Spectrum Sharing For Delay-Sensitive Applications With Continuing QoS Guarantees

Transcription

1 Spectrum Sharng For Delay-Senstve Applcatons Wth Contnung QoS Guarantees Yuanzhang Xao, Kartk Ahuja, and Mhaela van der Schaar Department of Electrcal Engneerng, UCLA Emals: Abstract We study a wreless network n whch multple users stream delay-senstve applcatons such as vdeo conferencng and vdeo streamng. Exstng spectrum sharng polces, whch determne when users access the spectrum and at what power levels, are ether constant (.e. users transmt smultaneously, at constant power levels) or weghted round-robn tme-dvson multple access (TDMA) (.e. users access the spectrum n turn, one at a tme). Due to mult-user nterference, constant polces have low spectrum effcency. We show that round-robn polces are neffcent for delay-senstve applcatons because the varous postons (.e. transmsson opportuntes) n a cycle are not created equal: earler transmsson opportuntes are more desrable snce they enable users to transmt wth lower delays. Specfcally, we show that (weghted) round-robn TDMA polces cannot smultaneously acheve hgh network performance and low transmsson delays. Ths problem s exacerbated when the number of users s large. We propose a novel framework for desgnng optmal TDMA spectrum sharng polces for delay-senstve applcatons, whch can guarantee ther contnung QoS (CQoS),.e. the desred throughput (and the resultng transmsson delay) startng from every moment n tme s guaranteed for each user. We prove that the fulfllment of CQoS guarantees provdes strct upper bounds on the transmsson delays ncurred by the users. We construct the optmal TDMA polcy that mzes the desred network performance (e.g. -mn farness or socal welfare) subject to the users CQoS guarantees. The key feature of the proposed polcy s that t s not cyclc as n (weghted) round-robn polces. Instead, t adaptvely determnes whch user should transmt next, based on the users remanng amounts of transmsson opportuntes needed to acheve the desred performance. We also propose a low-complexty algorthm, whch s run by each user n a dstrbuted manner, to construct the optmal polcy. Smulaton results demonstrate that our proposed polcy sgnfcantly outperforms the optmal constant polcy and round-robn polces by up to 6 db and 4 db n peak sgnal-tonose rato (PSNR) for vdeo streamng. I. INTRODUCTION A varety of bandwdth-ntensve and delay-senstve applcatons, such as multmeda streamng, gamng, and teleconferencng, are ncreasngly deployed over wreless networks. Such applcatons mpose huge challenges when deployed over wreless networks, n whch the users share the spectrum and cause nterference to each other. Hence, t s crucal to desgn spectrum sharng polces that provde delay-senstve users wth both hgh rates and low delays. The spectrum sharng polces studed n earler works [1] [5] requre the users to transmt at constant power levels all the tme 1. We call them constant (spectrum sharng) polces. 1 Although some spectrum sharng polces go through a transent perod of adjustng the power levels before the convergence to the optmal power levels, the users mantan constant power levels after the convergence. Constant polces are neffcent n many spectrum sharng scenaros wth strong mult-user nterference. Under strong mult-user nterference, ncreasng one user s power level sgnfcantly degrades the other users throughput, whch results n low spectrum effcency. The optmal way to share the spectrum should take nto account the users cross nterference. Ideally, we should assgn the users to several (perhaps overlappng) subsets based on ther cross nterference, such that the users n the same group have low cross nterference. Then the dfferent subsets of users can transmt n turn. However, the optmal assgnment of subsets requres knowledge about the cross channel gans among the users, whch s hard to get n a decentralzed spectrum sharng scenaro. In addton, snce the number of subsets grows exponentally wth the number of users, even wth the complete knowledge of cross channel gans, the computatonal complexty of searchng for the optmal parttonng s prohbtvely hgh. One way to manage nterference whle lmtng the nformatonal and computatonal costs s to smply let one user access the spectrum at one tme, as n e.g e MAC wreless networks [6]. Such polces are commonly known as tme-dvson multple access (TDMA) polces. Our focus n ths paper s on desgnng optmal TDMA polces for delay-senstve users. All the exstng TDMA polces are round-robn polces or ther varants (e.g. weghted round-robn polces) [6] [9]. In