Exact Probabilistic Analysis of the Limited Scheduling Algorithm for Symmetrical Bluetooth Piconets

Transcription

1 Exact Probablstc Analyss of the Lmted Schedulng Algorthm for Symmetrcal Bluetooth Pconets Gl Zussman *, Ur Yechal 2, and Adran Segall Department of Electrcal Engneerng Technon Israel Insttute of Technology, Hafa 32000, Israel 2 Department of Statstcs and Operatons Research School of Mathematcal Scences Tel Avv Unversty, Tel Avv 69978, Israel ury@post.tau.ac.l Abstract. Bluetooth enables wreless communcaton va ad-hoc networks. The basc network topology (pconet) s a collecton of slaves controlled by a master. A scatternet s a multhop network of pconets. Effcent pconet and scatternet communcaton requres desgn of ntra and nter-pconet schedulng algorthms. Thus, numerous schedulng algorthms have been recently proposed. Analytcal performance evaluaton of such algorthms has great mportance, snce t may provde nsght on ther desgn and optmzaton. However, due to nherent complextes of the Bluetooth Medum Access Control (MAC), the performance of these schedulng algorthms has been analyzed mostly va smulaton. Recently, Msc and Msc [6], [7], [8], [9], [20], [2], [22], [23] have clamed to provde exact analytc results regardng ntra and nter-pconet schedulng algorthms whch are based on the theory of M/G/ queue wth vacatons. In ths paper, we present alternatve analytc results regardng the lmted (pure round robn) schedulng algorthm whch are based on the theory of pollng systems. Usng these results we show that the results presented n [8] are ncorrect, as they are based on unsatsfed assumptons leadng to naccurate probablty generatng functons. Thus, n some cases [8] underestmates the ntrapconet delay by more than 50%. We also ndcate that for smlar reasons the results presented n [6], [7], [9], [20], [2], [22], and [23] seem to be ncorrect. Keywords: Bluetooth, Schedulng, Pollng, Queueng, Lmted, Personal Area Network (PAN) * Correspondng author - phone: , fax: Ths research was supported by a grant from the Mnstry of Scence, Israel.

2 Introducton Bluetooth s a Personal Area Network (PAN) technology, whch enables portable devces to connect and communcate wrelessly va short-range ad-hoc networks [4],[5]. The basc Bluetooth network topology (referred to as a pconet) s a collecton of slave devces operatng together wth one master. A multhop ad-hoc network of pconets n whch some of the devces are present n more than one pconet s referred to as a scatternet (see for example Fgure ). A devce that s a member of more than one pconet (referred to as a brdge) must schedule ts presence n all the pconets n whch t s a member (t cannot be present n more than one pconet smultaneously). Master Slave Brdge Master whch s also a Brdge Fgure. An example of a Bluetooth scatternet In the Bluetooth specfcatons [4], the capacty allocaton by the master to each lnk n ts pconet s left open. The master schedules the traffc wthn a pconet by means of pollng and determnes how the bandwdth capacty s to be dstrbuted among the slaves. Effcent scatternet operaton requres determnng the lnk capactes that should be allocated n each pconet, such that the network performance wll be optmzed [29],[30]. The requred lnk capactes should be allocated by nterpconet schedulng algorthms. These algorthms schedule the presence of the brdges n dfferent pconets. Numerous heurstc ntra and nter-pconet schedulng algorthms have been proposed (see for example [],[6],[7],[9],[0],[],[2],[24] and references theren). Analytcal performance evaluaton of ntra and nter-pconet schedulng algorthms has great mportance, snce t may provde nsght on ther desgn and optmzaton. We wll show that for a few ntra-pconet schedulng regmes, a pconet can be analytcally modeled as a pollng system. However, as mentoned n [7], due to the specal characterstcs of the Bluetooth Medum Access Control (MAC), A pollng system conssts of several queues served by a sngle server accordng to a set of rules (pollng scheme) [3, p. 95], [3], [26], [28].

