Opportunistic Scheduling with Reliability Guarantees in Cognitive Radio Networks

Transcription

1 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL 008 Opporunisic Scheduling wih Reliabiliy Guaranees in Cogniive Radio Neworks Rahul Urgaonkar, ichael J. Neely Universiy of Souhern California, Los Angeles, CA hp://www-scf.usc.edu/ urgaonka Absrac We develop opporunisic scheduling policies for cogniive radio neworks ha maximize he hroughpu uiliy of he secondary (unlicensed) users subjec o maximum collision consrains wih he primary (licensed) users. We consider a cogniive nework wih saic primary users and poenially mobile secondary users. We use he echnique of Lyapunov Opimizaion o design an online flow conrol, scheduling and resource allocaion algorihm ha mees he desired objecives and provides explici performance guaranees. Index Terms Cogniive Radio, Queueing Analysis, Resource Allocaion, Lyapunov Opimizaion I. INTRODUCTION Cogniive radio neworks have recenly emerged as a promising echnique o improve he uilizaion of he exising radio specrum. The key enabler is he cogniive radio [ ha can dynamically adjus is operaing poins over a wide range depending on specrum availabiliy. The main idea behind a cogniive nework is for he unlicensed users o exploi he spaially and/or emporally under-uilized specrum by ransmiing opporunisically. However, a basic requiremen is o ensure ha he exising licensed users are no adversely affeced by such ransmissions. In his paper, we develop opporunisic scheduling schemes ha maximize he hroughpu uiliy of he secondary (or unlicensed) users subjec o maximum collision consrains wih he primary (or licensed) users in a cogniive radio nework. A survey on he echnology, design issues and recen work in cogniive radio neworks is provided in [, [3. The problem of opimal specrum assignmen o secondary users in saic neworks is reaed in [4 [7 where i is assumed ha scheduling is aware of primary user ransmissions. Scheduling he secondary users under parial channel sae informaion is considered in [8, [9 which uses a probabilisic maximum collision consrain wih he primary users. In his paper, we use he echniques of adapive queueing and Lyapunov Opimizaion o design an online flow conrol, scheduling and resource allocaion algorihm for a cogniive nework ha maximizes he hroughpu uiliy of he secondary users subjec o a maximum rae of collisions wih he primary users. This algorihm operaes wihou knowing he mobiliy paern of he secondary users and provides explici performance bounds. Lyapunov Opimizaion echniques were This maerial is suppored in par by one or more of he following: he DARPA IT-ANET program gran W9NF , he Naional Science Foundaion gran OCE Fig.. 3 Channel 4 4 Channel 4 3 y Channel Channel 3 5 x Primary User Secondary User Access Poin Example cogniive nework showing primary and secondary users perhaps firs applied o wireless neworks in he landmark paper [0, where Lyapunov drif is used o develop a join opimal rouing and scheduling algorihm. This mehod has since been exended o rea problems of join sabiliy and uiliy opimizaion in general sochasic neworks in [ [4 and wireless mesh neworks in [5. The analysis presened in his paper applies he sochasic nework opimizaion framework of [4. The main conribuions of his work are described below. This paper: Develops hroughpu opimal conrol policies for cogniive neworks wih general inerference and mobiliy models. Inroduces he noion of collision queues ha are used o provide srong reliabiliy bounds in erms of he wors case number of collisions suffered by a primary user in any ime inerval. In paricular, he collision queue mehod here is adaped from he virual power queue echnique of [3. However, he collision queues developed here are designed o ensure reliabiliy consrains, raher han average power consrains. Differen from [3, his requires he inpus o he virual queues o be random collision variables ha can be evaluaed only afer packe ransmission has aken place. Develops easier o implemen consan facor approximaions o he opimal resource allocaion problem.

