Scheduling in Wireless Networks under Uncertainties: A Greedy Primal-Dual Approach

Scheduling in W ireless Networks under Uncertainti es: A Greedy Primal-Dual Approach Qiao Li qiaoli@cmu.ed u Departmen t of Electrical and C ompu ter Engin eering Carnegie Mellon Uni versity 5000 Forbes A ve., Pittsburgh, P A 15213 Rohit Negi negi@ece.cmu.ed u Departmen t of Electrical and Compu ter Engin eering Carnegie Mello n University 5000 Forbes A ve., Pittsburgh, P A 1521 3 Abstract —This paper proposes a dynamic primal-dual type algorithm to solv e the optimal scheduling problem in wireless networks subject to u ncertain p arameters, which are generated by stoc hastic netwo rk processes such as random packet arr iva ls, channel fading, and node mobilities. The algorithm is a gen- eralization of the well-k nown max-weight sched uling algorithm proposed by T assiulas et al. , wh ere only queu e length information is used f or computing the schedules when the arriv al rates are uncertain. Usin g the tech nique of fluid limit s, sample p ath con vergence of the algorithm to an arbitrarily close to op timal solution is prov ed , under the assumption that the Strong Law of Larg e Numbers (SLLN) applies to t he random processes which generate the uncertain parameters. The perf ormance of the algorithm is furth er verified by simulation results. Th e method may potentially be app lied to other applications where dynamic algorithms for conv ex problems with uncertain parameters are needed. I . I N T RO D U C T I O N Scheduling in wire less n etworks in volves efficiently allo- cating n etwork resource s am ong competin g network users in the pr esenc e of unce rtainties . Th ese u ncertainties m ay b e either due to unexpected events, such as link failures, or due to intricate cross-layer interac tions in wireless networks. For example, th e packet arriv al rates may be unknown (e.g. , [1], [2]), which depe nd on up per layer dynam ics such as ro uting and congestion control protocols. As another examp le, the wireless cha nnel statistics may be also unkn own (e. g., [2]), since they dep end on comp lex network events such as channel fading, power control a nd node m obilities. In the presen ce o f such unce rtain parameters, it may no longer be optimal to use th e static allo cation ap proach (e.g. , [3]), which p roduces per iodic sched ules by solving a static underly ing conve x o ptimization prob lem (which usually has exponential size) with estimated un certain par ameters. In p ar- ticular , if these uncer tain par ameters ar e slowly converging, or time varying, the estimated pa rameters may fail to trac k the chang es in their true value, which often leads to subo p- timal sche dules. F urther, it may be impractical to estimate the uncertain p arameters fo r large wir eless ne tworks, as the number o f the parameter s m ay grow fast (e.g ., exponen tially) with the size o f the network . For a simple illustratio n, consider a wireless n etwork with n links, such that each link rand omly switches on or off after ce rtain rand om numb er of time slots. In such case, a c omplete spec ification o f the network topology probab ilities may req uire a s large as 2 n − 1 parameters, which quickly becomes imp ossible to estimate as n grows. On the other h and, online algorithms (such as [ 1], [2], [4], [6], [7]) are more robust to the chang es to the unc ertain parameters (such as arrival r ates), since th ey use queu e length (which can a lso be interprete d as prices [9], [ 10]) info rmation for schedulin g in each tim e slot. For example, it has been shown that [4] such algo rithms can achieve network stability ev en if the “instantan eous rates” of the traffic vary arbitrar ily inside the network capacity region . Fur ther , co mpared to the estimation based approach , these online algorith ms may b e more scalable to the network size, in the sense that the dimension of the qu eue length vector co rrespond s to the number of co nstraints ( such as r ate constraint for each link ), which usu ally grows slowly , where as the n umber of uncer tain parameters c an g row very fast (e.g., exponen tially). In this pap er w e s olve a general class o f optimal wire- less ne twork sched uling problem s with uncertain p arameters, whose underly ing static p roblem is described by the convex optimization prob lem OPT in Section II. Essentially , we require that the structures of the con vex objecti ve fu nctions an d conv ex constraint func tions ar e known, excep t the values o f the u ncertain p arameters. These paramete rs will be gener ated by cer tain stochastic p rocesses and observed by the network gradua lly over tim e slots. W e propo se a g reedy primal-du al dynamic algorith m (Algorithm 1 in Section II I) to