TECHNICAL RESEARCH REPORT

Transcription

1 TECHNICAL RESEARCH REPORT Performance ssues of Bluetooth scatternets and other asynchronous TDMA ad hoc networks by Theodoros Salonds, Leandros Tassulas CSHCN TR 00 (ISR TR 005) The Center for Satellte and Hybrd Communcaton Networks s a NASAsponsored Commercal Space Center also supported by the Department of Defense (DOD), ndustry, the State of Maryland, the Unversty of Maryland and the Insttute for Systems Research. Ths document s a techncal report n the CSHCN seres orgnatng at the Unversty of Maryland. Web ste

2 Performance ssues of Bluetooth scatternets and other asynchronous TDMA ad hoc networks Theodoros Salonds and Leandros Tassulas Insttute for Systems Research Unversty of Maryland at College Park Abstract In ths paper we address a practcal performance ssue arsng n wreless ad hoc networks usng tme dvson multple access (TDMA). Ths ssue relates to the degradaton of the network ablty to allocate bandwdth when the usual assumpton of systemwde slot synchroncty does not hold. The problem s nvestgated for the case of Bluetooth, a new promsng TDMA wreless technology that enables the formaton of ad hoc networks called scatternets. A scatternet does not support a global slot synchronzaton mechansm. Instead, nodes are grouped n multple channels called pconets and each pconet uses a dfferent tme slot reference provded by a node desgnated as master. Traffc forwardng between pconets s performed n a tme dvson fashon by brdge nodes that are aware of the tme references of the pconets they partcpate n. Due to the nherent lack of global synchroncty, slots are wasted when brdges swtch pconet tme references. Ths translates to a bandwdth loss compared to an deally synchronzed scatternet. Whle the exstence of ths asynchroncty overhead has been reported n prevous work, there has been no effort to mnmze t or even evaluate ts possble extent. Nevertheless, these ssues are very mportant for the deployment of practcal TDMAbased ad hoc networks. We consder a scatternet that allocates bandwdth to ts lnks usng an asynchronous perodc conflctfree TDMA schedule. In ths settng the asynchroncty overhead s manfested as an ncrease n the mnmum perod requred to realze a demand allocaton when compared to a perfectly synchronzed system. We frst derve a general upper bound on the overhead of any scatternet and demand allocaton. Then we consder the problem of mnmzng the overhead gven a scatternet confguraton and demand allocaton on ts lnks. We cast ths as a combnatoral optmzaton problem and propose two algorthms for ts soluton. The frst algorthm reaches the optmal soluton usng exhaustve search. However ths approach can be computatonally prohbtve for large problem szes. To ths, we ntroduce a second practcal heurstc algorthm of polynomal complexty. Usng smulatons, the heurstc s shown to perform very well for problem szes where the optmal can be computed. For large problem szes we use the heurstc to nvestgate the effect of the varous system parameters on the generated overhead and evaluate ts performance wth respect to the derved general upper bound. I. INTRODUCTION A wreless ad hoc network s a collecton of nodes equpped wth rado nterfaces and form a multhop wreless nfrastructure wthout the ad of any centralzed admnstraton. Tme dvson multple access (TDMA) s a well known medum access scheme for determnstc bandwdth allocaton and qualty of servce provson n ad hoc networks. Accordng to TDMA, bandwdth can be allocated to the network lnks usng a schedule of perod T slots. At every slot of such a schedule, several lnks are actvated for transmsson such that no conflcts occur at the nted recevers. Then the amount of conflctfree slots a lnk gets wthn a perod T determnes ts allocated bandwdth. A central performance ssue arsng n a TDMAbased ad hoc network s the determnaton of the set of allocatons t can acheve. A demand lnk rate allocaton r =[r l ] (0» r l» ) s feasble f the network can allocate f l = br l Tc conflctfree slots to every lnk l wthout exceedng the system perod T. Ths decson problem s ntrnscally coupled wth an optmzaton one: Fnd a lnk schedule of mnmum perod that realzes slot allocaton f = [f l ]. If the soluton of ths problem s less that the system perod then the allocaton s feasble, otherwse t s not. The lnk schedule optmzaton problem has been studed for ad hoc networks usng a sngle broadcast channel [] and ones usng multple ponttopont channels at the physcal layer []. In the frst case the problem has been shown to be NPcomplete whle n the second t can be solved by a polynomal algorthm of hgh complexty. The works n [7][6] provde heurstc technques of lower complexty that compute suboptmal lnk schedules. The above performance studes as well as most proposed centralzed or dstrbuted TDMA protocols for slotted ad hoc networks assume that the tme slot boundares are provded by a global system clock. Ths systemwde synchronzaton mechansm s not always possble to acheve n the dstrbuted ad hoc network settng. Ths s the man reason that some current standardzed technologes for ad hoc networks such as WLAN use asynchronous random access schemes (80. DFWMAC). Nevertheless, random access cannot provde wth any bandwdth allocaton guarantees to the network. Bluetooth [] s a new TDMA wreless technology that enables the formaton of ad hoc networks called scatternets. Whle beng a slotted system, Bluetooth has the nterestng feature of not supportng a global slot synchronzaton mechansm. Instead, tme reference s provded locally for each lnk by one of the node ponts actng as master. A node that acts as slave to more than one lnks swtches tme reference when t needs to talk to a new master. When ths happens, a slot must be wasted by the slave for tunng to the tme reference of the lnk master. Ths phenomenon has been reported n works related to scatternet schedulng [9] [0] [] [] [] [5] as a source of overhead. However there has been no attempt to formally study ts effect n the ablty of the system to allocate bandwdth. Ths ablty s lnked to the determnaton of the feasble allocatons

3 regon, or equvalently, the soluton of the related lnk schedule optmzaton problem. Gven a demand allocaton, the mnmum perod acheved by an asynchronous TDMA system s expected to be greater than the one acheved by a perfectly synchronzed one. Ths s because the varous tme reference swtches over tme can have a cumulatve addtve effect on the overall mnmum perod requred by the asynchronous system. The ncrease n the mnmum perod s essentally the overhead ntroduced by the system asynchroncty. Based on the above observaton, we can use a two step approach to address the lnk schedule optmzaton problem for the asynchronous TDMA scattternet settng. The frst step assumes perfect synchronzaton and fnds a synchronzed lnk schedule of mnmum perod that realzes the demand allocaton. Bluetooth falls n the category of multchannel systems studed n [] and the algorthms and results theren can be used for ths purpose. The second step (whch s the contrbuton of ths paper), uses the optmal synchronzed schedule as a reference to fnd an asynchronous schedule of mnmum overhead. It turns out that the overhead deps on the order of lnk actvatons n the reference synchronzed schedule. We ntroduce two algorthms for addressng ths problem. The frst algorthm derves a mnmum overhead asynchronous schedule for a specfc lnk actvaton orderng of the synchronzed schedule. It also has an upper bound for the overhead t generates for any possble nput orderng or scatternet confguraton. Usng ths algorthm t s possble to reach the optmal soluton by executng t over all possble orderngs. Ths leads to a problem of combnatoral nature that prohbts exhaustve search for large problem szes. To ths we ntroduce a second heurstc algorthm of polynomal complexty. The heurstc s shown to have excellent performance for problem szes where the optmal can be computed. For large problem szes we nvestgate the effect of the varous system parameters to the generated overhead and compare the heurstc performance to the derved upper bound. The rest of the paper s organzed as follows. Secton II s an ntroducton to the archtecture of Bluetooth scatternets and related work on ther schedulng. Secton III ntroduces a schedulng framework for allocatng bandwdth n the asynchronous scatternet settng by means of perodc conflctfree lnk schedules. Secton IV and V provde the formulaton of the asynchroncty overhead problem and the algorthms used for mnmzng t. Secton VI evaluates the algorthm performance and nvestgates the effect of varous factors affectng the asynchroncty overhead. Fnally, secton VII concludes the paper. II. PICONETS AND SCATTERNETS Every Bluetooth unt has an nternal natve system clock that determnes the tmng and hoppng of the rado transcever. The natve clocks of dfferent Bluetooth nodes are not synchronzed and dffer by a phase. Clock synchronzaton happens only locally when nodes are grouped n multple communcaton channels called pconets. In each pconet, one unt assumes the role of master whle the others act as slaves. The master defnes the pconet frequency hoppng sequence and provdes ts natve clock as the pconet tme reference. Wthn a pconet, a master controls access to the channel by pollng slaves accordng to a slotted Tme Dvson Duplex (TDD) protocol. Accordng to ths protocol, each pconet slot conssts of a masterslave par of halfduplex mnslots and therefore supports full duplex communcaton. All slaves know the slot start tme and the frequency hop of a pconet slot and lsten passvely for a poll by the master. The master then selects one of the slaves by pollng t durng the frst half of the pconet slot and then the slave can respond n the second half. Pconets can be nterconnected va brdge nodes to form a bgger ad hoc network known as a scatternet. Brdge nodes can tmeshare between multple pconets, recevng data from one pconet and forwardng t to another. There s no restrcton on the role a brdge node can play n each pconet t partcpates n. A brdge node can be a master n at most one pconet and slave n another (termed as M/S brdge) or a slave n all pconets (termed as S/S brdge). The Bluetooth technology standard [] has not yet specfed the way brdges should schedule ther vsts n dfferent pconets and there s currently an ntense research effort on ths topc. The emphass s on dstrbuted schedulng schemes and the approaches can be categorzed accordng to the degree of coordnaton they offer. Accordng to hard coordnaton schemes [][5], the lnk schedulng s performed n such a way that when a master polls a slave on a Bluetooth lnk, ths slave s guaranteed to be lstenng on ths pconet. Snce no transmsson conflcts exst, these schemes can potentally acheve strct bandwdth allocaton guarantees. However, there s an assocated mplementaton and communcaton complexty for mantanng the conflctfree property, especally when the scatternet becomes hghly dynamc. Soft coordnaton schemes [][0][] tradeoff perfectly conflctfree transmssons for lower complexty. The downsde here s that ths comes to a loss of the ablty to provde bandwdth guarantees. Whle there s stll a smplcty vs performance debate between the two approaches, the bandwdth loss due to pconet swtchng always exsts due to the asynchronous nature of Bluetooth. In the next secton we ntroduce a hard coordnaton schedulng framework for overhead mnmzaton. There are manly two reasons for dong ths. Frst, n ths case the overhead s naturally lnked to the ablty of the system to allocate bandwdth. Second and most mportant s that a coordnated schedulng approach s the best we can do for mnmzng the overhead and therefore provdes a useful pont of reference. III. SCATTERNET COMMUNICATION MODEL The scatternet s represented as a drected graph G(N;E). A drected edge (; ) E sgnfes that nodes and are wthn wreless range and they have establshed a Bluetooth lnk where s the master and the slave. Each fullduplex pconet slot supports bdrectonal communcaton ntated by the master node: Durng the frst half of the slot the master polls and durng the second half the slave responds f polled by the master. Each Bluetooth node has the followng operatng restrctons:

