Delay Analysis of Maximum Weight Scheduling in Wireless Ad Hoc Networks

Transcription

1 Deay Anaysis o Maximum Weigh Scheduing in Wireess Ad Hoc Neworks The MIT Facuy has made his arice openy avaiabe. Pease share how his access beneis you. Your sory maers. Ciaion As Pubished Pubisher ong Bao e, K. Jagannahan, and E. Modiano. Deay anaysis o maximum weigh scheduing in wireess Ad Hoc neworks. Inormaion Sciences and Sysems, 009. CISS rd Annua Conerence on Insiue o Eecrica and Eecronics Engineers. hp://dx.doi.org/ /ciss Insiue o Eecrica and Eecronics Engineers Version Fina pubished version Accessed Sa Mar 30 0:40:33 EDT 019 Ciabe ink Terms o Use Deaied Terms hp://hd.hande.ne/171.1/5983 Arice is made avaiabe in accordance wih he pubisher's poicy and may be subjec o US copyrigh aw. Pease reer o he pubisher's sie or erms o use.

2 1 Deay Anaysis o Maximum Weigh Scheduing in Wireess Ad Hoc Neworks ong Bao e, Krishna Jagannahan, and Eyan Modiano Absrac This paper sudies deay properies o he weknown maximum weigh scheduing agorihm in wireess ad hoc neworks. We consider wireess neworks wih eiher onehop or mui-hop ows. Speciicay, his paper shows ha he maximum weigh scheduing agorihm achieves order opima deay or wireess ad hoc neworks wih singe-hop raic ows i he number o acivaed inks in one ypica schedue is o he same order as he number o inks in he nework. This condiion woud be saisied or mos pracica wireess neworks. This resu hods or boh i.i.d and Markov moduaed arriva processes wih wo saes. For he mui-hop ow case, we aso derive igh backog bounds in he order sense. Index Terms Maximum weigh scheduing, backog/deay bounds, capaciy region, order opima deay I. INTRODUCTION Wireess scheduing has been known o be a key probem or hroughpu/capaciy opimizaion in wireess neworks. The we-known maximum weigh scheduing agorihm has been proposed by Tassiuas in his semina paper 1] where he proved is hroughpu opimaiy. aer deveopmens in his area incude exension o his maximum weigh scheduing agorihm o wireess neworks wih rae/power conro ], 3], nework conro when oered raic is ouside he capaciy region 4], and oher scheduing poicies wih owercompexiy 5]-8]. Whie mos exising works in he area o sochasic nework conro ocused on hroughpu perormance o opima and subopima scheduing poicies, deay properies o mos scheduing poicies proposed or wireess ad hoc neworks remain unknown. In his paper, we sudy backog/deay properies o he maximum weigh scheduing agorihm in wireess ad hoc neworks. There are some recen works which invesigaed backog/deay bounds or he supopima maxima scheduing agorihm in wireess ad hoc neworks and maximum weigh scheduing agorihm in he downink/upink o ceuar neworks. Speciicay, in 13] Neey showed ha maxima scheduing achieves deay scaing o O (1/(1 ρ)) or raic inside he reduced sabiiy region derived in 8]. This reduced sabiiy region can be as sma as 1/I o he capaciy region, where I is he maximum number o inks in any ink inererence se which do no inerere wih one anoher. In 14], 15], Neey aso proved he order opima deay or he maximum weigh scheduing agorihm in he wireess ceuar upink/downink wih ON/OFF wireess inks. Noe ha he capaciy region in he ceuar seing can be expiciy This work was suppored by NSERC Posdocora Feowship and by ARO Muri gran number W911NF The auhors are wih Communicaions and Neworking Research Group, Massachuses Insiue o Technoogy, Cambridge, MA, USA. Emais: {ongbe, krishnaj, modiano}@mi.edu /09/$ IEEE 389 described which