Determination of the Multi-slot Transmission in Bluetooth Systems with the Estimation of the Channel Error Probability

Transcription

1 Determnaton of the Mult-slot Transmsson n Bluetooth Systems wth the Estmaton of the Channel Error Probablty K Won Sung and Chae Y. Lee Det. of Industral Engneerng, KAIST, Kuseong Dong, Taejon, Korea Emal: kwsung@kast.ac.kr, chae@kast.ac.kr Abstract Bluetooth s an oen secfcaton for a technology to enable short-range wreless communcatons that oerate n an ad-hoc fashon. Bluetooth uses frequency hong wth a slot length of 6 s. Each slot corresonds to a acket and mult-slot ackets of 3 or slots can be transmtted to enhance the transmsson effcency. However, the use of mult-slot acket may degrade the transmsson erformance under hgh channel error robablty. Thus, the length of mult-slot should be adjusted accordng to the current channel condton. Segmentaton and Reassembly (SAR) oeraton of Bluetooth enables the adjustment of the length of mult-slot. In ths aer, we roose an effcent mult-slot transmsson scheme that adatvely determnes the otmal length of slots of a acket accordng to the channel error robablty. We frst dscuss the throughut of a Bluetooth connecton as a functon of the length of a mult-slot and the channel error robablty. A decson crtera whch gves the otmal length of the mult-slot s resented under the assumton that the channel error robablty s known. For the mlementaton n the real Bluetooth system the channel error robablty s estmated wth the maxmum lkelhood estmator (MLE). A smle decson rule for the otmal mult-slot length s develoed to maxmze the throughut. Smulaton exerment shows that the roosed decson rule for the mult-slot transmsson effectvely rovdes the maxmum throughut under any tye of channel error correlaton.

2 Index Terms Bluetooth, Segmentaton and Reassembly, Channel Error Probablty, Maxmum Lkelhood Estmator I. Introducton Bluetooth [1] s an oen secfcaton for a technology to enable short-range, ont to multont wreless communcatons that oerate n an ad-hoc fashon. The Bluetooth secfcaton was develoed by the Bluetooth SIG (Secal Interest Grou), whch s an ndustry consortum wth more than 1800 member comanes. The objectve of the Bluetooth technology s the desgn of low ower, small szed, and low cost rado that can be embedded n ortable devces such as PDAs, moble hones, notebook comuters, and so on. More detaled revew of Bluetooth can be found n []. Bluetooth oerates n the unlcensed ISM (Industral, Scentfc, and Medcal) band at.4 GHz. Snce ISM band s unlcensed, Bluetooth devces must coexst wth exstng technologes such as IEEE wreless LAN and Home RF. Thus a frequency ho transcever s aled to combat nterferences. A set of 79 ho carrers s emloyed at 1 MHz sacng. The nomnal ho duraton s 6 s whch concdes wth a length of the sngle slot. Each slot corresonds to a acket, and addtonally mult-slot ackets of 3 or slots can be transmtted to enhance the transmsson effcency. In cases of mult-slot ackets, they are sent on a sngle ho carrer. Fgure 1 dects the mult-slot transmsson. As the length of mult-slot ncreases, the robablty of the acket error ncreases under the same channel condton and the damage by the channel error becomes crtcal. Thus, the use of mult-slot acket may degrade the erformance under the hgh channel error robablty. Therefore, the length of mult-slot should be adjusted wth the current channel condton. A Bluetooth connecton may exerence a dynamc change of channel condton because a large number of ad-hoc Bluetooth connectons may coexst n the same transmsson area wthout any mutual coordnaton. The mact of the nterference generated from other ad-hoc

