Exploiting Wireless Channel State Information for Throughput Maximization

Transcription

1 Exploiing Wireless Channel ae Informaion for Throughpu Maximizaion V Tsibonis, L Georgiadis, L Tassiulas Absrac We consider he problem of scheduling packes over a number of channels wih ime varying conneciviy Policies proposed for his problem eiher sabilize he sysem when he arrival raes are wihin he sabiliy region, or opimize an objecive funcion under he assumpion ha all channel queues are sauraed We address he realisic siuaion where i is no known apriori wheher he channel queues are sauraed or no, and provide a scheduling policy ha maximizes he weighed sum of channel hroughpus We employ a bursiness-consrained channel model ha allows us o dispense of saisical assumpions and simplifies he proofs Index Terms cheduling, eerminisic nework calculus, Qo in wireless neworks I INTROUCTION The primary moivaion of his work is he problem of scheduling he ransmissions of muliple daa flows sharing he same wireless channel The relaive delay olerance of daa applicaions, ogeher wih he bursy raffic characerisics, opens up he poenial for scheduling ransmissions so as o opimize hroughpu Given he above consideraions, we examine a ime-sloed parallel queue sysem wih a single server The condiion of he associaed channel of every queue varies wih ime beween on and off saes In every ime slo only one packe can be ransmied from a given queue, if he associaed channel is in he on sae and he queue is non empy The main resul of his paper is he design of a scheduling policy ha allocaes he server o he queues in such a way ha he weighed sum of channel hroughpus is maximal A relaed approach along hese lines is proposed in [4], where he auhors idenify opimaliy properies for scheduling downlink ransmissions o daa users, in CMA neworks or arbirary-opology neworks, he problem of admission conrol and rae allocaion o he users so ha cerain qualiyof-service requiremens are me, is invesigaed A mahemaical programming formulaion is obained for deermining he opimal ransmission schedule The effec of wireless channels on he performance of ransmission proocols such as TCP is examined hrough simulaions in [5] The auhors conclude ha channel-sae scheduling can lead o significan improvemen in channel uilizaion The problem of scheduling wireless channels wih ime varying conneciviy has been addressed in he pas in several V Tsibonis and L Georgiadis are wih Arisole Universiy of Thessaloniki, Thessaloniki, GREECE s: vsib@egnaiaeeauhgr, leonid@engauhgr L Tassiulas is wih he Universiy of Thessaly, Greece and Universiy of Maryland, College Park leandros@glueumdedu differen conexs In [12], opimal scheduling for a wireless sysem consising of muliple queues and a single server is sudied The arrival processes o he queues are assumed iid Bernoulli The wireless channels can be in he on or in he off sae according o iid Bernoulli processes The auhors derived he sysem sabiliy region; moreover, hey showed ha he policy ha among he queues whose channel is on serves he one wih he longes queue, sabilizes he sysem whenever he arrival raes are wihin he sabiliy region In [13], Tassiulas considered a sysem ha generalizes he one in [12] in he following aspecs irs, a nework wih arbirary opology is considered econd, he opology is represened by a hidden Markov model insead of an independen and idenically disribued (iid) process Third, anicipaive scheduling policies are aken ino consideraion ourh, muliple link ransmission raes are considered In ha conex, afer he characerizaion of he region of achievable hroughpus, a ransmission scheduling policy is proposed, ha achieves all hroughpu vecors achievable by any anicipaive policy The problem of scheduling ransmissions over a wireless channel wih ime-varying ransmission raes is considered in [11], [6], [9] and [8] The problem of providing a scheduling policy ha sabilizes he sysem whenever he arrival-rae vecor lies wihin he sabiliy region is deal in [11] and in [8] In [11], a finie se of channel saes is assumed and every channel can be in one of hese saes Wih each sae here is an associaed daa rae, represening he rae a which he queue is served if seleced for ransmission The arrival processes o he queues are assumed muually independen, ergodic, Markov chains wih counable sae space Under hese assumpions i was shown ha he scheduling