Jointly optimal transmission and probing strategies for multichannel wireless systems

Transcription

1 Jontly optmal transmsson and probng strateges for multchannel wreless systems (Invted Paper) Sudpto Guha, Kamesh Munagala, and Saswat Sarkar Dept. of Computer and Informaton Scences, UPenn, Phladelpha, PA 94. Emal: Dept. of Computer Scence, Duke Unversty, Durham, NC 778. Emal: Dept. of Electrcal and Systems Engneerng, UPenn, Phladelpha, PA 94. Emal: Abstract We consder a wreless system wth multple channels when each channel has several dfferent transmsson states. Dfferent states are assocated wth dfferent probabltes of successful transmssons. In such networks, we are faced wth makng transmsson decsons n the presence of partal nformaton about channel states. hs (typcally probablstc) nformaton about any channel can be refned by sendng control packets n the channels. In presence of multple alternatve channels, ths process of probng every channel to fnd the best one s onerous and resource consumng. here s a natural tradeoff between the resource consumed n probng and the estmate of channel state we can obtan. he desred tradeoff can be attaned by udcously determnng whch and how many channels to probe and also whch channel to transmt. We present adaptve algorthms for provably approxmatng the desred tradeoffs wthn constant factors. I. INRODUCION Future wreless networks are lkely to provde each node access to a large number of channels. A channel can for example be a frequency n a frequency dvson multple access (FDMA) network, or a code n a code dvson multple access (CDMA) network, or an antenna or a polarzaton state (vertcal or horzontal) of an antenna n a devce wth multple antennas (MIMO). Several exstng wreless technologes, e.g., IEEE 8.a [], IEEE8.b [8], IEEE8.h [] propose to use multple frequences. For example, IEEE 8.a protocol has 8 channels for ndoor use and 4 channels for outdoor use n the 5GHz band, whle the IEEE 8.b protocol has channels n the.4ghz band. he potental deregulaton of the wreless spectrum s lkely to enable the use of a sgnfcantly larger number of frequences. Due to sgnfcant advances n devce technology, laptops wth multple antennas (antenna arrays) ncorporated n the front ld, and devces wth smart antennas have already been developed, and the number of such antennas are lkely to sgnfcantly ncrease n near future. hs ncrease n the number of channels s expected to sgnfcantly enhance network capacty and enable several new bandwdth-ntensve applcatons as multple transmssons can now proceed smultaneously n a vcnty usng dfferent channels, and the probablty (at any gven tme) of exstence of at least one channel wth acceptable transmsson qualty sgnfcantly ncreases. he man challenge, however, n explotng multple channels s that a node has only lmted nformaton about the transmsson qualty of the ndvdual channels whch stochastcally vary wth tme. Presumably, a node s transmsson decsons wll become closer to optmal as the avalable nformaton about ts channels ncreases. However, the bandwdth and the energy expended n acqurng such nformaton also ncreases wth the amount of nformaton acqured. Note that a node usually probes n a channel by transmttng a control packet n the channel, and the recever nforms the sender about the qualty of the channel n a response packet (e.g., the RS and CS packet exchange n IEEE 8.). he exchange of control packets consumes addtonal energy, and prevents other neghborng users from smultaneously utlzng the channel. hus, each probe s assocated wth a cost. Owng to the probng costs, the amount of nformaton a node acqures about ts channels becomes an mportant decson varable. Before each transmsson, a node needs to determne how many and whch channels t wll probe and also the sequence n whch these channels wll be probed (probng polcy). Note that dependng on the avalable hardware (e.g., avalablty, or lack thereof, of multple network nterface cards, or compatble transmsson crcuts to approprately dstrbute the power across the antennas), a node may, or may not, be able to smultaneously transmt n multple channels. In ths paper, we consder the scenaro where a node can transmt n only one channel n a tme slot and transmts one packet n each slot. Based on the outcomes of the probes, a node must select one of the avalable channels (channel selecton polcy), whch need not be those that t has probed. An mportant performance goal n such networks s to desgn a ontly optmal probng and channel selecton polcy that maxmzes a system utlty whch s the dfference between the probablty of successful transmsson and the expected probng cost before each transmsson. Loosely, ths utlty functon represents the gan or the proft of the sender f the sender receves credt from the recever for each packet t delvers successfully and needs to addtonally compensate the wreless provder for each probe packet t transmts. We frst enumerate the challenges n desgnng the optmal polcy. We consder a sngle node wth access to n channels. he optmal polcy needs to probe adaptvely,.e., the result he sender may have to share wth the provder part of the credt t receves from the recever for each successfully delvered packet; then the credt we are consderng here s the credt remanng after the sharng process.

