Asynchronous TDMA ad hoc networks: Scheduling and Performance

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1 Asynchronous TDMA ad hoc networks: Schedulng and Performance Theodoros Salonds and Leandros Tassulas, Department of Electrcal and Computer Engneerng and Insttute of Systems Research Unversty of Maryland, College Park, Maryland, USA Department of Computer and Communcaton Engneerng Unversty of Thessaly, Volos, Greece Abstract A common assumpton of TDMA-based wreless ad hoc networks s the exstence of network-wde slot synchronzaton. Such a mechansm s dffcult to support n practce. In asynchronous TDMA systems, each lnk uses a local tme slot reference provded by the hardware clock tck of one of the node ponts. Inevtably, slots are wasted when nodes swtch tme slot references. Ths restrcts the rate allocatons that can be supported when compared to a perfectly synchronzed system. To address ths practcal performance ssue we frst ntroduce a general framework for conflct-free schedulng n asynchronous TDMA networks. We then propose schedulng algorthms that target overhead mnmzaton whle ensurng upper bounds on the generated overhead. Through extensve smulatons the algorthm performances are evaluated n the context of Bluetooth, a wreless technology that operates accordng to the asynchronous TDMA communcaton paradgm. I. INTRODUCTION An ad hoc network s a collecton of nodes equpped wth rado nterfaces that forms an all-wreless communcaton nfrastructure. Tme dvson multple access (TDMA) s a wellknown medum access method for determnstc bandwdth allocaton and qualty of servce (QoS) provson n ad hoc networks. Accordng to TDMA, bandwdth can be allocated to the network lnks usng a schedule of perod T system slots. Durng every slot, several lnks are actvated for transmsson such that no conflcts occur at the nted recevers. The number of conflct-free slots each lnk receves wthn a system perod determnes ts allocated bandwdth. A central performance ssue that arses n a TDMA-based ad hoc network s determnaton of the set of allocatons that can be acheved. Consder a set of -to- sessons sharng the network. The sum of ther requested rates over the lnks they traverse creates a demand allocaton for each lnk. A lnk rate allocaton r =[r l ] ( r l ) sfeasble f the network can allocate τ l = r l T system conflct-free slots to every lnk l wthout exceedng the system perod. Determnaton of feasblty s ntrnscally coupled wth an optmzaton problem: to fnd a lnk schedule of mnmum perod that realzes slot allocaton τ =[τ l ]. If the mnmum perod does not exceed T system slots, then the allocaton s feasble. Optmal lnk schedulng n wreless ad hoc networks has been nvestgated for varous nterference constrants [], []. These studes (along wth most proposed centralzed or dstrbuted TDMA protocols for slotted ad hoc networks) assume the slot boundares are provded by a global system clock. System-wde synchronzaton mechansm s not always possble to mplement n the dstrbuted ad hoc network settng. Bluetooth [3] s a new TDMA wreless technology that enables the formaton of ad hoc networks called scatternets. Bluetooth has the nterestng feature of not supportng a global slot synchronzaton mechansm. Instead, tme slot reference s provded locally for each lnk by one of the node ponts actng as master, whle the other pont acts as slave. In ths asynchronous TDMA settng, slots are wasted when nodes swtch tme references as slaves. Ths phenomenon has been reported n the scatternet schedulng lterature [4], [5], [6], [7], [8] as a source of overhead. However, no formal study has examned ts effect on the system s ablty to allocate bandwdth. Ths ablty s lnked to the determnaton of the regon of feasble allocatons or, equvalently, to the soluton of the related lnk schedule optmzaton problem. Due to the slots wasted for tme reference algnment, the mnmum perod requred for realzng a gven allocaton wll be greater than the one requred by a perfectly synchronzed system. Ths ncrease can be seen as overhead due to system asynchroncty. Based on ths observaton, we can use a twostep procedure to address the optmal lnk schedulng problem for asynchronous TDMA ad hoc networks. The frst step nvolves fndng a mnmum-perod synchronzed schedule for the demand allocaton at hand. The second step, our contrbuton, utlzes the optmal synchronzed schedule to fnd an asynchronous schedule of mnmum overhead. The amount of overhead deps on the order by whch lnks are actvated n the reference synchronzed schedule. We frst ntroduce an algorthm that derves a mnmum-overhead asynchronous schedule for a specfc orderng. The generated overhead s always upper-bounded regardless of orderng or network confguraton. Usng ths algorthm, t s possble to determne the optmal soluton by searchng over all possble or-

2 derngs. Ths leads to a combnatoral problem where exhaustve search s not feasble for large problem szes. To ths, we ntroduce a heurstc algorthm of reduced complexty. The heurstc performs excellent for problem szes where an optmal soluton can be computed. When ths s not possble, we nvestgate the effect of varous system parameters on the generated overhead and use the upper bound as the performance measure. The paper s organzed as follows: Secton II ntroduces a conflct-free schedulng framework for asynchronous TDMA ad hoc networks. In sectons III, IV and V the problem s ntroduced and formulated and the overhead mnmzaton algorthms are presented. Secton VI evaluates the algorthms performances n varous scenaros. Secton VII places the proposed framework n the specfc context of Bluetooth. Secton VIII concludes the paper. II. ASYNCHRONOUS TDMA COMMUNICATION MODEL Every wreless node has a hardware clock that determnes the tmng of the rado transcever. The clocks of dfferent nodes are not synchronzed and no mechansm exsts for synchronzng them under a global tme reference. The ad hoc network s