Structural Properties of LDP for Queue-Length Based Wireless Scheduling Algorithms

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1 Structural Propertes of LDP for Queue-Length Based Wreless Schedulng Algorthms V. J. Venataramanan and Xaojun Ln Abstract In ths paper, we are nterested n wreless schedulng algorthms for the downln of a sngle cell that can mnmze the queue-overflow probablty. Assumng that a samplepath large-devaton prncple holds for the baclog process, we frst study structural propertes of the mnmum-cost-pathto-overflow for a class of schedulng algorthms collectvely referred to as the α-algorthms. For a gven α, the α-algorthm pcs the user for servce at each tme that has the largest product of the transmsson rate multpled by the baclog rased to the power α. We show that when the overflow metrc s approprately modfed, the mnmum-cost-to-overflow under the α-algorthm can be acheved by a smple lnear path, and t can be wrtten as the soluton of a vector-optmzaton problem. Usng ths structural property, we then show that when α approaches nfnty, the α-algorthm asymptotcally acheves the largest value of the mnmum-cost-to-overflow under all schedulng algorthms. I. INTRODUCTION Ln schedulng s an mportant functonalty n wreless networs due to both the shared nature of the wreless medum and the varatons of the wreless channel over tme. In the past, t has been demonstrated that, by carefully choosng the schedulng decson based on the channel state and/or the demand of the users, the system performance can be substantally mproved (see the references n [). Most studes of schedulng algorthms have focused on optmzng the long-term average throughput of the users. Smlarly, n the class of stablty problems, the goal s to fnd schedulng algorthms that can stablze the networ at gven offered loads, whch also ensures that the long-term average servce rate s no less than the arrval rate of each user. An mportant result along ths drecton s the development of the socalled throughput-optmal algorthms [2. An algorthm s called throughput-optmal f, at any offered load that any other algorthm can stablze the system, ths algorthm can stablze the system as well. Therefore, a throughput-optmal schedulng algorthm s optmal f we only mpose stablty constrants,.e., t can stablze the system over the largest set of offered loads. Whle stablty (and ensurng that the long-term servce rate s no smaller than the arrval rate) s an mportant frstorder metrc of success, for many delay-senstve applcatons t s far from suffcent. Note that a stablty objectve ensures that the pacet delay does not ncrease to nfnty. For realtme applcatons such as voce and vdeo, we often need to ensure a stronger condton that the pacet delay can be upper Ths wor has been partally supported by the Natonal Scence Foundaton through awards CCF , CNS , and CNS The authors are wth School of ECE, Purdue Unversty, West Lafayette, IN Emal: {vvenat,lnx}@ecn.purdue.edu. bounded wth hgh probablty. One approach to quantfy the requrements of these delay-senstve applcatons s to enforce constrants on the probablty of queue overflow. In other words, we need to guarantee that the probablty of each user s queue exceedng a gven threshold s no greater than a target value. In ths paper, we are nterested n schedulng algorthms that are optmal the above type of delay constrants. We focus on the downln of a sngle cell n a cellular networ. The base-staton serves multple users. Due to nterference, the base-staton can only serve one user at a tme. We assume that perfect channel nformaton s avalable at the base-staton. The ultmate queston that we attempt to answer s the followng: Is there a delay-optmal algorthm n the sense that, at any gven offered