2 of a probe determnes the channels to be probed subsequently. For example, consder channels wth possble states (,, ), each of whch s assocated wth a dfferent transmsson qualty. Clearly, the probng termnates f a probed channel s n the hghest state. Now, let a probed channel be n the ntermedate state (state ). hen the subsequent probes should be lmted to channels that have hgh probabltes of beng n the hghest state. However, f all channels that have been probed n a slot are n the lowest state, then the channels that have hgh probabltes of beng n the ntermedate state may also be subsequently probed. Furthermore, the channel selecton decson depends on the outcome of the probes and also the expectaton and uncertanty of the transmsson qualty of the channels that have not been probed. he optmal polcy s therefore a decson tree over n varables. he tme to compute the optmum decson tree usng a nave optmzaton whch evaluates all the decsons trees over n varables s therefore clearly exponental n the problem sze. Next, the space requred to store the optmum tree wll also be exponental n the problem space as ths requres storage of decsons assocated wth all the branches. In a companon paper [], we have showed that for two state channels the optmum polcy can be computed and stored n polynomal complexty. In ths paper, we show that for an arbtrary number of states the optmal net gan can be approxmated wthn a factor of usng a smple approxmaton algorthm (Secton IV), and when the number of states s the approxmaton rato can be mproved to (Secton V). he computaton and storage complextes of our polces are polynomal n the number of channels. We revew the related work n Secton II and defne the system model n Secton III. II. RELAED WORK Opportunstc selecton of channels wth complete knowledge of channel states has been comprehensvely nvestgated over the last decade (e.g., [8]). he ont optmzaton of the reward obtaned from nformed selectons and the cost ncurred n acqurng the requred nformaton however remans largely unexplored. Recently, Kanoda et. al. [] and J et. al. [] consdered scenaros where a node probes multple channels and selects a channel based on the outcomes of the probes. hey consder only statstcally dentcal channels wth equal probng costs and assume that a node can transmt n only a channel that has been probed. hus, ther problem reduces to a decson of how many channels to probe whch s equvalent to that of the well-nvestgated optmal stoppng tme problem [5]. Optmzng the order of evaluaton of random varables so as to mnmze the cost of evaluaton ( ppelned flters ) has been nvestgated n several dfferent contexts lke dagnostc tests n fault detecton and medcal dagnoss, optmzng conunctve query and ont orderng n data-stream systems, web servces, and sensor networks [6], [4], [], [7], [5], [], [6], [7], [4]. However our work s dfferent from all the above (ncludng [], []) n that, we allow a node to transmt n a channel even f the channel has not been probed. Furthermore, we allow for channels wth dfferent dstrbutons of the transmsson qualtes and dfferent probng costs whch s not consdered n [], [], and consder multstate channel models whch ppelne flters seldom consder. hese complcatons sgnfcantly alter the decson ssues and the optmal solutons. III. SYSEM MODEL AND PROBLEM DEFINIION A sender U has access to n channels whch are denoted as channels,,..., n, each of whch has K possble states,,..., K. We assume that tme s slotted. In any slot channel s n state wth probablty p ndependent of ts state n other slots and the states of other channels n any slot. In any slot, U transmts a data packet n one channel, and f the channel s n state, the transmsson s successful wth probablty r. hus, r s the reward assocated wth state. Wthout loss of generalty we assume r < r < < r K. For smplcty, we also assume that r = ; all analytcal results can however be generalzed to the scenaro where r >. Whenever U probes a channel, t pays a cost of c. Probng dfferent channels may ncur dfferent costs as the probng process for dfferent channels may nterfere wth the channel access of dfferent number of users (based on geometry and allocaton of channels). We now formally defne the polces and the performance metrcs. Defnton.: A probng polcy s a rule that, gven the set of channels the sender has already probed n a slot (whch would be empty at the begnnng of the slot) and the states of the channels probed n the slot, determnes (a) whether the sender should probe any more channels and (b) f the sender probes addtonal channels whch channel t should probe next. he sender knows the state of a channel n a slot f and only f t probes the channel n the slot. Defnton.: A selecton polcy s a rule that selects a channel for the transmsson of a data packet n a slot on the bass of the states of the probed channels, after the completon of the probng process n the slot. he selecton polcy can select a channel even f t has not been probed n the slot, and n that case, the channel s referred to as a backup channel. Defnton.: he probng cost s the sum of the costs of all channels probed n the slot. he probng cost s clearly a random varable that depends on the probng polcy and the outcomes of the probes (as the sender may probe subsequent channels dependng on the outcomes of the prevous probes). he expected probng cost s the expectaton of ths random varable and depends on both the probng polcy and the channel statstcs. Defnton.4: In any slot, the transmsson reward s f the packet s successfully transmtted n the slot and otherwse. Agan, the transmsson reward n any slot s a random varable that depends on the probng and selecton polces and the states of the channels n the slot. he expected transmsson reward depends on the probng and selecton polces and the channel statstcs.

