Distributed Algorithms for the Optimal Design of Wireless Networks

Transcription

1 Unversty of Pennsylvana ScholarlyCommons Publcly Accessble Penn Dssertatons Dstrbuted Algorthms for the Optmal Desgn of Wreless Networks Ychuan Hu Unversty of Pennsylvana, Follow ths and addtonal works at: Part of the Electrcal and Electroncs Commons Recommended Ctaton Hu, Ychuan, "Dstrbuted Algorthms for the Optmal Desgn of Wreless Networks" (2013). Publcly Accessble Penn Dssertatons Ths paper s posted at ScholarlyCommons. For more nformaton, please contact lbraryrepostory@pobox.upenn.edu.

2 Dstrbuted Algorthms for the Optmal Desgn of Wreless Networks Abstract Ths thess studes the problem of optmal desgn of wreless networks whose operatng ponts such as powers, routes and channel capactes are solutons for an optmzaton problem. Dfferent from exstng work that rely on global channel state nformaton (CSI), we focus on dstrbuted algorthms for the optmal wreless networks where termnals only have access to locally avalable CSI. To begn wth, we study random access channels where termnals acqure nstantaneous local CSI but do not know the probablty dstrbuton of the channel. We develop adaptve schedulng and power control algorthms and show that the proposed algorthm almost surely maxmzes a proportonal far utlty whle adherng to nstantaneous and average power constrants. Then, these results are extended to random access multhop wreless networks. In ths case, the assocated optmzaton problem s nether convex nor amenable to dstrbuted mplementaton, so a problem approxmaton s ntroduced whch allows us to decompose t nto local subproblems n the dual doman. The soluton method based on stochastc subgradent descent leads to an archtecture composed of layers and layer nterfaces. Wth lmted amount of message passng among termnals and small computatonal cost, the proposed algorthm converges almost surely n an ergodc sense. Next, we study the optmal transmsson over wreless channels wth mperfect CSI avalable at the transmtter sde. To reduce the lkelhood of packet losses due to the msmatch between channel estmates and actual channel values, a backoff functon s ntroduced to enforce the selecton of more conservatve codng modes. Jont determnaton of optmal power allocatons and backoff functons s a nonconvex stochastc optmzaton problem wth nfntely many varables. Explotng the resultng equvalence between prmal and dual problems, we show that optmal power allocatons and channel backoff functons are unquely determned by optmal dual varables and develop algorthms to fnd the optmal soluton. Fnally, we study the optmal desgn of wreless network from a game theoretcal perspectve. In partcular, we formulate the problem as a Bayesan game n whch each termnal maxmzes the expected utlty based on ts belef about the network state. We show that optmal solutons for two specal cases, namely FDMA and RA, are equlbrum ponts of the game. Therefore, the proposed game theoretc formulaton can be regarded as general framework for optmal desgn of wreless networks. Furthermore, cogntve access algorthms are developed to fnd solutons to the game approxmately. Degree Type Dssertaton Degree Name Doctor of Phlosophy (PhD) Graduate Group Electrcal & Systems Engneerng Frst Advsor Roch Guern Keywords Dstrbuted algorthms, Wreless networks Ths dssertaton s avalable at ScholarlyCommons:

3 Subject Categores Electrcal and Electroncs Ths dssertaton s avalable at ScholarlyCommons:

4 DISTRIBUTED ALGORITHMS FOR THE OPTIMAL DESIGN OF WIRELESS NETWORKS Ychuan Hu A DISSERTATION n Electrcal and Systems Engneerng Presented to the Facultes of the Unversty of Pennsylvana n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy 2013 Supervsor of Dssertaton Alejandro Rbero, Assstant Professor of Electrcal and Systems Engneerng Graduate Group Charperson Saswat Sarkar, Professsor of Electrcal and Systems Engneerng Dssertaton Commttee Roch Guern, Professsor of Computer Scence and Engneerng Nkos Sdropoulos, Professsor of Electrcal and Computer Engneerng Steven Weber, Assocate Professsor of Electrcal and Computer Engneerng