policy, called he exponenial rule, makes he queues sable if here exiss any policy ha can do so In [8], he auhors consider he problem of power and server allocaion in a muli-beam saellie downlink which ransmis daa o differen ground locaions over ime varying channels The auhors esablish he sabiliy region of he sysem and develop a power allocaion policy, which sabilizes he sysem whenever he sysem is sabilizable and when he arrival and channel sae processes are iid In [9] and [6] he problem of developing a scheduling policy for efficien channel uilizaion is considered for he case ha all he queues are unsable In [9] he sae of a channel is modeled by a sochasic process, which represens he level of performance of he given channel A scheduling policy is provided which maximizes he average sysem performance given ha a predeermined ime-fracion assignmen

2 is achieved when all he queues are unsable In [6], he auhors consider a base saion serving daa-users The feasible raes of he users vary over ime according o some saionary discreeime sochasic process A scheduling policy ha explois he variaions in he channel condiions and maximizes he minimum hroughpu is developed The main conribuion of his paper is ha i sudies a sysem wih ime-varying conneciviy when he arrival raes do no belong necessarily in he sabiliy region of he sysem This is an imporan siuaion ha arise in pracice, since he channel parameers and he arrival raes may no be known apriori, or may vary over ime In such a case, scheduling policies proposed before for maximizing he hroughpu may fail, and he sysem may have a raher erraic behavior In he curren work we consider he scheduling problem of maximizing he weighed sum of user hroughpus We provide a scheduling policy ha is opimal under any arrival raes In he mos general case, under he opimal policy we propose, some queues will be sable while ohers will operae in sauraion uch a dynamic behavior makes he analysis of he sysem raher difficul Insrumenal in he analysis of our policy was he adopion of a bounded bursiness model for he variabiliy of he channel inspired by bursiness consrained raffic models ha have been used over he las several years in he analysis of rae-conrolled communicaion neworks [2] The paper is organized as follows In ecion II, he raffic and channel model is inroduced pecifically, he consrains on he arrival and slo availabiliy processes are given In ecion III we provide he problem formulaion and define he scheduling policy In ecion IV we provide he opimaliy proof of he proposed policy Conclusions and suggesions for furher work are discussed in ecion V A Noaions and Convenions Before we proceed, we discuss some of he noaions and convenions ha we use hroughou he paper es of numbers are denoed by calligraphic capial leers In paricular we define N = {1,,N} A subse of a se is denoed by and a sric subse by In several places we will use ses as subscrips or argumens, say () To simplify noaion and if here is no possibiliy for confusion, insead of ({i 1,, i k }) we wrie (i 1,,i k ) Also, we wrie P y i o denoe P i y i If =, hen we define P x i =0 Also, l i=k i = if k>lthe cardinaliy of a se is denoed by If =[x ij ] and Y =[y ij ] are marices, hen Y ( < Y) means ha x ij y ij (x ij <y ij ) for all i and j inally, by T we denoe he ranspose of ome of he raher echnical proofs in ecions III and IV ha are no essenial for undersanding he main ideas of he argumens, are omied due o lack of space These proofs can be found a he sie hp://genesiseeauhgr /ITE_AUTH_UNIVERITY/ITE_TIVIION /users/georgiadis/english/personalpages /ConferencePapers/TimeVarInfoc03pdf II TRAIC AN CHANNEL MOEL We consider a sysem consising of N channels Wih each channel here is an associaed queue holding packes ha are o be ransmied over he given channel Packes are of fixed size and ime is divided in slos of uni lengh, equal o he ransmission ime of a packe lo 1 refers o he inerval ( 1,] In he inerval ( 1,] (slo ), a i () new packes join queue i o be ransmied over he corresponding channel A he beginning of slo, ie, a ime 1, one packe among hose already in one of he N queues may be chosen for ransmission a slo The number of packes from queue i ransmied in slo is b i () (herefore, b i () is eiher 0 or 1) and he number of packes in queue i a ime 0 is q i () Therefore he number of packes a queue i, i N, evolves wih ime according o he