4 ffl (R.): A slave cannot be lstenng as slave to more than one pconet at the same slot. ffl (R.): A node cannot poll as master and be lstenng to another pconet as slave at the same slot. ffl (R.): A master must not poll more than one slave at the same slot. The frst two constrants are due to the sngle rado transcever n Bluetooth unts, whle the thrd s due to the requrement of conflctfree communcaton n both drectons of a fullduplex pconet slot. These constrants mply that durng a slot, a Bluetooth node can use at most one of ts adacent lnks ether as master (transmt a poll durng the frst half of the slot and lsten for a response n the second half ) or slave (start lstenng for a poll and respond durng the second half of the slot f a poll s receved). A. Conflctfree bandwdth allocaton n scatternets Each node allocates slots to ts adacent lnks by mantanng a local lnk schedule S of perod T system. Each slot entry n S corresponds to a fullduplex pconet slot of duraton of.5ms. The local schedule s wth respect to the node s own natve clock tck and the node uses t to determne ts communcaton acton for the duraton of every slot entry: t can ether be actve on a lnk (actng as master or slave) or reman dle. The slot boundares of dfferent local schedules are not algned n tme. Lnk pont nodes mantan a relatve phasewth respect to each other n order to know whch slot postons overlap n ther local schedules. If node mantans a relatve phase ff = wth respect to, then slot poston p n S overlaps wth slot postons p, p n S.Ifff =then p n S overlaps wth p and p +n S.Aff =0ndcates that the nodes happen to be perfectly synchronzed. The relatve phase mantaned at the other lnk pont s ff = ff. Communcaton s successful on a lnk (; ) only f both sdes assgn tmeoverlappng slots n ther local schedules. The assgnment must be such that when the master starts pollng n slot p of S, the slave must have assgned slots ff (+ff ) ff (+ff ) p + and p + n S for lstenng to ths master. In general, for successful conflctfree communcaton on x consecutve fullduplex pconet slots, the master must allocate x slots n ts local schedule for pollng, whle the slave must allocate at least x +tmeoverlappng slots n ts local schedule for tunng to the pconet of ths master. Therefore, certan slots n a local schedule may not be used for communcaton but for algnng to dfferent pconet tme references. The postons of such slots are the ones where the node must swtch to a new pconet and act as a slave 4. Due to the extra swtchng slots needed by the slaves, each lnk pont wll allocate a dfferent number of slots for ths lnk n ts local schedule. The slots where transmssons take place on the lnk are the ones allocated by the master. More Accordng to the TDD pollng protocol, f the master polled more than one slaves durng the frst half of the slot, these slaves would automatcally respond n the second half causng a recepton collson at the master Each halfduplex slot lasts 0:65ms The relatve phases can be acqured durng the Bluetooth lnk establshment. 4 If the node operates as master n all of ts adacent lnks, there are no swtchng slots n ts local schedule specfcally, f f! s the number of slots each node has allocated for lnk (; ) n ts local schedule S, the slot allocaton f =(f ) realzed by the asynchronous network lnk schedule S s determned by: f =ρ f! f node s the master of lnk (; ) () f! f node s the master of lnk (; ) Fgure llustrates a scatternet where nodes A and C act as S A S B S C A B C Fg.. A smple scatternet topology wth two pconets. Nodes A and C act as masters of B (B s an S/S brdge). Each node operates accordng to a local perodc schedule of T =9slots. The swtchng slots n B s local schedule are marked n red. masters and B acts as a (S/S) brdge. The asynchronous schedule realzes a slot allocaton of slots per lnk over a perod of 9 slots. Two slots n S B are used for swtchng between the pconets of masters A and C. IV. THE ASYNCHRONICITY OVERHEAD A synchronzed system wll need a smaller (or at least equal) perod than an asynchronous one for realzng the same slot allocaton. When global synchroncty s not present, brdge nodes need extra swtchng slots n ther local schedules for supportng the same slot allocaton n ther adacent lnks. Ths may force an ncrease n the overall system perod for realzng the same allocaton. Fgure llustrates a representatve example where a slot allocaton of slots per lnk s requred for the scatternet topology of Fgure. If the scatternet were synchronzed, ths allocaton could be realzed by a lnk schedule of mnmum perod of 6 slots. However, both asynchronous schedules of Fgure need a larger perod for realzng ths allocaton. Ths s due to the extra swtchng slots needed by brdge node B. S A S B S C T=8 (a) S A S B S C T=9 T= Fg.. Two asynchronous schedules realzng slot allocaton (; ) for the scatternet of Fgure. Another observaton derved from ths example s that for the same demand allocaton, the brdge node needs a dfferent (b)

5 4 amount of swtchng slots n the asynchronous schedules (a) and (b), depng on the order lnks are actvated. In (a) the brdge node swtches pconets only once durng the perod (and thus needs only two addtonal swtchng slots wth respect to the reference synchronzed system,) whle n (b) t swtches pconet every other slot yeldng a hgher requred system perod of slots. Accordng to the above example, the asynchroncty overhead for realzng a gven slot allocaton can be defned as an ncrease n perod wth respect to a perfectly synchronzed system. Also, the amount of overhead deps on the lnk actvaton order. The problem then s to fnd an asynchronous schedule and lnk actvaton order of mnmum overhead. The followng sectons provde wth a formulaton of ths problem and an approach for solvng t. A. Synchronzed lnk schedules and ther nstances Consder a scatternet that s synchronzed on a slot bass. Durng a slot, we say that a lnk s actvated for fullduplex communcaton f both ponts have assgned t to ths slot n ther local lnk schedules. A set of lnks that can be actvated at the same slot wthout transmsson conflcts at the nted recevers s called a lnk actvaton set. Accordng to the schedulng constrants S: S:, along wth the fact that Bluetooth s a multchannel system, a lnk actvaton set conssts of lnks that do not have common node ponts 5. A synchronzed lnk schedule S ~ of perod T ~ s a perodc sequence of lnk actvaton set nstances (A ;:::; A k ; :::; A ~T ). Let M be the set of all dstnct lnk actvaton sets n the network topology and M ( S) ~ Mbe the dstnct lnk actvaton sets that appear n schedule S. ~ Each lnk actvaton nstance A k corresponds to an element of M ( S). ~ Then S ~ can be compactly represented by the dstnct sets M a n M ( S) ~ and the number of nstances ff of each set M a wthn the schedule perod T ~ : ~S = f(m ff ; ff ): M ff M;ff=; ;:::;M ( S); ~ () ff f; :::; Tgg ~ The number of slots allocated to each lnk (; ) durng a perod ~T s gven by the slot allocaton vector ~f : ~f =X ff ff I((; ) M ff ); 8 (; ) E () where I(:) s an ndcator functon that evaluates to one when ts argument s true and to zero otherwse. The synchronzed schedule ~ S = f(m ff ; ff )g has ~ T! nstances that result from all possble permutatons of the lnk actvaton set nstances A. The synchronzed schedule nstance ~S (ß) correspondng to permutaton ß s gven by: ~S (ß) =(A ß() ;:::;A ß( ~T )): (4) where ß s a mappng of the nstance ndexes ß : f; :::; ~ Tg! f;:::; ~ Tg. Fgure llustrates two nstances of a synchronzed schedule for the topology of Fgure. 