3 the operaton model of most schedulng regmes dffers from those of classcal pollng models. Accordngly, n the past most of the proposed schedulng algorthms have been evaluated va smulaton. Recently, Msc and Msc [6],[7],[8],[9],[20],[2],[22],[23] have clamed to provde exact analytc results regardng the performance of varous ntra and nter-pconet schedulng regmes. Ther analyss s based on the theory of M/G/ queue wth vacatons (ntroduced n [4], see also [27]) and to the best of our knowledge s the only avalable analytcal modelng attempt regardng the performance of Bluetooth schedulng algorthms. In partcular, n [8] closed form solutons for the lmted (pure round robn) and exhaustve schedulng algorthms are exhbted. In ths paper, we focus on the lmted schedulng algorthm and present alternatve analytc results. We show that the man results provded by Msc and Msc n [8], regardng ths algorthm, are ncorrect, as they are based on unsatsfed assumptons leadng to ncorrect probablty generatng functons. We also ndcate that for smlar reasons the results presented n [6],[7],[9],[20],[2],[22], and [23] seem to be ncorrect. To the best of our knowledge, the results presented n ths paper are the only correct analytc results regardng the performance of Bluetooth schedulng algorthms. We model pconets operated accordng to the lmted schedulng algorthm as a TDMA (Tme Dvson Multple Accesses) system and as a -lmted pollng system, and derve exact analytc results. Accordngly, t s shown that the oversght n [8] s that t gnores mportant dependences ncorporated n the pconet operaton model when applyng the results of the M/G/ queue wth vacatons to the analyzed system. Then, we compare numercal results obtaned accordng to [8] wth numercal results computed by our model and show that [8] underestmates the delay n the system. Due to space constrants, we do not elaborate on the exhaustve schedulng algorthm. However, we note that n [3] we have shown that a pconet wth undrectonal traffc can be modeled as an exhaustve pollng system and outlned the complextes n analyzng a pconet wth b-drectonal traffc. Usng these results, we have ponted out that the results presented n [8] regardng the exhaustve algorthm are ncorrect. We are not aware of any attempt to study the performance of pconets wth non-symmetrcal traffc by a queueng theoretc approach, besdes our work n [3]. 2

4 The rest of the paper s organzed as follows. Secton 2 gves a bref ntroducton to the Bluetooth technology and descrbes related work, whle Secton 3 presents the model. In Sectons 4 we analyze the lmted schedulng algorthm and dscuss ts analyss n [6],[7],[8], and [9]. In Secton 5 we present numercal results and n Secton 6 we summarze the man results and dscuss future research drectons. 2 Background 2. Bluetooth Technology In a pconet one unt acts as a master and the others act as slaves (a master can have up to 7 slaves). Bluetooth channels use a frequency-hop/tme-dvson-duplex (FH/TDD) scheme n whch the tme s dvded nto 625-µsec ntervals called slots. The master-to-slave transmsson starts n even-numbered slots, whle the slave-to-master transmsson starts n odd-numbered slots. Masters and slaves are allowed to send,3 or 5-slot packets, whch are transmtted n consecutve slots. Packets can carry synchronous nformaton (voce lnk) or asynchronous nformaton (data lnk). Informaton can only be exchanged between a master and a slave,.e. there s no drect communcaton between slaves. A slave s allowed to start transmsson n a gven slot f the master has addressed t n the precedng slot. The master addresses a slave by sendng a data packet or a -slot POLL packet (f t has no data to transmt). The slave must respond by sendng a data packet or a -slot NULL packet (n case t has nothng to send). We shall refer to the master-to-slave communcaton as downlnk and to the slave-tomaster communcaton as uplnk. An example of the TDD scheme n a pconet wth n slaves s gven n Fgure 2. Master Slave Slave 2 Slave n Fgure 2. An example of the TDD scheme n a Bluetooth pconet Tme (slots) In ths paper we concentrate on networks n whch only data lnks are used. 3

5 The master schedules the traffc wthn a pconet accordng to an ntra-pconet schedulng algorthm. Two well-known ntra-pconet schedulng algorthms were analyzed n [8]: Lmted Round Robn (Pure Round Robn [7]) The master communcates wth the slaves accordng to a fxed cyclc order. At most a sngle packet s sent n each drecton (downlnk and uplnk) every tme a master-slave queue par s served. Exhaustve Round Robn The master communcates wth the slaves accordng to a fxed cyclc order. The master does not swtch to the next master-slave queue par untl both the downlnk (master-to-slave) and the uplnk (slave-to-master) queues are empty. In ths paper, we focus on the lmted algorthm. In a scatternet, a unt (referred to as a brdge) can partcpate n two or more pconets, on a tmesharng bass, and even change ts role when movng from one pconet to another. Namely, a brdge can be a slave of a few masters or a master n one pconet and a slave n another pconet. Fgure above llustrates an example of a scatternet ncludng brdges from these two types. The presence of a brdge n dfferent pconets has to be controlled by an nter-pconet schedulng algorthm. 2.2 Related Work The Bluetooth specfcaton does not ndcate any method for scatternet formaton. Thus, several scatternet formaton algorthms have been recently proposed (see [5] for a through revew of these algorthms). For nstance, Basagn and Petrol [2] propose the BlueMesh algorthm that constructs a scatternet topology wth multple paths between any par of nodes. A dfferent approach s taken by Chassern et al. [8] who formulate the topology constructon problem as a centralzed optmzaton problem and descrbe procedures for nserton and removal of nodes from the scatternet. Another algorthm whch s based on the assumpton that each node knows the locaton of tself and ts neghbors s presented by L et al. [5]. Once the scatternet s constructed, the lnk capactes that should be allocated n each pconet, such that the network performance wll be optmzed, have to be determned. Accordngly, n [29] and [30] the scatternet capacty assgnment problem s formulated and optmal and heurstc algorthms for ts soluton are proposed. The soluton of the capacty assgnment problem s a desrable nput to ntra and nter-pconet schedulng algorthms. 4