represented as a drected graph G(N,E). A drected edge from node to node sgnfes that and are wthn range and communcate on a lnk where has been assgned the role of master and the role of slave. The system s slotted and carres pont-to-pont traffc each transmsson slot carres a packet destned to a sngle outgong lnk. The tme slot reference of each lnk s provded locally by the hardware clock of the master node pont. Each slot supports full-duplex communcaton ntated by the master: Durng the frst part of the slot the master polls a slave; durng the second part a slave responds f polled by the master. Each node has a sngle rado transcever and can communcate (transmt or receve) to at most one lnk at a tme. Thus, nodes need to coordnate ther presence on lnks n mutual tme ntervals. Based on ts own hardware clock, each node dvdes tme n fxed-sze slots each equal to the duraton of a fullduplex communcaton slot. Transmssons on adacent lnks are coordnated usng a local lnk schedule S of perod T system slots. The local schedule determnes communcaton acton for the duraton of a slot: the node can ether be actve on a sngle lnk (start actng as master or slave) or reman dle. Local schedules of dfferent nodes are not necessarly tmealgned. Every node mantans a relatve phase φ wth respect to each adacent lnk (, ). If φ =, slot p n the local schedule S overlaps n tme wth slots (p, p) n the local schedule S.Ifφ =, then slot p n S overlaps wth slots (p, p +)ns. A relatve phase φ =ndcates that the hardware clocks of the ponts happen to be perfectly synchronzed. The relatve phase mantaned at the other lnk pont s φ = φ. Gven the relatve phases and master-slave role assgnment on lnk l, the lnk phase φ l s defned as the relatve phase of the master node pont. Communcaton s successful on a lnk l only f both node ponts assgn tme-overlappng slots n ther local schedules. The assgnment must be such that when the master starts pollng n slot p of ts local schedule, the slave must have assgned slots p + φ l(+φ l ) and p + φ l(+φ l ) n ts own local schedule for lstenng to ths master. For conflct-free communcaton on τ l consecutve slots on lnk l, the master must allocate τ l slots n ts local schedule for pollng whle the slave must allocate at least τ l +tme-overlappng slots for algnng to the tme reference of ths master. In general, an extra slot s needed every tme a node swtches to a new tme reference as slave. We assume that the system does not support capture f a node receves more than one transmssons at a specfc tme nstant all of them are lost and a conflct occurs. The nterference constrants determne whch lnks can transmt smultaneously wthout causng conflcts at the nted recevers. Accordng to prmary nterference constrants, lnks sharng common nodes cannot be actvated smultaneously. Ths s due to the sngle rado transcever (a node cannot transmt or receve smultaneously), as well as the pont-to-pont traffc requrement (each transmsson s addressed to a sngle lnk). Secondary nterference arses due to the broadcast nature of the wreless medum and occurs between dfferent transmtter/recever pars. Two smultaneous transmssons, one from node to node and one from node k to node l wll result to a secondary conflct f at least one of the recevers s wthn range of the ser of the other nted transmsson. The system may use a sngle channel or multple channels for communcaton. A channel can be mplemented as a dstnct frequency or spread spectrum code. In sngle-channel systems both prmary and secondary nterference exst. In mult-channel systems secondary nterference can be mtgated by assgnng to each lnk a channel derved by the (unque) dentty of the master node pont. In ths settng, every set of smultaneous transmssons satsfyng the prmary constrants wll occur at dfferent channels. Bluetooth s an nstance of a mult-channel asynchronous TDMA system where communcaton channels (termed as pconets) are mplemented as frequency hoppng sequences. When we refer to mult-channel systems, we wll assume that dfferent channels are orthogonal transmssons on a channel are correctly receved by a node lstenng on that channel despte any n-range transmssons that may be occurrng at dfferent channels. Recent studes [9] have shown that ths s a good approxmaton for the frequency hoppng channels used by Bluetooth. For both sngle-channel and mult-channel systems, a slot allocaton τ =[τ l ] realzed by the network TDMA schedule s the number of slots every lnk l transmts conflct-free durng T system slots and equals the number of slots allocated to the local schedule of the master pont. A network confguraton conssts of the ensemble of a network topology, lnk phases and master-slave lnk role assgnments. Fgure llustrates an example of a network confguraton and asynchronous schedule as well as the slot allocaton realzed by ths schedule. III. OPTIMAL TDMA SCHEDULING Gven a network confguraton t would be of nterest to determne feasblty of any gven demand slot allocaton. Ths problem s equvalent to fndng a lnk schedule of mnmum