5 Snce each pconet channel uses a dfferent frequency hoppng sequence we assume that there are no conflcts among transmssons that happen between dfferent colocated masterslave pars. S A S B S C T=6 (a) S A S B S C Fg.. Two nstances of synchronzed schedule S = f(fg; ); (fg; )gg for the scatternet of Fgure. The dstnct lnk actvaton sets are M = fg and and M = fg and each set has actvaton nstances wthn the 6 slot perod. B. Equvalent schedules Let ~f be the allocaton realzed by a synchronzed schedule ~S. An asynchronous schedule S (ß) realzng slot allocaton f s called equvalent to an nstance S ~ (ß) of S ~ f the followng condtons hold: ffl (E.): Each node actvates ts adacent lnks n S (ß) n the same order as n S ~ (ß). ffl (E.):f = ~f. ffl (E.): S (ß) s conflctfree and satsfes the above condtons usng the mnmum possble perod. As an example, consder the asynchronous schedule of Fgure and the synchronzed schedule nstance n Fgure (a). Both schedules realze a slot allocaton of slots per lnk (condton E.) and all nodes actvate ther adacent lnks n the same order n both schedules (condton E.). However, the two schedules are not equvalent accordng to the above defnton. Ths s because the asynchronous schedule of Fgure (a) also satsfes condtons () and () usng a smaller perod of 8 slots. In fact, ths s the equvalent asynchronous schedule of the synchronzed nstance of Fgure (a) because wth ths lnk actvaton order, brdge node B needs at least swtchng slots n ts local schedule to guarantee the tme overlap wth the master transmssons. Thus, an equvalent schedule s the mnmum perod asynchronous schedule for a gven orderng of lnk actvatons n a synchronzed reference schedule. C. An algorthm for fndng equvalent schedules In ths secton, we present an algorthm called EQUIVA LENT that takes as nput a scatternet confguraton 6 and a reference synchronzed schedule nstance S ~ (ß). The output s the equvalent asynchronous schedule S (ß) of S ~ (ß). The algorthm constructs S (ß) by teratng over the lnk actvaton nstances A ß() ; :::; A ß( ~T of ~ ) S (ß). Durng teraton k, the lnk actvaton set nstance A ß(k) s consdered. Let l be a lnk n A ß(k) and and be ts master and slave ponts respectvely. Also for each node n, let p (k ) n be the maxmum slot poston assgned so far n ts asynchronous local schedule S (ß) n. 6 The scatternet confguraton conssts of the topology graph, masterslave role assgnments on the lnks and relatve phases T=6 (b)

6 5 Frst the master must assgn 7 to lnk l the earlest slot poston n ts local schedule after p (k ) that does not overlap n tme wth the last assgned slot p (k ) of slave. There are three cases to consder when computng slot for the master: ffl Case A: Lnk (; ) was actvated n teraton k as well: In ths case the nodes local schedules are n synch due to the prevous teraton and the master smply allocates to lnk l the next slot: = p (k ) + (5) ffl Case B: Lnk (; ) was not actvated n teraton k and p (k ) > p (k ) : In ths case the master s local schedule s forward n tme wth respect to the slave s. The earlest nonoverlappng slot s agan gven by: = p (k ) + (6) ffl Case C: Lnk (; ) was not actvated n teraton k and p (k ) p (k ) : In ths case the slave s local schedule s consdered forward wth respect to the master, so the master must fnd the earlest possble nonoverlappng slot. Depng on the nodes relatve phase, the poston of slot s gven by: = p (k ) + ff ff + Then assgns slot unassgned slots between p (k ) ; ff f; 0; g (7) to lnk l. If there are any ntermedate and they are assgned as dle n S (ß). Once the master updates ts local schedule,the slave must compute the earlest unassgned slot n S (ß) that wll exceed n tme slot n S (ß). Depng on the nodes relatve phase the poston of ths slot s computed as: = ff ( + ff ) + ;ff f; 0; g (8) If there are any unassgned slots between p (k ) and, they are assgned to lnk l n S (ß). The same assgnment steps happen for the node ponts of every other lnk l n M ß(k). For every node n that dd not update ts local schedule durng teraton k, n = p (k ) n. At the of the teraton k, the algorthm keeps track of the asynchronous schedule forward progress f (k) whch s the maxmum progress over all local schedules after ths teraton: f (k) =max nn fp(k) n g (9) After ~ T teratons, all lnk actvaton set nstances M ß(k) have been added to the asynchronous schedule S (ß) and each node n 7 In dfferent teratons, the same node may have a dfferent role, depng on the actvated lnk. When the node assgns slots to a lnk, t also marks the role t wll play. If t s a master t wll be pollng whle f t s a slave t wll be tuned to the pconet of the lnk for the duraton of these slots. S A C A 5 D E 4 B A B C D E A 0 X B X 0 X C X 0 X X D X 0 X E X X 0 (a) Scatternet topology: Nodes C,D,E are masters whle A,B are S/S brdges. Phase matrx: Entres marked by X are nvald because no lnk has been formed between the correspondng nodes. S A S B S C S D S E T=0 (b) An nstance of a synchronzed schedule ~ S = f(f; g; ); (f5; g; ); (f; 4g; 4); (f; 4g; )g of (mnmum) perod ~ T =0. () () () S B S C () () () () (4) (5) (5) (6) (7) (7) (8) (8) (9) (0) ()()() () (5) (5) (5) (5) (5) (5) (6) (7) (8) (0) (0) (0)(0) () S D () () () (4) (5) (5) (6) (8) (8) (8) (8) (9) (0)() () () S E () (4) (5) (5) (6) (7) (8) (9) (9) (0) (0) ()() () () () () (4) (7) (7) (7) (7) (7) (9) (9) ()()() ()() (c) The numbers n parentheses ndcate the teraton where the slot was placed by the algorthm on each node s local schedule. Swtchng slots are ndcated by red. The equvalent schedule perod s determned at the 0 th teraton and s equal to 4. Two addtonal teratons are performed so that all nodes fll ther local schedules up to ths perod. Fg. 4. (k) (0) () () () (4) (5) (6) (7) (8) (9) (0) f(k) p A (k) p B (k) p C (k) p D (k) p E (k) (d) Evoluton of the n and progress f (k). An example of the EQUIVALENT algorthm executon has assgned p ( T ~ ) n slots n ts local schedule. The asynchronous schedule perod T (ß) must be set to the maxmum of these values, whch s the forward progress after the last teraton: T (ß) = f ( ~ T ) (0) Fnally, the algorthm restarts from M ß() and performs one or more extra teratons so that all nodes fll ther local schedules up to slot T (ß). When ths happens, all nodes use the frst T (ß) slots n ther local schedules to form an asynchronous schedule wth ths perod. The detaled algorthm pseudocode s provded n the Appx, whle Fgure 4 provdes an example of the algorthm

