A Comprehensive Analysis of Bandwidth Request Mechanisms in IEEE Networks

Transcription

1 A Comrehensve Analyss of Bandwdth Reuest Mechansms n IEEE Networks Davd Chuck, Kuan-Yu Chen and J. Morrs Chang Deartment of Electrcal and Comuter Engneerng Iowa State Unversty, Ames, Iowa 500, USA Emal: {chuang, kychen, morrs}@astate.edu Abstract IEEE standard s consdered as one of the most romsng technologes. Bandwdth reservaton s emloyed to rovde ualty of servce (QoS guaranteeng servces. A reuest/grant scheme s defned n the IEEE standard. There are two tyes of bandwdth reuest (BR mechansms, uncast ollng and contenton resoluton, defned n the standard. As secfed, connectons belongng to schedulng classes of ertps, nrtps and BE have otons to make BRs va both mechansms deendng on the schedulng decson made by the base staton (BS. However, most research works only assume one of them s avalable and do not take both of them nto account. A comrehensve study of both mechansms s crtcal for the BS to make an arorate decson for those connectons to acheve better system erformance. To the best of our knowledge, ths s the frst attemt to analyze ths ssue. There are two major contrbutons resented n ths aer. Frst, a comrehensve study of both BR mechansms n terms of bandwdth utlzaton and delay s rovded. Addtonally, we roose two ractcal erformance objectves: when the exected delay or target bandwdth utlzaton s gven, how does the BS to make schedulng decson such that the erformance of the other metrc (ether delay or bandwdth utlzaton s otmzed? As our second contrbuton, we roosed two schedulng algorthms to fnd the combnaton of both mechansms to meet our objectves. The smulaton results show that our schedulng algorthms can always hel the BS make schedulng decson to reach better system erformance. I. INTRODUCTION The IEEE standards (e.g., [] are consdered as one of crtcal broadband wreless access (BWA technologes n the forth generaton (4G networks. The Worldwde Interoerablty for Mcrowave Access (WMAX, based on ths famly of standards, s desgned to facltate servces wth hgh transmsson rates for data and multmeda alcatons n metrooltan areas. The hyscal (PHY and medum access control (MAC layers of WMAX have been secfed n the IEEE standard. Many advanced communcaton technologes such as OFDM/OFDMA and MIMO are embraced n the standards. Suorted by these modern technologes, WMAX s able to rovde a large servce coverage, a hgh seed data rate and ualty of servce (QoS guaranteeng servces. Because of these features, WMAX s consdered to be a romsng alternatve for last mle BWA. In order to rovde the QoS guaranteeng servces, bandwdth reservaton s adoted n the WMAX network. A reuest/grant bandwdth allocaton s emloyed for reservng bandwdth. The subscrber staton (SS s reured to reserve the suffcent amount of bandwdth from the base staton (BS before any data transmssons. The amount of reserved bandwdth can be reserved or adjusted by the SS va sendng bandwdth reuests (BRs. There are two tye of bandwdth reuests secfed n the IEEE standard: uncast ollng and contenton resoluton. In uncast ollng, the BS allocates a small ece of bandwdth to the target SS. Ths small ece of bandwdth s on the to of reserved bandwdth and contans one or more transmsson oortuntes (TxOPs deendng on the schedulng olcy that the BS enforces. These TxOPs are called uncast ollng TxOPs n ths aer. The target SS can use these TxOPs to send BRs. Moreover, for smlcty, we assume that the uncast ollng TxOP s only used for transmttng a BR. Contenton resoluton, on the other hand, reures that each SS contends a TxOP ndeendently to transmt a BR. The BS schedules an amount of bandwdth, dvded nto several TxOPs, for a grou of SSs to send BRs. These TxOPs are called contenton TxOPs. If the attemt of contenton s faled, then the SS enters nto the back-off rocedure to reare next attemt untl reachng the maxmum number of attemts. Each tye of BR mechansms (.e. uncast ollng or contenton resoluton has ts own advantages and dsadvantages. In uncast ollng, the uncast ollng TxOPs are allocated exclusvely for the target SS. It guarantees that ths SS has oortuntes to make BRs successfully. Therefore, the delay of the SS to transmt a BR can be bounded wthn a certan range. However, because of the exclusve usage, the allocated uncast ollng TxOPs are wasted f the target SS does not make BRs. Ths may reduce the bandwdth utlzaton of the system. In contenton resoluton, on the other hand, the allocated bandwdth s shared by a grou of SSs. The SS contends wth each other n order to get a contenton TxOP for the BR. In the contenton resoluton, each SS contends for a contenton TxOP actvely. Therefore, the SS erforms the contenton rocedure only f the SS wants to transmt a BR. It may lead to hgher bandwdth utlzaton. However, Each SS cannot be guaranteed to have contenton TxOPs for sendng BRs. Thus, the delay to reuest bandwdth can not be ensured. Suort for QoS s a fundamental art of the IEEE MAC-layer desgn. When the servce data unt arrves n the IEEE MAC layer, the classfcaton rocess s erformed. The classfcaton rocess s the rocess whch mas the servce data unt to the arorate schedulng class based on the QoS constrants of the servce data unt. Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