3 Bluetooth connectons s studed n [3-]. Moreover, the use of unlcensed ISM band such as IEEE wreless LAN generates another source of nterference to the Bluetooth connecton. The nfluence of wreless LAN to Bluetooth connectons s nvestgated n [6, 7]. Consderng the frequent change of the wreless channel condton, the estmaton of the channel error robablty s necessary to determne an effectve length of a mult-slot. However, the estmaton of the channel error robablty s dffcult because mult-slot ackets wth dfferent lengths are transmtted durng a sngle Bluetooth connecton. Besdes, the estmaton should be smle enough to be run on Bluetooth devces that requre low ower consumton. In Bluetooth, the determnaton of the mult-slot length s erformed by the Segmentaton and Reassembly (SAR) oeraton. Smle SAR schemes that enhance the lnk utlzaton are roosed n [8, 9]. Also, mult-slot transmsson s used for Bluetooth conet schedulng n [10]. However, the channel error s not consdered n these works. SAR schemes roosed n [11, 1] consder the channel error robablty. In these works, the best length of the acket to be transmtted s redcted from the error robablty of currently used acket. However, the methods cannot rovde an accurate estmator when the lengths of ackets change frequently. In ths aer, an accurate error robablty estmator s resented such that each Bluetooth devce estmates the error robablty from ts ast transmsson hstory. The maxmum lkelhood estmator (MLE) s used to estmate the channel error robablty from the hstory of the mult-slot errors. We roose an effcent mult-slot transmsson scheme that adatvely determnes the otmal length of slots of a acket that reflects the channel error robablty. We frst consder the throughut of a Bluetooth connecton as a functon of the mult-slot length and the channel error robablty. Then the decson crtera whch gves the otmal length of the mult-slot s resented wth the assumton that the current channel error robablty s known. The channel error robablty s then estmated wth the MLE, and a smle decson rule for the otmal length of mult-slot s develoed. The roosed decson rule for the otmal length of the mult-slot s so smle that any Bluetooth devce can emloy the rule wth low ower consumton.

4 Ths aer s organzed as follows. In Secton, the throughut of Bluetooth s dscussed and the decson crtera for the otmal lengths of the mult-slots are resented. In Secton 3, the channel error robablty s estmated wth MLE and a smle decson rule of mult-slot transmsson s develoed. The erformance of the roosed mult-slot decson rule s demonstrated n Secton 4 and the concluson s resented n Secton. II. Throughut of Bluetooth In Bluetooth, user data are transmtted based on acket. The nformaton of the uer layer s fragmented nto ackets wth the lengths of 1, 3, or slots. The duraton of one slot s 6 s, whch corresonds to one ho duraton. Mult-slot occues multle of one slot duraton. However, the actual mult-slot duratons are aroxmately 0 s shorter than the resectve multle of the hong duraton to allow for syntheszer re-tunng [3]. Snce the tme requred for re-tunng cannot be used for data transmsson, the er-slot ayload sze of 1-slot acket (7 bytes/slot) s smaller than that of 3-slot acket (61 bytes/slot) and -slot acket (67.8 bytes/slot). We ntroduce the concet of mn-slot to take the above asect nto account. The duraton of a mn-slot s half of one slot duraton. A mult-slot acket whch conssts of slots s assumed to consst of 1 mn-slots. A fast, unnumbered ARQ scheme s used n the Bluetooth. An ACK or a NAK s returned n resonse to the recet of revously receved acket. Packets are retransmtted untl an ACK s returned. The ACK and the NAK s assumed to occuy one mn-slot n ths aer. Let us defne burst_set() to be the set of mult-slot of slots and an ACK or a NAK. In other words, burst_set() conssts of mn-slots. The concet of the burst_set() s dected n Fgure. We also defne and as the error robablty of one mn-slot and a burst_set(), resectvely. Let N () denote the number of ayload mn-slots n a burst_set(), and N t () be the total number of mn-slots durng the transmsson erod of one burst_set(). As an examle n the burst_set() of Fgure, N ( ) = 9 and N ( ) = 1. t

5 The tme requred for frequency re-tunng s consdered as one mn-slot both after the transmsson of the ayload and the ACK/NAK. Then, t s clear that ( ) = 1and N t ( ) = +. Let N b () be the number of reeated burst_set()s for the transmsson of a burst_set(). It ncludes the re-transmsson n case of error. Also, let T (, ) be the throughut of burst_set() wth mn-slot error robablty. Then, the throughut of Bluetooth s gven by N N ( ) 1 T (, ) = = (1) N ( ) E[ N ( )] ( + ) E[ N ( )] t b b The transmsson of a burst_set() s successful only when all mn-slots n the burst_set() s successfully transmtted. Thus, s gven as follows: = 1 (1 ) () Snce N b () follows geometrc dstrbuton wth success robablty 1 ), ( 1 E[ N b ( )] = (3) 1 Therefore, the throughut becomes ( 1)(1 ) T (, ) = ( + ) (4) As shown n Equaton (4), the throughut s a functon of the length of the mult-slot and the mn-slot error robablty. If a channel has no error,.e., when =0, we have T ( 0,1) = 0., T ( 0, 3) = 0. 6, and T ( 0, ) = These values are the maxmum

