Externalities in wireless communication: A public goods solution approach to power allocation. by Shrutivandana Sharma

Transcription

1 Externaltes n wreless communcaton: A publc goods soluton approach to power allocaton by Shrutvandana Sharma SI 786 Tuesday, Feb 2, 2006

2 Outlne Externaltes: Introducton Plannng wth externaltes Power allocaton: problem formulaton and soluton Suggestons

3 Externalty Defnton (Economcs) An externalty s present whenever the utlty of a consumer or the producton possbltes of a frm are drectly affected by the actons of other agents n the economy. Examples Fshery s productvty affected by emssons from an upstream ol refnery ary could not concentrate on her homework because her roommates played the musc very loud Tom s room gets heated f Brad, n the adjacent room, sets hgh temperature Note: Externaltes can ether be detrmental or desrable Specal case Producton of or contrbuton for Publc goods

4 Wreless communcaton scenaro Tapan John Chm Greg Oops! dsconnected H Hello! H!! Hello! Sgnal Interference Lan Shrut

5 Outlne Externaltes: Introducton Plannng wth externaltes Power allocaton: problem formulaton and soluton Suggestons

6 Plannng wth externaltes Jean-Jacques Laffont and Patrck Sant Perre, 979 The model Termnology frms, =,2,...,, a manager and an RC (=0) q productve resources, l consumpton goods, Input constrant = q x R l y R l Consumpton prce vector π R Producton cost C( y) - convex Utlty U ( y ) = π y C ( y ) - concave x w prvate nformaton Envronment: ( x, y ): = (( x, y),( x2, y2),...,( x, y ),( x+, y+ ),...,( x, y)) Producton program: ( xy, ): = (( x, y),( x2, y2),...,( x, y)) Techncally possble program set: S : = {( x, y) ( x, y) f( x, y ), ( x, y ) B } Sem-feasble program set: S0 : = {( x, y) x } w = Feasble program set: S S = I = 0

7 Plannng wth externaltes, ctd Objectve (O) max U ( ) y ( xy, ) S = Allocaton mechansm (A). ( n) ( n) ( n ) The manager proposes an arbtrary program ( x, y ) and a penalty parameter τ + > 0 2. The frms and the RC maxmze ther utltes modfed by the penalty and send the optmal answers to the manager 3. The manager computes the average and proposes t wth a new penalty parameter x ( n+ ) = 0 4. The manager also computes the weghted tme averages and stores them 5. The process repeats from step () wth ( x, y ) argmax U ( y ) x x y y τ *( n+ ) *( n+ ) ( n) 2 ( n) 2 = ( n+ ) + ( xy, ) S *( n+ ) = ( n+ ) *( n+ ) x y = y + + z ( x, y ) n+ ( n+ ) ( k + ) *( n+ ) *( n+ ) = τ ( n+ ) σ k = z = 0 ( n+ ) n= n+ = n+ ( n+ ) ( n+ ) ( k) z σ = + τ = 0 k =

8 Plannng wth externaltes, ctd Theorem z ( n) The sequence of programs converge towards the optmal soluton f the followng assumptons are satsfed. Assumptons. The sets S ( n ), = 0,,..., and hence S are z convex and compact + 2. The penalty parameters satsfy,. ( n+ ) ( n) 0 < τ τ for all n. ( n) lmτ = 0. n lm σ N ( N) ( n) = lm τ = N N n = Remarks Informaton s decentralzed No assumpton on utlty functons except concavty Users do not cheat and follow the rules as set up The proof reles on the convexty of sets S whch n some cases may be volated due to externaltes Speed of convergence s not known

9 Outlne Externaltes: Introducton Plannng wth externaltes Power allocaton: problem formulaton and soluton Suggestons

10 A wreless network R3 R T h 23 h 2 R2 T4 p 2 T2 h 24 h 22 R4 p 2 h 24 T3 Note: A transmtter plus recever par s referred to as a user

11 Why consder power allocaton? Wreless lnk s prone to nterference If everybody uses maxmum avalable power Interference also ncreases Battery lfe reduces Unrestrcted use of power prohbts new users to enter the system Dfferent applcatons requre dfferent QoS whch depends on the accuracy of the receved data. The accuracy / error depends on the transmsson power and nterference.

12 Power allocaton problem System model spread spectrum users sharng bandwdth B Utlty from power allocaton s a functon of receved SINR ph U( γ); γ = N0 + ( p j jh j ) B Interference temperature constrant = p h 0 P Devce constrants: p max [0, P ] A manager governs the power allocaton Objectve (O2) max max U( γ ) st.. p [0, P ] =

13 Problem formulaton n the framework of Laffont, 979 Analoges Powers (,,..., ) p p2 p Allocaton goods Input-Outputs (( x, y ),...,( x, y )) γ = N 0 ph + B = ( p ) jhj ph 0 j P Externaltes Resource constrants Producton plan determned by envronment = x w Devce constrants, Channel gans, Applcaton Prvate nformaton Producton technology, Utlty Base staton Controller anager

14 Power allocaton problem, ctd Soluton If utlty functons are concave n power, the optmal soluton of problem O2 can be obtaned by allocaton mechansm A Assumng concavty n power restrcts the doman of applcablty Remarks If SINR s also consdered as an allocaton varable- the class of utlty functons can be generalzed allocaton sets become non convex, mechansm A cannot be used. Current work Extendng mechansm A n the context of power allocaton to overcome the dffcultes due to non-convextes n the producton sets. generalze the applcablty of current soluton

15 Questons?

16 Thank You!