2 2 As secfed n the IEEE standard, only connectons belongng to three schedulng classes (.e. extended real tme ollng servce (ertps, non-real tme ollng servce (nrtps and best effort (BE are allowed to have otons to choose between uncast ollng and contenton resoluton for make BRs. Because of the features of each BR mechansm, A schedulng decson made by the BS for the connectons n these schedulng classes to transmt BRs may affect the overall bandwdth utlzaton and delay. For examle, uncast ollng may result low bandwdth utlzaton when the robablty of a SS to make a BR s low. Smlarly, the contenton resoluton may leads to a large number of collsons when the robablty that a SS makes BRs s hgh. The motvaton of ths research s how does the BS schedule those two tyes of BR mechansms to serve the SS whle mantanng good system erformance?. An arorate decson made by the BS s needed n order to acheve the desred erformance objectves. Thus, the mact of ths research s to hel the BS make schedulng decsons between two tyes of BR mechansm secfed n the standard n order to meet our erformance objectves. There are two roosed erformance objectves consdered n ths aer: Maxmzng the bandwdth utlzaton whle satsfyng the desred delay. 2 Mnmzng the delay whle mantanng the target bandwdth utlzaton. To acheve the erformance objectves resectvely, two schedulng algorthms are roosed n Secton V: M AX U (for the frst objectve and M IN D (for the second objectve. Many research works related to those two BRs mechansms are only focused on the otmzaton of one tye of BRs mechansms based on the assumton that only one tye of BR mechansms s avalable to be used. A comrehensve study consderng both mechansms s desred for the BS to schedule the connectons whch are allowed to send BRs va both mechansms. In ths aer, we rovde mathematcal analyss for both BR mechansms. Based on the analyss, we roosed two schedulng algorthms for erformance objectves to hel the BS make an arorate schedulng decson such that the system can have better erformance. The rest of aer s organzed as follows. An overvew of IEEE and the related work are rovded n Secton II and Secton III, resectvely. Our mathematcal analyss of both BRs mechansms s gven n Secton IV. In Secton V, we ntroduce the objectves and roosed schedulng algorthms. Secton VI resents the smulaton and Secton VII concludes our dscusson. II. OVERVIEW OF IEEE A IEEE network s comosed by a number of SSs and at least one BS. There are two oeratonal modes, ont-to-multont (PMP and mesh, defned n the IEEE standard. Ths aer s focused on the PMP mode whch defnes that transmssons are only allowed between the BS and SSs. All transmssons can be classfed nto downlnk (DL and ulnk (UL transmsson based on the drecton of transmssons. The DL transmsson s defned as the transmsson from the BS to a SS. Conversely, the UL transmsson s the transmsson n the ooste drecton. Accordng to the IEEE standard, the BS s resonsble for schedulng both UL and DL transmssons. All schedulng behavor s exressed n a MAC frame. The structure of a MAC frame defned n the IEEE standard can be dvded nto UL subframe and DL subframe. The UL subframe s for UL transmssons. Smlarly, the DL subframe s for DL transmssons. In a IEEE network, all SSs should be coordnated by the BS. All coordnatng nformaton ncludng burst rofles and offsets s resded n the DL and UL mas, whch are broadcasted at the begnnng of the MAC frame. The IEEE network s connecton-orented. It reures SSs to establsh connectons wth the BS before any data transmssons. In order to suort wde varety of alcatons, the IEEE standard classfes all traffcs nto fve schedulng classes based on the dfferent QoS reurements: Unsolcted Grant Servce (UGS, Real Tme Pollng Servce (rtps, Extended Real Tme Pollng Servce (ertps [?], Nonreal Tme Pollng Servce (nrtps and Best Effort (BE. The mechansm to make BRs for each schedulng class has been secfed n the IEEE standard. For examle, a fxed amount of bandwdth s gven to UGS connectons and BRs are rohbted to be made for ths tye of connectons. All connectons n other schedulng classes (.e. rtps, ertps, nrtps and BE are allowed to make BRs va uncast ollng oortuntes. However, ertps, nrtps and BE connectons are the only connectons whch are allowed to reuest bandwdth va contenton resoluton. The oeraton rocedure of uncast ollng defned n the IEEE standard s straght forward. The BS allocates an extra ece of bandwdth to the target SS. Ths extra ece of bandwdth can be consdered as one or more uncast ollng TxOPs. The target SS makes bandwdth reuests by utlzng these TxOPs. Snce these TxOPs are exclusvely allocated to ths artcular SS, t can ensure that ths SS has oortuntes to reuest bandwdth f needed. However, the drawback s that these TxOPs are wasted f ths SS does not make BRs. The contenton resoluton, on the other hand, s not TxOPsguaranteed. It means that the SS may not have oortuntes to transmt BRs due to the falures of contenton. The BS schedules a few contenton TxOPs for a grou of SSs. Each SS wthn ths grou s reured to contend for a contenton TxOP wth each other n order to transmt a BR. Note that each contenton TxOP can only carry one BR. If the SS fals n the contenton rocedure, t enters nto the back-off rocedure for rearng the next attemt. In ths aer, the bnary exonental back-off (BEB algorthm [] s emloyed as the back-off rocedure. The ntal back-off wndow sze and the maxmum back-off wndow sze are controlled by the BS and secfed n the UL ma. The value of contenton wndow sze s reresented as a ower-of-two value. For examle, a value of 4 ndcates that the contenton wndow sze s 6. The oeraton rocedure of contenton resoluton s summarzed as Fg.. When a SS tends to contend a TxOP, t selects a random number from 0 to W, where W s the current back-off wndow sze. Ths random number s called back-off counter and ndcates the number of contenton TxOPs that the SS shall defer before transmttng. The number of contenton Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

3 3 Fg.. The oeraton of contenton resoluton TxOPs s determned by the BS and may be dfferent n each frame. If the back-off counter does not reach zero wthn a contenton erod. Its countdown should be frozen at the end of the contenton erod and resume at the begnnng of the next comng contenton erod. When the back-off counter reaches zero, the SS attemts to send a BR. It s ossble that there are more than one SS whch back-off counter reaches zero at the same tme. It means that there are more than one SSs tryng to send a BR n the same TxOP. In ths case, collson haens. Snce t s not ractcally ossble for SSs to sense the UL channel to detect a collson, the SS can only know the success of BR transmsson f t receves a resonse from the BS n the form of bandwdth grant wthn a fxed number of subseuent UL ma messages. If the SS fals to receves the resonse, t consders that the BR was not delvered successfully. The SS shall double ts back-off wndow sze f the current contenton wndow sze s smaller than the maxmum back-off wndow sze whch s controlled by the BS. The SS selects a fresh random number from 0 to W, where the W ndcates the new backoff wndow sze, and reeat the deferrng rocedure descrbed above. The SS can attemt to transmt BRs untl the maxmum number of retres s reached. III. RELATED WORK Many research works related to uncast ollng and contenton resoluton have been roosed n the lterature. In [3], an adatve ollng scheme for ON/OFF traffc was roosed to mrove the bandwdth utlzaton for uncast ollng. Durng ON erods, ollng ntervals are fxed and short, whle durng OFF erods ollng ntervals are lengthened exonentally. Therefore, adatve ollng reduces the sgnalng overhead wthout sgnfcantly comromsng delay erformance. A Markov chan model for uncast ollng s roosed n [4]. The authors roosed the Markov chan analyss whch ams to mnmze average ollng delay whle ncreasng network throughut. Based on the QoS reurements of each schedulng class, the rortes can be gven between schedulng classes. However, ths obtans reward only from hgh-class servces because the rorty does not dfferentate the rortes of nodes. Contenton resoluton has been dscussed not only n IEEE but also n IEEE A classc Markov Chan model to analyze contenton resoluton n IEEE 802. has been roosed n [7]. Because the bandwdth reservaton s emloyed n the IEEE standard, t s not ractcal for the SS n the IEEE network to sense the medum status. Instead, the SS n the IEEE network wats a fxed number of subseuent UL mas for recevng the resonse from the BS before enterng nto the back-off rocedure. By consderng ths dfference, a Markov model of contenton n the IEEE network s roosed n [6]. Ths model conssts of two tyes of states: back-off states and watng states. The former llustrate the contenton rocedure. The latter reresent the status that the SS wats for the resonse from the BS before enterng nto the back-off rocedure. The arameters that control the contenton resoluton n the IEEE network such as back-off start/end values have been nvestgated n [2]. Moreover, the connectons belongng to three tyes of schedulng classes (.e. ertps, nrtps and BE are able to jon the contenton resoluton. The connecton n each schedulng class has ts own QoS reurements. However, there are no rortes emloyed n the contenton resoluton snce the BS fxes the ntal and maxmum back-off wndow and each SS n the system uses the same value for all connectons. In order to dstngush the rortes between the connectons n dfferent schedulng classes, a modfed contenton resoluton rocess s roosed [8] to mrove the system erformance ncludng end-to-end delay and throughut by assgnng dfferent ntal wndow sze to the connecton n dfferent schedulng class. The research works summarzed above rovde the nvestgaton of ether uncast ollng or contenton resoluton. However, the connectons n the schedulng classes of ertps, nrtps and BE are allowed to use both bandwdth reuest mechansms (.e. uncast ollng and contenton resoluton. Unfortunately, none of the research works shown above take ths oton nto consderatons. Ther research s based on the assumton that only one bandwdth reuest mechansm s avalable. A research work consderng both BR mechansms s resented n [0]. The authors frst comare two bandwdth reuest mechansms secfed n the standard. Ther results demonstrate that the contenton resoluton outerforms uncast ollng when the robablty of makng bandwdth reuests rate s low. However, the authors do not rovde detaled analyss for each tye of bandwdth reuest mechansms. Moreover, the schedulng algorthms to hel the BS make schedulng decsons are desred. In ths aer, two major contrbutons are ncluded. Frst, a comrehensve study of both BR mechansms s rovded. We erform the erformance analyss of each bandwdth reuest mechansm n terms of bandwdth utlzaton and delay. Second, two erformance objectves are roosed. In order to acheve each of our roosed erformance objectves, two schedulng algorthms are roosed to reach them ndvdually. The smulaton results resented n Secton VI show that our schedulng algorthms can also have the better erformance Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

4 4 whle the corresondng erformance objectves are satsfed. IV. ANALYTICAL MODELING In ths secton, we analyze the erformance of each BR mechansm n terms of the bandwdth utlzaton and the delay of delverng a BR. The network model used for analyzng both BR mechansms s comosed by a BS resdng at the center of geograhcal area and N SSs randomly dstrbuted n the servce coverage of the BS. Each SS serves one dentcal varable bt rate (VBR traffc, based on the traffc model ntroduced n [5], whch s classfed as a BE connecton wth the average robablty P r to transmt bandwdth reuests. Addtonally, we assume that each SS transmts at most one BR durng the exected delay. Ths assumton s reasonable snce there s no maxmum delay reurement n BE connectons and our objectve s to make that the average delay s no more than the exected delay. Although ggybackng defned n the IEEE standard s another way for SSs to transmt BRs, however, t s otonal and not able to carry all tyes of BRs. Conseuently, we do not consder ggybackng n ths aer. A. Uncast ollng We begn our analyss of uncast ollng by nvestgatng the mnmum average number of uncast ollng TxOPs allocated n each frame and the average delay of transmttng a BR. For ease of reference, a lst of mortant notatons are summarzed n Table I. Notaton N N M T P r U Descrton Total number of SSs Number of SSs wth uncast ollng TxOPs Number of frames er second The mnmum average number of uncast ollng TxOPs er