2 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL 008 II. NETWORK ODEL We consider a cogniive radio nework consising of primary users and N secondary users as shown in Fig.. Each primary user has a unique licensed channel and hese are orhogonal in frequency and/or space. Thus, he primary users can send daa over heir own licensed channels o heir respecive access poins simulaneously. The secondary users do no have any such channels and opporunisically ry o send heir daa o he access poins by uilizing idle primary channels. Such opporuniies are called specrum holes. We consider a ime-sloed model. The primary users are assumed o be saic. However, he secondary users could be mobile so ha he se of channels hey can access can change over ime. In a imeslo, a secondary user can access a subse of he primary channels poenially depending on is curren locaion. This informaion is concisely represened by an N binary channel accessibiliy marix H() = h nm () N where: if sec. user n can access channel m in slo h nm () = 0 else For example, he channel accessibiliy marix for he example nework in Fig. is given by: H() = We assume ha he mobiliy process for he secondary users is such ha he resuling H() process is arkovian and has a well defined seady sae disribuion. Le S() = (S (), S (),..., S ()) represen he curren primary user occupancy sae of he channels. Here, S i () 0, (for i,,..., ) wih he inerpreaion ha S i () = 0 if channel i is occupied by primary user i in imeslo and S i () = if primary user i is idle in imeslo. We assume ha exacly packe can be ransmied over any channel in a imeslo. A secondary user can aemp ransmission over a mos channel subjec o he consrains in H(). This ransmission is successful only when he channel is no being used by is primary user or any oher secondary user. If a secondary user ransmis on a channel which is busy, here is a collision and boh packes are los. To capure he inerference ha a secondary user ransmission may cause on oher channels, for all n,,..., N, m,,...,, we define I nm as he se of channels ha secondary user n inerferes wih when i uses channel m. We include m in he se I nm. We furher define he following indicaor variables (o be used laer): Inm k if k Inm k,,..., = 0 else Clearly, Inm m = for all m, n. Under his inerference model, he following wo condiions are necessary for a We assume ha muli-user deecion/inerference cancelaion is no available so ha if he secondary user aemps o ransmi is own daa when some oher user is also ransmiing, here is enough inerference a he access poin and no daa is successfully received. ransmission by secondary user n on channel m in slo o be successful: ) S m () = ) For all oher secondary users i ransmiing on a channel j,,...,, we have m / I ij (where i,,..., N \ n) This inerference model is general enough o capure scenarios in which he channels may no be orhogonal wih respec o he secondary user ransmissions alhough hey are orhogonal for he primary user ransmissions. In mos pracical siuaions, he cardinaliy of he inerference ses I nm would be small. An imporan special case is when he channels are indeed orhogonal for all secondary user ransmissions, so ha I nm = m n. We assume ha he ses I nm are fixed. In a more general model, hese ses could also change over ime (possibly depending on he secondary user locaion). Our formulaion can be exended o rea hose cases. We assume ha he primary user channel occupancy sae process S() evolves according o a finie sae ergodic arkov chain on he space 0,. I is furher assumed o be independen of he secondary user mobiliy process H(). The channel sae informaion available o he secondary users is described by a probabiliy vecor P() = (P (), P (),..., P ()) where P i () is he probabiliy ha primary user i is idle in imeslo and is defined as P m () = E S m () S( ) m,,...,. We assume ha his informaion is obained eiher by sensing he channels, or hrough a knowledge of he raffic saisics of he primary users, or a combinaion of hese. This models siuaions where he exac channel sae may no be available o he secondary users (e.g. due o limiaions in carrier sensing). These probabiliies capure he inheren sensing measuremen errors associaed wih any primary ransmission deecion algorihm. Inuiively, he closer P() is o S(), he smaller he chances of collisions. Each primary user m receives exogenous daa a a rae ν m packe/slo and can olerae a maximum ime average rae of collisions given by ρ m, where ρ m is a known consan. A. Queueing Dynamics and Conrol Decisions Each secondary user n,,..., N receives daa according o an i.i.d arrival process A n () ha has rae λ n packes/slo. We assume ha he maximum number of arrivals o any secondary user n is upper bounded by a consan value A max every imeslo. This daa arrives a he ranspor layer and flow conrol decisions on how many packes o admi o he nework layer are aken by each secondary user. We assume ha here are no ranspor layer buffers and add/drop decisions are aken immediaely. Le U n () be he backlog in he nework layer queue of secondary user n a he beginning of imeslo. Le R n () be he number of new packes admied ino his queue in slo. Define µ nm () as he number of aemped packe ransmissions when a conrol acion allocaes channel m o In addiion, predicion based echniques could be also be used o ge his informaion.

