On the Exact Analysis of Bluetooth Scheduling Algorithms

Transcription

1 On the Exact Analyss of Bluetooth Schedulng Algorth Gl Zussman Dept. of Electrcal Engneerng Technon IIT Hafa 3000, Israel Ur Yechal Dept. of Statstcs and Operatons Research School of Mathematcal Scences Tel Avv Unversty, Tel Avv 69978, Israel Adran Segall Dept. of Electrcal Engneerng Technon IlT Hafa 3000, Israel Abstract Effcent communcaton n Bluetooth scatternets requres desgn of ntra and nter-pconet schedulng algorth, and therefore numerous algorth have been proposed. Analytcal performance evaluaton of such algorth has great mportance, snce t may provde nsght on ther desgn and optmzaton. However, due to complextes of the Bluetooth MAC, the performance of these algorth has been analyzed mostly va smulaton. We present exact analytc results regardng the exhaustve and lmted (pure round robn) schedulng algorth n pconets wth undrectonal traffc. We show that, surprsngly, n symmetrcal pconets wth only uplnk traffc, the mean delay s the same for both algorth. Ths result s extended for Tme- Dvson-Duplex syste wth arbtrary packet lengths. We also demonstrate the dffcultes n analyzng the performance of the exhaustve algorthm n a pconet wth b-drectonal traffc. Our results regardng the exhaustve algorthm do not conform wth those presented by Msc and Msc n [7], [8], [9], [], [], [3], [4], [] where they have clamed to provde exact analytc results. We show that [9] actually presents approxmate results, as t neglects mportant dependences ncorporated n the pconet operaton model. Consequently, [9] underestmates or overestmates the delay, n some cases by more than 0%. The results presented n [7], [8], [0], [], [], [3], [4], and [] are based on smlar models, and therefore seem to be approxmate results. Keywords-Schedulng, Pollng, Queueng, Exhaustve, Lmted, Tme Dvson Duplex, Bluetooth, Personal Area Network (PAN) I. INTRODUCTION Bluetooth s a Personal Area Network (PAN) technology, whch enables portable devces to connect and communcate wrelessly va short-range ad-hoc networks [4], []. The basc 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 Fg. ). 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). 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 Master Slave Brdge Master whch s also a Brdge Fgure. An example of a Bluetooth scatternet 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 [30], [3]. The requred lnk capactes should be allocated by nter-pconet schedulng algorth. These algorth schedule the presence of the brdges n dfferent pconets. Numerous ntra and nter-pconet schedulng algorth have been proposed (see [], [6], [7], [9], [0], [], [], [6], and references theren). Analytcal performance evaluaton of ntra and nterpconet schedulng algorth has great mportance, snce t may provde nsght on ther desgn and optmzaton. However, as mentoned n [7], due to the specal characterstcs of the Bluetooth Medum Access Control (MAC), the performance of these algorth has been analyzed mostly va smulaton. In ths paper we show that a pconet wth undrectonal traffc operated accordng to the exhaustve schedulng algorthm can be modeled as an exhaustve pollng system and derve exact analytc and numercal results regardng ntrapconet delays. It should be noted that those results also apply to pconets wth asymmetrcal traffc. We also show that a pconet wth undrectonal traffc operated accordng to the lmted (pure round robn) schedulng algorthm can be modeled as a -lmted pollng system and derve exact analytc results. Then, we show a surprsng result: n a pconet wth only up- A pollng system conssts of several queues served by a sngle server accordng to a set of rules (pollng scheme) [3, p. 9], [3],[7],[9].

2 lnk traffc n whch all arrval rates are statstcally equal, the mean delays for the lmted and exhaustve regmes are equal. Ths result s extended for any arbtrary Tme-Dvson-Duplex system, operated n a smlar manner to a Bluetooth pconet, n whch the packets are not necessarly, 3, and slots long (as requred by the Bluetooth specfcatons [4]). Namely, t holds for any dstrbuton of packet lengths. We argue that when the traffc s b-drectonal, t see that there s no closed form expresson for the probablty generatng functon of the tme (n slots) to exhaust the queues at the master and a gven slave. Recently, Msc and Msc [7], [8], [9], [], [], [3], [4], [] 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 [8]). Snce our results do not comply wth these results, we show that the closed form solutons exhbted n [9] for the exhaustve ntra-pconet schedulng algorthm are actually approxmate solutons, as they are based on unsatsfed assumptons leadng to naccurate probablty generatng functons. We show that the probablty generatng functon of the tme (n slots) to exhaust the queues at the master and a gven slave s sgnfcantly dfferent from the functon derved n [9]. In addton, t s argued that the M/G/ queue wth vacatons model cannot be used drectly to analyze the system. We also ndcate that the results presented n [7], [8], [0], [], [], [3], [4], and [] are based on smlar models, and therefore seem to be approxmate results. Fnally, we provde numercal results regardng the exhaustve and the lmted algorth. These results demonstrate the dfference between the results derved n [9] and the exact results. We note that n [3] we have derved exact analytc results for a pconet wth b-drectonal traffc operated accordng to the lmted schedulng algorthm by dentfyng ts equvalence to a -lmted pollng system. It has been shown that the analyss of the lmted schedulng algorthm n [9], based on a drect applcaton of results from the M G queue wth vacatons model to the pconet system, gnores mportant statstcal dependences that exst n the pconet operaton model. Recently, Morand et al. [6] have presented an approxmate analyss of the lmted schedulng algorthm for a pconet wth asymmetrcal traffc. Ther analyss supports our concluson n [3] that the analyss of the lmted algorthm n [9] does not provde exact results. To the best of our knowledge, the results presented n ths paper and n [3] are the only avalable correct exact analytc results regardng the performance of Bluetooth schedulng algorth. We are not aware of any exstng exact results regardng the performance of pconets wth asymmetrcal traffc derved by a queueng theoretc approach. The rest of the paper s organzed as follows. Secton II gves a bref ntroducton to the Bluetooth technology and descrbes related work, whle Secton III presents the model. In Secton IV we analyze the exhaustve and lmted schedulng algorth n pconets wth undrectonal traffc and n general Tme-Dvson-Duplex syste. In Secton V we analyze the exhaustve algorthm n a pconet wth b-drectonal traffc and dscuss the analyses n [7] - []. Sectons VI and VII present numercal results and summarze the man results. Master Slave Slave Slave n II. BACKGROUND A. 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 6-µsec ntervals called slots. The master-to-slave transson starts n evennumbered slots, whle the slave-to-master transson starts n odd-numbered slots. Masters and slaves are allowed to send,3 or -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 transson 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 (f t has nothng to send). We refer to the master-to-slave communcaton as downlnk and to the slave-to-master communcaton as uplnk. An example of the TDD scheme n a pconet wth n slaves s gven n Fg.. The master schedules the traffc wthn a pconet accordng to an ntra-pconet schedulng algorthm. Varous ntra-pconet schedulng algorth have been recently proposed. For example, n the lmted (pure round robn) algorthm, the master communcates wth the slaves accordng to a fxed cyclc order and at most a sngle packet s sent n each drecton every tme a master-slave queue par s served. In the exhaustve algorthm, the master communcates wth the slaves accordng to a fxed cyclc order and t does not swtch to the next masterslave queue par untl both the downlnk and the uplnk queues are empty. In a scatternet, a unt (referred to as a brdge) can partcpate n two or more pconets, on a tme-sharng bass, and even change ts role when movng from one pconet to another. Namely, a brdge can be a slave of a few masters (referred to as slave/slave brdge) or a master n one pconet and a slave n another pconet (referred to as master/slave brdge). Fg. above llustrates an example of a scatternet ncludng brdges Fgure. 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 from these two types. The presence of a brdge n dfferent pconets has to be controlled by an nter-pconet schedulng algorthm. B. Related Work The Bluetooth specfcaton does not ndcate any method for scatternet formaton. Thus, several scatternet formaton algorth have been recently proposed (see [] for a through revew of these algorth). For nstance, Basagn and Petrol [] 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. []. 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 [30] and [3] the scatternet capacty assgnment problem s formulated and optmal and heurstc algorth for ts soluton are proposed. The soluton of the capacty assgnment problem s a desrable nput to ntra and nterpconet schedulng algorth. Few of the frst smple ntra-pconet schedulng algorth were proposed and evaluated va smulaton n []. Snce then, numerous ntra-pconet schedulng algorth 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 algorth. In [7] and [9] the performance of ntra-pconet schedulng algorth 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 shown that a pconet wth bdrectonal traffc operated accordng to the lmted schedulng algorthm s equvalent to a -lmted pollng system. Morand et al. [6] present an approxmate analyss of the lmted schedulng algorthm n a pconet wth asymmetrcal traffc that s based on a renewal process. Among other reasons, the results are approxmate, snce the tool of probablstc routng [3] s employed whle t s assumed that the varous resultng flows are ndependent. Accordng to [6], the assumpton of ndependent flows, although provdng good results at low traffc load, leads to substantal match wth the smulaton results as the system gets close to stablty lmt. 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. [] descrbe nter-pconet schedulng algorth 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 smallscale scatternets. Ths algorthm uses the low power mode hold to establsh rendezvous ponts adjusted to the traffc patterns. Fnally, Racz et al. [6] descrbe an ntra and nterpconet schedulng algorthm n whch the nodes assgn rendezvous ponts wth ther peers, such that the sequence of these ponts follows a pseudo random process. III. THE MODEL To facltate our cla we descrbe the pconet model presented n [9] and 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 and 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 sze (number of packets n a burst) s denoted by G b (x). We wll show that the results presented n [9] 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 slots are p, p 3, and p, respectvely. The PGF of the packet length s denoted by G p (x). Its mean and second moment are denoted by L = p+ 3p3 + p and L = p+ 9p3 + p. The access delay s defned as the tme a packet has to wat n the uplnk queue before t s served (ts mean s denoted by W a ). In [9], 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. 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. Smplfyng the model n [9], we assume that the master s the fnal destnaton of all packets generated at the slaves (.e. the master does not route packets between slaves). We also assume that packets are generated at every downlnk queue accordng to a Posson arrval process. Due to the 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 [9] should also hold for ths scenaro. However, we show that ths s not the case. Broadenng the symmetrcal model presented n [9] where the arrval rates to the uplnk and the downlnk queues are equal, we also consder pconets n whch these arrval rates dffer. Thus, we consder three dfferent cases n whch packets are generated at the uplnk and downlnk queues accordng to a Posson arrval process: Symmetrcal pconet The arrval rate to every downlnk and uplnk queue s λ. 3