round-robn polces, tme slots are dvded nto cycles of a fxed predetermned length, and each user transmts n fxed predetermned postons wthn each cycle. The cyclc nature of round-robn polces smplfes the mplementaton, but mposes restrctons that render round-robn polces neffcent for delay-senstve applcatons. For delay-senstve applcaton, not all the transmsson opportuntes (.e. postons) n a cycle are created equal: the earler transmsson opportuntes (TXOPs) are more desrable because they result n hgher chances to delver packets on tme, pror ther delay deadlnes [6] [1]. To ensure that the user s rate and delay constrants are met, round-robn polces need a long cycle, and a careful sharng of TXOPs n a cycle. Frst, a long cycle s necessary. Suppose that the cycle length s the shortest possble, namely equal to the number of users (as n standard round-robn polces). Then the user allocated to the last TXOP suffers severely from delay. We can compensate ths user for ts delay by havng a longer cycle and allocatng some of the extra TXOPs to t. However, a long cycle results n an exponentally ncreasng (n the cycle length) number of possble polces to choose from. Second, a careful sharng of TXOPs s necessary (see Fg. 1 for an llustraton of the followng dscusson). Suppose that the cycle length s twce the number of users, and that each user gets two postons n a cycle. For farness, no user should get two advantageous (.e. earler) TXOPs. A

2 low delay, but unfar for user 4: far, but hgh delay for user 1: Fg. 1. Two smple round-robn schedules wth cycle length 8 for 4 users. The frst one has low delay of 4 for all 4 users, but unfar sharng of transmsson opportuntes (TXOPs) (.e. user 4 gets later TXOPs). The second one has a far sharng of TXOPs, but ncurs hgh mum delay of 7 for user 1. TABLE I. COMPARISON WITH RELATED WORKS. Spectrum CQoS Delay System or ndvdual effcency guarantee guarantee performance acheved Constant [1] [5] Low No Yes Nether Round-robn Ether for small # of users Hgh No Yes [6] [9] Nether for large # of users Proposed Hgh No Yes Both possble far sharng may ensure that the user who gets an earler TXOP s also allocated to a later TXOP (e.g. one user should get the frst and the last TXOPs n a cycle). However, such a far sharng s neffcent n terms of ndvdual performance: the user who gets the frst and the last TXOPs n a cycle wll experence hgh delay between consecutve transmssons. As we wll llustrate n our motvatng example (Secton IV) and by smulatons (Secton VII), round-robn polces cannot smultaneously acheve hgh system performance (e.g. mn farness) and fulfll the guarantees n terms of transmsson delays requred by the delay-senstve users. Ths becomes even more dffcult to do wthn reasonable computatonal complexty when the number of users s large. In ths paper, we propose a framework for desgnng optmal TDMA spectrum sharng polces for delay-senstve applcatons. We defne a novel qualty-of-servce (QoS) metrc, called contnung QoS (CQoS) guarantees. CQoS guarantees requre a user s average throughput startng from every pont n tme to be hgher than a threshold. CQoS guarantees are strcter requrements than conventonal QoS guarantees whch only guarantee the average throughput startng from the begnnng. We wll prove that fulfllng CQoS guarantees results n upper bounds on transmsson delays. We propose a systematc desgn methodology, whch constructs the optmal TDMA polcy that mzes the system performance (e.g. farness) subject to the users CQoS guarantees. The key feature of the proposed polcy s that t s not cyclc as n round-robn polces. Instead, t adaptvely determnes whch user should transmt accordng to the users remanng amounts of TXOPs needed to acheve the target throughput. We propose a low-complexty algorthm, whch can be run by each user n a dstrbuted manner, to construct the optmal polcy. Smulaton results show that our proposed polcy sgnfcantly outperforms the optmal constant polcy [1] [5] and round-robn polces n peak sgnal-to-nose rato (PSNR) for vdeo streamng, by up to 6 db and 4 db respectvely,. Fnally, we summarze the comparson of our work wth the exstng works n Table I. The rest of the paper s organzed as follows. In Secton II, we descrbe the system model. Then n Secton IV, we motvate our proposed polcy by showng the neffcency of roundrobn polces through a motvatng example. We formulate the polcy desgn problem n Secton V, and solve t n Secton VI. Smulaton results are presented n Secton VII. Fnally, Secton VIII concludes the paper. II. SYSTEM MODEL We consder a wreless network wth N users. The set of users s denoted by N {1, 2,...,N}. Each user has a transmtter and a recever. The channel gan from user s transmtter to user j s recever s g j. Each user chooses a power level p from a compact set ˆP. We assume that ˆP, namely user can choose not to transmt. We also assume that the users need to comply wth some nterference temperature constrants (ITCs) measured at K locatons n the network. Dependng on dfferent scenaros, the ITCs can be mposed by prmary users n a cogntve rado network or the base staton n a femtocell network. The channel gan from user s transmtter to the kth locaton s g k. Each user knows the channel gan {g k } K k=1 to each measurement locaton and the nterference temperature lmt {I k } K k=1 at each locaton. Hence, each user s set of admssble power levels s P = {p ˆP : g k p I k, k =1,...,K}. (1) For convenence, we also defne user s mum admssble power level as P p P p. Remark 1: Our system model s general enough to model many wreless communcaton networks. It can model wreless ad hoc networks where N users transmt n the unlcensed spectrum (e.g. the 2.4 GHz frequency band) wthout ITCs (K =). It can also model the uplnk (the recevers are colocated) and the downlnk (the transmtters are co-located) of a cellular network wth possble ITCs mposed by base statons n nearby cells. It can also model cogntve rado networks (or femtocell networks) wth N secondary users (or femtocells) sharng the spectrum wth K prmary users (or K =1base staton) mposng ITCs at ther recevers. We denote the jont power profle of all the users by p =(p 1,...,p N ). Snce the users cannot jontly decode ther messages and can only treat other users nterference as nose, each user s nstantaneous throughput under the jont power profle p s [1] [5] r (p) =log 2 (1+ p g j N,j pjgj+σ ), (2) where σ s the nose power at user s recever. We wrte each user s mum throughput as r log 2 (1 + p g /σ ), whch s acheved when user transmts at the mum power level and the other users do not transmt. The system s tme slotted at t =, 1,... We assume that the users are synchronzed as n [1] [5] (e.g. by usng a global clock from the global postonng system (GPS)). We wrte each user s transmsson polcy as π : N + P, where π (t) s user s transmt power level at tme t. The spectrum sharng polcy s then the collecton of all the users transmsson polces, denoted by π = (π 1,...,π N ). In a constant polcy, we have π(t) =p const for all t N +.In a TDMA polcy, we have π(t) =1, where s the l- norm that represents the number of nonzero elements n a vector. Remark 2: In ths paper, we wll focus on TDMA polces. In other words, we do not consder constant polces, and

3 some non-constant polces that are not TDMA. Constant polces are known to be nferor to TDMA polces when the mult-user nterference s strong. However, non-constant, non- TDMA polces may acheve better performance than TDMA polces, by allowng the subset of users wth weak mutual nterference to transmt smultaneously. However, determnng the optmal subsets requres knowledge about the cross channel gans among the users, whch s hard to get n a decentralzed spectrum sharng scenaro. In addton, snce the number of subsets grows exponentally wth the number of users, a nave exhaustve searchng for the optmal parttonng has prohbtvely hgh complexty. Hence, as n [6] [9], we wll focus on the desgn of TDMA polces n ths paper. Each user s (dscounted) average throughput s defned as R (π) =(1 δ) t= δt r (π(t)), (3) where δ [, 1) s the dscount factor that models the delay-senstvty of a user [5][1]. A more delay-senstve user dscounts the future throughput more (.e. has a smaller dscount factor), because t has more urgency to transmt. We use the dscounted average throughput nstead of the average throughput, because for delay-senstve applcatons, the packets need to be transmtted as soon as possble to avod mssng ther deadlnes [1]. In [9], we also studed the case where the users have dfferent dscount factors but have no CQoS constrants. Extensons to the case wth heterogeneous dscountng and CQoS constrants are nterestng future work. III. CONTINUING QOS GUARANTEES The wdely-used QoS guarantee [1] [1] s that the average throughput s above some guaranteed fracton γ avg of the mum possble throughput r, namely 2 R (π) γ avg r. (4) The above QoS guarantee does not provde suffcent guarantees for TDMA polces: even f a user s average throughput (startng from the begnnng) R (π) s hgh, t may get a extremely low throughput startng from certan pont n tme, because t may not get suffcent TXOPs after certan pont. 3 Such an ntuton wll be llustrated n the motvatng example n Secton IV. In ths paper, we propose contnung QoS guarantees, whch ensure that at every pont n tme, a user s future throughput s guaranteed to be above some desred mnmum requrement. Such contnung guarantees are mportant for delay-senstve users. We formally defne contnung QoS guarantees as follows. Frst, we defne the contnuaton throughput at tme t as R t(π) =(1 δ) τ= δτ t r (π(τ)), (5) whch s the dscounted average throughput startng from tme t. Note that R (π) = R (π). Then, the contnung QoS guarantees can be wrtten as CQoS: R t (π) γ cont r, t =, 1,... (6) 2 In the case of constant polces [1] [5], such QoS guarantee reduces to the requrement on the nstantaneous throughput, namely r (p const ) γ avg r, because the users choose the same power profle at any tme. 3 Constant polces do not have such a problem. If they fulfll the average throughput requrements, the throughput wll be hgh enough startng from every pont n tme. However, t s dffcult for them to fulfll the average throughput requrement n the frst place, due to mult-user nterference. Amount of data Amount of data actually transmtted Amount of data transmtted at the average rate R Amount of data transmtted at the CQoS guaranteed rate γ cont r Low R 1 Hgh R 9 Small delay d (9)=1 1 Large delay d (1)= Tme slot Fg. 2. Relatonshp of delay and CQoS guarantees of user. The sold curve wth square data ponts s the amount of data transmtted; each jump n the curve corresponds to a transmsson. The two straght lnes through the orgn are the amount of data transmtted as f the throughput was R and γ cont r, respectvely. At each tme t, f the contnuaton throughput R t s hgher, the user needs to transmt more after tme t. Hence, the correspondng delay d (t) s lower. To dfferentate from CQoS, we wll call the commonly-used QoS guarantee n (4) as average QoS (AQoS) guarantee n the rest of ths paper. We can see that the CQoS guarantee contans the requrement for the average throughput R (π). Hence, we wll assume that γ cont s redundant. <γ avg. Otherwse, the AQoS guarantee A byproduct of the CQoS guarantees s that once they are satsfed, we can also provde upper bounds on the transmsson delays of each user. Frst, we defne user s transmsson delay at any tme t as Transmsson Delay: d t (π) mn τ>t {τ t : π (τ) > }. In words, the transmsson delay d t (π) s the mnmum wat tme untl the next transmsson. An upper bound on the transmsson delays are crtcal for delay-senstve applcatons. As we wll prove n Theorem 3, each user s CQoS guarantee leads to an upper bound on ts mum delay sup t d t (π). We also llustrate the relatonshp of delay and CQoS guarantees n Fg. 2. We can see that the delay s determned by the dfference between the traffc arrval rate and the guaranteed rate (.e. CQoS). A hgher CQoS results n lower delay. IV. A MOTIVATING EXAMPLE We provde a motvatng example to llustrate the mportance and mpact of the CQoS guarantees, and to show the advantage of the proposed optmal TDMA polcy over round-robn TDMA polces, n terms of both the performance and the computatonal complexty. Consder a smple network wth four symmetrc users. They have the same mum throughput normalzed to 1 bts/s/hz (.e. r =1, ), and the same dscount factor of δ =.83. The system performance metrc s the -mn farness (.e. the mnmum of all the users throughput). A. CQoS Guarantees and System Performance We frst llustrate the tradeoff between CQoS guarantees and the system performance. Intutvely, CQoS guarantees requre that a user has suffcently many transmsson opportuntes every once n a whle. In other words, the transmsson

4 TABLE II. ROUND-ROBIN TDMA POLICIES CANNOT ACHIEVE BOTH GOOD PERFORMANCE AND GOOD CQOS. Cycle length L =4 L =5 L =6 L =7 Rates (bts/s/hz) CQoS (bts/s/hz) TABLE III. PERFORMANCE LOSS AND COMPLEXITY OF ROUND-ROBIN TDMA POLICIES, UNDER CQOS GUARANTEES γ cont =.1. Cycle length L =4 L =5 L =6 L =7 Proposed Worst user s rate CQoS guarantee fulflled fulflled fulflled volated fulflled Performance loss (compared to proposed) 4% 32% 25% N/A # of polces delay at any pont n tme should be small. For example, for round-robn TDMA polces wth cycle length L =8, the one that mzes CQoS guarantees (or mnmzes transmsson delay) has a cycle of It s not dffcult to see that, any other polces (we consder the polcy wth a cycle of as the same snce the users are symmetrc) wth L =8wll have a mum transmsson delay hgher than 4, and wll have a worse CQoS. However, the polcy wth cycle s not far: user 4 always transmts at later postons n the cycle, and hence wll experence a very low average rate. The polcy that acheves the best -mn farness (.e. the worst user s rate s mzed) has a cycle of , because user 4 wll get two postons n the mddle of the cycle. However, such a polcy has a worst-case transmsson delay of 7 (for user 1). In other words, user 1 has a low CQoS (e.g. ts throughput startng from tme slot 2 s very low, because t needs to wat untl tme slot 8 to transmt). We llustrate the tradeoff between the CQoS guarantees and the system performance (.e. -mn farness) for round-robn TDMA polces n Table II. We can see that wth the ncrease of the cycle length, round-robn TDMA polces acheve better performance, but wll have worse CQoS guarantees. In Table III, we llustrate the performance loss of roundrobn TDMA polces compared to the proposed optmal TDMA polcy. We fnd the optmal round-robn polces of dfferent cycle lengths subject to a CQoS guarantee of.1 bts/s/hz. The proposed polcy acheves the optmal farness (.e..25 bts/s/hz for all 4 users), and outperforms roundrobn polces by at least 2%. B. Computatonal Complexty Remarkably, not only s the proposed optmal polcy much more effcent than round-robn polces, t s much easer to compute. To get a hnt of why ths s so, note that n a roundrobn polcy, the user s performance s determned not only by the number of slots n a cycle but also by the postons of the slots snce users are dscountng ther future throughput (due to delay senstvty). For a gven number of users N and a gven cycle length L, the number of non-trval round-robn schedules 4 s greater than N L N. So searchng among these schedules wll be totally mpractcal even f L s moderately larger than N - but n order to acheve effcency close to the optmal polcy, the cycle length L must be much larger 4 Non-trval schedules are the ones n whch each user gets at least one tme slot n a cycle. than N. For nstance, for the 4-user case above, achevng energy effcency wthn 1% of the optmal polcy requres that the cycle length be at least 7, and requres searchng among the thousands (84) of dfferent nontrval schedules of cycle length 7. Even ths small problem s computatonally ntensve. For a moderate number of users - say 1 - and a cycle length of 2 - we need to search more than ten bllon (.e. 1 1 ) schedules, whch s completely ntractable. However, we wll propose a smple algorthm to compute the optmal polcy whose complexty grows lnearly wth the number of users. We wll compare the complexty formally n Secton VI- C and Table IV. Throughout ths paper, we dscuss the complexty of desgnng round-robn polces (e.g. how many polces to search), nstead of the complexty of mplementng them. It s easy to mplement round-robn polces; however, t s computatonally complex to determne the optmal polcy before the mplementaton. V. FORMULATION OF THE POLICY DESIGN PROBLEM We am to desgn a TDMA spectrum sharng polcy π that mzes the system performance, defned as a functon of the users throughput, W (R 1 (π),...,r 1 (π)). We assume that W ( ) s ncreasng and strctly concave n each argument R. Such a defnton of system performance s general enough to nclude the objectve functons adopted n most exstng works [1] [1] as specal cases. One example of system performance, whch wll be used n our smulatons, s the (normalzed) mn farness defned as W (R 1 (π),...,r 1 (π)) = mn (π) R r. In addton, we wll mpose the AQoS and CQoS guarantees. To sum up, we can formally defne the polcy desgn problem as follows Desgn Problem: W (R 1 (π),...,r 1 (π)) (7) π s.t. AQoS: R (π) γ avg r,, CQoS: R(π) t γ cont r,, t. For the polcy desgn problem (7) to be feasble, we requre that N γavg 1. VI. SOLVING THE POLICY DESIGN PROBLEM In ths secton, we solve the polcy desgn problem (7). Our proposed soluton (llustrated n Fg. 3) has two phases: an offlne phase mplemented before run-tme, whch determnes the optmal operatng pont (.e. each user s target average throughput), and a low-complexty onlne phase mplemented at run-tme, whch determnes the transmsson schedule that acheves the optmal operatng pont whle fulfllng AQoS and CQoS guarantees. A. Offlne Phase The Optmal Operatng Pont Before run-tme, the users solve the followng problem n a dstrbuted manner to determne the optmal operatng pont: r = arg r W (r 1,...,r N ) (8) s.t. N r /r =1, r γ avg r,. In (8), the lnear equalty N r /r = 1 comes from the requrement that the polcy s TDMA. Intutvely,

5 t ensures that the total fracton of all the users transmsson opportuntes sum up to 1. Note that the CQoS guarantees are not present n (8). They wll be taken care of n our schedulng polcy descrbed n the next subsecton. Algorthm 1 The Optmal Operatng Pont Selecton (OOPS) algorthm run by user. Requre: AQoS γ avg, precson ε 1: Set λ =, λ =1, λ = λ. 2: Solve W r = λ r for r, set r r } 3: Broadcast r /r {r,γavg, and receve rj /r j from users j 4: whle j N r j /r j > 1 do 5: λ 2 λ, λ λ 6: Solve W r = λ r for r, set r {r,γavg r } 7: Broadcast r /r, and receve rj /r j from users j 8: end whle 9: whle j N r j /r j 1 >εdo 1: λ λ+ λ 2 11: Solve W r = λ r r } for r, set r {r,γavg 12: Broadcast r /r, and receve rj /r j from users j 13: f j N r j /r j < 1 then 14: λ λ 15: else 16: λ λ 17: end f 18: end whle ) 19: Normalze r r / ( j N r j /r j AQoS CQoS Input: system performance metrc W(R 1,,R N) OOPS algorthm optmal operatng pont LDF schedulng Output: optmal schedulng (wth transmsson delay guarantees) π Fg. 3. Illustraton of our proposed desgn framework. The operatons n blue and n red are done by the polcy desgner and the decentralzed users, respectvely. Algorthm 2 The Longest-Dstance-Frst (LDF) schedulng algorthm run by user. Requre: normalzed operatng ponts {rj /r j } j N, dscount factor δ Intalzaton: t =, dstances α j () = rj /r j, j N repeat Fnd the user wth the largest dstance { } mn arg α j(t) j N f = then Transmt at power level P end f Updates dstances α j (t +1) for all j N as follows: α (t +1) = α (t) δ ( 1 δ 1), r αj(t) j (t +1) = δ, j t t +1 untl We propose a dstrbuted optmal operatng pont selecton (OOPS) algorthm (descrbed n Algorthm 1) to solve (8). We can prove that the dstrbuted Algorthm 1 converges to the optmal operatng pont lnearly 5 at rate 1 2. Theorem 1: The optmal operatng pont r that solves (8) can be found by each user runnng the dstrbuted OOPS algorthm (Algorthm 1), whch converges lnearly at rate 1 2. Proof: See [12, Appendx A]. B. Onlne Phase The Optmal Transmsson Schedule After fndng the optmal operatng pont r, we need to determne the transmsson schedule that acheves t. Importantly, the transmsson schedule should fulfll the CQoS guarantees, whch s the major challenge of our soluton. We propose a dstrbuted onlne longest-dstance-frst (LDF) schedulng algorthm (descrbed n Algorthm 2). Algorthm 2 has a nce nterpretaton of longest dstance frst schedulng. At each tme slot t, the user wth the largest dstance to target (.e. α j (t) n Algorthm 2) transmts n ths 5 Followng [11, Sec ], we defne lnear convergence as follows. Suppose that the sequence {x k } converges to x. We say that ths sequence converges lnearly at rate c, fwehavelm k x k+1 x x k x = c. tme slot 6. The algorthm updates the dstances n the correct way, such that the optmal operatng ponts are acheved. Theorem 2 proves the desrable propertes of the proposed LDF schedulng algorthm. N j N γcont j Theorem 2: For any dscount factor δ, f each user N runs the dstrbuted LDF schedulng algorthm, we have each user s average throughput up to tme t converges to ts optmal operatng pont lnearly at rate δ, namely (1 δ) t τ= δτ r τ r r δ t+1 ; each user fulflls ts CQoS guarantee, namely R t (π) γcont r, t =, 1, 2,... Proof: See [12, Appendx B]. Remark 3: Although the CQoS guarantees do not drectly appear n Algorthm 2, they mpose a constrant on the dscount factor δ used n Algorthm 2. Theorem 2 proves that the algorthm, gven a proper nput of dscount factor (namely N j N γcont j δ CQoS guarantees. ), wll construct a polcy that fulfll the 6 Tes can be broken arbtrarly. In Algorthm 2, we choose the user wth the smallest ndex. Specfcally, when arg j N α j (t) returns a set of ndces, we choose the mnmum one.

6 TABLE IV. Polcy Constant polces [1] [5] Round-robn TDMA (cycle length L) Proposed COMPARISON OF COMPUTATIONAL COMPLEXITY. Computatonal complexty NP-hard to fnd the optmal p const Offlne: N L N polces to search Onlne: Offlne: N O(log 2 1/ε) Onlne: O(N) Theorems 1 and 2 establsh the convergence results of our proposed scheme. Theorem 1 proves that the process of fndng the optmal operatng ponts converges n logarthmc tme, and Theorem 2 proves that the LDF schedulng acheves the optmal operatng ponts n logarthmc tme. Hence, the overall convergence speed s fast. Moreover, Theorem 2 ensures that the CQoS guarantees are fulflled. As we have dscussed before, a byproduct of the CQoS guarantees s the upper bounds on the transmsson delays, whch are provded n Theorem 3 N j N γcont j Theorem 3: For any dscount factor δ, f each user N runs the dstrbuted LDF schedulng algorthm, we have each user s mum transmsson delay s upper bounded, namely sup d t (π) t log γcont ; log δ at each tme t, each user s transmsson delay s upper bounded, namely d t (π) log α (t), log δ where α (t) s user s dstance from target at tme t calculated n Algorthm 2. Proof: See [12, Appendx C]. Theorem 3 gves us the upper bound of the mum transmsson delay, as well as fner upper bounds of transmsson delays at each tme t based on the user s dstances from target α (t) (calculated n Algorthm 2). Note that the upper bound on the mum delay, namely log γ cont / log δ, s decreasng n the CQoS, because we have γ cont < 1 and δ<1. C. Computatonal Complexty and Message Exchange We compare the computatonal complexty of the exstng solutons and our proposed soluton, and dscuss the amount of message exchange n our soluton. 1) Computatonal Complexty: For constant polces, fndng the optmal power profle p const s NP-hard n general [1]. Ths s due to the nonconvexty of the problem: the throughput functon s not jontly concave n the power profle because of the nterference. For round-robn TDMA polces, the number of polces to search s lower bounded by N L N. To ensure a good performance, the cycle length needs to be large, whch means that the number of polces grows exponentally wth the number of users N. Hence, t may take a long tme to fnd the optmal round-robn TDMA polcy before run-tme, although they are easy to mplement at run-tme. In contrast, n our proposed soluton, the OOPS algorthm converges n logarthmc tme before run-tme, and the complexty of the onlne LDF schedulng s low (.e. each user only needs to update the dstances based on smple analytcal formula). 2) Message Exchange: In our proposed soluton, the message exchange happens only before run-tme. The total amount of message exchange (.e. the broadcast of r /r ) s N O(log 2 1/ε). There s no message exchange at run-tme. VII. SIMULATION RESULTS We demonstrate the performance gan of our proposed TDMA polcy over exstng polces. Throughout ths secton, we use the followng system parameters. The nose powers at all the users recevers are normalzed as db. The mum transmt powers of all the users are 2 db. Wthout loss of generalty, we normalze the drect channel gans to 1, namely g =1,, and generate the cross channel gans randomly accordng to the dstrbuton g j CN(,.5), j. The system performance s measured by the (normalzed) -mn farness mn R /r, namely we am to mze the worst user s (normalzed) throughput. At the optmal -mn farness, each user s normalzed average throughput R (π)/r cannot exceed 1 N. Hence, we let each user s AQoS guarantee to be wthn 1% of ts mum normalzed throughput, namely γ avg =.9 N,. In most smulatons, we wll vary the CQoS guarantees (whch are equal cross users). Gven each CQoS guarantee, we choose the mnmum dscount factor specfed by Theorem 2, namely δ =. In other N(1 γ cont ) words, we evaluate the performance of the most delay-senstve applcatons. A. Performance Under Dfferent CQoS Guarantees We frst fx the number of users to be N =4, and ncrease the CQoS guarantees from.1 to.22. Note that the AQoS guarantee under N =4s.225. Hence, a CQoS guarantee of.22 s close to the AQoS guarantee. In Fg. 4, we show the optmal -mn farness (.e. mn R /r ) acheved by dfferent polces. We can see that under all CQoS guarantees, our proposed polcy can acheve the optmal -mn farness of.25. In contrast, the optmal constant polcy acheves at least 5% away from the optmal -mn farness when the CQoS guarantee s small, and becomes nfeasble when the CQoS guarantee exceeds.13. For round-robn polces, we search all the polces up to cycle length 9 (there are non-trval polces wth cycle length 9) and choose the optmal one under each CQoS guarantee. We can see that the performance of round-robn polces decreases to 2% away from the optmal performance before t becomes nfeasble at CQoS of.19. Next, we nvestgate the mum number of users that can be supported by each polcy under dfferent CQoS guarantees. We ncrease the CQoS guarantees from.5 to.2. Note that theoretcally, the mum number of users that can possbly be supported s 1 (because we need N γ cont γ cont 1). In Fg. 5, we can see that the mum numbers of users supported by our proposed polcy are the same as the theoretcal upper bounds at most of the tme. At certan CQoS guarantees (e.g. γ cont =.5), the theoretcal upper bound (e.g. 2 users when γ cont =.5) can be acheved only when

7 Normalzed Max Mn Farness Proposed Round robn Constant CQoS Fg. 4. Comparson of the -mn farness acheved by dfferent polces under dfferent CQoS guarantees. Maxmum Numbers of Users Theoretcal upper bound Proposed Round robn Constant CQoS Fg. 5. Comparson of the mum number of users that can be supported by dfferent polces under dfferent CQoS guarantees. the dscount factor s 1. However, snce we consder delaysenstve applcatons (.e. δ < 1), we can support slghtly fewer users (e.g. 19 users when γ cont =.5). In contrast, the other two polces can support much fewer users. At a CQoS guarantee of.5, we roughly double the number of users accommodated, compared to the other two polces. In other words, we can utlze the spectrum much more effcently whle fulfllng CQoS guarantees. B. Performance for Vdeo Transmsson Fnally, we evaluate the performance of dfferent polces for wreless vdeo transmsson. In the performance evaluaton, we use the PSNR, whch s commonly-used as performance metrc for vdeo qualty. In the experment, We consder a network wth 4 users, and use the classc Foreman and Coastguard vdeo sequences. In Table V (for Foreman vdeo sequence) and Table VI (for Coastguard vdeo sequence), we show the worst-case PSNR acheved by dfferent polces under dfferent CQoS guarantees. We can see that our proposed polcy mproves the PSNR of the constant polcy and the round-robn polcy by up to 6 db and 4 db, respectvely. Moreover, when the CQoS guarantees ncrease, the other two polces become nfeasble. VIII. CONCLUSION In ths paper, we studed spectrum sharng among users wth delay-senstve applcatons. We proposed a novel performance metrc, namely contnung QoS guarantees, to ensure the performance of delay-senstve applcatons. We desgned the optmal TDMA polcy that mzes the system performance subject to the CQoS guarantees, and proposed low- TABLE V. IMPROVEMENT OF PSNR IN FOREMAN SEQUENCE OVER CONSTANT AND ROUND-ROBIN POLICIES UNDER DIFFERENT CQOS GUARANTEES. CQoS guarantee Constant 31 db nfeasble nfeasble nfeasble Round-robn 37 db 35 db 34 db nfeasble Proposed 38 db 38 db 38 db 38 db Improvement over Constant 6dB Improvement over Round-robn 1dB 3dB 4dB TABLE VI. IMPROVEMENT OF PSNR IN COASTGUARD SEQUENCE OVER CONSTANT AND ROUND-ROBIN POLICIES UNDER DIFFERENT CQOS GUARANTEES. CQoS guarantee Constant 29 db nfeasble nfeasble nfeasble Round-robn 34 db 32 db 32 db nfeasble Proposed 36 db 36 db 36 db 36 db Improvement over Constant 6dB Improvement over Round-robn 1dB 3dB 4dB complexty dstrbuted algorthms for the users to construct the optmal polcy. Our proposed polcy sgnfcantly outperforms exstng constant polces and round-robn polces, n terms of the system performance (e.g. -mn farness), the number of users accommodated whle fulfllng ther QoS guarantees, as well as the computatonal complexty of desgnng the optmal polces. When appled to vdeo streamng, our proposed polcy can acheve performance mprovement of up to 6 db and 4 db, compared to constant polces and round-robn polces. REFERENCES [1] C. W. Tan and S. H. Low, Spectrum management n multuser cogntve wreless networks: Optmalty and algorthm, IEEE J. Sel. Areas Commun., vol. 29, no. 2, pp , 211. [2] J. Huang, R. A. Berry, and M. L. Hong, Dstrbuted nterference compensaton for wreless networks, IEEE J. Sel. Areas Commun., vol. 24, no. 5, pp , May 26. [3] N. Gatss, A. G. Marques, G. B. Gannaks, Power control for cooperatve dynamc spectrum access networks wth dverse QoS constrants, IEEE Trans. Commun., vol. 58, no. 3, pp , Mar. 21. [4] S. Sorooshyar, C. W. Tan, M. Chang, Power control for cogntve rado networks: Axoms, algorthms, and analyss, IEEE/ACM Trans. Netw., vol. 2, no. 3, pp , 212. [5] R. Etkn, A. Parekh, and D. Tse, Spectrum sharng for unlcensed bands, IEEE J. Sel. Areas Commun., vol. 25, no. 3, pp , 27. [6] M. van der Schaar, Y. Andreopoulos, and Z. Hu, Optmzed scalable vdeo streamng over IEEE a/e HCCA wreless networks under delay constrants, IEEE Trans. Moble Comput., vol. 5, no. 6, pp , June 26. [7] D. Pradas, M. A. Vazquez-Castro, NUM-based far rate-delay balancng for layered vdeo multcastng over adaptve satellte networks, IEEE J. Sel. Areas n Commun., vol. 29, no. 5, May 211. [8] P. Dutta, A. Seetharam, V. Arya, M. Chetlur, S. Kalyanaraman, J. Kurose, On managng qualty of experence of multple vdeo streams n wreless networks, n Proc. IEEE Infocom, 212. [9] Y. Xao and M. van der Schaar, Spectrum sharng polces for heterogeneous delay-senstve users: A novel desgn framework, n Proc. Allerton, 213. [1] M. van der Schaar and F. Fu, Spectrum access games and strategc learnng n cogntve rado networks for delay-crtcal applcatons, Proc. of IEEE, Specal ssue on Cogntve Rado, vol. 97, no. 4, pp , Apr. 29. [11] S. Boyd and L. Vandenberghe, Convex Optmzaton. New York: Cambrdge Unv. Press, 24. [12] Y. Xao and M. van der Schaar, Appendx, Avalable at: