Distributed Cross-Layer Optimization of Wireless Sensor Networks: A Game Theoretic Approach

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1 Distributed Cross-Layer Optimization o Wireess Sensor Networks: A Game Theoretic Approach Jun Yuan and Wei Yu Eectrica and Computer Engineering Department, University o Toronto {steveyuan, weiyu}@commutorontoca Abstract This paper proposes a distributed optimization ramework or wireess mutihop sensor networks base on a game theoretic approach We show that the cross-ayer optimization probem can be decomposed into two subprobems corresponding to two separate ayers (the physica and the appication ayers) o the overa system By modeing each subprobem as a noncooperative game, we aim to sove the noncovex appication-ayer rateaocation and physica-ayer power-aocation subprobems in a distributed manner Further, we prove the existence, uniqueness, and stabiity o the Nash equiibria or both games under certain suicient conditions Finay, we show that by using a set o dua variabes as the market prices to coordinate the physica ayer suppy and the appication ayer demand, the overa optimization process strikes a right baance between the two ayers in an overa cross-ayer design I INTRODUCTION PSrag repacements Wireess sensor networks have a wide range o appications, such as miitary security, traic contro, and environmenta monitoring A sensor network consists o a arge number o sensors depoyed in a ied Each sensor makes a oca observation o some underying physica phenomenon, quantizes its observation, and transers the data back to a centra estimation oice (ie, ) Due to the imited transmission power, sensors that are ar away rom the deiver their quantization data through a mutihop network as shown in Fig 1 The goa o the sensor network design is to estimate the underying physica phenomenon as accuratey as possibe under the network resource imitation Thus, the sensor network probem can be ormuated as a network optimization probem, in which the objective is to the minimize the overa distortion, ie, the dierence between the true underying ied and its estimation at the However, due to the partia observation at each sensor, the overa estimation error at is a couped and nonseparabe unction o a sensors data rates In addition, due to the shared nature o the wireess medium, geographicay cose transmissions oten interere with each other Because o the intererence, the traditiona bit-pipe assumption on the ogica ink capacity no onger hods We address the above issues in this paper by considering the undamenta perormance imits o sensor networks We adopt a separate source-channe coding mode and use inormation theoretica concepts such as rate-distortion region and capacity region to gain insights into the undamenta tradeos in wireess sensor network design In our previous paper 1], Fig 1 Source Sensor Networks we showed that the overa network optimization probem may be decomposed in the dua domain into two disjoint subprobems: a power contro subprobem at the physica ayer and a source coding subprobem at the appication ayer A set o dua variabes can then be used to coordinate the interaction between the ayers This paper ocuses on eicient and distributed soutions to each o these subprobems At a irst gance, neither subprobem appears to be easy to sove due to the inherent noninearity and nonconvexity o the probems Further, even i an agorithm or inding the goba soution is avaiabe, it may not be amendabe to distributed impementation As reaistic sensor network depoyment oten encounters variations in both source statistics and physica ayer channe characteristics, rea-time and distributed agorithms are desired In this paper, we adopt a game-theoretic approach to sove each subprobem Game theory has been widey appied to communications probems in the iterature ], 3], 4], 5] However, existing ormuations tend to ocus on the physica ayer excusivey In this paper, we generaize the appication o game theory to mutipe ayers in a cross-ayer design o wireess sensor networks Our main contributions are the oowing: We ormuate a power contro game at the physica ayer and a source coding game at the appication ayer Both games can be impemented in a distributed ashion We prove suicient conditions under which both games have unique and stabe Nash equiibria We generaize the pricing mechanism or the games by showing that: i) the interaction between two games can be coordinated by shadow prices (ie, dua variabes), where the aw o demand and suppy appies ii) the socia optimum o each subprobem can be achieved by proper design o tax/price in the games