6 Few of the frst smple ntra-pconet schedulng algorthms were proposed and evaluated va smulaton n []. Snce then, numerous ntra-pconet schedulng algorthms talored to optmze varous performance metrcs (e.g. delay, throughput, and energy consumpton) and desgned for dfferent traffc patterns (e.g. TCP traffc) have been proposed. For example, Capone et al. [7] and Har- Sha et al. [0] study by smulaton the delay n varous deal and practcal ntra-pconet schedulng algorthms. In [7] and [9] the performance of ntra-pconet schedulng algorthms when the transport layer s TCP s also studed. In addton, Bruno et al. [6] propose an algorthm that dynamcally adapts the pollng frequency to the traffc patterns. Fnally, n [3] we have analyzed the ntra-pconet exhaustve schedulng and modeled a pconet wth undrectonal traffc as an exhaustve pollng system. The schedulng problem s much more complcated n a scatternet. Usually an nter-pconet schedulng algorthm establshes rendezvous ponts n whch brdges can swtch between pconets. For example, Baatz et al. [] and Johansson et al. [2] descrbe nter-pconet schedulng algorthms whch use the Bluetooth low power mode snff to establsh recurrng rendezvous ponts. On the other hand, Har-Sha et al. [0] descrbe an nter-pconet schedulng algorthm talored for small-scale scatternets. Ths algorthm uses the low power mode hold to establsh rendezvous ponts adjusted to the traffc patterns. Fnally, Racz et al. [24] descrbe an ntra and nter-pconet schedulng algorthm n whch the nodes assgn rendezvous ponts wth ther peers, such that the sequence of these ponts follows a pseudo random process. 3 The Model In ths secton we present a model for a Bluetooth pconet whch s based on the model descrbed n [8]. We use a smlar notaton. The number of nodes s denoted by m (accordngly, the number of slaves s m ). We assume that each node has an nfnte buffer. It s assumed that the traffc nto each node s a compound Posson process generatng bursts (batches) of packets accordng to a Posson arrval process wth rate λ (bursts/slot). The probablty generatng functon (PGF) of the burst (batch) sze (number of packets n a burst) s denoted by G b (x). Its mean and second factoral moment are denoted by B and (2) B. It s 5

7 assumed that the dstrbuton of the burst sze s geometrc. We wll show that the results presented n [8] are ncorrect even for the smplest case n whch the traffc s non-bursty.e. the burst sze s always. To that end, n the rest of the paper we assume that G b (x) = x. The probabltes of a packet length beng, 3, or 5 slots are p, p 3, and p 5, respectvely. Accordngly, the PGF of the packet length s G p (x) = p x + p 3 x 3 + p 5 x 5 and the mean s denoted by L = p + 3p + 5p5. 3 In [8], t s assumed that all packets wthn a burst have the same destnaton node. Furthermore, a burst generated at a gven node s ntended to one of the other (m ) nodes wth probablty / (m ). As a node, the master generates traffc ntended for the slaves and n addton routes packets between the slaves. Under these assumptons, the burst arrval rate to each uplnk (slave-to-master) queue s λ u = λ and the burst arrval rate to each downlnk (master-to-slave) queue s λ d = λ (.e. the model s symmetrcal wth respect to the slave devces). Notce that the arrval process to the uplnk queues s Posson whereas the arrval process to the downlnk queues, beng dependent on the schedulng regme, s, n general, not Posson. In [8], three related performance ndcators are defned: Access delay The tme a packet has to wat n the uplnk queue before t s served (denoted by W a ). Queueng delay at the master The tme a packet has to wat at a downlnk queue before t s served (denoted by W m ). End to end delay The tme from the moment a packet arrves to the uplnk queue at the slave, to and the moment t s receved at another slave (denoted by W e ). The mean values of the delay functons descrbed above (W a, W m, and W e ) are denoted by W a, W e. In Secton 4 we present analytc models for the computaton of the delay n a pconet. Smplfyng the model n [8], we assume that the master s the destnaton of all packets generated at the slaves (.e. the master does not route packets between slaves). On the other hand, we assume that packets are generated at every downlnk queue accordng to a Posson arrval process wth rate λ. Due to the W m Snce W e s defned n [8] as the sum of W a and W m, t s actually only the total tme a packet spends n the master s and the slave s queues not ncludng the servce tmes. 6