3 3 A C (-) (+) 3 (-) (a) S A S B S C S D T= (b) Fg.. (a) Network confguraton: Drected edges denote master-slave relatonshps. Nodes A and D act as masters on all ther adacent lnks, B s slave on lnks,4 and master on lnk 3 and C acts as slave on all ts lnks. The numbers n parentheses denote lnk phases. As an example, snce lnk has a lnk phase of (-), slot p n the local schedule S A of master A must overlap wth slots (p, p) n the local schedule S C of slave C. (b) The asynchronous TDMA schedule refers to a mult-channel system: only lnks and 4 can transmt smultaneously on the channels defned by nodes A and D, respectvely. Slots where nodes swtch tme reference as slaves are marked n red. The realzed slot allocaton s τ =(τ,τ,τ 3,τ 4 )=(3, 3, 3, 4).. perod that realzes the demand allocaton at hand. If the mnmum perod does not exceed the system perod T system the demand allocaton s feasble. Although certan demand allocatons can be realzed by schedules of greater perod than the mnmum, the mnmum perod soluton s optmal n the sense that t can detect all feasble allocatons. When global slot synchronzaton exsts, the problem of fndng the mnmum-perod TDMA lnk schedule for a demand allocaton n an arbtrary topology s NP-complete, for both sngle channel [] and mult-channel [] systems. References [], [] propose effcent heurstcs for ths problem. Alternatvely, optmal solutons exst for restrcted topologes. For sngle channel systems, tree topologes can be optmally scheduled [3]. For mult-channel systems, schedulng lnks s equvalent to colorng edges n a mult-graph where the multple edges between two node ponts map to the slot requrement of the correspondng lnk. If the network topology s bpartte the optmal soluton can be reached usng mnmum edge-colorng algorthms for bpartte mult-graphs [4]. It s not straghtforward to apply graph colorng technques to the asynchronous TDMA settng. For example, n multchannel systems there exsts no one-to-one mappng of the slot demand per node par n the network topology graph to multple edges for ths node par n the correspondng mult-graph: n the asynchronous system, each lnk slot demand s the number of slots that should be allocated n the local schedule of the master pont; however the slave pont must allocate addtonal slots n ts local schedule for tme reference algnment. Also, as wll be evdent later n the dscusson, the number of addtonal slots requred n the slave local schedules deps on the order lnks are actvated n the local schedules of the masters. Our approach s based on the observaton that the addtonal slots needed by the slaves yeld an ncrease n perod wth respect to the mnmum perod of a perfectly synchronzed system. Ths perod ncrease s an overhead nduced by the system asynchroncty. The lnk schedule optmzaton problem B (-) 4 D for the asynchronous system s translated to an overhead mnmzaton problem: Frst, a synchronzed lnk schedule that realzes the demand allocaton s computed. Usng ths schedule as a reference we seek an asynchronous schedule of mnmum overhead. If we can fnd a reference synchronzed schedule of mnmum perod, a mnmum overhead asynchronous schedule s a mnmum-perod asynchronous schedule. When the reference schedule perod s sub-optmal, a mnmum-overhead asynchronous schedule s stll useful: the resultng perod wll be compared to T system for determnng feasblty of the demand allocaton at hand. Therefore, mnmum-overhead schedules allow detecton of a greater number of feasble allocatons. The amount of overhead deps on the orderng of lnk actvatons n the reference synchronzed schedule. Consder the 3-node confguraton of Fgure where node B s slave to both nodes A and C and where the demand allocaton s 3 slots for each lnk. Frst, let us assume exstence of slot synchronzaton. Snce each node can communcate to only a sngle lnk at a tme, the demand allocaton can be realzed by a mnmum-perod schedule of 6 slots. In ths schedule, each lnk s actvated 3 tmes by assgnng concurrent slots n the ponts local schedules. Fgure (b) llustrates two possble nstances of the mnmumperod schedule, each usng a dfferent orderng of lnk actvatons. S A S B S C T=6 (I) S A S B S C Fg.. A B (a) Network confguraton: B s a slave to both A and C. C S A S B S C T=6 (II) (b) Synchronzed system: Two possble synchronzed schedule nstances realzng slot allocaton (τ,τ )=(3, 3) n a mnmum perod of 6 slots T=8 I S A S B S C T= (c) Asynchronous system: Depng on the order of lnk actvatons, slot allocaton (τ,τ )=(3, 3) s realzed by schedules of dfferent mnmum perods. An example of the asynchroncty overhead Fgures (c)-i and (c)-ii are two asynchronous schedules II

4 4 where lnks are actvated n the order of Fgures (b)-i and (b)- II, respectvely. Both asynchronous schedules need a greater perod than 6 slots to realze the demand allocaton: n Fgure (c)-i node B swtches tme reference only once per lnk yeldng a perod of 8 slots; n Fgure (c)-ii node B s forced to swtch tme reference every slot, yeldng a perod of slots. In the example of Fgure, t s possble to determne by nspecton the lnk orderng and asynchronous schedule that yeld mnmum overhead (Schedule (c)-i). However, for arbtrary confguratons and demand allocatons a systematc approach s needed. We frst ntroduce an algorthm that fnds a mnmum overhead asynchronous schedule for a fxed orderng of lnk actvatons n the reference synchronzed schedule. Ths algorthm can be used to determne the mnmum-overhead schedule over all possble orderngs va exhaustve search. The followng sectons descrbe n detal our approach for the soluton of ths problem. IV. EQUIVALENT SCHEDULES A lnk actvaton set conssts of lnks that can smultaneously transmt wthout conflcts to the nted recevers. A synchronzed lnk schedule S of perod T s a collecton of lnk actvaton sets {A k : k T }. Asynchronzed schedule nstance S (π) s a perodc sequence of a specfc orderng π of the lnk actvaton sets of S: S (π) =(A π(),..., A π( T ) ). () where π s a mappng of the ndces {,..., T } {,..., T }. Let S (π) be a synchronzed schedule nstance realzng allocaton τ. For the orderng of lnk actvatons n S (π), allocaton τ can be realzed by more than one asynchronous schedules, each havng a dfferent perod. Consder the synchronzed schedule nstance of Fg. (b)-i that realzes allocaton (τ,τ )=(3, 3) by actvatng each lnk n 3 consecutve slots. For ths orderng of lnk actvatons, the asynchronous schedule of Fg. (c)-i realzes the same allocaton usng a perod of 8 slots. If slave B spent 5 slots nstead of 4 lstenng on lnk, the same demand allocaton would also be realzed wth ths orderng of lnk actvatons but the overall perod of the resultng asynchronous schedule would be 9 slots nstead of 8. We defne an asynchronous schedule S (π) to be equvalent to a synchronzed schedule nstance S (π) f the followng condtons