7 6 operaton. The algorthm possesses two mportant propertes, summarzed by the followng theorems: Theorem : The asynchronous schedule S (ß) derved by algorthm EQUIVALENT s ndeed equvalent to the reference synchronzed schedule nstance S (ß). Theorem : If T ~ s the perod of the reference synchronzed schedule nstance, the perod T (ß) of any equvalent asynchronous schedule s upper bounded by T. ~ The proofs can be found n the Appx. Theorem states that an equvalent asynchronous schedule can have an overhead of at most ~ T slots. Ths mples the followng statement for feasblty of allocatons n scatternets: Corrolary on feasblty: Consder an (asynchronous) scatternet operatng wth a perod T system and a gven demand allocaton f.iff can be realzed by a synchronzed schedule S ~ of mnmum perod T ~ (f )»bt system =c, then f s guaranteed to be feasble by the scatternet. The proof s n the Appx. The corollary establshes that a scatternet usng the EQUIVALENT algorthm can realze at least half of the allocatons that can be realzed under perfect synchronzaton. Also for allocatons f for whch the condton ~ T (f )» bt system =c holds, any synchronzed schedule nstance wll generate an asynchronous schedule that realzes ths allocaton. If the condton does not hold for f, then we must solve the optmzaton problem addressed n the next secton. V. OPTIMAL EQUIVALENT SCHEDULES A. Optmal algorthm Algorthm EQUIVALENT generates an asynchronous schedule of mnmum perod T (ß) for a fxed orderng ß of lnk actvaton sets n the synchronzed schedule. The optmal asynchronous schedule s the one that has the mnmum perod over all possble permutatons ß. Ths schedule can be found f we execute EQUIVALENT for all synchronzed schedule nstances ~S (ß) and selectng the equvalent schedule S (ß) of mnmum perod. However an exhaustve search over the T ~! orderngs makes ths approach prohbtve even for small T ~. 8 The problem search space can be reduced f we only consder synchronzed schedule nstances where the ff lnk actvaton nstances of each dstnct lnk actvaton set M ff are scheduled n consecutve slots. Ths s because there are no swtchng slots generated by EQUIVALENT when A k = A k and the overhead s zero for ths transton. Thus, f M ( S) ~ s the set of dstnct lnk actvaton sets n the synchronzed schedule S,we ~ only need to search M ( S)! ~ synchronzed schedule nstances nstead of T ~!. However, for large problem szes even M ( S) ~ can be prohbtvely large for exhaustve search. In ths case we use the heurstc approach descrbed next. 8 (Accordng to [8], enumeratng 0! possbltes would about 76 years and for 0!, years! B. A heurstc algorthm MIN PROGRESS s a heurstc algorthm for overhead mnmzaton that conssts of two phases. The frst phase determnes an orderng ß h of the dstnct lnk actvaton sets n M ( S). ~ The second phase frst forms a reference synchronzed schedule where lnk actvaton sets are ordered accordng to ß h and the ff nstances of M ff are actvated n consecutve slots. Then EQUIVALENT s used to generate an equvalent asynchronous schedule for ths reference synchronzed schedule. We now descrbe phase I that selects permutaton ß h. An asynchronous schedule s constructed usng only the dstnct lnk actvaton sets nstead of all ther nstances. The sets are added to the asynchronous schedule n the same way as nstances are added n EQUIVALENT. Upon ntalzaton, an arbtrary set M a s selected from M ( S) ~ and added to the asynchronous schedule. Let Uset be the set of all unassgned lnk actvaton sets and U (k ) ts current contents at the start of teraton k. The addton of each set M ff of U (k ) would generate a forward progress f (ff; k) for the asynchronous schedule. The algorthm selects the lnk actvaton set of mnmum forward progress. If more than one sets generate ths progress, one of them s arbtrarly selected and added to the asynchronous schedule. Let M ff k be ths set. Then the kth entry of ß h s set to ff k. At the of the teraton k, M ff k s removed from the Uset ((U k = U (k ) fm ff k )g). The same steps are performed untl the Uset becomes empty after ~ T teratons. At that pont ß h contans the ndces of the actvaton sets that were selected by the algorthm teratons. The complexty of MIN PROGRESS s domnated by phase I whose complexty deps on the number of dstnct actvaton sets M ( ~ S) n the reference synchronzed schedule. Durng teraton k, U (k ) = M ( ~ S) k sets are consdered for addton n the asynchronous schedule. Thus, the total number of lnk actvaton sets consdered over all the teratons s (M ( ~ S) )+(M ( ~ S) )+:::+ = M ( ~ S)(M ( ~ S) )=, yeldng a complexty of O(M ( ~ S) ). VI. PERFORMANCE EVALUATION A. Factors affectng the overhead We are nterested n evaluatng the performance of the proposed algorthms n vew of the factors that affect the asynchroncty overhead. The overhead s frst related to the scatternet topology structure. In general, denser topologes are expected to produce more overhead snce more lnks mean more pconet swtches. Performance s also affected by the way masterslave roles are assgned n the scatternet topology graph. For example f node B n Fgure s assgned as master nstead of S/S brdge for nodes A and C, the overhead s always zero snce there s only a sngle tme reference n the system. For a specfc scatternet confguraton the overhead deps on the demand allocaton at hand. A parameter specfc to the demand allocaton s the rato M ( S)= ~ T ~ of dstnct lnk actvaton sets to the perod T ~ of the optmal reference schedule. A small rato s desrable because overhead s generated only durng the transtons between dstnct actvaton sets n the synchronzed schedule. Another related parameter s the perod ~T of the synchronzed schedule. Larger perods may allow for smaller M ( S)= ~ T ~ ratos and therefore less generated overhead.