signiicany eases he backog/deay anaysis. Average backog bounds were derived or maximum weigh scheduing in severa works ], 4], 10], 9]. These backog bounds were obained by bounding he maximum ransmission raes and he number o arrivas in each ime so which are, hereore, no igh in genera. There are some oher works which invesigae exponens or he ais o queue backogs in he wireess ceuar seing 11], 1]. In his paper, we consider a wireess ad hoc nework wih eiher one-hop and mui-hop raic ows. We show ha average deay or he case o one-hop raic ows scaes as O (1/(1 ρ)) i we can consruc a se o disinc schedues o cover he nework where he number o acivaed inks in each o hese schedues is o he same order as he number o nework inks. This condiion woud be saisied or mos pracica arge-scae wireess neworks. This deay scaing hods or boh i.i.d and Markov moduaed raic arriva processes wih a mos wo saes. These resus are saed in Proposiions 4 and 5 o he paper. To he bes o our knowedge, hese are he irs deay opima resus or he maximum weigh scheduing agorihm in wireess ad hoc neworks. For wireess ad hoc neworks wih muihop raic ows, we aso derive a igh backog bound which scaes as O (N/(1 ρ)) where N is he number o wireess nodes. The remaining o his paper is organized as oows. Deay anaysis or singe-hop raic ows is presened in secion II. In secion III, we derive backog bounds or wireess neworks wih muihop raic ows. II. ANAYSIS OF SINGE-HOP FOW CASE A. Sysem Modes and Assumpions We mode a wireess ad hoc nework as direced graph G = (V,E) where V is he se o wireess nodes and E is he se o wireess inks. Suppose he cardinaiies o V and E are N and, respecivey. We consider singe-hop raic ows in his secion. Daa rom a ows raversing a paricuar ink is buered a he corresponding ransmier o he ink. Assume ime is soed wih ixed-size so inervas. For now, raic arriving o source nodes o singe-hop ows is assumed o be independen and idenicay disribued (i.i.d) over ime. Assume ha packes arriving during ime so can ony be ransmied rom ime so +1 a he earies. e denoe by A () he number o packes arriving a ink in ime so and μ () he number o packe ransmied on ink in ime so. For simpiciy, assume ha μ () =1i ink is schedued in ime so, oherwise μ () =0. In he remaining o his paper, we wi use r o describe a coumn vecor wih eemens r denoing quaniies such as queue engh, schedued inks,

3 ec. The queue evouion or he ow a ink can be wrien as oows: ( +1)= () μ ()+A (). (1) Assume ha ony backogged inks are schedued, i can be veriied ha his queue evouion equaion hods or arbirary () and μ (). Regarding he scheduing, we consider he we-known maximum weigh scheduing agorihm which is known o achieve he capaciy region 1]. The maximum weigh scheduing agorihm deermines he opima schedue μ () based on he ink queue backogs as oows: μ () = argmax ()μ () () μ() S where S denoes he se o a possibe easibe schedues according some inererence consrains. In his secion, we are going o derive he deay bound or his scheduing poicy assuming ha arriva raic is sricy wihin he capaciy region so ha he maximum weigh scheduing agorihm wi sabiize he nework 1]-3]. B. Backog/Deay Anaysis or i.i.d Arriva Traic In his subsecion, we obain a deay bound or he aoremenioned scheduing scheme using he yapunov dri echnique 1]-3]. Traic arriving o a ransmiing buer o wireess ink is assumed o be i.i.d over ime wih average arriva rae λ. In he oowing, we use a resu which was saed in 13]. emma 1: (Theorem 1 rom 13]) e () be he queue backog vecor