frame The exected delay The robablty of each SS to send BR Bandwdth Utlzaton of uncast ollng TABLE I LIST OF NOTATIONS FOR UNICAST POLLING Assume N s the total number of SSs assgned wth uncast ollng TxOPs, where 0 N N. Snce t s not necessary to schedule an uncast ollng TxOP to each SS n every frame, we focus on the mnmum number of uncast ollng TxOPs whch should be scheduled er frame n order to acheve the exected delay. Assume that the robablty of the SS to make a BR s unformly dstrbuted between two consecutve uncast ollng TxOPs. In order to mantan the exected delay, denoted as T, the BS has to schedule at least one uncast ollng TxOP to the SS n every 2T. Conseuently, the mnmum average number of uncast ollng TxOPs assgned to each frame can be exressed as: M N 2T ( where M stands for the mnmum average number of uncast ollng TxOP scheduled n each frame. Because of the nature of uncast ollng, the uncast ollng TxOP s wasted f the assgned SS does not transmtted a BR. Therefore, the bandwdth utlzaton of uncast ollng s same as the robablty of a SS to transmtted a BR (.e. U = P r. B. Contenton Resoluton Notaton N N c T c P r S E W S W E R b(, r Descrton Total number of SSs Number of SSs wth contenton TxOPs The number of frames er second The mnmum average number of TxOPs for contenton resoluton er frame The target delay of contenton resoluton The robablty of a SS to send BR Back-off start value Back-off end value Probablty of a unsuccessful transmsson Intal back-off wndow sze Maxmum back-off wndow sze Maxmum number of retres Probablty of the BS to accet a BR A back-off state n -th attemt wth random back-off counter r w (, t A watng state n the branch of collson/ w 2 (, t non-collson n -th attemt and the SS has wated for t frames f τ the robablty of falures The robablty of a SS to transmt a BR n a randomly chosen TxOP T The exected delay before/ after the T 2 contenton wndow sze reaches the W E. T w The maxmum number of subseuent UL-MAP messages that a SS wats for a resonse from the BS TABLE II LIST OF NOTATIONS FOR CONTENTION RESOLUTION We analyze the contenton resoluton n IEEE network by usng a 2-D Markov Chan (MC model n Fg 2. As shown n the fgure, each SS attemts to transmt a BR untl the number of attemts reaches the maxmum retry lmt R. If the SS cannot transmt a BR successfully n R attemts, ths BR shall be dscarded. A lst of mortant notatons are summarzed n Table II. Accordng to the secfcaton of contenton resoluton rocedure descrbed n the IEEE standard, the SS shall select a random value wthn ts back-off wndow. Ths random number ndcates the number of contenton TxOPs that the SS shall defer before transmttng a BR. After the contenton transmsson, the SS has to wat for a fxed number of subseuent UL mas before enterng nto the back-off rocedure. Therefore, the contenton resoluton rocedure s classfed nto two lanes: back-off lane and watng lane. Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

5 5 Transmsson Comleted Start 0, Tw - - 0, - 0, 0-0,0 0, 0,2 0,Ws- 0, Tw 0, 0, 0 -, Tw - -, -, 0 -,0,,2,Ws-, Tw,, 0 -, Tw - -, -, 0 -,0,,2,Ws-, Tw,, 0 - E-S, Tw - - E-S, - E-S, 0 - E-S,0 E-S, E-S,2 E-S,WE- E-S, Tw E-S, E-S, 0 - R-, Tw - - R-, - R-, 0 - R-,0 R-, R-,2 R-,WE- R-, Tw R-, R-, 0 - Watng lane Dscard Back-off lane Fg. 2. Markov Chan model for contenton resoluton The back-off lane descrbes how the SS transmts a BR (.e. BEB n ths aer. After transmttng a BR, the SS should wat for the resonse from the BS. The watng lane s used to reresent ths watng erod. In Fg. 2 all states n back-off lane and watng lane are denoted as ellses and rectangles, resectvely. In back-off lane, each back-off state, denoted as b(, r, reresents the -th attemt of sendng a BR wth a randomchosen back-off counter r. Ths 2-D MC modelng s ossble f we assume an ndeendent and constant robablty of an unsuccessful reuest,, for each attemt. It s ntutve that ths assumton results more accurate as long as the backoff wndow sze, W, and the number of SSs wth contenton resoluton TxOPs, N c, get larger. The correctness of ths assumton has been roven n [6]. We refer to as the condtonal collson robablty [7]. A SS starts to transmt a BR when ts back-off counter euals to 0, regardless of the back-off stage. Once the ndeendence s assumed, s suorted to be a constant value. After a BR s transmtted, the SS enters nto watng lane whch reresents that the SS wats for a resonse from the BS. Accordng to the IEEE standard, the SS should consder that the transmsson was faled f t does not receve a resonse from the BS wthn the number of subseuent UL-MAP messages secfed by the arameter of Contentonbased Reservaton Tmeout. Here, we use T w to reresent the maxmum number of subseuent UL-MAP messages that the SS can wat before enterng nto the back-off rocedure. There are two ossbltes that the SS cannot receve a resonse wthn T w subseuent UL-MAPs: the BR s collded wth another BR sent from other SSs. 2 the BR s rejected by the BS. Based on these two ossbltes, the watng states are classfed nto two branches: collson and non-collson. The states n collson branch and non-collson branch are reresented by w (, t and w 2 (, t, resectvely, where s the -th attemt and t s the number of subseuent UL-MAP Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

