throughput observed when the channe is in state ψ λ (λ 0 λ ) is the time fraction that the channe spends in state. (Here in what foows, with with.) Note that we shoud count the transferred payoad bits ony after a successfu competion of transmitting a whoe packet, but not after each fragment transmission, because a packet transmission process can end with the packet rejection in spite of the fact that some fragments of the packet can be transferred successfuy. Thus, simiary to [9], [10], the throughput S is determined by the formua S L s T s K() 8Q 0 (, ), (4) where T s e σ st s c T c, T s T c are the mean durations of a virtua sot, the successfu coisiona sots, respectivey, starting when the channe is in state. K( ) is the number of fragments, which the packet of ength is subdivided to, that is, K( ) is the minima integer not ess than L f. Q 0 (, ) is the probabiity that an (,, )-attempt carried out in a successfu sot competes successfuy the whoe packet transmission. The duration of the coisiona sot is the sum of the transmission time of the ongest frame invoved in the coision the EIFS interva. Let d( ) H H MAC 8 V be the transmission time of a DATA frame incuding a fragment of ength PHY MAC headers transmitted in time H H MAC, respectivey, where V is the channe rate (in bits per a second) adopted for a frame parts except of the PHY header transmitted with rate V. Then the duration of a coision, where exacty stations are invoved in, is equa to d ( ) δ e if ( ony if): (i) each of these stations tries transmitting a fragment of ength 0 {,..., }, (ii) at east one of these stations transmits a fragment of ength. So the mean duration T c of such coisions starting, when the channe is in state, is: L f T c [ d ( ) δ e ][D(, ) D(, ) ], (5) r where D(, ) is the probabiity that a chain with the first fragment of ength 0 is transferred in the current 206

4 attempt, i.e., D(L f, ) D(, ) r max(r) 0 L f +r with < L f. [Here max ( ) is the integer part of the ratio (L ) L f.] Since ( ) π τ ( τ ) is the probabiity that exacty of stations transmit in a sot starting when the channe is in state, we obtain the foowing formua for the mean duration of a coisiona sot: T c ( c ) 2 π T c, (6) or, after simpe transformations of (5) (6), L f T c δ e ( c ) r d( ) { ( ) ( ) s [D(, ) D(, )]}, (7) where ( ) { τ [ D(, )]}. The case of a successfu sot is more compicated, we consider it in the next section. IV. Successfu Sot Now we study a successfu sot starting when the channe is in state. At the beginning of this sot, ony one station makes an attempt of transmission that is an (,, )- attempt with probabiity. This attempt is concuded successfuy, i.e., with successfu transfer of a whoe packet of ength, with probabiity Q 0 (, ) if none of the frames exchanged between the sender receiver in this process is distorted by noise, that is, L where K() T s L K() T (, )Q 0(, ) (8) T e (,, )[Q (, ) Q (, )], T (, ) ( )[ d (L f ) δ] ACK d( 0 ) δ d is the average duration of successfu (,, )-attempt, ACK is the ACK transmission time, 0 is the ength of the ast fragment. T e (,, ) ( )[ d (L f ) ACK δ] d( ) ( ACK δ)θ (,, ) δ e is the successfu sot average duration taken under the condition that the first ( < ) fragments of the packet remainder were transmitted successfuy, whie the -th fragment which ength is fais. Q (, ) Q (, ) are the probabiities that the condition hods the channe state changes or does not after the sot competion. [Here further, we determine the probabiities separatey for cases of changing (subscript ) not changing (subscript ) the channe state.] At ast, Θ (,, ) is the probabiity that the faiure happens just because of the ast ACK frame distortion. Let us find these probabiities. A transmission of a frame of ength f bytes (incuding the MAC -byte MAC header), which starts when the channe is in state, is competed successfuy with probabiities where I that is, ν 0 (f) exp{ λ (H f V ) µ µ f} ν 0 (f) µ f I H 0 µ h λ H I (f), λ λ exp{ V [µ µ (H )]}, I λ H µh λ H (µ [ h µ h ) ] λ H (µ µ ) with λ H (µ µ ) I λ H µh otherwise. I (f) is defined simiary with the substitution of f for, f V for H, µ µ for µ µ, respectivey. The transmission fais with probabiities ν (f) exp{ λ (H f V )}[ exp{ µ µ f}] ν (f) ν 0 (f) ν 0 (f) ν (f). Now et us consider the process of transmitting a fragment of ength, incuding the possibe receipt of the response ACK frame. Let the process start when the channe is in state. Then it succeeds with probabiities η 0 ( ) ν 0 (f r )[γ ν 0 ( ACK ) γ ν 0 ( ACK )] ν 0 (f r )[γ ν 0 ( ACK ) γ ν 0 ( ACK )] η 0 ( ) ν 0 (f r )[γ ν 0 ( ACK ) γ ν 0 ( ACK )] ν 0 (f r )[γ ν 0 ( ACK ) γ ν 0 ( ACK )], where f r MAC, γ γ exp{ λ δ}, ACK is the ACK ength. The process fais with probabiities η ( ) ν (f r ) ν 0 (f r )[γ ν ( ACK ) γ ν ( ACK )] 207

5 η ( ) η 0 ( ) η 0 ( ) η ( ). Note that the process fais because of DATA distortion with probabiities ν (f r ), if the channe state does not change, ν (f r ), if the channe passes from state to. Now we can find the sought probabiities Q, Q, Θ : Q (, ) ( ) [η ( )γ e η ( ) γ e ] ( ) [η ( ) γ e η ( )γ e ], Q (, ) ( ) [η ( ) γ e η ( )γ e ] ( ) [η ( )γ e η ( ) γ e ], Θ (,, ) Θ (, ) Θ (, ) Q (, ) Q (, ), where γ e γ e exp{ λ δ e }, ( ),,,, η 0 (L f )γ η 0 (L f ) γ η 0 (L f )γ η 0 (L f ) γ. Moreover, Θ Θ are determined simiary to Q Q, substituting the appropriate functions ν with argument MAC for a functions η. At ast, the probabiity of a successfu (,, )-attempt competing the whoe packet transmission is obviousy equa to Q 0 (, ) the channe appears in state competion with probabiities [Q (, ) Q (, )], or upon this attempt Q 0 (, ) ( ) [η 0 ( 0 )γ d η 0 ( 0 ) γ d ] ( ) [η 0 ( 0 ) γd η 0 ( 0 )γd ] Q 0 (, ) Q 0 (, ) Q 0 (, ), respectivey, where γ d γ d exp{ λ δ d }. Thus, we have found a components of (8), the throughput S can be found by (4) (8) if the transmission beginning probabiities τ 0 τ the probabiity distribution { } are known. V. Transmission Rejection Probabiities In this section, we study the process of transmitting a packet of ength by a station. This process starts when the packet is chosen from the queue ends with either the successfu transmission of the packet or its rejection. Let f be the mean numbers of the packet transmission attempts virtua sots in which the considered station defers from transmission during this process, these attempts sots being taken into account ony if the channe is in state at their beginnings. Then τ L f L (f ). (9) To find the distribution { },..., K(), we can use the foowing formua: f L u K(u) v uf uv, (10) where f is the mean number of (,, )-attempts carried out with transferring the packet of ength. Moreover, we wi aso seek the averaged probabiity re of packet rejection taking pace when the s-counter attains its imiting vaue s. This probabiity can be found from the foowing sum: re L re (), (11) where re () is the probabiity of rejecting a packet of ength. Let us start with ooking for f. We can write it in the foowing form: f 0 ψ p F [ s, K(),, ], (12) where ψ p is the probabiity that the packet transmission process starts when the channe is in the initia state s, it is easy to show that ψ p ( Φ ) (2 Φ 0 Φ ), (13) where Φ is the probabiity that, at the end of a packet transmission process, the channe appears in the same state as at the process beginning. Function F ( s,, r, s) represents the mean number of the packet transmission attempts that remain to be perform under the foowing conditions: (i) fragments remain to be transferred; (ii) the station has just passed to the backoff state with the contention window equa to W ( r ); (iii) at the moment, the channe is in state s ; 208

6 (iv) the vaue of s-counter is s. Obviousy, the function is cacuated recursivey: { F ( s,, r, s) 0 q [ r, s ] f 0 1( s < s ) u0 cc Q u (, )]F (u,, 2 cc Q u (, )F (u,, [ [ cc Q c u ( c ) r, s ) 1( ) ]} r, ), (14) where r r with r < r r r otherwise. Here in what foows, we use the Booean function 1(condition) which takes the vaue 1 if the condition within brackets hods. Moreover, in (14), f 0 1( ), cc cc ( τ ) is the probabiity of the current attempt faiure due to a coision, during which the channe passes from state to state u with probabiity Q c u ( c ) Qc ( c ), where Q c ( ) γ e u ( γ e u γe ) exp{ λ } c is the mean duration of this coision. The duration is determined simiary to (5) (7): c δ e ( cc ) π ˆc, where L f ˆc d ( )[D(, )] rr 1 k + d( ){[D(, )] [D(, )] } is the mean duration of such coision, where exacty of other stations coide with the considered stations. After transformations, we have c d( δ ) e cc [ L f ( cc ) rr 1 k + d( )[ ( ) cc ] ( ) ( )]. At ast, q [ r, s ] is the probabiity that the channe appears in state upon competion the backoff time of the considered station, which current contention window is W ( r ), this probabiity is taken under the condition that the channe was in state s at the beginning of the backoff. It is easy to show that q [ r, s ] W ( r ) W (n r) b0 (b), where (b) b,,,, is the probabiity that the channe passes from state to for a virtua sot, during which the given station does not transmit. Obviousy, ˆe λ σ ˆcQ c ( ˆT c ) ˆsQ s, where ˆe, ˆs, ˆc, ˆT c are defined simiary to e, s, c, T c with substitution of for. Moreover, [ ] K() L Q s Q 0 (, ) Q (, ) is the probabiity that the channe does not change its state during a successfu sot. To find, f v, re (), we can aso adopt formua (12), using functions W [ s, K(),, ], F v [ s, K(),, ], R [ s, K(),, ], respectivey, instead of F ( ), whie Φ L [, K(),, ]. (15) Functions W ( ), F v ( ), R ( ), ( ) with arguments s,, r, s are defined, in turn, by (14) modified as foows: For W ( ), f 0 is excuded the item ( r ) W (n r) b W ( r ) W ( r ) (b ) starts the right part of (14). For F v ( ), we substitute f (, ) 1( )1( ) for f 0 the operator 1( ) for 1( ). For R ( ), we repace f 0 by For ( ), f 0 0 (,, s) 1( s s ){ cc cc [η ( ) η ( )]}. is repaced by 0 (,, s) cc Q 0 (, ) 1( s s ){ cc Q c ( c ) cc Q (, )}. Thus, we can cacuate the throughput S the averaged packet rejection probabiity re, using the foowing iterative procedure. Firsty, we cacuate Q (, ) Q 0 (, ) for a possibe,,,,..., K(),,...,. Then we define initia vaues for τ, Ψ p, S, re, with a possibe,, cacuate their modified vaues by (3) (12). If not both reative differences of initia modified vaues of S re are ess than a sma pre-defined imit, then we set new initia vaues of τ, Ψ p, S, re, equa to 209

7 TA. Vaues of protoco parameters Sot time, σ 20 µs PHY header, 24 bytes MAC header, MAC 34 bytes PHY header transfer time, H 192 µs MAC header transfer time, H MAC 25 µs Length of ACK, ACK 14 bytes ACK transfer time, ACK 202 µs SIFS 10 µs DIFS 50 µs EIFS 364 µs Channe rate, V 11 Mbps Retry imit, s 7 Minima contention window, W 0 32 Maxima contention window, W max 1024 hafsums of their modified od initia vaues repeat the cacuation. We do not prove exacty the convergence of this iterative technique due to its compexity. In practice, numerous exampes of adopting the technique with various vaues of Wi-Fi ad hoc LAN parameters have shown that it provides very fast convergence to the soution high speed of cacuating the vaues of estimated performance indices. It takes few seconds to cacuate S re when executing its program impementation on Inte Pentium IV 3.0 GHz. VI. Numerica Resuts To vaidate our mode, we have compared its resuts with that obtained by GPSS (Genera Purpose Simuation System) simuation [11]. The object of our numerica investigations was a saturated Wi-Fi ad hoc LAN consisting of stations. The vaues of protoco parameters used to obtain numerica resuts for the anaytica mode simuation were the IEEE b defaut vaues [7] for the Long Preambe mode summarized in Tabe I. Moreover, the payoad size (in bytes) is samped uniformy from the set {,..., 2 }. In our simuation mode, we took into account of a rea features of the MAC protoco did not adopt the assumptions used in anaytica modeing described in Section III. In each run (it took about an hour on the average) of the simuation mode, we observed vaues of the measured performance indices stopped the simuation when their fuctuations became quite sma (within 0.5%). In Figs. 3 4, we present some resuts of studying the throughput the averaged rejection probabiity with varying the BER fixed 2. Figs. 3b 4b correspond to the case of uncorreated channe faiures, whie the curves in Figs. 3a 4a have been obtained for the case of correated faiures with the foowing channe mode parameters: λ 0 5, λ 5λ 0, µ 0 4 BER, µ 28 BER. (State 0 is assumed to be the Good one.) Here further in the section, we study packet transmission without fragmentation (curves n/f ), with fixed fragmentation (curves f ) when the fixed fragmentation threshod L f 56 bytes is adopted, with optima fragmentation (dotted curves o.f ) when the threshod is chosen optimay, using our method depending on vaues of channe parameters. (The optimization criterion is the maxima throughput.) Moreover, we assume that µ. µ,,. Curves n/f f have been obtained by both our anaytica method (soid curves) GPSS simuation (dashed curves). Let us note a high accuracy of the anaytica mode: the errors never exceed 3% with throughput estimation 6% with rejection probabiity estimation. Further, we see that both throughput rejection probabiity are much more sensitive to the BER growth with uncorreated faiures than with correated ones, especiay in the case of non-fragmented transmission: with very high BER.5 4, we have S. Mbps re. 9 for uncorreated faiures, S 2. Mbps re. 7 for correated ones. Packet fragmentation aows reaxing the noise infuence makes the performance measures shown with correated uncorreated faiures coser to each other: for the same high BER the fixed fragmentation, we have S.9 Mbps re. 33 with uncorreated faiures, S 2.3 Mbps re. 29 with correated ones. The optima fragmentation provides ony sighty improving the throughput (no more than by 10% in the shown cases) with respect to the maximum of throughput vaues obtained with the fixed fragmentation non-fragmented transmission. Now et us investigate in detai how the correation parameters affect the throughput fragmentation efficiency. Fig. 5 shows the throughput versus the Byte Error Rate µ 0 observed in the Good state with fixed BER 4, λ Famiies (a) (b) of curves correspond to (a) λ λ 0 (b) λ 5λ 0, in each of these famiies, soid, dashed dotted curves are reated to non-fragmented transmission, the fixed fragmentation, the optima fragmentation, respectivey. Comparing the considered cases, we see that the ess Good state BER the Bad state mean duration, the higher throughput fragmentation is ess effective. Fig. 6 concuding this investigation shows fragmentation efficiency areas in the foowing cases: (1) BER 4, 5, λ 0 5 ; (2) BER 5 5, 5, λ 0 5 ; (3) BER 5 5, 2, λ 0 5 ; (4) BER 5 5, 5, λ