load, the algorthm can acheve the smallest probablty of queue-overflow. Note that f we mpose a qualty-of-servce (QoS) constrant on each user n the form of an upper bound on the queueoverflow probablty, then the above optmalty condton wll also mply that the algorthm can support the largest set of offered loads the QoS constrant. The above queston has well-nown to be a dffcult one. The closest answer n the lterature s provded n [3, where the author studed the problem n a large-devaton settng, and showed that the so-called exponental-rule s delayoptmal n the case wth two-users. In a related result, t was shown that for the case when the servce rate s fxed, the largest-weghted-delay-frst (LWDF) algorthm s delayoptmal n a large-devaton settng [4, [5. To the best of our nowledge, the general case for wreless networs wth an arbtrary number of users s stll open. Note that to study the queue-overflow probablty, t s natural to use the large-devaton theory because the overflow probablty of nterest s often very small [6, [7. The queue-overflow probablty can then be mapped to the decay-rate of the taldstrbuton of the queue, and the delay-optmal schedulng algorthm wll correspond to the one that maxmzes ths delay-rate. Large-devaton theory has been successfully appled to wrelne networs (see, e.g., [8 [3) and to wreless schedulng algorthms that only use the channel state to mae the schedulng decsons [4 [6. However, when appled to wreless schedulng algorthms that use also the queue-length to mae schedulng decsons, ths approach encounters a sgnfcant amount of techncal dffculty. Note than many schedulng algorthms of nterest are of ths latter flavor,.e., they choose the user to serve based on both the channel state and the queue baclog. For example, the maxweght algorthm that s nown to be throughput-optmal [7

2 serves at each tme the user wth the largest product of the queue length and the servce rate. Intutvely, ths class of queue-length-based schedulng algorthms wll have a lower queue-overflow probablty compared to the queue-unaware algorthms because they mae an effort to suppress longer queues. Indeed, the wor n [8 has analytcally shown the superorty of queue-length-based schedulng algorthms over queue-unaware algorthms for a symmetrc case wth ON- OFF channels. However, the techncal dffculty assocated wth the queue-length-based schedulng algorthms s that the statstcs of the servce-rate process for each user s unnown (because now the servce-rate process s tghtly coupled wth the baclog process, the channel varatons, and the arrval process). In order to apply the large-devaton theory to queue-length-based schedulng algorthms, one has to use sample-path large-devaton, and formulate the problem as a mult-dmensonal calculus-of-varatons (CoV) problem for fndng the mnmum-cost path to overflow. The decay-rate of the queue-overflow probablty then corresponds to the cost of ths path, whch s referred to as the mnmum cost to overflow. Unfortunately, for many schedulng algorthms of nterest, ths mult-dmensonal calculus-of-varatons problem s very dffcult to solve. In the lterature, only some restrcted cases have been solved: Ether restrcted problem structures are assumed (e.g., symmetrc users and ON-OFF channels [8), or the sze of the system s very small (only two users) [9. Due to the above dffculty, the queston of fndng the optmal wreless schedulng algorthms under delay constrants becomes very challengng. In ths paper, we provde a number of results along ths drecton. Assumng that a sample-path large-devaton prncple holds, we study the structural property of the mnmum-cost-path-to-overflow for a class of queue-lengthbased schedulng algorthms. In partcular, we show that when the form of the