6 throughuts wth burst_set(1), burst_set(3), and burst_set(), resectvely. However, as the channel error robablty ncreases, the throughut decreases. Moreover, the throughut degrades fast for long mult-slot ackets because the robablty of the acket error ncreases as the number of mn-slot ncreases. Fgure 3 shows the throughut of mult-slot transmsson comuted by the Equaton (4). In the fgure, t s clear that the otmal length of the mult-slot that gves the maxmum throughut vares accordng to the mn-slot error robablty. Hence, an arorate burst_set() should be selected deendng on the channel error robablty. In the fgure, burst_set() gves the best throughut when the mn-slot error robablty s less than When the error robablty les between 0.04 and 0., burst_set(3) shows the best throughut. For the error robablty whch exceeds 0., the best throughut s obtaned by burst_set(1). From the fgure, a decson crtera for the selecton of the otmal burst_set() can be obtaned as n Table 1. Note that the mn-slot error robablty =0.04 and =0. corresonds to bt error rates of and , resectvely from the equaton Number of Bts ( 1 ) = 1 BER and the aroxmate number of ayload bts of 80 n a mnslot. III. Determnaton of the Mult-slot Transmsson The decson crtera shown n the Table 1 s based on the assumton that the mn-slot error robablty s known. However, ths s not the case n the real world. In the real Bluetooth envronment, we can only measure the errors of the burst_sets. Therefore, mn-slot error robablty should be estmated from the burst_set error rate. The estmaton of the mn-slot error robablty s dffcult due to dfferent tyes of transmtted burst_set()s durng a sngle Bluetooth connecton. However, to obtan an accurate mn-slot error robablty the nformaton of all three tyes of burst_set()s should be emloyed. For examle, suose that 3 of 10 burst_set(1)s, of 6 burst_set(3)s, and 3 of

7 burst_set()s have errors, then what s the mn-slot error robablty? Note that error robablty of each burst_set() s dfferent even wth the same mn-slot error robablty due to the dfferent number of mn-slots. Moreover, the error robablty changes dynamcally by other nterferers such as other Bluetooth connectons, wreless LAN, or Home RF. Thus the estmaton of mn-slot error robablty should be executed erodcally. To satsfy the frequent estmaton of the mn-slot error robablty of the Bluetooth devces wth low ower consumton, a smle estmaton rocess s requred. We roose a smle mn-slot error robablty estmaton scheme by emloyng the ast transmsson hstory of each Bluetooth devce. Maxmum lkelhood estmator s emloyed for the estmaton. Let N be the number of total transmtted burst_set()s used for the hstory nformaton to estmate the mn-slot error robablty. Also, let F be the number of faled burst_set()s among N. Frst, the lkelhood functon of s consdered. The number of faled burst_set()s among N follows a bnomal dstrbuton wth robablty. Therefore, the lkelhood functon of s gven by L N F N F ( ) = F (1 ) () Further, the transmsson of a burst_set() s ndeendent of other burst_set()s. Thus, the lkelhood functon of mn-slot error robablty becomes L N = N N (6) 1 F 1 N1 F1 3 F3 N3 F3 F ) N F ( ) 1 (1 1) 3 (1 3 ) (1 F 1 F 3 F The MLE ˆ of the mn-slot error robablty s the value whch maxmzes the lkelhood functon.