hold: (E.): Every node actvates ts adacent lnks n S (π) n the same order as n S (π). (E.): S (π) realzes the same allocaton as S (π). (E.3): S (π) satsfes (E.) and (E.) n mnmum perod. Thus, an equvalent schedule S (π) of a synchronzed schedule nstance S (π) s an asynchronous schedule that yelds mnmum overhead for the orderng of lnk actvatons n S (π). We now present an algorthm called EQUIVALENT that takes as nput a network confguraton and a reference synchronzed schedule nstance S (π) and outputs the equvalent asynchronous schedule S (π) of S (π). EQUIVALENT constructs S (π) ncrementally by teratng over the lnk actvaton sets of S (π). Durng teraton k, let l be a lnk n actvaton set A π(k) and and be ts master and slave ponts. Also let p (k ) and p (k ) be the last assgned slot postons n the local schedules S (π) and S (π), respectvely =, n N). (p () n Frst, master determnes the earlest possble slot assgned to lnk l n S (π). There are three possble cases: to be Case A: Lnk l was actvated n teraton k : The local schedules are n synch and node can allocate to lnk l the next slot: = p (k ) + () Case B: Lnk l was not actvated n teraton k and p (k ) >p (k ) : The master s local schedule s consdered forward n tme wth respect to the slave s. The earlest slot s agan: = p (k ) + (3) Case C: Lnk l was not actvated n teraton k and p (k ) p (k ) : The slave s local schedule s consdered forward n tme wth respect to the master, so the master must fnd the earlest unassgned slot n S (π) whose start tme exceeds the tme of slot p (k ) n S (π) : Then assgns slot = p (k ) + φ l φ l +,φ l {,, } (4) slots exst between p (k ) S (π). to lnk l. If any ntermedate unassgned and, they are assgned as dle n Once the master updates ts local schedule, slave determnes as the earlest unassgned slot n S (π) whose tme exceeds the tme of n S (π). Depng on the lnk phase φ l the poston of ths slot s computed as: = + φ l( + φ l ),φ l {,, } (5) If there are any unassgned slots between p (k ) and, they are assgned to lnk l n S (π). The same assgnment steps are performed for every lnk l n A π(k). For every node n not consdered durng teraton k, n = p n (k ). At the of teraton k, the forward progress f(k) s defned as: f(k) = max n N {p(k) n } (6) After T teratons, the asynchronous schedule perod T (π) s set to the forward progress f( T ). Then, startng agan from A π(), a few extra teratons are performed untl all nodes assgn ther local schedules up to slot T (π). Upon termnaton, all nodes use the frst T (π) slots n ther local schedules to form an asynchronous schedule wth ths perod. An example of the

5 5 algorthm operaton s llustrated n Fgure 3. C (-) A 5 (+) D 4 (+) (-) (+) E 3 B Proposton : The worst-case computatonal complexty of EQUIVALENT s O(N T ). Proof: See Appx IX-D. For any network confguraton and any lnk actvaton orderng π, EQUIVALENT possesses two mportant propertes, summarzed by the followng theorems: S A 9 (a) Network confguraton A B C D E T= (b) Reference synchronzed schedule nstance of perod T = slots, realzng allocaton (τ,τ,τ 3,τ 4,τ 5 ) = (6,, 5, 5, 3) () () () S B S C () () () (3) (4) (5) (5) (6) (7) (7) (8) (8) (9) () ()()() () (5) (5) (5) (5) (5) (5) (6) (7) (8) () () ()() () S D () () (3) (4) (5) (5) (6) (8) (8) (8) (8) (9) ()() () () S E Fg (3) 5 (4) (5) (5) (6) (7) (8) (9) (9) () () ()() () () () (3) (4) (7) (7) (7) (7) (7) (9) (9) (9) ()() ()() (c) Numbers n parentheses ndcate teraton where the slot was assgned on each node s local schedule. Swtchng slots are shaded. The equvalent schedule perod (=4) s determned at the th teraton. Two addtonal teratons are performed so that all nodes assgn ther local schedules up to ths perod. (k) () () () (3) (4) (5) (6) (7) (8) (9) () f(k) p A (k) p B (k) p C (k) p D (k) p E (k) (d) Evoluton of n and f(k). An example of the EQUIVALENT algorthm executon Theorem : The asynchronous schedule S (π) derved by EQUIVALENT ncurs mnmum overhead for the lnk actvaton orderng correspondng to S (π). Theorem : If T s the perod of the reference synchronzed schedule, the perod T (π) of any equvalent asynchronous schedule s upper bounded by T. The proofs can be found n Appces IX-A and IX- B. Theorem states that the maxmum overhead of an equvalent schedule s T slots. Ths leads to the followng statement for feasblty of allocatons n asynchronous TDMA: Corollary on feasblty: Consder an asynchronous TDMA ad hoc network operatng wth a perod T system and a demand allocaton τ. If τ can be realzed by a synchronzed schedule of perod T T system /, then τ s feasble by the asynchronous system. The proof appears n Appx IX-C. The corollary asserts that EQUIVALENT can realze at least half the allocatons that are feasble under perfect synchronzaton. If the condton T T system / holds for a demand allocaton, any reference synchronzed schedule nstance can be used to generate an asynchronous schedule realzng ths allocaton. Otherwse, we must solve the optmzaton problem addressed next. V. COMPUTING OPTIMAL ASYNCHRONOUS SCHEDULES A. Optmal algorthm The optmal asynchronous schedule can be determned by executng EQUIVALENT for all T! synchronzed schedule nstances S (π) and selectng the equvalent schedule of mnmum overhead. Such an exhaustve search s prohbtve even for small values of T. A lnk actvaton set may appear multple tmes n the reference synchronzed schedule. The search space can be reduced f we consder only reference schedules where all nstances of each lnk actvaton set are scheduled n consecutve slots no swtchng slots are generated by EQUIVALENT when A π(k ) = A π(k) ; the overhead s zero durng such a transton. If M( S) s the set of dstnct lnk actvaton sets appearng n the reference schedule, we only need to search M( S)! schedule nstances nstead of T!. Unfortunately, even M( S) can be prohbtvely large for exhaustve searches. In ths case we resort to the heurstc algorthm ntroduced n the next secton.

6 6 B. MIN PROGRESS MIN PROGRESS s a heurstc for overhead mnmzaton that conssts of two phases. Phase I determnes an orderng π h of the dstnct lnk actvaton sets n M( S). Phase II nvolves two steps: frst, a synchronzed schedule nstance s formed, where dstnct lnk actvaton sets are ordered accordng to π h and the nstances of each set are actvated n consecutve slots. Second, ths synchronzed schedule nstance s nput to EQUIV- ALENT to generate the fnal asynchronous schedule. We now descrbe Phase I that selects π 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 lnk actvaton set of M( S) s added to the asynchronous schedule. Let U (k ) be the set of all unassgned lnk actvaton sets at the start of teraton k (U () = M( S)). The addton of each set M α of U (k ) wll generate a forward progress f(α, k) for the asynchronous schedule. The algorthm selects the lnk actvaton set yeldng mnmum forward progress, wth tes beng broken arbtrarly. Let M α k be the selected set. Then the k-th entry of π h s set to α k and set M α k s removed from the U- set. The same steps are repeated untl the U-set becomes empty after M( S) teratons. Phase I can be exted to select and nsert multple lnk actvaton sets per teraton, accordng to a horzon parameter h. Durng teraton k, all possble h-set blocks n the U-set and all possble orderngs (h!) of the lnk actvaton sets wthn each h-set block are consdered. The block and orderng that yelds mnmum forward progress s selected and added to the asynchronous schedule. The selected block s removed from the U-set and the next teraton s performed. Depng on whether h dvdes M( S) or not, the algorthm wll termnate M( S) M( S) n h or h +teratons, respectvely. For the mnmum horzon value (h =), each block conssts of a sngle actvaton set. Durng teraton k, the remanng M( S) k actvaton sets n he U-set are tested. Therefore, only M( S) k= k = ( M( S) )( M( S) +) tests or, equvalently, O(N M( S) ) operatons are performed n ths case. Increasng the horzon h s expected to mprove performance because more orderngs are tested per teraton. However, ths comes at an expense of computatonal complexty. For the maxmum horzon value (h = M( S) ) MIN PROGRESS s essentally the optmal algorthm t ncludes a sngle teraton where a block of M( S)! orderngs must be exhaustvely tested. The depence of complexty on h s summarzed by the followng proposton: Proposton : For h> and fxed, the worst-case computatonal complexty of MIN PROGRESS s O(N M( S) h+ ). Proof: See Appx IX-E Gven a specfc nput reference schedule, the horzon h must be carefully selected for tractablty. Accordng to MIN PROGRESS, the maxmum number of ( M( S) ) h blocks must be consdered n the frst teraton. The horzon h must be selected small enough to allow exhaustve enumeraton of ths number, as well as exhaustve enumeraton of h! orderngs per block. The algorthm performance wth respect to h wll be nvestgated n the experments secton that follows. VI. PERFORMANCE EVALUATION A. Factors affectng the overhead We are nterested n evaluatng performance n vew of the factors that affect the asynchroncty overhead. The overhead s frst related to the topology structure. In general, denser topologes are expected to produce hgher overhead because more lnks wll translate to a hgher number of tme reference swtches. Performance s also affected by the master-slave role assgnments. In the example of Fgure, f node B s assgned as master to nodes A and C, the overhead s zero due to the sngle tme reference n the system. For a specfc network confguraton the overhead also deps on the demand allocaton at hand. A parameter specfc to the demand allocaton s the rato M( S) 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. B. Expermental settng Performance must be evaluated for a varety of network confguratons and optmal reference synchronzed schedules. As mentoned n secton III, determnaton of optmal synchronzed schedules s n general an NP-complete problem. However, for bpartte topologes n mult-channel systems, the mnmum perod equals the maxmum node utlzaton: T (τ ) = max N l L() τ l. (7) where L() s the set of adacent lnks to node. Thus, n ths case, optmal reference synchronzed schedules of perod T can be constructed by generatng arbtrary conflct-free schedules where at least one node transmts durng the entre perod. In the experments we consder N -node mult-channel bpartte networks wth N / nodes per bpartte set. Ths provdes a baselne topology of N /4 lnks. We use the restrctve parameters B max and f to generate varous topologes from the baselne. The channel degree parameter B max s an upper bound on the number of channels a node can partcpate as slave. Such a constrant would arse n practce to avod excessve overhead. We also restrct the number of lnks where a node can act as master to 7. Ths restrcton s specfc to Bluetooth, a mult-channel asynchronous TDMA system. Combned wth B max, ths provdes an upper bound of B max +6to the overall lnk degree of each node n the topologes we consder. The densty parameter f ( f ) generates topologes

7 7 where an arbtrary f % lnks of the baselne topology reman ntact whle the rest have been removed. Gven a topology constructed as above, asynchroncty s ntroduced by ) master-slave role assgnments on the lnks and ) arbtrary phase dfferences on the hardware clocks of the nodes n the network. Accordng to the lnk role assgnments, a node may act as master to all ts adacent lnks (termed as master ) or act as slave to all ts adacent lnks (termed as S/S brdge ), or act as master to some lnks and slave to others (termed as M/S brdge ). C. Performance of MIN PROGRESS wth respect to optmal Sx -node bpartte topologes ( masters and S/S brdges) of varyng densty are consdered n ths experment. For each topology we randomly generate reference synchronzed schedules of perod T = 7. Ths perod allows exhaustve search and determnaton of the optmal asynchronous schedule. Fgure 4 compares the resultng optmal and MIN PROGRESS perods. For each topology, the perods are averaged over all reference schedules. Usng a horzon h =, MIN PROGRESS exceeds the optmal by less than one slot