overflow threshold s approprately modfed, at least one of the mnmum-cost-path-to-overflow s lnear. Ths result allows us to convert the calculusof-varatons problem (of sample-path large-devaton) to a vector-optmzaton problem. Usng ths structure property, we then show the man result of the paper that, as one of the parameters approaches nfnty, these class of queue-lengthbased schedulng algorthms wll asymptotcally acheve the largest mnmum-cost-to-overflow among all schedulng algorthms. As an mmedate corollary of ths result, we can show that wth the ON-OFF channel model, the max-weght schedulng algorthm s optmal. The rest of paper s organzed as follows. We frst present the system model and the class of queue-lengthbased schedulng algorthms (referred to as α-algorthms) n Secton II. In Secton III, we provde an upper bound on the mnmum-cost-to-overflow for any schedulng algorthm. We then study the structural propertes of the mnmum-costpath-to-overflow for α-algorthms n Secton IV. Then n Secton V, we prove the man result that, as the parameter α approaches nfnty, ths class of schedulng algorthms asymptotcally acheve the largest possble value of the mnmum-cost-to-overflow. Then we conclude. II. THE SYSTEM MODEL AND ASSUMPTIONS We consder the downln of a sngle cell n whch a basestaton serves N users. We assume a slotted system, and we assume that the state of the channel at each tme slot s..d from one of M possble states. Let C(t) denote the state of the channel at tme t =,2,..., and let p j = P[C(t) = j, j =,2,...,M. Let p = [p,...,p M. We assume that the base-staton can serve one user at a tme. Let Fm denote the servce rate for user when t s pced for servce at state m. We assume that data for user arrves as flud at a constant rate λ. Let λ = [λ,...,λ N. Let X (t) denote the baclog of user at tme t, and let X(t) = [X (t),...,x N (t). In general, the decson of pcng whch user to serve s a functon of the global baclog X(t) and the channel state C(t). Let U(t) denote the ndex of the user pced for servce at tme t. The evoluton of the baclog for each user s then gven by X (t + ) = [X (t) + λ {C(t)=m,U(t)=} Fm + () where [ + denotes the projecton to [0,+ ). Note that N {C(t)=m,U(t)=} = snce only one user can be served at a tme. One partcular class of schedulng algorthms that we wll study are collectvely referred to as the α-algorthms, where α s a parameter that taes values from the set of natural numbers. Gven α, the behavour of the algorthm s as follows. When the baclog of the users s X(t) and the state of the channel s C(t) = m, the algorthm chooses to serve the user for whch the product X α (t)f m s the largest. If there are several users that acheve the largest X α (t)f m together, one of them s chosen arbtrarly. It s well-nown that ths class of algorthms are throughput-optmal,.e. they can stablze the system at the largest set of offer-loads λ [7. Consder the system when t s operated at a gven offered load and s stable under a gven schedulng algorthm. In ths paper, we are nterested n the probablty that the maxmum baclog among the users exceeds a certan threshold B,.e., P[max X (0) B. (2) Note that the probablty n (2) s equvalent to a delayvolaton probablty when the arrval rates λ are constant, because the two types of events are related by (see [8, [20) P[Delay at ln d = P[X (0) λ d. In ths paper, we wll be nterested n schedulng algorthms that mnmze (2), at a gven offered λ. The problem of calculatng the exact probablty P[max X (0) B s often mathematcally ntractable. In ths paper, we are nterested n usng large-devaton technques to compute estmates of ths probablty. Specfcally,

3 we are nterested n those cases when the followng lmt exsts. lm B B logp[max X (0) B = I 0 ( λ). (3) (We wll dscuss how to compute I 0 ( λ) usng samplepath large-devaton and the correspondng assumptons n Secton II-A). Note that f Equaton (3) holds, t mples that, when B s large, the overflow probablty can be approxmated as P[max X (0) B exp( BI 0 ( λ)). Thus, the schedulng algorthm that mnmzes the overflow probablty corresponds to the one that maxmzes the decayrate I 0 ( λ). A. Sample-Path Large Devaton We next descrbe the sample-path large-devaton settng used to compute I 0 ( λ). We follow the conventon n [8, [2. Use B > 0 also as a scalng factor. For a large enough T, defne the scaled emprcal measure process on the tme nterval [, 0 as s B j (t) = B B(T +t) l=0 {C(l)=j}, for t = B T, = 0,...,BT, and by lnear nterpolaton otherwse. Note that, n the above defnton, we have scaled both the tme and the magntude. The quantty s B j (t) can be nterpreted as the sum of the (scaled) tme n [,t that the system s at state j. Further, t s easy to chec that M j= sb j (t) = t + T for all t [,0. Let sb (t) = [s B (t),...s B M (t). Analogously, defne the scaled baclog process as, x B (t) = B X (B(T + t)) for t = B T, = 0,...,BT, and by lnear nterpolaton otherwse. Let x B (t) = [x B (t),...,x B N (t). Note that the baclog process x B (t) s related to the emprcal measure process s B j (t) by = x B (t + B ) [ x B (t) + λ (4) B + (s B j (t) s B j (t B )) {U(B(T+t))=}Fm. Thus, gven a partcular ntal condton x B (), Equaton (4) defnes a mappng f B from the emprcal measure process s B (t) to the baclog process x B (t). Further, although we have assumed s B (t) to be pecewse lnear to begn wth, the defnton of the mappng f B can be naturally extended to all absolute contnuous functons s B (t). For any φ 0 and M j= φ j =, defne = M j= φ j log φj p j. The sequence of emprcal measure processes s B (t) s nown to satsfy a sample-path large devaton prncple [7, p76 wth large-devaton rate-functon Is T ( s( )) gven as follows: I T s ( s( )) = 0 H( φ(t) p)dt, f s(t) s absolute contnuous and component-wse nondecreasng on [,0, s() = 0, and M j= s j(t) = t + T for all t; where φ(t) = d dt s(t) (Note that φ(t) s well defned almost everywhere on [, 0 snce s(t) s absolute contnuous on [, 0). Otherwse, I T s ( s( )) = +. Such a large-devaton prncple means that, for any set Γ of trajectores on [, 0 that s a contnuty set [7, p5 accordng to the essental supremum norm [7, p76, p352, the probablty that the sequence of emprcal measure processes s B (t) falls nto Γ must satsfy lm B B logp[ sb ( ) Γ = nf s( ) Γ IT s ( s( )). (5) In ths paper, we assume that a sample-path large-devaton prncple also holds for the sequence of baclog processes x B (t). Specfcally, we adopt the followng assumptons: A) As B, the sequence of mappngs f B has a lmtng mappng f that also maps any absolute contnuous emprcal measure process s(t) to a baclog processes x(t). B) The mappng f s unque and s contnuous wth respect to approprately-chosen topologes of the space of emprcal measure processes and the space of the baclog processes. C) The sequence of mappngs f B are exponentally equvalent to f [7, p30. If these assumptons hold, then for any sequence of baclog processes that start from x B () = 0, we can nvoe the contracton prncple [7, p3 and obtan a samplepath large-devaton prncple for the sequence of baclog processes x B (t) wth large-devaton rate-functon gven by: { 0 } Ix T ( x( )) = nf H( φ(t) p)dt { s( ): x( )=f( s( ))} where φ(t) = d dt s(t), and the nfmum s taen over all emprcal measure processes s( ) that map (under the mappng f) to the same baclog process x( ) gven that x() = 0. (We refer the readers to [2 for cases when these assumptons hold.) Defne an overflow metrc as a functon h( x) such that h( 0) = 0, h(b x) = Bh( x), and h( x) s component-wse ncreasng. An overflow metrc of the form h( x) = max x, wll be consstent wth the queue-overflow threshold defned earler. However, later we wll also use other overflow

4 metrcs. The event of queue overflow s then represented by h( x B (0)). As B, we have, lm B B logp[h( xb (0)) = nf{ix T ( x( )) over all trajectores x( ) that go from x() = 0 for some T > 0 to h( x(0)) = }. (6) The trajectory that attans the nfmum n (6) s often called the most lely path to overflow. The value of the nfmum tself s often called the mnmum cost to overflow. Note that I 0 ( λ) n (3) corresponds to (6) when h( x) = max x. In the rest of the paper, our goal s to fnd schedulng algorthms that can acheve the largest value of I 0 ( λ) (.e., the largest value of the mnmum-cost-to-overflow) at a gven offered load λ. III. AN UPPER BOUND ON I 0 ( λ) We frst provde an upper bound on I 0 ( λ) over all schedulng algorthms. Then, n Secton V, we wll show that the α-algorthm asymptotcally acheves ths upper bound as α, and hence s asymptotcally optmal. A. Defntons Gven a schedulng algorthm A (e.g., an α-algorthm ), and an overflow metrc h( ), let Ψ A be the set of all possble trajectores under schedulng algorthm A. Precsely, each element of Ψ A s a trplet ( φ( ), x( ),T) such that T > 0, φ(t) = d dt s(t) where s( ) s an nstance of the emprcal measure process, x() = 0, and x(t), t [,0, s the correspondng baclog process governed by the schedulng algorthm A. For ease of exposton, we use F(Ψ A,h) to denote the calculus-of-varatons problem n (6),.e., F(Ψ A,h) nf φ(t),t 0 H( φ(t) p)dt ( φ( ), x( ),T) Ψ A (7) h( x(0)) = (8) x() = 0. (9) In partcular, we use F(Ψ A,max) to denote the case when h( x) = max x. Let Ψ A Ψ A be defned as follows Ψ A = { ( φ( ), x( ),T) Ψ A such that } d φ(t) dt = 0,.e., t contans all trajectores that correspond to a lnear emprcal measure process s(t). We can smlarly defne F(Ψ A,h) where the constrants set Ψ A n (7) s replaced by Ψ A. We defne and let ẃ( φ) mn φ, x max(x ) [ x = µ m 0, I opt nf φ λ + µ mfm for all N µ m = φ m for all m, (0) ẃ( φ). The followng theorem states that I opt s an upper bound on I 0 ( λ) for any schedulng algorthm. Theorem : For any schedulng algorthm A, F(Ψ A,max) I opt. Proof: Frst, note that by defnton, Ψ A Ψ A. Ths fact leads to the concluson that F(Ψ A,max) F(Ψ A,max) snce the constrant set n the optmzaton problem on the rght hand sde s smaller. Thus, t suffces to show the followng F(Ψ A,max) I opt = nf φ ẃ( φ). Consder any trajectory ( φ(t), x(t),t) n the feasble regon of F(Ψ A,max). Recall that φ(t) s a constant by defnton of Ψ A. Denote φ(t) = φ. By (9), x() = 0. Further, by the queue-evoluton equaton (4) we have the followng nequalty, x (0) T[λ M µ mfm +, where by µ m we denote the average fracton of tme n [,0 that the user s served and the channel state s m. Fnally, by (8) we now that max x (0) =. We thus have = max x (0) T max ([λ µ mfm + ) Tẃ( φ) T H( φ p) ẃ( φ). Note that T s precsely the cost of the trajectory. Snce ths nequalty holds for all trajectores n Ψ A, we have F(Ψ A,max) I opt. IV. STRUCTURAL PROPERTIES OF THE MINIMUM-COST-PATH-TO-OVERFLOW FOR α -ALGORITHMS We next turn our attenton to the α-algorthms. Our ultmate goal s to show n Secton V that the α-algorthms asymptotcally acheve the mnmum-cost-to-overflow equal to I opt. In ths secton, we frst derve some structural Ths upper bound s equvalent to the one n [3.

5 propertes of the mnmum-cost-path-to-overflow under α- algorthms. Note that the calculus-of-varatons problem n (6) and (9) wth the overflow metrc h( x) = max x s often very dffcult to solve. In general, the mnmum-cost-path-tooverflow may not be of a smple lnear form. The trc that we use here s to modfy the overflow metrc to one that s talored to the schedulng algorthm. In partcular, for the α-algorthm, we use the overflow metrc h( x) = V α ( x) ) α+. Note that V α ( x) s well-nown to be the Lyapunov functon for provng that the α-algorthm s throughput-optmal. Thus we wll refer to V α ( x) as the Lyapunov overflow metrc, and refer to h( x) = max x as the max-queue overflow metrc. The connecton between V α ( ) and max x wll be clear n Secton V. Wth the overflow metrc V α ( x), the calculus-of-varatons problem for fndng the mnmum-cost-to-overflow s represented by F(Ψ α-algo,v α). A. A Lower bound on the mnmum-cost-to-overflow We frst provde a lower bound on F(Ψ α-algo,v α). We start wth a property of the lmtng mappng f that maps the emprcal measure process s(t) to the baclog process x(t). Note that accordng to the defnton of x(t) and s(t), they are both Lpschtz-contnuous, and hence are dfferentable almost everywhere. For any tme t such that both x(t) and s(t) are dfferentable, the followng propertes can be shown: N There must exst µ m(t) 0 such that µ m(t) = φ m (t) and ẋ (t) = [λ N the notaton [u + v = µ m(t)fm + x (t), where we have used { u f v > 0 or u 0 0 otherwse. Recall that φ m (t) = d dt s m(t) can be vewed as the fracton of tme the system s n state m n an nterval [Bt,B(t+δt) mmedately after t. µ m(t) can then be vewed as the fracton of tme that user s served and the system s n state m wthn such an nterval. In addton, the followng lemma can be shown. Lemma 2: µ m(t) = 0 f x α (t)fm < max x α (t)fm. Proof: Ths can be shown by notng that f x α (t)f m < max x α (t)f m, then for all suffcently large B, (x B (s))α F m < max (x B (s))α F m holds for an nterval s [Bt,B(t + δt) mmedately after t. Hence, user wll not be pced for transmsson over ths entre nterval. We can thus show that µ m(t) = 0. We now use the Lyapunov functon approach n [22 to derve a lower bound on F(Ψ α-algo,v α). Frst, defne a local rate functon of x(t) as: l( x, y) = nf φ y = [λ µ m 0 and µ mfm + x for all N µ m = φ m for all m µ m = 0 f x α F m < max x α F m. Note that l( x, y) denotes how lely the trajectory x( ) can move n the drecton d dt x(t) = y mmedately after t, gven x(t) = x. Usng Lemma 2 we thus have F(Ψ α-algo,v α) nf T 0 l( x(t), x(t))dt ( φ( ), x( ),T) Ψ A V α ( x(0)) = x() = 0. Further, lettng V (t) = V α ( x(t)), we can defne the local rate-functon of V (t) as Then, l V (v,w) = nf x, y F(Ψ α-algo,v α) nf T l( x, y) V α ( x) = v [ T x V α( x). y = w. 0 l v (V (t), V (t))dt V () = 0,V (0) =. () Note that the rght-hand-sde s a one-dmenson calculusof-varatons problem that s much easer to solve. For α- algorthms, f y and µ m satsfy the constrants of l( x, y), then [ T. y = = x V α( x) ( N ) α α+ [ N ) α x α (λ [ α+ N x α λ µ mfm) φ m max x α Fm. (2) Hence, the local rate-functon of V (t) can be rewrtten as l V (v,w) = nf φ, x

6 ) α α+ [ N ) α+ x α λ φ m max x α Fm = v. = w It s easy to show that l V (v,w) s ndependent of the value of v,.e., l V (v,w) = l V (,w) for all v 0. Let l(w) l V (,w). Then the soluton to the calculus-ofvaratons problem on the rght-hand-sde of () s gven by [6, p520 J mn w 0 w l(w) = mn φ, x,w 0 w H( φ p) [ N x α λ φ m max x α Fm = w ) α+ =. (3) We thus obtan the followng result. Lemma 3: The mnmum cost to overflow F(Ψ α-algo,v α) must be no smaller than J. B. Attanablty of the Lower-bound J In ths subsecton, we show that the lower bound J s attanable wth a smple lnear trajectory s(t) = (t + T) φ, t. Note that the soluton of (3) wll produce a φ (t s easy to verfy that such a φ always exsts). If ths φ can n fact map to a trajectory that starts from x() = 0 and overflows at t = 0, then the mnmum-costto-overflow F(Ψ α-algo,v α) must be no larger than the cost of ths trajectory J 2 = TH( φ p). Further, f J 2 = J, then we can conclude that F(Ψ α-algo,v α) = J. We next show that ths s ndeed the case. Towards ths end, we frst show that for each lnear emprcal measure process s(t) = (t+t) φ, t, there exsts a unque trajectory x(t) startng from x() = 0. We wll need the followng lemma. Lemma 4: (a) Gven φ, the optmal values of the followng two problems are the same. a( φ) = max x 0 [ N x α λ N, φ m max x α Fm and b( φ) = mn y 0 y α+ y = [λ µ m 0, ) α+ Fmµ m + for all N µ m = φ m for all m. (b) The optmzer x for a( φ) and the optmzer y for b( φ) are both unque and they satsfy x = γ y for some γ > 0. Further, f the optmzer x 0, then x and y are the only vectors that satsfy the followng condtons: there exst µ m 0 such that N µ m = φ m, y = [λ M N (x )α+ =, and F mµ m +, x = γy for some γ > 0, µ m = 0 f (x ) α Fm < max (x ) α Fm. Lemma 4 can be proved by showng that b( φ) s the dual problem of the optmzaton problem a( φ) wth an approprate change of varables. The varables µ m of b( φ) are the Lagrange multplers. Due to lac of space, we omt the proof here and the detals are provded n our techncal report [23. We now show that, f the emprcal measure process s(t) s lnear, then the queue trajectory x(t) must also be lnear, and t must solve b( φ) n Lemma 4. For ease of exposton, we start the tme from t = 0 (nstead of t = ). Lemma 5: Let x(0) = 0 and s(t) = tφ for t 0. Then the correspondng queue trajectory x(t) under the α-algorthm must satsfy the followng: (a) The queue trajectory s lnear,.e., for each, x (t) = x t for some x 0. N (b) There must exst µ m 0 such that µ m = φ m, and µ m = 0 f x α (t)fm < max x α (t)fm for all t. In other words, the queue trajectory x(t) s consstent wth the schedulng rule. (c) x s the unque mnmzer of b( { φ). Proof: Let Ω( φ) = λ λ = M µ mfm, } N µ m = φ m,µ m 0. Note that Ω( φ) would have been the capacty regon (for stablty) f the channel state dstrbuton was φ. Recall that (from (2)) dv (t) dt [ Vα ( x(t)) = x T. d dt x(t)

7 = ) α [ α+ N (t) x α (t)λ φ m max x α (t)fm If λ Ω( φ), dv (t) we wll have dt < 0 whenever V (t) = V α ( x(t)) > 0. Hence, startng from x(0) = 0, we must have V (t) = 0 and x(t) = 0 for all t 0. Therefore, part (a) holds wth x = 0 for all. Part (b) then trvally holds. Part (c) follows from Lemma 4 snce the mnmzer of b( φ) for ths case s y = 0. On the other hand, f λ / Ω( φ), then for all x(t) 0, by settng x (t) = x (t) P [ N (t) α+, we have dv (t) N = x α (t)λ φ m max x α (t)fm dt [ N α+ and (t) =. dv (t) dt We thus have a( φ) and V (t) ta( φ). Ths shows that ta( φ) upper bounds the maxmum growth of V (t). On the other hand, let µ m be the average fracton of tme n [0,t that user s pced and the channel state s m. Then N µ m = φ m for all m, and x (t) t[λ M µ mfm +. Hence, V (t) = V α ( x(t)) tb( φ). However, by Lemma 4, a( φ) = b( φ). We thus have V (t) = V α ( x(t)) = ta( φ) = tb( φ),.e., there s only one possble trajectory V (t) gven that s(t) = tφ. Further, we have V α ( x(t) t ) = b( φ). x(t).e., t optmzes b( φ). Snce the optmzer of b( φ), denoted by x, s unque, we thus have x(t) = t x. Ths shows parts (a) and (c). Part (b) follows from part (b) of Lemma 4. Proposton 6: The mnmum cost to overflow F(Ψ α-algo,v α) s equal to J. Proof: Let φ, w, and x denote the soluton to J. If we use s(t) = (t + T) φ, t as the underlyng emprcal measure process, and let the queue process start from x() = 0 where T = /w, then there s a lnear trajectory accordng to Lemma 5,.e., x(t) = (t + T) x, where x s the mnmzer of b( φ ). Further, by the structure of J, w 0, and thus [ N w = max x α λ φ m max x α Fm x 0 N =.. The rght hand sde s equal to a( φ ), whch s also equal to b( φ ). Hence, V (0) = V α (T x ) = Tb( φ ) = w w =. In other words, the lnear emprcal measure process s(t) = (t + T) φ, t, ndeed drves the queue from x( w ) = 0 to overflow at t = 0. Hence, F(Ψ α-algo,v α) TH( φ p) = w H( φ p) = J. Then, usng Lemma 3, F(Ψ α-algo,v α) J, the result then follows. Hence, we conclude that the mnmum-cost-to-overflow F(Ψ α-algo,v α) s attanable by a smple lnear trajectory whose cost s gven by J = nf φ b( φ). V. ASYMPTOTICAL OPTIMALITY OF α-algorithms In ths secton, we return to the orgnal overflow metrc h( x) = max x and we wll establsh that, n the lmt as α, the α-algorthm asymptotcally acheves the largest mnmum-cost-to-overflow equal to I opt gven n Secton III. We wll use some of the results and notatons from Secton IV. In partcular, to emphasze the dependence of b( φ) on α, we rewrte b α ( φ) = b( φ) here: b α ( φ) mn φ, x V α ( x) x = [λ µ m 0, µ mfm + for all N µ m = φ m for all m. (4) In Secton IV, we have shown that F(Ψ α-algo,v α) = nf φ b α( φ). Earler, n Secton III, we showed that I opt s an upper bound on the mnmum-cost-to-overflow for all schedulng algorthms. We now show the followng. Theorem 7: lm α F(Ψ α-algo,max) I opt. Proof: Frst, t s easy to show that F(Ψ α-algo,max) F(Ψ α-algo,v α). Ths s true because f a trajectory overflows accordng to the max-queue overflow metrc,.e., max x (t) =, then t must have already overflowed accordng to the Lyapunov overflow metrc snce max x (t) = V α ( x(t)). Usng Proposton 6, we then have F(Ψ α-algo,max) nf φ b α ( φ). We wll now show that lm α nf φ b α( φ) = I opt nf φ ẃ( φ), whch then completes the proof. Observe that b α ( φ) n (4) and ẃ( φ) n (0) both have the same constrant set. The followng nequalty s easly establshed. N α+ max (x ) V α ( x) max(x ) for all x 0.

8 Hence, N α+ ẃ( φ) bα ( φ) ẃ( φ). Note that ths mples that b α ( φ) > 0 ẃ( φ) > 0. Let Q = { φ such that ẃ( φ) > 0}. It s suffcent to show lm nf α φ Q b α ( φ) = nf φ Q ẃ( φ). Now, for all φ n Q, the followng holds ẃ( φ) H( φ p) b α ( φ) N α+ ẃ( φ). Tang nfmum across the nequaltes over the set Q, we get nf φ Q ẃ( φ) nf φ Q b α ( φ) N α+ nf φ Q ẃ( φ). Lettng α, N α+. The result of the Lemma then follows. Combnng Theorem and Theorem 7, we conclude that the α-algorthm asymptotcally acheves the largest possble value of the mnmum-cost-to-overflow. A. Systems wth ON-OFF Channels Consder the scenaro where F m can tae ether the value 0 or a postve constant C. Ths scenaro corresponds to a wreless system wth ON-OFF channels and the ON-rates for all users are the same. In ths case, for any α > 0, x α F m max x α Fm x Fm max x Fm. Hence, the α-algorthms (for any α ) are equvalent to the max-weght algorthm (.e., α = ). Usng the result n ths paper, we mmedately reach the followng corollary. Corollary 8: For the above ON-OFF channel model, the max-weght schedulng algorthm acheves the largest mnmum-cost-to-overflow I opt. VI. CONCLUSION In ths paper, we study wreless schedulng algorthms that can mnmze the queue-overflow probablty. Assumng that a sample-path large-devaton prncple holds for the baclog process, we frst establsh a structural property of the mnmum-cost-path-to-overflow for the class of α- algorthms. Specfcally, when the overflow metrc s approprately modfed, we show that the mnmum-cost-tooverflow under the α-algorthm can be acheved by a smple lnear path, and t can be wrtten as the soluton of a vectoroptmzaton problem. Usng ths structural property, we then show that when α approaches nfnty, the α-algorthm asymptotcally acheves the largest value of the mnmumcost-to-overflow under all schedulng algorthms. For future wor, we plan to study condtons under whch the sample-path large-devaton prncple holds. 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