5 ABSTRACT DISTRIBUTED ALGORITHMS FOR THE OPTIMAL DESIGN OF WIRELESS NETWORKS Ychuan Hu Alejandro Rbero Ths thess studes the problem of optmal desgn of wreless networks whose operatng ponts such as powers, routes and channel capactes are solutons for an optmzaton problem. Dfferent from exstng work that rely on global channel state nformaton (CSI), we focus on dstrbuted algorthms for the optmal wreless networks where termnals only have access to locally avalable CSI. To begn wth, we study random access channels where termnals acqure nstantaneous local CSI but do not know the probablty dstrbuton of the channel. We develop adaptve schedulng and power control algorthms and show that the proposed algorthm almost surely maxmzes a proportonal far utlty whle adherng to nstantaneous and average power constrants. Then, these results are extended to random access multhop wreless networks. In ths case, the assocated optmzaton problem s nether convex nor amenable to dstrbuted mplementaton, so a problem approxmaton s ntroduced whch allows us to decompose t nto local subproblems n the dual doman. The soluton method based on stochastc subgradent descent leads to an archtecture composed of layers and layer nterfaces. Wth lmted amount of message passng among termnals and small computatonal cost, the proposed algorthm converges almost surely n an ergodc sense. Next, we study the optmal transmsson over wreless channels wth mperfect CSI avalable at the transmtter sde. To reduce the lkelhood of packet losses due to the msmatch between channel estmates and actual channel values, a backoff functon s ntroduced to enforce the selecton of more conservatve codng modes. Jont determnaton of optmal power allocatons and backoff functons s a nonconvex stochastc optmzaton problem wth nfntely many varables. Explotng the resultng equvalence between prmal and dual

6 problems, we show that optmal power allocatons and channel backoff functons are unquely determned by optmal dual varables and develop algorthms to fnd the optmal soluton. Fnally, we study the optmal desgn of wreless network from a game theoretcal perspectve. In partcular, we formulate the problem as a Bayesan game n whch each termnal maxmzes the expected utlty based on ts belef about the network state. We show that optmal solutons for two specal cases, namely FDMA and RA, are equlbrum ponts of the game. Therefore, the proposed game theoretc formulaton can be regarded as general framework for optmal desgn of wreless networks. Furthermore, cogntve access algorthms are developed to fnd solutons to the game approxmately.

7 Contents 1 Introducton Background Random access wreless channels Random access wreless networks Roadmap Dstrbuted algorthms for optmal random access channels Problem formulaton Adaptve algorthms for optmal random access channels Problem decomposton and ts dual Adaptve algorthms usng stochastc subgradent descent Structure of the optmal prmal soluton Numercal results System wth nstantaneous power constrant System wth average power constrant Summary Appendces Proof of Proposton v

8 3 Dstrbuted algorthms for optmal random access networks Problem formulaton Optmal operatng pont Problem approxmaton Dstrbuted stochastc learnng algorthm Stochastc subgradent descent Network operaton, layers, and layer nterfaces Message passng Successve convex approxmaton Feasblty and optmalty Proof of Theorem Numercal results Summary Appendces Proof of Lemma Proof of Lemma Optmal wreless communcatons wth mperfect CSI Pont-to-pont channels Ergodc rate optmzaton Optmal power allocaton and channel backoff functons Onlne learnng algorthms Orthogonal frequency dvson multplexng Optmal soluton Onlne learnng algorthms Random access v

9 4.3.1 Optmal soluton Onlne learnng algorthms Numercal results Pont-to-pont channel Downlnk OFDM channel Uplnk RA channel Summary Appendces Proof of null dualty gap of problem (4.7) Cogntve access algorthms for wreless communcatons Cogntve access algorthm for multple access channel Multple access channel wthout power control Multple access channel wth power control Cogntve access algorthm for wreless networks Numercal results Summary v

10 Lst of Tables 5.1 β for channels wth dfferent asymmetry levels v

11 Lst of Fgures 2.1 An example multple access channel wth n = 20 nodes communcatng wth a common access pont (AP). Nodes are randomly placed n a 100 m 100 m square and the AP s located at the center of the square. Nodes labels represent ndexes and dstances to the AP. Subsequent numercal experments use ths realzaton of the random placement Convergence of the proposed algorthm to near optmal utlty wth nstantaneous power constrans but no average power constrants. Throughput utlty of the proposed adaptve algorthm and of the optmal offlne scheduler are shown as functons of tme for one realzaton and for the ensemble average of realzatons. In steady state the adaptve algorthm operates wth mnmal performance loss wth respect to the optmal offlne scheduler. A utlty gap smaller than 10 s acheved n about 350 teratons. Power constrant p nst = 100 mw, step sze ɛ = 0.1, capacty achevng codes v