8 7 B. Generatng reference synchronzed schedules and scatternet topologes The performance of the algorthms must be evaluated over a varety of scatternet confguratons and demand allocatons. Demand allocatons must be provded by optmal reference synchronzed schedules. Determnng the mnmum perod lnk schedule for a specfc allocaton s a hard problem even for synchronzed systems. Haek and Sasak [] proved that for the class of multchannel systems where synchronzed scatternets belong, the problem can be solved by a polynomal algorthm. However ths algorthm s of very hgh complexty and hard to mplement n practce. Whle [6] provdes a practcal heurstc wth good propertes for the soluton of ths problem, we want to ensure that an optmal synchronzed schedule s an nput to our algorthms. The work n [] also showed that f the network topology graph s bpartte, the mnmum synchronzed schedule perod ~T (f ) equals the maxmum node utlzaton mposed by f : ~T (f ) = max N X N() f : () where N () the set of onehop neghbors of. The above result provdes wth a straghtforward method for generatng mnmum perod synchronzed schedules for bpartte scatternets 9 : If ~ T s the desred perod of the reference synchronzed schedule, we only need to generate an arbtrary conflctfree schedule of perod ~ T and ensure that there s at least one node wth utlzaton equal to ~ T. In the followng experments we consder bpartte topology graphs consstng of N = nodes per bpartte set. Ths provdes a baselne topology of N =4 possble lnks. A Bluetooth node cannot partcpate n more than seven actve lnks at a tme. As a result the maxmum node degree n the network s 7. Ths restrcts the maxmum number of lnks that can be smultaneously establshed n the baselne topology. A network desgner or scatternet topology constructon algorthm may wsh keep the maxmum number of pconets a node should partcpate to B max» 7. The less B max, the less pconets a brdge needs to vst and therefore the less the potental overhead. We also use a parameter f (0» f» ) for tunng the densty of the scatternet topologes. Ths parameter s used to generate topologes where f% of the lnks have been arbtrarly removed from the baselne graph and s used to nvestgate the effect of topology densty when there s no restrcton on B max. Gven a topology graph constructed as above, asynchroncty can then be ntroduced by a lnk phase matrx and masterslave role assgnment. For each lnk n the scatternet the role assgnment determnes ts master and slave node ponts. Ths s the most general role assgnment method snce after the assgnment each node may have a dfferent role on ts adacent lnks 0. 9 Bpartte topologes arse very frequently n the Bluetooth settng. For example, a scatternet that uses only S/S brdges (.e. brdges that act only as slaves n the pconets they partcpate n) s by defnton bpartte. 0 Ths method can also capture the more specal case where the scatternet conssts only of masters and S/S brdges. C. Performance of MIN PROGRESS wth respect to optmal Sx 0node bpartte topologes of ncreasng densty are used for ths experment. These topologes consst of 0 masters and 0 S/S brdges. For each topology we randomly generate 00 reference schedules of small perod T ~ = 7. Ths perod allows the use of exhaustve search on all possble (7!) nstances of each schedule and the determnaton of the optmal soluton. Fgure 5 compares the average computed optmal perod and the one computed by MIN PROGRESS. For almost all topologes, the perod computed by the heurstc exceeds the optmal by less than one slot on the average, whle n topology 5 the optmal s exceeded by : slots T T T opt T h T opt B max Fg. 5. Each bar graph corresponds to a dfferent topology, where densty ncreases by varyng B max from to 7. The reference perod s 7 slots. The optmal T opt and the heurstc T h asynchronous perods of each bar are averages over 00 reference schedules. Also notce that both the optmal and the heurstc asynchronous perods ncrease as B max ncreases. For B max =7 both come very close to the worst case upper bound whch s ~ T = 4. Ths s because the small perod ~ T s comparable to B max and brdge nodes wth ths degree wll need to swtch pconet at every slot thus causng hgh overhead. D. Performance evaluaton for large problem szes In the followng experments we evaluate the performance of MIN PROGRESS for larger topologes and reference perodc schedules. We use an N = 00 baselne bpartte topology (50 nodes per bpartte set) where all nodes have the maxmum possble degree of seven adacent lnks. For each experment the overhead s plotted as the %ncrease n the synchronzed system perod T ~ due to asynchroncty. If T h s the perod computed by MIN PROGRESS, ths quantty s T h T ~. A value of 00% denotes that the heurstc reaches an ~T overhead equal to the worst case upper bound. ) Effect of topology densty: Gven the baselne N = 00 bpartte topology, the parameters B max and f are used to derve topologes of varable densty. For each set of parameters (B max ;f), we generate 0 topologes and for each topology 00 arbtrary reference optmal synchronzed schedules of perod T ~ ( T ~ s also a parameter). Lnk phases and roles are randomly assgned for each topology. Snce the optmal algorthm consders only the dstnct lnk actvaton sets M ( ~ S) nstead of all the ~ T actvaton set nstances, we ensure that for each generated schedule ~ S, M ( ~ S) = ~ T =7:

9 8 For each set of parameters (B max ;f) and ~ T the fgures llustrate the average %overhead over all generated topologes and reference synchronzed schedules. Fgure 6 nvestgates the effect of B max (f = :0). For a fxed ~ T, the overhead consstently ncreases as the maxmum number of pconets per brdge ncreases. For ~ T =8, when B max s restrcted to the overhead s 5% but when there s no restrcton (B max = 7), t reaches 60%. The overhead decreases for larger perods. Thus for B max =7, whle the overhead reaches a 60% for T =8 slots, t reduces to 0% for T =896slots. Whle ths decrease s more drastc for smaller perods (.e from 8 to 56 slots), t s less n the transton for larger perods (.e from 448 to 896 slots). Ths mples that there may stll be a nonneglgble overhead even f the reference synchronzed system operates wth a large perod. Smlar trs arse n Fgure 7 where the topology densty s vared wthout enforcng a partcular B max. Ths shows that the overhead wll generally ncrease wth the scatternet densty regardless of whether a B max s enforced or not n the system. % Heurstc Overhead T=8 T=56 T= T=4 T=448 T= B max Fg. 6. Heurstc overhead as B max and ~ T vary. f s set to : T=8 T=56 T= T=4 T=448 T=896 the mnmum perod ~ T.) BN(f )=fn : arg max N X N() f g: () We conecture that maxmum overhead wll be generated f the followng condtons hold for a demand allocaton f max n Ψ ( T ~ ) and at least one of the bottleneck nodes n BN(f max ): ffl P: In addton to maxmum utlzaton, the node must have the maxmum number of adacent lnks n the network. ffl P: The node has been assgned as an S/S brdge. ffl P: Allocaton f max s such that the node s requested to allocate an equal number of slots to ts adacent lnks. The ntuton n the above condtons s that a maxmum utlzaton node wll need to be consdered at every teraton of an overhead mnmzaton algorthm. Also snce ths s a node of maxmum degree and acts as an S/S brdge t wll vst the maxmum possble number of pconets n the system (B max ). If the requested slots are evenly dstrbuted for ths node, then we can show that the overhead wll be maxmzed under the worst permutaton f ts adacent lnk actvatons. Accordng to [4][5], a maxmn far allocaton n a synchronzed multchannel wreless ad hoc network maxmzes the utlzaton of the maxmum degree nodes n the network. If at least one of these nodes s also assgned as an S/S brdge, then the above condtons hold for at least one node n the network. Fgure 8 compares the heurstc overhead resultng from a maxmn far synchronzed schedule and the average heurstc overhead over 00 other schedules realzng arbtrary allocatons. (The maxmn far schedule s computed usng the algorthm n [5]). Each pont n the bar graphs s the average of the overheads generated by the scatternet topologes of Fgures 6 and 7. Each bar graph corresponds to a dfferent synchronzed schedule perod T ~. As expected, the average heurstc over % Heurstc Overhead f Fg. 7. Heurstc overhead as f and ~ T vary. B max s set to 7 %Heurstc Overhead Average MMF ) Effect of demand slot allocaton: The prevous experments nvestgated the algorthm performance averaged over arbtrary demand allocatons and scatternet topologes. A natural queston s whether there exsts a scatternet role assgnment and/or demand allocaton for whch the generated asynchroncty overhead s maxmzed. In ths secton we make a frst attempt to nformally classfy such worst case nstances and then test our ntuton va smulatons. Let G(N;E) be a bpartte topology graph and Ψ ( ~ T ) the set of all allocatons realzed by a synchronzed schedule of mnmum perod ~ T. For any allocaton f of Ψ ( ~ T ), let BN(f ) be the set of bottleneck nodes that get the maxmum utlzaton under f. (Snce the graph s bpartte, the maxmum utlzaton equals Fg. 8. Comparng the heurstc overhead for maxmn far allocaton and the average overhead generated by arbtrary allocatons. Both overheads have been averaged over all topologes consdered n Fgures 6 and 7 head for arbtrary allocatons decreases as the system perod ncreases. However, the one due to the maxmn far allocaton does not change sgnfcantly and t s n the order of 80% for all cases. Ths shows that the overhead can be very hgh for the allocatons we dentfed even f we use an overhead mnmzaton algorthm such as MIN PROGRESS. A counterntutve result s that the overhead remans constant even f the perod ~T ncreases. Nevertheless, t wll always be less than the upper bound ~ T gven by Theorem.

10 9 VII. CONCLUSIONS In ths paper we addressed for the frst tme the problem of mnmzng the pconet swtchng overhead n Bluetooth scatternets. Ths overhead arses due to slots wasted when brdge nodes synchronze to the dfferent pconet tme references. Whle the problem was studed n the Bluetooth context, the results apply to any wreless ad hoc network usng slotted TDMA access and multple local tme references nstead of a global synchronzaton mechansm. It was shown that ths overhead can sgnfcantly affect the bandwdth allocaton ablty of a scatternet f no measures are taken to mnmze t. To ths we ntroduced two schedulng algorthms that am for overhead mnmzaton whle ensurng that the generated overhead has an upper bound regardless of the scatternet or demand allocaton at hand. The frst algorthm reaches the optmal soluton but cannot be appled to large problem szes because t reles on exhaustve search. For large problem szes a heurstc algorthm was devsed and through smulatons t was shown to have excellent performance. We also dentfed certan condtons on demand allocatons and scatternet confguratons for whch the overhead can be hgh even f an overhead mnmzaton algorthm s run. We outlned the general propertes of such allocatons and verfed our ntuton through smulatons. A formal study of the exact nature of these allocatons s an nterestng future work drecton. Both the optmal and heurstc overhead mnmzaton algorthms are centralzed and can be used n settngs where global nformaton s avalable. More mportant though s the fact that that they can provde desgn nsghts and be used as a reference performance measure for dstrbuted overheadaware scatternet schedulng protocols. Fnally, we beleve that the dervaton of a smlar overhead mnmzaton framework for softcoordnaton scatternet schedulng schemes s another challengng open research ssue. REFERENCES [] Bluetooth Specal Interest Group,Specfcaton of the Bluetooth system, ver.0b., October 000. [] E. Arkan, Some complexty results about packet rado networks. IEEE Trans. Inform. Theory, Vol. IT0 pp , July 984. [] B. Haek and G. Sasak, Lnk Schedulng n Polynomal Tme. IEEE Trans. Inform. Theory, No 5, Vol. 4, 988. [4] R. Ramaswam and K. Parh, Dstrbuted Schedulng of Broadcasts n a Rado Network. In Proc. IEEE INFOCOM 89, Ottawa, Ont., Canada, Apr [5] A. Ephremdes and T. V. Truong, Schedulng Broadcasts n Multhop Rado Networks. IEEE Trans. Commun., vol. 8, no. 4, Aprl 990. [6] M. Post, P. Sarachk and A Kershenbaum, A Based Greedy Algorthm for Schedulng Multhop Rado Networks. 