Efficient Power Allocations in Wireless ARQ Protocols

Transcription

1 Efficient Pwer Allcatins in Wireless ARQ Prtcls Naveen Arulselvan and Randall Berry Department f Electrical and Cmputer engineering Nrthwestern University, Evanstn, IL, 1. Abstract In this paper, efficient pwer allcatin strategies fr ARQ prtcls perating in wireless envirnments are studied. Pwer as well as transmissin rate is ptimally adapted by the sender based n channel state infrmatin (CSI) btained thrugh feedback, while guaranteeing quality-f-service (QOS) cnstraints such as average thrughput r average delay. The pwer plicies adpted fr basic ARQ prtcls are characterized, and a cmparisn f their perfrmance is studied. Numerical results are presented fr a Rayleigh fading channel. Keywrds Wireless Netwrks, Retransmissin prtcls, errr cntrl, pwer cntrl INTRODUCTION Pwer cnservatin is a key cncern fr wireless technlgies. Prtcls deplyed in wireless netwrks must udiciusly utilize the available pwer as well as deliver reliable cmmunicatins. In this wrk, we study pwer efficiency in the cntext f link-level errr cntrl fr a wireless netwrk. Specifically, we cnsider varius autmatic repeat request (ARQ) prtcls, which may be used t prvide reliability t delay-tlerant applicatins. Fr these prtcls, we investigate the trade-ffs between the average transmitted pwer and varius QS parameters such as average thrughput and delay. We als present a cmparisn between the perfrmance f Stp-and-Wait and G-Back-N prtcls. There is a grwing awareness that a strnger cupling between traditinal netwrk layers can be beneficial in wireless settings [2]. In this wrk, we explre such a cupling between the physical layer and link-layer retransmissin prtcls. We fcus n the perfrmance f different ARQ prtcls when physical layer parameters, such as transmissin rate and pwer can be adapted, based in part n the available CSI. Here, CSI culd be the exact r average fade level r any ther meaningful descriptin f the physical layer. Several ther appraches alng these lines have been investigated in the literature. In [7],[8],[9], transmissin plicies t minimize the average energy expended are studied. An adaptive algrithm t ptimally adust the packet lengths is examined in [4]. CHANNEL MODEL We cnsider the situatin shwn in Fig. 1, in which the sender transmits packets ver a narrw-band blck-fading Transmitter θ n ACK/NAK channel θ n Retransmissin Receiver Figure 1. Feedback based errr cntrl channel with additive nise. T simplify ur discussin, we assume the blck length is equal t the packet transmissin time. A discrete-time mdel is used, where each time unit crrespnds t the blck length. After every time step, the transmitter receives new CSI, independent f whether a transmissin attempt was made r nt. We assume this CSI takes values frm the finite set. The actual fade level f the channel depends cnditinally n the current CSI. The channel is assumed t have a fixed prpagatin delay,, after which the receiver acknwledges each packet as crrectly received r in errr. If in errr, the packet is retransmitted accrding t the ARQ prtcl. Initially, we assume that the transmissin rate is fixed at bits per secnd, and that the transmitter can adapt the transmissin pwer used t send each packet. Let dente the pwer allcated when the CSI has value. The prbability a packet arrives in errr depends n the transmitted pwer and the available CSI,. We express this via a functin. This functin is assumed t be decreasing in ; a specific example is given in the next sectin. A case where the sequence f CSI values are mdelled as a statinary, ergdic Markv chain is cnsidered. Let dente the steady-state prbability distributin f. The steady-state average prbability f a successful transmissin is given by! $# %'&( ) Similarly, the average pwer expended is fund t be, + -$ Prblem Frmulatin Fr mst cmmn ARQ prtcls, the needed success prbability! can be determined given a required average thrugh- (1) (2)

2 % w ž put f /.. Fr instance, cnsider a G-Back-N prtcl with large enugh windw size 132)465 s that the transmitter never idles if packets are available [6]. In this case, the needed success prbability is given by! /. :<; 1 2 4=5. where? is the number f data bits in each packet, is the frame size. Given such a relatin, the pwer allcatin prblem can be frmulated as, ACB$D E s.t F ) %G&(HI- ) SRTU -PO>Q KJMLNI. where J L /. is a functin that depends n the ARQ prtcl emplyed. Fr instance, JWVYX)ZH[\.! is given as in (3) fr a G-Back-N prtcl. Fr a Stp-and-wait prtcl [6], J 2^] I. /.II@ ; _?M Fr a Selective Repeat prtcl with large enugh windw size, Ja` Let -Le/. dente the slutin t (4) as a functin f the required thrughput., fr a given prtcl. The functin, -Le/. then describes the trade-ff between pwer and thrughput fr that prtcl. We nte that the minimum energy per bit slutin investigated in [9] can be interpreted as a particular value f -Le/.. RAYLEIGH FADING CHANNELS We assume each packet is transmitted ver a Rayleigh fading channel with Additive White Gaussian Nise (AWGN). The pwer spectral density f the nise prcess is fhg and the bandwidth f the channel is i Hz. Let C be the channel fading T 5k when in the channel state. We assume the CSI sequence ^ prvides the crrespnding expected fade levels, i.e, ' mlnc C T6 fr all Let p tu)v qsr be the transmitted SNR when the current CSI is We mdel the prbability a packet is drpped using a capacity vs. utage framewrk [5]. Specifically, a packet is drpped if the Shannn capacity f channel realizatin during that particular blck falls belw the infrmatin rate, this event being referred t as an utage. The utage prbabilities in the w channel states can be calculated as, SR T6 I x m%'&3yz {~}&d Cƒ (3) (4) (5) (6) (7) Tw State Channel Cnsider a channel with tw CSI states, with steady-state C prbabilities and H and with expected gains C. The pwer allcatin prblem in (4) can be frmulated in terms f the transmissin SNRs (equivalently pwers) in each state. The ptimal pwer plicy is characterized belw as a functin f the average thrughput requirement /.. Prpsitin 1. When JMLe/., the transmissin pwers are H D' ˆ where is a slutin in C DaŠŒ Ž 8\, ZN ^5 C H D IJ L I. & D Š n 8 ZŒ 5 Fr J L I/. Prf:, the pwer must be split in bth the channel states t satisfy the cnstraint in (4). Using the cnstraints, and have the abve frm fr sme š œ Q %. That must satisfy (9) fllws the first rder ptimality cnditins fr (4). Prpsitin 2. When J L Ic. Ÿ pwers are D J L. ˆ KQ When JdLN/. Z either as in (8) r (). = When J L I/. Ÿ Z states can be shwn t be sub-ptimal. When JMLNIc. Z Y Z (8) (9) the transmissin (), the pwer plicy adpted is, transmitting in bth the channel, the pwer plicy is either f (8) r (). The exact nature f the slutin seems t be specific t the prblem. Cmparisn f Stp-and-Wait and G back N We study the pwer required fr a G-Back-N and Stp-and- Wait prtcl with w equally likely CSI values. The G- Back-N prtcl will transmit in 1(2 465 slts befre sliding it s windw, where 1(2 465 is the number f packets needed t fill the channel. In this same time, a Stp-and-Wait prtcl will have transmitted in ne slt. Hence, we cmpare the ttal energy per 1(2 465 slts, i.e. ` Ic. and 132)465 VYX)ZH[/.. Prpsitin 3. There exists a threshld /.I1 2)465 such that, R `. pÿ 1( VYX Z[.. Ÿ =5 (11) Prf: In the pwer allcatin prblem fr the G-back-N prtcl, if 1 2 4=5 is replaced with, the equality cnstraint can be rewritten as, $# yz { n& Š - C ª «J VYX)ZH[.

3 % w }»» º r» º r ž Q i These are success prbabilities in the varius channel states and hence nn-negative. Then, as a cnsequence f the multinmial therem, yyz {~}& & % h G O>J VYX)ZH[ I. The success prbability functin is mntnically increasing. Therefre, fr Stp-and-Wait t be mre pwer efficient than G-Back-N, it is sufficient t have, w+jd` (I. Ÿ Iw+J VYX)ZH[ I. ± I ) Substituting the prtcl cnstants frm (3) and (5), it fllws that the inequality is true fr sme /.I132)465 Q Therefre, SR ` I. Ÿ 1+-V=X)Z[.. Ÿ =5. Thus, with a lw enugh pwer requirement, Stp-and-Wait will have a higher thrughput than G-Back-N. This can be cntrasted with a wire-line netwrk, in which G-Back-N will always have a higher thrughput. The reasn fr this is that G-Back-N sends all packets in the current windw, befre sliding back. When the pwer is lw and errr rates are high, successive packets are thrwn ut, leading t a lss f energy. We nte that as in a wireline netwrk, a Selective Repeat prtcl will always have a higher thrughput than either G-Back-N r Stp-and-Wait. In Fig. 2, the trade-ff between pwer and thrughput fr a Stp-and-Wait prtcl, is cmpared with a G-Back-N prtcl emplying a windw size f 2. The link rate is Kbps. In this case until abut percent thrughput requirement, Stp-and-wait is mre pwer efficient than G- Back-N. As the thrughput requirements increase, Stp-and- Wait prves t be less efficient due t the verhead invlved in waiting fr an ACK. Timesharing Strategy The thrughput versus pwer plt shwn in Fig. 2 exhibits a distinct dip. This arises ut f the nature f the success (cmplimentarily ² ²³ ² ) functin, ǴI I m%&ah I, which is fund t be cnvex fr lw SNR values and cncave everywhere else. Over the regin where 'I I is cnvex, thrughput may be imprved by using a timesharing scheme. In the timesharing scheme we suspend transmissin with prbability!, if the alltted transmissin SNR falls belw a threshld. Otherwise, the transmissin SNR is fixed at the threshld value. Chsing the ptimal timesharing scheme yields the mdified success functin: µ ṕi C ¹(º ri¼u½ 'I )¾ Z I if, Ÿ Z Z if (12) Figure 3 shws the pwer allcatin curves fr the mdified as well as riginal success functin, pltted n a linear scale. As expected, the average thrughput btained increases ver Thrughput (kbps) 8 7 Pwer allcatins fr G Back N and Stp and Wait Stp and Wait GBN Windw size = 2 equal pwer GBN windw size = Pwer (dbm) Figure 2. GBN vs SW - Rayleigh fading channel a certain pwer range. Als ntice, the thrughput vs pwer curve is nw cncave thrughut. Rate Variatin In additin t pwer variatin, we nw assume that the transmitter adapts the number f infrmatin bits per packet, while keeping the keeping the transmissin time f each packet the same. Therefre, bth rate as well as pwer are adapted based n the CSI. We assume the maximum rate f transmissin is. Let c be the rate alltted when the CSI has value. T simplify ur discussin, we cnsider transmissin rates small cmpared t the channel bandwidth, (i.e), & %À (13) The analysis that ensues can be extended fr higher rates f transmissin as well. Using (13) in (7) we set, ) E%'&(yYz {šá\& c xiãâ SR Nte if cd ÄQ, then clearly d ÅQ and by cnventin Q E%. Fr a Stp-and-Wait prtcl perating a ver a tw state i.i.d. fading channel with gains C, the pwer allcatin prblem in (4) can be mdified with the prtcl cnstant, JMÆ 465 ¼UÇ ` (. É This is expressed in terms f the packet transmissin time,. A similar cnstraint can be frmulated fr G-back-N prtcls as well.[1] As an example, we cnsider a Stp-and-Wait prtcl perating ver a channel with 1db and x x db. The maximum transmissin rate is set t kbps. Figure 4 shws the pwer allcatin scheme when packet sizes are ;. T6

4 :! & 1 Optimized pwer allcatin Timesharing Timesharing scheme fr Stp and Wait Pwer Variatin Rate plus Pwer Variatin Pwer vs Thrughput 25 Thrughput (kbps) Thrughput (kbps) Pwer (linear scale) Pwer (dbm) Figure 3. Timesharing Figure 4. Optimal Rate plus Pwer adaptatin adusted in additin t transmissin pwer. The plt shws that in this case there is marginal benefit in adapting bth rate and transmissin pwer. DELAY ANALYSIS In sme applicatins we may need t guarantee an average delay fr each packet, while efficiently utilizing pwer. We study the case where packets arrive frm a Pissn surce at an average rate Ê packets per time slt. Here each time slt is assumed t include the packet transmissin time as well as rund trip prpagatin delay, i.e ne time slt equals 8~; secnds. The packets are buffered at the transmitter and cnstrained t have an average packet delay. The packet lsses are assumed t be i.i.d. The transmitter s queue then can be mdelled as a M/G/1 queue with the service time distributin Ë given as IË ÌN1 ; %! )%Í&! )Î fr all Ì [3]. S, ËÏ E% ; 1Ð %'&! Ë E% ; I1 ; 1 %'&!! (14) (15) Here! is the average prbability f a successful transmissin. The average packet delay, given by the Pllaczek-Khinchin frmula is Á Ë ; Ê Ë )% &ƒê Ë Â (time slts) (16) Fr Stp-and-Wait r G-back-N perating ver a Gaussian channel with tw equally likely and independent states, the pwer allcatin prblem can be written as ACB$D s.t ; M UÑ I ; UÑ H O Q /O>Q JdÒ ¼ ÓÔÇ LNÊ (17) Ntice that the structure f this prblem is similar t (4). As in (4), we are restricting urselves t pwer allcatins that depend nly n the channel state ; fr the delay-cnstrained case we nte that such a restrictin is nt ptimal. Here JdÒ ¼ ÓÕÇ LŒIÊ is specific t the prtcl emplyed and can again be interpreted as the average prbability f a successful transmissin. Fr a G-Back-N prtcl emplying a windw size 1, using (14),(15) and (16) it can be shwn that JdÒ ¼ ÓÔÇ VYX t IÊ 6 Ö ; 1 & 1 ; U Ø ÖÙ; I1Ú & I1Û& % (18) Setting 1Ü E%, the prtcl cnstant fr Stp-and-Wait can be btained. The slutin frm fr a tw state i.i.d. fading channel is as discussed fr the thrughput case. Figure 5 gives the pwer plicy adpted by the Stp-and- Wait prtcl described in the previus sectins, fr varius arrival rates. This is cmpared with an equal pwer splitting apprach. Ntice fr small delay, an equal pwer is nearly ptimal. There seems t be cnsiderable benefit in emplying this scheme, especially when the delay requirement is relaxed and arrival rates are lw. CONCLUSIONS This paper discussed ptimal pwer allcatin schemes fr ARQ prtcls perating ver fading channels and cn-

5 Pwer (dbm) 8 7 Pwer allcatins fr Stp and Wait with different arrival rates lambda=.1 lambda=.5 lambda=.8 Equal pwer lambda=.1 Equal pwer lambda=.5 Equal pwer lambda= Delay (time units) Figure 5. Effect f Arrival rate strained by QOS requirements. The ptimal pwer plicy enhances perfrmance as cmpared t a simple equal-pwer transmissin scheme. The relative perfrmance f the plicies adpted by Stp-and-Wait and G-Back-N prtcls was als studied. It was shwn that Stp-and-Wait is pwer efficient ver a range f thrughput requirements. This is an interesting result in the cntext f wireless channels, particularly because Stp-and-Wait is always less efficient in a wireline scenari. Finally, the pwer plicy adpted while cnstrained by delay requirements was cnsidered. Results similar t the thrughput case were discussed. REFERENCES [1] N. Arulselvan. Efficient Pwer Allcatins in Wireless ARQ Prtcls. Masters prect, Nrthwestern University, June. [2] E. Biglieri et.al. Fading channels: infrmatin-theretic and cmmunicatin aspects. IEEE trans. n IT, Vl. 44, N. 6, Octber [3] R. Gallager and D. Bertsekas. Data Netwrks. Prentice-Hall, [4] E. Mdian. An adaptive algrithm fr ptimizing the packet size used in wireless ARQ prtcls. Wireless Netwrks, Vl.5, pp , [5] L. Ozarw et.al. Infrmatin theretic cnsideratins fr cellular mbile radi. IEEE trans. n Vehi. Tech., vl 43, pp , May [6] J. Walrand. Cmmunicatin Netwrks. McGraw Hill, [ Ò Ýe, [7] D. Zhang and K. Wasserman. Energy efficient data cmmunicatin ver fading channels. WCNC, chicag,. [8] D. Zhang and K. Wasserman. Transmissin schemes fr time-varying wireless channels with partial state bservatins. INFOCOM, 2. [9] M. Zrzi and R. Ra. Energy cnstrained errr cntrl fr wireless channels. IEEE Persnal Cmm., vl. 4, pp , Dec