8 assumpton regardng the Posson arrval process, the analyss of ths scenaro s smpler than the analyss of the scenaro descrbed above n whch the master does route packets. Therefore, the results regardng the access delay obtaned n [8] should also hold for ths scenaro. However, we show that these results do not hold even for the smplfed scenaro. Furthermore, t s argued that the results obtaned n [8] for the more complcated model, whch ncludes ntra-pconet routng by the master, suffer from the assumpton that the arrval of traffc to the downlnk queues s Posson, ndependent of the uplnk traffc. 4 Analyss of the Lmted (Pure Round Robn) Regme In ths secton we show that a pconet operatng accordng to the lmted regme and n whch all packets are slot long s equvalent to a TDMA system. Then, we consder a standard pconet wth packet szes of, 3, and 5 slots, and formulate t as a -lmted pollng system. We obtan exact analytcal results whch are dfferent from those derved n [8] and ndcate that [8] apples the results of the M/G/ queue wth vacatons wthout consderng the dependences between the queues. 4. Analyss as a TDMA system Consder a pconet operated n the lmted regme n whch all packets are slot long (.e. p = ). In such a pconet, a sngle slot s allocated to each downlnk and uplnk n every cycle. Therefore, the pconet can be analyzed as a TDMA system [3, p. 94] wth a cycle length of 2(m ) slots. Every slot n the cycle s allocated to one of the 2(m ) downlnks and uplnks. The computaton of the delay n a TDMA system s based on the analogy wth the M D queue wth determnstc servce tme and vacaton length both equal to 2(m ) [3, p.94]. Accordngly, applyng [3] eq. (3.58), where the number of queues s 2(m ) and the total arrval rate s 2(m )λ, we obtan the mean access delay ( a W ) and the mean queueng delay at the master ( W m ) (n slots): W a m = W m =. () 2( m ) λ Although n [8] t s assumed that the system s symmetrcal, we also derve analytc results for a non-symmetrcal system. 7

9 Notce that under the assumptons that the master does not route packets and the arrval process to the master s Posson, the mean access delay (queueng delay at the slave) s equal to the mean queueng delay at the master. For ths smple schedulng regme, the result regardng the access delay obtaned n eq. (5) n [8] concdes wth our result (). The model presented n [8] s symmetrcal wth respect to the slaves (.e. the arrval rates to all master and slaves queues are the same). However, n a TDMA system, the queung behavor of one user s ndependent of the queung behavor of other users. Thus, analytc results can be easly obtaned for a non-symmetrcal pconet. We denote the arrval rate to slave as λ u and the arrval rate to the master of packets ntended for slave by λ d. We denote the access delay of packets n the uplnk queue of slave by W a and the queueng delay at the master of packets ntended to slave by W. Snce n ths m (determnstc) case every lnk can be ndependently analyzed as an M D queue, the mean access delay (n slots) s agan derved from eq. (3.58) n [3]: 2( m ) λ u m W a = 2( m ) + 2( m ) =. (2) 2 ( 2( m ) λ ) 2 2( m )λ u u A smlar equaton descrbes the mean queueng delay at the master of packets ntended to slave ( W ). In ths case W m replaces W a and λ d replaces λ u. m 4.2 Analyss as a -lmted Pollng system Snce n most Bluetooth applcatons the length of the packets vares, we now consder a standard pconet operated n the lmted regme wth,3, and 5-slot packets. We show that such a pconet can be modeled as a -lmted pollng system wth 2(m ) queues and present a closed form expresson for the mean access delay. In a pconet operated accordng to the lmted schedulng regme, even f the master has nothng to send to a specfc slave, one slot s used durng the downlnk communcaton (by the POLL packet). Smlarly, even f the slave has nothng to send, one slot must be used durng the uplnk communcaton In a -lmted pollng system, at each vst of the server to a queue only the frst packet n the queue s served. The server ncurs a swtchover tme when t shfts from one queue to another [3, p. 20], [3], [26]. 8

10 (by the NULL packet). Thus, n order to model the pconet as a -lmted pollng system, we defne the swtchover tme to each of the queues as slot. Accordngly, when data packets are sent, some of the data s actually sent durng the swtchover tme. Therefore, the servce tme of a -slot data packet s defned as 0 slots, the servce tme of a 3-slot packet s 2 slots, and the servce tme of a 5-slot packets s 4 slots. Recall that we focus on symmetrcal systems n whch the arrval rates to all queues are equal. By applyng the model for a symmetrcal lmted gated pollng system 2 descrbed n [3, p. 20] we can obtan the mean watng tme of a packet n a queue. The watng tme n [3] s defned as the tme a packet wats untl ts servce starts (.e. the tme untl the end of the swtchover that precedes t). In our model, the frst slot of a data packet s consdered as the swtchover tme. Thus, n order to obtan the mean watng tme n a pconet,.e. the mean access delay, one has to deduct slot from the expresson for the watng tme n [3], eq. (3.77). Accordngly, applyng [3] eq. (3.77), where the number of queues s 2(m ), the total arrval rate s 2(m )λ, the swtchover tme s one slot wth zero varance, the traffc ntensty s ρ = 2( m ) λ( L ), and the second moment of the servce tme (denoted n [3] as 2 X ) s 4p 3 + 6p 5, and deductng tme unt (.e. slot), we obtan the mean access delay ( W a ) and the mean queueng delay at the master ( W W m ) (n slots): { λ } + ( m ) + 2 ( p + 6 p ) 3 5 a = W m = 2( m ) λl. (3) Notce that n ths system t must hold that 2( m ) λl <. Thus, the maxmal allowable arrval rate s gven by λmax = (2( m ) L). We shall refer to 2( m ) λl as the load n the system. The result presented n (3) was verfed by a smulaton model based on OPNET (for more detals regardng the smulaton, see [0]). Moreover, n a system n whch only -slot packets are used (.e. p =, p 3 = 0, and p 5 = 0), the result presented n (3) reduces to (). 2 The problem of computng exact mean delays n general -lmted pollng systems has not been resolved yet [3]. The system s referred to as the lmted gated pollng system, snce only a message that s found n the begnnng of the swtchover tme s served. 9