9th Annu. Conf. on Informaton Scences and Systems, Johns Hopkns Unv., March 985. [7] J Slvester, Perfect Schedulng n Multhop Broadcast Networks In Proc. 6th Int. Conf. on Computer Communcatons, London, England, Sept. 98. [8] B. Korte and J. Vygen, Combnatoral Optmzaton. Sprnger Verlag, 99. [9] G. Mklos et al, Performance Aspects of Bluetooth Scatternet Formaton. Proceedngs of IEEE/ACM MobHoc, Boston, MA, Aug [0] N.Johansson, F. Alrksson, U. Jonsson, JUMP mode a dynamc wndowbased schedulng framework for Bluetooth scatternets. Proceedngs of IEEE/ACM MobHoc, Long Beach CA, Oct. 00. [] A. Racz, G. Mklos, F. Kubnszky, A. Valko A Pseudo Random Coordnated Schedulng algorthm for Bluetooth Scatternets. Proceedngs of IEEE/ACM MobHoc, Long Beach CA, Oct. 00. [] Smon Baatz, Matthas Frank, Carmen K uhl, Peter Martn, Chrstoph Scholz, Bluetooth Scatternets: An Enhanced Adaptve Schedulng Scheme. Proceedngs of Infocom 00, New York, 00. [] N. Johansson, U. Korner, L. Tassulas, A dstrbuted schedulng algorthm for a Bluetooth scatternet. In Proc. Of the 7th Internatonal Teletraffc Congress, ITC 7. Salvador da Baha, Brazl, Sep. 00. [4] L. Tassulas and S. Sarkar, Maxmn Far Schedulng n Wreless Networks. Proceedngs of Infocom 00, New York, 00. [5], Dstrbuted onlne schedule adaptaton for balanced slot allocaton n Bluetooth scatternets and other ad hoc network archtectures. APPENDIX A: Proof of Theorems and Proof of theorem : We need to show that the followng condtons are satsfed: ) Nodes must actvate the lnks n the same order n both ~S (ß) and S (ß). ) The two schedules must realze the same slot allocatons. ) The asynchronous schedule must be conflctfree and of mnmum perod. Condton s satsfed because the lnk actvaton set nstances are added to S (ß) n a sequental manner. When a lnk l =(; ) s consdered at teraton k, the master assgns only one slot to lnk l. Thus the lnk masters assgn n ther local schedules a number of slots equal to the number of slots that are assgned to l n the synchronzed schedule. By consderng the defnton of allocaton n an asynchronous schedule (equaton ()), condton holds as well. Regardng condton, when a lnk l s consdered on teraton k, equatons (7) and (5) for ensure that the master assgns the earlest possble slot n ts local schedule that does not overlap n tme wth the last assgned slot p (k ) of slave. Then, equaton (8) for ensures that the slave wll assgn the smallest possble number of tme overlappng slots wth respect to. Smlarly, every other pont node for a lnk of teraton k progresses n ts local schedule by the mnmum number of slots that guarantee a conflctfree transmsson. Thus, at every step k, the forward progress f (k) = max nn fp(k) n s the mnmum possble. Snce ths property holds for all steps k, t also holds for f ( ~ T ) whch s by defnton the perod of the resultng asynchronous schedule. Proof of Theorem : For ths proof we need the followng lemmae: Lemma : For every masterslave lnk (; ) let L (k) = maxf ; g. Then the followng nequaltes hold: L (k) L (k) L (k ) 0; 8k =; ;::; T: ~ () L (k )» ; 8k where lnk (; ) s actvated: (4) Proof: When lnk (; ) s actvated n teraton k, both nodes and assgn slots n ther local schedule and therefore L (k) L (k ) g >. If nodes and are not nvolved n any lnk actvaton durng teraton k, then L (k) not updated. Therefore n general L (k) = L (k ) snce the p and p are L (k ). We now prove the upper bound. Let lnk (; ) where master s and slave s be actvated n teraton k. If ths s the case

11 0 then due to equaton (8), and therefore L (k) =. We now dstngush dfferent cases that arse when the lnk (; ) s actvated n teraton k: ffl Lnk (; ) was actvated n teraton k : Equaton (8) was used n teraton k and therefore p (k ) = p (k ) + ff (+ff ) p (k ). Therefore L (k ) = p (k ). From equatons (7) and (8), Snce L (k) = L (k) = p (k ) ++, we fnally have that ff (+ff ). L (k ) =» : (5) ~TX k= (f (k) f (k ))» ~TX k= () () f (0)=0 f ( T ~ )» T ~ T 4 (ß) =f ( ~T ) () T (ß)» ~ T ffl ffl Lnk (; ) was not actvated n teraton k and p (k ) > p (k ) : In ths case L (k ) = p (k ). Also from equatons (7) and (8) we have that L (k) = = p (k ) ff (+ff ) ++. Therefore, L (k) L (k ) ff ( + ff ) =+» : (6) Lnk (; ) was not actvated n teraton k and p (k ) p (k ) : In ths case L (k ) = p (k ). Applcaton of equatons (7) and (8) yelds L (k) +and then: p (k ) L (k) = = L (k ) =» : (7) The proof s complete. Proof of Corrolary on scatternet feasblty: From Theorem, for any ß: T (ß) (f )» T ~ (f ) (0)» (bt system =c) ()» T system : () For all cases L (k) L (k )» and the proof s complete. Lemma : The followng property holds for the forward progress f (k) for every teraton k: 0» f (k) f (k )» ; 8k =; ::; ~ T (8) Proof We use contradcton. Suppose there s an teraton k for whch f (k) f (k ) >. Snce f (k) s strctly greater than f (k ) the ncrease n the forward progress was contrbuted by at least one lnk l = (; ) n the lnk set A ß(k) that was actvated durng ths teraton. Ths means that L (k) = f (k). From Lemma t holds that: L (k ) L (k), L (k ) f (k) (9) and from the hypothess we have that f (k ) < f (k). Therefore t must be that L (k ) > f (k ). We arrve at a contradcton snce by the defnton of these quanttes ths mples that maxfp (k ) ;p (k ) g > max n g. Proof of Theorem Startng from Lemma we have that: nn fp(k ) Theorem states that T (ß) (f ) s the mnmum perod that can be generated by lnk actvaton orderng ß. Snce the mnmum perod s less than or equal to the system perod, the allocaton f s feasble. APPENDIX B: Algorthm pseudocodes Procedure EQUIVALENT nput output : G(N;E);Φ =[ff ]; ~ S (ß) =[A ß(k) ]; ß; ~ T : S eq =[S n ]; n N: The asynchronous equvalent schedule of ~ S (ß) T eq : The perod of S eq. local : p =[ n ]; f =[f (k)]; k Intalzaton: f (0) = 0, p 0 n =0; 8n N; begn for k =to T ~ do AddLnkActvatonSet(G;Φ;k;A ß(k) ; p;f(k); S eq ); T eq = f ( T ~ ) ; AddLnkActvatonSet(G; Φ; T ~ +;A ß() ; p; f( T ~ + ); S eq );