on the average, whle n topology 5 t exceeds the optmal by.3 slots on the average. ) Effect of horzon: In ths set of experments, we use - node bpartte topologes ( masters and S/S brdges) and vary the densty parameter f (B max =7) and reference perod T. Fgure 5 plots the overhead of MIN PROGRESS usng up to 3 actvaton sets per block (h= to h=3). For all scenaros, the overhead decreases as h ncreases. The mprovement s always more drastc from h = to h = than from h =to h =3. Usng h =nstead of h = appears benefcal for larger perods and denstes (bar graphs f =.6,.9 n Fg. 5(c) and Fg. 5(c)), wth a maxmum overhead reducton of 3% at T = and f =.9. %Overhead f h= h= h=3 4 T T opt T T h T opt (a) T =8slots, B max =7, f vares h= h= h= %Overhead B max Fg. 4. MIN PROGRESS vs. optmal. Each bar graph corresponds to a dfferent -node bpartte network confguraton where densty ncreases by varyng B max from to 7. The reference synchronzed schedule perod s 7 slots. The optmal perod T opt and the MIN PROGRESS (h =) perod T h of each bar are averages of reference synchronzed schedules f The optmal and MIN PROGRESS perods ncrease wth B max and for B max = 7 they both approach 4 slots, the upper bound of EQUIVALENT. The hgh overhead stems from B max beng equal to the small reference perod T : S/S brdges wth such a channel degree need to swtch tme reference almost every slot regardless the lnk actvaton order n the reference schedule. D. Performance of MIN PROGRESS for large problem szes For each parameter set (N,B max,f, T ) we generate topologes and, for each topology, arbtrary reference synchronzed schedules. For each (N,B max,f, T ), the overhead s averaged over the correspondng topologes and reference schedules and s plotted as the %ncrease n the reference perod T. If T h s the perod computed by MIN PROGRESS, ths quantty s equal to T h T, wth % denotng that T MIN PROGRESS yelds perod T, the upper bound of EQUIVALENT. We proceed by nvestgatng the varous factors that affect the performance of MIN PROGRESS. %Overhead (b) T = slots, B max =7, f vares. h= h= 9 h= f (c) T = 448 slots, B max =7, f vares. Fg. 5. Effect of the choce of horzon for varyng topology denstes and reference perods (N =, B max =7). For the lowest densty consdered, MIN PROGRESS performs

8 8 smlarly for all h. Overall, a horzon h =seems to provde a good performance/complexty trade-off at hgher reference perods and topology denstes, whle a horzon h =appears suffcent at low topology denstes. ) Effect of phase and role assgnments: Consder a topology graph G(N,E). Snce for every lnk l, φ l can be -,, or, there are 3 E possble lnk phase assgnments n the network. Also, there are E possble master-slave lnk role assgnments. In ths experment, we consder -node bpartte topologes. For a specfc topology and reference synchronzed schedule, we measure the standard devaton of the generated overhead of MIN PROGRESS for a sample of arbtrary phase (or role) assgnments. Then, for each parameter set (f, T ) we plot the average standard devaton over the correspondng topologes and reference schedules T ref =8 T = ref T =448 ref 8, the overhead s 5% when B max =but reaches 6% when B max = 7. The overhead decreases as the reference perod ncreases. At B max =7the overhead reduces to 3% for T = 896 slots. Whle ths decrease s more drastc for transtons between smaller perods (e.g. from 8 to 56 slots), t s less for larger perods (e.g. from 448 to 896 slots). Ths ndcates that a non-neglgble overhead may stll exst even f the system uses a large perod. Smlar trs arse n Fgure 9 where B max = 7 and only parameter f s used to vary the topology densty. The overhead ncreases wth network densty regardless of the number of tme references n whch each node partcpates. % Heurstc Overhead T=8 T=56 T= T=4 T=448 T=896 %Overhead devaton B max Fg. 8. Overhead of MIN PROGRESS (h =) for -node networks as B max and T vary (f =.) f Fg. 6. Average overhead standard devaton due to lnk phase varablty for -node networks and varous values of f and T (h =, fxed lnk roles per. topology) T ref =8 T = ref T ref =448 % Heurstc Overhead T=8 T=56 T= T=4 T=448 T=896 %Overhead devaton f Fg. 9. Overhead of MIN PROGRESS(h =) for -node networks as f and T vary (B max =7) f Fg. 7. Average overhead standard devaton due to lnk role assgnment varablty for -node networks and varous values of f and T (h =, fxed lnk phases per topology). For every (f, T ), role varablty (Fg 7) produces hgher standard devaton than phase varablty (Fg 6) the dfference never exceeds %. Apart from ths dfference, both fgures have smlar propertes: For a fxed densty the standard devaton appears nsenstve to T less than.5% changes are observed. However, for every T, the standard devaton decreases as the densty ncreases. Ths ndcates that the overhead devates less from a certan mean as the number of lnks per localty ncreases; therefore varablty n phase and role assgnments affect the algorthm performance to a lesser extent n ths case. 3) Effect of densty: Here, a -node (5 masters, 5 S/S brdges) baselne bpartte topology s used. Fgure 8 llustrates the effect of B max on the overhead of MIN PROGRESS. For fxed T the overhead consstently ncreases wth B max.at T = 4) Effect of demand allocaton: The prevous experments nvestgated the algorthm performance averaged over arbtrary demand allocatons and topologes. A natural queston that arses next s whether there exsts a network confguraton and demand allocaton for whch the generated overhead s maxmzed. In ths secton we make a frst attempt to nformally classfy such worst-case nstances and then test our ntuton through smulatons. Let the topology be bpartte and Ψ ( T ) be the set of all allocatons realzed by a synchronzed schedule of mnmum perod T. For any allocaton τ n Ψ ( T ), let BN(τ ) be the set of nodes that receve maxmum utlzaton T under τ. BN(τ )={n : arg max τ }. (8) N N() We conecture that maxmum overhead wll be generated f the followng condtons hold for a demand allocaton τ max n Ψ ( T ) and at least one of the bottleneck nodes n BN(τ max ): P: In addton to maxmum utlzaton, the node has maxmum lnk degree.