11 4.3 Examnaton of the Analyss as an M/G/ Queue wth Vacatons [8] The theory of M/G/ queue wth vacatons s used n [8] n order to analyze a pconet operated n the lmted regme n whch the packets are,3, and 5 slots long. We now brefly descrbe ther analyss and pont out that ther mstake s the drect use of the model of M/G/ queue wth vacatons [4],[27] to analyze a -lmted pollng system, wthout takng nto consderaton the dependences between the queues. We note that f ther analyss was correct, the analyss of symmetrcal -lmted pollng systems would have been much smpler than those presented n [3] and [26]. Recall that, for smplcty, we assume that the traffc s non-bursty (G b (x) = x) and that packets are generated at every master-to-slave (downlnk) and slave-to-master (uplnk) queue accordng to a Posson arrval process wth arrval rate λ. The pconet servce cycle tme X c s defned n [8] as the tme (number of slots) requred for the master to serve all the slaves once. The PGF, the mean, and the second moment of the cycle tme are denoted by 2 GX ( x ), X c, and X c. It s clamed that snce the model s symmetrc, t s suffcent to c consder a sngle master-slave channel (accordngly, we refer to the consdered slave as slave ). It s mentoned that the probablty that a downlnk queue s not empty s that an uplnk queue s not empty s downlnk and uplnk communcatons are presented as: Pu Pd = λ X c and that the probablty = λ X c. In [8], eq. () and (2), the PGFs of the duratons of the ( ) 3 G ( x) = p p + ( p ) x+ p p x + p p x d d d d 3 d 5 5, (4) ( ) 3 G ( x) = p p + ( p ) x+ p p x + p p x u u u u 3 u 5 5, (5) whle the PGF of the cycle tme s presented as: ( ) m X ( ) = ( ) ( ) c d u. (6) G x G x G x The model of M/G/ queue wth vacatons s used n [8]. Accordng to ths analyss, the servce tme of a sngle uplnk queue (.e. of the queue at slave ) n the vacaton model s equal to the cycle tme (X c ). A vacaton starts when the master polls slave and fnds ts uplnk queue to be empty. As we Accordng to the model of M/G/ queue wth multple vacatons, the server begns a vacaton every tme the system becomes empty. If the server returns from a vacaton to fnd the system not empty, t starts workng mmedately and contnues untl the system becomes empty agan. If the server returns from a vacaton to fnd an empty system, t begns another vacaton mmedately [27]. 0

12 understand, the vacaton s composed of the NULL packet returned by slave, the servce of the other m 2 uplnk and downlnk queues, and the transmsson of a packet to slave n the downlnk queue. V l denotes the duraton of the vacaton perod and ts PGF, mean, and second moment are denoted by 2 G ( x), V l, and V l. The PGF of the duraton of the vacaton perod s provded n [8], eq. (3): Vl ( ) m 2 V ( ) = ( ) ( ) ( ) l d d u. (7) G x xg x G x G x Fnally, the access delay s derved from the watng tme n a batch arrval system M [x] G wth multple vacatons ([27, p. 43], eq. (3.2a)). Snce we assume that the arrval process s non-bursty (.e. (2) Gb ( x ) = x, B =, B = 0), eq. (5) n [8] reduces to [27, p. 23], eq. (2.4a): W a ( λ X c ) 2 2 c l λ X V = + 2 2V l (8) We now descrbe a few crucal problems n the model presented above. For the clarty of presentaton, the descrpton of each solated problem gnores the exstence of other problems.. The cycle length descrbed n (6) s napproprate for use as a servce tme n a vacaton model. Consder the case n whch the server returns to slave from a vacaton or completes servce there (.e. the master completes a cycle) and at least one packet s found n the uplnk queue of slave. In such a case, the server wll not take a vacaton and a servce perod wll start. Accordng to [8], the PGF of the servce tme s defned as the PGF of the cycle tme (X c ) presented n (6). It s composed of the duraton of the communcaton n the uplnk queue of slave and the duratons of the communcaton n the rest of the uplnk and downlnk queues. When the servce starts there s obvously a data packet n the queue of slave. Hence, the PGF of the duraton of the uplnk communcaton of slave s G p (x) (defned n Secton 3) and not G u (x) (presented n eq. (5)). Therefore, one of the terms n eq. (6) should be replaced. Consequently, eq. (6) cannot be used as a servce tme n a vacaton model. 2. The uplnk and downlnk communcaton perods composng a cycle are not ndependent. The PGF of the cycle length ( G X ( x) ) presented n (6) s a multple of the PGFs of the uplnk (Gu(x)) and the c downlnk (G d (x)) communcaton perods. Thus, an underlyng assumpton n the dervaton of Ths problem becomes crtcal for small pconets n whch G Xc ( x) s composed of only a few components.