equaion q i () =(q i ( 1) b i ()) + + a i (), where (x) + =max{0,x} efine a () = P a i() and b () = P b i() A slo, channel i may or may no be available for ransmission of queue i packes If he channel is available for ransmission, we say ha he channel is in he on sae We define for N, 6=, c () = 1 if a leas one channel in is on in slo 0 oherwise and c () 0 or example, igure 1 shows he channel availabiliy for 3 channels during 15 ime slos According o he figure c {1} () =1for 1 12 andzeroelsewhere c {3} () = 1 for 2 7, 9 14 and zero elsewhere c {1,2} () =1for 1 15 and zero elsewhere c {1,3} () =1for 1 14 and zero elsewhere c {1,2,3} () =1for 1 15 and zero elsewhere ince only one packe may be ransmied in one slo, we have b () = 1 if b i () =1for one of he channels in 0 oherwise Transmission over channel i make ake place (b i () =1) only if he channel is in he on sae and hence, b () c () (1) If x() is any of he quaniies defined above, we denoe (s, ) = x(τ) τ=s+1 We make he following assumpions regarding he raffic and slo availabiliy processes Traffic Model: A i (s, ) is (σ U i, σl i, α i)-consrained, ie, i holds for any s 0, α i ( s) σ L i A i (s, ) α i ( s)+σ U i, (2) where σ L i 0, σ U i 0, α i 0

3 rom (4), (5), (6) we conclude ha () saisfies he following relaions for any subses T and of N ( ) =0, (T ) (), T, (T )+() (T )+ ( T ) (7a) (7b) (7c) ig 1 Channel availabiliy ( on sae) for 3 channels Parameer α i is he packe arrival rae o queue i A ( ie, α i = lim i(0,) ) for ransmission over he corresponding channel We allow he possibiliy ha α i =, in order o include he case ha some of he queues are infinie for 1 Noe ha i follows from he definiion ha if A i (s, ) is (σ U i, σl i, α i)-consrained for i, hen A (s, ) is P σu i, P σl i, P α i -consrained Channel Availabiliy Model: C (s, ), is (θ U, θ L,())-consrained, ie, i holds for any s 0, ()( s) θ L C (s, ) ()( s)+θ U, (3) where θ L 0, θ U 0 We also use he convenion ( ) =θ L = θ U =0 We refer o he inequaliies in (2) and (3) as bursiness consrains Wih he excepion of α i,i N, he parameers used in he previous models are assumed finie The definiions for he raffic model are sandard, see eg, [2], [3], [1] We elaborae on he lo Availabiliy Model rom (3) i follows ha C (0,) lim = (), (4) ha is, () is equal o he long-erm fracion of ime ha a leas one of he channels in is in he on sae Also, from he definiion of c () we have ha for any subses, T,of N, and for every, i holds, and hence c T () c (), if T c ()+c T () c T ()+c T (), C T (s, ) C (s, ), if T (5) C (s, )+C T (s, ) C T (s, )+C T (s, ) (6) The las propery is known as he submodulariy propery As an example, suppose ha he channel availabiliy paern in ig 1, is repeaed indefiniely, ie, we have a periodic channel availabiliy process Consider he firs channel, ie, = {1} I holds ½ 1, for 1 12 c {1} () = 0, for and c {1} ( + 15) = c {1} (), for every ime-slo 1 Therefore we have ¹ º» ¼ s s 12 C {1} (s, ) 12, or ( s) 12 C {1}(s, ) 12 ( s) In conjuncion wih definiion (3), he above inequaliy saes ha C {1} (s, ) is (θ U {1}, θ L {1},(1))-consrained, wih θ U {1} = θ L {1} = 12 and (1) = 12/15, ie, (1) is equal o he long-erm fracion of ime ha he firs channel is on imilarlywehavehac (s, ) is (θ U, θ L,())-consrained and according o he figure or = {3}, θ U {3} = θ L {3} =12and (3) = 12/15 or = {1, 2}, θ U {1,2} = θl {1,2} =0and (1, 2) = 1 or = {1, 3}, θ U {1,3} = θ L {1,3} =14and (1, 3) = 14/15 or = {1, 2, 3}, θ U {1,2,3} = θ L {1,2,3} = 0 and (1, 2, 3) = 1 We close his secion wih a few commens on he adoped raffic and channel models The assumpion ha channels can be in wo saes only is made for echnical reasons and we make no claims o pracicaliy This assumpion is adoped in order o simplify he siuaion and ge a beer insigh ino he problem a hand The adoped bursiness consrained models provide a clear descripion of sysem dynamics and make possible he analysis of he sysem using basically elemenary (alhough no sraighforward) echniques Compared o inroducing saisical assumpions for hese models, here are boh advanages and disadvanages Noe ha he saionariy assumpion is no needed in our model, alhough he exisence of averages is implied On he oher hand deerminisic raher han sochasic bounds on process flucuaions are imposed III PROBLEM ORMULATION Consider a scheduling policy π ha a he beginning of slo, ie, a ime 1, decides which packe (if any) o ransmi o one of he channels ha are on in slo Le r π i = lim inf Bi π(0,), be he hroughpu of channel i under policy π