9 9 P: The node s a S/S brdge. P3: Allocaton τ max s such that the node s requested to allocate an equal number of slots to ts adacent lnks. A maxmum utlzaton node wll be consdered at every teraton of an overhead mnmzaton algorthm. Also, snce ths s a node of maxmum degree and acts as a S/S brdge, t wll vst the maxmum possble number of tme references (B max ) as slave. If lnk demands are equal for ths node, we can show that the overhead wll be maxmzed under the worst orderng of lnk actvatons. A maxmn far allocaton n a synchronzed mult-channel wreless ad hoc network maxmzes utlzaton of the nodes wth maxmum lnk degree [5], [6]. If at least one of these nodes s also assgned as a S/S brdge then condtons P-P3 wll hold. Fgure compares the MIN PROGRESS overhead resultng from a maxmn far reference schedule and the average MIN PROGRESS overhead over arbtrary schedules. (The algorthm n [5] s used to compute the reference maxmn far schedules). The average MIN PROGRESS overhead decreases as the system perod ncreases. The overhead for the maxmn far schedule however, does not change sgnfcantly n the order of 8% for all cases. Ths ndcates that the overhead can be very hgh for the allocatons we dentfed even f we use an overhead mnmzaton algorthm such as MIN PROGRESS. Counter-ntutvely, the overhead remans hgh even f the reference perod ncreases. Nevertheless, t s always less than the upper bound gven by EQUIVALENT Average MMF master n a sngle pconet and slave n others (a M/S brdge) or act as slave n multple pconets (a S/S brdge). Accordng to the above descrpton, a Bluetooth scatternet s an nstance of a mult-channel asynchronous TDMA ad hoc network where channels are mplemented as frequency hoppng sequences and each node can act as master to at most seven adacent lnks. The scatternet schedulng problem s the problem of coordnatng the node vsts n dfferent pconets n an effcent manner and s currently a subect of ntense research effort. Emphass s placed on dstrbuted schemes and the approaches can be categorzed accordng to the degree of coordnaton they offer. Hard coordnaton schemes [8], [5] provde determnstc allocatons va conflct-free schedulng; however, a certan degree of mplementaton complexty and communcaton overhead s needed to mantan the conflct-free property under topology dynamcs. Soft coordnaton schemes [5], [6], [7] trade off perfectly conflct-free transmssons for lower complexty. The downsde s loss of the ablty to provde bandwdth allocaton guarantees. Whle there s stll a smplcty v.s. performance debate between the two approaches, the overhead of brdges swtchng to the dfferent pconet tme references always exsts. In ths paper we ntroduced a hard-coordnaton framework for overhead mnmzaton for two reasons. Frst, n ths case, the overhead s naturally lnked to the ablty of the system to allocate bandwdth. Second, a useful pont of reference s establshed snce conflct-free schedulng s the best we can do to mnmze overhead. The dervaton of a smlar overhead mnmzaton framework for soft coordnaton schemes would be an nterestng research avenue to pursue. %Heurstc Overhead Fg.. MIN PROGRESS overhead for maxmn far allocatons vs. average MIN PROGRESS overhead. For each reference perod, both quanttes are averaged over all topologes consdered n Fgures 8 and 9 VII. THE CASE OF BLUETOOTH Bluetooth s a new wreless technology based on frequency hoppng. Bluetooth nodes can be grouped n dstnct communcaton channels called pconets. Each pconet s a frequency hoppng sequence derved from the dentty of one of the pconet members actng as master. The master provdes ts own hardware clock as the pconet tme slot reference and arbtrates transmssons by pollng the rest of the pconet members (slaves) usng a Tme Dvson Duplex (TDD) protocol. Bluetooth restrcts the maxmum number of pconet members to eght. Pconets can be nterconnected va brdge nodes to form a larger ad hoc network known as a scatternet. Brdges can tmeshare between multple pconets, recevng data from one pconet and forwardng t to another. A brdge may act as VIII. CONCLUSIONS In ths paper, we addressed for the frst tme the problem of mnmzng overhead n TDMA wreless ad hoc networks that use multple local tme slot references nstead of a sngle global tme slot reference. Ths overhead arses due to slots wasted when nodes synchronze to the dfferent local tme slot references and manfests as loss of supported allocatons wth respect to a perfectly synchronzed system. The problem was cast and addressed usng a generc framework and the results can be drectly appled for the case of Bluetooth, a wreless technology operatng accordng to the asynchronous TDMA communcaton paradgm. It was demonstrated that the overhead can sgnfcantly affect the ablty of a network to allocate bandwdth f no measures are taken to mnmze t. We ntroduced two schedulng algorthms that am to mnmze overhead whle ensurng that the generated overhead has an upper bound regardless of the network confguraton 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 such cases an effcent heurstc was devsed. We also dentfed and verfed, through smulatons, certan condtons on demand allocatons and network confguratons for whch the overhead can be hgh even f an overhead mnmzaton algorthm s run. Further nvestgaton of the exact nature of such condtons s an nterestng research drecton.

10 Both optmal and heurstc algorthms are centralzed and can operate n settngs where global nformaton s avalable. More mportantly, they can be used to provde desgn nsghts and serve as a reference performance measure for any overheadaware dstrbuted approaches that wll follow. REFERENCES [] E. Arkan. Some complexty results about packet rado networks. Proc. IEEE Transactons on Informaton Theory, 3:68 685, July 984. [] B. Haek and G. Sasak. Lnk schedulng n polynomal tme. Proc. IEEE Transactons on Informaton Theory, 34:9 97, September 988. [3] Bluetooth Specal Interest Group. Specfcaton of the bluetooth system, verson.. In [4] G. Mklos, Z. Turany, A. Valko, and P. Johansson. Performance aspects of bluetooth scatternet formaton. In Proc. ACM MOBIHOC, Boston, MA, August. [5] N. Johansson, F. Alrksson, and U. Jonsson. Jump mode - a dynamc wndow-based schedulng framework for bluetooth scatternets. In Proc. ACM MOBIHOC, Long Beach CA, October. [6] A. Racz, G. Mklos, F. Kubnszky, and A. Valko. A pseudo random coordnated schedulng algorthm for bluetooth scatternets. In Proc. ACM MOBIHOC, Long Beach CA, October. [7] S. Baatz, M. Frank, C. Kuhl, P. Martn, and C. Scholz. Bluetooth scatternets: An enhanced adaptve schedulng scheme. In Proc. IEEE INFO- COM, New