13 GX c ( x ) s that all the downlnk and uplnk communcaton perods are ndependent. However, the length of a communcaton perod depends on the exstence of a packet n the correspondng queue (n case the queue s empty, the length s slot). The exstence of a packet depends on the lengths of the precedng uplnk and downlnk communcaton perods, snce long perods ncrease the probablty of a packet arrval. Hence, the lengths of the perods composng a cycle do not seem to be ndependent, and therefore the dervaton of the cycle length n (6) s ncorrect. 3. The cycle tme depends on the length of the vacaton or the cycle that precedes t. In [8] the vacaton model descrbed by Takag [27] s used n order to compute the mean access delay, presented n eq. (8). One of the mportant assumptons made n [27, p. ] s that: Messages arrve n the system accordng to a Posson process of fxed rate and have servce tmes wth ndependent and dentcal dstrbuton. These servce tmes are ndependent of the arrval process, and each servce tme s ndependent of the sequence of vacaton perods that precede that servce tme. However, the servce tme n the model descrbed n [8] s taken as the cycle tme (X c ) and s composed of uplnk and downlnk communcaton n m queues. The length of each of the communcaton perods depends on the exstence of packets n the correspondng queue. If the cycle follows a long vacaton or cycle, the probabltes that the queues are not empty wll ncrease, and thereby the probablty of a long cycle wll ncrease. On the other hand, f the cycle follows a short vacaton or cycle, the probablty of a short cycle wll ncrease. Thus, the requrement for ndependent servce tmes and for servce tmes, whch are ndependent of the vacaton lengths, does not seem to hold n the model presented n [8]. Therefore, the use of the delay presented n eq. (8) s problematc. In Secton 4., we have analyzed a pconet operated n the lmted regme n whch all packets are slot long. In such a pconet the duraton of a communcaton perod s determnstc ( slot), and therefore, there are no dependences between the cycle and vacaton lengths as well as wthn a cycle. Thus, as we have mentoned, for ths smple regme the results regardng the access delay obtaned n eq. (5) n [8] concdes wth our result (). However, due to the problems descrbed above, n a We note that t seems that the mean cycle length derved from combnng (4),(5), and (6) s correct. However, for the vacaton model the second moment of the cycle length s also requred. 2

14 standard pconet (n whch the packets are,3, and 5 slot long) the results presented n [8] dffer consderably from our results. Recall, that we assume that the master s the destnaton of all packets generated at the slaves (.e. the master does not route packets). On the other hand, n [8] t s assumed that the master routes some of the traffc between the slaves. Accordngly, the mean value of the end-to-end delay s defned as the sum of the access delay and the queueng delay at the master ( W e = W a + W m). A method for computng the mean queueng delay at the master ( W traffc, m ) s descrbed at the end of Secton 2 n [8]. For non-bursty W m computed accordng to that method s equal to W a descrbed n (8) (.e. W a W = m ). Thus, t seems that an underlyng assumpton n [8] s that the arrval process of packets to the master from the slaves can be treated as Posson. Ths assumpton probably follows the analyss of pollng systems wth probablstc routng (e.g. [25]). The analyss of such systems s based on an mportant assumpton that the servce tmes of a packet n dfferent queues are ndependent. However, ths s not the case n a pconet (for example, a -slot packet sent from the slave to the master cannot become a 3-slot packet when t s forwarded from the master to another slave). Thus, the computaton of the end-to-end delay s naccurate not only because of the naccuraces n the computaton of the access delay but also due to the assumptons made regardng the arrval process of packets whch requre routng. Fnally, we note that napproprate assumptons, smlar to the ones ndcated n ths secton, also appear n [6], [7], [9], [20], [2], [22], and [23]. For example, In [6] the lmted schedulng algorthm s analyzed n a smlar methodology to the analyss descrbed n [8]. For nstance, eq. (5) n [6] presents the mean watng tme. However, the calculaton of ths watng tme gnores the dependency between the servce tme and the vacaton length precedng t (see tem 3 above). Moreover, accordng to eq. (7) n [6] the maxmal allowable arrval rate s gven by λ0 (2( )( )) = m L. If the arrval rate approaches ths value, the cycle length defned n [6] approaches nfnty. Snce the longest possble cycle s composed of only 5-slot packets, the maxmal cycle length s 0(m ) slots, and therefore, the result n [6] s unreasonable. Recall that 3