12 2.3 Average transmsson rates (bts/s/hz) n 500 tme slots,.e., r (500) as defned n (2.40), for all termnals. The optmal offlne scheduler and the proposed adaptve algorthm yeld smlar close to optmal average rates. The varaton n acheved rates s commensurate wth the varaton n average sgnal to nose ratos (SNRs) due to dfferent dstances to the access pont. For the network n Fg.2.1 and the pathloss and power parameters used here, average sgnal to nose ratos vary between 0.4 and 10. Instantaneous power constrant p nst = 100 mw, step sze ɛ = 0.1, capacty achevng codes Average transmsson probabltes n 500 tme slots for all termnals. Offlne and adaptve optmal schedulers shown. Despte dfferent channel condtons all termnals transmt wth a smlar probablty close to 1/n = Ths s consstent wth the use of a logarthmc,.e., proportonal far, utlty. Instantaneous power constrant p nst = 100 mw, step sze ɛ = 0.1, capacty achevng codes Steady state optmalty gap between proposed adaptve algorthm and optmal offlne scheduler as a functon of step sze ɛ. Values of ɛ between 10 2 and 10 3 shown. As the step sze decreases, the optmalty gap decreases. The optmalty gap can be made arbtrarly small by reducng ɛ. Instantaneous power constrant p nst = 100 mw, capacty achevng codes Prmal and dual objectves when nstantaneous and average power constrants are n effect. One realzaton and ensemble average of realzatons shown. As tme grows the dualty gap decreases, eventually approachng a small postve constant and mplyng near optmalty of the acheved rates. Instantaneous power constrant p nst = 100 mw, average power constrant p avg = 5 mw, step sze ɛ = 0.1, adaptve modulaton and codng wth M = 4 modes wth rates τ 1 = 1 bts/s/hz, τ 2 = 2 bts/s/hz, τ 3 = 3 bts/s/hz, and τ 4 = 4 bts/s/hz and transtons at SNRs η 1 = 1, η 2 = 4, η 3 = 8, and η 4 = x

13 2.7 Average power consumpton for termnals 3 and 13,.e., p 3 (t) and p 13 (t) as defned n (2.41). Average power constrants p avg = 5 mw are satsfed as tme grows. Power p 3 (t) consumed by Termnal 3 s smaller than the allowed budget p avg 3 due to unfavorable channel condtons. Termnal 13 adheres to ts power budget after approxmately 600 teratons. Parameters as n Fg Instantaneous power allocatons p (t) for termnals = 3 and = 13 plotted aganst the channel realzaton h (t). Notce that the channel axes scales are dfferent n (a) and (b). In both cases, no power s allocated when channel realzatons are bad. Termnal 3 uses only the AMC mode wth the lowest rate τ 1 = 1 bts/s/hz, whle Termnal 13 uses two modes wth rates τ 2 = 2 bts/s/hz and τ 3 = 3 bts/s/hz. Ths happens because Termnal 13, beng closer to the AP, has a better average channel than Termnal 3. Parameters as n Fg Layers and layer nterfaces. The stochastc subgradent descent algorthm n terms of layers and layer nterfaces. Layers mantan prmal varables a k (t), rk j (t), c j(t), p j (t), q j (t) as well as auxlary varables p (t), x j (t), and y (t) whle multplers λ k (t), µ j(t), ν j (t), α j (t), β (t) and ξ (t) are assocated wth nterfaces between adjacent layers. Prmal varables can be easly computed based on multplers from nterfaces to adjacent layers and dual varables are updated usng nformaton from adjacent layers Queue dynamcs. Termnal operates by controllng queues n dfferent layers based on operatng ponts a k (t), rk j (t), c j(t), p j (t) and q j (t). In the transport layer and the network layer, each flow k has a queue. In the lnk layer and the physcal layer, each outgong lnk (, j) mantans a queue. In ths partcular example, there are two flows k 1 and k 2 and there are two neghborng nodes j 1 and j 2. Packets for flow k 1 are marked red whle packets for k 2 are n blue x

14 3.3 Message passng. (a) Termnal begns by transmttng dual varables λ k (t) and ν j (t) to all neghbors j N (). (b) It then computes and shares k N () ν k(t) wth all j N (). Ths nformaton, along wth locally avalable multplers, s then used to perform the prmal teratons assocated wth all the layers n Fg.3.1. (c) Termnal passes prmal varables y (t) and rj k (t) to all neghbors j N (). (d) It then evaluates and broadcasts k N () y k(t) to j N (). Dual updates assocated wth the layer nterfaces n Fg.3.1 are now performed usng these and locally accessble prmal varables. We proceed to (a) for the next teraton Connectvty graph of a network wth n = 15 termnals randomly placed n a square wth sde L = 100 meters. Termnals can communcate wth neghbors whose dstances are wthn 30 meters. The numbers on each edge shows the dstance (n meters) between two communcatng termnals Feasblty. After about 500 steps, all constrants are satsfed n an ergodc sense wthn 10 2 tolerance. The average rate constrant takes the longest tme to be satsfed. Ths s because the transmsson rate on lnk T T j depends not only on schedules and powers of T but also on those of T j and neghbors of T j. Ths requres nformaton to be receved from, and propagated to, 2-hop neghbors (a) Optmalty. As tme grows, prmal and dual objectves approach each other. (b) Correlaton between Q 1 (t) and Q 6 (t). At the begnnng, there s sgnfcant correlaton between Q 1 (t) and Q 6 (t). But as tme grows, the correlaton vanshes and becomes neglgble Optmal routes for flow 1 (from T 1 to T 2 ) and flow 2 (from T 8 to T 11 ) x

15 4.1 Optmal power allocaton functon P (ĥ) (left) and channel backoff functon B (ĥ) (rght) for sngle user pont-to-pont channel. Curves shown for channel state nformaton (CSI) varance σ 2 e = 0.1, σ 2 e = 0.1, and σ 2 e = 0, correspondng to perfect CSI. As CSI varance ncreases power allocaton s more conservatve for small channel values. When the CSI varance s large, the backoff functon selects codes of a hgher rate than what s dctated by the channel estmate. Channel coeffcent follows a complex Gaussan dstrbuton CN (0, 2), average power budget P 0 = 1, and channel condtonal pdf m h ĥ as n (4.9) Convergence of average transmsson rate (left) and average power consumpton (rght) for Algorthm 2. Average transmsson rate as a functon of tme s shown for Algorthm 2 and cases n whch only the backoff functon s optmzed meanng p(t) = P 0 or only the power allocaton functon s optmzed mplyng b(t) = ĥ(t). Jont optmzaton yelds substantal ncrease of average communcaton rate. Average power budget P 0 = 1, constant step sze ɛ = 0.01, and channel estmaton error σ 2 e = Rate (left) and power (rght) convergence of Algorthm 3. Sum of average transmsson rates s shown for Algorthm 3 and two suboptmal solutons. One case uses a backoff functon wth fxed outage probablty 0.01 and the other case optmzes power allocaton only mplyng b f n(t) = ĥf n(t). Jont optmzaton yelds substantal ncrease of average communcaton rate. Average power budget P 0 = 4, constant step sze ɛ = 0.01, and channel estmaton error σ 2 e = x

16 4.4 Rate (left) and power (rght) convergence of Algorthm 4. Proportonal far utlty of average transmsson rates s shown for Algorthm 4 and two suboptmal solutons n whch only the backoff functon meanng p n (t) = P 0,n or only the power allocaton functon mplyng b n (t) = ĥn(t) are optmzed. Jont optmzaton yelds substantal ncrease of average communcaton rate. Average power budget P 0,n = 1 (rght), constant step sze ɛ = 0.01, and channel estmaton error σe 2 = Comparson of the expected sum rate utlty acheved by the optmal FDMA (ρ = 1), the optmal RA (ρ = 0) and the proposed algorthm (ρ {0, 0.1, 0.2,, 1}). The total number of termnals s n = The expected sum rate utlty acheved by the proposed algorthm normalzed by that acheved by the optmal FDMA for n = 10 and n = 50. The horzontal lne s 1/e The expected sum rate utlty acheved by the proposed algorthm for dfferent ρ. For all cases, the expected utlty ncreases as the total number of termnals grows Comparson of the average sum rate utlty of the network acheved by the optmal FDMA (ρ = 1), the RA (ρ = 0) and the proposed algorthm (ρ {0, 0.1, 0.2,, 1}). The network connectvty graph s the same as the one shown n Fg x

17 Chapter 1 Introducton Optmal desgn s emergng as the future paradgm for wreless networkng. The fundamental dea s to select operatng ponts as solutons of optmzaton problems, whch, nasmuch as optmzaton crtera are properly chosen, yeld the best possble network. Results n ths feld nclude archtectural nsghts, e.g., 9, and protocol desgn, e.g., 13, 22, but a drawback shared by most of these works s that they rely on global channel state nformaton (CSI);.e., the optmal varables of a termnal depend on the channels between all pars of termnals n the network. Whle avalablty of global CSI s plausble n certan stuatons, t s unlkely to hold f tme varyng fadng channels are taken nto account. In ths case, dstrbuted algorthms n whch termnals operate based on locally avalable CSI are more practcal. The focus of ths thess s to develop dstrbuted algorthms for the optmal desgn of wreless networks. When only local CSI s avalable, operatng varables of each termnal are selected as functons of local CSI. Ths further leads to the selecton of random access as the natural medum access choce. Indeed, f transmsson decsons depend on local channels only and these channels are random and ndependent for dfferent termnals, transmsson decsons can be vewed as random and resultant lnk capactes as lmted by collsons. In ths chapter, we present an overvew of random access channels and networks that wll be used n the rest of the thess. 1

18 1.1 Background Random access wreless channels Consder a multple access channel wth n termnals contendng to communcate wth a common AP. Tme s dvded n slots dentfed by an ndex t. We assume a backlogged system,.e., all termnals always have packets to transmt n each tme slot. The tme-varyng nonnegatve channel h (t) R + between termnal and the AP at tme t s modeled as block fadng for ths to be true the length of a tme slot has to be comparable to the coherence tme of the channel. Channel gans h (t 1 ) and h (t 2 ) of termnal at dfferent tme slots t 1 t 2 are assumed ndependent and dentcally dstrbuted (..d.) wth pdf m h ( ). Channel gans h (t) and h j (t) of dfferent termnals j are also assumed ndependent. Channels are assumed to have contnuous pdf. Ths latter assumpton holds true for models used n practce, e.g., Raylegh, Rcan and Nakagam 14, Ch. 3. We assume each termnal has access to ts channel gan h (t) at each tme slot t. Whle there are varous alternatves to obtan channel state nformaton, the smplest would be for the AP to send a beacon sgnal at the begnnng of each tme slot. Ths beacon sgnal would serve the double purpose of provdng a reference for channel estmaton as well as a synchronzaton sgnal. Based on ts channel state h (t), node decdes whether to transmt or not n tme slot t by determnng the value of a schedulng functon q (t) := Q (h (t)) : R + {0, 1}. Node transmts n tme slot t f q (t) = 1 and remans slent f q (t) = 0. Notce that each termnal has a dfferent schedulng functon and that schedules q (t) are determned based on the CSI of each node ndependently of other termnals. Although each node has access to ts local CSI h (t), the underlyng pdf m h ( ) s unknown. Besdes channel access decsons, termnals also adapt transmsson power to ther channel gans through a power control functon P (h (t)) : R + 0, p nst, where p nst R + s a constant representng the nstantaneous power constrant of node. By usng ths functon, termnal 2

19 adjusts ts transmsson power P (h (t)) n response to h (t). Smlar to q (t), we defne p (t) := P (h (t)), representng the power allocated to node n tme slot t. If node transmts n tme slot t, p (t) and h (t) jontly determne the transmsson rate through a functon C (h (t)p (t)) : R + R +. The exact form of C (h (t)p (t)) depends on how the sgnal s modulated and coded at the physcal layer. Examples consdered later n the thess nclude capacty-achevng codes and adaptve modulaton and codng (AMC). Wth capacty-achevng codes, C (h (t)p (t)) takes the form ( C (h (t)p (t)) = B log 1 + h ) (t)p (t), (1.1) BN 0 where B and N 0 are the channel bandwdth and the power spectral densty of the channel nose, respectvely. Wth AMC, there are M transmsson modes avalable. The mth mode affords communcaton rate τ m and s used when the sgnal to nose rato (SNR) h (t)p (t)/bn 0 s between η m and η m+1. The rate functon s therefore C (h (t)p (t)) = M m=1 ( τ m I η m h ) (t)p (t) η m+1, (1.2) BN 0 where I( ) denotes the ndcator functon. To keep the analyss general we do not restrct C (h (t)p (t)) to take ether specfc form. It s only assumed that C (h (t)p (t)) s a nonnegatve ncreasng functon of the product of h (t) and p (t) that takes fnte values for fnte arguments. These assumptons are satsfed by (1.1) and (1.2) and are lkely to hold n practce. Snce termnals contend for channel access, transmsson of termnal n a tme slot t s successful f and only f q (t) = 1 and q j (t) = 0 for all j. If the transmsson of termnal s successful, ts transmsson rate s determned by C (h (t)p (t)). As as consequence, the nstantaneous transmsson rate for termnal n tme slot t s r (t) = C (h (t)p (t)) q (t) n j=1,j 1 q j (t). (1.3) Assumng an ergodc mode of operaton, qualty of servce s determned by the long term be- 3