in ime so and ( ()) be a yapunov uncion. Aso, deine a one-sep yapunov dri as oows: { Δ() = E ( ( + 1)) ( ()) } (3) where he expecaion E(.) is aken over he randomness o queue backogs () and sysem dynamics given he queue backogs (). I he yapunov dri saisies hen we have im sup 1 1 τ=0 Δ() E {g()} E {()} (4) E {()} im sup 1 1 τ=0 E {g()}. (5) The proo o his proposiion oows direcy by appying he queue evouion reaion in (1), and is omied or breviy. As shown in 1], he capaciy region coincides wih { he } convex hu o a possibe easibe schedues. e S = Ri be he se o a possibe schedues where one paricuar schedue R i is a coumn vecor o dimension wih he -h eemen equa 1 i ink is schedued and equa 0 oherwise. For any arriva rae vecor λ sricy inside he capaciy region, we have λ β iri (9) where denoes he cardinaiy o se S, β i < 1 and <, denoes boh a reguar inequaiy and an eemenwise inequaiy. We have he oowing reaion N λ = r λ β i r Ri r μ β i < r μ (10) where.] r denoes he vecor ransposiion and μ is he opima schedued vecor given backog vecor. I can be veriied ha hese resus hod by using he reaions in (9) and (). Now, we sae a bound on he oa queue backogs or i.i.d. arriva raic in he oowing proposiion. Proposiion : Assume ha he arriva rae vecor λ is sricy inside he capaciy region so ha here exiss a vecor ɛ such ha λ+ ɛ is inside he capaciy region where ɛ is a vecor wih a eemens equa o ɛ. Aso, assume ha a arriva processes on a wireess inks have bounded second momens. Then, he nework is sabe and he oa average queue backog can be bounded as λ o + E { A () } λ ɛ (11) where λ o = λ is he oa ink arriva raes. Proo: Using (10) or backog vecor (), wehave ] r ] ] r () λ + ɛ () μ (). (1) Hence, ] r ] r ] r () λ () μ () () ɛ. (13) Now, consider he oowing quadraic yapunov uncion ( ()) = (). (6) We have he oowing resu or he yapunov dri. Proposiion 1: The yapunov dri saisies he oowing reaion or any ime so : Δ() =E {B()} + E { ()(A () μ ())} (7) where B() =A () μ ()] = A () +A ()μ ()+μ (). (8) /09/$ IEEE 390 Noe ha he second erm o (7) can be wrien as E { ()(A () μ ())} = E { ()(λ μ ())} =E { () ] r ( λ μ() ) }. (14) Using (13) and (14) in (7) wih μ() represening an opima schedued vecor, we have Δ() E {B()} () ] r ɛ

4 3 = E {B()} ɛ E { ()}. (15) Using he resu in emma 1 in (15), we have im sup 1 1 τ=0 E { (τ)} B ɛ (16) where B = 1 1 im sup τ=0 E {B(τ)}. From (8), using he ac ha μ () 1 and arriva processes have bounded second momens, we have B <. Thereore, he queueing nework is srongy sabe. Because i evoves according o an ergodic Markov chain wih counabe sae space, he imiing ime averages o queue backogs equa o he corresponding seady sae averages. To cacuae B, we noe ha under he sabiiy condiion we have 1 1 im sup τ=0 μ (τ) = λ. Aso, noe ha μ () = μ () because μ () = {0, 1} depending on wheher ink is schedued in ime so or no. As a consequence, B can be wrien as B = = λ o + E { A () } λ + λ E { A () } λ. (17) Because ime average imis o queue backogs are equa o heir seady sae averages, using (17) he inequaiy (16) can be rewrien as λ o + E { A () } λ. (18) ɛ Hence, he proposiion is proved. 1) Deay Bound: Appying ie s aw o (11), we can obain a deay bound as oows: W 1+ 1 { λ o E A () } ] λ. (19) ɛ Now, in order o undersand he scaing o his deay bound, we need o deermine he reaionship beween he raic oading acor ρ and he parameer ɛ. e us denoe by Λ he capaciy region. Assume ha he arriva rae vecor λ =(λ1,λ,,λ ) r is sricy inside he capaciy region Λ, hen here exiss a oading acor ρ<1 such ha λ ρλ. (0) In he oowing, we sae a deay bound by choosing a sraighorward oading acor ρ as a uncion o ɛ. Proposiion 3: I arriva rae is in ρ-scaed capaciy region as described in (0), he average oa deay can be bounded as 1+ 1 { λ o E A () } ] ] λ W. (1) (1 ρ) In he specia case where he arriva process on each wireess ink is Poisson, we have W (1 1 λ o λ ). () 1 ρ /09/$ IEEE 391 Proo: The proo oows by using he ac ha we can choose ɛ = (1 ρ) 1 where 1 is an a-one vecor wih dimension such ha λ+ ɛ Λ or any λ ρλ. Speciicay, by pugging ɛ = (1 ρ) ino he deay bound in (19), we can obain (1). Now, we show ha λ+ ɛ Λ or ɛ = (1 ρ). Noe ha or any λ ρλ, we can wrie λ = ρ β ir i where β i < 1. Deine e i be a vecor o dimension wih a zeros excep a one a he i-h posiion. I can be easiy seen ha any e i (i =1,,,) represens a easibe schedue (wih ink i being acivaed). Aso, noe ha e i = 1. Hence, we have he oowing resu 1 ρ 1 +ρ β i Ri = 1 ρ e i +ρ β iri Λ. (3) When he arriva processes are Poisson, we have E { A () } = λ + λ. Using his reaionship in he deay bound (1), we obain (). Noe ha he erm 1 λ o E E { A () } is ypicay O(1) or any raic saisying A () A max. In ac, in such cases 1 we have λ o E E { A () } A max. Hence, he deay bound saed in Proposiion 3 is ypicay O(/(1 ρ)). C. Tigher Deay Bound In he oowing, we sae a igher deay bound under speciic assumpions which can be achieved by expoiing underying inererence consrains and nework opoogy. Proposiion 4: Assume ha he arriva rae is in he ρ- scaed capaciy region as described in (0). Aso, assume ha we can ind a se o easibe schedues, namey Ψ = { s i,,,,t}, saisying he oowing assumpions For any schedue s i Ψ, i ink is acivaed in s i hen ink is no acivaed in any oher s j Ψ or j i (i.e., any ink shoud beong o one and ony one schedue in he se Ψ). e E be he se o inks acivaed by a schedues in Ψ, hen E = E where reca ha E is he se o a nework inks (i.e., he union o acivaed inks by a schedues covers he whoe nework). e K i denoe he number o acivaed inks in schedue s i and K min = min i K i. Then, we have he oowing deay bound W 1+ 1 { λ o E A () } ] λ. (4) K min (1 ρ) / Beore proving his proposiion, we noe ha or wireess neworks such ha K min = O(), proposiion 4 impies ha he nework deay ypicay scaes as O(1/(1 ρ)). This condiion woud hod i he nework opoogy is suicieny sparse and uniorm so ha he mos baanced se o schedues Ψ (i.e., amos a schedues in Ψ have he same number o acivaed inks in he order sense) saisies K min = O(). Noe ha his condiion woud be saisied or mos pracica wireess neworks because a ypica schedue woud acivae mos inks in he nework. We wi provide one such nework exampe aer he proo. Proo: The proo or his proposiion oows he same ine as ha or proposiion 3. However, a igher deay bound

5 4 is achieved in his proposiion by consrucing ɛ rom he se o schedues Ψ each o which has a eas K min acivaed inks. Now, consider he oowing inear combinaion o easibe schedues whose oucome ies inside he capaciy region (1 ρ) T K i T j K s i + ρ β iri Λ. (5) j Thereore, he resu saed in proposiion 4 oows by pugging ɛ =(1 ρ) K min ino he deay bound (19). In he oowing, we provide a simpe exampe where he assumpions o he proposiion hod. Exampe: Consider a grid nework and one-hop (primary) inererence mode or he sake o simpiciy as being shown in Fig. 1. In his igure, we aso show how o consruc a se o easibe schedues Ψ ha covers he whoe nework graph (again, each schedue has he same ink paern). To anayze is deay bound, assume ha he size in one dimension o he grid nework is H inks, hen i can be veriied ha =H(H+1). From he consruced se o schedues Ψ as shown in his igure, we have K min =(H +1) H/. Thereore, using he resu in proposiion 4, he deay can be bounded as H(H +1) 1+ 1 { λ o E A () } ] ] λ W (H +1) H/ (1 ρ) 1+ 1 { λ o E A () } ] ] λ (1 ρ) which scaes ypicay as O(1/(1 ρ)). a ime. In order o obain deay bound or his case, we wi use one resu proved in 13] which is saed in he oowing emma. emma : (rom secion V.A o 13]) Deine C = E {A ( 1)A ()} or E 1 and C =0or E.For a ink, wehave E { ()A ()} E { ()} λ + C. δ + σ Now, we sae deay bounds or he case o ime-correaed arrivas in he oowing proposiion. Proposiion 5: I he arriva raic is wihin he ρ-scaed capaciy region and he assumpions in proposiion 4 are saisied, hen he nework is sabe and he average deay can be bounded as W where B =1+ 1 λ o E B + C K min (1 ρ)/. (6) E { A () }, C = 1 λ o E 1 C σ + δ. (7) The proo oows by using resus in emma and Proposiion 1 so i is omied or breviy. The erm 1 λ o E E { A () } is ypicay O(1) or any raic saisying A () A max. In ac, in such cases we have B 1+A max. I is no very diicu o see ha or E 1, we have C λ λ max where λ max < 1 is he maximum condiiona rae over a inks and saes. Hence, we can obain he oowing deay bound W 1+A max + max E1 {λ max /(σ + δ )} K min (1 ρ)/ which scaes as O(1/(1 ρ)) or K min = O(). (8) Fig. 1. Grid neworks wih one-hop (primary) inererence mode. D. Anaysis or Time-Correaed Arrivas wih Two Saes Here, assume ha arriva process A () or inks is eiher i.i.d. or moduaed by a discree ime saionary and ergodic Markov chain Z () having wo saes (i.e., saes 1 and ). e σ and δ be ransiion probabiiies rom sae 1 o sae and rom sae o sae 1, respecivey. For each ink, deine he condiiona average arriva raes λ (m) as oows: λ (m) = E {A () Z () =m}. Now, e denoe by E 1 E as he se o inks wih imecorreaed arrivas where λ (1) λ (). Aso, assume ha arriva raic o any oher inks in E = E E 1 is eiher i.i.d or ime-correaed wih wo saes saisying λ (1) = λ (). Assume ha he moduaing Markov chains o a arriva processes are saionary so ha or a inks we have E {A ()} = λ or /09/$ IEEE 39 III. ANAYSIS OF MUTIHOP FOW CASE A. Sysem Modes and Assumpions We consider he same nework mode as secion II. We assume ha here is se o muihop ows F where ow F has a ixed roue rom a source node s() o a desinaion node d(). We denoe he se o inks and nodes on he roue o ow as () and R(), respecivey. For simpiciy, we assume ha packe arrivas o source nodes o a ows are i.i.d sochasic processes. We denoe he queue engh o ow a node n a he beginning o ime so as n() and he number o packes arriving a he source node o ow as A s() (). Noe ha daa packes o any ow are deivered o he higher ayer upon reaching he desinaion node, so d() ()=0. In addiion, e μ n() be he number o packes o ow ransmied rom node n aong ink (n, m) o is roue which is buered a node m i m d(). Again, we assume ha μ n() =1i we acivae ink (n, m) on he roue o ow and μ n() =0, oherwise. Given he roues or a ows, he maximum weigh scheduing agorihm is used or daa deivery 1]. Speciicay, he scheduing is perormed in every ime so as oows:

6 Each ink (n, m) inds he maximum dierenia backogs as oows: { w nm () = max n () m() }. (9) :(n,m) () Based on cacuaed ink weighs, a maximum weigh schedue is ound as μ () = argmax w nm ()μ nm (). (30) μ() S (n,m) For any schedued ink, one packe is ransmied rom he buer o he ow achieving he maximum dierenia backog. The queue evouions can be wrien as n( +1)= n() μ n()+πn() (31) where his equaion hods because μ n() =1ony i n() 1 (i.e., we do no schedue inks wih empy queues). Aso, πn() is he number o packes arriving o queue n() in ime so which can be wrien as π n() = { A n (), i n = s() μ n 1 (), oherwise. (3) B. Backog Bound The queue backog bound is saed in he oowing proposiion. The igher backog bound wi be deveoped aer ha. Proposiion 6: Assume ha arriva processes or a ows are i.i.d over sos and have bounded second momens. Aso, assume ha arriva rae ies wihin he ρ-scaed capaciy region. Then, 1) The oa average queue backogs can be bounded as =1 n R() n() θ max max (N + D). (1 ρ) where ρ is a oading acor, max = max{ F } (), θ max is he maximum number o ows raversing any ink in he nework, N is oa number o nework nodes, and D = { ( ) } E A s() () λ + λ. (33) =1 ) For Poisson arriva process, he backog can be bounded as θ max max (N +λ o ) F λ n() (1 ρ) =1 n R() where λ o = F λ. Proo: Consider he oowing yapunov uncion ( ) () = ( n () ) =1 n R() (34) where again he expecaion E(.) is aken over he randomness o queue backogs () and sysem dynamics given he queue backogs (). The yapunov dri can be wrien as oows: ( Δ() = E n ( +1) ) ( n () ) ] =1 n R() /09/$ IEEE 393 = E π n () μ n() ] (35) =1 n R() +E n() πn() μ n() ].(36) =1 n R() Now, consider (35) we have E π n () μ n() ] N + B 1() (37) =1 n R() { ] where B 1 () = =1 E A s() () μ s() }. () This inequaiy hods because we have F,s() n π n () μ n() ] 1. Now, using (3), we can manipuae (36) as oows: E n() πn() μ n() ] =1 n R() = μ nm()wnm + s() ()λ (38) =1 where μ nm() corresponds o he maximum weigh schedue wih queue backogs () and w nm = max :(n,m) () n () m() ] +. Noe ha we have wrien down n() m() ] + insead o n () m() ] because inks wih negaive weigh wi no be schedued by he maximum weigh scheduing agorihm. Noe ha we can rewrie =1 s() ()λ as oows: s() ()λ = =1 =1 (n,m) () λ n () m() ] :(n,m) () max n () m() ] + (39). :(n,m) () Suppose ha λ = (λ ) F is sricy inside he capaciy region. Then, here exiss a vecor ɛ such ha λ+ ɛ is si inside he capaciy region. This impies ha here exiss a vecor o ink raes (μ nm ) Co(S) such ha 1] (λ + ɛ) = λ + θ nm ɛ μ nm (40) :(n,m) () :(n,m) () where Co(S) represens he convex hu o a easibe ink schedues and θ nm denoes he number o ows raversing ink (n, m). Using (38), (39), (40), we can rewrie (36) as (36) μ nm()+ λ wnm ɛ :(n,m) () λ ( μ nm()+μ nm θ nm ɛ) w nm θ nm w nm. (41) 5

7 6 Now, we have he oowing n() () (n,m) () () (n,m) () (n,m) () n () m() ] + w nm. (4) Noe ha a simiar bounding echnique or (n,m) () n() used in he above inequaiy has been empoyed in 16] o prove he nework sabiiy. We insead aim a obaining a igh upper-bound o queue backogs as a uncion o he oading acor ρ, so he yapunov dri anaysis in his paper is very dieren rom ha in 16]. Using (4), we have =1 (n,m) () max =1 (n,m) () n() () =1 w nm = max (n,m) () w nm θ n,m w nm. (43) where max = max { F } (). Using (37), (41) and (43), he yapunov dri can be bounded as Δ() N + B 1 () ɛ max =1 n R() n(). (44) I he arriva processes or a ows have bounded second momens hen B 1 () is bounded. Under his condiion, he yapunov dri wi be negaive when he oa backog becomes arge enough. Hence, he nework is sabe and he ime average imis o queue backogs are equa o heir seadysae averages. Aso, i is no very diicu o see ha he ime average imi o B 1 () is equa o D. Thereore, by using emma 1 and subsiuing