3 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL secondary user n in slo. Noe ha in his model µ nm () 0, m, n. Also, here is a successful ransmission on channel m only when he necessary condiions specified earlier are me. Then he queueing dynamics of he secondary user n is described by: U n ( + ) = max[u n () µ nm ()S m (), 0 + R n () where () µ nm () 0, m, n () µ nm () h nm () m, n (3) 0 µ nm () n (4) µ nm () = j= i= i n N Iij m µ ij () = 0 m, n (5) 0 R n () A n () (6) Here, inequaliy (3) represens he consrain imposed by he channel accessibiliy marix H(). Inequaliy (4) represens he consrain ha a secondary user can be allocaed a mos channel. (5) represens he second necessary condiion for successful ransmission expressed in erms of he Inm k variables. In he special case of orhogonal channels, his simplifies o he consrain ha a channel can be allocaed o a mos secondary user, i.e., N 0 µ nm () m (7) B. Discussion of Nework odel The above nework model considers access poin based neworks wih saic (or locally mobile) licensed and fully mobile unlicensed users. Examples of real neworks ha can be modeled like his include Wi-Fi, cellular and mesh neworks wih boh licensed and unlicensed users. In such neworks, he licensed users may no schedule heir ransmissions and hus send a any ime hey desire. The unlicensed users mus make an effor o opporunisically use he specrum holes wihou inerfering oo much wih he licensed users, and hence need sophisicaed scheduling mechanisms. A axonomy of differen approaches o specrum sharing in cogniive neworks is provided in [3. The nework model used in his paper falls ino he specrum overlay approach o specrum sharing. III. AXIU THROUGHPUT OBJECTIVE Le r n denoe he ime average rae of admied daa for secondary user n, i.e., r n = lim R n (τ) Le r = (r,..., r N ) denoe he vecor of hese ime average raes. We define he following collision variables for each primary user m,..., : if here was a collision wih he primary user C m () = in channel m in slo 0 else Le c m denoe he ime average rae of collision for primary user m, i.e., c m = lim C m (τ) Le θ,..., θ N be a collecion of posiive weighs. Then he conrol objecive is o design a flow conrol and scheduling policy ha yields ime average rae r ha solves he following opimizaion problem: aximize: N θ n r n Subjec o: 0 r n λ n n,..., N c m ρ m m,..., r Λ (8) Here, Λ represens he nework capaciy region for he nework model as described above. I is defined as he se of all inpu rae vecors λ = (λ,..., λ N ) of he secondary users for which a scheduling sraegy exiss ha can suppor λ (wihou flow conrol) subjec o he consrains imposed by he nework, including he c m ρ m consrain. The noion of nework capaciy for general neworks wih ime varying channels and energy consrains is formalized in [, [3, [4 where i is shown o be a funcion of he seady sae nework opology disribuion, channel probabiliies, and ime average ransmission raes. Le r = (r,..., rn ) denoe he opimal soluion o he opimizaion problem defined above. In principle, i can be solved if all sysem parameers, including he se Λ are known in advance. However, in pracice, his region may no be known o he nework conroller (e.g., because he mobiliy paerns of he secondary users are unknown) and he above maximizaion problem mus be done for inpu raes eiher inside or ouside of he capaciy region. Even if all sysem parameers are known, he opimal soluion may be difficul o implemen as i may require cenralized coordinaion among all users. We nex presen an online conrol algorihm ha overcomes all of hese challenges. IV. OPTIAL CONTROL ALGORITH We now presen he Cogniive Nework Conrol Algorihm (CNC), a cross-layer conrol sraegy ha can be shown o achieve he opimal soluion r o he nework opimizaion problem presened earlier. I operaes wihou knowledge of wheher he inpu rae is wihin or ouside of he capaciy region Λ. Furher, i provides deerminisic wors case bounds on he maximum backlog a all imes and he maximum number of collisions wih a primary user in a given ime inerval. These are much sronger han probabilisic performance guaranees.

4 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL Finally, i offers a conrol parameer V 0 ha enables an explici rade-off beween he average hroughpu uiliy and delay. This algorihm is similar in spiri o he backpressure algorihms proposed in [3, [5 for problems of energy opimal neworking in wireless ad-hoc and mesh neworks. The algorihm is decoupled ino wo separae componens. The firs componen performs opimal flow conrol a he ranspor layers and is implemened independenly a each secondary user. The second componen deermines a nework wide resource allocaion every slo and needs o be solved collecively by he secondary users. In addiion o he acual queue backlog U n (), his algorihm uses a se of collision queues X m () for each channel m. These queues are virual in ha hey are mainained purely in sofware. These are used o rack he amoun by which he number of collisions suffered by a primary user m exceeds is ime average collision consrain rae ρ m. We assume ha he secondary users are aware of he X m () value for each channel m ha hey can access a ime. We define he collision queue X m () for channel m as follows: X m ( + ) = max[x m () ρ m, 0 + C m () (9) where C m () is he collision variable for channel m as defined in he previous secion. The above equaion represens he queueing dynamics of a single server sysem wih inpu process C m () and consan service rae ρ m. This sysem is rae sable if and only if he service rae exceeds he inpu rae, i.e., c m = lim C m (τ) ρ m This is precisely he collision consrain in he uiliy opimizaion problem saed earlier. Thus, if our policy sabilizes all collision queues as defined above, he maximum average rae of collisions will mee he required consrain. This echnique of urning ime average consrains ino queueing sabiliy problems was inroduced in [3 where i was used for saisfying average power consrains. A. Cogniive Nework Conrol Algorihm (CNC) Le he flow conrol and resource allocaion decision under he CNC algorihm be Rn CNC () and µ CNC nm () respecively. These are deermined according o he following algorihm, which uses a consan parameer V 0 o affec a performance/delay radeoff: Flow Conrol: A each secondary user n, choose he number of packes o admi Rn CNC () as he soluion o he following problem: inimize: R n ()[U n () V θ n Subjec o: 0 R n () A n () (0) This problem has a simple hreshold-based soluion. In paricular, if he curren queue backlog U n () > V θ n, hen Rn CNC () = 0 and no new packes are admied. Else, if U n () V θ n, hen Rn CNC () = A n () and all new packes are admied. Noe ha his can be solved separaely a each user and does no require knowledge of θ n weighs of oher users. Resource Allocaion: Choose a resource allocaion µ CNC nm () ha solves he following problem: ax: [ µ nm () U n ()P m () X k ()( P k ())Inm k k= Subjec o: consrains (), (3), (4), (5) () Afer observing he oucome of his allocaion a he end of he slo, he virual queues are updaed as in (9) based on he feedback received abou a collision wih a primary user or a successful ransmission. Noe ha only collisions wih a primary user affec (9), collisions beween secondary users do no affec he virual collision queues. The above problem is a generalized aximum Weigh ach ( problem where he weigh for a pair (n, m) is given by U n ()P m () ) k= X k()( P k ())Inm k. This is he difference beween he curren queue backlog U n () weighed by he probabiliy ha primary user m is idle and he weighed sum of all collision queue backlogs for he channels ha user n inerferes wih if i uses channel m. The weigh for a collision queue is he probabiliy ha he corresponding primary user will ransmi. Noe ha if his difference is nonposiive, hen he link (n, m) can be removed from he decision opions, simplifying scheduling. This problem is hard o solve in general, hough consan facor approximaions exis ha are easier o implemen. We discuss hese in Sec.VI. For he case when all channels are orhogonal from he poin of view of secondary users (which means a secondary user ransmission on a channel does no cause inerference o oher channels), I nm = m so ha Inm m =, Inm k = 0 k m. Then he above maximizaion simplifies o he following problem: aximize: [ µ nm () U n ()P m () X m ()( P m ()) Subjec o: consrains (), (3), (4), (7) () The above maximizaion requires solving he aximum Weigh ach (W) problem on an N biparie graph of N secondary users and channels. This problem can be solved in polynomial ime, hough his may require cenralized conrol. We discuss simpler consan facor approximaions in Sec.VI. Also, we consider a cell pariioned nework in he simulaions of Sec.VII for which a full maximum weigh mach can be implemened in a disribued manner. B. Performance Analysis We now characerize he performance of he CN C algorihm. This holds for general secondary user mobiliy processes ha are described by finie sae ergodic arkov Chains. Theorem : (Algorihm Performance) Assume ha all queues are iniialized o 0. Suppose all arrivals A n () are upper bounded so ha A n () A max for all n,. Also suppose he H() and P() processes are arkovian and have a well defined seady sae disribuion. Then, implemening

5 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL he CNC algorihm every slo for any fixed conrol parameer V 0 sabilizes all real and virual queues (hereby saisfying he maximum ime average collision consrains) and yields he following performance bounds: ) The wors case queue backlog for all secondary users n is upper bounded by a finie consan U max for all : U n () U max = V θ max + A max (3) where θ max = max n,...,n θ n ) For all m, such ha P m (), le ɛ > 0 be such ha P m () ɛ. 3 Then, he wors case collision queue backlog for all channels m is upper bounded by a finie consan X max : ( ɛ) X m () X max= U max + (4) ɛ Furher, he wors case number of collisions suffered by any primary user m is no more han ρ m T + X max over any finie inerval of size T. 3) The ime average hroughpu uiliy achieved by he CNC algorihm is wihin B/V of he opimal value: lim inf N θ n E R n (τ) N θ n r n B V (5) where B = B + C U + C X + N log V + log V and where B, C U, C X are consans (defined precisely in (8), (8), (9)). The consans C U and C X are deermined by he sochasics of he mobiliy and channel sae probabiliy processes and i is shown in Appendix A ha hese are O(log V ) when hese processes evolve according o a finie sae ergodic arkov model. Therefore, by par (3) of he heorem, he achieved average hroughpu uiliy is wihin O(log V/V ) of he opimal value. This can be pushed arbirarily close o he opimal value by increasing he conrol parameer V. However, his increases he maximum queue backlog bound U max linearly in V, leading o a uiliy-delay radeoff. The above bounds are quie srong. In paricular, he maximum collisions bound in par () gives deerminisic performance guaranees ha hold for any inerval size. This is quie useful in he conex of cogniive neworks since i implies ha he licensed users are guaraneed o suffer a mos hese many collisions. Probabilisic guaranees (e.g. [8) do no provide such bounds. We nex prove he firs wo pars of Theorem. Proof of par (3) uses he echnique of Sochasic Lyapunov Opimizaion and is provided in he nex secion. Proof of par (): Suppose ha U n () U max for all n,..., N for some ime. This is rue for = 0 as all queues are iniialized o 0. We show ha he same holds for ime +. We have cases. If U n () U max A max, hen from (), we have U n (+) U max (because R n () A max for all ). Else, if U n () > U max A max, hen U n () > V θ n + A max A max = V θ n. Then, he flow conrol par of he algorihm chooses R n () = 0, so ha by (): U n ( + ) U n () U max 3 Such an ɛ exiss for any finie sae ergodic arkov Chain. This proves (3). Proof of par (): Suppose ha X m () X max for all m,..., for some ime. This is rue for = 0 as all queues are iniialized o 0. We show ha he same holds for ime +. Firs suppose P m () =. Then, by definiion, here is no collision wih he primary user in channel m in slo so ha C m () = 0. Then, from (9), we have X m ( + ) X max. Nex, suppose P m () <. We again have cases. If X m () X max, hen from (9), we have X m ( + ) X max (because C m () ( ɛ) for all ). Else, if X m () > X max = U max ɛ, hen X m ()ɛ > U max ( ɛ). This implies X m ()( P m ()) > U max P m () U n ()P m () for all n,..., N. Then, since Inm m =, he resource allocaion par of he algorihm chooses µ nm () = 0 for all n. This would yield C m () = 0 (since no collision akes place wih primary user m), so ha by (9): This proves (4). X m ( + ) X m () X max V. STOCHASTIC LYAPUNOV OPTIIZATION Le Q() = (Q (),..., Q K ()) be a vecor process of queue lenghs for a discree ime sochasic queueing nework wih K queues (possibly including some virual queues like he collision queues defined in he previous subsecion). Le L(Q) be any non-negaive scalar valued funcion of he queue lenghs, called a Lyapunov funcion. Define he Lyapunov drif () as follows: () =E L(Q( + )) L(Q()) Suppose he nework accumulaes rewards every imeslo (where rewards migh correspond o uiliy measures of conrol acions). Assume rewards are real valued and bounded, and le he sochasic process f() represen he reward earned during slo. Le f represen he arge reward. The following resul (a varian of relaed resuls from [3, [4) specifies a drif condiion which ensures ha he ime average of he reward process f() is close o meeing or exceeding f. Theorem : (Delayed Lyapunov Opimizaion wih Rewards) Suppose here exis finie consans V > 0, B > 0, d > 0, and a non-negaive funcion L(Q) such ha E L(Q(d)) < and for every imeslo > d, he Lyapunov drif saisfies: hen we have: () V E f() B V f (6) lim inf E f(τ) f B V Proof : Inequaliy (6) holds for all > d. Summing boh sides over τ d,..., yields: E L(Q()) E L(Q(d)) B( d) V ( d)f + V E f(τ) τ=d

6 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL Rearranging erms, dividing by, and using non-negaiviy of L(Q) yields: ( d)f ( d)b V E L(Q(d) V E f(τ) The resul follows by aking limi as. We now use Theorem o prove par (3) of Theorem. This is done by comparing he Lyapunov drif of he CNC algorihm wih ha of a saionary randomized algorihm ha makes conrol decisions every slo purely as a funcion of he curren channel sae informaion P() and H(). We firs obain an expression for he Lyapunov drif under any conrol policy for our cogniive nework model. A. Lyapunov Drif Le Q() = (U (),..., U N (), X (),..., X ()) represen he collecion of all real and virual queue backlogs in he cogniive nework. We define he following Lyapunov funcion: L(Q()) = [ N U n() + Xm() Using queueing dynamics () and (9), he Lyapunov drif () under any conrol policy (including CN C) can be compued as follows: () B E where N ( U n () E X m ()(ρ m C m ()) ) µ nm ()S m () R n () B = N(A max + ) + ρ m + (7) (8) The collision variable C m () can be expressed in erms of he conrol decisions µ ij () and channel sae S() as follows: C m () = N i= j= µ ij ()Iij m [Ui()>0( S m ()) (9) where [Ui()>0 is an indicaor variable of non-zero queue backlog in secondary user i. This follows by observing ha a collision wih he primary user occurs in channel m if he primary user is busy (i.e. S m () = 0) and if µ ij () = for some secondary user i wih non-zero backlog using channel j ha inerferes wih channel m. We will find i useful o define he following relaed variable: Ĉ m () = N i= j= µ ij ()Iij m ( S m ()) (0) For a given conrol parameer V 0, we subrac he N reward meric V E θ nr n () from boh sides of he drif inequaliy (7) and use he fac ha Ĉ m () C m () o ge he following: N () V E θ n R n () B N ( E U n () ) µ nm ()S m () R n () N E X m ()(ρ m Ĉm()) V E θ n R n () B. Opimal Saionary, Randomized Policy () We now describe he saionary, randomized policy ha chooses conrol acions only as a funcion of P() and H() every slo. We have he following fac: Opimal Saionary, Randomized Policy: For any rae vecor (λ,..., λ N ) (inside or ouside of he nework capaciy region Λ), here exiss a saionary randomized scheduling policy n nm ha chooses feasible allocaions R (), µ () for all n,..., N, m,..., every slo as a funcion of he channel sae informaion P() and H() and yields he following seady sae values: E Rn () = rn () µ n = lim E µ nm (τ)s m (τ) rn (3) ĉ m = lim Ĉ E m () ρ m (4) where r = (r,..., rn ) is he soluion o (8). The above fac can be proven using echniques similar o he ones used in [, [3 for showing he exisence of capaciy achieving saionary, randomized policies ha make conrol decisions independen of queue backlog and is omied for breviy. We now prove an imporan propery of he CN C algorihm. Claim: Given a paricular backlog Q(), he CNC algorihm minimizes he righ side of inequaliy () over all alernae feasible decisions ha can be made on slo, including he saionary, randomized policy. Proof : By changing he order of summaions and using (0), he righ side of () can be expressed in a more convenien form: N B ρ m E X m () + E R n ()(U n () V θ n ) E [ µ nm () U n ()S m () X k ()( S k ())Inm k k= (5) Noe ha E S m () S( ) = P m () m. By wriing he las wo erms on he righ hand side as an ieraed expecaion by condiioning on he queue backlog and S( ), i can be seen ha CNC chooses conrol decisions (0) and () ha minimize hese erms for every possible value of he backlog and S( ), so ha he acual expecaion is also minimized.

7 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL Using his fac, we have: N CNC () B N ( E U n () µ E X m ()(ρ m N V E θ n R n nm () ()S m () R AT ĈST m ()) n ) () (6) x y Primary User Secondary User In Appendix A, we show ha his can be expressed as: N CNC () B N V θ n rn This is in a form ha fis (6). Thus, applying Theorem proves (5). VI. DISTRIBUTED IPLEENTATION Here we discuss consan facor approximaions o he resource allocaion problem () ha are easier o implemen in a disribued nework. We focus on he orhogonal channel case in which a secondary user ransmission on a channel does no cause inerference o oher channels. As noed earlier, in his case, he resource allocaion problem () reduces o a aximum Weigh ach (W) problem on an N biparie graph beween N secondary users and channels. An edge exiss beween nodes n and m of his graph if h nm () =, i.e., if secondary user n can access channel m in slo. The weigh of his edge is given by (U n ()P m () X m ()( P m ())). While he W problem can be solved in polynomial ime in a cenralized way, here we are ineresed in simpler implemenaions. In paricular, we use he idea of Greedy aximal ach Scheduling. Similar maximal scheduling algorihms for consan facor guaranees have been invesigaed in recen works [6 [8, [4. A maximal mach is defined as any se of edges (m, n) ha do no inerfere wih each oher such ha adding any new edge o his se necessarily violaes a maching consrain. A Greedy aximal ach can be achieved as follows: Firs selec he edge (m, n) wih he larges posiive weigh and label i acive. Then selec he edge wih he second larges posiive weigh (breaking ies arbirarily) ha does no conflic wih an acive edge and label i acive. Coninue in he same way, unil no more edges can be added. I is no difficul o see ha his final se of edges labeled acive has he desired maximal propery. A Greedy aximal ach can be compued wih much less overhead as compared o he aximum Weigh ach. I can be shown ha using maximal maches insead of he maximum weigh mach every slo can sill suppor any rae wihin Λ. In paricular, resource allocaions µg mn () chosen according o greedy maximal maches have he following Fig.. Example cell-pariioned nework used in simulaion propery: [ µ G nm () U n ()P m () X m ()( P m ()) [ µ nm() U n ()P m () X m ()( P m ()) where µ nm() is he opimal soluion o (). Using his, we have he following resul: Theorem 3: (Performance Bound for Orhogonal Channels wih Greedy aximal ach Scheduling) The ime average hroughpu uiliy achieved by he CN C algorihm wih Greedy aximal ach Scheduling is wihin B/V of N θ nr n: lim inf N θ n E R n (τ) VII. SIULATIONS N θ n r n B V We simulae he CNC algorihm on an example cogniive nework consising of 9 primary users and 8 secondary users as shown in Fig.. We consider a simple cell-pariioned nework wih one primary user per cell. The primary users are saic and each has is own licensed channel ha can be used by hem simulaneously. A secondary user can only aemp o ransmi on he channel associaed wih he primary user in is curren cell. The secondary users move from one cell o anoher according o a arkovian random walk. In paricular, a he end of every slo, a secondary user decides o say in is curren cell wih probabiliy β, else decides o move o an adjacen cell wih probabiliy β/4 (where β = 0.5 for he simulaions). If here is no feasible adjacen cell (e.g, if he previous cell is a corner cell and he new chosen cell does no exis), hen he user remains in he curren cell. I can be shown ha he resuling H() process forms an irreducible, aperiodic arkov Chain where he seady sae locaion disribuion is uniform over all cells. The channel sae process S m () for each primary user m is governed by an ON/OFF arkov chain wih symmeric

8 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL Toal Average Congesion (log scale) V = V = V = 5 V = 0 V = 00 No Flow Conrol Throughpu (packes/slo) V = V = V = 5 V = 0 V = Inpu Rae (packes/slo) Inpu Rae (packes/slo) Fig. 3. Toal average congesion vs. inpu rae for differen values of V Fig. 4. Achieved hroughpu vs. inpu rae for differen values of V ransiion probabiliies beween he ON and OFF saes given by 0. m. The maximum collision rae ρ m = 0. m. New packes arrive a he secondary users according o independen Bernoulli processes, so ha a single packe arrives i.i.d. wih probabiliy λ every slo. We assume here are no ranspor layer sorage buffers, so ha all packes ha are no immediaely admied o he nework layer are necessarily dropped. Flow conrol is performed according o (0) (wih θ n = n) and resource allocaion decisions are made every slo according o (). In his paricular cell-pariioned nework srucure wih one channel per cell, he maximum weigh mach is same as a greedy maximal mach so ha he opimal resource allocaion decisions can be made in each cell in a disribued way. In Fig. 3 we plo he average oal occupancy (summing all packes in he queues of he secondary users) versus he inpu rae λ. Each daa poin represens a simulaion over 500, 000 imeslos, and he differen curves correspond o values of he flow conrol parameer V,, 5, 0, 00, and he case V = (no flow conrol) is also shown. In his case, he average oal occupancy increases wihou bound as he inpu rae approaches nework capaciy. The verical asympoe which appears roughly a λ = 0.85 packes/slo corresponds o his value. Fig. 4 illusraes he achieved hroughpu versus he raw daa inpu rae λ for various V parameers. The achieved hroughpu is almos idenical o he inpu rae λ for small values of λ, and he hroughpu sauraes a a value ha depends on V, being very close o he 0.85 capaciy level when V is large. Also, i was found ha all real and virual queue backlogs are always bounded by he maximum values given in (3) and (4). In paricular, ɛ = 0. for his nework, so ha X m () X max = 4U max + = 4V + 5. Finally, he maximum average rae of collisions was very close o he arge ρ m = 0.. APPENDIX A DELAYED LYAPUNOV DRIFT Here, we use delayed queue backlogs o express he Lyapunov drif of he CN C algorihm in a form ha fis (6). Throughou, he backlogs U n () and X m () represen he values associaed wih he CN C algorihm. Recall ha Rn () and µ nm () denoe he resource allocaion decisions under he saionary, randomized policy inro- duced in Sec. V. We use he following sample pah inequaliies. Specifically, for all > d, we have for each secondary user queue U n () and for each collision queue X m (): U n ( d) + da max U n () U n ( d) d X m ( d) + d X m () X m ( d) dρ m These follow by noing ha he queue backlog a ime canno be smaller han he queue backlog a ime ( d) minus he maximum possible deparures in duraion ( d, d). Similarly, i canno be larger han he queue backlog a ime ( d) plus he maximum possible arrivals in duraion ( d, d). Using hese in (6) and using E Rn () = rn (from ()), we ge: N CNC () B + C U + C X N ( E U n ( d) µ E X m ( d)(ρ m nm where C U and C X are given by: ()S m () R AT ĈST m ()) V n ) () N θ n rn (7) C U = dn + da maxn (8) C X= d ( + ρ m) (9) Using ieraed expecaions, we have he following: N E U n ( d) µ ()S m () = nm N E U n ( d) E E X m ( E AT d)ĉst m () µ nm ()S m () T ( d) = (30) X m ( d) E Ĉ m () T ( d) (3)

9 PROC. IEEE INFOCO, PHOENIX, AZ, APRIL where T ( d) = (H( d), S( d), Q( d)) represens he composie sysem sae a ime ( d) and includes he opology sae and queue backlogs. By he arkovian propery of he H(), S() (and herefore P()) processes, any funcionals of hese saes converge exponenially fas o heir seady sae values (his is formalized in Appendix B). Since he policy makes conrol decisions only as a funcion of P() and H(), he resuling allocaions are funcionals of hese arkovian processes. Thus, here exis posiive consans α, α and 0 < γ, γ < such ha: E µ nm ()S m () T ( d) µ n α γ d ĈST AT E m () T ( d) ĉm + α γ d where µ n, ĉm are he seady sae values as defined in (3), (4). Using hese, he above can be wrien as: E µ nm ()S m () T ( d) rn α γ d (3) ĈST AT E m () T ( d) ρ m + α γ d (33) Thus, using (3), (33) in (30), (3), inequaliy (7) can be expressed as: N CNC () B + C U + C X N + E U n ( d)α γ d + E X m ( d)α γ d V N θ n rn B + C U + C X + NU max α γ d + X max α γ d V N θ n rn (using (3), (4)) Define d = log(αumax) log(/γ ), d = log(αxmax) log(/γ ). Then choosing d = max(d, d ), we have: N CNC () B + C U + C X + N log V + log V V N θ n rn (34) Since U max and X max are O(V ), we have d O(log V ). APPENDIX B CONVERGENCE OF ARKOV CHAINS Le Z() be a finie sae, discree ime ergodic arkov chain. Le S denoe is sae space and le π i i S be he seady sae probabiliy disribuion. Then, for all inegers d 0, here exis consans α, γ such ha: P rz() = j Z( d) = i π j αγ d (35) where α 0 and 0 < γ <. This implies ha he arkov chain converges o is seady sae probabiliy disribuion exponenially fas (see [9). Le f(z()) be a posiive random funcion of Z() (negaive case can be reaed similarly). Define f = j S π jm j where m j = E f(z()) Z() = j. Then: E f(z()) Z( d) = i = j S E f(z()) Z() = j P rz() = j Z( d) = i j S m j (π j + αγ d ) (using (35)) f + sm max αγ d where m max = max j S m j and s = cards. This shows ha funcionals of he saes of a finie sae ergodic arkov chain converge o heir seady sae value exponenially fas. REFERENCES [ J. iola. Sofware radios: wireless archiecure for he s cenury. John Wiley & Sons Inc, 000. [ I. F. Akyildiz, W-Y Lee,. C. Vuran and S. ohany. NeX generaion/dynamic specrum access/cogniive radio wireless neworks: a survey. Compuer Neworks Journal (Elsevier), vol. 50, pp. 7-59, Sep [3 Q. Zhao and B.. Sadler. A survey of dynamic specrum access: signal processing, neworking, and regulaory policy. IEEE Signal Processing agazine, vol. 4, no. 3, pp , ay, 007. [4 C. Peng, H. Zheng, and B.Y. Zhao. Uilizaion and fairness in specrum assignmen for opporunisic specrum access. AC/Springer ONET, vol., issue 4, Aug [5 L. Cao, H. Zheng. Disribued specrum allocaion via local bargaining. Proc. IEEE SECON, Sep [6 W. Wang, X. Liu and H. Xiao. Exploring opporunisic specrum availabiliy in wireless communicaion neworks. Proc. of IEEE VTC, Sep [7 Y.T. Hou, Y. Shi, and H.D. Sherali. Opimal specrum sharing for mulihop sofware defined radio neworks. Proc. IEEE INFOCO, ay 007. [8 Y. Chen, Q. Zhao, and A. Swami. Join design and separaion principle for opporunisic specrum access. under submission o IEEE Transacions on Informaion Theory [9 Q. Zhao, L. Tong, A. Swami, and Y. Chen. Decenralized cogniive AC for opporunisic specrum access in ad hoc neworks: A PODP framework. IEEE Journal on Seleced Areas in Communicaions (JSAC): Special Issue on Adapive, Specrum Agile and Cogniive Wireles Neworks, vol. 5, no. 3, pp , April, 007. [0 L. Tassiulas and A. Ephremides. Sabiliy properies of consrained queueing sysems and scheduling policies for maximum hroughpu in mulihop radio neworks. IEEE Transacaions on Auomaic Conrol, vol. 37, no., pp , Dec. 99. [. J. Neely, E. odiano, and C. E Rohrs. Dynamic power allocaion and rouing for ime varying wireless neworks. IEEE Journal on Seleced Areas in Communicaions, vol. 3, no., pp , Jan [. J. Neely, E. odiano, and C. Li. Fairness and opimal sochasic conrol for heerogeneous neworks. Proc. IEEE INFOCO, arch 005. [3. J. Neely. Energy opimal conrol for ime varying wireless neworks. IEEE Transacions on Informaion Theory, vol. 5, no. 7, July 006. [4 L. Georgiadis,. J. Neely, L. Tassiulas. Resource allocaion and crosslayer conrol in wireless neworks. Foundaions and Trends in Neworking, Vol., no., pp. -49, 006. [5. J. Neely and R. Urgaonkar. Cross layer adapive conrol for wireless mesh neworks. Ad Hoc Neworks (Elsevier), vol. 5, no. 6, pp , Aug [6 X. Lin and N. B. Shroff. The impac of imperfec scheduling on crosslayer rae conrol in wireless neworks. Proc. IEEE INFOCO, arch 005. [7 P. Chaporkar, K. Kar, and S. Sarkar. Throughpu guaranees hrough maximal scheduling in wireless neworks. Proc. of 43rd Alleron Conference, Sep [8 X. Wu and R. Srikan. Bounds on he capaciy region of mulihop wireless neworks under disribued greedy scheduling. Proc. IEEE INFOCO, April 006. [9 S. Ross. Sochasic Processes. John Wiley & Sons, Inc., New York, 996.