20 havor of r (t). Ths mples that system performance s determned by the ergodc lmts 1 r := lm t t 1 = lm t t t r (u) u=1 t C (h (u)p (u)) q (u) u=1 n j=1,j 1 q j (u). (1.4) Assumng ergodcty of schedules q (t) = q (h (t)) and power allocatons p (t) = p (h (t)), the lmt r can be wrtten as a expected value over channel realzatons, r = E h Q (h )C (h P (h )) n 1 Q j (h j ), (1.5) j=1,j where we have defned the vector h = h 1,, h n T groupng all channels h. An mportant observaton here s that snce termnals are requred to make channel access and power control decsons ndependently of each other, Q (h ) and P (h ) are ndependent of Q j (h j ) and P j (h j ) for all j. Ths allows us to rewrte r as r = E h Q (h )C (h P (h )) n j=1,j 1 Ehj Q j (h j ). (1.6) In addton to nstantaneous power constrants p (t) p nst, termnals adhere to average power constrants p avg R + as n, e.g., 8. Ths average power constrant restrcts the long term average of transmtted power that we ether wrte as an ergodc lmt or as an expectaton over channel realzatons, 1 p := lm t t t q (u)p (u) = E h Q (h )P (h ). (1.7) u= Random access wreless networks Consder an ad-hoc wreless network consstng of J termnals ndexed as = 1,... J. Network connectvty s modeled as a graph G = (V, E) wth vertces v V := {1,..., J} representng the J termnals and edges e = (, j) E connectng pars of termnals that can communcate wth each other. Denote the neghborhood of termnal as N () := {j (, j) E} and defne the nterference neghborhood of the lnk (, j) as the set of nodes M (j) := N (j) {j}\{} whose transmsson 4

21 can nterfere wth a transmsson from to j. The network supports a set K := {1,..., K} of end-to-end flows through multhop transmsson. The average rate at whch k-flow packets are generated at s denoted by a k. Termnal transmts these packets to neghborng termnals at average rates r k j and, consequently, receves k-flow packets from neghbors at average rates rk j. To conserve flow, exogenous rates a k and endogenous rates rk j at termnal must satsfy a k j N () ( r k j rj k ), for all V, and k K. (1.8) Further denote the capacty of the lnk from j as c j. Snce packets of dfferent flows k are transmtted from to j at rates rj k t must be rj k c j, for all (, j) E. (1.9) k K Unlke wrelne networks where c j are fxed, lnk capactes n wreless networks are dynamc. Smlar to what we dd n Secton 1.1.1, let tme be dvded nto slots ndexed by t and denote the channel between and j at tme t as h j (t). The channel s assumed to be block fadng and channel gans h j (t) of lnk (, j) are assumed ndependent and dentcally dstrbuted wth probablty dstrbuton functon (pdf) m hj ( ). For reference, defne the vector of termnal outgong channels h (t) := {h j (t) j N ()} and the vector of all channels h(t) := {h j (t) (, j) E}. Denote ther pdfs as m h ( ) and m h ( ), respectvely. Based on the channel state h (t) of hs outgong lnks, termnal decdes whether to transmt or not on lnk (, j) n tme slot t by determnng the value of a schedulng functon q j (t) := Q j (h (t)) {0, 1}. If q j (t) = 1, termnal transmts on lnk (, j) and remans slent otherwse. Further defne q (t) := Q (h (t)) := j N () Q j(h (t)) to ndcate a transmsson from to any of hs neghbors. We restrct to communcate wth, at most, one neghbor per tme slot mplyng that we must have q (t) {0, 1}. We emphasze that q j (t) := Q j (h (t)) depends on local outgong channels only and not on global CSI. Further note that termnals have access to nstantaneous local CSI h (t) but underlyng pdfs m h ( ) are unknown. Besdes channel access decsons, termnals also adapt transmsson power to local CSI through 5

22 a power control functon p j (t) := P j (h (t)) takng values n 0, p nst. Here, pnst represents the j maxmum allowable nstantaneous power on lnk (, j). The average power consumed by termnal s then gven as the expected value over channel realzatons of the sum of P j (h ) over all j j N (),.e., p E h P j (h )Q j (h ), (1.10) j N () where we also relaxed the equalty constrant to an nequalty, whch can be done wthout loss of optmalty. If termnal transmts to node j n tme slot t, p j (t) and h j (t) determne the transmsson rate through a functon C j (h j (t)p j (t)) whose form depends on modulaton and codng. Due to contenton, a transmsson from to j at tme t succeeds f a collson does not occur. In turn, ths happens f: () Termnal transmts to j,.e., q j (t) = 1. () Termnal j s slent,.e., q j (t) = 0. () No other neghbor of j transmts,.e. q l (t) = 0 for all l N (j) and l. Recallng the defnton of nterference neghborhood M (j) and that f a transmsson occurs ts rate s C j (h j (t)p j (t)) we express the nstantaneous transmsson rate from to j n tme slot t as c j (t) := c j (h (t)) = C j (h j (t)p j (t))q j (t) l M (j) 1 q l(t). Assumng an ergodc mode of operaton, the capacty of lnk j can then be wrtten as c j = E h C j (h j P j (h ))Q j (h ) l M (j) 1 Q l (h l ). (1.11) Because termnals are requred to make channel access and power control decsons ndependently of each other, Q j (h ) and P j (h ) are ndependent of Q lm (h l ) and P lm (h l ) for all l. Snce Q l (h l ) := m N (l) Q lm(h l (t)) by defnton, t follows that Q j (h ) s also ndependent of Q l (h l ) for all l. Ths allows us to wrte the expectaton of the product on the rght hand sde of (1.11) as a product of expectatons, ) c j E h C j (h j P j (h ) Q j (h ) l M (j) 1 E hl Q l (h l ), (1.12) 6

23 where we also relaxed the equalty constrant to an nequalty, whch can be done wthout loss of optmalty 1. The operatng pont of a wreless network s characterzed by varables a k, rk j, c j, p and functons P j (h ), Q j (h ). Besdes (1.8)-(1.12), these varables are subject to certan box constrants. Admsson varables, have lower and upper bounds due to applcaton layer requrements,.e., a mn a k a max. Smlarly, routng varables, lnk capactes, and termnal power budgets cannot be negatve and are also subject to gven upper bounds,.e., 0 r k j rmax j, 0 c j c max j, and 0 p p max. Furthermore, accordng to defnton, P j (h ) and Q j (h ) can only take values from 0, p nst and {0, 1}, respectvely. For notatonal smplcty, we defne j vectors x := { p, a k j, rk j, c j : j N () } and P (h ) := {P j (h ), Q j (h ) : j N ()} to group all the varables related to termnal and summarze these box constrants as {x, P (h )} B wth B := x, P (h ) amn a k a max, 0 rj k rj max, 0 c j c max j, 0 p p max, 0 P j (h ) p nst j, Q j (h ) {0, 1}, Q (h ) {0, 1}. (1.13) 1.2 Roadmap Our frst nvestgaton focuses on random access channel where termnals contend for communcatng wth a the central AP. Ths models the physcal layer of the wreless random access network we shall study later on. We develop adaptve schedulng and power control algorthms for random access n a multple access channel where termnals acqure nstantaneous channel 1 If we have channel recprocty h j (t) = h j (t), the dervaton of (1.12) from (1.11) s no longer vald snce power control and channel access functons of neghborng nodes wll have common arguments mplyng that Q j (h ) and Q j (h j ) would not be ndependent. The general methodology used here seems applcable but s beyond the scope of the present paper. 7

24 state nformaton but do not know the probablty dstrbuton of the channel 16. In each tme slot, termnals measure the channel to the common access pont. Based on the observed channel value, they determne whether to transmt or not and, f they decde to do so, adjust ther transmtted power. We show that the proposed algorthm almost surely maxmzes a proportonal far utlty whle adherng to nstantaneous and average power constrants. These results are presented n Chapter 2. We then generalze the algorthm proposed for random access channel to wreless multhop networks where each node determnes ts operatng pont usng ts local CSI dstrbutedly 17. Snce the assocated optmzaton problem s nether convex nor amenable to dstrbuted mplementaton, a problem approxmaton s ntroduced. Ths approxmaton s stll not convex but t has zero dualty gap and can be solved and decomposed nto local subproblems n the dual doman. The soluton method s through a stochastc subgradent descent algorthm that operates wthout knowledge of the fadng s probablty dstrbuton and leads to an archtecture composed of layers and layer nterfaces. Wth lmted amount of message passng among termnals and small computatonal cost, we show that the proposed algorthm converges almost surely n an ergodc sense. These results are presented n Chapter 3. Both above proposed algorthms requre termnals to adapt transmsson parameters such as power and rate to tme-varyng channel condtons to mprove system s overall performance. Although accurate CSI s essental to acheve ths goal, perfect CSI s rarely avalable n practce due to estmaton errors and, perhaps more fundamentally, to feedback delay. Our next topc s to develop algorthms to handle mperfect CSI n the transmsson over wreless channels 18. In partcular, we consder three types of wreless channels, namely sngle user pont-to-pont block fadng channels 15, multuser downlnk orthogonal frequency dvson multplexng (OFDM) 38, and multuser uplnk random access (RA) 29, where the transmtter adapt transmtted power and codng mode to mperfect channel estmates n order to maxmze expected throughput subject to average power constrants. To reduce the lkelhood of packet losses due to the msmatch 8