ime average imis o queue backogs by heir seady-sae averages, we have =1 n R() n max(n + D). (45) ɛ Now, o undersand he scaing o his backog bound, we need o ind ɛ as a uncion o he oading acor ρ as beore. Suppose arriva rae vecor λ is inside he ρ-scaed capaciy region, and e λ be a vecor wih (n, m)-h eemens equa λ nm = :(n,m) () λ. Then, we have λ = ρ β iri (46) or some non-negaive β i such ha β i < 1. e θ max = max {} θ nm where reca ha θ nm is he number o ows raversing ink (n, m). We wi show ha λ (1) = λ + ɛ Λ or ɛ = (1 ρ)/(θ max ). Now, e us consruc a vecor λ () wih is (n, m)-h eemens equa o () λ nm = :(n,m) () λ(1). Then, we have ( ) λ () = λ 1 ρ + θ nm θ max where ρ β iri + 1 ρ ( 1 ρ θ max θ nm ) e nm Λ (47) denoes a vecor wih (n, m)-h eemens equa o he quaniy inside he bracke. Subsiue ɛ =(1 ρ)/(θ max ) ino (45), we can obain par 1) o he proposiion. To prove par ) o he proposiion, we need some manipuaion o D or he Poisson arriva process. { ( ) } Speciicay, or Poisson process we have E A s() () = λ + λ. Subsiue his ino (33), we have D = λ o =1 λ. Pug his ino he bound in par 1), we obain par ) o he proposiion. I can be shown ha he average backogs derived in his proposiion scae as O(N/(1 ρ)) or max = O(1). In ac, using he simiar idea as ha o proposiion 4, a igher backog bound can be obained. Speciicay, i he arriva processes saisy assumpions in proposiion 6 and he assumpions o proposiion 4 hod, hen he oa queue backogs can be bounded as =1 n R() n θ max max (N + D) K min (1 ρ)/. (48) This backog bound ypicay scaes as O(N/(1 ρ)) or max = O(1) and K min = O(). REFERENCES 1]. Tassiuas and A. Ephremides, Sabiiy properies o consrained queueing sysems and scheduing poicies or maximum hroughpu in muihop radio neworks, IEEE Trans. Auomaic Conro, vo. 37, no. 1, pp , Dec ] M. Neey, E. Modiano, and C. Rohrs, Power aocaion and rouing in muibeam saeies wih ime-varying channes, IEEE/ACM Trans. Neworking, vo. 11, no. 1, pp , Feb ] M. Neey, E. Modiano, and C. Rohrs, Dynamic power aocaion and rouing or ime varying wireess neworks, IEEE INFOCOM ] M. Neey, E. Modiano, and C. i, Fairness and opima sochasic conro or heerogeneous neworks, IEEE INFOCOM ] E. Modiano, D. Shah, and G. Zussman, Maximizing hroughpu in wireess neworks via gossiping, ACM SIGMETRICS ] S. Sanghavi,. Bui, and R. Srikan, Disribued ink scheduing wih consan overhead, ACM SIGMETRICS 007, June ] X. in and N. Shro, The impac o imperec scheduing on cross-ayer rae conro in wireess neworks, IEEE INFOCOM ] P. Charporkar, K. Kar, and S. Sarkar, Throughpu guaranees hrough maxima scheduing in wireess neworks, Aeron 005, Sep ] Y. Yi, A. Prouiere, and M. Chiang, Compexiy o wireess scheduing: Impac and radeos, ACM Mobihoc, May ] G. Gupa and N. B. Shro, Deay anaysis o scheduing poicies in wireess neworks, Asiomar Conerence on Signas, Sysems, and Compuers, Oc ] A. Soyar, arge deviaions o queues under os scheduing agorihms, Aeron 006, Sep ] V. J. Venkaaramanan and X. in, Srucura properies o DP or queue-engh based wireess scheduing agorihms, Aeron 007, Sep ] M. Neey, Deay anaysis or maxima scheduing in wireess neworks wih bursy raic, IEEE INFOCOM ] M. Neey, Order opima deay or opporunisic scheduing in muiuser wireess upinks and downinks, Aeron 006, Sep ] M. Neey, Deay anaysis or max weigh opporunisic scheduing in wireess sysems, Aeron 008, Sep ]. Bui, R. Srikan, and A. Soyar, Nove archiecure and agorihms or deay reducion in back-pressure scheduing and rouing, IEEE INFOCOM /09/$ IEEE 394