3 Defnton.5: he expected net gan of the sender, denoted smply as gan, s the dfference between the expected transmsson reward and expected probng cost. hs depends on the probng and selecton polces and the channel statstcs. Problem Defnton: Gven {c }, {r } and {p } fnd a probng and selecton polcy so as to maxmze the expected gan for ndependent channels. Let OP denote the denote gan of the optmal polcy. Snce we are consderng the ndependent channel model, the optmal probng and selecton polces n a slot need not depend on the decsons and the observatons n other slots. Also, the optmal polces reman the same n all slots, though the specfc choces made by each polcy may be dfferent n dfferent slots dependng on the outcome of the probes. In [], we showed that the optmal probng and channel selecton polcy for two state channels can be computed n polynomal tme. In ths paper we consder K state channels when K. IV. OPIMAL POLICIES WHEN K We frst show that the optmal polcy n the class of polces that does not transmt on an unprobed (backup) channel can be computed n polynomal tme (Subsecton IV-A). We then obtan a polcy that may transmt n a backup channel, but s guaranteed to attan at least the maxmum gan n the class of all polces that may or may not use backup (Subsecton IV- B). A. Optmal Algorthm wthout Backups We present an optmal polynomal tme algorthm for multstate channels when no backup channel s allowed to be used. We frst ntroduce the followng defnton. Defnton 4.: Defne r [u] = / v:u v p vr v v:u v p v and p [u] = v:u v p v. Defne H u = Φ for{ all u > K. Recursvely, startng from H K, defne H u = } v:v>u H v and r [u] c p > r [u] u. Assume c / p [u] = + when p [u] =. OPNOBKUP Consder each H u n decreasng order of u startng from u = K. Wthn each H u probe n non-ncreasng order of r [u] c p [u], and stop f any channel s found to be n state u or above. Select the channel whch s n the hghest state among all probed channels. We now present the ntuton behnd OPNOBKUP. Note that once a sender observes that a probed channel s n state u t can not ncrease ts gan any further by dscoverng another probed channel n state u or lower. hus, subsequently t probes only the channels for whch the ncremental gan ( r [u + ] p [u + ] r u ) s less than the cost c,.e., the channels n H v, v > u. he probng sequence n each H u naturally follows an ncreasng order of the ncremental gans. heorem 4.: he expected gan of OPNOBKUP s maxmum among all strateges that do not use a backup. Proof: he proof follows mmedately from Lemmas 4. and 4.. Lemma 4.: he optmum polcy probes only channels n v>u H v, after t observes a channel to be n state u. Further f there an s un-probed channel n v>u H v and the best state seen so far s u, then probng that channel mproves the expected gan. Proof: he proof s mmedate for u = K, where no further probng s needed. Consder H v. Snce v>u H u+ we know that r [u + ] p [u + ] r u c. But r [u + ] p [u + ] r u s the expected gan (over the already seen channel n state u), and ths s less than the cost of probng the channel. Clearly t s suboptmal to probe such a channel after we have seen a channel at state u. Lkewse consder n H v for some v > u. he expected gan (reward mnus cost) of probng s r [v] p [v] c r u > r v r u >. he optmum cannot therefore stop n a state u f any channel from H v for some v > u s left unprobed. Lemma 4.: he optmum polcy must probe the channels of H u n non-ncreasng order of r [u] provded t has not seen a channel n state u or better so far. Further f v > u then the optmum polcy must probe all the channels of H v before probng any channel n H u. Proof: We wll prove by nducton, frst on u (startng from u = K+) and then on the number of unprobed channels of H u remanng n a partcular sequence/run of the optmum polcy. he base case s u > K and there s nothng to prove. Assume that we are n some nductve case u. We assume there s some channel n H u whch s unprobed and the best state seen so far s worse than u; otherwse there s nothng to prove for u. Among those (unprobed) channels of H u let be the channel wth the largest r [u] value. Suppose the optmum polcy at the current pont s to probe some contradctng the hypothess. If we fnd the channel n state u or