15 we have shown that the maxmal allowable arrval rate s gven by λmax = (2( m ) L). If the arrval rate approaches ths value, the cycle length defned n [6] approaches 2( m ) L. Oddly, although both [6] and [8] deal wth the lmted schedulng regme, the vacaton length descrbed n eq. (4) n [6] dffers from the vacaton length descrbed n eq. (3) n [8]. In both cases the use of the vacaton model s problematc dsregardng the exact vacaton length. The analyss of the lmted schedulng algorthm n [7] and [9] s very smlar to the analyss n [8]. The only dfference s that n [7] and [9] t s assumed that the master does not generate traffc and ts only role s to route packets between the slaves. The analyss of the exhaustve regme n [8] s also based on the theory of M/G/ queue wth vacatons. In [3] we show that the PGF of the tme to exhaust the queues derved accordng to [8] dffers from the correct PGF. Moreover, we argue that due to the reasons dscussed above, the use of the model of M/G/ queue wth vacatons n order to analyze the exhaustve regme s napproprate. Snce the analyss of ntra and nter-pconet schedulng algorthms n [20],[2],[22], and [23] s mostly based on the analyss of the exhaustve regme, t can be shown that smlar errors appear n the analyss of these algorthms. 5 Numercal Results In ths secton we present exact numercal results computed accordng to the analyss n Secton 4.2. Then, n order to demonstrate the dfference between the results presented n [8] and the exact results, we calculate numercal results accordng to the analyss n [8] and compare them to results computed by our eq. (3). We show that n some cases the results obtaned accordng to [8] underestmate the mean access delay by more than 50%. Notce that n [6], Msc and Msc menton that The dfference between analytcal and smulaton results s qute small, although t tends to ncrease slghtly wth packet arrval rate λ. The dfference becomes notceable at hgh arrval rates, where mean watng tme s above 0 to 5 (slots). It seems that the nconsstency between ther smulaton results and ther analytc results s due to the naccuracy of the analytc results. In order to obtan W a accordng to [8], we have used (4) (8) and computed X c G' X c () =, 2 c X c X c 2 X = G'' () + G' (), V = G ' (), V = G '' () +G' (). l V l l V l V l 4

16 Fgure 3 llustrates the exact mean access delay ( W a ) (computed accordng to (3)) n pconets wth varous numbers of slaves n whch the probabltes of, 3, and 5-slot packets are equal ( p = p 3 = p 5 = /3). The fgure presents the delay (n slots) as a functon of the load n the system (defned n Secton 4.2 as 2( m ) λl ) Delay {slots} Load m=2 m=3 m=4 m=5 m=6 m=7 m=8 Fgure 3. The exact mean access delay (computed accordng to (3)) n pconets operated accordng to the lmted schedulng algorthm n whch p = p 3 = p 5 = /3 Fgure 4 compares the mean access delay ( W a ) computed accordng to [8] to the mean access delay computed accordng to our model (.e. accordng to (3)) when all packets are 5 slots long ( p 5 =, L = 5 ). The delay (n slots) s depcted as a functon of the system load n a pconet wth 2 slaves (m = 3) and n a pconet wth 4 slaves (m = 5). The fgure demonstrates that the results obtaned n [8] sgnfcantly underestmate the access delay Delay {slots} Load [8] m=3 eq. (3) m=3 [8] m=5 eq. (3) m=5 Fgure 4. The mean access delay derved accordng to [8] and the exact mean access delay (computed accordng to (3)) n pconets wth 2 and 4 slaves n whch all packets are 5 slots long 5

17 As we have mentoned, the exact results derved n [6] are slghtly dfferent from the results derved n [8]. Thus, we have computed the mean access delay accordng to [6] and compared t to our results. In all cases we have checked, the mean delay obtaned accordng to [6] s lower than the exact mean delay. Moreover, n Secton 4.3 we have mentoned that the cycle length (X c ) s napproprate for use as a servce tme n a vacaton model. In order to evaluate the effect of ths problem (and gnorng all other ssues dscussed n Secton 4.3), we have redefned the PGF of the servce tme as: ( 2 ( ) ( ) ( ) ( ) ( ) m S = p d d u, (9) G x G x G x G x G x ) and computed the mean access delay usng ths corrected model. Fgure 5 compares the mean access delay ( W a ) computed accordng to [8] to the mean access delay computed accordng to our model, n a pconet wth 4 slaves n whch probabltes of, 3, and 5-slot packets are equal ( p = p 3 = p 5 = /3, L = 3 ). It s seen that the delay values are hgher than the values obtaned n [8]. The fgure also presents the access delay computed accordng to [6]. Although the results obtaned accordng to [6] dffer from the results obtaned accordng to [8], they also underestmate the delay. In addton, the fgure presents the access delay computed usng the corrected servce tme as presented n (9). Ths correcton slghtly ncreases the delay. Yet, due to the other problems, dscussed n Secton 4.3, the computed delay s stll lower than the exact delay Delay {slots} Load eq. (3) Corrected servce tme - eq. (9) [8] [6] Fgure 5. The exact mean access delay (obtaned accordng to (3)), the mean access delay derved usng the corrected servce tme (9), the mean access delay derved accordng to [8], and the mean access delay obtaned accordng to [6] n a pconet wth 4 slaves n whch p = p 3 = p 5 = /3 6