4 Half-symmetrcal pconet The arrval rate to all the downlnk queues s the same (denoted by λ d ) and the arrval rate to all the uplnk queues s also the same (denoted by λ u ), but λ d λ u. d Asymmetrcal pconet The arrval rates to each of the downlnk and uplnk queues are not necessarly equal. We denote the arrval rate to the uplnk queue at slave as λ u and the arrval rate nto the master of packets ntended for slave by λ. The mean access delay of packets n the uplnk queue of slave s denoted by IV. W a. PICONETS WITH UNIDIRECTIONAL TRAFFIC In ths secton we focus on pconets wth only undrectonal (e.g. slave-to-master) traffc. We show that when such a pconet s operated accordng to the exhaustve schedulng algorthm, t can be modeled as an exhaustve pollng system. Then, we show that when the pconet s operated accordng to the lmted (pure round robn) schedulng algorthm t s equvalent to a -lmted pollng system. Based on these observatons, we obtan exact results for half-symmetrcal pconets and show that the values of the access delay are equal for both schedulng algorth. Fnally, we extend ths result to a general Tme-Dvson-Duplex system n whch packets are not necessarly, 3, and slots long. A. Analyss of the Exhaustve Algorthm Consder a pconet wth undrectonal traffc (.e. wth only slaves-to-master or only master-to-slaves traffc) operated accordng to the exhaustve schedulng algorthm. Frst, we analyze a half-symmetrcal pconet (.e. a pconet n whch λ u = λ > 0 and λ d = 0). Then, we show that an asymmetrcal pconet (n whch the arrval rates to the uplnk queues are not necessarly equal) can be analyzed n a smlar manner. Snce λ d = 0, when the master communcates wth a partcular slave t sends only POLL packets. The slave reples wth data packets untl ts queue s empty. Then, t sends a NULL packet whch sgnals the end of the exhaustve communcaton wth that partcular slave. At frst glance, t see that ths regme can be modeled as an exhaustve pollng system n a straghtforward manner. Namely, the servce tme can be defned as the packet length plus slot (for the precedng POLL packet) and the swtchover tme can be defned as slots (the POLL-NULL exchange). However, f a data packet arrves to an empty uplnk queue durng the transson of the POLL packet, the slave starts transmttng t n the next slot. Consequently, n ths straghtforward model a packet that arrves durng the frst half of the total -slot swtchover tme actually cancels the swtchover. Thus, an alternatve modelng s requred n order to model the pconet as an exhaustve pollng system. To ths end, we The termnaton of the master-slave exchange wth a POLL-NULL exchange comples wth the assumptons made n [9]. In an exhaustve pollng system, at each vst of the server to a queue, all the packets n that queue are served. The server ncurs a swtchover tme when t shfts from one queue to another [3, p. 98], [3],[7],[9]. defne the servce tme of a k-slot data packet (k =,3,) as (k + ) slots whch are composed of the k slots of data, augmented by the followng POLL packet. Hence, the servce tme of a -slot packet s defned as slots, for 3-slot packet t s 4 slots, and for -slot packet t s 6 slots. The swtchover tme s defned as slots, composed of the NULL packet endng the exchange wth a partcular slave and the POLL packet startng the exchange wth the next slave. For a half-symmetrcal pconet (λ u = λ > 0, λ d = 0), we apply the model for a symmetrcal exhaustve pollng system descrbed n [3, p. 98]. Accordngly, usng eq. (3.69) n [3], where the number of queues s (m ), the arrval rate s (m )λ, the swtchover tme s two slots wth zero varance, the traffc ntensty s ρ = ( m ) λ ( L + ), and the second moment of the servce tme s 4p + 6p p, we obtan the mean access delay (n slots): W a { λ p p 3 }. () ( m ) + 4 ( + 3 ) = ( m ) λ( L + ) In ths system t must hold that λ < (( m )( L + )). We shall refer to ( m ) λ ( L + ) as the load n the undrectonal exhaustve system. In a pconet wth a sngle slave (m = ) there s no dfference between the exhaustve and the lmted schedulng algorth. As a specal case, consder a pconet wth undrectonal traffc of -slot packets (.e. p = ) operated accordng to the lmted regme. Its mean access delay s gven n eq. () n [3] as: W a ( m ) =. () ( m ) λ It readly follows that for such a pconet ( λu = λ, m =, and p = ), () and () concde. The result presented n () was also verfed by a smulaton model based on OPNET (for more detals regardng the desgn of the smulaton model 3, see [0]). For example, we have computed by smulaton the average access delay n pconets wth 7 slaves for 0 dfferent load values (for each load value, the results have been computed after 30,000 slots). We have examned 3 dfferent packet length dstrbutons: () all packets are -slot long, () all packets are slots long, and (3) p = 0., p 3 = 0.6, and p = 0.. In the worst case, the dfference between the smulaton and exact results was less than.%. In Secton IV.C we wll show that, surprsngly, () also holds for a half-symmetrcal pconet (n whch λ u = λ > 0 and λ d = 0) operated accordng to the lmted schedulng algorthm. A half-symmetrcal pconet wth only downlnk traffc (.e. the arrval rate to each downlnk queue s λ d = λ and to each uplnk queue s λ u = 0) can be modeled as an exhaustve pollng system, n a smlar manner. However, the operaton model of such a pconet should be precsely defned. For example, snce traffc flows only from the master to the slaves so that the 3 Notce that n [0] the delay s defned as the tme untl the whole packet s receved by the destnaton. u 4

5 master has complete nformaton on the status of ts downlnk queues, there s no reason to send a POLL packet n order to end a master-slave exchange. On the other hand, n case all queues are empty, the master should transmt POLL packets untl a packet arrves to one of ts downlnk queues. We now consder an asymmetrcal pconet wth only uplnk traffc (.e. the arrval rate to each downlnk queue s λ d = 0, the arrval rate to each uplnk queue s λ u > 0, and the arrval rates to the uplnk queues are not necessarly equal). Such a pconet can be analyzed n a smlar manner to a halfsymmetrcal pconet wth only uplnk traffc. Namely, t can be modeled as an asymmetrcal exhaustve pollng system composed of (m ) queues, wth -slot swtchover tme and wth servce tme of (k + ) slots for a k-slot data packet (k =,3,). Accordngly, the PGF of the servce tmes n each uplnk queue can be computed n a smlar way to the models for exhaustve pollng syste descrbed n [3], [7] and [9]. Then, the mean access delay n each uplnk queue can be obtaned by solvng (m ) 3 equatons. Snce m 8, the computatonal complexty s neglgble. We note that results can be obtaned even for the case n whch the probabltes of a packet length beng, 3, or slots vary n dfferent uplnk queues. Fnally, we note that n Secton VI, we shall numercally compare the results derved n ths secton to the results derved n [9] and show that [9] sgnfcantly underestmates the access delay n case of undrectonal traffc. B. Analyss of the Lmted Algorthm Consder a pconet wth undrectonal traffc operated accordng to the lmted (pure round robn) schedulng algorthm. Frst, we analyze a half-symmetrcal pconet (.e. a pconet n whch λ u = λ > 0 and λ d = 0) and derve exact results. Then, we show that an asymmetrcal pconet can be modeled n a smlar manner. In [3] we have shown that a pconet wth b-drectonal traffc operated accordng to the lmted schedulng algorthm s equvalent to a -lmted pollng system wth (m ) queues. We have defned 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 k-slot data packet has been defned as (k ) slots. In order to obtan exact results, we have focused on symmetrcal pconets, snce the problem of computng exact delays n general -lmted pollng syste has not been resolved yet [3]. Accordngly, n [3], eq. (3) the access delay s presented as: W a { λ } + ( m ) + ( p3 + 6p ) = ( m ) λl. (3) Consder a pconet operated accordng to the lmted regme n whch there s only uplnk traffc. In such a pconet, the master contnuously sends POLL packets to the slaves. Even f the slave has nothng to send, one slot must be used durng the uplnk communcaton (by the NULL packet). At frst glance, t see that such a pconet can be modeled as a -lmted pollng system wth (m ) queues n a smlar manner to the modelng of a pconet wth b-drectonal traffc n [3]. Namely, the swtchover tme can be defned as slots that are composed of the POLL packet and the frst slot of the data packet or of the POLL-NULL exchange. As a result, the servce tme of a k-slot packet should be defned as (k ) slots. Accordng to ths modelng a packet that arrves to an empty uplnk queue durng the frst half of the swtchover tme (.e. durng the transson of the POLL packet) wll be served mmedately after the swtchover tme. On the other hand, a packet that arrves to an empty uplnk queue durng the second half of the swtchover tme (.e. durng the transson of the NULL packet) wll not be served mmedately (as opposed to the stuaton n a - lmted pollng system). Thus, an alternatve modelng s requred. We defne the begnnng of the swtchover to a queue as the moment n whch the slave starts transmttng a NULL or a data packet. A queue servce ends at the moment n whch the master completes the transson of the POLL packet ntended to the next slave. Accordngly, we defne the swtchover tme to each of the queues as slots: In case the slave sends a 3 or -slot data packet, the swtchover slots are the frst slots of the packet. In case the slave sends a -slot data packet or a NULL packet, these slots are composed of the packet sent by the slave, augmented by the followng POLL packet. Consequently, when data packets are sent, some of the data s actually sent durng the swtchover tme. Therefore, the servce tme of a k-slot data packet s defned as (k + = k ) slots. Note that ths mples that a -slot packet has a servce tme of 0 slots. Gven the above, we now focus on half-symmetrcal syste n whch the arrval rates to all uplnk queues are equal (.e. λ u = λ > 0 and λ d = 0). By applyng the model for a symmetrcal lmted gated pollng system descrbed n [3, p. 0] 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, f the queue s not empty, the swtchover ends slots after the actual servce of a Bluetooth packet starts. Thus, n order to obtan the mean watng tme n a pconet,.e. the mean access delay, one has to deduct slots from the expresson for the watng tme n [3], eq. (3.77). Accordngly, we apply [3] eq. (3.77), where the number of queues s (m ), the total arrval rate s (m )λ, the swtchover tme s two slots wth zero varance, the traffc ntensty s ρ = ( m ) λ ( L ), and the second moment of the servce tme s 4p 3 + 6p. Deductng tme unts (.e. slots), we obtan the mean access delay ( W a ) (n slots): W a { λ } { λ p 3 } = (4) + ( m ) + 4 ( p ) ( m ) + 4 ( p + 3 ) = ( m ) λ( L + ) ( m ) λ( L + ) Notce that n ths system t must hold that ( m ) λ( L + ) <. 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. 0], [3], [7]. The system s referred to as the lmted gated pollng system, snce only a packet that s found n the begnnng of the swtchover tme s served.

6 As a specal case, consder a half-symmetrcal pconet wth undrectonal traffc of -slot packets (.e. p = ) operated accordng to the lmted regme. Its mean access delay has been derved n [3] and t s gven by (). It readly follows that for such a pconet ( λu = λ and p = ), (4) concdes wth (). Moreover, n a pconet wth a sngle slave (m = ), there s no dfference between the exhaustve and the lmted schedulng algorth. Indeed, t readly follows that for m =, the result presented n (4) concdes wth (), whch presents the mean access delay n a pconet operated accordng to the exhaustve algorthm. Fnally, we note that a half-symmetrcal pconet n whch λ d = λ > 0 and λ u = 0 can be modeled as a -lmted pollng system, n a smlar manner to the modelng presented above. The derved queung delay at the master downlnk queue s the same as the delay presented n (4). Smlarly, an asymmetrcal pconet wth undrectonal traffc can be modeled as a - lmted pollng system wth (m ) queues. In order to obtan approxmate results for such a pconet, one of the approxmaton methods revewed n [3] can be used. C. Exhaustve and Lmted Algorth Equal Delays The access delay n the exhaustve algorthm (.e. the result obtaned n ()) s dentcal to the access delay n the lmted algorthm (.e. the result obtaned n (4)). That s the mean access delay n a half-symmetrcal pconet wth only uplnk traffc s the same for the exhaustve and the lmted schedulng algorth. We have no ntutve explanaton to ths surprsng phenomenon. It s well known [7] that n the classcal symmetrcal pollng syste, where swtchover tme s ncurred whenever the server moves from one channel to the next, the mean delay n the exhaustve regme s smaller than ts counterpart n the lmted regme. In a pconet wth only uplnk traffc operated accordng to the lmted regme, when two adjacent slaves queues are nonempty, no real swtchover tme s ncurred. Real swtchover tmes are pad only when a slave has nothng to transmt. On the other hand, when the pconet s operated accordng to the exhaustve algorthm, real swtchover tme s ncurred only at the end of a slave-master sesson. Snce we deal wth symmetrc cases, ths may (partally) explan the equalty between the mean access delays n both algorth. We now extend the result presented above to general syste operated accordng to the Tme-Dvson-Duplex (TDD) mechansm. Consder an arbtrary TDD system composed of a master and a few slaves, operated n a smlar manner to a Bluetooth pconet. Namely: The channel s slotted. A slave s allowed to start transson 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. The slave must respond by sendng a data packet or a -slot NULL packet. The man dfference between ths TDD system and the Bluetooth pconet s that masters and slaves are allowed to send data packets of any length (not necessarly, 3, and -slot packets). We assume that there s only uplnk traffc and that the arrval rates to all uplnk queues are equal. In ths secton, the probablty of a packet length beng n slots long s denoted by p n. Accordngly, the mean packet length s L = pn n= n and the second moment of the packet length dstrbuton s L = pn. n= n Smlarly to the analyss n Secton IV.A, t can be shown that the consdered TDD system operated accordng to the exhaustve schedulng algorthm s equvalent to the symmetrcal exhaustve pollng system, analyzed n [3, p. 98]. In the equvalent pollng system, the servce tme of a k slot packet (of the TDD system) s defned as k + slots and the swtchover tme s defned as two slots wth zero varance. Accordngly, usng the notaton of [3], the mean servce tme s X = L +, the mean swtchover tme s V =, the varance of the swtchover tme s σ V = 0, and the second moment of the servce tme s: n n= X = ( n+ ) p = L + L+. () Usng [3], eq. (3.69) where the number of queues (denoted n [3] by m) s set as (m ), the total arrval rate (denoted n [3] by λ) s set as (m )λ, and the traffc ntensty s ρ = ( m ) λ ( L + ), we obtan the mean access delay (n slots): W a = ( m ) λ( L+ ) ( m ) λ( L + L+ ) + ( m ) ( m ) λ( L+ ). (6) Smlarly to the analyss n Secton IV.B, t can be shown that the consdered TDD system operated accordng to the lmted schedulng algorthm s equvalent to the symmetrcal lmted gated pollng system, analyzed n [3, p. 0]. In the equvalent pollng system, the servce tme of a k slot packet (of the TDD system) s defned as (k ) slots and the swtchover tme s defned as two slots wth zero varance. Accordngly, X = L, V =, σ = 0, and V n n= X = ( n ) p = L L+. (7) Usng [3], eq. (3.69) where the number of queues s set as (m ), the total arrval rate s set as (m )λ, and ρ = ( m ) λ ( L ), we obtan the watng tme n the pollng system (n slots): ( m ) λ( L L+ ) W = + ( m ) λ( L ) ( m ) λ ( m ) + ( m ) λ( L ) λ ( m ) λ( L ) ( m ) λ As mentoned n Secton IV.B, n order to obtan the mean access delay n the TDD system operated accordng to the lmted algorthm, there a need to deduct tme unts from the (8) 6

7 watng tme obtaned n (8). Ths deducton yelds the result obtaned n (6), and therefore: The access delay n the exhaustve and lmted schedulng algorth s equal for any gven packet length dstrbuton. V. PICONETS WITH BI-DIRECTIONAL TRAFFIC In ths secton we consder pconets wth b-drectonal traffc and outlne the complextes n dervng the PGF of the tme (number of slots) to exhaust the queues at the master and a gven slave. Then, we show that the PGF of the tme to exhaust the queues derved accordng to [9] dffers from the correct PGF and argue that the use of the model of M/G/ queue wth vacatons n order to analyze the exhaustve regme leads to no more than approxmate results. A. Analyss of a Sngle Channel Analyzng the performance of schedulng regmes such as the exhaustve, gated, and globally gated [9] n a pconet wth b-drectonal traffc requres obtanng the PGF of the exchange tme of a sngle master-slave queue par (channel). Ths analyss s sgnfcantly complcated by the TDD mechansm and the use of POLL and NULL packets by the master and the slaves. In order to demonstrate the dffcultes n analyzng the exhaustve regme, we dscuss a less complcated case, namely a sngle master-slave channel n a pconet, operated n the gated regme, and pont out the obstacles n analyzng the TDD mechansm. In the gated regme, only the packets that are found n the uplnk and downlnk queues when the master starts servng the master-slave queue par are transmtted. If the number of downlnk packets exceeds the number of uplnk packets, the slave sends NULL packets as a response to some data packets. On the other hand, f the number of uplnk packets exceeds the number of downlnk packets, the master sends some POLL packets n order to allow the slave to reply wth data packets. We assume that at the end of the master-slave exchange, the slave has to respond wth a NULL packet to a POLL packet. Fg. 3 llustrates the operaton of a gated regme n a pconet composed of a master and two slaves. When the master starts transmttng the packet ntended to slave, there are 3 data packets (of szes 3,, and slots) n the downlnk queue and a sngle data packet n the uplnk queue, and therefore the slave reples wth NULL packets to the second and thrd data packets. Followng the last data packet, the master sends a POLL packet whch sgnals the end of the master-slave exchange. Ths packet s repled by a NULL packet. When the master starts transmttng a packet ntended to slave, there s a sngle data packet (of sze 3) n the downlnk queue and 3 Master Slave Slave Data Packet POLL Packet NULL Packet Tme (slots) Fgure 3. An example of the operaton of the gated regme n a pconet composed of two slaves In some cases ths POLL-NULL exchange s redundant. data packets (of szes 3, 3, and ) n the uplnk queue, and therefore the master sends POLL packets n order to allow the slave to reply wth ts data packets. Let X G denote the total tme (number of slots) requred for the exchange duraton of a sngle master-slave channel n the gated regme. Namely, t s the number of slots requred to serve all packets present n both downlnk and uplnk queues at the nstance when the master starts servng the queue par plus slots (the last POLL-NULL exchange). The PGF and the mean of X G are denoted by X G( x) and X G. For smplcty, we assume that all packets are slot long (p = ) and that packets have been accumulated n both queues for some T slots before the gated servce starts. We defne U and D as the number of packets accumulated n the uplnk and downlnk queues, respectvely, durng T slots (U,D~Posson(λT)). Thus, gven that p =, X G equals twce the maxmum of U and D plus slots. Namely, t s a functon of the maxmum of two Posson random varables. Accordngly, the PGF of the tme to serve a sngle master-slave channel s gven by: U D + ( ) max(, ) m G ( ) Prob(max(, ) ) m= 0 X x = E x = x x U D = m (9) where Prob(max( U, D) = m) = m m j m λt ( λt) λt ( λt) λt ( λt) (0) e e + e m! j= 0 j! m! Unfortunately, t appears that n vew of eq. (0) there s no closed form expresson for (9). We note that accordng to (9), the mean tme to serve a sngle master-slave channel s gven by: X = E( max( U, D) + ) = E(max( U, D)) +. () G It s clear that the exact analyss of the exhaustve regme s more complcated. B. Examnaton of the Analyss n [9] Analyss of the performance of the exhaustve regme n a pconet wth b-drectonal traffc s exhbted n [9]. It s based on stages: () the dervaton of the PGF of the tme to exhaust a sngle master-slave queue par, and () modelng the pconet as an M/G/ queue wth vacatons. We now brefly descrbe the analyss there and show that the derved PGF of the tme to exhaust a queue par dffers from the correct PGF. We wll not elaborate on the nd stage naccuraces, snce they are smlar to those dscussed for the lmted regme n [3]. Although the regme analyzed n [9] s referred to as exhaustve, t has some of the characterstcs of the gated regme. Accordng to [9] the master generates traffc and routes traffc generated by the slaves. Thus, n ths exhaustve regme, when the master serves a partcular master-slave channel, the packets present n the downlnk queue at the begnnng of an exchange wll be transmtted. Moreover, the packets generated by the master durng the exchange, whch are ntended for the partcular slave, wll also be transmtted. Thus, whle the overall mean 7

8 arrval rate to the downlnk queue s λ, the arrval rate to the downlnk queue durng the exchange perod s only λ / (m ). Ths follows snce durng the exchange perod, packets generated by other slaves accumulate n ther uplnk buffers. Denote the tme (number of slots) to empty both channel queues for a partcular slave n the exhaustve regme as X. Its PGF and mean are denoted by X ( x) and X. In [9], eq. (7), the PGF of X s presented as: ( λu + λd ) Xc( x)( Gb( GP( x)) ) λ λ X ( x)( G ( x) ) u d c b X ( x) = e e x () where X c (x) denotes the PGF of the cycle length and s presented as: X c (x) = (X (x)) m-. Let q j be the probablty that the cycle length s j slots so j that X c( x) = q j = jx. Assumng that the traffc s non-bursty (G b (x) = x) and that all packets are slot long (G p (x) = x), () reduces to: j j ( λu + λd ) qx j ( x ) λu λ d qx j ( x ) j= j= X ( x) = e e x, (3) If we assume that the arrval rates are symmetrcal (λ u = λ d = λ), then by dfferentatng (3) the mean tme to serve both queues s: X λ = +. Ths result s ndependent of the number of slaves or the cycle length, and therefore see unreasonable. Thus, we beleve that Msc and Msc [9] meant that the PGF of X s a functon of the mean cycle length ( X c ) and not of the PGF of the cycle length (X c (x)). Namely: ( λu + λd ) Xc ( Gb( GP( x)) ) λu λd Xc ( Gb( x) ) X ( x) = e e x. (4) We now show that even ths corrected equaton s naccurate. Assumng that the traffc s non-bursty, all packets are slot long, and the cycle length s T, eq. (4) reduces to: ( λu+ λd ) T( x ) λ λ T( x ) u d X ( x) = e e x. () As we understand, () s composed of 3 ndependent components. The frst component s the PGF of the number of packets arrvng accordng to a Posson arrval process wth arrval rate (λ u + λ d ) durng T slots. We suspect that ths component was ntended to represent the transson tme of uplnk and downlnk data packets. The second component s the PGF of the number of packets arrvng durng T slots, accordng to a Posson arrval process wth arrval rate λ u λ d. We suspect that ths component was ntended to represent the POLL or NULL packets sent by the devce that has the smaller number of data packets. The last component (x ) represents the POLL-NULL transson at the end of the exchange. As argued n Secton V.A, t appears that there s no closed form expresson for the PGF of the number of POLL or NULL We note that usng X c s consstent wth eq. () n [], whch presents the PGF of the tme to exhaust a master-brdge queue par as a functon of the duraton n whch packets accumulate n the master and the brdge and not as a functon of the PGF of ths duraton. packets sent by the devce whch has the smaller number of data packets (.e. there s no closed form for max(u,d) mn(u,d)). Clearly, ths dfference does not follow the Posson dstrbuton as mpled by (). Ths can be demonstrated by assumng that the arrval rates are symmetrcal (λ u = λ d = λ). Even n ths case a few POLL or NULL packets wll be sent durng every master-slave channel servce. However, accordng to (), the mean tme to serve both queues reduces to: X = λt +. Thus, n such a case, no POLL or NULL packets are sent durng a master-slave channel exchange (except for the last packets). Ths s of course ncorrect. Numercal results that demonstrate the dfference between () and the results derved n Secton V.A are presented n Secton VI. Although [9] analyzes the exhaustve regme, t see that t does not take nto consderaton all possble complcated scenaros. For example, t s not clear whether the analyss consders the scenaro n whch at the begnnng of the master-slave exchange the master has more data packets than the slave and durng the slave transson of NULL packets, several data packets arrve at the slave s uplnk queue. In such a scenaro the slave wll send a few NULL packets and the master wll send a few POLL packets. Smlarly to the analyss of the lmted regme, the rest of the analyss of the exhaustve regme n [9] drectly uses the model of M/G/ queue wth vacatons, wthout takng nto consderaton the dependences between the queues (a more elaborate dscusson s presented n [3]). Thus, even f the PGF of the tme to exhaust both queues (.e. (4)) was correct, the rest of the analyss s naccurate. In Secton VI, we compare numercal results obtaned accordng to [9] for the case n whch (4) see to hold, wth exact results obtaned accordng to the analyss n Secton IV.A. It s shown that the drect use of the vacaton model leads to approxmate results. Fnally, we note that napproprate assumptons, smlar to the ones ndcated n ths secton, also appear n [7], [8], [0], [], [], [3], [4] and []. For example, The analyss of the exhaustve schedulng algorthm n [8] and [] s very smlar to the analyss n [9]. In [7] the exhaustve schedulng algorthm s analyzed n a somewhat dfferent methodology. However, the model of M/G/ queue wth vacatons s drectly used and results n problematc dependences as dscussed n [3]. In [] a scatternet composed of two pconets connected by a brdge whch s a slave of the two masters s analyzed. The ntra-pconet schedulng algorthm s exhaustve and ts analyss s smlar to the analyss n [9] (for example, eq. (3) n [] s dentcal to eq. (7) n [9]). Furthermore, each master exchanges packets wth the brdge accordng to an exhaustve regme. The PGF of the length of ths exchange s derved n [], eq. () and t does not take nto consderaton the complextes dscussed n Secton V.A. The analyss n [3] and [] s very smlar to the analyss n []. In [0] the performance of scatternets composed of two pconets connected through a brdge s analyzed. The master exchanges packets wth the brdge accordng to an 8

9 exhaustve regme. The performance of the scatternets s analyzed for exhaustve and lmted ntra-pconet schedulng algorth. Although the ntra-pconet exhaustve schedulng algorthm s analyzed n a somewhat dfferent methodology than the analyss descrbed n [9], the model of M/G/ queue wth vacatons s stll drectly used. In Secton VI, we wll present numercal results that demonstrate the dfference between the exact results and the results obtaned accordng to [0]. Delay {slots} m= m=3 m=4 m= m=6 m=7 m=8 VI. NUMERICAL RESULTS In ths secton we present exact numercal results computed accordng to the analyss n Sectons IV.A and IV.B. Then, we compare numercal results calculated accordng to the analyss n [9] wth numercal results based on our analyss. We show that n some cases [9] sgnfcantly underestmates the tme to empty a master-slave queue par. Moreover, we show that n very smple scenaros, the mean access delay obtaned accordng to [9] and [0] consderably dffers from the exact mean access delay. Fg. 4 llustrates the exact mean access delay ( W a ), computed accordng to (), n a half-symmetrcal pconet wth only uplnk traffc (.e. λ u = λ, λ d = 0) operated accordng to the exhaustve regme. The fgure presents the delay (n slots) as a functon of the load n the undrectonal exhaustve system (defned n Secton IV.A as ( m ) λ( L + )) n pconets wth varous numbers of slaves n whch the probabltes of, 3, and -slot packets are equal ( p = p 3 = p = /3). Snce the results presented n () and (4) are the same, the fgure actually also presents the delay n pconets operated accordng to the lmted regme. Under the assumptons made n Sectons V.A and V.B, n a symmetrcal pconet, the mean tme to empty both queues accordng to [9] (.e. λt + ) dffers from the mean tme obtaned n Secton V.A (.e. E(max(U,D) + ). In order to demonstrate ths dfference, we have obtaned by smulaton average values of max(u,d) for a cycle of length T. Fg. llustrates the rato of the average value of max(u,d) to λt for dfferent values of λt. We have also computed the rato of the average tme to exhaust a master-slave channel, obtaned va smulaton accordng to our analyss n Secton V.A, to the mean tme to exhaust a channel, obtaned accordng to [9]. Fg. 6 presents ths rato when packets are all, all 3, or all slots long. The rato s presented as a functon of the mean number of packets arrvng durng a cycle (λt). It can be seen that the results obtaned n [9] underestmate the tme to empty both queues. We note that for every value of λt, we have obtaned 300,000 dfferent values of U and D n order to compute the average value of max(u,d) and the average tme to exhaust a channel. In order to obtan W a accordng to [9], we have used eq. (7) - (0) n [9]. As mentoned n Secton V.B, we assume that the PGFs n eq. (7) and (8) n [9] are functons of the mean cycle length ( X c ) and not of the PGF of the cycle length (X c (x)). Rato Rato Load Fgure 4. The exact mean access delay (computed accordng to ()) n halfsymmetrcal pconets wth only uplnk traffc, operated accordng to the exhaustve or lmted schedulng algorth, n whch p = p 3 = p = / λτ Fgure. The rato of the average value of max(u,d) to λt 0 4 λτ 6 8 -slot packets 3-slot packets -slot packets Fgure 6. The rato of the average tme to exhaust a master-slave queue par obtaned va smulaton, accordng to our analyss n Secton V.A, to the mean tme to exhaust a queue par obtaned accordng to [9] n pconets n whch packets are all, all 3, or all slots When the pconet s composed of a sngle slave the lmted regme and the exhaustve regme are equvalent. Fg. 7 presents the rato of the mean access delay computed accordng to the analyss of the exhaustve regme n [9] to the mean access delay computed accordng to our lmted model of a pconet wth b-drectonal traffc (.e. accordng to (3)) n a symmetrcal system wth a sngle slave. The fgure presents the rato as a functon of the load n the lmted system (defned n [3] as ( m ) λl ) for 3 cases: () when all packets are slot long, () when the probabltes of, 3, and -slot packets are equal, and (3) when all packets are slots long. We note that even f one uses the results presented n [9] for a pconet wth a sngle slave, the calculated mean access delay n the exhaustve regme s usually much hgher than the calculated mean access 0 9

10 delay n the lmted regme. For example, when all the packets are slots long and the arrval rate s λ = 0.08 (packets/slot), the rato between calculated mean access delay n the exhaustve and the lmted regmes (both accordng to [9]) s It see that the PGF of the tme to exhaust both queues derved n [9] (.e. (4)) holds only for undrectonal flows. Fg. 8, compares the mean access delay computed accordng to the analyss of the exhaustve regme n [9] when λ d = 0 to the mean access delay computed accordng to our analyss of a pconet wth undrectonal traffc operated accordng to the exhaustve algorthm (.e. accordng to ()). The fgure presents the delay (n slots) as a functon of the load n the undrectonal exhaustve system ( (m ) λ( L + ) ) n halfsymmetrcal pconets wth slaves. It can be seen that the results obtaned n [9] underestmate the access delay. Fnally, we note that n [0] the ntra-pconet exhaustve schedulng algorthm s analyzed n a somewhat dfferent methodology than n [9]. Thus, Fg. 9 presents the rato of the exact mean access delay to the mean access delay computed accordng to [0], n half-symmetrcal pconets wth only unlnk traffc (λ u = λ, λ d = 0) n whch the probabltes of, 3, and -slot packets are equal. VII. CONCLUSIONS Ths work presents an analytcal study of some versons of the exhaustve and lmted schedulng algorth for Bluetooth pconets, and examnes the analytcal study of the exhaustve algorthm n [9]. Frst, we have analyzed pconets wth undrectonal traffc. We have modeled the exhaustve schedulng algorthm as an exhaustve pollng system and the lmted schedulng algorthm as a -lmted pollng system. Based on ths modelng, we have derved exact analytc results and shown that, surprsngly, n half-symmetrcal pconets wth undrectonal traffc, the values of the mean access delay are the same for the lmted and exhaustve regmes. Ths result has been extended for general Tme-Dvson-Duplex syste wth arbtrary packet lengths. Then, the complexty of analyzng the gated schedulng regme n pconets wth b-drectonal traffc has been descrbed, ndcatng that the correspondng analyss of the exhaustve regme s even more complex. As a consequence, t has been shown that the results presented n [9] regardng the exhaustve schedulng algorthm are actually approxmate results. Moreover, t has been argued that the results presented n [7], [8], [0], [], [], [3], [4], and [] seem to be approxmate results. Fnally, we have provded numercal results and llustrated the dfference between our exact results and those presented n [9]. Rato p= p=p3=p=/3 p= Load Fgure 7. The rato of the mean access delay derved accordng to the analyss of the exhaustve regme n [9] to the exact mean access delay (derved accordng to (3)) n symmetrcal pconets wth b-drectonal traffc and a sngle slave Delay {slots} Rato [9] p= eq. () p= [9] p=p3=p=/3 eq. () p=p3=p=/3 [9] p= eq. () p= Load Fgure 8. The mean access delay derved accordng to [9] and the exact mean access delay (derved accordng to ()) n half-symmetrcal pconets wth only uplnk traffc, composed of slaves, and operated accordng to the exhaustve regme m= m=3 m=4 m= m=6 m=7 m= Load Fgure 9. The rato of the exact mean access delay (obtaned accordng to ()) to the mean access delay derved accordng to [0] n half-symmetrcal pconets wth only uplnk traffc, operated accordng to the exhaustve regme, n whch p = p 3 = p = /3 Eq. (0) n [9], whch presents the access delay (as a functon of λ), does not explctly dstngush between the uplnk and downlnk flows (t assumes that λ u = λ d ). Therefore, we have valdated that t s dentcal to eq. (9) n [] whch explctly separates the uplnk and downlnk flows (λ u and λ d ) and used t to compute the access delay when λ d = 0. In order to compute results regardng only ntra-pconet exhaustve schedulng accordng to [0], we have used eq. ()-(7),(9),(3)-(33) n [0] and assumed that G r (x) (defned n eq. () n [0]) equals. Moreover, we have assumed that λ u = λ and λ d = 0. Future research wll focus on the nterestng phenomenon, dentfed n ths paper, regardng the equalty of the delays n the exhaustve and lmted regmes n Tme-Dvson-Duplex syste wth undrectonal traffc. Moreover, 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 Secton V.A), t see that there s no closed form expresson for the delay under such regmes. 0