4 Given coss c i,i N,c 1 c 2 c N 0, our objecive is o deermine a policy such ha he weighed sum of hroughpus c i ri π, N is maximal Assume ha he channel sae a a given slo is known o he scheduler a he beginning of ha slo and consider he following policy cheduling Policy π Wih queue i associae an index I i (q) of he form ½ q if q (N +1 i) T I i (q) = (N +1 i) T if q>(n +1 i) T where T > 0 A ime, consider he nonempy queues whose channel is on Among hese queues, le i be he one wih larges index I i (q i ()) (if here are muliple such queues selec one randomly) Transmi a packe from queue i a slo +1 Our objecive is o show ha for T large enough, policy π maximizes he weighed sum of hroughpus, irrespecive of wheher he overall sysem is sable or no Before we proceed, i is worh observing he following Only he order of he coss c i,i N, no he acual values deermine policy π This siuaion is similar o he well-known µc-rule in queueing heory The raffic and channel model parameers deermine how large T should be chosen In oher words, he policy depends on hese parameers only hrough T Alhough esimaes of T can be obained hrough he analysis ha follows, in pracice he raffic and channel parameers may no be known beforehand Of course, one can pick very large values of T bu his implies larger delays and slower convergence Hence, developmen of adapive mehods for deermining T seems a more appropriae plausible way of choosing T A Achievable Throughpu pace and Relaed Linear Opimizaion Problem Consider ha he sysem operaes under an arbirary scheduling policy rom (1), he definiions of B (s, ), C (s, ) and (4) we have for any N, C (0,) () = lim B π lim inf (0,) P = lim inf = lim inf Bπ i (0,) Bi π(0,) r π i (8) In addiion, he fac ha A i (0,) Bi π (0,) and (2) imply ha for any i N, i holds 0 ri π α i (9) rom (8) and (9) we see ha he maximum weighed sum of hroughpus ha can be achieved by any scheduling policy, canno exceed he value of he following opimizaion problem: Linear Opimizaion Problem N max c i x i, x i=1 where for N = {1, 2,, N} x i (), N (10a) x i α i,i=1,,n (10b) x i 0, i=1,, N (10c) and () saisfies (7a), (7b), (7c) Le N k = {1,, k}, and N 0 = I can be shown ha he soluion o he previous opimizaion problem is given recursively by ( ( x k =min α k, min (k ) )) x i, (11) N k 1 for k =1,,N The proof is omied Our objecive in he nex secion is o show ha he scheduling policy π achieves he hroughpus defined by (11) and herefore is opimal IV OPTIMALITY PROO ince we deal only wih policy π in his secion, in order o simplify he noaion we eliminae π from all relaed noaions, eg, we use r i in place of ri π Before going ino he deails of he proof, we give he general idea of he approach In he general case, i can be shown ha under π, a subse U of he queues will grow o infiniy, while he res of he queues will receive he maximum possible hroughpu, ie, we have r i = α i, i N U The queues in N U are called sable queues, while hose in U unsable I can be proved ha for any sable queue i, we have r i = x i To deermine he hroughpus of he unsable queues we firs show ha for T large enough, each of he sable queues flucuaes in a cerain range around kt, 0 k N This fac and he manner he indices are used o deermine he scheduling decisions, implies ha r k = x k for all unsable queues We menion ha in he course of he proof, he fac ha r i = x i, is esablished by saring from he smalles indices and moving o he larges, raher han by firs proving he resul for he sable and hen for he unsable queues The following lemma will be useful in he sequel Lemma 1: or any subses 1, 2 of N and any i holds q i () q i () 1 2 q i ( 1) q i ( 1) + α i + σ U i ( 2 )+θ U 2 1

