Stability and Capacity of Large Aloha Systems with Retransmission Gain and Poisson New Arrival

Transcription

1 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March Sably and Capacy of arge Aloha Syses wh Reranssson Gan and Posson New Arrval Y Sun Deparen of Elecrcal Engneerng The Cy College of Cy Unversy of New York New York NY 003 E-al: ysun@ee-al.engr.ccny.cuny.edu Absrac We consder large Aloha syses n whch he ean arrval rae and channel bandwdh end o nfny wh her rao kep a consan. In such large Aloha syses he channel uncerany s capured by a noon of effecve nerference and hus he packe success probably depends only on he ean copose arrval per denson and s ndependen of he nuber of users ha rans packes n a parcular slo. I s shown ha wh Posson new arrval he nubers of backlogged packes ransed packes and successful packes n each slo n he lng syse all are Posson dsrbued. Moreover her eans he backlog offered load and seady hroughpu per denson can be obaned by a deernsc erave equaon. The sably and he capacy are analyzed. To consder guaraneed QoS ranssson- and delay-consraned capaces are hen defned. All MAC perforance easures are obaned n closed fors nsead of bounds. I s shown ha he sably and capacy are ndependen of packe ranssson probably hough he backlog and packe delay depend on he ranssson probably. The resuls are appled o large CDMA Aloha syses ha eploy rando ndependen equprobable spreadng sequences. Nuercal exaples for he large CDMA syses eployng MF decorrelaor and MMSE deecors are presened. I s shown ha exploaon of reranssson dversy can subsanally ncrease capacy of large CDMA Aloha syses. I. INTRODUCTION Convenonal pure Aloha assues very sple physcal layer such ha recever s ncapable of deodulang ulple packes say n collson channels. Ths assupon s no realsc for praccal councaon syses lke CDMA syses ha are purposely desgned for uluser access. Ghez Verdú and Schwarz [] analyzed sably and capacy of sloed Aloha syses ha are capable of ulpacke recepon. I s shown ha f he average nuber of successful packes wh nfnely any ransed packes s greaer han he Ths work s n par prepared hrough collaborave parcpaon n he Collaborave Technology Allance for Councaons & Neworks sponsored by he U.S. Ary Research aboraory under Cooperave Agreeen DAAD ean arrval rae he syse s sable. To prove hroughpu perforance by sgnal processng echnque Tong Zhao and Mergen [2] exploed ulpacke recepon capably n desgn of edu access conrol (MAC) proocols. Adreddy and Tong [3] recenly exploed channel saes n a rando access syse. Transsson probably of each user s deerned by he channel sae so ha beer MAC layer perforance can be obaned. Perforance of councaon syses s nose and nerference led. I s nevable for a packe o fal n deodulaon and hen o be reransed. Due o varaon of nose nerference and channel fadng wh e reranssson of he sae packe provdes reranssson dversy gan for packe deodulaon. Tsasans e al. [4] suded nework-asssed dversy n wreless rando access syses. Reranssson dversy for CDMA channels n packe-swched syses was exploed n [5]-[8]. I s evdenced [5]-[8] ha reranssson dversy s poran par of nework resources. Whle exploaon of reranssson dversy subsanally proves syse perforance ncreased cos n copuaonal coplexy and requred eory s gnorable copared wh ha reranssson dversy s no exploed [6][7]. Seady hroughpu and packe delay for large CDMA rando access syses wh reranssson dversy over fadng ulpah channels were analyzed n [8]. In he analyss consdered are large CDMA syses n whch he nuber of copose arrval packes and processng gan end o nfny wh her rao kep consan. Resuls of large CDMA syses n [9] are eployed. I s shown ha he packe success probably n he large syse seng s deernsc and dencal n each slo hus grealy splfyng perforance analyss a MAC layer. I s deonsraed ha reranssson dversy can sgnfcanly ncrease hroughpu and reduce packe delay. However he analyss s based on he assupon ha he syse s already n a seady sae and he copose arrval s Posson. I s unknown n wha condon he syse can ener such a seady sae. Our objecve n hs paper s o analyze sably and capacy of large sloed Aloha syse wh reranssson dversy gan and Posson new arrval. arge rando access syses wh reranssson gan and arbrary arrval processes are recenly suded n