8 = max L( ) (7) Thus, ˆ s the soluton of the followng equaton. dl( ) = 0 d (8) In the Aendx, t s roved that Equaton (8) has a unque soluton for 0 < < 1. Now, usng the decson crtera of Table 1 and the MLE ˆ of the mn-slot error robablty, we develo a decson rule for the mult-slot transmsson as n Fgure 4. In the fgure, F (x) s the numerator of d ln L( x), whch aears n Equaton (13) n the dx Aendx. In the range of 0 < < 1, F( 1 ) s monotoncally ncreasng functon, and F ( 1 ˆ) = 0. Therefore, f F ( 1 ) > 0, then ˆ les between 0 and. In other words, f F ( ) > 0, ˆ s less than 0.04, whch means that burst_set() gves the best erformance among the three burst_set()s. In the same way, f F ( 1 0.) < 0, ˆ s larger than 0.. Thus we should select burst_set(1) to obtan the hghest throughut. In other cases, burst_set(3) should be selected. The decson of burst_set() deends on N burst_set()s used for the hstory nformaton n the roosed decson rule. The nfluence of N on the erformance of the roosed mult-slot transmsson scheme s dscussed n the next secton. Wth the decson rule n Fgure 4, the otmal length of mult-slot can be easly obtaned. The comutaton of the roosed decson rule s so smle that t can be used wth small amount of comutng ower n any Bluetooth devce n real tme.

9 IV. Smulaton Results Smulaton exerments are erformed to evaluate the erformance of the roosed mult-slot transmsson scheme. A conet that conssts of one master and one slave s consdered n ths aer. In case of a mult-slave conet, the roosed scheme can be emloyed for each master-slave connecton. We assume that the acket to be transmtted s generated contnuously n downlnk (from master to slave). In ulnk, only ACK or NACK s transmtted wthout ggybackng. FEC s not used n the exerments. In wreless envronment t s generally consdered that the channel error undergoes correlaton. Thus, we consder the correlaton of the channel errors between the successve mn-slots n the smulaton exerments. The channel error model emloyed n [13] s consdered for the correlaton. The channel error of mn-slot s assumed to follow two-state dscrete Markov chan. Let s be the robablty that the transmsson of next mn-slot s successful gven that the current mn-slot has an error. Also, let e be the robablty that the transmsson of next mn-slot has an error gven that the current mn-slot has been successfully transmtted. Then, the robablty transton matrx M s gven as follows: M 1 = s e e 1 s (9) Let P S and P E resectvely be the steady-state robablty that the transmsson of a mnslot succeeds or fals. Then from (9), P S and P E s comuted as P S s = and s + e P E e = (10) s + e Note n the above equatons that P E s equvalent to the mn-slot error robablty. Clearly, as + converges to zero, error robabltes of successve mn-slots are more s e

10 correlated. When = 1, the consecutve mn-slots are uncorrelated [13]. s + e In order to determne the channel correlaton, t s mortant to nvestgate a roer Markov chan. By arttonng the range of receved sgnal to nose rato (SNR) nto a fnte number of ntervals, a Markov chan can be constructed. Let 0 = A 0 < A 1 < A = be the thresholds of receved SNR. Then a channel s sad to be n state s k f the receved SNR s n the nterval [A k, A k+1 ]. In a tme-varyng channel envronment, s s aroxmated by the rato of crossng rate from s 0 to s 1 dvded by the orton of state s 0. Smlarly, e s aroxmated by the rato of crossng rate from s 1 to s 0 dvded by the orton of state s 1. Detaled dscusson on the desgn of the Markov chan s gven n [14] wth references. In the smulaton, the followng three cases of correlaton are examned: 1. s + e = 1.0: uncorrelated. s + e = 0.: moderately correlated 3. s + e = 0.1: strongly correlated. The smulaton results of above three cases are shown n Fgure, 6, and 7, resectvely. In each case, the mn-slot error robablty s changed from 0 to 0.3. Note that mn-slot error robablty of 0.3 roughly corresonds to bt error robablty of The throughut n the three fgures llustrates that the roosed mult-slot decson rule s well suted both for the correlated and the uncorrelated error cases. The roosed scheme forms the envelo of the maxmum throughut for all three cases. In Fgure, the throughut by the roosed transmsson scheme almost concdes wth the maxmum throughut, and the otmal burst_set() crosses at =0.04 and =0. whch s dentcal to the analyss n Secton. It llustrates that the roosed mult-slot decson rule selects an arorate burst_set() accordng to the current channel error robablty when the channel errors are uncorrelated. Fgure 6 shows the throughut wth moderately correlated errors. Burst_set() gves the best erformance n the range of , and burst_set(3) shows the hghest throughut n the range of When the channel error s strongly correlated as n Fgure 7, burst_set() always gves the hghest throughut. Also, the overall throughut by the

