Excursions of Max-Weight Dynamics
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1 Excursios o Max-Weigh Dyamics Joh N. Tsisiklis (wih Arsala Shariassab ad Jamal Golesai, Shari U. Workshop o The Nex Wave i Neworkig Research i hoor o Jea Walrad Simos Isiue, Berkeley Sepember 27 Sepember 28
2 Jea... Syle Tase
3 Excursios o Max-Weigh Dyamics Joh N. Tsisiklis (wih Arsala Shariassab ad Jamal Golesai, Shari U. Workshop o The Nex Wave i Neworkig Research i hoor o Jea Walrad Simos Isiue, Berkeley Sepember 27 Sepember 28
4 Oulie The Max-Weigh policy The quesio A broader class o sysems A geeral heorem Specializig MW Fluid limis Sae
5 Max-Weigh Policy [Tassiulas, Ephremides, 992] Q Geeralizes serve he loges queue Example: Nework wih iererece cosrais (,,, : serve queues 2 ad 4 Q 2 S: iie se o possible service vecrs [Techicaliy: 2 S, se some compoes zero: agai i S] ( 2 argmax 2S Q T ( ( 2 argmax Q T ( I 2 µ2s R Q( + Q(+A(+(R Imi (,Q(
6 The viewpoi Q Q 2 Schasic perurbaio o simple (luid deermiisic sysem perurbaio: lucuaios o A i ( Exploi properies o k he luid sysem piecewise cosa k dri kapplek Deermiisic, i a all imes Schasic A i ( o-expasive: kq( q (k applekq( q (k [Subramaia, 2]
7 x( x ( + x ( + u( x ( + o-expasiveess x ( + u( Wha does buy x ( buy us? Wha does o-expasiveess x( x( + x( + u( Wha does buy us?us? x( + o-expasiveess x( x ( x( Wha does buy o-expasiveess o-expasive: k(x (yk kx yk x( + x( + x( x( x ( + x ( + u( o-expasive: k (x (yk k x ( + x ( + u( kx( x (k k x (x ( + + u( + x ( x ( + u( x ( x( e:(x he k (x (yk x I+x(, (x xkx +,yk he kx( x ( x ( I (x O( x ( x( o-expasive: k (x (yk kx x ( Ikx( x( (x x (k x+, he kx( x (k u( ku( k O( kx( x (k u( o-expasive: k (x (yk kx yk I (x x +, he o-expasive: k (x kx o-expasive: k (x (yk (yk x (k kxyk yk kx( x (k kx( u( kx( x (k ku( p k p O( kx( x (k u( I (x x +, he kx( O( O( k k kx(x (k x (k ku( ku( O(O(p h: O( Wish: kx( x (k u( O( Wish: p k k O( kx( x (k C x (k max C u( Wish: kx( max u( k< k< k p O( kx( x (k C max u( Wish: k< o-expasiveess o eougho eough guaraee his k his o-expasiveess guaraee o-exp kx( x (k C max u( Wish: k< o-expasiveess o eough guaraee his k
8 MW subgradie low Igore boudary e ecs or he mome Replace by heir meas Q (I R (Q apple 2 argmax Q T (I R Q T 2S (Q max apple apple Q T (I R Q T Q : piecewise liear, covex, iiely may pieces
9 Z Subgradie Flow Sesiiviy Theorem (Coiuous Time h (x max x T i µ i + b i ẋ 2 F (x, a.e. i x( x( + sem Z Z ( Z d ( 2 F (x( [o-expasive dyamics, exisece, uiqueess] Perurbed sysem x( x( + Z ẋ 2 F (x+u ( d + U( ( 2 F ( x( Give sysem F, here exiss cosa C F s.. i x( x(, he: 2 x( x( apple C F apple ku( k, apple 8 U(, x( Easy? resul ails i has iiiely may pieces; or i co. di ereiable ad sricly covex
10 Proo idea w Y Oe dimesioal sysem wih dri w ( dimesioal sysem F Y Use iducio o dimesio Need separae argume whe i he viciiy o criical pois
11 uid soluio q(: coiuous ime Back MW ochasic process Q(: discree ime Back MW Back MW MW Back Fluid soluio Back soluio MW Fluid q(: coiuous-ime erurbed because cumulaive 6 Fluid Fluid soluio soluio q(: q(: coiuous-ime coiuous-ime Fluid soluio q(: coiuous-ime Schasic process Q( Schasic pro oudary e ecs: Q( mus say limi o-egaive Low-hagig rui: Fluid assume Schasic process Q( Schasic process Q( Schasic process Q( perurbed because cumulaive 6 iscree ime perurbed be perurbed because cumulaive 6 6 perurbed because cumulaive becauseq( cumulaive 6 perurbed boudary e ecs: mus say o-egaive Low-hagig rui: Fluid limi-egaive assume boudary e ecs: Q( mus say p boudary e boudary e ecs: Q( mus say o-egaive A( boudary e ecs: Q( mus say o-egaive O( discree ime, as discree ime ow-hagig rui: Fluid limi assume discree discree ime discreeime ime T Low-hagig p A( +A( O( as p, Q( q T +A( O( Q( q( A( C + k k + max A(k < as, k ow-hagig rui: Fluid limi assume T Q( q Q( rui: q(q( C limi + k kassume + max A(k Low-hagig Fluid q(, <as k q T Q( A(k, as Q( q he I T k Q( q(, as T A(, as Q( T q(, as he q T I Q( q
12 Srog Sae Space Collapse Srog Sae Space Collapse ( A ( A(, rae A A(, rae AASrog ( A(, rae ( A(, ( A(, rae A rae Assum Sae Space Collapse CollapseA ( Srog Sae Space Collapse A(, rae A ( A(, rae ( Sae > A > Srog (, Space Collapse (, P, as P A (, as >> ( ( (, A ( P,P (, A(, rae AA (, P, asas A ( A(, rae (, P A A (( > >,as P, as (, A ( A(, rae (, P A ( > as, (, P (, A ( > A (, > as P, as (, exp( : large deviaios priciple p (, ( >,deviaios aspriciple priciple priciple large deviaios A exp( : large (, exp( : large deviaio (, exp( : large (, exp( : deviaios (, exp( : large deviaios priciple (, exp( : large deviaios priciple (, exp( : large (, deviaios exp( :priciple large deviaios priciple : Ivaria se o luid model : Ivaria se o luid model + : Ivaria se o luid model : Ivaria se o luid model : Ivaria se o luid model priciple 2 (, exp( : large deviaios : Ivaria se o luid model + : Ivaria se o luid model + 22 : Ivaria se o luid model + 2 deviaios priciple Srog sae di usio scalig: Suppose (, 2 : Ivaria se o luid model Srog sae di usio scalig: Suppose (, Srog sae di usio scalig: 2 Srog sae probabiliy, di usio Suppose (quie se weak ; sar I Wih Srog high Q(scalig: / says sae : Ivaria o luid model 2 Srog sae di usio scalig: Suppose (, Srog sae di usio scalig: Suppose Srog (quie sae di usio scalig: weak ; sar I. Wih high probabiliy, Q( / says 2 2 uiormly weak;, or sar Suppose (, (quie I (quie weak ; sar I Wih high probabiliy, (quie weak ; sar weak I Wih high probabiliy, Q(I2 2 /Q( Hisry: (quie ; sar (quie weak ; sar I Wih high probabiliy, Q( say uiormly, or. Hisry: uidsuppose model + sae Srog di usio (, (quie weak; sar Iscalig: 2 uiormly, or. uiormly by2 ad,wischik or (22 uiormly., or cojecured Hisry: uiormly, or. Srog sae di usio scalig: (, cojecured by ad Wischik Hisry: Suppose (, Q( 2(quie weak; sar(22 ISuppose Wih high probabiliy, / says uiormly,. Hisry: Hisry: Lyapuov-based i, Zhog (2, or i.i.d. bouded 2 cojecured by ad cojecured by(22 adprobabiliy, Wischik (22 Q( / s Hisry: 2 Hisry: (quie ; sar IWischik high ollapse high di usio scalig: weak Lyapuov-based i Wih, Zhog (2, or probabiliy, Q( / says uiormly, i.i.d..b Wih Hisry: i Lyapuov-based i, or Zhog (2, or i.i.d by ad Wischik (22 uiormly cojecured by ad Wischik (22 sar Lyapuov-based Zhog (2, i.i.d. bouded 2,, or /. cojecured Our low-hagig rui uie weak; I cojecured by ad cojecured by ad Wischik (22 Wih high probabiliy, Q( says uiormly, Wis cojecured by ad Wischik (22 Lyapuov-based probabilisic par, o he isi elemeary, Zhog (2, or i.i.d i.i.d. Hisry: bouded : Zhog (2, Lyapuov Our low-hagig rui Our low-hagig rui Lyapuov-based i Sha Lyapuov-based i, Zhog (2, or i.i.d. boude Lyapuov-based i, Zhog (2, or i.i.d. bouded Our low-hagig geeral rui 2 /low-hagig probabilisic par o. he is elemeary Our rui Q( says uiormly, cojecured by ad Wischik (22 probabilisic par ew is elemeary loger imepar scales probabilisic o o he is elemeary geeral Our par low-hagig rui is elemeary probabilisic o ew Our low-hagig rui Our low-hagig rui Our low-hagig rui Lyapuov-based i, Zhog (2, or i.i.d. boud loger ime scales geeral geeral probabilisic par o ishe is elemeary par oo he elemeary probabilisic geeral probabilisic par he is elemeary probabilisic par o he loger ime scales loger ime scales geeral geeral Our low-hagig geeral loger ime scales rui geeral
13 Exesios, geeralizaios Weighed max-weigh max Q T W (I R W : diagoal, posiive weighs Backpressure policies rouig o loger ixed joi rouig ad schedulig, akes i accou weighs o desiaio queues MW-? (Ope max(q T W (I R, >
14 Cograulaios, Jea
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