2 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March []. Much hgher seady hroughpu and specral effcency can be obaned by opally choosng ranssson probably wh explong reranssson dversy. Capacy of large wreless CDMA ad hoc neworks wh reranssson gan s analyzed n [2]. I s shown ha when all nodes n he 2-D plane shall a coon CDMA channel wh a fxed bandwdh per node and he power decays wh he ransng dsance of a power greaer han wo he ranspor capacy of an ad hoc nework s scalable o s sze. II. CAPACITY OF ARGE AOHA SYSTEMS A. Syse odel and prelnares e A be he nuber of new packes arrvng durng packe slo. Assue ha {A} 0 are..d. rando varables wh Posson dsrbuon of ean λ. Consder channel bandwdh W and b perod T b. For a CDMA syse he processng gan s approxaely equal o N = WT b. Assue ha λ and N boh end o nfny wh her rao kep a consan λ/n = α <. The processng gan s also he denson of sgnal space and α s he ean arrval rae per denson. All users are dencal. Alhough he ean arrval rae ends o nfny he new packe arrval probably for each user s nfnesal such ha each user eher has no packe or has one packe n ranssson or backlogged. Tha s a user never has wo or ore packes. Users are synchronzed so ha packe ransssons sar a he begnnng of each slo. A he end of each slo users ha ransed a packe n he slo learn f her packes were successfully receved. Unsuccessful (or backlogged) packes wh prevous unsuccessful ransssons are reransed n each subsequen slo wh probably γ (0 ] ndependen of prevous ransssons oher packes n ranssson and backlog. New packes are consdered backlogged packes wh zero prevous ranssson and wll be ransed wh probably γ sarng n he frs slo followng her arrval. Wh K users ransng packes n a slo a packe of s h reranssson s deodulaed wh he packe success probably p(k/n) = p[k/(wt b )] ndependen of recepon of oher ransed packes and ndependen of backlogged packes. The packe success probably can be specfcally gven when physcal channel paraeers are gven such as n a CDMA syse eployng a uluser deecor [5]-[7]. In hs paper we ephasze he effec of he nuber of reransssons he oal nuber of ransed packes K n a slo bandwdh W (or processng gan N) and ranssson probables γ on he packe success probably and perforance hough oher physcal and MAC layer paraeers can also be consdered wh a ore coplcaed analyss. We are neresed n he large syses n whch boh he nuber of ransng users and he processng gan (or bandwdh) end o nfny wh her rao kep consan. Throughou hs paper p(k/n) s a connuous funcon of K/N s no decreasng wh respec o and s onooncally decreasng wh K/N. In a fne-sze rando access syse he packe success probably depends on he nuber of ransng users K ha s a rando process dynacally varyng slo by slo. The randoness of packe success probably akes dffcul o analyze he sably and seady perforance a MAC layer. However as we wll see he analyss for large rando access syses wh Posson new arrval process s racable. The racably reles on he followng leas ha lnk he MAC layer perforance o he physcal layer perforance n our large rando access syses. ea [7][8]: e K be a Posson rando varable wh ean λ. If λ N and λ/n = α (0 ) where α s consan hen K/N α n probably. When appled o our large rando access syses ea saes ha f he nuber of ransed packes n a slo s Posson dsrbued hen he nuber of ransed packes per denson K/N s asypocally consan as s ean and denson boh end o nfny wh her rao kep a consan. Snce he packe success probably s a connuous funcon of K/N we have he followng corollary. I pons ou ha