5 Proof: This is immediae from he bursiness consrains on he arrival and slo availabiliy processes ollowing he definiion in [12], we call a scheduling policy Longes Conneced Queue (LCQ) irs, he scheduling policy which among he queues whose channels are on, selecs he one wih he larges number of packes (if here are a leas wo such queues, pick one arbirarily) In he nex lemmas we use he following noaion or N, denoe by () he lh maximum of {q i ()} i and by π l () a permuaion of he indices in such ha q π l () () = (), l {1,, } Hence q(1) () =max i {q i ()} and () =min i {q i ()} Also, le (l) () = l j=1 π j () (ie, he se conaining he l larges queues among he queues in ) and (l) () = (l) () We use he erm a se G has prioriy a ime whena ime, policy π chooses one of he packes of he queues in G (if any) for ransmission, provided ha he associaed channel is on In order o avoid cluering he noaion in he proofs, in he following we will use he symbol O, o denoe a finie nonnegaive quaniy ha depends only on he parameers of he arrival and slo availabiliy processes, and N As will be clear from he proofs, in principle O can be explicily compued - eg, in Lemma 1, O = P 1 α i + P 1 σ U i + ( 2 )+θ U Lemmas 2 and 4 are used o deermine he range around kt in which each of he sable queues flucuaes Lemma 2: uppose ha here are numbers H 0, Φ > 0, and queue ses L N, N L, such ha he following hold a) The se G() ={i : q i () >H}, has highes prioriy a ime among he queues in N L b) If max i {q i ()} H + Φ, he queues in he se G() are served according o he LCQ policy c) or any i holds q ( ) α i (L ) (L) (12) d) q i (0) H for all i, hen, here is a number O such ha, if Φ > O, i holds max {q i()} H + O, for all 0 i Proof: Le = Also define y l () = (l) () (q i () H) + We observe ha for 2 l 1, i holds y l 1 ()+y l+1 () = 2y l ()+ () H and q (l+1) + () H +, (13) y 2 () =2y 1 () + q (2) () H + q (1) () H +, (14) y 1 () =y () q () () H + (15) Consider a ime where () >Hand le 0 1 be he larges ime before, such ha (l) () 6= (l) ( 0 1) or ( 0 1) H (for l = only he second siuaion makes sense) Time 0 is well defined because of assumpion d) of he lemma In he ime inerval [ 0,], he se (l) () remains he same and all he queues in his se are bigger han H, ie, nonempy Provided ha Φ is large enough (as will be seen i suffices o be O), his se of queues has prioriy over he queues in N L and uses all he available slos in [ 0,] ince hese slos are a leas C (l) L ( 0) ( 0,) C L ( 0,), we have B (l) ( 0) ( 0,) C (l) L ( 0) ( 0,) C L ( 0,) eing () = (L ) (L), we conclude q i () = q i ( 0 )+A (l) (l) (0) (l) ( 0) ( 0,) (0) B (l) (0)( 0,) q i ( 0 ) + (l) ( 0) α i (l) ( 0) (l) ( 0) (l) ( 0) ( 0 )+O q i ( 0 )+O (16) In he firs inequaliy above we used he bursiness consrains on he arrival and channel availabiliy processes In he second inequaliy we used assumpion c) of he lemma Because of he way 0 is defined, we have ha for i (l) ( 0) i holds q i () >Hand q i ( 0 ) >H Therefore by subracing H, l imes from boh sides of inequaliy (16) we obain (q i () H) + (l) ( 0) (l) ( 0) (q i ( 0 ) H) + + O (17) Hence, y l () y l ( 0 )+O (18) Le 2 l 1 We will show now ha y l ( 0 ) 1 2 y l 1( 0 )+ 1 2 y l+1( 0 )+O (19) Noe ha by definiion ( 0) > H and herefore from equaion (13) i holds y l ( 0 )= 1 2 y l 1( 0 )+ 1 2 y l+1( 0 )+ 1 2 ( 0) H 1 q (l+1) ( 0 ) H + (20) 2 We consider wo cases Case 1 ( 0 1) H Then, here mus be an index i 0 (l) ( 0) such ha q i0 ( 0 1) H and ( 0) q i0 ( 0 ) Using Lemma 1 wih 1 = {i 0 }, 2 =, we conclude ha q i0 ( 0 ) q i0 ( 0 1) + O H + O

6 Hence, ( 0) H O (21) rom (21) and (20) (aking also ino accoun ha q (l+1) ( 0 ) H + 0) follows (19) Case 2 (l) ( 0) 6= (l) ( 0 1) and ( 0 1) >HThen, here mus be indices i 0 (l) ( 0), j 0 (l) ( 0 ), such ha q i0 ( 0 1) q j0 ( 0 1) and ( 0) q (l+1) ( 0 ) q i0 ( 0 ) q j0 ( 0 ) By using Lemma 1 wih 1 = {i 0 }, 2 = {j 0 }, we conclude ha q i0 ( 0 ) q j0 ( 0 ) q i0 ( 0 1) q j0 ( 0 1) + O O Hence, ince have q (l+1) ( 0) q (l+1) ( 0 ) O (22) + (l+1) ( 0 ) H q ( 0 ) H, from (20) we y l ( 0 ) 1 2 y l 1( 0 )+ 1 2 y l+1( 0 ) ( 0) q (l+1) ( 0 ) which in conjuncion wih (22) shows (19) imilarly, we have wih an analogous definiion of 0, y 1 () 1 2 y 2 ( 0 )+O, (23) y () y 1 ( 0 )+O (24) If () H, hen we have from (13), (14) and (15) ha (19) as well as (23) and (24) sill hold wih 0 = ix now a ime and define y l =max y l () < (25) rom (18), (19), (24) and (23) i holds for 2 l 1, ha for any, 0, y l () 1 2 y 1 l y l+1 + O, (26) and y 1 () 1 2 y 2 + O (27) y () y 1 + O (28) Therefore we have for 2 l 1, and 1 y l 2 y 1 l y l+1 + O, (29) 1 y 1 2 y 2 + O, (30) y y 1 + O (31) The above inequaliies can be wrien in marix form as: (I B) Y O, (32) where Y = T y 1 ()y,andi is he uniy marix, O is a marix whose elemens are of ype O and 0 1/ /2 0 1/2 0 0 B = 0 1/2 0 1/ /2 0 1/ ince he row sums of B are all less han or equal o 1 and he sum of he firs raw is 1/2, ie, less han 1, i follows from he Perron-robenius Theorem [10], ha