2 II MULTIPLE GAMES IN CROSS-LAYER OPTIMIZATION A Optimization Framework and Decomposition In a mutihop wireess sensor network, the design goa is to minimize the tota distortion by jointy optimizing source coding and power aocation Adopting the setup in 1], the joint optimization probem can be written as: minimize α T d (1) subject to s R(d), c C(p), Ac s where α is a vector representing the reative emphasis on dierent eements o the distortion vector d; s is a set o source rates at each node; c is a set o ink capacities; and p is the power consumption vector R(d) is a undamenta concept in source coding, caed rate-distortion region The constraint s R(d) modes the inter-dependence o the distortion on the source rates C(p) is a undamenta concept in channe coding, caed capacity region The constraint c C(p) modes the inter-dependence o the ink capacity vector on the power consumption The ast inequaity Ac s reects the act that the source rate at each node must be ess than the ink capacity support Here, A is an N L node-incident matrix with N nodes and L inks 1 Using muti-commodity ow (routing) mode 6], the matrix eements can be written as: { 1 i n is the start node or ink a n = 1 i n is the end node or ink 0 ese Appying dua decomposition technique 1], the joint optimization probem can be urther decouped into two distinct subprobems A power contro subprobem at the physica ayer maximize { µ T c } c C(p) and a source coding subprobem at the appication ayer: { } minimize α T d + λ T s s R(d) (3) where µ is reated to the dua variabe λ by the ink price consistency equations : µ T = λ T A The Lagrange mutipiers λ and µ have the interpretation o being the shadow prices coordinating the appication ayer demand and physica ayer suppy Mathematicay, the shadow price λ can be updated by subgradient method with stepsize ν λ This update reects the aw o demand and suppy ] + λ (k+1) = λ (k) + ν (k) λ (s Ac) (4) The main point o this paper is that both subprobems can be soved rom a game theory perspective, where eicient soution and distributed impementation are possibe, as is shown in the next subsections 1 For simpicity, we consider a network with ony one and set its corresponding node-ink row as the ast row in A In the ormuation o the optimization probem, this ast row may be deeted without oss o generaity because it is ineary dependent o previous rows () B Physica Layer: Power Contro Game The physica ayer subprobem addresses the transmission intererence among nearby sensors Using an intererence mode, we expicity write down the capacity region (more precisey the achievabe region) constraints o () as oows: max p µ c (5) st c = og (1 + SINR ) G SINR = p j G j p j + σ 0 p p,max where c is the capacity o ink ; SINR is the signa to intererence and noise ratio o ink ; G and σ are the ink gain and noise respectivey, G j is the intererence gain rom ink j to ink, and p is ink s power action that has a power constraint p,max Because o the intererence structure, the power contro subprobem (5) is a nonconvex optimization probem, which is inherenty diicut to sove In this paper, we expore ways o approximating the optima soution using game theory Inspired by the work o Saraydar, Mandayam, and Goodman 3], we introduce a tax mechanism into the game so that the payers wi have an incentive to inteigenty avoid intererence More precisey, we ormuate a power contro game (PCG) at the physica ayer, under which each ink payer maximizes its own payo unction ( ) max Q PHY G p = µ og 1 + p j G jp j + σ t p st 0 p p,max (6) where t is the tax rate or ink, and p is the action or ink More power ink uses, more intererence it wi produce to others; thereore more tax (ie, t p ) it has to pay One sensibe choice o the tax rate is the oowing: t = s µ sc s p (7) = µ s G s G ss p s (G ss p s + s j s G sjp j + σs)( j s G sjp j + σs) where t is the rate at which other users achievabe data rates decrease with an additiona amount o power In genera, not every game has a Nash equiibrium, neither is the equiibrium necessariy stabe One o our contributions is the oowing suicient condition, under which the game is ensured to converge to a unique and stabe Nash equiibrium Deinition 1: The stricty diagona dominance (SDD) condition hods, i the channe gain G satisies: G > G j, (8) Theorem 1: I channe gain satisies the stricty diagonay dominant condition, given the tax rates, the power contro game (6) aways converges to a unique and stabe Nash equiibrium Proo: We prove it in Appendix

3 PSrag repacements Next, we propose the power contro game agorithm Agorithm 1: Power Contro Game (PCG) Agorithm 1) Initiaize p (0), t (0) Set k = 0 ) Set p (τ0) = p (k) Set i = 0, iterativey update p (τi) as oows: p (τi+1) = µ 1 G t (k) j p (τi) j G j + σ θ1 θ θ = θm Fig H n 1 y 1 y N n N Quantizer q 1 Quantizer q N Distributed Source Coding u 1 ns1 u N nsn NETWORK ˆθ project p (τi+1) into power constraint interva 0, p,max ] Repeat unti p (τi) converges Set p (k+1) = p (τi) 3) Update tax rate t t (k+1) = G bcm SINR (k+1) SINR (k+1) bcm = µ G p (k+1) 1 + SINR (k+1) 4) Return unti convergence The power update in step () is the best response o ink payer given the tax rate and his assessment o others action Next, in step (3), the tax rates are updated according to (7) As the tax rates converge, the power contro game Agorithm 1 converges to a unique and stabe Nash equiibrium Such power aocation equiibrium strikes a baance between maximizing rate and minimizing intererence Furthermore, the PCG agorithm can be impemented in a distributed ashion Speciicay, inspired by the work o 7], we propose a two-phase message passing mechanism in step (3) At the irst phase, each ink cacuates its broadcast message (ie, bcm ) by oca inormation (ie, µ, SINR, G, p ); and broadcasts to the network At the second phase, each ink coects broadcast messages rom others, and computes the tax rate t, where the intererence term (ie, G ) can be estimated, or exampe, by piots For simpicity, we present the power contro game or a scenario in which each ink consists o a singe channe The same idea can be extended to the cases in which each ink consists o mutipe physica channes The proposed agorithm is simiar to an agorithm proposed by Huang, Berry, and Honig in 5] However, the authors o 5] ocus on a utiity og(sinr(p)), whie our anaysis ocus on og(1 + SINR(p)), which is more reaistic in ow SINR scenarios In addition, the proo o convergence is aso dierent The authors o 5] use the supermoduar game theory, whie we prove the convergence (Theorem 1) rom the earning theory o games C Appication Layer: Source Coding Game The source coding subprobem characterizes the interaction among sensor rates and estimation distortion Consider an environment sensing appication depicted in Fig The underying physica phenomenon is denoted as θ, which is a vector o independent random variabes N sensors are depoyed in The contro overhead due to message passing shoud not be negected Such overhead may have impact on the scaabiity issue o sensor network However, rigorous overhead anaysis is out o the scope o this paper the ied, each making a oca (and possiby partia) observation o θ, whie being corrupted by independent observation noise n i The observation channe is characterized by a matrix H At each sensor i, the noisy observation y i is quantized into a codeword u i The quantized inormation rom a sensors is transmitted back through the network to a remote centra oice (ie, ) with source rates (s 1, s N ) At the remote, the decoder irst jointy decodes the codewords u, then estimates the source The source estimation is denoted as ˆθ The perormance criterion is to minimize the mean squared error, ie, D(ˆθ, θ) = ˆθ θ Foowing the setup in 8], we tacke the source coding subprobem using rate-distortion theory The rate-distortion region in (3) can be expicity written as oows: min w α T d + N λ(i)s i (9) i=1 st α T d = tr (R θ ) tr ( R θ H T (HR θ H T + Rw 1 ) 1 HRθ T ) ( ) 1 + σ s i = og si w i 1 σni w i 0 w i 1 σni, σsi = h T i R θ h i In this paper, we assume equa weights on distortion eements (ie, α = 1 ), thereore, the irst equaity is the MMSE estimation distortion Here, R θ is the covariance matrix o underying phenomenon; R w is a diagona matrix with the w i as ith diagona eement We urther deine a variabe w, which has the interpretation o quantization eort, ie, the arger the w, the smaer the distortion The second equaity characterizes the dependence o the source rate s i on the quantization eort w i The arger the w, the higher the source rates Each sensor s observation noise has a variance σni ; ht i is the ith row o the channe observation matrix H Source coding subprobem aims to ind an optima baance between distortion and source rate Here, we introduce a price mechanism into source coding game (SCG) such that it is easy or nodes to make a good tradeo in a distributed manner More precisey, we approximate the source rate as a inear unction o quantization eort with a price indicator m i, thus, each payer maximizes its payo: max w i Q AP P i = tr ( R θ H T (HR θ H T + R 1 w ) 1 HR T θ λ i m i w i st 0 w i 1/σni (10) where m i is the price indicator showing how expensive it is )

4 to quantize the source according to the oowing: m i = s i w i = σ si σ ni 1 + σsi w + i 1 σni w i Next, we present the source coding game agorithm Agorithm : Source Coding Game (SCG) Agorithm (11) 1) Initiaize ) At round (k + 1), payers sequentiay update their best response Repeat rom i = 1 to i = N 1 Each sensor i updates w i as oows w (k) i = argmax wi Q APP i Update rate price m i m (k+1) i = σ si 1 + σ si w(k) i + σ ni 1 σ ni w(k) i 3 Broadcast w i 3) Repeat () unti convergence The update strategy in step () is the best response or each payer i given its assessment o other payers action It is aso possibe to impement the agorithm in a distributed manner by a message passing mechanism 7] This is done in step (3) o the SCG agorithm: each sensor broadcasts its quantization eort (w i ), thereore, the best response can be cacuated ocay in step (1) We urther proceed to examine the convergence and goba optimaity o the source coding game agorithm According to the work o 9], the eect o quantizer q i in Fig can be modeed as a Gaussian random variabe with zero mean and variance σqi This variance is reated to the quantization eort and observation noise, ie, σqi = 1/w i σni We deine the oowing condition Deinition : The source coding optimaity (SCO) condition hods, i any o the oowing is true: σni σ si (1) σni < σ si, and σ qi 0, σ si + ] σ ni σsi σ σ ni (13) ni where σsi = ht i R θh i This is a reasonabe set o conditions because o the oowing The irst condition characterizes a scenario in which the sensor noise variance is arger than the variance o the underying physica phenomenon The second condition characterizes a scenario in which the sensor noise variance is smaer and the quantization variance is smaer than sensor noise variance times a constant which is arger than 1 (Note that a high resoution quantizer has a sma quantization variance) In most practica quantizer design, the quantization resoution is amost aways set to be beow the sensor noise Thus, the SCO condition amost aways hods in a we designed sensor network Theorem : The source coding game Agorithm converges to a unique and stabe Nash equiibrium that is the goba optimum or subprobem (9), provided that the SCO hods Proo: Due to the space imitation, we briey outine the proo According to the deinition o rate price m i, we caim that the Nash equiibrium o the source coding game (10) is precisey the oca optimum o (9) It can be shown that under the SCO condition, the optimization probem o (9) is convex by checking that the Hessian is positive semideinite Thereore, (9) has a unique oca optimum that is gobay optima Hence, the Nash equiibrium is unique and gobay optima III PRIMAL-DUAL ALGORITHM AND ILLUSTRATION A Distributed Prima-Dua Agorithm In this section, we present a distributed prima-dua agorithm, which iterativey executes power contro game and source coding game, and updates shadow prices Agorithm 3: Distributed Prima-Dua Agorithm 1) Initiaize λ (0) ) At time (k), given the price λ = λ (k), set µ T = λ T A PCG Agorithm 1 c G, p G SCG Agorithm s G, d G 3) In dua domain, update λ using the oowing rue: + λ (k+1) = λ (k) + ν (k) λ (sg Ac )] G (14) 4) Return to step unti convergence The dua price update (14) reects the aw o demand and suppy For exampe, when the appication ayer demand s G is greater than the physica ayer suppy Ac G, the price wi increase Furthermore, the price update can be accompished in a distributed way, because the λ i update requires ony the oca source coding rate s i and corresponding income and outcome ink capacities Thereore, combined with the distributed impementation o the games, the entire primadua agorithm can be impemented in a distributed manner B Simuation Exampe We simuate an exampe o a wireess sensor network in Fig 3(a) to iustrate the main ideas The underying physica phenomenon to be observed is a two dimension Gaussian vector with an identity covariance matrix For the sake