achieve the optimal sched uling asympto tically . Using the novel techn ique of fluid limits [12], op timality can be guaran teed unde r the assumption that a ll the network proce sses g enerating the uncertain par ameters satisfy SLLN ( see details in Section II) . Note that this assumption is quite mild, since we can guaran tee optimality as long as these pr ocesses conver ge, no matter how slowly the co n v er gence happen . Thus, intuiti vely , our algorithm can automa tically tra ck the conver gence of these processes and correct the mistakes which are made within a ny finite time h istory . Our algorith m is a genera lization o f the well-kn own max- weight algorithm [1], which was shown to be throu ghpu t optimal fo r i. i.d ar riv al pro cesses. Our algo rithm is related to, but different from the utility- optimal sched uling algor ithm b y Neely [ 2], which achieves the o ptimal schedu ling by clev erly transform ing the p roblem into op timizing the time-average of the utility (a nd constraint) fu nctions, to which a dual-typ e algorithm ap plies. Our algor ithm is also different fro m the primal-du al algorith m by Neely [5 ] since we use a scaled queue length (by 1 /t ) in the scheduling, which c orrespon ds to the approximated grad ient. Stolyar [6] also prop osed a primal- dual type sch eduling algorithm, and proved its optimality using a fluid limit obtained from a different scaling. Since th e fluid scaling in [6] is taken over different systems, it is hard to relate the optimality in th e fluid limits to the one in the original system. Finally , o ur algor ithm can be u sed as a MA C layer solution for th e general f ramework of cross-layer o ptimization problem (e.g., [4], [7], [8], [10]) for wireless networks. The organization of the fo llowing sections is as fo llows: In Section II we describ e th e queueing network model as well as the optimization p roblem O PT , an d in Sectio n II I we d escribe the schedu ling algo rithm. Section IV pr oves the o ptimality of the algorith m, Sec tion V illu strate the algor ithm perf ormance in simulation, and finally Section VI conclu des this paper . I I . N E T W O R K M O D E L A N D P RO B L E M F O R M U L AT I O N In this section we d escribe the q ueueing network and propo se the optim ization prob lem OPT . W e first intro duce the queu eing network mo del. A. Qu eueing Network W e con sider th e scheduling prob lem at th e med ium access (MA C) layer of a m ulti-hop wireless network, wh ere the network is modeled as a set o f n links. W e a ssume a time- slotted network m odel, and in each time slot t , the n etwork is in on e of the following states: M , { 1 , 2 , . . . , M } . The network state can be u sed to mode l network topology , chan nel fading, and user mo bility , etc. W e fu rther assume that th ese network states can be measured by th e u ser nodes 1 , which are a ssumed to be equip ped with chan nel mon itoring devices. W e associate each n etwork state m ∈ M with a finite set of resou rce allocation mod es Ξ ( m ) = { ξ ( m ) 1 , ξ ( m ) 2 , . . . } , where each mode ξ ( m ) k ∈ Ξ ( m ) correspo nds to a configu ration of network resource allocation, such as carrier and frequency selection in OFDM systems, spreadin g codes choice in CDMA systems and time slots assignment in TDMA systems. Denote y as the u ncertain p arameters, which are gener ated by the stochastic proc ess Y ( t ) , which is a cumulative vector process w hose time av erage conv erges to y . Spec ifically , the assumptions on Y ( t ) are : 1) it is subject to SLLN, i.e. , with probab ility 1 (w .p.1), lim t →∞ Y ( t ) /t = y (1) and 2) it has un iformly bo unded incr ement in ea ch time slot: k Y ( t ) − Y ( t − 1) k ≤ Y max , ∀ t > 0 (2) 1 Note that although the netwo rk states are global, they often all ow (approxi- mate) deco mpositions (e.g., [2]) according to eithe r the geographic structure or channe l orthogon ality , in which case one only needs to measure local network states. where Y max > 0 is a finite con stant. For specific examples, consider th e cumulative network state proce ss M m ( t ) : S m ( t ) = t X τ =1 1 { m ( τ )= m } (3) where m ( t ) is the network state at time slot t , and 1 {·} is the indicato r functio n, i.e., 1 { true } = 1 an d 1 { false } = 0 . T hus, SLLN imp lies that w .p.1, lim t →∞ 1 t S m ( t ) = lim t →∞ 1 t t X τ =1 1 { m ( τ )= m } = π ( m ) , ∀ m ∈ M As a nother example, c onsider the external p acket arrival pr o- cess A ( t ) , w hich is a n × 1 vector r epresenting the cu mulative external packet arriv als du ring the first t tim e slots. Similar ly , SLLN imp lies that w .p.1, lim t →∞ A ( t ) /t = a , w .p.1 (4) Further, we requir e that the m aximum packet arriv a ls in any time slot ar e unifor mly b ound ed: k A ( t ) − A ( t − 1) k ≤ A max , ∀ t > 0 (5) where A max > 0 is a finite c onstant. W e finally describe th e qu eueing system model. The queu- ing dynamics o f th e network is m odeled as f ollows: Q ( t ) = Q (0) + A ( t ) − R ( m ( t )) D ( t ) (6) where Q ( t ) is the queue length vecto r at tim e slot t , and R ( m ( t )) is the n × n routing matrix, such that R ii ( m ( t )) = 1 , and R ij ( m ( t )) = − 1 only if link i serves as the next h op for link j at time slot t , as specified by certain ro uting p rotocols, otherwise R ij ( m ( t )) = 0 . Note that the routin g matrix R ( m ( t )) is a func tion o f the network state, and therefore SLLN imp lies lim t →∞ 1 t t X τ =1 R ( m ( τ )) = X m ∈M π ( m ) R ( m ) (7) D ( t ) is a n × 1 vector represen ting the cum ulativ e packet departur es du ring the first t time slots, which are d etermined by the r esource allocation m odes as specified by the scheduler in eac h time slot. Specifica lly , at each time slot t with network state m , if the scheduler cho oses a resour ce allocation mode ξ ( m ) k , ther e is an associated d eparture vector G ( m ) k , whose each entr y G ( m ) ki correspo nds to th e nu mber of packets transmitted suc cessfully b y lin k i . Note that the choic e of resource allocatio n mod e ξ ( m ) k is subject to th e co nstraint that Q ( t ) − G ( m ) k  0 , so that th e queu e len gths never b ecome negativ e. Note that this constrain t ca n be easily satisfied in general systems. F o r exam ple, if the allocation mode ξ ( m ) k correspo nds to indepe ndent sets of the interf erence grap h (see, f or example, [7]), one can simply tr ansmit the sub set of links with nonem pty qu eues, which are still independ ent. In a compact f orm, we can express the dep arture proc ess as D ( t ) = X m ∈M G ( m ) T ( m ) ( t ) , (8) where G ( m ) is a matrix wh ose columns are G ( m ) k , and T ( m ) ( t ) is a vector whose each entry T ( m ) k ( t ) corresp onds to the number o f time slots that re source allocation m ode ξ ( m ) k is chosen durin g th e first t time slots. A basic requ irement o n the scheduler is that it shou ld achieve rate stability [12], i.e ., lim t →∞ D ( t ) / t = a (9) so that th e depar ture ra te of ea ch link is equal to the arrival rate, a s required by the u nderlyin g static optimization proble m OPT , which we formulate in the next subsectio n. B. Op timization Pr oblem In th is section we introduce the optimization problem OPT , which is implicitly solved by the op timal schedulers. The problem OPT is as follows OPT : OPT: min x f ( x ; y ) (10) s.t. h ( x ; y )  0 (11) x ( m )  0 , 1 T x ( m ) = 1 , ∀ m ∈ M (1 2) In the above f ormulation , x ( m ) as a resour ce allocation vecto r when the network state is m . That is, each en try x ( m ) k is the asymptotic time fr action (assum ing the limit exists for now) that resour ce allo cation mo de ξ ( m ) k is chosen , durin g th e time slots where the network state is m . Thus, x ( m ) is subject to the simplex constraint (12). x = ( x (1) , x (2) , . . . , x ( M ) ) is a big vector r epresenting the total resou rce allo cation vector as specified by the schedule r . f ( x ; y ) is a general conve x cost function o f variable x , and h ( x ; y ) is a vecto r of g eneral conv ex constrain t functio ns o f variable x . The ad ditional parameter y r epresents the uncer tain parameters, which is valid unde r the assumptio n th at the corresp onding pro cesses are subject to SLLN. Finally , we assume that both f ( x ; y ) and h ( x ; y ) are continu ously differentiable as functions of variables ( x , y ) . The formu lation of OPT is quite general, which can be used to model various ap plications in th e literature. For examp le, if we want to minimize the total transmission p ower , we can choose y = ( π , p ) , an d ch oose the co st f unction as follows f ( x ; y ) = X m ∈M π ( m ) p ( m ) T x ( m ) (13) where π ( m ) is the time fraction tha t the network state is m , and p ( m ) is a power vector wh ere each entr y p ( m ) k correspo nds to the power consump tion wh en resource alloc ation mode ξ ( m ) k is chosen at network state m . Th us, th e cost fu nction in (13) can be interpreted as th e average p ower con sumption by th e scheduler . N ote tha t we can also en code the power constrain t into h ( x , y ) by cho osing y = ( π , p ) and then cho osing h ( x ; y ) = X m ∈M π ( m ) P ( m ) x ( m ) − p (14) where P ( m ) is a diagonal matrix wher e each diagon al entry P ( m ) kk correspo nds to th e power consum ption wh en reso urce allocation m ode ξ ( m ) k is chosen when the n etwork state is m , an d p is the p ower constraint vector . In this case, (14) is equiv alent to requiring a constraint of p on th e average transmission p ower . In ord er to encode th e ne twork stability constraint, w e can c hoose y = ( π , a ) and then choo se h ( x ; y ) = a − X m ∈M π ( m ) R ( m ) G ( m ) x ( m ) (15) Thus, (15) requir es that the average external an d internal arriv als shou ld be less than the average departu res, in which case the n etwork is rate stable. I I I . S C H E D U L I N G A L G O R I T H M In this section we will d escribe the algorithm to solve OPT . As a standard ap proach in solv ing constrained conve x optimization p roblems [ 11], we tran sform OPT into anothe r static “penalized” pr oblem, PEN , to which our sched uling algorithm can directly apply . Based on this, we then introduce the schedu ling algorith m which solves PEN and, therefore, also solves OPT . A. T ransformed Pr ob lem Assuming that OPT is strictly feasible, we first ch ange the constraints in ( 11) as follows h ( x ; y ) + z = 0 , ǫ 1  z  z max 1 (16) where ǫ > 0 is a small scalar , and z max > 0 is a sufficiently large con stant such that the ineq uality a nd equality constraints are equiv alent. Den ote f ⋆ ǫ as the op timal co st when the constraint is changed to (1 6). Thus, the optimal value o f OPT is f ⋆ 0 . W e have th e following sensitivity lemma statin g that f ⋆ ǫ is a g ood a pprox imation of f ⋆ 0 with sufficiently small ǫ . Lemma 1 ([11]): Den ote λ 0 and λ ǫ as two Lagr angian multipliers for f ⋆ 0 and f ⋆ ǫ , respectively . W e have | f ⋆ 0 − f ⋆ ǫ | ≤ ǫ max( k λ ǫ k 1 , k λ 0 k 1 ) (17) W e next de fine the transformed p roblem a s f ollows: PEN: min x , z g ( x , z ; y ) = f ( x ; y ) + β p ( x , z ; y ) s.t. ǫ 1  z  z max 1 x ( m )  0 , 1 T x ( m ) = 1 , ∀ m ∈ M where β is a large co nstant to co ntrol appr oximation a ccu- racy , and p ( x , z ; y ) corr esponds to the pen alty term , which correspo nds to various stan dard pe nalty f unctions [ 11], e. g., p ( x , z ; y ) = 1 α k h ( x ; y ) + z k α (18) for α > 1 . In particular, the stand ard L yapun ov drift analysis (e.g., [ 1], [2], [ 4], [7]) co rrespon ds to th e case α = 2 . Denote ( x ⋆ p , z ⋆ p ) as a solution of PEN . W e h av e the fo llow- ing result h olds: Lemma 2 : f ( x ⋆ p ; y ) ≤ f ⋆ ǫ . Pr oof: Denote ( x ⋆ o , z ⋆ o ) as a solution of OPT with con- straint in (11) replaced by (16). W e h av e f ( x ⋆ p ; y ) ≤ f ( x ⋆ p ; y ) + β p ( x ⋆ p , z ⋆ p ; y ) ( a ) ≤ f ( x ⋆ o ; y ) + β p ( x ⋆ o , z ⋆ o ; y ) ( b ) = f ⋆ ǫ where ( a ) is because ( x ⋆ p , z ⋆ p ) solves PEN , and ( b ) is because ( x ⋆ o , z ⋆ o ) satisfy the constraint ( 16). Th us, the claim ho lds. In the following we will fo cus on solvin g PEN , since Lemma 1 and Lemma 2 guaran tee that PEN achieve an objective fu nction value which is arbitrarily c lose to the optimal in O PT . W e next d escribe the schedulin g alg orithm. B. A lgorithm D escription The pr oblems OPT and PEN are static. On the oth er hand, the network is dyn amic, and must be described by tim e serie s. Therefo re, befo re describ ing the algorithm, we n eed to define dynamic counterp arts of the static variables x , y and z . W e first define empirical resourc e allocation variable x ( m ) ( t ) = T ( m ) ( t ) 1 T T ( m ) ( t ) , ∀ m ∈ M (19 ) i.e., each en try x ( m ) k ( t ) cor responds to the time fraction th at resource a llocation mode ξ ( m ) k is chosen d uring th e fir st t time slots, when the n etwork state is m . No te that we hav e x ( m ) ( t )  0 , 1 T x ( m ) ( t ) = 1 , ∀ m ∈ M (20) Thus, x ( m ) ( t ) can be interpreted as the empirical value of x ( m ) which is defin ed in PEN . Similarly , we den ote the empirical value o f the uncertain parameter y as y ( t ) = Y ( t ) /t (21) i.e., y ( t ) is form ed by directly taking the average of the process Y ( t ) . Further, d efine the em pirical value of z as z ( t ) = Z ( t ) /t (22) where the c umulative pro cess Z ( t ) is d efined by Z ( t ) = t X τ =1 u ( τ ) (23) and u ( τ ) is co mputed by the sch eduler in A lgorithm 1 . Finally , we in troduce some notation s. D enote ∇ m and ∇ z as the g radient o perator with respect to variables x ( m ) and z , respectively . Fur ther , with an abuse of notation , we use the following abbreviated notation s: f ( t ) , f ( x ( t ); y ( t )) p ( t ) , p ( x ( t ) , z ( t ); y ( t )) g ( t ) , g ( x ( t ) , z ( t ); y ( t )) = f ( t ) + β p ( t ) The algo rithm is d escribed as in Algorith m 1. Essentially , the a lgorithm updates the variables x ( t ) and z ( t ) by com put- ing descent direction s v ( m ) ( t ) and u ( t ) in Step 1 and Step 2 , respectively , where v ( m ) ( t ) is an all-zero vector except an one Algorithm 1 Optim al Scheduling Step 1. At each time slot t with network state m , cho ose allocation m ode ξ ( m ) k , where k ∈ arg min j  ∇ m f ( t ) + β ∇ m p ( t )  j (24) Step 2. Cho ose variable u ( t ) such that u i ( t ) =  ǫ if ( ∇ z p ( t )) i ≥ 0 z max else (25) and update variables x ( t ) , y ( t ) an d z ( t ) accordin gly . at the k -th entry . Further note that constraint Q ( t )  G ( m ) k can be satisfied implicitly with regular cost a nd pen alty function s, i.e., assuming the co st for transmitting a set of link s is always no smaller th an th at of transmitting any of its sub sets. From the defin ition of x ( t ) and z ( t ) , these pro cesses are naturally updated as follows x ( l ) ( t ) = x ( l ) ( t − 1) , l 6 = m x ( m ) ( t ) = x ( m ) ( t − 1) + 1 1 T T ( m ) ( t ) ( v ( m ) ( t ) − x ( m ) ( t − 1)) z ( t ) = z ( t − 1 ) + 1 t ( u ( t ) − z ( t − 1 )) Thus, Algorithm 1 can be viewed as a stochastic gr adient algorithm fo r PEN , wh ere the rando mness comes fro m the time varying functions f ( t ) and p ( t ) , which are subject to the changes in uncertain parameters y ( t ) . The optimization of (24) requires tracking the v ar iables x ( t ) and y ( t ) , in gen eral. However , in applications the structu re of the cost f unction f ( x ; y ) and p enalty fun ction p ( x ; y ) often allows a m uch simpler comp utation. For example, in the im portant case o f o ptimal power sched uling, where the cost fu nction is formulated as (13) and the con straint is as (15) with the ty pical value α = 2 , we have ∇ m f ( t ) + β ∇ m p ( t ) = π ( m ) ( t )( p ( m ) + β t ( R ( m ) G ( m ) ) T Q ( t )) where π ( m ) ( t ) = 1 T T ( m ) ( t ) P l ∈M 1 T T ( l ) ( t ) is the empirical time frac tion of network state m . Thu s, the op timization in (24) e ssentially only require s the qu eue length info rmation (n ote that π m ( t ) becomes an irrelevant scaling factor in the o ptimization) . In particular, if we are o nly interested in th e rate stability , i.e., setting the objectiv e f unction as f ( x ; y ) = 0 , the optimization in (2 4) is e quiv alen t to k ∈ arg min j  G ( m ) T ( R ( m ) T Q ( t ))  j (26) which is the same a s th e ma x-weight back -pressure algorithm propo sed by [1]. W e finally co nclude th is section by th e fol- lowing lemma, which form ally shows, essentially , th e descen t proper ty o f Algo rithm 1. Lemma 3 : Th e fo llowing p roperties h old f or Algorithm 1: 1) v ( m ) ( t ) solves the f ollowing pro blem GRAD-X: min v ( m ) ∇ m g ( t ) T v ( m ) (27) s.t. v ( m )  0 , 1 T v ( m ) = 1 2) u ( t ) solves the following pro blem GRAD-Z: min u ∇ z g ( t ) T u (28) s.t. ǫ 1  u  z max 1 Thus, the variables ( v , u ) computed by Algorithm 1 ca n be interpreted as the points in th e fea sible region of PEN which achieves the min imum inner pro duct with the corresp onding (stochastic) gradien ts. Pr oof: For 1 ), note that GRAD-X is a L inear Pro- grammin g (LP) problem over a simplex, and therefore the solution c an be obtained a t a vertex [11] with the minimum directional der i vati ve. For 2), note that GRAD-Z is an LP over a hyper cube, and therefore the solution is o btained at the bou ndary poin ts. T hus the claim f ollows by notin g that ∇ z g ( t ) = ∇ z p ( t ) , since f ( · ) is not a fun ction of z . I V . O P T I M A L I T Y P R O O F In this section we will p rove the op timality of Algorithm 1. T here are two issues to consider : 1) W e nee d to show th at Algorithm 1 a chieves the optimality of OPT asymp totically , and 2) W e need to sho w that Algorithm 1 i s feasible for OPT , i. e., constrain t (11) can no t be vio lated. W e first b riefly introdu ce fluid limits, which serves as the key techniqu e for the optimality proof . A. F luid Limits W e extend the do main of all pro cesses to co ntinuou s time by linear interpolatio n, and define the flu id scaling of a f unction l ( t ) as l r ( t ) = l ( rt ) /r (29) where l can be function s T , Y and Z . It can b e shown that these scaled function s are unifor mly Lipschitz-con tinuous. Thus, acco rding to the Arzela-Ascoli Theorem [13], any sequence of function s which is indexed by { r n } ∞ n =1 , i.e., ( T r n , Y r n , Z r n ) , contains a subsequen ce { r n k } ∞ k =1 which conv erges uniform ly on co mpact sets to a set o f absolutely continuo us f unctions ( and, th erefore, differentiab le almo st ev erywher e [1 3]) ( ¯ T , ¯ Y , ¯ Z ) . Define any such limit as a fluid limit. (Note that fluid limits are denoted by a bar .) W e next state some prop erties of the fluid limits. Lemma 4 : Th e pr ocesses in any flu id limit satisfies the following: For any t > 0 , we have w .p.1, ǫ 1  ¯ Z ( t ) /t  z max 1 (30) and the f ollowing proper ties hold w .p.1: For all t ≥ 0 ¯ Y ( t ) = y t (31) 1 T ¯ T ( m ) ( t ) = π ( m ) t ∀ m ∈ M (32) Pr oof: (30) fo llows fr om Algo rithm 1, wh ere each u i ( t ) is ch osen between ǫ and z max . (31) and (32) follows direc tly from th e (functio nal) SLLN. W e next define the resou rce allocation variables a nd a uxil- iary variables in fluid limit as follows (o ne can compare with (19) a nd ( 22) fo r similar ities) ¯ x ( m ) ( t ) = 1 π ( m ) t ¯ T ( m ) ( t ) (33) ¯ z ( t ) = ¯ Z ( t ) /t (34) Similarly , define th e f ollowing variables as th e co unter parts of v ( m ) ( t ) and u ( t ) in Algorithm 1: ¯ v ( m ) ( t ) = ˙ ¯ T ( m ) ( t ) /π ( m ) (35) ¯ u ( t ) = ˙ ¯ Z ( t ) (36) W e hav e the following lem ma ho lds, which states that both ( ¯ x ( t ) , ¯ z ( t )) a nd ( ¯ v ( t ) , ¯ u ( t )) are feasible po ints fo r PEN . Lemma 5 : For any flu id limit an d t > 0 , we have 1) ( ¯ x ( t ) , ¯ z ( t )) is fe asible fo r PEN : ǫ 1  ¯ z ( t )  z max 1 (37) 1 T ¯ x ( m ) ( t ) = 1 , ¯ x ( m ) ( t )  0 , ∀ m ∈ M (38) 2) ( ¯ v ( t ) , ¯ u ( t )) is also feasible fo r PEN : ǫ 1  ¯ u ( t )  z max 1 (39) 1 T ¯ v ( m ) ( t ) = 1 , ¯ v ( m ) ( t )  0 , ∀ m ∈ M (40) 3) Th e der iv atives of ¯ x ( m ) ( t ) an d ¯ z ( t ) are ˙ ¯ x ( m ) ( t ) = ( ¯ v ( m ) ( t ) − ¯ x ( m ) ( t )) /t (41) ˙ ¯ z ( t ) = ( ¯ u ( t ) − ¯ z ( t )) /t (42) Pr oof: F o r 1), (37) f ollows f rom ap plying (30) to th e definitions of ¯ z ( t ) , and (3 8) follows from applyin g (3 2) to the definition of ¯ x ( m ) ( t ) . Similar ly we ca n prove 2), by notin g that ǫτ 1  ¯ Z ( t + τ ) − ¯ Z ( t )  z max τ 1 for any t ≥ 0 and τ > 0 . 3) f ollows f rom d irect calcu lation. W e are now read y to pr ove the optim ality of Algo rithm 1. B. Op timality Pr oof For the ease of pr esentation, we use ¯ g ( t ) , g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) (43) as a short-h and n otation ( note that they are the fun ctions in fluid limits), with an abuse o f n otation. W e next establish the following key technica l lemma, which, essentially , extends the optimality p roperty in L emma 3 to the fluid limits. Lemma 6 : Le t a fluid limit ( ¯ T , ¯ Y , ¯ Z ) and m ∈ M , t > 0 be g iv en. The following pr operties ho ld: 1) ¯ v ( m ) ( t ) solves the following pr oblem GRAD-XBAR: min ¯ v ( m ) ∇ m ¯ g ( t ) T ¯ v ( m ) (44) s.t. ¯ v ( m )  0 , 1 T ¯ v ( m ) = 1 2) ¯ u ( t ) solves th e fo llowing pr oblem GRAD-ZBAR: min ¯ u ∇ z ¯ g ( t ) T ¯ u (45 ) s.t. ǫ 1  ¯ u  z max 1 Thus, the o ptimality in L emma 3 still holds in fluid limits. W e first outline th e proo f. For 1), since G RAD-XBAR an LP over a simplex, th e op timum must co rrespond to th e vertices with the smallest gradient. Thus, it is su fficient to prove that any resourc e allocation mo de j will have ¯ v ( m ) j ( t ) = 0 if th ere is a k such th at ( ∇ m ¯ g ( t )) j > ( ∇ m ¯ g ( t )) k (46) which follows f rom the op timality shown in Lemma 3 along a conv ergent subsequen ce. For 2) , we will pr ove that f or any feasible p oints ¯ u of (45), we have ∇ z ¯ g ( t ) T ¯ u ( t ) ≤ ∇ z ¯ g ( t ) T ¯ u (47) which also fo llows fro m the optimality in L emma 3 along a conv ergent subsequ ence. Pr oof: For the clarity of p resentation, the pro of is moved to the Append ix. Based on th e above lemma, we are now ready to prove th at Algorithm 1 achieves the o ptimal cost in th e fluid lim it. Lemma 7 : For any flu id limit, we hav e fo r all t > 0 , g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) = g ⋆ (48) where g ⋆ = f ( x ⋆ p ; y ) + β p ( x ⋆ p , z ⋆ p ; y ) . T hus, the optim ality is achieved in the fluid limit. W e first ou tline the pro of. Note that it is always tr ue that g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) ≥ g ⋆ (49) since ( ¯ x ( t ) , ¯ z ( t )) are always feasible points o f PEN . Thus, the claim h olds if we can prove the rev erse direction. This can b e done by defining a prope r “L yapun ov” f unction L ( t ) = tg ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) (50) and show that ˙ L ( t ) ≤ g ⋆ , by using the pro perties in Lemma 6 and the conv exity o f fu nction g ( · ) . Pr oof: Conside r the “L y apunov” func tion as in ( 50) in any flu id limit. From Lemm a 5 we kn ow that fo r any t > 0 , ( ¯ x ( t ) , ¯ z ( t )) are feasible for PEN , a nd theref ore we have L ( t ) ≥ g ⋆ t due to the defin ition of g ⋆ . On the other hand, ˙ L ( t ) = ¯ g ( t ) + t ˙ ¯ g ( t ) = ¯ g ( t ) + t ∇ z ¯ g ( t ) T ˙ ¯ z ( t ) + t X m ∈M ∇ m ¯ g ( t ) T ˙ ¯ x ( m ) ( t ) ( a ) = ¯ g ( t ) + t ∇ z ¯ g ( t ) T ( ¯ u ( t ) − ¯ z ( t )) + X m ∈M ∇ m ¯ g ( t ) T ( ¯ v ( m ) ( t ) − ¯ x ( m ) ( t )) ( b ) ≤ ¯ g ( t ) + t ∇ z ¯ g ( t ) T ( z ⋆ p − ¯ z ( t )) + X m ∈M ∇ m ¯ g ( t ) T ( x ( m ) ⋆ p − ¯ x ( m ) ( t )) ( c ) ≤ g ( x ⋆ p , z ⋆ p ) ( d ) = g ⋆ where ( a ) is obtain ed by sub stituting the eq uation in Lemma 5, ( b ) is fro m Lem ma 6, i.e., ¯ v ( m ) ( t ) and ¯ u ( t ) are solutions of GRAD-XBAR and GRAD-ZBAR , respectively . ( c ) is d ue to the conv exity of f unction g ( · ) , and ( d ) is b ecause ( x ⋆ p , z ⋆ p ) is th e solution o f PEN , by definition. Thus, we have L ( t ) = L (0) + Z t 0 ˙ L ( τ ) dτ ≤ g ⋆ t (51) from wh ich we conclude that L ( t ) = g ⋆ t . Having established the optimality in the fluid limit, we are now able to prove optimality in th e original system. The following the orem states that Algorithm 1 achie ves the optimal cost in th e original network. Theor em 1: ( optimal cost ) In the origina l network , the following holds w .p.1: lim sup t →∞ f ( x ( t ); y ( t )) ≤ f ⋆ ǫ (52) Pr oof: Suppose that it is not true. Th en there is a sequence { t n } ∞ n =1 such that lim n →∞ f ( x ( t n ); y ( t n )) > f ⋆ ǫ (53) From the Arzela-Ascoli T heorem [1 3], there is a subsequen ce { t n k } ∞ k =1 which conv erges to a fluid limit. Lemm a 7 im plies lim k →∞ f ( x ( t n k )) ≤ lim k →∞ g ( x ( t n k )) (54) ( a ) = g ( ¯ x (1)) (55) ( b ) = g ⋆ ≤ f ⋆ ǫ (56) where ( a ) follows from the fact that f or all m ∈ M , x ( m ) ( t n k ) = T ( m ) ( t n k ) / 1 T T ( m ) ( t n k ) (57) = ( T ( m ) ) t n k (1) / 1 T ( T ( m ) ) t n k (1) (58) → ¯ T ( m ) (1) / 1 T ¯ T ( m ) (1) as k → ∞ (59) = ¯ x ( m ) (1) (60) and that g ( · ) is continu ous. ( b ) is because o f Lemm a 7. Th us, we have a con tradiction, and the cla im ho lds. In the next subsection we will continu e to prove th e fea- sibility result, nam ely , the lim it po ints of x ( t ) pro duced by Algorithm 1 ar e indeed feasible f or OPT . C. F easibility P r oof Note that Algorithm 1 is d esigned to solve PEN . Thus, in order to prove that the scheduler pr oduce f easible points for OPT , we need the fo llowing lemma, which conn ects the objective fun ction value in PEN to the co nstraint in OPT . Lemma 8 : Th e following proper ties h old fo r PEN : For large enoug h β , we hav e k h ( x ⋆ p ; y ) + z ⋆ p k ≤ ǫ/ 2 (61) for any solution ( x ⋆ p , z ⋆ p ) . Pr oof: For the ease of pr esentation, we only con sider the penalty functio n as (18), althou gh the pro of c an b e easily extended to gen eral cases. Note that fro m Lem ma 2 we have β α k h ( x ⋆ p ; y ) + z ⋆ p k α ≤ f ⋆ ǫ − f ( x ⋆ p ; y ) (62) Thus, (61) holds by choosing sufficiently large β . Finally , we conclu de this section by th e following theorem, which states th at th e limit p oints produced b y Algorithm 1 are always feasible for the original p roblem O PT . Th is, co mbined with Theo rem 1, proves the optim ality of Algor ithm 1 for OPT . Theor em 2: ( feasibility ) For sufficiently large β , we have lim sup t →∞ h i ( x ( t ); y ( t )) ≤ 0 (63) for any co nstraint fun ction h i ( x ; y ) in h ( x ; y ) . Pr oof: Su ppose that this is n ot true. Then there exist a sequence { t n } ∞ n =1 such that lim n →∞ h i ( x ( t n ); y ( t n )) > 0 (64) From Arzela-Ascoli Th eorem, ther e is a subsequ ence { t n k } ∞ k =1 which conv erges to a flu id limit. Thus, we have k h ( ¯ x (1); ¯ y (1)) + ¯ z (1) k ≥ h i ( ¯ x (1); ¯ y (1)) + ¯ z i (1) ( a ) = lim k →∞ h i ( x ( t n k ); y ( t n k )) + z i ( t n k ) ( b ) ≥ lim k →∞ h i ( x ( t n k ); y ( t n k )) + ǫ ( c ) ≥ ǫ where ( a ) can b e argued similarly as in th e proo f of Theorem 1, ( b ) is because for any i and t > 0 we have z i ( t ) > ǫ , due to Algorith m 1, and ( c ) is b ecause o f the assumption in (64). But acc ording to Lemm a 7, ( ¯ x ( t )) , ¯ z ( t )) solves PEN , and therefor e Lemma 8 implies that k h ( ¯ x (1); ¯ y (1)) + ¯ z (1) k ≤ ǫ / 2 (65) Contradiction ! T herefore the c laim holds. Thus, Alg orithm 1 prod uces feasible po ints fo r OPT , an d achieves a cost whic h is arbitrarily close to f ⋆ , by proper ly selecting parameters β an d ǫ . V . S I M U L A T I O N In this section we verify the perf ormanc e o f Algorithm 1 throug h a simulation in a r andom wireless n etwork wh ere th e network is as s hown in Fig .1. There are 7 lin ks in the network, where square no des d enote the transmitters, and rou nd no des denote the receiv ers. W e simulate a special case of OPT , the f ollowing minimum power schedulin g problem , which we denote as POW : POW : min x p T x (66) s.t. a − G x  0 (67) x  0 , 1 T x = 1 (68) where p is a power vector whose each element p k correspo nds to th e power consum ption when the independen t set G k is chosen. In the simulation we choose p k = k G k k 2 . Here, a correspo nds to the arriv al r ate vector , which is assum ed to be the only u nknown par ameter in the network. Thu s, (67) correspo nds to the ra te stability constraint. −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transmitter Receiver Fig. 1. An example wireless net work with 7 links, where square nodes are transmitt ers, and round nodes are rec ei vers. 10 0 10 1 10 2 10 3 10 4 10 5 0.2 0.4 0.6 0.8 1 Time Slots Arrival Rate Empirical Average 1 Empirical Average 2 Expected 10 0 10 1 10 2 10 3 10 4 10 5 0 10 20 30 40 Time Slots Average Cost Average cost 1 Average cost 2 Optimal cost Fig. 2. Con verg ence results of averag e cost with sources with differe nt con verg ence behavi ors. Fig. 2 shows the conv ergence results o f the co st functio n (the b ottom sub- figure) with slowly c on verging sources (the top sub-figu re) after a simu lation of 10 5 time slots. In the simulation, we cho ose ǫ = 1 0 − 3 and β = 5 × 1 0 3 . It can be ob served fro m the top sub- figure that our alg orithm achieves the op timal co st. Fur ther, by comparin g the conver - gence results of the co st an d the arri val p rocesses, we can conclud e that Algor ithm 1 can track the uncertain parameter a dy namically . In the simulation , it is fur ther observed that the ma ximum qu eue length in the network is aro und 10 2 , so that th e constraint in (67) is clearly satisfied. V I . C O N C L U S I O N In this pap er we for mulated a general class o f sch eduling problem s in wireless networks with u ncertain para meters, subject to the constra int th at these paramete rs can be obtain ed from the em pirical a verage v alues o f certain s tochastic network processes. W e prop osed a class of primal-dual typ e schedulin g algorithm s, and showed its o ptimality as well as feasibility using fluid limits. A P P E N D I X P R O O F O F L E M M A 6 Pr oof: W e first p rove 1). Let a sequ ence of fu nctions ( T r n , Y r n , Z r n ) be given, which con verge to a flu id limit ( ¯ T , ¯ Y , ¯ Z ) . In the fluid limit, suppo se that there is time t > 0 , m ∈ M and resour ce allocation mod es j, k such th at ( ∇ m ¯ g ( t )) j ≥ ( ∇ m ¯ g ( t )) k + ǫ (69) where ǫ > 0 is a small constant. Then, since ∇ m ¯ g ( t ) is a continuo us fu nction of v ariable t , there is δ 1 > 0 such th at for all τ ∈ ( t − δ 1 , t + δ 1 ) , we have ( ∇ m ¯ g ( τ )) j ≥ ( ∇ m ¯ g ( τ )) k + ǫ / 2 (70) Further, since ∇ m ¯ g ( τ ) , ∇ m g ( ¯ x ( τ ) , ¯ z ( τ ); ¯ y ( τ ))) is con - tinuous as a fun ction of variables ( ¯ x ( τ ) , ¯ y ( τ ) , ¯ z ( τ )) (a nd therefor e is absolutely continuou s when restricted to a compact local region ), th ere is an ǫ ′ > 0 such that k ( x , y , z ) − ( ¯ x ( τ ) , ¯ y ( τ ) , ¯ z ( τ )) k ≤ ǫ ′ implies th at  ∇ m g  x , z ; y )  j ≥  ∇ m g ( ¯ x ( τ ) , ¯ z ( τ ); ¯ y ( τ ))  k + ǫ / 4 for all τ ∈ ( t − δ 1 , t + δ 1 ) . Now we defin e ( x r n ) ( m ) ( τ ) , T ( m ) ( r n τ ) / 1 T T ( m ) ( r n τ ) (71) y r n ( τ ) , Y ( r n τ ) /r n τ (72) z r n ( τ ) , Z ( r n τ ) /r n τ (73) Then, the definitio n o f fluid limits implies that there exists N ∈ N and δ 2 > 0 such that for all n > N and τ ∈ ( t − δ 2 , t + δ 2 ) , k ( x r n ( τ ) , y r n ( τ ) , z r n ( τ )) − ( ¯ x ( τ ) , ¯ y ( τ ) , ¯ z ( τ )) k < ǫ ′ (74) Thus, by taking δ = min ( δ 1 , δ 2 ) w e have  ∇ m g  x r n ( τ ) , z r n ( τ ); y r n ( τ )  j ≥  ∇ m g ( x r n ( τ ) , z r n ( τ ); y r n ( τ )  k + ǫ / 4 for a ll τ ∈ ( t − δ, t + δ ) . Further, by comparin g th e above definitions of x r n ( τ ) , y r n ( τ ) and z r n ( τ ) to that of x ( τ ) , y ( τ ) and z ( τ ) in (19), (21) an d (22), r espectiv ely , we conclude that they are essentially the same, except a dif ference in time scale, i.e., x r n ( t ) = x ( r n t ) . Thus, the following holds in the original system: for any n > N and all τ ∈ ( r n ( t − δ ) , r n ( t + δ )) ,  ∇ m g ( x ( τ ) , z ( τ ); y ( τ )  j ≥  ∇ m g ( x ( τ ) , z ( τ ); y ( τ )  k + ǫ / 4 Therefo re, accordin g to Lemma 3, ξ ( m ) j is ne ver chosen in any time slot d uring ( r n ( t − δ ) , r n ( t + δ )) , and we hav e tha t T ( m ) j ( τ ) is a constant during ( r n ( t − δ ) , r n ( t + δ )) , from which we co nclude th at ˙ ¯ T ( m ) j ( t ) = 0 . Therefo re, ¯ v ( m ) j ( t ) = 0 following the definition that ¯ v ( m ) j ( t ) = ˙ ¯ T ( m ) j ( t ) /π ( m ) . W e n ext p rove 2). Le t ¯ u be given as a feasible po int of GRAD-ZBAR and ǫ > 0 b e gi ven. Since ∇ z g ( · ) is a continuo us fun ction of t , there is δ > 0 and N ∈ N su ch that f or n > N and all τ < ( t − δ, t + δ ) , the following holds: k∇ z g ( x r n ( τ ) , z r n ( τ ); y r n ( τ )) − ∇ z g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) k < ǫ (75) Further, note th at Lemma 3 imp lies th at for any time slot in ( r n ( t − δ ) , r n ( t + δ )) , we have ∇ z g ( x ( τ ) , z ( τ ); y ( τ )) T u ( τ ) ≤ ∇ z g ( x ( τ ) , z ( τ ); y ( τ )) T ¯ u Thus, applyin g ( 75) to th e above inequality we have ∇ z g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) T u ( τ ) ≤ ∇ z g ( ¯ x ( t ) , ¯ z ( t ); ¯ y ( t )) T ¯ u + cǫ for all τ ∈ ( r n ( t − δ ) , r n ( t + δ )) , wher e c > 0 is a proper constant. 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