York, NY, June. [8] U. Korner N. Johansson and L. Tassulas. A dstrbuted schedulng algorthm for a bluetooth scatternet. In Proc. Internatonal Teletraffc Congress (ITC), Salvador da Baha, Brazl, September. [9] A. Kumar and R. Gupta. Capacty evaluaton of frequency hoppng based ad-hoc systems. In Proc. ACM SIGMETRICS, Cambrdge, MA, June. [] I. Holyer. The np-completeness of edge colorng. Proc. SIAM Journal of Computng, :69 97, 98. [] M. Post, P. Sarachk, and A. Kershenbaum. A based greedy algorthm for schedulng multhop rado networks. In Proc. Annual Conference on Informaton Scences and Systems (CISS), Johns Hopkns Unv., March 985. [] S. Ramanathan. A unfed framework and algorthm for channel assgnment n wreless networks. In Proc. IEEE INFOCOM, Kobe, Japan, September 997. [3] S. Ramanathan and E. Lloyd. Schedulng algorthms for multhop rado networks. Proc. IEEE Transactons on Networkng, :66 77, Aprl 993. [4] N. Alon. A smple algorthm for edge-colorng bpartte multgraphs. Informaton Processng Letters. [5] T. Salonds and L. Tassulas. Dstrbuted on-lne schedule adaptaton for balanced slot allocaton n bluetooth scatternets and other ad hoc network archtectures. Techncal Report, Insttute of Systems Research (ISR), Unversty of Maryland, TR -4: [6] L. Tassulas and S. Sarkar. Maxmn far schedulng n wreless networks. In Proc. IEEE INFOCOM, New York, NY, USA, October. IX. APPENDIX A. Proof of Theorem We need to show that the reference synchronzed schedule S (π) and the derved asynchronous schedule S (π) satsfy the followng condtons: ) Nodes actvate the lnks n the same order n both schedules. ) Both schedules realze the same slot allocaton. 3) Schedule S (π) s conflct-free and has the mnmum possble perod for the orderng π of lnk actvatons. Condton s satsfed because the lnk actvaton set nstances are added to S (π) n a sequental manner. Also, when a lnk l =(, ) s added 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 assgned to l n the synchronzed schedule. Snce a slot allocaton of an asynchronous schedule s defned as the number of conflct-free slots n the local schedules of the master node ponts, condton also holds. Regardng condton 3, when a lnk l s consdered on teraton k, equatons (4) and () 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 (5) 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 conflct-free transmsson. Thus, at every step k, the forward progress f(k) = max n N {p(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. B. Proof of Theorem To prove Theorem, we frst need to establsh the followng lemmae: Lemma : For every master-slave lnk (, ) let L (k) = max{, }. Then the followng nequaltes hold: L (k) L (k) L (k ), k =,,.., T. (9) L (k ), k wherelnk(, ) s actvated. () Proof: When lnk (, ) s actvated n teraton k, both nodes and assgn slots n ther local schedule and therefore L (k) L (k ) >. If nodes and are not nvolved n any lnk actvaton durng teraton k, then L (k) = L (k ) snce the p and p are not updated. Therefore n general L (k) 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 then due to equaton (5), and therefore L (k) =. We now dstngush 3 dfferent cases that arse when the lnk (, ) s actvated n teraton k: Lnk (, ) was actvated n teraton k : Equaton (5) was used n teraton k- and therefore p (k ) = p (k ) φ l (+φ l ) p (k ) equatons (4) and (5), L (k) =. Therefore L (k ),wefnally have that L (k) = p (k ) = p (k ) ++ φ l(+φ l ) +. From. Snce L (k ) =. () Lnk (, ) was not actvated n teraton k and p (k ) > p (k ) : In ths case L (k ) = p (k ). Also from equatons (4) and (5) we have that L (k) p (k ) ++ φ l(+φ l ). Therefore, L (k) = = L (k ) =+ φ l( + φ l ). ()

11 Lnk (, ) was not actvated n teraton k and p (k ) p (k ) : In ths case L (k ) = p (k ). Applcaton of equatons (4) and (5) yelds L (k) = = +and then: p (k ) For all cases L (k) L (k) L (k ) L (k ) =. (3) and the proof s complete. Lemma : The followng property holds for the forward progress f(k) for every teraton k: f(k) f(k ), k =,.., T (4) 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 π(k) that was actvated durng ths teraton. Ths means that = f(k). From Lemma t holds that: L (k) L (k ) L (k) L (k ) f(k) (5) 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 max{p (k ),p (k ) } > max n N {p(k ) n }. Proof of Theorem Startng from Lemma we have that: T k= (f(k) f(k )) The proof s complete. T k= () f()= f( T ) T T (π) =f( T ) T (π) T C. Proof of corollary on scatternet allocaton feasblty From Theorem, for any π: T (π) (τ ) T (τ ) (6) ( T system / ) (7) T system. (8) Theorem states that T (π) (τ ) s the mnmum perod that can be generated by lnk actvaton orderng π. Snce the mnmum perod s less than or equal to the system perod, the allocaton τ s feasble. D. Proof of Proposton Durng teraton k of EQUIVALENT the lnk actvaton set A π(k) s added to the asynchronous schedule. Addton of each lnk l of A π(k) requres a constant number of arthmetc operatons: Checkng whether lnk l =(, ) s n A π(k ) : Ths operaton can be performed by nspectng f slots p (k ) and p (k ) have been assgned to l n the local schedules S (π) and S (π), respectvely. (two comparsons). Comparng p (k ) Updatng and wth p (k ) (one comparson). (two addtons). Snce A π(k) s a matchng n the network topology graph, t conssts of at most N/ lnks, the sze of a perfect matchng. Therefore, nserton of a lnk actvaton set A π(k) requres O(N) operatons. EQUIVALENT requres T teratons to determne the perod of the asynchronous schedule T (π), as well as a certan number of addtonal teratons untl all nodes fll ther local schedules up to T (π). Due to the schedule perodcty, there wll be no more than T extra teratons. Therefore, EQUIVALENT requres at most T teratons (O( T )). Snce each teraton requres O(N) operatons, the complexty of EQUIVALENT s O(N T ). E. Proof of proposton Let M be the number of dstnct actvaton sets n the reference synchronzed schedule (M = M( S) ). The complexty of MIN PROGRESS s determned by the complextes of Phases I and II: ) Complexty of Phase I: Durng teraton k, ( ) M (k )h h blocks are consdered and, for each block, h! orderngs of actvaton sets are tested. Depng on whether h dvdes M or not, the last teraton wll consst of a sngle block of h or (M modh) actvaton sets, respectvely. Testng each orderng of actvaton sets nvolves nserton of h actvaton sets to the asynchronous schedule. Therefore, the total number of nsertons C I throughout the executon of Phase I s gven by: where C I = M h k= r(m,h) = ( M (k )h h ) h! h + r(m,h) h (9) { (Mmodh)! f Mmodh otherwse After some algebrac manpulatons, equaton (9) yelds: C I = h M h k= () h ((M ) hk)+r(m,h) h () =