o simpicity, we assume that ony the nearest two nodes (eg sensor 1, ) are active in sensing the ied, whie the rest nodes act as reays The mode or the physica ayer is an intererence channe, where each ink consists o mutipe subchannes We use the proposed prima-dua agorithm to ind an optima soution or the joint source coding and power contro probem in the mutihop network Both source coding game agorithm and power contro game agorithm are impemented in a distributed ashion Fig 3(b) shows the convergence process between the appication ayer source coding game and the physica ayer power contro game At the beginning, the appication ayer demand o source rates is high, whie the physica ayer suppy o ink capacities is ow During the iterations, the shadow prices as shown in Fig 3(c) coordinate both physica ayer and appication ayer moving toward the market equiibrium Under this market equiibrium, a ink capacities exacty

5 λ Convergence Process: Source rate vs Support rate PHY Support rate APP Source rate (a) Shadow Price: λ (c) Distortion (b) Distortion Convergence Fig 3 (a) Sensor network topoogy; (b) Convergence process between source rates and capacity support, (c) Convergence process o dua variabe λ, (d) Convergence process o network utiity, ie, estimation distortion support the source rates (ie, Ac = s) as shown in Fig 3(b) Finay, Fig 3(d) iustrates the graceu convergence o the overa distortion IV CONCLUSIONS In this paper, we tacke the genera noninear and nonconvex optimization probem or wireess sensor networks rom a game theoretic perspective The incorporation o game theory in a cross-ayer ramework aows the overa network optimization probem to be soved approximatey in a distributed manner Appendix: Proo o Theorem 1 We irst prove the existence o Nash equiibrium (NE) The action proie set o payer is a nonempty compact convex set, ie, p 0, p,max ] Q PHY is continuous in p, and Q PHY is stricty concave in p According to Proposition 03 10], the power contro game (6) has at east one pure NE The best response o ink payer is: BR (p) = µ 1 G j p j + σ (15) t G j j BR (p) = BR (p) = (d) G j G (16) According to the deinition o SDD (8), G j < 1 BR (p) G p j j < 1 (17) Thereore the best response is contractive Due to Theorem 34 in 11], the game has a unique Nash equiibrium Deinition 3: The dynamic stabiity (DS) matrix o the game is a square matrix, whose (, j)th entry is deined as oows: DS (,j) = BR (p),, j = 1,, L (18) According to Gersgorin theorem 1], a the eigenvaues o the dynamic stabiity (DS) matrix are ocated in the region L z DS(,) =1 DS(,j) Because the diagona eement o DS are a zeros, the region can be urther simpiied L z DS(,j) G L j = { } z < 1 =1 G Thereore, a the eigenvaues o DS a into the unit circe According to 13], a game is asymptoticay stabe, i the absoute vaue o eigenvaues o the dynamics stabiity matrix are a ess than one Hence, the power contro game is asymptoticay stabe, and aways converges to a unique, stabe Nash equiibrium under SDD REFERENCES 1] W Yu and J Yuan, Joint Source Coding, Routing and Resource Aocation or Wireess Sensor Networks, IEEE Internationa Conerence on Communications (ICC), May 005 ] R D Yates, A Framework or Upink Power Contro in Ceuar Radio Systems, IEEE J Seect Areas Commun, vo 13, pp , ] C Saraydar, N B Mandayam, and D J Goodman, Eicient Power Contro via Pricing in Wireess Data Networks, IEEE Trans Commun, vo 50, no, pp , 00 4] W Yu, G Ginis, and J Cioi, Distributed Mutiuser Power Contro or Digita Subscriber Line, IEEE J Seect Areas Commun, vo 0, no 5, pp , 00 5] J Huang, R Berry, and M L Honig, Distributed Intererence Compensation or Wireess Networks, IEEE J Seect Areas Commun, vo 4, no 5, pp , May 006 6] D Bertsekas and R G Gaager, Data Networks, Prentice Ha, ] M Chiang, Baancing Transport and Physica Layers in Wireess Mutihop Networks: Jointy Optima Congestion Contro and Power Contro, IEEE J Seect Areas Commun, vo 3, no 1, pp , Jan 005 8] J Yuan and W Yu, Joint Optimization o Source Coding and Power Aocation in Sensor Networks, 3rd Biennia Symposium on Communications, May 006 9] W R Bennett, Spectra o Quantized Signas, Be Syst Tech J, vo 7, pp , ] M J Osborne and A Rubinstein, A course in game theory, MIT Press, ] J W Friedman, Game Theory with Appications to Economics, Oxord University Press, ] R Horn and C Johnson, Matrix Anaysis, Cambridge Univ Press, ] D Fudenberg, The Theory o Learning in Games, MIT Press, 1998 =1