Optimal Energy and Delay Tradeoffs for Multi-User Wireless Downlinks

Transcription

1 IEEE PROCEEDINGS OF INFOCOM, APRIL Optmal Energy and Delay Tradeoffs for Mult-User Wreless Downlnks Mchael J. Neely Unversty of Southern Calforna mjneely Abstract We consder the fundamental delay tradeoffs for mnmzng energy expendture n a mult-user wreless downlnk wth randomly varyng channels. Frst, we extend the Berry- Gallager bound to a mult-user context, demonstratng that any algorthm that yelds average power wthn O(1/V ) of the mnmum power requred for network stablty must also have an average queueng delay greater than or equal to Ω( V ). We then develop a class of algorthms, parameterzed by V, that come wthn a logarthmc factor of achevng ths fundamental tradeoff. The algorthms overcome an exponental state space exploson, and can be mplemented n real tme wthout a- pror knowledge of traffc rates or channel statstcs. Further, we dscover a super-fast schedulng mode that beats the Berry- Gallager bound n the exceptonal case when power functons are pecewse lnear. Index Terms queueng analyss, stablty, optmzaton, stochastc control, asymptotc tradeoffs, satellte communcaton I. INTRODUCTION In ths paper we consder the fundamental tradeoff between energy and delay n a mult-user wreless network. We focus on the case of a wreless downlnk that transmts to N dfferent users over N tme varyng channels (Fg. 1). Transmsson rates depend on current channel condtons and current power allocaton decsons. We assume that tme s slotted and that channel condtons can be measured every tmeslot. The goal s to allocate power n reacton to current channel states and current queue backlogs to stablze the system whle mnmzng energy expendture and mantanng low delay. Ths objectve s mportant for satellte and wreless downlnks, as well as for wreless nodes that transmt to neghbors wthn a larger ad-hoc network. It s crucal to understand the fundamental performance lmts of such systems, as these lmts must be pushed to ther maxmum to support the demands that wll be placed on future networks. The fundamental energy-delay tradeoff was characterzed by Berry and Gallager n [1] for the specal case of a sngle queue that stores data for transmsson over a sngle fadng channel. Average energy for such a system can be mproved by accumulatng data for more effcent future transmsson, at the expense of ncreasng queueng congeston and delay. It was shown n [1] that, subject to strct convexty assumptons on the rate-power curve of the system, any set of algorthms that yeld average power wthn O(1/V ) of the mnmum power requred for stablty (for ncreasngly large postve numbers V ), must also have average queueng delay greater Ths materal s based on work supported n part by the Natonal Scence Foundaton under grant OCE λ 1 λ 2 λ Ν Energy Delay Fg. 1. A wreless downlnk wth multple nput streams, wth an assocated energy-delay tradeoff curve. than or equal to Ω( V ). 1 Ths demonstrates that any effort to reduce the power expended to support ncomng traffc wll necessarly ncrease delay. Further, an algorthm for achevng ths fundamental tradeoff was proposed, based on the concept of buffer parttonng. Related work on mnmzng energy n a statc wreless node wth known arrval tmes and a sngle transmtter s consdered n [2] [3], and a problem wth a statc lnk but stochastc arrvals s treated usng flter theory n [4]. Smlar problems of mnmzng energy subject to delay constrants or mnmzng delay subject to energy constrants are treated for sngle lnk satellte and wreless systems n [3] [5] [6] usng stochastc dfferental equatons, dynamc programmng, and Markov decson theory. Whle such technques mght mprove the delay coeffcent n comparson to the Berry- Gallager algorthm, they cannot overcome the fundamental Ω( V ) tradeoff curve. However, there has been lttle work to extend ths theory of energy-delay tradeoffs to mult-user networks. Ths s largely due to the complexty exploson assocated wth ncreasng the number of queues beyond one. Indeed, the number of backlog vectors and channel state vectors both ncrease exponentally wth the number of queues n the system, makng dynamc programmng approaches and Markov decson theory approaches prohbtve. Indeed, even for the case of a sngle data lnk, t s dffcult to mplement the proposed stochastc algorthms ([1] [3] [5] [6]), as these algorthms requre schedulng polces to be pre-computed based on full knowledge of the nput rate and the steady state channel probabltes. Whle t may be possble to estmate these statstcs n the sngle lnk case when the number of channel states s relatvely small, t s not practcal to envson estmatng the exponentally growng 1 The notaton Ω( V ) denotes a functon that ncreases at least as fast as a square root functon.

2 IEEE PROCEEDINGS OF INFOCOM, APRIL number of parameters n the mult-user case, nor s t practcal to envson solvng the correspondng optmzaton problems even f all of these parameters were known exactly. The complexty exploson problem s one of the major obstacles that we overcome n ths paper. We do so by combnng the concept of buffer parttonng wth the recently developed theory of performance optmal Lyapunov schedulng [7] [8] [9]. Specfcally, [7] [8] [9] develops Lyapunov technques for treatng stablty and performance optmzaton smultaneously (extendng the Lyapunov stablty results developed, for example, n [12]-[16]). These results are appled n [7] to develop smple stablzng algorthms for general mult-hop wreless networks that yeld average power wthn O(1/V ) of the mnmum power requred for stablty (for any control parameter V > 0). However, the resultng end-to-end delay of these algorthms was shown to grow lnearly wth V, even for the specal case of a sngle data lnk. Thus, the algorthms do not yeld the optmal delay tradeoff n the sngle lnk case, suggestng that mproved tradeoffs mght be possble. In ths paper, we extend the theory of stochastc optmal networkng to treat optmal delay tradeoffs. Frst, we extend the Berry-Gallager bound to mult-user systems, establshng that the Ω( V ) tradeoff curve also apples n these more general scenaros. Next, we construct a smple algorthm that acheves the fundamental energy-delay tradeoff to wthn a logarthmc factor. Ths s the frst algorthm to acheve such performance for mult-user systems. Furthermore, the algorthm does not requre knowledge of nput rates or channel statstcs, and s smple to mplement n real tme for systems wth any number of users. The technque ntroduces a novel method of drft steerng that mantans a set of vrtual queues wth Lyapunov coeffcents that turn ON or OFF based on congeston thresholds. For smplcty of exposton, we focus on the case of a sngle downlnk wth N channels, although the technque readly extends to the more general scenaro of mult-node, mult-hop networks usng the backpressure technques developed n [7]-[9], [12], [16]. Our analyss further reveals an mportant exceptonal case where super-fast convergence s possble. In partcular, f power curves have certan pecewse lnear propertes, such as when power allocaton s restrcted to usng ether zero power or full power at any server, then our algorthm can be modfed to acheve smlar energy performance wth delay that grows only logarthmcally n the V parameter. Ths demonstrates that t s possble to outperform the Berry-Gallager bound n systems wth pecewse lneartes. Prevous work n the area of capacty and stable schedulng for mult-user networks s found n [10]-[16], and energy effcency s consdered n [17]-[21], [7]. Most work n network optmzaton s closely ted to statc optmzaton theory and convex dualty, ncludng [22]-[26] for statc networks and [17], [27]-[29] for stochastc gradent algorthms and flud lmt models. A Lyapunov method for performance optmzaton s developed n [9] [8] [7] whch yelds strateges smlar to those suggested by gradent optmzaton approaches, and also yelds explct performance and delay bounds. Ths paper bulds on the Lyapunov method to acheve optmal delay tradeoffs, and makes a sgnfcant contrbuton to the feld by developng new schedulng algorthms that go beyond the classcal gradent methods of optmzaton theory. An outlne of ths paper s as follows. In the next secton, we defne the system model and prove the Ω( V ) lower bound on delay for any algorthm mplemented on systems wth strct convexty propertes. In Secton IV we revew the man features of the orgnal Berry-Gallager algorthm from [1] and outlne the complexty challenges assocated wth the multdmensonal problem. In Secton V we ntroduce Lyapunov drft theory and present the control algorthm. Performance analyss and extensons to super-fast schedulng are treated n Sectons VI and VII. II. PROBLEM FORMULATION Consder a wreless downlnk wth N tme-varyng channels, each for a dfferent wreless user. The system operates n slotted tme wth slots normalzed to one unt. We let S(t) = (S 1 (t),..., S N (t)) represent the vector of current channel states for each lnk durng slot t, for t 0, 1, 2,...