he eigenvalues of B are all smaller han 1 in absolue value Therefore, he marix (I B) 1 has nonnegaive elemens Hence, we can muliply (32) wih (I B) 1 o ge Y (I B) 1 O, or Therefore, Hence lim y l () O, l =1,, (33) y l () =supy l () =y l O, l =1,, (34) max {q i ()} H = q (1) i () H q (1) () H + = y1 () y 1 O and he lemma follows The nex lemma shows ha when he inequaliies in (12) are srenghened o sric inequaliies, hen he queues sizes are bounded by H + O afer some ime slo, under any finie iniial condiion on he queue sizes Lemma 3: uppose ha condiions a) and b) of Lemma 2 hold and in addiion, for any, i holds α i <(L ) (L) Then under any iniial queue sizes q i (0) <, i, hereis a finie ime τ 0 such ha max {q i()} H + O, for all τ 0 i Proof: The proof is omied The nex lemma provides condiions under which i is known ha he queue sizes of cerain se of queues do no fall below cerain hreshold afer some ime slo Is proof is analogous o he proof of Lemma 3 Lemma 4: If here is a posiive number H and queue ses L N, N L, such ha he following hold a) The queues in L always have packes o ransmi and have higher prioriy han he queues in G() = {i : q i () H} b) The queues in he se G(), are served according o LCQ policy c)or any subse, wih = i holds α i >(L ) (L ), hen here i a ime τ 0 such ha min i {q i ()} H O, for all τ 0

7 Lemma 5: Consider he vecor {x k } k N defined for k = 1,,N, by he recursion ( ( x k =min α k, min (k ) )) x i N k 1 ig 2 Pariioning of he index se Nex we need o examine in more deail he srucure of he opimal linear programming soluion (11) According o (11), x k may ake values less han or equal o α k An index such ha x k = α k is called sable, while an index such ha x k < α k, unsable We herefore have ha for a sable index k, forany N k 1 α k (k ) x i (35) imilarly, for an unsable index k, for any se k N k 1 such ha x k = (k k) x i, k i holds (k k ) k x i < α k (36) The general srucure of he vecor {x k } k N is as follows The se of indices is pariioned ino index ses Ii s, i = 1,, l s, Ii u,i=1,, l u such ha Indices in he se ls i=1 Is i are sable Indices in he se l u i=1 I u i are unsable Index se Ii x,x {s, u} consiss of successive inegers If i>jhen all indices in Ii x,x {s, u} are larger han he indices in Ij x igure 2 shows an example of he pariion of he index se for N =10channels or convenience in he discussion we assume ha for a given i, he indices in Ii s are smaller han he indices in Ii u Hence, for consisency, if index 1 is unsable, we define I1 s = If k I u j hen define b I k = j i=1 Is i enoe by u 1 <u 2 < < u L,L= l u i=1 Ii u he unsable indices In he following, we assume ha sric submodulariy holds, ie, whenever 6= and 6=, i holds, ()+ () >( )+ ( ), (37) herefore, whenever ()+ () = ( )+ ( ), hen eiher or This assumpion is no essenial bu simplifies he proofs a) or any se I s 1 i holds α i ( ) b) Wih index k I u j,j=1,, l u, here is an associaed index se c k such ha c k N k 1, x i = k c k, k c k for all N k 1 k k b x i (k ) x i, k b and for all b k, x k = k k b x i <(k ) k b x i c) u i 1 b ui 1 b ui, 2 i L d) c ui = i 1 j=1 u j f ui, where f ui b I ui, 1 i L e) If k is an unsable index, hen for any se H k I b k e k i holds α i k b k k k b f) If k is he larges index in I u j hen for any se bik Ij+1 s f k i holds α i k c k k c k Proof: a)le I1 s and le k be he larges index in Then, since {k} N k 1, according o he definiion of x k and Is 1 we have or, α k (k ( k)) k α i α i () (38) b) According o he definiion of x k,hereisase k N k 1 such ha for all N k 1, x k = (k k ) k (k ) x i (39) Le k 1 N k 1 be anoher se such ha x k = k k 1 x i 1 k x i

8 ince by definiion k k 1 x i (k k 1 k ) x i, k 1 k 1 k (k k ) x i (k 1 k k ) x i, k 1 k k we have by adding and cancelling summaion erms, (k k )+(k k) 1 (k k 1 k )+(k k 1 k )= (k k 1 k )+((k k) 1 (k k )) By submodulariy, (k k )+ k k 1 = (k k 1 k )+((k k) 1 (k k )) This equaliy and he sric submodulariy propery imply ha eiher k k 1 or 1 k k Le now b k be he se wih smalles cardinaliy among hose ha saisfy (39) By definiion, his se saisfies par b) of he lemma Assume for he momen ha par c) holds up o value i Par d) for i =1follows from he definiions, and for i 2 i is immediae from c) We will also show ha e) and f) hold To prove e), le bi ui e ui = bi ui b ui N ui 1 Then, (u i b ui ) (u i b ui ) bui, b ui hence, aking ino accoun ha b ui =, (u i b ui ) (u i b ui ) (40) To prove f) assume ha u i is he larges index in Ij u and Ij+1 s 6= Le biui Ij+1 s (u i b ui ) and le i k be he larges index in If i k Ij s hen I b ui b ui and (40) holds If i k Ij+1 s,hensinceu i b ui ( i k ) N ik 1, and u i ui b ( i k )=, we have from (35), α ik (i k u i b ui ( i k )) α j u i b ui i k = (u i b ui ) (u i b ui ) i k α j, where we used he fac ha by par b) of he lemma, u i b ui = (u i b ui ) Tha is, α j (u i b ui ) (u i b ui ) I remains o prove par c) Assume ha par c) holds up o index i 1 Assume also ha for index u m 1, 1 <m<ii holds u m 1 b um 1 b ui We will show hen ha i also holds, u m b um b ui, which will prove par c), provided ha we also show ha u 1 b u1 b ui (41) The proof of (41) will be oulined a he end, since i is basically a rewording of he argumen for general m ince b ui u m b um N ui 1, u i ui b ui b Tha is u i b ui u m b um = u i b ui u m um b b ui (u m b um ) b ui u m b u m u m b um = u i b ui u m um b (u m b um ) b ui (u m b u m ) b ui (u m b um ) (u i b ui )+(u m b um ) (u i b ui u m b um ) u i ui b u m um b (42) = bui u m um b, (43) where in (42) he inequaliy is sric if u i ui b u m um b 6= and u m um b u i ui b 6= Equaliy (43) follows from he fac ha since i > m, u i u m um b = We also have since b ui b um N um 1, u m um b bum = u m bui um b u m bui um b b ui b u m u m ( b ui b u m) + x u m, where in he las equaliy we used he fac ha u m / b ui b um ince by par b), x u m = u m um b P b u, m we conclude ha u m bui um b, (44) u m ( b ui b um ) where in (44), by par b) of he lemma, he inequaliy is sric if b ui b um b um

9 Assume now ha u m / b ui Then b ui u m um b b ui b um efine b E m = bui b um = u m 1 um 1 b ince by he inducive assumpion we have u m 1 b um 1 b ui b um, we can wrie = b ui (u m b u m ) = bem = bem b ui b u m + + u m 1 b um 1 u m 1 b um 1 ince u m / b ui and u i / b um, from he sric submodulariy and (43) i follows ha > bui um b be m u m 1 um 1 b (45) Observe ha by he inducive hypohesis we have ha E b m bi um 1 e um 1 or E b m bium 1 Ij+1 s e um 1 depending on wheher u m 1 is no, or is, he larges elemen in Ij uin eiher case have by e) and f) = α j u m 1 b um 1 E m b be m be m u m 1 um 1 b = bui um b u m 1 um 1 b (46) Inequaliy (45) conradics (46) and herefore we conclude ha u m b ui This implies ha b ui (u m b um )=u m bui um b rom (43) and (44) i follows ha u m bui b um u m ( b ui b u m) u m bui um b, wih equaliies holding only if b um b ui To complee he proof we need o prove (41) or his we follow essenially he same argumens as above, by replacing m wih1andu m 1 c um 1 wih he empy se Accordingoparsb)andc)ofLemma5, b ui can be wrien as b ui = i 1 j=1 u j e ui, where e ui I b ui Le e ui, hence i 1 j=1 u j b ui Applying par b) of Lemma 5 and aking ino accoun ha x i = α i when i e ui,wehave i j=1u j ui e x j i 1 l=1 u l e ui < i j=1u j i 1 l=1 u l α j α j, or i j=1u j ui e i j=1u j < e ui eing = e ui in he previous inequaliy we have he following corollary Corollary 6: If we se U i = i j=1 u j, hen for any e ui, 6=, i holds U i e ui U i eui < The nex heorem provides he range wihin which each of he queues in N flucuaes Theorem 7: Under policy π,ifq i (0) = 0, i N, hen for T large enough, he queues in L i=1 Iu i end o infiniy Moreover, seing e u0 =, we have for a number M = O, α i α i max {q i ()} (N +1 u j )T + M for all 0, e uj e uj 1 min {q i ()} (N +1 u j )T M for all τ 0, e uj e uj 1 and if i / e ul, max ls j=1 Ij s e ul q i () M for all I suffices o ake T>2M In he course of he proof of Theorem 7 we also prove he main resul of his paper, ha is, Theorem 8: The hroughpus achieved by policy π for T large enough, saisfy he resurcive equaions ( ( x k =min α k, min (k ) )) x i, N k 1 for k =1,,N Proof: In Lemma 2 se = I s,h=(n +1 u 1 1)T, Φ = T, and L = Condiions a), b) of he lemma hold because of he definiion of policy π rom Lemma 5 we have ha condiion c) holds as well ince condiion d) of he lemma also holds by he assumpion ha q i (0) = 0, i N, we conclude ha for all 0, max {q i ()} (N +1 u 1 )T + O (47) i I s 1 provided ha T O ince B i (0,)=A i (0,) q i (), from (47) we conclude ha for i I1, s B i (0,) A i (0,) r i = lim = lim = α i = x i (48) ince b u1 = e u1 I1 s we have from (36) and he definiion of I s ha 1 (u 1 e u1 ) α i < α u1 (49) u1 e Observe ha q j () =A u1 e u1 (0,) B u1 e u1 (0,) u 1 e u1 A u1 e u1 (0,) C u1 e u1 (0,) O, u 1 e u1 α i (u 1 e u1 )

10 and aking ino accoun (49) we conclude ha lim q j () = u 1 e u1 ince by (47) q i () is finie for i e u1 I1, s we conclude ha lim q u1 () = According o Corollary 6, for any subse e u1 i holds α i > u 1 u1 e u 1 eu1 Noice ha since e u1 I1, s by he definiion of he queue indices I i (q) he queues in e u1 are served according o he LCQ policy when hey are smaller han H = (N +1 u 1 )T Moreover, since lim q u1 () =, i holds ha I u1 (q u1 ()) = H, for larger han or equal o some ime 0 Hence, for 0 queue u 1 has prioriy over he queues in e u1 whenever max i {q i ()} H Therefore, we can apply Lemma 4 wih L = u 1, = e u1 and H = (N +1 u 1 )T o conclude ha here is some ime τ such ha for i e u1, and for all τ 1 0, i holds min {q i ()} (N +1 u 1 )T O (50) i e u1 rom Lemma 5, e) i follows ha for any subse E 1 = I1 s e u1 (E 1 = I1 s Is 2 e u1 if u 1 is he only elemen in I1 u ) i holds α i u 1 e u1 u 1 u1 e Applying now Lemma 2 wih = E 1,H=(N +1 u 2 )T, Φ = T and L = u 1 e u1, we conclude ha for all 0, max {q i ()} (N +1 u 2 )T + O (51) i E 1 provided ha T O PicknowT large enough so ha T O > O (52) Then since u 2 >u 1, i holds (N +1 u 1 ) T O > (N +1 u 2 ) T + O (53) Inequaliies (50), (51) and (53) and he fac ha q u1 (), ( ie, I u1 (q u1 ()) = (N +1 u 1 ) T, for τ 1 0)implyha he queues in u 1 e u1 have higher prioriy over he res of he queues for τ 1 0 and ha hey are nonempy Therefore, he queues in u 1 e u1 use all he available channel slos ince B u1 e u1 (τ 1 0,)=C u1 e u1 (τ 1 0,), we conclude ha B u1 e u1 (0,) C u1 e u1 (0,) lim = lim Taking ino accoun (48) and he fac ha C u1 e u1 (0,) lim = u 1 u1 e, we conclude ha lim (B u1 (0,)/) exiss and B u1 (0,) r u1 = lim = u 1 u1 b α i, bu1 ha is, r u1 = x u 1 Using similar argumens we can show he claims for he res of he queue indices V CONCLUION We presened a policy for scheduling packes for ransmission over channels wih ime varying conneciviy We showed ha he proposed policy maximizes he weighed sum of channel hroughpus under any packe arrival raes We adoped a bursiness-consrained channel model for he analysis This model faciliaes he analysis, while a he same ime allows us o dispose of saisical assumpions In his paper, we resriced ourselves o he case where he channels are in one of wo saes, on and off A subjec of furher work is o generalize our approach o include he case of muli-rae channels Anoher subjec of ongoing work is o generalize our approach o include more general opimizaion funcions urhermore, he consideraion of packe delays, in addiion o hroughpu, is a pracical maer ha needs o be addressed REERENCE [1] C Chang, Performance Guaranees in Communicaion Neworks, pringer Verlag, 2000 [2] R L Cruz, A calculus of delay Par I: Nework elemen in isolaion", IEEE Trans Inform Theory, vol 37, pp , Jan 1991 [3] R L Cruz, A calculus of delay Par II: Nework analysis, IEEE Trans Inform Theory, vol 37, pp , Jan 1991 [4] Bedekar, A, Bors, C, Ramanan, K, Whiing, PA, Yeh, M, ownlink cheduling in CMA aa Neworks, Proc IEEE Globecom 1999, pp [5] P Bhagwa, P Bhaacharya, A Krishna, Tripahi, Enhancing Throughpu over Wireless LANs using Channel ae ependen Packe cheduling, Proc IEEE Infocom 97, April 1997 [6] em Bors, Phil Whiing, ynamic Rae Conrol Algorihm for HR Throughpu Opimizaion, IEEE Infocom 2001 [7] M Groschel and L Lovasz, A chrijver, Geomeric Algorihms and Combinaorial Opimizaion, pringer-verlag, 1991 [8] Michael Neely, Eyan Modiano and Charles Rohrs, Power and erver Allocaion in a Muli-Beam aellie wih Time Varying Channels, IEEE Infocom 2002, New York, June, 2002 [9] in Liu, Edwin K P Chong, Ness B hroff, Transmission cheduling for Efficien Wireless Nework Uilizaion, Infocom 2001, Anchorage, Alaska, pp , 2001 [10] E enea, Non-negaive Marices and Markov Chains, pringer-verlag 1980 [11] anjay hakkoai and Alexander olyar, cheduling for Muliple flows haring a Time-Varying Channel: The Exponenial Rule, To appear in Transacions of he AM, A volume in memory of Karpelevich, Providence, RI: American Mahemaical ociey, 2001 [12] L Tassiulas and A Ephremides, ynamic erver Allocaion o Paraller Queues wih Randomly Varying Conneciviy, IEEE Trans on Informaion Theory, vol 39, no 2, March 1993 [13] L Tassiulas cheduling and performance limis of neworks wih consanly changing opology, IEEE Transacions on Informaion Theory, Vol 43, No 3, pp , May 1997