25 between channel estmates and actual channel values, a backoff functon s further ntroduced to enforce the selecton of more conservatve codng modes. Jont determnaton of optmal power allocatons and backoff functons s a nonconvex stochastc optmzaton problem wth nfntely many varables that despte ts lack of convexty s part of a class of problems wth null dualty gap. Explotng the resultng equvalence between prmal and dual problems, we show that optmal power allocatons and channel backoff functons are unquely determned by optmal dual varables. Ths affords consderable smplfcaton because the dual problem s convex and fnte dmensonal. We further explot ths reducton n computatonal complexty to develop teratve algorthms to fnd optmal operatng ponts. These results are presented n Chapter 4. So far the dstrbuted algorthms we developed are based on local CSI only (ether perfect or mperfect). In practce, termnals may have knowledge about channels of neghborng nodes n addton to local CSI. Ths motvates us to nvestgate wreless networks where each termnal has a dfferent belef about the global channel states and adapts ts transmsson polcy to the belef. In ths settng, frequency dvson multple access (FDMA) and channel aware random access (RA) are two specal cases where perfect global and local CSI are avalable, respectvely. To fnd solutons for general cases, we formulate the problem as a Bayesan game n whch each termnal maxmzes the expected utlty based on ts belef. We show that optmal solutons for both FDMA and RA are equlbrum ponts of the game. Therefore, the proposed game theoretc formulaton can be regarded as general framework for multuser wreless communcatons. Furthermore, we develop a cogntve access algorthm that solves the problem approxmately. These results are presented n Chapter 5. 9

26 Chapter 2 Dstrbuted algorthms for optmal random access channels In ths chapter, we consder wreless random access channels n whch termnals contend for access to a common access pont (AP) as ntroduced n Secton To explot favorable channel condtons termnals adapt ther transmtted power and access decsons to the state of the random fadng channels lnkng them to the AP. The challenges n developng ths adaptve scheme are that termnals have access to ther own channel state nformaton (CSI) only, and that the probablty dstrbuton functon (pdf) of the fadng channel s unknown. Our goal s to develop a dstrbuted learnng algorthm to determne optmal transmtted power and channel access decsons relyng on local CSI only. The dea of adaptng medum access and power control to CSI has been extensvely explored n wreless communcatons. Early references dealng wth power adaptaton on the uplnk of multuser systems focus on centralzed power control schemes where the AP collects channel states for all termnals to select the one to be scheduled. In, e.g., 19, the AP schedules the termnal wth the best channel gan wth a power adapted to the channel condton. Smlar deas 10

27 have also been used for schedulng and resource allocaton n broadcast downlnk channels, see e.g., 3, 11, 23. Although these centralzed schemes explot multuser dversty, they requre sgnfcant nformaton exchange between termnals and the AP; a problem exacerbated when the number of users s large. To avod ths overhead, recent work ntegrates channel adaptaton nto random access protocols. Explotng the dea of algnng schedules to good channel opportuntes, 29 develops a dstrbuted channel-aware Aloha protocol n whch termnals transmt only when ther channel gans exceed pre-defned thresholds. Ths algorthm s shown to be asymptotcally optmal n the sense that the only performance loss compared to a centralzed scheme s due to user contenton. Under smple collson models, t has been shown that dstrbuted threshold-based schedulers wth properly desgned thresholds maxmze total throughput of a network wth homogeneous users and total logarthmc throughput n the case of heterogeneous users 50. Smlar thresholdbased decentralzed adaptve random access schemes have been nvestgated for other types of networks wth dfferent packet recepton models, see e.g., 1, 6, 25, 27, 30, 46, 51. To compute the optmal thresholds, however, termnals are assumed to know the probablty dstrbuton of ther fadng channels. