Saturation Throughput - Delay Analysis of IEEE DCF in Fading Channel

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1 Saturaton hroughut - Delay Analyss o IEEE 8. DCF n Fadng Channel Zoran Hadz-Velkov Ss. Cyrl and Methodus Unversty Faculty o Electal Engneerng Skoje, Macedona zoranhv@et.uk.edu.k Abstract In ths aer, we analytcally analyzed the act o an error-rone channel over all erorance easures n a tracsaturated IEEE 8. WLAN. We calculated staton s transsson robablty by usng the oded Markov chan odel o the backo wndow sze that consders the rae-error rates and axal allowable nuber o retranssson attets. he rae error rate has a sgncant act over theoretcal throughut, ean rae delay, and dscard robablty. he eak throughut o a WLAN s nsenstve o the axal nuber o retransssons. Dscard robabltes are nsenstve to the staton access ethod, Basc or RS/CS. I. INRODUCION In recent years, IEEE 8. networks are becong a redonant technology or wreless connectvty n local areas. IEEE 8. standard [] has been develoed to rovde hgh bandwdth to oble users n ndoor envronents. However, the rado channel ntroduces sgncant colexty to the desgn and erorance analyss o the WLANs. hs s rarly due to ultath adng, whch roduces hgh error rates, deendng on the channel condtons, sgnal rates, and staton oblty. he IEEE 8. erorance, rarly the throughut, has been studed n nuber o aers both analytcally [-8] and by sulaton, but none o the consder ean rae delays and rae-loss ratos based on an analytcal odel. Further, u to author s knowledge the act o rae-error rates also has not be consdered analytcally. In ths aer, we rovde an analyss o all erorance easures o the IEEE 8. Dstrbuted Coordnaton Functon (DCF) n saturaton n non-deal channel condtons. By extendng the Markov chan odel ro [7], we were able to roduce analytcal solutons or the eak syste caacty, ean rae delays, and dscard robabltes n a saturated WLAN exosed to an error-rone rado channel. II. PERFORMANCE ANALYSIS A. Moded Markov chan odel For the urose o the analyss, we need to deterne the transsson robablty τ o each staton n a randoly chosen slot te. hus, we used the dscrete Markov chan odel ro [7], whch relates only to statons wth ersstuntl-success retranssson strategy n deal channel Bors Sasenovsk Ss. Cyrl and Methodus Unversty Faculty o Electal Engneerng Skoje, Macedona borss@et.uk.edu.k condtons. In ths aer, we extend ths odel by takng the rae-error robablty P nto account. Addtonally, we consder the nte nuber o retranssson attets ( + + ) ater whch the rae s dscarded ro the transt queue and a new rae s adtted n the queue. Let us consder nte nuber o statons n the network, n. In saturatng condtons, ater successul transsson or dscard o a rae, each staton has edately a new rae avalable or transsson,.e. ts queue s always assued to be non-ety. Under such condtons, t s reasonable to assue that ater erorng Carrer Sensng, the staton wll nd the channel occued, re-enter backlog condton, and edately start executng Collson Avodance rocedure,.e. the bnary exonental backo algorth. hus, startng wth the very rst transsson attet, the staton tres to access the channel ater erorng rando backo. he nte-state odel o each staton s reresented by the twodensonal Markov chan o ts backo wndow sze, whch s dected n Fg.. A current state (, k) o a staton s deterned by the current value o ts backo ter k (, W ) ater t suered revous unsuccessul transsson attets (row n Fg. ). Startng wth the very rst transsson attet (backo stage ), the ntal value o the backo ter s unorly chosen n the range between and W. Ater the staton enters backo stage, ts backo ter s rentalzed to a rando value between and W (slots). Ater + + unsuccessul retranssson attets, the rae s droed ro queue. Untl the -th retranssson attet, the axal backo ter W ncreases by actor o, ater whch t s rozen to W untl the + + retranssson when the rae s successully transtted or dscarded,.e. W, W W, +, () where W s the ntal contenton wndow. he backo ter s decreented by n each consecutve slot. However, the slot duraton ders: t s an dle slot - the slot lasts σ µs; the slot s occued - t can be a successul transsson slot, an unsuccessul transsson slot due to rae error, or a collson slot. In the begnnng o a busy slot, each backlogged staton decreents ts ter by one and then the ter s reezed untl the channel becoes dle /3/$7. 3 IEEE

2 / W ( ) / W ( )/W,,,, W, W / W -, -, -,, W-, W- / W / W,,,, W, W through the robablty o staton to be n state (, ) as ollows: b, b,, < + (4) whch yelds to: b, b,, < + (5) Snce transsson occurs only n states (, ), the robablty τ o a staton transttng n a randoly chosen slot can be exressed as: / W+,,,, W, W / W,,,,, W W / W,,,,, W Fgure. Fnte-state staton odel n saturaton based on Markov chan o the backo wndow sze / W W We denote the transton robablty ro one stage to another (e.g. ro row to row n Fg. ) by. It s also the robablty o an unsuccessul (re)transsson attet seen by a test staton as ts rae s beng transtted on the channel. he unsuccessul (re)transsson attet can haen due: collson o ths staton wth at least one o the n reanng statons, occurrng wth robablty, τ, () n ( ) and/or an errored rae, occurrng wth robablty P (due to the channel adng and/or nose). Snce both events are ndeendent, the robablty can be exressed as: ( )( P ) + P P. (3) In case o an unsuccessul transsson attet, ater backo ter exry n state (, ), the staton oves n any state on row (, k) wth robablty /W. Followng a successul transsson (occurrng wth condtonal robablty ) whle the observed staton s n stage (, + ), a new acket s adtted n the queue, the staton returns n backo stage, and ts backo ter unorly selects any nteger value n the range (, W ) wth robablty ( )/W. I the staton reaches backo stage +, and once ts backo ter reaches, ts rae can be successully or unsuccessully transtted. In both cases, a new rae s adtted n the queue and the staton returns n backo stage, and ts backo ter s unorly chosen n the range (, W ) wth robablty /W. Let the statonary dstrbuton o the chan be b,k, denotng the robablty o the staton to be n state (, k). he robablty o staton to be n state (, ) can be exressed + τ b, b, b,. (6) hen, robablty b,k or < (), can be gven sly as and b,k as W W b, k b, b,, < ( + W W W b, k ( ) b j, + b, W j W W j W ) b, + W j W W ( b W ), (7), b. (8), W Fro (7) and (8), b,k can generally be exressed: b W b,, ( ). (9) W, k + Noralzng the statonary dstrbuton o the chan to, and usng () and (9), we have: W W W b, k b, + W + b, k k W b b + + +,, ( W + ) + ( W + ) b, [ ( ) ]( ) + ( ) ( )( ) + W () + ( )( ) Fnally, we attan the robablty τ: τ ( )( + ( )( + ) + W[ ( ) ) ( + )] () Equatons (3) and () reresent a non-lnear syste wth sngle soluton, whch we solve usng Matheatca. It s obvous that there s a sngle soluton o τ or each n, W,,, and P,.e. τ (n, W,,, P ). he odel ro Fg. and () are generalzatons o the odel ro [7]. Actually, one can obtan the corresondng results ro [7] by solvng the non-lnear syste (3) () n the secal case or and P. +

3 B. Saturaton hroughut Followng a slar reasonng ro [7], we can exress the noralzed saturaton throughut o IEEE 8. DCF wthn a sngle WLAN cell n an error-rone channel as ollows: S ax Ps Ptr ( P ) E[ L] ( Ptr ) σ + Ptr Ps ( P ) s + Ptr ( Ps ) c + Ptr Ps P e () In (), E[L] s the average rae ayload sze, although to establsh uer erorance lt n the nueal analyss, we assued all generated ackets are xed and axzed so that E[L] L 3 octets. P tr s the robablty o at least one transsson n the observed te slot, P tr ( τ) n. hus, the robablty o an ety slot s P tr. P s s the robablty o a sngle successul transsson gven at least one staton (out o n statons) s transttng, P s nτ( τ) n- /P tr. he robablty o successul transsson n a slot te s denoted by P tr P s ( P ), the unsuccessul transsson robablty due to sultaneous transsson n the sae slot (.e. collson) s P tr ( P s ), and the unsuccessul transsson robablty due to errored rae s P tr P s P. s s the average te the channel s sensed busy by each staton because o a successul transsson, c s the average te the channel s sensed busy durng a collson, and e s the average te t s sensed busy ro a rae whch suered transsson errors. Assung the duraton announceents (contaned n the reable/header art o the rae) are always successully receved by all statons, and the rae errors can occur only n the reanng art o the rae, t s clear that e s. he values o s and c der deendng on the network access ode (gven below or Basc and RS/CS) and addtonal network oeratng araeters (able I) ABLE I. RELEVAN NEWORK PARAMEERS Paraeter Deault Channel Rate Mbs PHY Preable 44 sybols PHY Header 48 sybols MAC header 34 octets ACK 4 octets + PHY re/hdr RS octets + PHY re/hdr CS 4 octets + PHY re/hdr SIFS µs DIFS 5 µs Slot_e σ µs 5 Intal contenton wndow W 8 basc s 6.4 µs basc c 948. µs s 589. µs c 56.5 µs bas Basc: s PHYre / hdr + MAChdr + L + SIFS + ACK + DIFS bas c PHYre / hdr + MAChdr + L + DIFS rts / cts s RS + SIFS + CS + SIFS + PHYre / hdr RS/CS: + MAChdr + L + SIFS + ACK + DIFS rts / cts c RS + DIFS C. Frae Dscard Probablty he robablty o a rae dscard P D s actually the robablty o occurrence o consecutve + + unsuccessul retranssson attets, ater whch the rae s dscarded, a new rae s adtted n the transt queue, and the staton returns to backo stage. hus, + n + PD [ ( P )( τ ) ]. (3) Note that dscard robabltes are nsenstve to the utlzed staton s access ethod (Basc or RS/CS). D. Mean Delay Now let us concentrate on a sngle staton to deterne the average delay d o each rae ro the oent the backo rocedure s ntated untl rae s successul transsson. Durng the backo deer slots o the observed staton n the -th stage, the robablty o transsson o at least one o the n reanng statons slots s, whle the robablty o exactly one transsson ro one o the n reanng statons (gven at least one o the s transttng) s: s ( n ) τ ( τ ) n. (4) In each backo stage (, ), the ntal value o the backo ter has ean o (W )/, so that the average deerral nterval beore a retranssson attet s (W )/ slots. he average nuber o consecutve dle slots n dle between two consecutve busy slots o the n reanng statons can be calculated as: n dle ( ). (5) hus, a sngle renewal cycle between two consecutve transssons o the n reanng statons ncludes ultle consecutve dle slots and an occued slot,.e. n n dle + / slots. Snce each occued slot can be a successul, a raeerrored, or a collded slot, the average duraton o a renewal cycle s ndle + s ( P ) s + ( s ) c + s σ P. (6) Snce the retranssson attet o observed staton n backo stage s on the average receded by (W )/ n (W ) / renewal cycles o n reanng statons, the average te between two consecutve retransssons o the observed staton s (W ) /. he average elased te tct, beore the test e 3

4 staton akes ts ( + )-th retranssson attet (row n Fg. ), can be calculated as: tct, k Wk + +. (7) Wk ( + ) k where s the average duraton whle the observed staton tsel occues the channel durng each unsuccessul retranssson attet: ( P ) c + P c + ( ) P e c + ( ) P e. (8) Ater substtuton o () nto (7), we have: tct, ( + ) +, + W. (9) + ( ), + Fnally, the average rae delay untl successul transsson s: d + ( ) ( tct, + s ). () By ntroducng (9) nto (), the latter can easly be solved n closed-or. In the secal case, () attans a ore sler or: d + ( s ) + W ( ) ( )( ) () Addtonally, to calculate d or and P, one needs to substtute and c nto (). he reason s that the rae delay n each staton orgnates ro the backo deer erods n a sgncant orton, as well as ro the erods when the staton artcates n collsons. Obvously, due to the ersstent-untl-success retranssson strategy ( ), there are no rae losses at all,.e. P D. We ehasze the grahs ro Fg. and all the ollowng results reer to Mbs transsson rate (able I), whle corresondng syste araeters ust be used accordng to IEEE 8.b standard or rates o,, and 5.5 Mbs. Addtonally, the ntal contenton wndow s 8 (W 8), whle the retranssson attet threshold (ater whch the ntal backo wndow s rozen) s set to 5,.e. 5. S ax (Mbt/s) d (s) basc n 3 (a) basc III. NUMERICAL RESULS he throughut and delay erorance vs. n or secal case and P s dected n Fg.. As exected, the saturaton throughut decreases wth the ncrease o nuber o contendng statons n Basc access ode, whle t reans stable n RS/CS access ode. he ean delay ncreases n both access odes, although t s lower n RS/CS access ode snce only the short RS raes artcate nto the collsons. However, as coared to the rato between the length o the useul rae and the RS rae, the derence s not as excessve as one would exect n 3 (b) Fgure. Network erorance n ersst-untl-success retranssson strategy and an error-ree channel; (a) throughut, (b) delay he axal allowable nuber o retranssson attets ( + + ) has nor act over the axu achevable throughut S ax. However, ts nluence over ean delay d and dscard robablty P D cannot be dsregarded. 4

5 Fg. 3 dslays the ean rae delay and dscard robabltes n uncton o, wth P. and.5 as curve araeters, and n 3. It s reasonable to exect that ean delay d wll ncrease (Fg. 3a) and the rae losses P D would decrease (Fg. 3b) as we allow or hgher nuber o retranssson attets. In Fg. 3b, the curves or Basc and RS/CS access odes or gven P concde. IEEE 8.b WLAN n uncton o both the SNR and the delay sread o the ultath-aded channel. S ax (Mbt/s) d (s) 5 P. (basc) P. () P.5 (basc) P.5 () 4 3 (basc) (basc) () () P D (%) (a) P. (basc) P.5 (basc) d (s).. P (a) 5 5 (basc) (basc) () () (b) Fgure 3. Inluence o retranssson threshold over erorance: (a) delay, (b) dscard robablty he act o rae-error rates P over all erorance easures s dected n Fg. 4; aears as araeters n the grahs ( and ), whle n 3. Increasng P ro. to, throughut degrades towards (Fg. 4a), and dscard robablty ncreases towards % (Fg. 4c). Saturaton ean delay also ncreases wth the ncrease o P, however as P aroaches near, the ean delay sharly decreases to because alost every rae s dscarded (Fg. 4b). At ths ont, we note that the grahs o P vs. channel sgnalto-nose rato SNR n [9] can be used to calculate the corresondng curves o the three erorance easures or the P D (%).. P (b) (basc) (basc).. P (c) Fgure 4. Iact o rae-error rate P over erorance: (a) saturaton throughut, (b) delay, (c) dscard robablty 5

6 IV. CONCLUSION Based on our oded dscrete Markov odel or the backo wndow sze wth nte nuber o retranssson attets, we calculated the transsson robablty n a slot te o each IEEE 8. staton oeratng n saturated condtons n an error-rone rado channel. he transsson robablty roves to deend on the nuber o contendng statons, the rae error robablty, and the network oeratng araeters. Usng the transsson robablty, we rovde analytcal solutons or all relevant erorance easures o a sngle IEEE 8. cell: network throughut, and staton s ean delay and dscard robablty o the DCF n saturaton or both access odes, Basc and RS/CS. Increasng the axal nuber o retransssons ater whch a rae s dscarded ro staton s queue, the delay ncreases and dscard robablty decreases, whle the saturaton throughut s ractcally unchanged or both access odes. Conversely, the rae-error rate P sgncantly acts only the throughut or. < P <., whle or hgh error rates - all erorance easures are aected. Fnally, let s ehasze that n ths aer we only consdered the act o the rae-error rate. he ultath adng channel ntroduces addtonal colexty aectng wreless networks erorance, e.g. the cature eect. Reer to [8] or ore detaled saturaton throughut analyss o IEEE 8.b DCF under cature REFERENCES [] IEEE Standard or Wreless LAN Medu Access Control and Physcal Layer Seccatons, IEEE 8.b, Nov [] H.S. Chhaya and S. Guta, Perorance odelng o asynchronous data transer ethods o IEEE 8. MAC rotocol, Wreless Networks, vol. 3.,. 7-34, 997 [3] K. C. Huang and K.C. Chen, Intererence Analyss o nonersstent CSMA wth hdden ternals n ultcell wreless data networks, Proc. IEEE PIMRC, oronto, Set. 995, [4].S. Ho and K.C. Chen, Perorance evaluaton and enhanceent o the CSMA/CA MAC rotocol or 8. wreless LAN, Proc. IEEE PIMRC, ae, Oct. 996, [5] A. Zahed and K. Pahlavan, Natural Hdden ernal and the Perorance o the Wreless LANs, Proc. IEEE 6 th Int. Con. on Unv. Pers. Co, , 997. [6] F.Cal, M. Cont, and E. Gregor, IEEE 8. wreless LAN: Caacty analyss and rotocol enhanceent, Proc. INFOCOM, San Francsco, Mar. 998 [7] J. Banch, Perorance Analyss o IEEE 8. Dstrbuted Coordnaton Functon, IEEE JSAC, vol. 8, No. 3, Mah, [8] Z. Hadz-Velkov and B. Sasenovsk, On the Caacty o IEEE 8. DCF wth Cature n Multath-aded Channels, Int. Journal o Wreless Inoraton Networks, vol. 9, No. 3, Kluwer Acadec, July [9] C. Heegard et al. Hgh-Perorance Wreless Ethernet, IEEE Councatons Magazne, vol. 39, no., Nov.,