}. These states can represent current fadng levels, attenuaton, and/or nose levels assocated wth the channel durng slot t. It s assumed that channels hold ther states for the duraton of a tmeslot, but potentally change on slot boundares. For smplcty, we assume there are a fnte number of channel states S, and that the channel process s..d. from one tmeslot to the next. 2 We defne π S as the occurrence probablty for each channel state S. These channel probabltes determne the capacty regon of the network [16], but are not necessarly known to the downlnk controller. Every tmeslot, the controller observes the current channel states and chooses transmsson rates by allocatng power as a vector P (t) = (P 1 (t),..., P N (t)) subject to an nstantaneous power constrant P (t) Π, where P (t) represents the power allocated to lnk durng slot t, and Π s a compact set that specfes the collecton of admssble power vectors. For example, a system wth only a peak power constrant P peak can be modeled wth a power set Π consstng of all nonnegatve power vectors P that satsfy N P P peak. Addtonal constrants on nstantaneous power transmsson can be ncorporated smply by modfyng the power set Π. Transmsson rates for each lnk are determned by the current channel state vector S(t) and the current power allocaton vector P (t) accordng to a general rate-power curve µ( P, S) = (µ 1 ( P, S),..., µ N ( P, S)). We assume that µ( P, S) s a contnuous functon of the power vector for each channel state vector S. Let µmax represent a bound on the maxmum transmsson rate of a sngle lnk, maxmzed over all channel states S and all power vectors P Π. Data arrves n packetzed form accordng to N random processes, and each packet s stored n one of N nternal queues accordng to ts destnaton (see Fg. 1). We let A(t) = (A 1 (t),..., A N (t)) represent the vector of new packet arrvals every slot, where A (t) s the number of bts that arrve for user durng slot t. We assume arrval vectors A(t) are..d. over slots, and defne the rate vector λ = (λ 1,..., λ N ), where λ = E A (t)} represents the arrval rate to queue n unts 2 Extensons to non-..d. systems can be treated va technques n [9] [16].

3 IEEE PROCEEDINGS OF INFOCOM, APRIL of bts/slot. For smplcty, we assume that λ > 0 for all. Let U (t) denote the unfnshed work n queue at slot t, representng the backlog of bts watng to be transmtted over channel. Let r(t) = µ( P (t), S(t)) represent the transmsson rate vector at slot t. Queue dynamcs thus proceed accordng to the equaton: U (t + 1) = max[u (t) r (t), 0] + A (t) (1) where r (t) = µ ( P (t), S(t)), and P (t) Π. The goal of the network controller s to allocate power subject to the power constrants so that all queues are stablzed and, deally, average power s mnmzed. It turns out that mnmum power cannot be acheved wthout nfnte average delay, and hence our precse objectve s to stablze the system wth average power that can be pushed arbtrarly close to the mnmum power requred for stablty, wth an optmal delay tradeoff. Note that the above formulaton for a wreless downlnk s qute general and can equally model sngle-hop networks wth multple nodes and N data lnks. The only dfference n the network case s that queues and transmtters are dstrbuted over the dfferent nodes of the network, and so extra coordnaton mght be requred for control decsons. A. Example Rate-Power Functons In the specal case where there s only a peak power constrant and lnks are ndependent wth no nter-channel nterference, rate-power functons have the form: µ( P, S) = (µ 1 (P 1, S 1 ),..., µ N (P N, S N )) (2) where power s allocated so that P (t) P peak. In cases when there s nter-channel nterference, transmsson rates on each lnk depend on the full vector of channel states and power allocatons. For example, under a sgnal-to-nterference rato model, the rate functons are gven by: µ ( P, S) = f (SINR ( P, S)) (3) where SINR ( P, S) s the sgnal-to-nterference-plus-nose rato on lnk when power vector P s allocated under channel state S, and f ( ) s any functon of SINR. The power set Π mght specfy further system constrants, such as allowng transmssons over at most one channel durng any slot and/or restrctng allocatons to ether full power or zero power. B. The Mnmum Energy Functon Here we descrbe the mnmum power requred to stablze all queues of the downlnk system descrbed above. In [9] [16], the capacty regon Λ s defned as the closure of the set of all nput rate vectors λ stablzable under some power allocaton algorthm that conforms to the power constrant P (t) Π. Throughout ths paper, we assume that the nput rate vector s strctly nteror to the capacty regon Λ, and so the system s stablzable. In [7], the mnmum average power requred for stablty s shown to be the soluton to an optmzaton problem assocated wth a statonary randomzed power allocaton strategy. Below we present a generalzed statement that consders mnmzng the tme average of a power cost functon h( P (t)). We assume that h( P ) s nonnegatve and contnuous n the power vector P. We defne the average power cost as follows: 1 t 1 h av= lm E h( P t } (τ)) τ=0 The mnmum average power problem corresponds to a cost functon: h( P ) = N P Alternate cost metrcs, such as second moments of power, can be modeled as desred by choosng dfferent h( P ) functons. Theorem 1: If the functons h( P ) and µ( P, S) are contnuous n the power vector P, then the mnmum average power cost h av s gven by the soluton to the followng optmzaton S problem (defned n terms of auxlary probabltes γ k and power vectors P S k for all S and for k 1,..., N + 2}): 3 Mnmze: h av = N+2 S π S k=1 γ S k h( P S k ) (4) Subject to: µ av= S π N+2 S k=1 γ S k µ( P S k, S) λ S P S k Π, γ k 0 for all k, S N+2 k=1 γ S k = 1 for all S Thus, the mnmum power cost for stablty s acheved among the class of statonary polces that measure the current channel state S(t) and then randomly allocate a power vector S P S k wth probablty γ k. Ths result s proven n [7] for the case when h( P ) = P by showng that no stablzng algorthm can yeld average power lower than h av, but that stablzng algorthms can be constructed wth average power that s arbtrarly close to h av. The fact that a probablstc combnaton of N +2 power vectors s used for every channel state S follows by extendng the dmensonalty of the system from N to N +1 (due to the sngle mnmum power objectve) and usng Caratheodory s theorem, as descrbed n [7]. The proof for general h( P ) functons s smlar to [7] and s omtted for brevty. Note that the mnmum power cost requred for stablty depends on the traffc statstcs only through the nput rate vector λ. We thus defne the mnmum energy functon Φ( λ) as the mnmum power cost requred to stablze the nput rate vector λ, consderng all concevable algorthms. That s, Φ( λ) = h av, where h av s the soluton of the above optmzaton problem. An equvalent but more compact way to defne Φ( λ) s as follows. Defnton 1: The mnmum energy functon Φ( λ) s defned as the value h av that acheves the mnmum over the class of all feasble statonary randomzed power allocaton algorthms yeldng: h av = E h( P } (t)) E µ( P (t), S(t)) } λ 3 The same holds more generally for rate-power curves µ( P, S) that are upper sem-contnuous n the power vector, and cost functons h( P ) that are lower sem-contnuous.

4 IEEE PROCEEDINGS OF INFOCOM, APRIL It s not dffcult to show that any statonary randomzed algorthm that satsfes the above can be modfed to yeld average transmsson rates that are exactly equal to the nput rate vector λ. Thus, for any vector λ nsde the capacty regon Λ, there s a partcular power allocaton polcy wth power allocatons P (t) = (P1 (t),..., PN (t)) and transmsson rates r (t) = (r1(t),..., rn (t)) that are ndependent of queue backlogs, and that yeld: E h( P } (t)) = Φ( λ) (5) E r (t)} = λ (6) where the expectaton s wth respect to the random channel state vector and the randomzed control decsons. Whle such a polcy exsts for all vectors λ, t would be dffcult to construct such a polcy, as that would nvolve solvng the optmzaton problem (4) and would requre full knowledge of the rate vector λ and the channel probabltes π S. The mnmum energy functon Φ( λ) has an mportant convexty property for all vectors λ n the capacty regon Λ, as descrbed by the followng lemma. Lemma 1: The functon Φ( λ) s convex over λ Λ, and s non-decreasng n each entry λ ( 1,..., N}). Proof: Omtted for brevty. It s known that any convex mult-varable functon s twce dfferentable almost everywhere [30]. Hence, throughout ths paper we shall assume that the Φ( λ) functon s twce dfferentable at the pont λ of nterest. III. THE FUNDAMENTAL ENERGY-DELAY BOUND Here we extend the Berry-Gallager bound to mult-user systems. Specfcally, we show that under a strct convexty assumpton on the mnmum energy functon Φ( λ), any sequence of polces (parameterzed by ncreasng postve numbers V ) that yeld average power cost h av wthn O(1/V ) of the mnmum average cost requred for stablty must also have average delay greater than or equal to Ω( V ). Our proof closely follows the work n [1] for the sngle user case, and n partcular n ths secton we shall restrct attenton to the same class of admssble strateges: Defnton 2: A sequence of schedulng strateges, parameterzed by ncreasng postve numbers V that tend to nfnty, s admssble f 1) The polces make statonary (and possbly randomzed) power allocaton decsons based on the current queue backlog and channel state vectors U(t) and S(t). 2) For each V, the correspondng polcy stablzes the system and forms an ergodc Markov chan wth steady state queue backlog dstrbuton π( U). Furthermore, the steady N } state average backlog E U s fnte for all V and ncreases to nfnty as V. 3) There exst postve values θ 1, θ 2 such that for all tmeslots t and for each 1,..., N}, we have: [ P r A (t) µ ( P (t), S(t)) θ 2 U(t) ] θ 1 The thrd assumpton above states that there s a postve probablty that the backlog of any partcular queue ncreases by at least θ 2 durng a sngle tmeslot, regardless of the current backlog value. Ths assumpton holds whenever there s a nonzero probablty of an outage on channel durng a partcular tmeslot (that s, havng a channel state that yelds zero data rate on channel for every power allocaton), or whenever there s a non-zero probablty that A (t) > µ max. The frst two assumptons on statonarty and ergodcty smplfy the proof of the fundamental bound at the expense of slghtly reducng the class of schedulng polces consdered. However, the assumptons themselves are not very restrctve, n that algorthms that yeld optmal energy-delay tradeoffs can be formulated accordng to Markov decson theory, whch leads to statonary polces that base decsons on the system state vectors U(t) and S(t). Theorem 2: (Mult-User Berry-Gallager Bound) If the nput rate vector λ s strctly nteror to the capacty regon Λ and f the mnmum cost functon Φ( λ) has a postve defnte matrx of partal dervatves 2 Φ( λ) at the pont λ, then any sequence of admssble polces that yeld average power cost wthn O(1/V ) of Φ( λ) must have average delay greater than or equal to Ω( V ). Proof: See Appendx A. The postve defnte assumpton n the above theorem ensures that the second order terms of the Taylor seres expanson of Φ( λ + ɛ) are non-zero about the pont λ. Ths condton holds for a wde class of systems wth nonlnear rate-power curves, and s related to the noton of strct convexty. It s not dffcult to show that Φ( λ) s strctly convex whenever the feasble power set Π ncludes only a peak power constrant and when h( P ) = P and rate-power curves have the form specfed n (2) wth functons µ (P, S ) that are strctly concave n P for each lnk 1,..., N}. However, n systems wth only a dscrete set of power allocaton optons (such as systems that can allocate ether zero power or full power), the correspondng Φ( λ) functon s pecewse lnear and hence the energy-delay propertes of such systems are not necessarly governed by the Berry-Gallager bound. IV. BUFFER PARTITIONING To ntroduce the concept of buffer parttonng, we frst revew the basc Berry-Gallager threshold algorthm developed n [1] for the case of a sngle queue wth a sngle tme varyng data lnk (.e., the case N = 1), and for the case h( P ) = P. For such a system, let U(t) represent the current unfnshed work n the queue, let A(t) represent the arrvals on slot t, and let λ = E A(t)} represent the arrval rate. Defne Φ(γ) as the mnmum average power requred to support an average transmsson rate of γ. The Berry-Gallager algorthm swtches between two dfferent transmsson polces dependng on a buffer occupancy threshold. Specfcally, the buffer s parttoned nto two halves accordng to a postve threshold Q (where Q > µ max ). When U(t) < Q, power s allocated usng the statonary polcy that yelds an average transmsson rate of λ ɛ and an average power expendture of Φ(λ ɛ) (for some specfed value of ɛ such that 0 < ɛ < λ). When U(t) Q, power s allocated accordng to the statonary polcy that yelds an average transmsson rate of λ + ɛ and an average power expendture of Φ(λ + ɛ).

5 IEEE PROCEEDINGS OF INFOCOM, APRIL ε(u) μ max L ε -ε Q (a) R U U 2 Q ε ε (b) ε ε ε ε ε ε Q U 1 Fg. 2. (a) A parttonng of the sngle dmensonal state space nto Left and Rght regons for a sngle lnk system, wth drft drectons llustrated. (b) An example parttonng of a 2-user system wth the mult-dmensonal drft vectors shown. The backlog vector s currently n the upper left regon. Thus, for ths algorthm the queue backlog U(t) only affects power allocaton by determnng whch of the two transmsson polces s used. Once a partcular polcy s determned, power s allocated based only on the current channel state S(t). Now defne r(t) as the amount of bts transmtted by the queue on tmeslot t, and defne the drft as the expected dfference between A(t) and r(t) gven the current backlog level U(t). Note that the drft s equal to ɛ when U(t) Q, and s equal to ɛ when U(t) < Q, wth the exceptonal case when the queue s near empty and the drft s equal to ɛ(u) ɛ (see Fg. 2a). Ths s an edge effect that can only occur on the nterval 0 U < µ max, where t s possble that r(t) may be less than the scheduled transmsson rate r(t) = µ(p (t), S(t)) due to lttle or no data beng present n the queue. The Berry- Gallager algorthm s desgned to acheve the optmal energydelay tradeoff curve for the sngle lnk case (through sutable choces of the Q and ɛ parameters). Ths performance s due to the b-modal transmsson polcy, and cannot be acheved by polces that have un-modal drft propertes [1]. A. Mult-Dmensonal Buffer Parttonng Here we extend the buffer parttonng concept to a multuser context. We consder the mnmum energy functon Φ( λ) that corresponds to any contnuous and non-negatve power cost metrc h( P ). The algorthm we develop n ths subsecton s not meant to be a practcal control strategy. Rather, t s ntended to hghlght the challenges and desgn prncples assocated wth the mult-user problem. A practcal control strategy s developed n the next secton based on these prncples. Recall that all backlog vectors U take values wthn the N dmensonal state space [0, ) N. Consder parttonng ths state space nto 2 N regons accordng to the buffer threshold parameter Q. Specfcally, we defne H( U) as a vector wth bnary entres, where the th entry depends on whether U s to the rght or left of the Q threshold: H (U ) = L f U < Q R f U Q H( U) =(H 1 (U 1 ),..., H N (U N )) In ths way, the vector H( U(t)) ndcates whch of the 2 N regons currently contans the backlog vector U(t) (see Fg. 2b). Consder now an algorthm that swtches between 2 N dfferent transmsson polces dependng on the current regon of the backlog vector. The drft of each polcy s desgned to push each component of the backlog vector closer to the Q threshold. Specfcally, we consder a drft parameter ɛ such that 0 < ɛ < mn 1,...,N} λ }, and defne the vector ɛ( H) as follows: ɛ (H ) = ɛ f H = L +ɛ f H = R ɛ( H) =(ɛ 1 (H 1 ),..., ɛ N (H N )) Assume that λ + ɛ( H) Λ for all H. Whenever H( U(t)) = H, the algorthm uses the statonary polcy that allocates power based only on the current channel state S(t) to yeld an expected transmsson rate of λ+ ɛ( H) wth an average power expendture of Φ( λ + ɛ( H)). Such a polcy s guaranteed to exst by (5) and (6). Specfcally, f P (t) and r (t) represent the actual power vectors and transmsson vectors used by the polcy on a gven tmeslot t, we have (compare wth (5), (6)): E h( P (t)) H( U(t)) = H } = Φ( λ + ɛ( H)) (7) E r (t) H( U(t)) = H } = λ + ɛ( H) (8) Note that ths algorthm requres the pre-computaton of 2 N dfferent transmsson polces, one for each regon H. Each ndvdual polcy s computed by solvng the optmzaton problem (4) for the correspondng rate vector λ + ɛ( H), and each such optmzaton requres a-pror knowledge of the exponental number of channel state probabltes π S. Thus, such a polcy cannot be practcally mplemented n a real network. However, t s useful to consder the performance of ths strategy as an ad to analyzng the performance of the more practcal algorthm that we develop n the next secton. Consder an mplementaton of the polcy and let α H (t) represent the probablty that the backlog vector s wthn the regon H at a gven tmeslot t. The expected transmsson rate on lnk durng slot t s thus: E r (t)} = α H (t)e r (t) H( U(t)) = } H H = H α H (t)(λ + ɛ (H )) = λ + H α H (t)ɛ (H ) (9) Note that: α H (t)ɛ (H ) = ɛ α H (t) H H ɛ (H )=ɛ} α H (t) H ɛ (H )= ɛ} = ɛ [ α R (t) α L (t) ] (10) where we defne α R (t) and αl (t) as the probablty that U (t) s to the rght and to the left of the Q threshold, respectvely. Specfcally: α R (t) =P r[u (t) Q], α L (t) =P r[u (t) < Q] Usng (10) n (9), we have for all 1,..., N}: E r (t)} = λ + ɛ [ α R (t) α L (t) ] (11)

6 IEEE PROCEEDINGS OF INFOCOM, APRIL Smlarly, we can use (7) to compute the expected power expendture on slot t: E h( P } (t)) = α H (t)φ( λ + ɛ( H)) (12) H Because Φ( λ) s convex and twce dfferentable at λ, t follows by the mult-dmensonal Taylor theorem that: Φ( λ + ɛ( H)) Φ( λ) + N Φ( λ) λ ɛ (H ) + Nɛ 2 β (13) for a fxed value β > 0. In cases when Φ( λ) s twce dfferentable about the neghborhood ( ɛ, ɛ) N of the pont λ, the value of β s gven by: β = max δ ( ɛ,ɛ) N 2 Φ( λ + δ) 2 where 2 Φ( λ + δ) represents the matrx norm of the matrx of second partals. Usng (13) n (12) yelds: E h( P } (t)) [ H α H (t) Φ( λ) + Nɛ 2 β + ] N Φ( λ) λ ɛ (H ) = Φ( λ) + Nɛ 2 β + N Φ( λ) λ H α H (t)ɛ (H ) = Φ( λ) + Nɛ 2 β + N Φ( λ) λ ɛ [ α R(t) αl (t)] (14) where the last equalty follows from (10). To gan ntuton, assume the system s ergodc and tme averages exst. Takng tme averages of (14) and (11) thus yelds the followng bound on tme average power cost and the followng expressons for tme average transmsson rates: h av Φ( λ) + Nɛ 2 β + r = λ + ɛ [ α R α L ] N Φ( λ) ɛ [ α R α L ] (15) λ for 1,..., N} (16) where α R, α L are tme average probabltes for backlog n queue beng ether rght or left of the threshold. Note that for any stable system, the tme average transmsson rates are greater than or equal to the arrval rates, so that λ r. By (16), ths mples that ɛ [ ] α R α L 0 for all. Because the partal dervatves Φ( λ)/ λ are also non-negatve, the fnal summaton term n (15) s non-negatve and hence cannot be neglected. However, note that the tme average transmsson rate on any lnk can only exceed the arrval rate due to edge effects. Indeed, the amount of bts delvered over any lnk on a gven tmeslot t s exactly equal to the offered transmsson rate r (t) whenever U (t) µ max. Hence, defnng α E as the fracton of tme that 0 U (t) < µ max, we have for all : λ tme average bt output rate of queue r α E µ max (17) whch holds because the transmsson rate on any lnk s always less than or equal to µ max. From (17) and (16) we have: ɛ[α R α L ] α E µ max (18) It follows from (18) and (15) that f the edge probabltes α E are less than or equal to O(ɛ 2 ), then h av Φ( λ) O(ɛ 2 ). Fortunately, the system s desgned to have a postve drft (away from the near empty edge regon) whenever queue backlogs are below the Q threshold. Hence, the edge probabltes can be made arbtrarly small by ncreasng the value of Q. Ths yelds mproved energy performance at the expense of ncreasng average queue occupancy and average delay. By settng Q greater than or equal to Ω( 1 ɛ log( 1 ɛ )) and defnng V = 1/ɛ 2, we can show that the resultng algorthm s ndeed ergodc and comes wthn a logarthmc factor of achevng the optmal energy-delay tradeoff curve of Theorem 2. Rather than provng ths result, n the next secton we use these deas to prove a smlar result for a more practcal control strategy. V. THE TRADEOFF-OPTIMAL CONTROL ALGORITHM To construct a more practcal strategy, we use the concept of Lyapunov drft. Lyapunov drft theory has been useful n the development of stablzng control algorthms for wreless networks [12]-[16], and recent extensons that treat stablty and performance optmzaton smultaneously are developed n [9] [8] [7]. Here we extend the theory to treat performance optmzaton wth near-optmal delay tradeoffs. The basc dea s to defne a Lyapunov functon that measures current queue congeston, and to make greedy decsons to mnmze the Lyapunov functon every tmeslot based on current queue states and channel states. Such greedy decsons do not requre knowledge of channel statstcs or arrval rates, and hence offer a potental means of overcomng the complexty exploson problem descrbed n the prevous secton. A. Algorthm Desgn Strategy We shall consder the followng Lyapunov functon L( U) consstng of a sum of exponentals: L( U) = [ ] e ω(u Q) + e ω(q U) 2 (19) where Q s a postve buffer threshold and ω s a postve value affectng the rate of exponental ncrease of the Lyapunov functon. We assume that Q > µ max as before. Ths Lyapunov functon reaches ts mnmum value L( U) = 0 when U = Q for all 1,..., N}, and ncreases exponentally when any of the U components devates from Q ether to the rght or to the left. Rather than usng the quadratc Lyapunov functons as n [12]-[16], [7], ths exponental Lyapunov functon s chosen to ensure a suffcently small probablty that queue backlog s wthn the near-empty edge regon. Allocatng power to mnmze the expected change n ths Lyapunov functon from one slot to the next usually creates a postve drft n the th queue when U (t) < Q and a negatve drft n the th queue when U (t) Q. However, ths s not always the case, and more structure s needed to acheve the same energy savngs as the mult-modal drft algorthm of the prevous secton. In partcular, we wll fnd t s crucal to ensure that nequaltes of the type (18) are satsfed.to ths

7 IEEE PROCEEDINGS OF INFOCOM, APRIL end, we defne α E(t) = P r[u (t) < µ max ], and defne tme average probabltes as follows: α L (t) = 1 t α E (t) = 1 t t 1 t 1 τ=0 αl (τ), αr (t) = 1 t τ=0 αr (τ), (τ), for all 1,..., N} t 1 τ=0 αe Note by defnton that α L (t) + α R (t) = 1 (because backlog s ether to the rght or to the left of the Q threshold), and α E (t) α L (t) (because f backlog s n the near empty edge regon, then t must also be left of the Q threshold). Smlarly defne the tme average transmsson rates r (t) as follows: r (t) = 1 t 1 E r (τ)} t τ=0 We have the followng Lemma: Lemma 2: For any queueng system descrbed by the update equaton (1), f the followng condtons are satsfed for all 1,..., N}: lm nf ( r (t) λ ɛ [ α R (t) α L (t) ]) 0 (20) then the followng nequalty holds: [ ɛ lm sup α R (t) α L (t) ] µ max lm sup α E (t) Proof: The proof s gven n Appendx B. For ntuton, the reader can compare the above lemma to nequaltes (16)-(18) from the prevous secton. To ensure that constrants (20) are satsfed, we use the noton of a vrtual queue developed n [7]. For each, we defne a vrtual queue X (t) wth the followng update equaton: X (t + 1) = max[x (t) (r (t) + ɛ1 L (t)), 0] + A (t) + ɛ1 R (t) (21) Ths update equaton s dentcal to the equaton representng a dscrete tme queue wth nputs and transmsson rates as shown n Fg. 3. Defnton 3: A dscrete tme queue wth an unfnshed work process U(t) s strongly stable f: 1 t 1 lm sup E U(τ)} < t τ=0 Lemma 3: If the queues X (t) are strongly stable, then the tme average condtons of (20) are satsfed for all. Proof: The proof of the lemma follows drectly from the fact that f a queue wth a bounded transmsson rate s strongly stable then the lm nf of the dfference between the tme average transmsson rate and the tme average arrval rate s greater than or equal to zero (see [31]). A (t) + ɛ1 R (t) X (t) r (t) + ɛ1 L (t) Fg. 3. An llustraton of the vrtual queue X (t) assocated wth update equaton (21). One of the objectves n our dynamc control algorthm shall be to stablze both the actual and vrtual queues of the system. We hghlght the fact that ths cannot be done va tradtonal Lyapunov stablty theory. Indeed, Lyapunov theory s typcally used by comparng a dynamc queue-length aware control strategy to a statonary queue-length ndependent control strategy (see [16] [9] for a detaled dscusson of ths). However, the nputs and server processes of the vrtual queues X (t) are hghly dependent on the queue state of the system, as they are affected by queue backlogs U (t) through the ndcator functons 1 L (t) and 1R (t). Our soluton to ths state-dependent control problem represents another sgnfcant contrbuton of the paper. B. The Dynamc Control Algorthm The desgn prncples we have developed lead to the followng dynamc control algorthm. The algorthm s mplemented for any control parameter V > 0, and for gven postve parameters ω, Q, ɛ (to be determned later as functons of V ). Tradeoff Optmal Control Algorthm (TOCA): The network controller performs the followng operatons every tmeslot t: 1) Observe the current backlog and channel state vectors U(t), S(t) and allocate a power vector P (t) = P, where P solves the followng problem: where: Mnmze: Subject to: V h( P ) N W (t)µ ( P, S(t)) P Π W (t) =1 R (t) [ ωe ω(u(t) Q) + 2X (t) ] +1 L (t) [ ωe ω(q U(t)) + 2X (t) ] 2) Data s transmtted wth rates r(t) = µ( P (t), S(t)). 3) Observe the current arrvals A (t) and update the vrtual queues X (t) accordng to (21) (usng the transmsson rates r (t) from step 2). Note that the algorthm bases decsons only on the current system state, and does not requre knowledge of traffc rates or channel statstcs. The weghts W (t) contan terms that swtch ON or OFF dependng on whether U (t) s to the left or rght of the Q threshold. Ths abrupt change n the weght functons effectvely steers the drft so that queue backlogs spend the approprate amount of tme n each regon. The above algorthm requres only O(1) multply-addexponentate operatons per lnk n order to update the vrtual queue backlogs and to compute the weghts W (t). The power allocaton optmzaton s the most complex part of the algorthm, although t can be solved easly for many systems, ncludng for systems wth concave rate-power curves wth the structure gven n (2), and for systems where there s only a fnte (and small) number of power allocaton optons. Further note that queues do not need to be physcally located n the same node, and hence the same algorthm and analyss apples to sngle-hop networks wth multple nodes. However, ths would requre dstrbuted mplementaton of the power allocaton optmzaton, whch s smple n the case of separable cost functons h( ) and separable power curves as n

8 IEEE PROCEEDINGS OF INFOCOM, APRIL (2), but may requre more structured multple access schemes for systems wth nter-channel nterference [7] [16]. A smlar buffer parttoned strategy can be desgned to treat mult-hop networks usng backpressure [7]-[9], [12], [16], although we omt ths topc for brevty. C. Performance To smplfy performance analyss, we assume that arrvals are bounded by a constant A max every slot, so that A (t) A max for all t. Defne δ max = max[µ max, A max ] as the maxmum change n the backlog of any queue durng a sngle tmeslot. Further assume that the arrval rate matrx λ s strctly nteror to the capacty regon Λ and has all postve entres. Defne ɛ max as the largest value ɛ such that ɛ λ for all 1,..., N}, and such that λ + ɛ Λ (where ɛ s a vector wth all entres equal to ɛ). Theorem 3: (TOCA Performance) Suppose that Φ( λ) s twce dfferentable at the pont λ. For any V > µ max and for any ω > 0, ɛ > 0 chosen such that ɛ < ɛ max and satsfyng: ωδ max e ωδmax ɛ/δ max (22) the TOCA algorthm mplemented wth these parameters stablzes all vrtual and actual queues of the system. Furthermore: ( ) 1 (a) N U Q + 1 ω log D+V hmax/n ωɛ/2 (b) h av Φ( λ) ND V + Nβɛ2 Φ( + lm sup λ) λ ɛ [ α R (t) α L (t) ] [ ɛ (c) lm sup µ max α R (t) α L (t) ] where D =e ω(µmax+amax Q) + ω(δ max + ɛ) +(A max + ɛ) 2 + (µ max + ɛ) 2 h max = max P Π h( P ) ND+V h max ωɛ/2 e ω(µmax Q) and where the tme averages are defned: 1 N U = 1 t 1 lm sup t τ=0 E } 1 N N U (τ) (23) 1 t 1 h av= lm sup t τ=0 h( E P } (τ)) (24) Theorem 3 s perhaps best nterpreted through the followng corollary: Corollary 1: Under the assumptons of Theorem 3, for any V > 0, f we choose: ω= ɛ e ɛ/δmax (25) δ 2 max and ɛ=1/ V, Q= 6 ω log(1/ɛ), then TOCA yelds:4 1 U O( V log(v )) N h av Φ( λ) O(1/V ) It follows that per-bt delay s also less than or equal to O( V log(v )). Ths holds regardless of whether or not 4 Throughout ths paper, the log( ) functon denotes the natural logarthm. 2 Φ( λ) s postve defnte. If t s postve defnte, then performance s governed by the Berry-Gallager bound of Theorem 2, and hence the TOCA algorthm yelds performance wthn a logarthmc factor of the optmal energy-delay tradeoff curve. In Secton VII we show that delay can be mproved beyond the Berry-Gallager bound when Φ( λ) s pecewse lnear (the case when the postve defnte assumpton fals). The proof of Corollary 1 follows by showng that the defnton of ω gven n (25) satsfes the nequalty n (22), and that ω = O(1/ V ), e ωq = O(1/V 3 ) (proof omtted for brevty). The proof of Theorem 3 s presented n the next secton. D. Expermental Observatons Smulatons of TOCA reveal that average queue backlog s very close to Q, whch s not surprsng because the drft s desgned to push backlog towards ths value. We note that the value Q = 6 ω log(1/ɛ) n Corollary 1 was chosen only to ensure a suffcently small probablty that queue backlog s n the near empty edge regon. However, our analyss was conservatve, and a varety of smulatons revealed that average queue backlog was n the edge regon only for the frst one or two tmeslots out of a total duraton of 10 mllon slots. In practce, a constant factor mprovement n average queue backlog can be obtaned by approprately reducng the value of Q, wthout sgnfcantly affectng edge probablty or average power expendture (results omtted for brevty). VI. PERFORMANCE ANALYSIS To prove Theorem 3, we frst present the man results concernng Lyapunov drft wth performance optmzaton from [9] [8] [7], presented here n a modfed form. Consder any dscrete tme system that evolves accordng to a Markov chan wth state space Z(t). Let Ψ( Z) be a non-negatve functon of the state space vector. We call Ψ( Z) a Lyapunov functon, and defne the condtonal Lyapunov drft ( Z(t)) as follows: ( Z(t)) =E Ψ( Z(t + 1)) Ψ( Z(t)) Z(t) } Lemma 4: (Lyapunov Drft) If there exst random processes f(t) and g(t) such that for every tmeslot and for all possble values of Z(t), the Lyapunov drft satsfes: ( Z(t)) E f(t) g(t) Z(t) } (26) then: lm sup 1 t 1 t τ=0 E g(τ)} lm sup 1 t 1 t τ 0 E f(τ)} Theorem 4: (Lyapunov Optmzaton) Assume the Z(t) state space represents a set of queue backlog values. If there exst values ɛ > 0, B > 0, V > 0, non-negatve and upper bounded cost functons c (t) and c(t), and a non-negatve functon q( Z) such that every tmeslot and for all Z(t) values the Lyapunov drft sastsfes: ( Z(t)) + V E c(t) Z(t) } B ɛq( Z(t)) + V E c (t) Z(t) }

9 IEEE PROCEEDINGS OF INFOCOM, APRIL then tme average performance satsfes: q( Z) B + V c ɛ c c + B/V where q( Z), c, and c are the lm sup expected tme averages of ther correspondng processes. Lemma 4 follows by takng expectatons of (26) and summng the telescopng seres, and Theorem 4 follows from the lemma (see [9] [7] for detals). The man dea of the theorem s that the dfference between c and a target value c can be made arbtrarly small accordng to the control parameter V, wth a correspondng ncrease n the tme average of q( Z(t)) that s lnear n V. A. Computng the Drft For the dynamc system under the TOCA algorthm, consder the Lyapunov functon Ψ( U, X) = L( U) + J( X), where L( U) s the exponental Lyapunov functon of (19) assocated wth the actual system queues, and J( X) = X2 s a quadratc Lyapunov functon assocated wth the vrtual queues. For ease of notaton, we defne the state varable Z(t) =[ U(t), X(t)] and defne ( Z(t)) = L ( Z(t)) + J ( Z(t)) to be the drft of the Lyapunov functon Ψ( ), where L ( Z(t)) and J ( Z(t)) are drft components assocated wth the actual queues and vrtual queues, respectvely. Lemma 5: If ωδ max e ωδmax ɛ/δ max, then: (a) L ( Z(t)) Ne ω(µmax+amax Q) + Nω(δ max + ɛ 2 ) ω 1R (t)eω(u(t) Q) E r (t) A (t) ɛ 2 Z(t) } ω 1L (t)eω(q U(t)) E A (t) r (t) ɛ 2 Z(t) } (b) J ( Z(t)) N(A max + ɛ) 2 + N(µ max + ɛ) 2 2 1L (t)x (t)e r (t) A (t) + ɛ Z(t) } 2 1R (t)x (t)e r (t) A (t) ɛ Z(t) } Proof: Part (b) of the lemma follows by squarng the vrtual queue equatons (21) and usng a standard quadratc Lyapunov drft argument [9] (calculaton omtted for brevty). Part (a) s proven n Appendx C by usng the dynamc equaton (1). By defnng D = e ω(µmax+amax Q) + ω(δ max + ɛ 2 ) + (A max + ɛ) 2 + (µ max + ɛ) 2 and usng Lemma 5, we have: ( Z(t)) + V E h( P (t)) Z(t) } DN ω 1R (t)eω(u(t) Q) E r (t) A (t) ɛ 2 Z(t) } ω 1L (t)eω(q U(t)) E A (t) r (t) ɛ 2 Z(t) } 2 1L (t)x (t)e r (t) A (t) + ɛ Z(t) } 2 1R (t)x (t)e r (t) A (t) ɛ Z(t) } +V E h( P (t)) Z(t) } The rght hand sde of the above drft expresson depends on the allocaton decson P (t) made by the controller at tme t (recall that r (t) = µ ( P (t), S(t))). The constructon of the TOCA algorthm from the prevous secton s now apparent: Allocatng power accordng to the TOCA algorthm at tmeslot t mnmzes the rght hand sde of the above drft expresson over all possble power allocatons P (t) Π. Indeed, t s clear from above that each weght W (t) s smply the sum of coeffcents multplyng the r (t) varables. It follows that ( Z(t)) s less than or equal to the resultng expresson when the power and rate decson varables P (t) and r (t) on the rght hand sde are replaced by the values P (t), r (t) correspondng to power and rates of the mult-modal statonary drft algorthm of Secton IV wth drft parameter ɛ. The P (t) and r (t) values depend only on the current regon of the U(t) vector wthn the set of 2 N regons of the backlog space, and satsfy (7) and (8). In partcular, we have the denttes: E r (t) A (t) U Q} = (λ + ɛ) λ = ɛ E A (t) r (t) U < Q} = λ (λ ɛ) = ɛ Usng decson varables r (t) and P (t) together wth these denttes n the rght hand sde of the above drft expresson makes the X (t) terms vansh, and furthermore we have: ( Z(t)) + V E h( P (t)) Z(t) } N D ω 1R (t)eω(u(t) Q) ɛ 2 ω 1L (t)eω(q U(t)) ɛ 2 +V E h( P (t)) Z(t) } (27) ωɛ N Now note that 2 1L (t) [ 1 e ω(u(t) Q)] 0 because U (t) < Q whenever 1 L (t) = 1. Smlarly, ωɛ 2 1R (t) [ 1 e ω(q U(t))] 0. Addng these nonnegatve terms to the rght hand sde of (27) and usng the fact that 1 L (t) + 1R (t) = 1 yelds: ( Z(t)) + V E h( P (t)) Z(t) } ND ωɛ [ 2 e ω(u (t) Q) + e ω(q U(t))] +V E h( P (t)) Z(t) } (28) where D= D + ɛω/2. The Lyapunov drft condton (28) s n the exact form for applcaton of Theorem 4, and hence: e ω(u Q) ND + V h max (29) ωɛ/2 e ω(q U) ND + V h max ωɛ/2 (30) h h + ND/V (31) where the overbar notaton denotes the lm sup expected tme average, as n Theorem 4. We now prove Theorem 3 drectly from these nequaltes. Proof: (Theorem 3 part (a)): By Jensen s nequalty and convexty of the functon e x, we have: e ω P 1 N N (U Q) 1 N N eω(u Q) Takng the log of both sdes and usng (29) yelds: ω 1 ) N N (U Q) log provng part (a) of Theorem 3. ( D+V hmax/n ωɛ/2

10 IEEE PROCEEDINGS OF INFOCOM, APRIL Proof: (Theorem 3 part (b)): Recall that P (t) s the power allocaton vector that would be chosen by the statonary randomzed polcy of Secton IV-A f U(t) s the observed backlog vector. By (14), we thus have for all t: E h( P } (t)) Φ( N λ) + Nɛ 2 β + Φ( λ) λ ɛ [ α R (t) α L (t) ] Summng over τ 0,..., t 1} and takng a lm sup yelds: h = lm sup 1 t 1 t τ=1 Φ( λ) + Nɛ 2 β N + lm sup E h (t)} Φ( λ) λ ɛ [ α R (t) α L (t) ] Usng ths nequalty together wth (31) proves part (b) of Theorem 3. Lemma 6: All actual and vrtual queues are strongly stable. Proof: The tme average bound n part (a) of the Theorem demonstrates strong stablty of all actual queues U(t). Smlarly, t can be shown that all vrtual queues X(t) are strongly stable. As an outlne of ths, note that nstead of substtutng the power allocaton polcy P (t) (whch yelds E r (t) U (t) < Q} = λ ɛ and E r (t) U (t) Q} = λ + ɛ), one can consder the polcy ˆP (t) that yelds E ˆr (t) U (t) < Q} = λ ɛ/2 and E ˆr (t) U (t) Q} = λ + ɛ + δ, where δ > 0 and satsfes λ + ɛ + δ Λ. Such a polcy yelds drft coeffcents of ɛ and 2δ, respectvely, multplyng the 1 L (t)x (t) and 1 R (t)x (t) terms. Proof: (Theorem 3 part (c)): Recall that α E(t) = P r[u (t) < µ max ]. Clearly we have: E e ω(q U(t))} E e ω(q U(t)) U (t) < µ max } α E (t) e ω(q µmax) α E (t) Summng over τ 0,..., t 1} and 1,..., N} yelds: 1 t 1 t N τ=0 E e ω(q U(τ))} N α E (t)e ω(q µmax) Takng the lm sup of both sdes and usng (30) yelds: N lm sup α E (t) ND + V h max e ω(µmax Q) (32) ωɛ/2 From Lemma 6 we know that all actual and vrtual queues of the system are stable. It follows from Lemma 3 n Secton V that condtons (20) are satsfed, and hence the result of Lemma 2 holds, placng a lower bound on the left hand sde of (32) and provng part (c) of Theorem 3. VII. BEYOND THE BERRY-GALLAGER BOUND Here we demonstrate a mode of super-fast convergence n the case when the mnmum energy functon Φ( λ) s pecewse lnear about the rate vector λ. Such cases are mportant and occur when there are only a fnte number of power or rate optons, such as when we can add ether full power to a sngle queue or no power to any queue, or when practcal codng schemes restrct to a fnte set of transmsson rates. In such cases, the Φ( λ) functon has a polyhedral structure wth a fnte number of vertces. We assume the λ pont s not a vertex. In ths case, the Taylor expanson of Φ( λ+ ɛ( H)) n (13) has a second order coeffcent β that s equal to zero whenever ɛ s wthn some range 0 ɛ ɛ max (for some value ɛ max ). Usng the fact that β = 0 and combnng the performance bounds n parts (b) and (c) of Theorem 3 yelds the followng bound on average power for any algorthm satsfyng the condtons of Theorem 3: h Φ( λ) ND V + max Φ( λ) λ } ( ) ND+V hmax ωɛ/(2µ max) e ωµmax e ωq (33) Ths leads to the followng corollary: Corollary 2: For the case β = 0, mplementng the TOCA algorthm wth parameters V, ɛ such that V > µ max, 0 < ɛ < ɛ max, and choosng ω = ɛ δ e ɛ/δmax, Q = 2 max 2 ω log(v ) yelds: U O(log(V )) h Φ( λ) O(1/V ) 1 N Proof: The proof of the average power bound follows mmedately from (33), and the proof of the backlog bound follows from part (a) of Theorem 3. One mght expect that the logarthmc delay bound s an artfact of the exponental Lyapunov functon that we have chosen, and that another Lyapunov functon (perhaps doubly exponental) mght yeld sub-logarthmc delay. However, ths s not the case. Indeed, below we present a smple example of a system wth a pecewse lnear Φ( λ) functon for whch t s mpossble to desgn an algorthm that acheves an energydelay tradeoff curve better than the one we have proven n the above corollary. Hence, our analyss captures the tghtest possble tradeoff. A. A Smple Example Consder a sngle queue wth a sngle nput stream of rate λ (n unts of packets/slot). Every tmeslot the controller decdes to ether transmt or reman dle, expendng one Watt of power when transmttng and 0 Watts when dle. The channel state vares n an..d. fashon between Good and Bad states, each equally lkely. Two packets can be transmtted n a sngle slot durng a Good channel state, whle only one packet can be transmtted durng a Bad channel state. It s not dffcult to show that the capacty regon for ths system s the set of all rates λ such that 0 λ 1.5, and for the objectve functon h(p ) = P the mnmum energy functon Φ(λ) s gven by the followng pecewse lnear curve: λ/2 f 0 λ 1 Φ(λ) = (λ 1) f 1 < λ 1.5 Intutvely, the mnmum energy polcy s to use as many of the Good channel states as possble, and transmt n Bad channel states only when absolutely necessary. Suppose the nput to the system s..d. and such that 2 packets arrve durng a tmeslot wth probablty p, and no packets arrve otherwse. The nput rate s thus λ = 2p (n unts of packets

11 IEEE PROCEEDINGS OF INFOCOM, APRIL per slot), and we assume that p = 1.25/2 so that λ = 1.25 n ths example. Thus, Φ(λ) = 0.75 Watts. Consder any control algorthm that stablzes the system, and for smplcty assume the algorthm s ergodc and yelds a well defned steady state. Usng standard nterchange arguments, t s not dffcult to show that any algorthm that does not transmt durng a Good channel state when the queue has at least two packets can be mproved n both energy and delay by transmttng the packets. Hence, we assume the polcy always transmts n such a scenaro. Defne λ g as the rate that the algorthm delvers packets to the destnaton, consderng only packets transmtted n the Good state when there are two or more packets n the queue. It follows that (1.25 λ g ) s the rate of all other packets, whch ncludes packets transmtted n Bad channel states and packets transmtted n Good channel states that are under-utlzed. We thus have: P = λ g (1.25 λ g) = 1.25 λ g /2 and hence P Φ(λ) = 0.5 λ g /2. Let U(t) represent the number of packets n the system at tme t, and let U represent the steady state average. Note that, n steady state, P r[u(t) 2U] 1/2 (by the Markov nequalty for non-negatve random varables U). Every tmeslot n whch the system has at least two packets, the queue ndependently decreases by two packets wth probablty (1 p)/2 (the probablty that no new packet arrves, and the channel state s Good so that two packets are transmtted). Defne T as the smallest nteger larger than U. The probablty that the system has fewer than two packets at a partcular tme t s thus greater than or equal to the probablty that U(t T ) 2U, and then havng T successve tmeslots when no packets arrve but channel states are Good. Thus: P r[fewer than two packets] 1 ( ) U+1 1 p = δ (34) 2 2 where we have defned δ as the lower bound on the probablty the system has fewer than two packets. Because a Good channel state arses every tmeslot wth probablty 1/2, and ths s ndependent of whether or not the queue has two or more packets at the begnnng of that slot, t follows that the fracton of unused or under-utlzed Good channel states s at least δ. Hence, λ g 2( 1 2 δ 2 ) = 1 δ, and so: P Φ(λ) = 0.5 λ g /2 = δ/2 0.5 (1 δ)/2 Defnng V =2/δ, we have P Φ(λ) 1/V. However, by defnton of δ n (34), we have: (U + 1) log((1 p)/2) = log(2δ) = log(4/v ) (35) and hence: U = log(v/4) log(2/(1 p)) 1 Ths proves that average delay ncreases logarthmcally n the V parameter whle the dstance to the mnmum energy level Φ(λ) s necessarly greater than or equal to 1/V. VIII. CONCLUSIONS We have establshed a fundamental tradeoff between energy and delay for mult-user wreless networks. Ths work extends the tradeoff results developed for a sngle lnk by Berry and Gallager, and demonstrates for the frst tme that the squareroot delay tradeoff s both necessary and achevable (to wthn a logarthmc factor) for general systems wth multple queues, multple users, and non-lnear power curves. Furthermore, we dscovered an mportant class of pecewse lnear systems that beat the tradeoff to acheve super-fast logarthmc delay. Our algorthms make use of a novel technque of Lyapunov drft steerng that swtches dscontnuously between dfferent weghts to drve average delay toward the optmal tradeoff curve. Ths approach overcomes an nherent state space exploson assocated wth delay optmzaton n mult-user systems. The resultng control algorthms are smple and do not requre knowledge of traffc rates or channel statstcs. Whle our analyss focused on the case of a mult-user downlnk, we note that these same technques can be used to solve multnode, mult-hop networkng problems, and are lkely to yeld fundamental mprovements n other networkng, control, and optmzaton contexts. APPENDIX A MULTI-USER BERRY-GALLAGER BOUND Here we prove Theorem 2. Wthout loss of generalty, we can assume that λ > 0 for all. Consder any polcy wthn a sequence of admssble polces of the type specfed n Secton III. Recall that such a polcy makes statonary (potentally randomzed) power allocaton decsons based only on the current backlog and channel state vectors U(t) and S(t). Let P ( U, S) be the random vector representng the power allocated under state U, S. Defne γ( U) =E µ( P ( U, S), S) U(t) = U } as the average transmsson rate vector when the queue backlog s U. Let π( U) represent the steady state dstrbuton of the U vector, and let E U } be the steady state average backlog n queue for 1,..., N}. Further, defne the expected queue drft δ( U) as follows: δ( U) = γ( U) λ (36) The value of δ( U) represents the expected dfference between transmsson rates and arrvals when the queue state s U. The followng lemma, whch s a modfed verson of a smlar lemma gven n [1], bounds the tal behavor of δ( U): Lemma 7: For any polcy wthn an admssble sequence of polces wth constants θ 1, θ 2 as defned n Secton III, and for any 1,..., N}, there exsts a value u such that:... δ ( U)dπ( U) θ 1θ2 2 (37) U U >u } 16E U } The proof of the lemma s almost dentcal to the proof of a smlar statement gven n [1], and s omtted for brevty. Now let h av represent the average power cost of the gven admssble polcy. Lemma 8: For any admssble polcy wth average power N cost h av and total average congeston U tot= E U }, we have: h av Φ( λ) Ω ( (1/U tot ) 2)