6 6 messages that the SS has wated after transmttng a BR. As mentoned, s the robablty of an unsuccessful reuest. Thus, the robablty to enter the branch of collson s also. It can obtan that the robablty of transton between all states n the branch of collson s due to the falure of the BR transmsson. Intutvely, the robablty to enter the states n the branch of non-collson s. It s ossble that the BS receves a BR successfully but rejects t due to the lack of rado resource or volatng ts schedulng olces. Suose s the robablty of the BS to accet a BR n each frame. It s reasonable to assume that s a constant for the watng states of ths 2-D Markov chan model. In fact, s controlled by the olcy of admsson control and s ndeendent of the oeraton of the MAC layer. By combnng these two factors whch may cause falures of BR transmssons (collson and rejecton by the BS, the robablty of falures, denoted as f, can be reresented as: f = ( ( T w (2 Here, s the robablty of enterng nto the branch of collson. ( ( T w denotes the robablty of enterng nto the branch of non-collson but no resonse receved from the BS. It leads to the followng observaton: b(, 0 = f b(, 0 0 < R (3 P {b(, k b(, k } = k (0, W 2 (0, m (4a P {b(, k b(, 0} = W k (0, W (0, E S (4b P {b(, k b(, 0} = f W E k (0, W E (E S, R (4c P {w (, 0 b(, 0} = P {w (, t w (, t } = (0, R, t (0, T w (4f P {b(, r w 2 (, T w } = (E S, R, r (0, W E (4k f (0, R (4d P {w 2 (, 0 b(, 0} = (0, R (4e P {w 2 (, t w 2 (, t } = P {b(, r w (, T w } = (0, R, t (0, T w (4g (0, E S, r (0, W (4h P {b(, r w (, T w } = (E S, R, r (0, W E (4 P {b(, r w 2 (, T w } = (0, E S, r (0, W (4j Based on euaton (2 and (3, the robabltes of transton between states shown n Fg 2 are summarzed n euaton (4a (4k. Euaton (4a reresents the countdown of backoff counter. Euaton (4b and (4c llustrate the robablty of enterng to each back-off state whle the wndow sze has and has not reached the maxmum wndow sze, resectvely. The robabltes of enterng nto the branch of collson and noncollson are shown n euaton (4d and (4e, resectvely. Euaton (4f and (4g are the robablty between states between the branch of collson and non-collson, resectvely. Euaton (4h and (4 exress that the SS enters nto the back-off rocedure from the branch of collson wth dfferent contenton wndow sze. Smlarly, euaton (4j and (4k exress that the SS enters nto the back-off rocedure from the branch of non-collson wth dfferent contenton wndow sze. Based on the sze of contenton wndow, the back-off states can be classfed nto two tyes; Tye : the sze of contenton wndow s smaller than W E. Tye 2: the sze of contenton wndow has reached W E. Suose b(, k and b(j, k 2 denote the back-off states n Tye and Tye 2, resectvely. Addtonally, w (, t and w 2 (, t stand for the watng states n the branch of collson and non-collson, resectvely. Suose P ds reresents the robablty that a SS dscards a BR because ths BR cannot be transmtted successfully n R attemts. Thus, The sum of robablty n all the states lus P ds must eual to as shown n euaton (5. By smlfyng euaton (5, we can derve b(0, 0 shown n euaton (6. = = E S R W S k =0 T w t =0 R T w t E S R W S k =0 b(, k w (, t j=e S k 2=0 R T w t =0 R T w t =0 R W E j=e S k 2 =0 w 2 (, t f b(r, 0 W E ( W S k b(, 0 b(, 0 W S ( W E k 2 b(j, 0 W E ( ( ( t b(, 0 f b(r, 0 { E S b(0, 0 = f ( W s 2 = R j=e S R b(0, 0 2 j f ( W E R 2( ( ( T w {[ 2 T w 2T w f f 2P R f b(j, k 2 } Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

7 7 b(0, 0 = 2 ] 2( ( ( Tw R f (2 f E S W S f 2 f E S 2 E S f R f W S 2 R f f {[ } 2 T w 2( ( ( T w R f (2 f E S W S f 2 f E S 2 E S f R f W S 2 R f f (5 ] } (6 The robablty that a SS transmts a BR n a randomly chosen contenton TxOP can be calculated as the sum of b(, 0, where 0 R. Ths robablty, denoted as τ, s exressed as: τ = = R b(, 0 b(0, 0 f (7 As shown n euaton (3, b(0, 0 s reresented as a functon of P f whch s a functon of resented n euaton (2. Thus, the value of τ stated n euaton (7 can be exressed as a functon of the condtonal collson robablty,, whch s unknown n our model. Agan, s the robablty that a collson occurs, whch s euvalent to the robablty of at least two SSs transmttng BRs at the same contenton TxOP. Thus, can be reresented as = ( τ (N c P r where τ s the robablty that a SS transmts a BR at the randomly chosen contenton TxOP shown n euaton (7. By usng euaton (7 and (8, we can solve these two unknown values, and τ, based on the known value of backoff start and end value (.e. S and E, the robablty of a SS to send a BR (.e. P r, the robablty of a BS to accet a BR (.e. and the number of SSs wth contenton TxOPs (.e. N c. To analyze the bandwdth usage of contenton TxOPs, t s necessary to fnd the bandwdth utlzaton, U c, whch s defned as the rato of the number of TxOPs whch delver BRs successfully to the total number of contenton TxOPs. To get ths rato, frst we nvestgate the robablty of transmsson, denoted as tx, whch s referred to the robablty that at least one SS transmttng a BR at a TxOP. Ths robablty can be obtaned as: tx = ( τ N c P r (9 The robablty of a successful transmsson, denoted as st, s the robablty that a BR s delvered successfully and (8 the BS grants ths BR. Ths robablty can be acheved by usng condtonal robablty that only one SS transmts a BR at a TxOP and the BS has enough bandwdth to serve ths BR under the condton that at least one transmsson s transmtted at ths TxOP. Therefore, the robablty of a successful transmsson can be addressed as: st = nτ( τ(n cp r tx ( ( T w (0 From euaton (9 and (0, the robablty of a TxOP whch delvers a BR successfully, reresented as sbr, s derved as: sbr = st tx = nτ( τ (NcP r ( ( Tw ( Intutvely, the robablty that a BR s delvered n a gven TxOP s euvalent to the robablty of a TxOP to be utlzed successfully. Conseuently, the bandwdth utlzaton of contenton TxOPs, U c s same as sbr. Although the maxmum delay reurement s not a necessary reurement for BE connectons, n ractce, we stll hoe the delay can be lmted nto certan bound whch s consdered as our exected delay, T c. Here, the delay s calculated as the tme dfference between the tme that the SS ntends to send a BR and the tme that the SS receves a resonse from the BS. One of the mortant factors to affect the delay s the number of contenton TxOPs scheduled by the BS n each frame. In ths aer, we focus on the relaton between the mnmum average number of contenton TxOPs assgned er frame (denoted as and the target delay (denoted as T c. Based on the contenton wndows sze, the exected delay can be calculated nto two sectons: E S and 2 E S < R, where s the -th attemt. Let T stand for the exected delay n the frst secton. It can be calculated as euaton (2. Smlarly, the delay of the second secton, denoted as T 2, can be derved as euaton (3. It s ntutve that the sum of the delay of two sectons s at most the target delay whch s reresented as T c. Moreover, n euaton (2 and (3, everythng s known excet and whch are the mnmum average number of contenton TxOPs assgned er frame and the robablty of an unsuccessful transmsson, resectvely. Therefore, we can use H(, to reresent the total delay as the sum of delay n these two sectons. By wrtng formally, t can be exressed as euaton (4. T = T 2 = E j=s T w j S f ( f [ ( ( j W m m=s R n=e S2 W j W m k=0 n f ( P f [ k W E W j k=0 T w W E k=0 k ] k (2 Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

8 8 where T w ( ( W E n d=e S 2 T b ] T b = E S m=s ( W m W E k=0 W m k=0 k k T w T w (3 T c T T 2 = H(, (4 V. SCHEDULING ALGORITHMS FOR PERFORMANCE OBJECTIVES Based on the analyss shown n secton IV, we roosed two schedulng algorthms to meet the two erformance objectves roosed n ths aer, resectvely: Maxmze the bandwdth utlzaton under the condton of satsfyng a gven target delay reurement (reresented as Fxed-delay-MAX-Utlzaton n the rest of ths secton 2 Mnmze the target delay when a gven bandwdth utlzaton as a constrant s gven (reresented as Fxed-Utlzaton-MIN-delay n the rest of ths secton To meet the frst objectve, a schedulng algorthm, called Maxmum Bandwdth Utlzaton Schedulng Algorthm (MAX-U, s roosed. It hels the BS to schedule the number of TxOPs and the number of artcatng SSs for each BR mechansm n order to maxmze the bandwdth utlzaton whle satsfyng the target delay. Smlarly, the schedulng algorthm roosed for the second objectve s called Mnmze Delay Schedulng Algorthm (MIN-D. It hels the BS to fnd the combnaton of TxOPs assgned for each BR mechansm such that the system delay s mnmzed whle mantanng the desred utlzaton. Note that both schedulng algorthms hel the BS schedule ether uncast ollng TxOPs or contenton TxOPs to each SS to acheve the corresondng erformance objectve. No SSs receve both tyes of TxOPs at the same tme. In ths aer, we only focus on the BR mechansms. Thus, the bandwdth utlzaton ndcated n ths aer s the bandwdth utlzaton of TxOPs assgned for both BR mechansms (.e. uncast ollng and contenton resoluton. Moreover, the TxOPs scheduled for each mechansm are only used for transmttng BR messages. A. MAX-U Ths algorthm s desgned to satsfy our frst erformance objectve: maxmze the bandwdth utlzaton whle satsfyng the fxed delay reurement. The flow of ths algorthm s shown n Fg. 3. Suose T D s the gven achevable target Algorthm MAX-U Inut: All varables secfed n Table I and II Outut: N s the number of SSs wth uncast ollng TxOPs. N c s the number of SSs wth contenton resoluton TxOP. M s the average number of TxOPs scheduled for uncast ollng er frame. s the average number of TxOPs scheduled for contenton resoluton er frame. For =0 to N do N, Nc N N Uncast Pollng: M M calculate by euaton ( Contenton Resoluton: a. Solve τ and by usng euaton (7 and (8 wth gven Nc. b. Mc Mc calculated by euaton (4 wth a known. c. sbr sbr calculated by euaton (. Fnalze: U t M P rm c sbr M M c End For U t Max{U t } wth Mn{M c M } M M, N N, M c, N c N c Return M N N c Ste Ste 2 Ste 3 Fg. 3. Fnd all combnatons of (N, N c such that N N c = N. For each (N, N c, calculate the corresondng (M, and the bandwdth utlzaton whle the target delay, T D, s satsfed. Return the (N, N c and the corresondng (M, such that the bandwdth utlzaton s maxmzed. The stes of MAX-U delay. Our objectve s to fnd the number of uncast ollng TxOPs and contenton TxOPs scheduled n each frame such that the bandwdth utlzaton s maxmzed. In ths algorthm, each SS s scheduled wth ether one of the BR mechansms: uncast ollng TxOPs and contenton TxOPs. Suose N c and N are the number of SSs scheduled wth contenton and uncast ollng TxOPs, resectvely, such that N c N = N. The objectve of the algorthm s to maxmze the bandwdth utlzaton whle acheve the gven target delay. For all combnatons of (N, N c, we calculate the corresondng value of M and for each combnaton and select a combnaton of (N, N c and the corresondng M and, whch can maxmze the bandwdth utlzaton. Note that the overall throughut may be hgher f we can mnmze the number of TxOPs assgned for BR mechansms because there s more bandwdth whch can be assgned for data transmssons. Therefore, f there are multle combnatons whch result the same maxmum bandwdth utlzaton, the one wth mnmum number of TxOPs (.e. M s selected. Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

9 9 Algorthm 2 MIN-D Inut: All varables secfed n Table I and II Outut: N s the number of SSs wth uncast ollng TxOPs. N c s the number of SSs wth contenton resoluton TxOPs M s the average number of uncast ollng TxOPs er frame. s the average number of contenton resoluton TxOPs er frame. For = 0 to N do N Nc N N K the set of all combnatons of (M, Mc such that euaton (5 s satsfed and M N. For j = to K Uncast Pollng: a. T j Calculated by euaton (. Contenton Resoluton: a. Solve τ and by usng euaton (7 and (8 wth gven Nc. b. Tc j Tc j Calculated by euaton 4. End For TD Mn{Max{T j, Tc j }} M M j Mc Mc j End For T D Mn{TD }, M M, N N, Mc, N c Nc Return M N N c Ste Fnd all combnaton of (N, N c such that N N c = N. Ste 2 (N, N c, fnd all corresondng (M, such that M ( P r ( sbr S u and M N. Ste 3 (N, N c, fnd the corresondng delay of each (M, and select one wth mnmum delay. Ste 4 (N, N c, set the delay as the corresondng delay of the cked (M,. Ste 5 Return the (N, N c wth mnmum delay. Fg. 4. B. MIN-D The stes of MIN-D Ths algorthm focus on achevng our second erformance objectve: mnmzng the delay whle satsfyng a gven bandwdth utlzaton reurement. Assume U t s the gven bandwdth utlzaton wth the fxed number of TxOPs, S t, for both BR mechansms (.e. M = S t. Thus, the number of unused TxOPs, denoted as S u, can be reresented as: S u = ( U t S t It s ntutve that the total unused TxOPs of both BR mechansms are at most S u. Formally, t can be exressed as: M ( P r ( sbr S u (5 Smlar to Algorthm, we exam all combnatons of (N, N c such that N N c = N. Our objectve s to fnd a combnaton of (N, N c wth the mnmum overall exected delay whle euaton (5 s satsfed. For each ar of (N, N c, there exst several ars of (M, whch satsfy the constrant stated n euaton (5. Suose M s the set of ualfed (M, for each ar of (N, N c. Therefore, we check all combnatons of (M, M and fnd the combnatons resultng the delay s mnmzed as our canddates. Here, delay s defned as max{t, T c }, where T and T c are the delay caused by uncast ollng and contenton resoluton, resectvely. Conseuently, for each ar of (N, N c, there are at least one ar of (M, as our canddates. Among these canddates, we ck one canddate wth the mnmum delay as our soluton for the schedulng decson. VI. NUMERICAL AND SIMULATION RESULTS A. System Set U In ths secton, we valdate the theoretcal results wth our smulaton results. The theoretcal results are made by the Matlab 2009a. The smulaton results are conducted by our smulator. The smulator s wrtten n C and followed the IEEE standard closely. Both analytcal and smulaton results are also comared wth two ordnary schemes: Uncast Pollng only. 2 Contenton Resoluton only. Table III summarzes the system arameters used n our numercal analyss and smulaton. In our smulaton, each SS serves one HTTP web browsng traffc [2] [3] whch s classfed as a BE connecton. In order to ncrease the varety of BE traffcs, the mean acket sze s randomly selected from 52 to 024 bytes. Because the mean traffc rate s fxed, the mean traffc rate can be calculated based on the selected mean acket sze. Parameters Value Number of BS Number of SS 200, 300, 400, 500 Frame Duraton 20 ms Modulaton BPSK, QPSK, 6QAM, 64QAM TTG/RTG 0 µs SSTG 4 µs Alcaton HTTP Traffc Tye VBR Schedulng Class BE Mean Packet Sze (byte Mean Traffc Rate 2Kbs B. MAX-U TABLE III SIMULATION PARAMETERS The target delay used n ths smulaton s second whch s the most common delay used for BE connectons. The results of bandwdth utlzaton under dfferent P r are shown n Fg. 5(a and Fg. 5(b. It s easy to observe that the results Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

10 0 of bandwdth utlzaton are smlar wth dfferent number of SSs. It shows that the bandwdth utlzaton does not strongly relate to the number of SSs n the system. The utlzaton of contenton resoluton only s always has around 35 % no matter what value of P r s. On the other hand, the utlzaton of uncast ollng only s very close to the value of P r. By these results, we can conclude that the uncast ollng can acheve better bandwdth utlzaton f P r s larger than Therefore, t s mossble to always reach the better erformance f only one BR mechansm s consdered. As shown n the fgures, our analytc and smulaton results are very close to each other. Ths valdates ths analyss resented n Secton IV. Addtonally, our results always acheve the better bandwdth utlzaton roduced by ether uncast ollng only or contenton resoluton only. For examle, n Fg. 5(a, our algorthm acheves around 35 % of the bandwdth utlzaton when P r = 0., whch s smlar to the one that contenton only acheves. However, uncast ollng only results 0 % of bandwdth utlzaton. When P r = 0.8, both uncast ollng and our algorthm reach 80 % of bandwdth utlzaton. The contenton only stll kees ts bandwdth utlzaton around 35 %. It s because our schedulng algorthm (.e. MAX-U can hel the BS schedule one tye of BR mechansms whch can acheve better erformance accordng to the current network status. It s worth to note that our schedulng algorthm (MAX- U schedules all SSs wth ether uncast ollng TxOPs or contenton TxOPs. The combnatons n between (.e. art of SS wth uncast ollng TxOPs and the rest of them wth contenton TxOPs do not exst. It s because the contenton resoluton can always gve 35 % of bandwdth utlzaton and t wll be chosen f the uncast ollng cannot contrbute as hgh bandwdth utlzaton as t does. On the other hand, the uncast ollng wll always be chosen when t can have more than 35 % n bandwdth utlzaton (.e. P r > 35 %. C. MIN-D Fg. 6(a and Fg. 6(b show the relatonsh between the exected delay and P r whle the target bandwdth utlzaton s 0.3 and 0.5, resectvely. From the fgures, we obtan that our schedulng algorthm (.e. MIN-D always cks a BR mechansm resultng better erformance (.e. shorter delay. For nstance, n Fg. 6(a, both uncast ollng and our algorthm reach 0 ms of delay when P r = 0.4. However, the contenton only kees the delay around 45 ms n all values of P r. In Fg. 6(a, there are no results for uncast ollng only when P r = 0. and 0.2. It s because the bandwdth utlzaton cannot acheve 0.3 f only uncast ollng s used. Smlarly, there are no results for contenton only n Fg. 6(b snce the contenton resoluton cannot reach 50 % of bandwdth utlzaton. Table IV shows the smulaton results of the schedulng algorthm n terms of the number of SSs and the number of TxOPs assgned to each BR mechansm. Here, the target bandwdth utlzaton s 0.3. It s worth to note that both BR mechansms are scheduled for BR transmssons when P r = 0.. It s because the erformance reurement (.e. U = 0.3 cannot be acheve f only one BR mechansms s consdered. Ths result shows an examle that the better erformance can be acheved by schedulng both tyes of BR mechansms. Fg. 5. Fg. 6. Bandwdth Utlzaton (% Bandwdth Utlzaton (% Contenton Only Uncast Pollng Only MAX-U Smulaton MAX-U Theoretcal Pr (% (a Number of SS = 200 Contenton Only Uncast Pollng Only MAX-U Smulaton MAX-U Theoretc Pr (% (b Number of SS = 500 Smulaton Results of MAX-U Exected Delay (ms Exected Delay (ms Pr (% (a U = 0.3 Contenton Only Uncast Pollng Only MIN-D Smulaton MIN-D Theoretcal Pr (% (b U = 0.5 Smulaton Results of MIN-D Uncast Pollng Only MIN-D Smulaton MIN-D Theoretcal Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.

11 P r N M N c TABLE IV SIMULATION RESULTS OF MIN-D VII. CONCLUSION Accordng to the IEEE standard, the connectons belongng to ertps, nrtps and BE are allowed to make bandwdth reuests (BRs va both BR mechansms (.e. uncast ollng and contenton resoluton. The mechansm that the BS schedules to those connectons may result dfferent system erformance because of the nature of each BR mechansm. However, most conventonal research works lmt the oton to consder only one tye of BR mechansms. A schedulng scheme by consderng both tyes of BR mechansms s desred for the BS n order to otmze the system erformance. Besdes, t s not necessary for BS to erform ether uncast ollng or contenton resoluton to all SSs wthn one frame. Instead, the BS needs to schedule the arorate number of contenton resoluton or uncast ollng TxOPs to the SS n order to meet the delay reurement. Therefore, the schedulng decson should be made n a mult-frame bass. In ths aer, we rovde the erformance analyss of each BR mechansms n terms of bandwdth utlzaton and exected delay. Based on the analyss, we take both BR mechansms nto account and roose two schedulng algorthms to hel the BS make the schedulng decson based on the current network status such that the corresondng erformance objectves are acheved. There are two erformance objectves roosed n ths aer: Maxmzng the bandwdth utlzaton under the condton that the target delay s satsfed. 2 Mnmzng the delay whle the desred bandwdth utlzaton s reached. Our numercal and smulaton confrm that the schedulng algorthms can always have the better erformance by schedulng the number of transmsson oortuntes to one of the BR mechansm. Addtonally, when the robablty of makng BR (.e. P r s 0., a hybrd decson (.e. Schedulng SSs wth two BR mechansms can conduct the mnmum delay whle satsfyng the desred bandwdth utlzaton. [5] Thomas G. Rbertazz Comuter Networks and Systems : Theory and Performance Evaluaton. Srnger-Verlag 990. [6] Yaser Pourmohammad, Farshd Agharebarast, Mahmood R. Mnhas, Hussen M. Alnuwer and Vctor C.M. Leung Analytcal Modelng of Contenton-Based Bandwdth Reust Mechansm n IEEE Wreless Networks IEEE Transacton on Vehcular Technology, Vol. 57, No. 5 Setember [7] Gusee Banch Performance Analyss of the IEEE 802. Dstrbuted Coordnaton Functon, IEEE Journal on Selected Areas n Communcatons, Vol. 8, No. 3, March [8] Jesús Delcado, Qang N, Francsco M. Delcado and Lus Orozco- Barbosa New Contenton Resoluton Schemes for WMAX, WCNC 2009,.-6 [9] Hossam Fattah and Hussen Alnuwer Performance Evaluaton of Contenton-Based Access n IEEE Networks wth Subchannelzaton, ICC 2009,.-6 [0] Qang N, Alexey Vnel, Yang Xao, Andrey Turlkov and Tao Jang Investgaton of Bandwdth Reuest Mechansms under Pont-to-MultPont Mode of WMAX Networks IEEE Communcaton Magazne, 32-38, May [] Robert M. Metcalfe and Davd R. Boggs, Ethernet: Dstrbuted Packet Swtchng for Local Comuter Networks, Communcatons of the ACM, vol. 9, no. 7,. 395 V 404, July 976. [2] M. Molna, P. Castell and G. Foodes, Web traffc modelng exlotng TCP connecton temoral clusterng through HTML-REDUCE, IEEE Network Magazne, vol. 4, no. 3,.46-55, May [3] Z. Sun, et al., Internet QoS and traffc modlng, IEEE Proceedngs Software, Secal Issue on Performance Engneerng, vol. 5, no. 5, , October, 2004 Davd Chuck receved the BS degree n Aled Math from Tatung Unversty, Tawan n 2004 and the MS degree n Comuter Scence from Illnos Insttute of Technology n He s currently workng toward hs Ph.D. degree n Comuter Engneerng at Iowa State Unversty. Hs research nterests nclude wreless networkng, network vrtualzaton, resource allocaton and game theoretcal study of networks. He also worked as a research ntern n NTT Network Innovaton Labs at Yokosuka, Jaan n Kuan-Yu Chen receved the BS degreed n electrcal engneerng from Natonal Chung Cheng Unversty, Chay, Tawan, n In 2008, he was a research assstant at Iowa State Unversty. He s currently workng toward hs MS degree n Comuter Engneerng at Iowa State Unversty. Hs research nterests nclude VoIP, resource allocaton, Qualty of Servce and WMAX technology. REFERENCES [] IEEE WG, IEEE Standard for Local and Metrooltan Area Network Part 6: Ar Interface for Fxed Boardband Wreless Access Systems IEEE Std [2] Alexander Sayenko, Oll Alanen and Tmo Hämälänen, On Contenton Resoluton Parameters for the IEEE Base Staton, GLOBECOM 2007, [3] Chun Ne, Muthaah Venkatachalam and Xangyng Yang Adatve Pollng Servce for Next-Generaton IEEE WMAX Networks, GLOBAECOM 2007, [4] Ben-Jye Chang, Chen-Mng Chou and Yng-Hsn Lang Markov chan analyss of ulnk subframe n ollng-based WMAX networks, Comuter Communcatons 3 (2008, J. Morrs Chang s an assocate rofessor at Iowa State Unversty. Dr. Chang receved hs Ph.D. degree n Comuter Engneerng from North Carolna State Unversty. Hs ndustral exerence ncludes ostons at Texas Instruments, Mcroelectronc Center of North Carolna and AT&T Bell Laboratores. He receved the Unversty Excellence n Teachng Award at Illnos Insttute of Technology n 999. Dr. Chang s research nterests nclude: Wreless Networks, Performance study of Java Vrtual Machnes (JVM, and Comuter Archtecture. Currently, he s a handlng edtor of Journal of Mcrorocessors and Mcrosystems and the Mddleware & Wreless Networks subject area edtor of IEEE IT Professonal. He s a senor member of IEEE. Authorzed lcensed use lmted to: Iowa State Unversty. Downloaded on March 02,200 at 7:25:07 EST from IEEE Xlore. Restrctons aly.