18 Fgure 6 presents the rato of the exact mean access delay to the mean access delay computed accordng to [8], n pconets wth varous numbers of slaves n whch the probabltes of, 3, and 5-slot packets are equal ( p = p 3 = p 5 = /3). Fgure 7 presents the same rato n pconets n whch all packets are 5 slots long ( p 5 = ). It can be seen that the results obtaned n [8] sgnfcantly underestmate the access delay n a pconet operated n the lmted regme Rato Load m=2 m=3 m=4 m=6 m=8 Fgure 6. The rato of the exact mean access delay (obtaned accordng to (3)) to the mean access delay derved accordng to [8] n pconets n whch p = p 3 = p 5 = / Rato Load m=2 m=3 m=4 m=6 m=8 Fgure 7. The rato of the exact mean access delay (obtaned accordng to (3)) to the mean access delay derved accordng to [8] n pconets n whch all packets are 5 slots long ( p 5 = ) 6 Conclusons Ths work presents an analytcal study of the lmted (pure round robn) schedulng algorthm for Bluetooth pconets, and examnes the analytcal study of ths algorthm n [8]. We have modeled a 7

19 pconet operated accordng to the lmted schedulng algorthm as a TDMA system and as a -lmted pollng system, and derved exact analytc results, whch dffer from the results obtaned n [8]. Then, we argued that the results presented n [6],[7],[9],[20],[2],[22], and [23] seem also to be ncorrect. Fnally, we have provded numercal examples that llustrate the consderable dfference between the exact results and those presented n [8]. Due to the nherent complextes n obtanng the PGF of the tme to exhaust the queues at the master and a gven slave n the gated and exhaustve regmes (presented n [3]), t seems that there s no closed form expresson for the delay under such regmes. Thus, a major future research goal s to obtan a good (at least approxmate) analyss of such regmes. Acknowledgments We would lke to thank Nr Naaman for helpful dscussons. We would lke to thank Ronen Kofman for hs help wth the smulatons. References [] S. Baatz, M. Frank, C. Kühl, P. Martn, and C. Scholz, Bluetooth Scatternets: An Enhanced Adaptve Schedulng Scheme, Proc. IEEE INFOCOM 02, June [2] S. Basagn and C. Petrol, Degree-Constraned Multhop Scatternet Formaton for Bluetooth Networks, Proc. IEEE GLOBECOM 02, Nov [3] D. P. Bertsekas and R. Gallager, Data Networks, Prentce-Hall Inc., New Jersey, 992. [4] Bluetooth Specal Interest Group, Specfcaton of the Bluetooth System Verson., Feb [5] J. Bray and C. Sturman, Bluetooth. connect wthout cables, Prentce Hall, 200. [6] R. Bruno, M. Cont, and E. Gregor, Bluetooth: Archtecture, Protocols and Schedulng Algorthms, Cluster Computng, Vol. 5, pp. 7-3, Apr [7] A. Capone, M. Gerla, and R. Kapoor, Effcent Pollng Schemes for Bluetooth Pcocells, Proc. IEEE ICC 0, June 200. [8] C. F. Chassern, M. A. Marsan, E. Barals, and P. Garza, Towards Feasble Dstrbuted Topology Formaton Algorthms for Bluetooth-based WPANs, Proc. HICSS-36, Jan [9] A. Das, A. Ghose, A. Razdan, H. Saran, and R. Shorey, Enhancng Performance of Asynchronous Data Traffc over the Bluetooth Wreless Ad-hoc Network, Proc. IEEE INFOCOM 0, Apr [0] L. Har-Sha, R. Kofman, G. Zussman, and A. Segall, Inter-Pconet Schedulng n Bluetooth Scatternets, Proc. OPNETWORK 2002, Aug [] N. Johansson, U. Korner, and P. Johansson, Performance Evaluaton of Schedulng Algorthms for Bluetooth, Proc. IFIP TC6 Int. Conf. on Broadband Communcatons, Nov [2] P. Johansson, R. Kapoor, M. Kazantzds, and M. Gerla, Rendezvous Schedulng n Bluetooth Scatternets, Proc. IEEE ICC 02, Apr

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