Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs Perforane Analyss of the Bnary Exonental Bakoff Algorth for IEEE 8. Based Moble Ad Ho Networks Y-Hua Zhu, Xan-Zhong Tan, and Jun Zheng Shool of Couter Sene and Tehnology, Zheang Unversty of Tehnology Hangzhou, Zheang 33, Chna Eal:{yhzhu, txz}@zut.edu.n Natonal Moble Counatons Researh aboratory Southeast Unversty Nanng, Jangsu 96, Chna Eal: unzheng@seu.edu.n Abstrat In an IEEE 8. based oble ad ho network (MANET, a network node aesses a oon wreless hannel through the dstrbuted oordnate funton (DCF, whh s rovded at the edu aess ontrol (MAC layer of the IEEE 8. standard. The bnary exonental bakoff (BEB algorth, whh uses slotted ontenton wndows, lays an ortant role n the DCF. Ths aer develos a atheatal odel for analyzng the erforane of the BEB algorth, whh takes nto aount the aket loss robablty of a wreless lnk. Based on the develoed atheatal odel, we derve the bakoff robablty dstrbuton of a node, the average nuber of bakoffs of a node, the average sze of a ontenton wndow, and the average aket delay. Moreover, we attet to fnd the otal value of the ntal ontenton wndow sze, to whh a node resets ts ontenton wndow sze after a suessful transsson, n order to avod the osllaton of the ontenton wndow sze and thus axze the utlzaton of the wreless hannel. Index Ters IEEE 8. standard; bnary exonental bakoff; oble ad ho network I. INTRODUCTION THE IEEE 8. standard [ sefes the hysal layer (PHY and edu aess ontrol (MAC layer of a wreless loal area network (WAN. At the MAC layer, ostve aknowledgeent (ACK s used to aheve relable delvery of data akets between network nodes. The reever of a data aket has to transt an ACK to the sender of the aket f t reeves the aket suessfully. The aket s onsdered lost and the sender has to retranst the aket f the sender does not reeve the ACK before the retranssson ter of the transtted aket tes out. The MAC layer rovdes the dstrbuted oordnate funton (DCF and the ont oordnate funton (PCF to suort and ontrol aess to the transsson hannel, where PCF s otonal whle DCF s andatory. DCF allows ultle ndeendent nodes to nterat wth eah other wthout a entral ontroller, whh s tyally sutable for oble ad ho networks (MANETs. The arrer sensng ultle aess/ollson avodane (CSMA/CA rotool and the bnary exonental bakoff (BEB algorth are two key DCF oonents that are used to avod aket ollsons. Wth the BEB algorth, a ontenton wndow s dvded nto a seres of te slots of equal length, whh are nubered n,,, CW. Here, CW s the axu slot nuber, whh s ntalzed to a onstant, CW n, and norally kees a value n [CW n, CW ax [. To redue the robablty that a aket ollson ours, a node randoly ks a value n [, CW to set ts bakoff ter and starts to transt ts data n the orresondng slot. However, ollsons ay stll our when two or ore nodes k the sae bakoff slot. To address ths roble, eah te a node enounters a aket ollson, the sze of ts ontenton wndow s doubled unless CW ax s reahed [. Meanwhle, the node randoly ks a bakoff slot n the nreased ontenton wndow for ts next transsson. As a result, the aket ollson robablty an be redued at the exense of nreasng the delay of the node to aess the hannel beause the robablty that the node ks a larger bakoff slot s nreased. The CW of the nreased ontenton wndow s reset to the ntal value CW n when the node has a suessful transsson [. Obvously, the value of CW has a bg at on the erforane of the BEB algorth. Aordng to exstng work, the BEB algorth s far fro satsfaton n rovng the network erforane n ters of aket delay and throughut [[3, and the standard BEB algorth ay ause unneessary long delay and low hannel untzaton [4. Moreover, resettng CW to CW n after eah suessful transsson ay ause an osllaton of the ontenton wndow sze n a network wth a large nuber of nodes, and thus affet the stablty of the network erforane. Therefore, t s desrable not to reset CW to ts ntal value after eah suessful transsson n order to stablze the network erforane [5. In [6, t s shown that an arorately hosen fxed CW sze an aheve a throughut erforane lose to that of the standard BEB algorth. Ths rases an nterestng queston: What s the otal sze of a fxed CW sze to axze the hannel untzaton and avod the throughut osllaton for a gven network? whh s the otvaton for ths work. In ths aer, we analyze the erforane of the BEB algorth used n a MANET and attet to answer the above entoned queston. The an ontrbutons of the aer 978--684-3-8//$6. IEEE
Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs nlude the followng asets: A atheatal odel s develoed for dervng the aket ollson robablty of a node, the bakoff robablty dstrbuton of a node, the average nuber of bakoffs of a node, the average sze of a ontenton wndow, and the average aket delay. An otal value of the ntal ontenton wndow sze s roosed to avod the osllaton of the ontenton wndow sze and thus axze the utlzaton of the wreless hannel, whh s defned as the rato of the nuber of te slots n whh a node transts a aket wthout ollson to the total nuber of te slots. 3 It s found that unlke the aroxaton onluson ade n [7 the aket ollson robablty of a node wth the BEB algorth s a onstant regardless of the nuber of retransssons and the aket ollson robablty dereases as the sze of the ontenton wndow nreases when the nuber of nodes n the network s large. The reander of the aer s organzed as follows. Seton II revews related work. Seton III resents the atheatal odel and analyzes the erforane of the BEB algorth. Seton IV resents nueral results to nvestgate the aket ollson robablty and the otal ntal ontenton wndow sze for avodng osllaton of the ontenton wndow sze. In Seton V, we onlude the aer. II. REATED WORK The BEB algorth has reently reeved uh attenton and otvated a lot of researh work. In [, Cal et al. showed that, deendng on the network onfguraton, the BEB algorth gven n the IEEE 8. standard erfors far fro the theoretal throughut lt, and the throughut of the IEEE 8. rotool ould aroah the theoretal lt by arorately tunng the bakoff algorth. In [8, Nasr and Albalt roved the BEB algorth and roosed a hstory based adatve bakoff (HBAB algorth to rove the qualty of serve (QoS erforane based on the network status and ondton. Consderng that the bakoff araeters of the ollson avodane ehans n the latest verson of the IEEE 8. standard are far fro otal under heavy traff load or n an error-rone WAN envronent, Deng et al. roosed a dstrbuted algorth that allows a node to dynaally adust ts ontenton wndow based on the turn-round-te easureent of the hannel status [3. In [5, Ksentn et al. roosed a deternst ontenton wndow algorth (DCWA, whh s dfferent fro the BEB algorth n the IEEE 8. standard n that the DCWA nreases both the uer bound and the lower bound of the bakoff range. In [9, Gannoune nvestgated a non-lnear dyna adataton algorth that enables eah node to tune the sze of the nu ontenton wndow (CW n. In [, Ye and Tseng roosed a ulthan bakoff (MCB algorth, n whh a node s allowed to adat to dfferent ongeston levels by usng ore than one bakoff han together wth ollson events aused by nodes. In [, Nafaa rovded a sldng ontenton wndow (SCW algorth for eah network flow, whh s able to dynaally adust to hangng network ondtons but antan a searaton between dfferent serve lasses. In [, Ma and Roy ntrodued a ont ontenton wndow and transsson oortunty (CWTO adataton algorth to rove the aggregate throughut and network farness for ult-ell hgh densty WANs. Deste the above work, however, lttle work has analyzed the bakoff robablty dstrbuton of a node and studed the ontenton wndow sze osllaton roble n a MANET wth lossy wreless lnks, whh are the fouses of ths aer. III. PERFORMANCE MODE In ths seton, we develo a atheatal odel for analyzng the erforane of the BEB algorth based on the saturaton throughut assuton [7,.e., the transsson queue of eah node s always nonety. We defne the sze of a ontenton wndow as the nuber of te slots that the ontenton wndow ontans. Sne the nu and axu slot nubers n a ontenton wndow are and CW, resetvely, the sze of a ontenton wndow s equal to CW+. We assue that the sze of the ontenton wndow reahes CW ax after t s doubled tes, and let be the sze of the ontenton wndow after t s doubled tes (=,,,, where s the nu ontenton wndow sze,.e., =CW n +, and s the axu ontenton wndow sze,.e., =CW ax +. Meanwhle, we assue that there are n nodes n the network and use V to denote the set of nodes n the network. Under CSMA/CA, a node has to sense the wreless hannel before ts transsson. The BEB algorth s trggered when the node has sensed an dle erod of dstrbuted nter-frae sae (DIFS. Only when the bakoff ter of the node tes out s the node allowed to transt ts data. As desrbed earler, the ontenton wndow sze of a node s doubled unless s reahed every te the node enounters a ollson. As a result, dfferent nodes usually have dfferent ontenton wndow szes. We use W sze (v to denote the ontenton wndow sze of node v and defne G k ={v W sze (v= k } wth ardnalty G k (k=,,,. Thus, we have G n k k =, = G = V k = k, G Gk = Φ ( k. Obvously, a node belongs to one of G k (k=,,, at any te. Further, we defne that a node s n state f t has ontnuously erfored the BEB algorth for tes before a suessful transsson. We use node( to denote a node n state. Clearly, the ontenton wndow sze of node( satsfes: =, =,,, ; =, =+, +,.. As a result, the nodes n states,,, and - are nluded n G, G,, and G -, resetvely, whereas all the nodes n state, +, + are ontaned n G. et be the aket loss robablty of the wreless hannel and defne q=-. et, be the robablty that a aket ollson ours durng the aket transsson of node(, whh s trggered one the bakoff ter of node( tes out. Obvously, for any >, we have, =, due to =. Fg.
Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs gves the state transton dagra of a node, n whh a rle stands for a state, an arrow reresents a state transton, and the exresson on an arrow s the transton robablty orresondng to the event of a state transton. Next we derve robablty,. Clearly,, =-,n, where,n s the robablty that node( transts wthout a ollson,.e., none of the other nodes n the network ks the sae bakoff slot as the one node( ks. We all node A olldes wth node B f node A and node B k the sae bakoff slot. Fg. State transton dagra Fg. Dfferent bakoff slots et x be the bakoff slot ked by node( wth ontenton wndow sze, and,n (x be the robablty that any other node n the network does not ollde wth node( n slot x. Clearly, node( randoly ks a slot n the set {,,, -} wth robablty /. Thus, the robablty that node( ks a artular te slot x and other nodes do not ollde wth node( n the te slot s equal to the rodut of,n (x and /, whh results n the robablty that node( transts wthout a ollson n any ked slot,.e.,, n =, n ( x. ( Notng that the nterval [, - an be dvded nto the ntervals: [, -, [, -, [, -, and [ -, -, Eq. ( an be rewrtten as, n = [ ( x + ( x + +,n ( x. (,n,n x = x = x = To derve,n, let us observe Fg., fro whh we an obtan the followng onlusons: For any x< (.f. x n Fg., any node n G olldes wth node( wth robablty / as eah node n G randoly ks a slot n the set {,,, -}, whh leads to the robablty that all the nodes n G do not ollde G wth node( beng ( /, where,,, and. Partularly, for, ths robablty beoes G ( / beause node( s also n G,.e., there are G - nodes n G, whh are lkely to ollde wth node(. Thus, the robablty that all the nodes n G, G,, and G do not ollde wth node( n slot x an be alulated as G G G, n ( x = ( ( ( = = +. (3 G = (, {,,, } x = Sung Eq. (3 leads to,n ( x = = ( G. (4 For x< (.f. x n Fg., any node n G does not ollde wth node(, but eah node n G, G,, and G olldes wth node( wth robablty /, /,, and /, resetvely, whh rodues G, n x = ( = (, where x {, +,, } wth - eleents. Thus, we have,n ( x = ( ( G. (5 3 For x< (.f. x n Fg., any node n G and G does not ollde wth node(, but eah node n G, G 3,, and G olldes wth node( wth robablty /,, / 3,, and /, resetvely, whh leads to G, n x = ( (, where x, +,, }. Thus, we have {,n ( x = ( ( G. (6 By observng Eq. (4-Eq. (6, we obtan the followng relatonsh: k G,n( x = ( k k (, (7 k where - and G = n. = = k k =,,,
Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs Also, aordng to Eq. ( and Eq. (7, we have whh yelds G, n = ( k k ( k = k, (8 G, =, n = ( k k ( k = k, (9 where =,,,. et P be the robablty that a node s n state ( =,,. Thus, we have = P =. ( In Fg., we an obtan the followng set of flow-balanng equatons: P = P [ + (, P = P [ + (, P = P [ + (,, P + = P [ + (, P + = P + [ + (, whh are followed by P = P [ + (,, =,,, ;. ( P + = P [ + (, [ + (, =,,, Notng that < and, <, we have <+(-, <. Thus, fro the seond equaton n Eq. (, we have = P + = P = P [ + ( ( (,, = Eq. (-Eq. ( lead to P = + [ + ( = + ( (, [ + ( [ + (, [ + (,,,. ( (3 Thus, P, P,, P an be deterned based on Eq. (9, Eq. (3, and Eq. (. Fro Eq. ( and Eq. (, we an obtan the average ontenton wndow sze of a node,.e., avg = = P ( + P = = + ( ( P + = =, [ + ( = P +, [ + (,. (4 In addton, the average nuber of bakoffs before a suessful transsson s = N bakoff P, (5 = M whh s aroxately equal to P =, where M s a ostve nteger suh that P M s suffently lose to. It an be easly roved that the average nuber of a bakoff slot s (-/ f a node randoly ks a bakoff slot n a ontenton wndow wth sze,.e., the node ks a bakoff slot n,,, -. et τ be the duraton of a te slot used n the BEB algorth. Thus, the average delay for a node to suessfully transt a aket s T = avg delay τ. (6 IV. NUMERICA ANAYSIS In the revous seton, we derved the aket ollson robablty dstrbuton of a node, whh s gven by Eq. (9. Aordng to Eq. (9,, deends on G, G,, and G, whh are varyng wth the aket ollsons and are dffult to be exatly deterned beause the nodes n the network erfor the BEB algorth ndvdually. For ths reason, t s stll dffult to fnd the robablty dstrbuton for a gven n. However, we an onsder G, G,, and G as rando varables beause eah node ks ts te slot randoly. Therefore, to nvestgate the roertes of the derved robablty dstrbuton and the erforane of the BEB algorth, we randoly set G, G,, and G suh that G + G + + G = n. (7 We assue that CW n =5 and CW ax =3, whh orresonds to the frequeny ho PHY [. Thus, CW takes a value n the set {5, 3, 63, 7, 55, 5, 3}, whh are equvalent to =6 and =, =,,, (= 6. Usually, a real network has several tens of nodes. However, f the nuber of nodes n a network s too sall, t does not suffe to nvestgate the erforane of the BEB algorth. Ths s beause n that ase the robablty that two or ore nodes k the sae te slot that leads to a aket ollson would be very sall and as a result the ontenton wndow sze of a node ould rarely be doubled to reah the axu (.e., CW ax +=4. For ths reason, we onsder [, as the soe of n. In addton, we set =.,.4,.7, and.9, resetvely. Under the above assutons and settngs, we obtaned soe nueral results based on Eq. (4 to Eq. (6, whh are shown n Fg. 3 to Fg. 6. All the results are an average over 3 runs wth eah randoly rodung G (=,,, satsfyng Eq. (7.
Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs Fg. 3 shows the aket ollson robablty wth dfferent node states when n=. It s seen that the aket ollson robablty dereases as a node oves fro the state wth a saller ontenton wndow sze to the one wth a larger sze. That s, the aket ollson robablty dereases as the sze of the ontenton wndow s doubled. Ths agrees wth our ntuton that the aket ollson robablty an be redued by nreasng the ontenton wndow sze of a node. The robablty of aket ollson when n=.9.8.7.6.5.4.3.. 3 4 5 6 Node state Fg. 3 Paket ollson robablty Fgs. 4-6 show the ats of the nuber of nodes n a network (.e., n and the aket loss robablty of the wreless hannel (.e., on the average sze of a ontenton wndow (.e., avg, the average nuber of bakoffs (.e., N bakoff, and the average aket delay (T delay, resetvely. It s seen that avg, N bakoff, and T delay gradually nrease wth the nrease of n and/or, resetvely. Ths agrees wth our ntuton that the ore nodes a network has, the ore lkely a aket ollson ours, ausng the sze of a ontenton wndow, the nuber of bakoffs, and the delay to nrease; the hgher the aket loss robablty, the ore akets (nludng ACK akets are lost, ausng the node to nrease ts wndow sze and retranst. Also, the BEB algorth akes the ontenton wndow sze of a node osllate. For exale, n a network wth nodes (.e., n=, f the aket loss robablty =., the average ontenton wndow sze of a node s around (.f. Fg. 4. In ths ase, when a node has a suessful transsson, ts ontenton wndow sze s reset to =6 n the standard BEB algorth. Then, the node has to bakoff 3 tes before the next transsson. Sefally, t frst hanges ts ontenton wndow sze fro 6 to 3, then to 64 and to 8 so that t an k a slot wthout aket ollson. In soe ases, the osllaton of a ontenton wndow sze ould be larger. To avod the osllaton of the ontenton wndow sze, we an take advantage of Eq. (4 to fnd an otal sze ot and allow a node to reset ts ontenton wndow sze dretly to ot (.e., reset CW to ot - nstead of to (.e., reset CW to CW n. For exale, the osllaton n the above ase an be avoded f we allow the node to reset ts CW to 3 - (.e., 7 nstead of CW n (.e., 5. To ustfy ths la, we erfored a sulaton exerent usng MATAB, n whh we assued that n= and eah sulaton result was averaged over runs wth eah run lastng for a erod of 5 te slots. The average sze of the ententon wndow of a node, avg The average delay (μs, T delay The average nuber of bakoffs, N bakoff 7 6 5 4 3 =. =.4 =.7 =.9 4 6 8 4 6 8 The nuber of nodes, n 8 6 4 Fg. 4 The ontenton wndow sze vs. the nuber of nodes =. =.4 =.7 =.9 4 6 8 4 6 8 The nuber of nodes, n 35 3 5 5 5 Fg. 5 The nuber of bakoffs vs. the nuber of nodes =. =.4 =.7 =.9 4 6 8 4 6 8 The nuber of nodes, n Fg. 6 The average delay vs. the nuber of nodes Fg. 7 shows the at of the ntal sze of a ontenton wndow on the utlzaton of the wreless hannel. The hannel utlzaton value orresondng to a artular ntal sze value s obtaned by resettng the ontenton wndow sze of a node dretly to that ntal sze value one the node has a suessful transsson. It an be learly seen that the hannel utlzaton reahes the axu value f 8 s ked as the resettng
Ths full text aer was eer revewed at the dreton of IEEE Counatons Soety subet atter exerts for ublaton n the IEEE ICC roeedngs value of the ontenton wndow sze (.e., CW=7 when =.; whle the best ontenton wndow sze s 64 (.e., CW=63 for =.3. These ndate that based on Eq. (4 we an fnd an otal ontenton wndow sze that an axze the hannel utlzaton. Channel utlzaton.34.3.3.8.6.4...8.6 =. =. =.3 =.4 6 3 64 8 56 5 4 The ntal sze of a ontenton wndow Fg. 7 Channel utlzaton vs. the ntal ontenton wndow sze V. CONCUSION In a MANET, a node transts ts akets through ontenton for aessng a oon wreless hannel. The BEB algorth ntrodued n the IEEE 8. standard lays an ortant role n avodng aket ollsons and has a bg at on the utlzaton