8 Throughput, Mbps 4 3 n/f f (a) Correated faiures o.f Rejection probabiity (a) Correated faiures n/f o.f f BER BER n/f (b) Uncorreated faiures Throughput, Mbps 4 3 (b) Uncorreated faiures o.f Rejection probabiity n/f 2 f 0.05 o.f f BER BER Fig. 3. Throughput ersus R 10 4 Fig. 4. Rejection probabiity ersus R

9 Throughput, Mbps (a) (b) (1) (2) (3) (4) Fig. 5. Throughput ersus yte rror Rate µ in the Good state Fig. 6. Fragmentation ef ciency areas µ 0 µ 1 ersus λ 0 λ 1 For each of these cases, points ocated upper the reated curve form the fragmentation efficiency area, where fragmentation with optimay chosen threshod provides higher throughput than non-fragmented transmission. First of a, et us note that the bounding curves are not monotonic: fragmentation efficiency increases when the mean Bad state duration becomes both coser to much ess than the mean Good state duration. As we coud expect, the fragmentation efficiency area widens with the BER growth [curves (1) (2)]. Further, we see [curves (2) (3)] that the area widens aso with increasing the number N of active stations. At ast, we shoud mention that a joint growth of the Good Bad durations [curves (2) (4)] does not much affect the fragmentation efficiency area. VII. Concusions In this paper, a continuation of [2], [4], [10], [9], [12], [13], we have deveoped an anaytica method for estimating the throughput of a Wi-Fi ad hoc LAN operating under saturation in the presence of noise. Besides the throughput, our method aows evauating the probabiity of a packet rejection due to attaining the retry number threshod [16]. Unike in [2], [4], [12], [13], this method is usefu in estimating the LAN performance indices under packet fragmentation recommended in the stard [16] for reducing the infuence of noise. Moreover, it is the first anaytica method for Wi- Fi network performance evauation in case of correated faiures inherent to reaistic wireess channes. The faiures correation has been described with the modified two-states Gibert mode [6], where sojourn times in each of channe states are assumed to be exponentiay distributed. According to numerica resuts obtained by both the deveoped method simuation, our method is quite exact: the errors never exceed 3% with throughput estimation 6% with rejection probabiity estimation. This method provides a high speed of cacuating the vaues of performance indices, what has aowed us to perform the exhaustive search of optima fragmentation threshod to show how the fragmentation efficiency depends on faiures correation. As a future research activity, we propose extensions of this method to take into account a possibe presence of hidden stations as we as to consider to optimize a channe rate switching mechanism what promises to be effective in the case of correated faiures. Acknowedgment This work was partiay supported by Russian Science Support Foundation. References [1] G. Anastasi L. Lenzini, QoS Provided by the IEEE Wireess LAN to dvanced Data Appications: a Simuation Anaysis, Wireess Networks, vo. 6, no. 2, pp , [2] G. Bianchi, Performance Anaysis of the IEEE Distributed Coordination Function, IEEE Journa on Seected Areas in Communications, vo. 18, pp , March

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