better, the optmum polcy s to stop snce by the nducton hypothess on v > u all states n H v have been probed and there s no further beneft (n expectaton) possble by Lemma (4.). If we observe any worse state, we probe next by the nducton hypothess (snce the number of unprobed channels n H u decreases, we can apply the hypothess). he stuaton resembles a decson tree as n Fgure (a). he trees... u correspond to observng the ordered par ( = u, = u ) where u, u u. he square boxes denote that we wll defntely not probe anythng else. Now consder an alternate scenaro of probng as shown n Fgure (b) where s probed frst and then. he tree correspondng to the ordered par ( = u, = u ) s assgned approprately, on the branch correspondng to observng n

4 (a) Fg.. he decson trees of the Optmal polcy for u = (b) u and subsequently observng n u. he contrbutons to the gan from the trees,... u reman the same because n both the scenaros the probablty of probng these trees are the same. he expected gan from scenaro (a) (from not consderng etc.) s p [u] r [u] c + ( p [u])[ p [u] r [u] c ]. hs accounts for stoppng after probng as well as stoppng after probng and then. Note that the reward for probng s r [u] p [u] whch s the weghted reward from observng n states u or better. he expected gan n scenaro (b) s p [u] r [u] c + ( p [u])[ p [u] r [u] c ]. Now f H u then we have r [u] c > r [u] c whch s the condton that arses from volatng the non-ncreasng order. Otherwse H u mples r [u] c p r [u] u. But r [u] p > r [u] u snce H u. herefore n both cases we have r [u] p r [u] [u] c p. [u] But ths mples that p [u] r [u] c + ( p [u]) { p [u] r [u] c } p [u] r [u] + c ( p [u]) { p [u] r [u] c } = ( p [u] p [u] r [u] c p [u] r [u] + c ) > p [u] hus by consderng the scenaro (b), we ncrease the proft of the optmum soluton, whch s mpossble and we arrve at a contradcton. hus by nducton the lemma s true. B. Approxmaton Algorthm for the Backup Case We now consder the case that the optmal polcy can transmt n an unprobed channel, and present a polcy whch attans at least the optmal gan, and has a computaton complexty whch s polynomal n n. APPROXBKUP Let l denote the channel wth the hghest expected reward, r l []. Compute the gan R of OPNOBKUP. If R > r l [] then use OPNOBKUP, else do not probe any channels, and select l. heorem 4.4: he gan of algorthm APPROXBKUP s at least half the optmal gan. Proof: In the optmal polcy, let the expected gan from usng backups (gven a backup s used) be z and let α denote the probablty wth whch backups are used. hus, the total gan from the backups s αz. Let ALG denote the expected gan of APPROXBKUP. We frst have ALG z. () Now modfy the optmal polcy so that the backups are not used, but the rest of the polcy remans the same. Let OP denote the expected gan of ths polcy, and let x denote the expected gan of ths polcy gven that the optmal uses a backup. hen, OP OP = α(z x). hus, OP OP +αz. In addton, snce OPNOBKUP returns a soluton wth gan at least OP, we have ALG OP. herefore, ALG OP αz. () Combnng Equatons and, we have ALG OP. Note that the gan of APPROXBKUP s at least max( OP +α, OP max c ), where α s the probablty wth whch the optmal soluton uses backups [9]. hus, the approxmaton rato s better than the worst case bound n many cases. V. HREE SAE CHANNELS We present an mproved approxmaton for -state channels. Defnton 5.: A /-path n a decson tree s a sub-tree where the next acton s the same rrespectve of whether a probed channel s n state or. Note that / paths are not paths but behave lke paths; hence henceforth we wll not dstngush between a path and a / path. Recall that H = { (r r ) > c p }. and r [] = r p +r p whch s the expected beneft of usng the channel as a backup. Clearly, after havng seen a channel n state the subsequent sequence of actons are dentcal f a channel s observed n

5 state. hus, after a channel s seen n a state, the optmal decson tree becomes a path. he key techncal lemma n ths secton s the followng. Lemma 5.: If the optmum polcy uses a backup after seeng some channel n state, then there exsts another optmal polcy where the decson tree rooted at channel s a /-path endng n the backup and the two polces are otherwse the same. Proof: Consder a node (say node m) closest to the decson tree at whch some channel s probed and the path whch corresponds to the observaton that s n state uses a backup channel. Snce after a channel s observed n state, the decson tree becomes a / path, the backup channel wll be used, unless a channel s observed n state somewhere before. Note that the expected reward of ths backup s at least r. Let the decson tree that arses after probng and observng t to be n state be A. If we observe to be n state then the decsons form a path P. hs s shown n Fgure (a). Let the optmum polcy traverse node m wth probablty p. Let G be the condtonal expected gan of the optmum polcy f t does not traverse node m, G A be the condtonal expected gan of the optmum polcy f t traverses node m and s observed to be n state, and G P be the condtonal expected gan of the optmum polcy f t traverses node m and s observed to be n state. Clearly, OP = ( p)g +p(p G A +p G P +p r ). Now, consder a modfed polcy where the tree A s used n place of the path P f s observed n state at node m. We refer to the gan of ths polcy as OP. Clearly, OP = ( p)g + p(p G A + p G A + p r ). Snce OP OP, G A G P. Now, consder another polcy whch s obtaned by modfyng the optmal polcy as follows: path P s used nstead of tree A when s observed to be n state n node m. We refer to the gan of ths polcy as OP. Snce unless a channel s observed n state, P uses a backup, G P s the optmal gan of ths polcy gven that t traverses node m and s observed to be n state. hus, OP = ( p)g + p(p G P + p G P + p r ). Snce G P G A, OP OP. hus, the second modfcaton corresponds to an optmal polcy as well. Note that the second modfcaton s otherwse smlar to the orgnal optmum, but ts decson sub-tree rooted at node m s a / path. he result follows. Backup (a) (b) (c) Fg.. he frst two fgures show how the paths are formed, (c) shows the consequence of Lemma 5. Applyng the above lemma bottom-up on the optmal decson tree yelds the followng structure theorem, whose proof we omt. heorem 5. (Structure heorem): For three-state channels, there exsts an optmum polcy that uses a unque backup channel (f at all) on only one path. he structure theorem mples that the choce of the backup does not depend on the outcomes of the probes. Note that the unqueness of the path on whch a backup s used mples that the probablty of usng a backup s lkely to be small. Furthermore, ths theorem allows us to mprove the approxmaton guarantee to / by combnng the polces APPROXBKUP, OPNOBKUP and another polcy RESERVEBKUP, whch we descrbe next. Defnton 5.: Let P (l) denote the class of polces, each of whch (a) never probes l and (b) never use any channel other than l as a backup. he best algorthm n P (l) (over all choces of l) may stll be suboptmal, but wll gve us the desred approxmaton. Consder the followng algorthm. RESERVEBKUP(l) ) If r l [] r then use the polcy whch s optmal under the two state model among all polces that use l as backup (the two state model s obtaned by treatng state the same as state, and the optmal polcy n ths case has been obtaned n []). ) Otherwse (for the remander of the algorthm, r l [] < r ) sort the channels n H \ {l} n decreasng order of r c p. ) Probe the channels n H \ {l} n the above order. Stop f a channel s found to be n state, and select the channel. 4) If all channels n H \ {l} have been probed and f a channel has been observed n state, select that channel. 5) Otherwse let H(l) = { H {l} and r [] c p n order of decreasng > r l []}. Probe channels n H (l) r [] c p and stop f any channel s observed n states or, and select channel. 6) If all the channels probed so far are n state, use l as a backup. Lemma 5.: he algorthm RESERVEBKUP(l) s optmal for the class of polces P (l). Proof: Frst note f r l [] r then the best algorthm n the class P (l) wll use the backup as long as no state s observed n state. In effect the algorthm wll smply gnore state. hs reduces ths case to the two state problem wth backup (see []), and RESERVEBKUP uses ths soluton. hus t suffces to consder r l [] < r n the rest of the proof. Usng arguments smlar to those n the proofs of Lemma 4. and Lemma 4., we can show the followng. ) All channels n H \{l} must be probed unless a channel s observed to be n state.

6 ) he optmal polcy probes the channels n H \ {l} n decreasng order of r p. ) If some channel n H \ {l} s found to be n state, channels outsde H \ {l} need not be probed. Parts () and () prove that the actons n step () of the algorthm are optmal for algorthms n P (l). Now, consder step (4) of the algorthm. From part () and snce r l [] r, after all channels n H \ {l} are probed, f any channel has been found n state, the probng wll stop and the channel must be selected. hus, t follows that the actons n step (4) of the algorthm are optmal for algorthms n P (l). We now outlne the proof that the actons n step (5) are optmal for algorthms n P (l). Note that the algorthm executes step (5) only when all channels n H \ {l} are observed to be n state. Agan note that at ths stage f any channel s found to be n state or, the probng must stop and the channel must be selected. hus, from ths pont onwards, the best algorthm n P (l) must treat states and as the same. hus, the channels n effect have two states, but the reward n state depends on the channel. We can show that n ths case the optmum algorthm n P (l) must probe a channel n H (l) before usng the backup (as then the gan ncreases). Subsequently we show that two consequently probed channels must be n non-ncreasng order of r[] p, or the net gan can be ncreased by swtchng ther order. he result follows. Fnally, we prove that f a channel s probed ust before usng the backup and r[] p r l [] then the net gan does not decrease by elmnatng the probe of. Applyng ths condton recursvely, we establsh that only the channels n H (l) need to be probed before the backup s used. hus, the actons n step (6) are optmal for all algorthms n H (l). We now present the man approxmaton algorthm CHOICE: CHOICE G = Gan of usng the best backup channel (G = max r []). G = Gan of OPNOBKUP. G = Gan of RESERVEBKUP(l) for the best choce of l. Select the best of the above three solutons. heorem 5.4: he CHOICE algorthm gves a / approxmaton to the best adaptve probng polcy for -state channels. Proof: By the structure heorem (heorem 5.), the optmum polcy uses a unque backup (f at all). Let ths backup be l. Recall that the reward of usng the backup s r l []. Let p l denote the probablty wth whch the backup s used. Recall that OP denotes the optmal gan. Let ALG denote the expected gan of CHOICE. Equatons and hold (settng z = r l []) ust as n heorem 4.4. Now, modfy the optmum polcy so that l s removed from all places where t s probed (but t may stll be used as the backup). he probablty wth whch t s probed s at most p l, and the gan from probng t s at most r l []. herefore, the expected gan from probng t s at most ( p l ) r l []. Let the new expected gan be OP. We have OP OP ( p l ) r l []. Snce RESERVEBKUP(l) returns a soluton wth at least ths value, we have ALG OP ( p l ) r l [] () Addng Equatons,, and, we have ALG OP, mplyng a approxmaton. VI. ACKNOWLEDGEMENS he contrbuton of the authors were supported n part by NSF grants NCR 84, CCF 4 76, CNS and CNS REFERENCES [] I. 8.a Workng Group, Wreless LAN Medum Acces Control(MAC) and Physcal Layer(PHY) Specfcatons-Amendment :Hgh speed Physcal Layer n the 5GHz Band, 999. [] I. 8.h Workng Group, Wreless LAN Medum Acces Control(MAC) and Physcal Layer(PHY) Specfcatons-Amendment 5:Spectrum and ransmt Power Management Extensons n the 5GHz Band n Europe,. [] S. Babu, R. Motwan, K. Munagala, I. Nshzawa, and J. Wdom, Adaptve orderng of ppelned stream flters, n Proc. of the 4 ACM SIGMOD Intl. Conf. on Management of Data, June 4, pp [4] J. Burge, K. Munagala, and U. Srvastava, Orderng ppelned query operators wth precedence constrants, Submtted, 5. [5] Y. S. Chow, H. 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