11 burst_set() s ncreased comared to Fgure and 6. It seems to be due to the burstness of the channel errors that are concentrated on a few burst_sets. As the mn-slot error robablty becomes more correlated, mn-slot errors occur ntensvely n a few short-term erods. Consequently, the error robablty of burst_set() decreases and the throughut ncreases. Now, notce n Fgure 7 that 0 burst_sets ( N =0) are used for the hstory nformaton to estmate the mn-slot error robablty. Snce the amount of nformaton requred to estmate the error robablty largely deends on the correlaton among successve mn-slots, we nvestgate the effect of N on the throughut for the three cases of correlaton. In the exerments N s ncreased from 0 to 400 whle the steady-state mn-slot error robablty s fxed to 0.. Fgure 8 and 9 resectvely shows the case of no correlaton and moderate correlaton. It turns out that the throughut by the roosed scheme shows good erformances when N 100. Fgure 10 shows the result when the strong correlaton exsts among mn-slots. It s evdent that the large number of hstorcal burst_sets does not rovde an accurate estmaton when strong correlaton exsts among mn-slots. In other words, snce the error rate changes frequently by a short-term erod, the number of burst_sets requred for the hstorcal nformaton should be reduced to reflect the most recent trend of the error robablty. V. Concluson An effcent mult-slot transmsson scheme that adatvely determnes the otmal length of Bluetooth slots of a acket s develoed accordng to the channel error robablty. The concet of mn-slot and burst_set() are ntroduced to measure the throughut n the wreless channel. The throughut of burst_set() s obtaned as a functon of the mult-slot length and the mn-slot error robablty. The maxmum lkelhood estmator s emloyed to estmate the mn-slot error robablty of a Bluetooth connecton. A smle decson rule for the

12 otmal length of the mult-slot s develoed based on the error robablty estmaton. The roosed decson rule s smle enough to be mlemented n any Bluetooth devce wth small comutng ower n real tme. Smulaton exerment s erformed by assumng that the channel error of mn-slot follows two-state dscrete Markov chan. Three cases of mn-slot error correlaton are examned: uncorrelated, moderately correlated, and strongly correlated error. For all cases, the roosed mult-slot decson rule forms the envelo of the maxmum throughut comared to other fxed burst_sets. The effect of the amount of hstorcal nformaton requred to estmate the error robablty s also nvestgated. When strong correlaton exsts among mn-slots, relatvely small number of hstorcal burst_set()s rovdes better erformance due to the burstness of mn-slot errors. Aendx: Proof of the unqueness of the soluton of Equaton (8) From Equaton (6) we have L N N N 1 F 1 N1 F1 3 F3 N3 F3 F ) N F ( ) = 1 (1 1) 3 (1 3 ) (1 F 1 F 3 F, = 1 (1 ), 1 F1 where N, N F, and N 3 F3. When F = 0 for all, ˆ = 0 satsfes Equaton (8). Also, when F = N, ) = 1solves the equaton. Thus, the case of 0 < < 1 s consdered n the roof. Let 1 = x ( 0 < x < 1 ) and ( N1 F1 ) + 6( N 3 F3 ) + 10( N F ) = A. Then, by takng the natural log of L ( ) we have N1 N 3 ln L( x) = ln ln F + 1 F3 N + ln F + F1 ln(1 x ) + F 3 ln(1 x 6 ) + F ln(1 x 10 ) + Aln x (11)

13 By dfferentatng the above equaton, d ln L( x) dx F1 x 6F3 x x 1 x 10F x + 1 x = A x = A(1 x )(1 x 6 )(1 x ) F1 x (1 x )(1 x ) 6F3 x (1 x )(1 x 6 10 x(1 x )(1 x )(1 x ) 10 ) 10F x 10 (1 x )(1 x 6 ) (1) Note that the denomnator of Equaton (10) s not zero. Let the numerator of Equaton (10) be F (x). Then, F1 x F( x) = A 1 x 6F3 x 1 x F x 1 x F 6F 10F N N 3 10N + (13) 6 1 x 1 x 1 x = 10 Note that F 0) = N + 4N + 6N F 4F 6F 0 and lm F( x) =. Furthermore, ( x 1 df( x) 4F1 x = dx (1 x ) 16F3 x + 4 (1 x ) 3 36F x + 6 (1 x ) < 0 (14) Equaton (14) llustrates that F (x) s a monotoncally decreasng functon of x. Thus, d ln L( x) F ( x) = 0 has a unque soluton. Consequently, = 0 dx has a unque soluton for dl( ) 0 < x < 1 and = 0 d has a unque soluton for 0 < < 1.

14 References [1] Bluetooth Secal Interest Grou, Secfcaton of the Bluetooth System, Verson 1.1, Volume 1: Core, htt:// [] Haartsen J, The Bluetooth Rado System, IEEE Personal Communcatons, 000; 7(1): [3] Zurbes S, Consderatons on Lnk and System Throughut of Bluetooth Networks, IEEE Internatonal Symosum on Personal, Indoor, and Moble Rado Communcatons (PIMRC), 000: [4] Zurbes S, Stahl W, Matheus K, and Haartsen J, Rado Network Performance of Bluetooth, IEEE Internatonal Conference on Comuncatons (ICC), 000: [] Howtt I, Mutual Interference Between Indeendent Bluetooth Pconets, IEEE Transactons on Vehcular Technology, 003; (3): [6] Howtt I, WLAN and WPAN Coexstence n UL Band, IEEE Transactons on Vehcular Technology, 001; 0(4): [7] Cont A, Dardar D, Pasoln G, and Andrsano O, Bluetooth and IEEE 80.11b Coexstence: Analytcal Performance Evaluaton n Fadng Channels, IEEE Journal on Selected Areas n Communcatons, 003; 1(): [8] Kala M, Bansal D, and Shorey R, Data Schedulng and SAR for Bluetooth MAC, IEEE Vehcular Technology Conference (VTC), Srng, 000: [9] Das A, Ghose A, Razdan A, Saran H, and Shorey R, Enhancng Performance of Asynchronous Data Traffc over the Bluetooth Wreless Ad-hoc Network, IEEE INFOCOM, 001: [10] Cordero C, Abhyyankar S, and Agrawal P, A Dynamc Slot Assgnment Scheme for Slave-to-Slave and Multcast-lke Communcaton n Bluetooth Personal Area Networks, IEEE GLOBECOM, 003: [11] Km J, Lm Y, Km Y, and Ma JS, An Adatve Segmentaton Scheme for the Bluetoothbased Wreless Channel, Internatonal Conference on Comuter Communcatons and

15 Networks, 001: [1] Anjum F and Famolar D, A Robust and Adatve Algorthm for Enablng Effcent Communcaton n Bluetooth Networks, IEEE GLOBECOM, 00: [13] Lu S, Bharghavan V, and Srkant R, Far Schedulng n Wreless Packet Networks, IEEE/ACM Transactons on Networkng, 1999; 7(4): [14] Wang HS and Moayer N, Fnte-State Markov Channel A Useful Model for Rado Communcaton Channels, IEEE Transactons on Vehcular Technology, 199; 44(1):

16 Fgures and Tables Fg. 1 Mult-slot Transmsson µ Fg. Burst_set()

17 Fg.3 Throughut of Bluetooth Table 1 Decson crtera for the mult-slot transmsson Mn-slot error robablty Otmal Burst_set() 0 < 0.04 Burst_set() 0.04 < 0. Burst_set(3) 0. Burst_set(1)

18 If F ( ) > 0 Then select burst_set() Else f F ( 1 0.) < 0 Then select burst_set(1) Else Then select burst_set(3) Fg.4 Decson Rule for the Mult-slot Transmsson Fg. Throughut when channel errors are uncorrelated (N =100)

19 Fg.6 Throughut when channel errors are moderately correlated (N =100) Fg. 7 Throughut when channel errors are strongly correlated (N =0)

20 Fg. 8 Effect of N when errors are uncorrelated (=0.) Fg.9 Effect of N when errors are moderately correlated (=0.)

21 Fg.10 Effect of N when errors are strongly correlated (=0.)