of the wreless hannel. However, the BEB algorth s far fro satsfaton n rovng the network erforane n ters of delay and throughut. In ths aer, we develoed a atheatal odel for analyzng the erforane of the BEB algorth used n a MANET wth lossy wreless lnks. Based on the develoed odel, we an analyze the erforane of the BEB algorth and fnd the otal ontenton wndow sze, whh s used by a node to reset ts CW after a suessful transsson so that the osllaton n the ontenton wndow sze s avoded and the utlzaton of the wreless hannel s thus axzed. In addton, we found that when the nuber of nodes n a network s large (e.g., ore than the aket ollson robablty of a node wth the BEB algorth dereases as the sze of ts ontenton wndow s doubled, whh agrees wth the ntuton that the node has less hane to k the sae bakoff te slot than other nodes f the ontenton wndow sze of the node s larger than those of other nodes. We noted that Banh obtaned the aroxaton onluson that eah aket olldes wth onstant and ndeendent robablty at eah transsson attet and regardless of the nuber of retranssson attet based on the sulaton results [7. Ths onluson s true for a network wth a sall nuber of nodes (e.g., no ore than 5 n whh the robablty that two or ore nodes k the sae bakoff te slot s very sall and thus the sze of the ontenton wndow s rarely doubled, whh leads the aket ollson robablty to be very lose. However, t ay not hold true for a network wth large nuber of nodes (e.g., nodes. ACKNOWEDGEMENT Ths work was suorted by the Natonal Natural Sene Foundaton of Chna under Grants No. 679 and No. 675, by the Zheang Provnal Key Sene & Tehnology Proet of Chna under Grant No. 9C433, and by Zheang Provnal Natural Sene Foundaton under Grant No. Z455. REFERENCES [ AN MAN Standards Cottee of the IEEE Couter Soety, IEEE 8. Standard wreless AN edu aess ontrol and hysal layer sefatons, June 997. [ F. Cal, M. Cont, E. Gregor, Dyna tunng of the IEEE 8. rotool to aheve a theoretal throughut lt, IEEE/ACM Transatons on Networkng, vol. 8, no. 6, De.,. 785-799. [3 D.-J. Deng, C.-H. Ke, H.-H. Chen, et al, Contenton Wndow Otzaton for IEEE 8. DCF Aess Control, IEEE Transatons. on Wreless Counatons., vol. 7, no., De. 8,. 59-535. [4 Q. Pang, V.C.M. eung, and S. C. ew, Iroveent of WAN ontenton resoluton by loss dfferentaton, IEEE Transatons on Wreless Counatons, vol. 5, no., 6,. 365 365. [5 A. Ksentn, A. Nafaa, A. Guerou, and M. Na, Deternst ontenton wndow algorth for IEEE 8., n Pro. 5 IEEE 6th Internatonal Syosu on Personal, Indoor and Moble Rado Counatons (PIMR 5, Berln, Gerany, Set. 5,. 7-76. [6 K. Medeall and F. A. Tobag, On otzaton of CSMA/CA based wreless ANs: Part I - Iat of exonental bakoff, n Pro. 6 IEEE Internatonal Conferene on Counatons (ICC 6, vol. 5, Istanbul, Turkey, June 6,. 89 94. [7 G. Banh, Perforane analyss of the IEEE 8. Dstrbuted Coordnaton Funton, IEEE Journal on Seleted Areas n Counatons, vol. 8, no. 3, Marh,. 535-547. [8 Q. Nasr and M. Albalt, Iroved bakoff algorth for IEEE 8. networks, n Pro. 9 IEEE Internatonal Conferene on Networkng, Sensng, and Control, Okayaa, Jaan, Marh 6-9, 9,. -6. [9. Gannoune, A non-lnear dyna tunng of the nu ontenton wndow (CW n for enhaned serve dfferentaton n IEEE 8. ad-ho Networks, n Pro. 6 IEEE 63 rd Vehular Tehnology Conferene (VTC 6-Srng, Melbourne, Australa, May 6,. 66-7. [ S.-R. Ye and Y.-C. Tseng, A ulthan bakoff ehans for IEEE 8. WANs, IEEE Transatons on Vehular Tehnology, vol. 55, no. 5, Set. 6,. 63-6. [ A. Nafaa, A. Ksentn, and A. Mehaoua, Sldng ontenton wndow (SCW: Towards bakoff range-based serve dfferentaton over IEEE 8. wreless AN networks, IEEE Network, vol. 9, no. 4, July/August 5,. 45-5. [ H. Ma and S. Roy, Contenton wndow and transsson oortunty adataton for dense IEEE 8. WAN based on loss dfferentaton, n Pro. 8 IEEE Internatonal Conferene on Counatons (ICC 8, Beng, Chna, May 8,. 556-56.