for Posson dsrbued copose arrval he packe success probably n our large syses s asypocally a consan ndependen of nuber of ransed packes n a slo. Corollary : Suppose ha he nuber of ransed packes K n slo s Posson dsrbued wh ean λ. If λ N and λ/n = α (0 ) where α s consan hen p[k/n] p(α) n probably. ea 2 [3] (p. 32): Suppose ha we are observng evens and ha K he oal nuber ha occurs s a Posson rando varable wh ean λ. Suppose also ha each even ha occurs s ndependen of oher evens classfed as a ype even wh probably p I =... I = p =. Then K he nuber of ype evens ha occur are ndependen Posson rando varables wh respecve eans λp =... I. Furherore we can exend ea 2 o he followng lea for our large syses. ea 3: Suppose ha he nuber of ransed packes K n slo s Posson dsrbued wh ean λ. Suppose also ha wh gven K each packe s ndependen of oher evens classfed as ype wh I probably p (K/N) =... I p ( K / ) = = N where p (x) are connuous funcons of x. Then as λ and N end o nfny wh her rao kep a consan λ/n = α K he nuber of packes classfed as ype converge n dsrbuon o ndependen Posson

3 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March rando varables wh respecve eans per denson equal o αp (α) =... I. B. Dsrbuons of backlog and offered load Denoe by he nuber of backlogged packes a he begnnng of slo ha have been ransed es prevously. Denoe by T he nuber of packes ha are ransed he h e n slo. G = T s he nuber of backlogged packes ha had unsuccessful prevous ransssons before and are no ransed n slo. Denoe by S he nuber of successful packes and by R he nuber of unsuccessful packes aong he T ransed packes. Then T = S + G. We denoe = ( 2 3 ) T = (T T 2 T 3 ) S = (S S 2 S 3 ) Z + where Z + s he - fold nonnegave neger se. The rando process s he sae of he Aloha syse and s a Markov chan. s he oal backlog (he nuber of backlogged packes) T s he offered load (he nuber of ransed packes) and S s he hroughpu (he nuber of successful packes) n slo. The sae ranson equaon s (+) = G + A = T + A () (+)=G +R = T +R 2. (2) By T T S and S we denoe he eans of T T S and S per denson respecvely. Assue ha he syse sars o accep new arrval packes a slo 0 and here were no backlogged packes before. Tha s = 0 for 0. Then we have he followng heore. Theore : As λ and N wh λ/n = α T T S and S for and converge n dsrbuon o Posson rando varables wh her eans per denson equal o T T S and S ha are deerned by he followng deernsc erave equaons () = α ( + ) = α + ( γ ) ( ) (3) ( + ) = ( γ ) ( ) + [ p( T ( ))] γ ( ) = 2 (4) + ( + ) = [ p( T ( ))] γ ( ) ( + ) = 0 > + ( + ) = + = ( + ) = ( ) + α p( T ( )) γ ( ) (5) = T ( + ) = γ ( + ) = 2 + (6) T ( + ) = 0 > + + = T ( + ) = γ ( + ) (7) S ( + ) = γ p( T ( + )) ( + ) = + (8) S ( + ) = 0 > + + = S ( + ) = γ p( T ( + )) ( + ) (9) In a fne-sze syse he packe success probably s rando dependen on he rando nuber of ransed packes ha are copose of new arrval and prevously backlogged packes. Because of hs he backlog and offered load n each slo can no be kep Posson even hough he new arrval process s Posson. On he oher hand as shown above n large syses he lng packe success probably n each slo s a consan. Thus he backlog and offered load n each slo are kep o be Posson. Ths grealy splfes analyss of sably and capacy of large Aloha syses. C. Sably The sably of Aloha syses wh fne ean arrval rae can be defned as ha he oal nuber of backlogged packes s ergodc []. In a seady sae he ean of he nuber of backlogged packes s fne. As a reasonable exenson for large Aloha syses havng fne ean arrval rae per denson we defne he sably as ha he oal nuber of backlogged packes s ergodc and s ean per denson n a seady sae s fne. To analyze he sably we have he followng lea. ea 4: For and and T are () onooncally ncreasng wh and () onooncally ncreasng wh α; and () s onooncally decreasng wh each γ. Theore 2: If s upper bounded wh a ean arrval rae α' > 0 hen for all α α' he dsrbuons of T and T converge o he saonary Posson dsrbuons wh eans per denson sasfyng α = [ p( T )] (0) γ = = α [ p( T )] () γ = [ = = T = α p( T )] (2) T = α [ p( T )]. (3) = = In he seady sae he hroughpu per denson s equal o he T

4 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March ean arrval rae per denson α whch n ers of (3) s S ( T ) = T. (4) [ p( T )] = = Nex we derve he average nuber of slos and he average nuber of ransssons for a packe o succeed n a seady sae. Corollary 2: If he syse s n a seady sae wh offered load per denson T hen here exss n such ha p ( T ) > 0 for n. Theore 3: In he seady sae he average oal nuber of slos (delay) and he average nuber of reransssons for a packe o succeed are D = α and M = T α respecvely. As shown n Theore 2 and Corollary 2 he seady offered load per denson T and he average nuber of reransssons are rrelave o he ranssson probables and depend only on he ean arrval rae per denson. However he backlog and he wang e are dependen on he reranssson probables. Theore 4: The opu ranssson probables ha acheve he nu packe delay whle keepng he sae seady hroughpu are γ = for. The opu se of ranssson probables s surprsngly sple as shown n Theore 4 snce T D =. (5) α = γ We pon ou ha alhough he e-nvaran ranssson probables do no effec on he seady hroughpu and capacy s sll possble o desgn a e-varan sequence of ranssson probables o acheve hgher hroughpu a each parcular slo. However n hs case he nubers of backlogged and ransed packes are no longer Posson n general. When reranssson dversy s no exploed he packe success probably for any nuber of reransssons s dencal o ha of he frs ranssson p ( T ) = p( T ). In hs case we can ake γ = γ snce s unnecessary o use dfferen ranssson probably f all ransssons are reaed as new ransssons. I s sraghforward o oban he followng corollary. Corollary 3: In a seady sae for a syse whou exploaon of reranssson dversy = α[ p( T )] γ = α [ γp( T )] T = α p( T ) S = Tp( T ) D = [ γp( T )] M = p( T ). D. Capacy On he bass of Theore 2 we defne he capacy of he syse as he supreu of ean arrval rae per denson such ha he nuber of backlogged packes per denson s upper bounded C = sup{ α l ( ) < }. (6) α > 0 Corollary 4: If C s fne hen he syse s sable for all α < C and unsable for all α > C. If C s nfne hen he syse s always sable for all α <. Gven α we can apply he erave equaon of j o deerne nuercally f he syse s sable and o esae he capacy va he axu convergen hroughpu. In followng we dscuss soe nsances o oban he capacy whou nvokng he erave equaon. Proposon : If g l β g( β ) s fne hen C = where g( β ) = ( γ ) γ [ p( j β )] (7) = 0 = 0 j= and γ = n{ γ }. Corollary 5: If here s an neger J such ha p(j) l β p(jβ) > 0 hen C =. Corollary 5 s undersandable snce s condon ples ha for any offered load per denson a packe s alos sure o succeed as he nuber of reransssons ncreases rrelave o he offered load whch ples an nfne capacy. If no reranssson dversy s exploed g ( β ) = [ γp( β )]. In he defnon of capacy n (6) we allow he ean nuber D of slos for packe success o be arbrary. D gh be nolerably hgh even hough he capacy of a syse s large. To consder a axu olerable ean nuber D of slos for packe success (packe delay) we defne a delay-consraned capacy as C D sup{ α α < D}. (8) = α > 0 C D s he supreu of all ean arrval raes wh whch syse s sable and he ean nuber of slos needed for packe success s saller han D. Clearly when D = C = C. Moreover snce he ean arrval rae anans sably due o () CD = sup α [ p( T )] < D (9) α > 0 = γ = where T = l T ( ). Snce he ranssson probables are explcly presened n C D hs defnon can be used for edu access conrol. Anoher defnon of capacy can be one consderng a axu olerable ean nuber of reransssons

5 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March M needed for packe success ha s he ranssson consraned capacy C M sup{ α T α < M} (20) = α > 0 Snce he ergodcy for one of T and ples for he oher C M s he supreu of all ean arrval raes ha ake he syse sable and ake he ean nuber of reransssons needed for packe success saller han M. Obvously C M = C f M =. I follows fro (3) (4) and Theore 3 ha C = sup{ S ( T ) T S ( T ) M}. (2) M < T > 0 Snce S (T ) depends only on T and s rrelave o reranssson probables he ranssson consraned capacy C M s easer o be nuercally evaluaed han he delay consraned capacy. III. APPICATION TO ARGE CDMA SYSTEMS A. Packe success probably n large CDMA syses We consder large CDMA syses of whch he ean arrval rae and processng gan boh end o nfny wh her rao kep consan. near deecors are appled and he axu rao cobner (MRC) opally cobnes receved sgnals of he sae packe n ulple reransssons o acheve he axu sgnal o nerference rao (SIR). In hs large syse seng b error rae eployng MF decorrelang and MMSE deecors s obaned n [7][8] for Gaussan and fadng ulpah channels. When soe nerference bs are he sae bs wh ulple reranssson n he slos ha he b of neres s ransed wh ulple es he cross nerference of he reransed nerference bs decreases reranssson dversy gan. We can show ha f rando sgnaure sequences are used he cross nerference does no exs. In lgh of hs we consder ha all bs n each ranssson are spread by rando equprobable sgnaure sequences. We ce here he resul [7] n Gaussan channels wh perfec power conrol. For a packe of bs whou error correcng code he packe success probably n s h reranssson usng hese lnear deecors converge n probably o p T Q MF ( ) = (22) T + η p p dec MMSE ( T ) = [ Q( ( T ) η )] 2 T < T (23) T Q T T 2 ( ) = ( + ( ) η ) + 4η ( + ( ) η) 2 (24) respecvely where η denoes SNR. Snce he axu rao cobner s eployed he oal SIR wh reransssons s equal o es he SIR n one ranssson for all hese lnear deecors. The packe success probably for hese lnear deecors obvously sasfes he condon n Corollary 5 and hus he non-consraned capacy s nfne. In fac hs s rue even for a rando guess deecor wh BER equal o ½ bu s packe delay can be nolerably hgh. In wha follows we consder he consraned capaces. B. Nuercal exaple In he nuercal exaple we consder a syse ha has SNR = 8 db and packe sze 256 bs. The seady hroughpu versus average nuber of reransssons wh and whou exploaon of reranssson dversy s shown n Fgs. and 2 respecvely. As we can see exploaon of reranssson dversy subsanally proves he hroughpu. As ore copuaon cos s pad for a ore powerful uluser deecor greaer proveen s obaned. Throughpu usng he MMSE and decorrelang deecors s no onoonc wh respec o he nuber of reransssons. If he consraned nuber of reransssons s 50 and 00 he ransssonconsraned capacy and he correspondng nuber of reransssons and offered load wh and whou reranssson gan are lsed n Tables I and II respecvely. On he oher hand hroughpu usng he MF deecor s onooncally ncreasng wh he average nuber of reransssons. Hence he ranssson-consraned capacy s equal o he hroughpu a he consraned average nuber of reransssons. Exploaon of reranssson dversy can subsanally ncrease he capacy of large AOHA syses. IV. CONCUSIONS Analyss of sably and capacy for large Aloha syse wh Posson new arrval and reranssson dversy gan s racable. Closed-for forulas for hroughpu and packe delay n a seady sae are obaned whch are deerned by ranssson probably and physcal paraeers. These forulas can be ulzed o nuercally evaluae he capacy packe delay and average nuber of reransssons. For he syse of dencal users change of ranssson probables does no change seady hroughpu and offered load and hus does no change capacy. The opu ranssson probables n ers of nzaon of packe delay whle keepng he sae seady hroughpu and capacy are equal o one. Reranssson dversy s par of syse resources ha conrbues capacy of Aloha syses. Muluser deecon proves he syse perforance. ACKNOWEDGEMENT The auhor s graeful for dscusson wh Dr.. Tong on hs sudy.

6 2003 Conference on Inforaon Scence and Syses The Johns Hopkns Unversy March REFERENCES [] S. Ghez S. Verdú and S. C. Schwarz Sably properes of sloed aloha wh ulpacke recepon capably IEEE Trans. Auoa. Conr. vol. 33 no. 7 pp July 988. [2]. Tong Q. Zhao and G. Mergen Mulpacke recepon n rando access wreless neworks: fro sgnal processng o opal edu access conrol IEEE Coun. Mag. Nov. 200 pp [3] S. Adreddy and. Tong Explong decenralzed channel sae nforaon for rando access subed o IEEE Trans. on Inforaon Theory Nov [4] M. K. Tsasans R. Zhang and S. Banerjee Nework-asssed dversy for rando access wreless neworks IEEE Trans. on Sgnal Processng vol. 48 pp March [5] Y. Sun Nework dversy of rando access sloed CDMA neworks n Proc. 33rd Asloar Conference on Sgnals Syses and Copuers vol. 2 pp Pacfc Grove CA Oc [6] Y. Sun Muluser deecon usng nework dversy for rando access packe-swched CDMA neworks n Proc. 33rd Annual Conf. on Inforaon Scence and Syses vol. pp. TA3.-6 CISS'2000 Prnceon Unversy Prnceon NJ March [7] Y. Sun and. Ca Muluser deecon for packeswched CDMA neworks wh reranssson dversy subed o IEEE Trans. on Sgnal Processng. [8]. Ca Y. Sun and A. N. Akansu On asypoc perforance of CDMA rando access syses wh packe cobnng n fadng channels IEEE Trans. on Wreless Cou. (o appear). [9] J. Evans and D. N. Tse near uluser recevers for ulpah fadng channel IEEE Trans. Infor. Theory vol. 46 pp Sep [0] D. N. C. Tse and S. V. Hanly near uluser recevers: effecve nerference effecve bandwdh and user capacy IEEE Trans. Infor. Theory vol. 45 pp Mar [] Y. Sun and Y. Wang arge rando access syses wh reranssson dversy acceped n 2003 CTA Annual Syposu and CISS [2] Y. Sun Capacy of large wreless CDMA Ad Hoc neworks wh reranssson dversy n Proc. 36h Asloar Conf. on Sgnals Syses and Copuers Pacfc Grove CA Noveber (also acceped n 2003 CTA Annual Syposu). [3] S. M. Ross Sochasc Processes (2 nd ed.). New York: John Wley & Sons Inc Table I. Transsson-consraned capacy wh reranssson gan Recever C M M T MMSE Decorrelaor MF Table II. Transsson-consraned capacy whou reranssson gan Recever C M M T MMSE Decorrelaor MF S (packes/slo) MMSE Decorrelaor MF M (ransssons) Fg.. Seady hroughpu versus average nuber of ransssons wh exploaon of reranssson dversy. S (packes/slo) MMSE Decorrelaor MF M (ransssons) Fg. 2. Seady hroughpu versus average nuber of reransssons whou exploaon of reranssson dversy. The sae condon as n Fg The vews and conclusons conaned n hs docuen are hose of he auhor and should no be nerpreed as represenng he offcal polces eher expressed or pled of he Ary Research aboraory or he U.S. Governen.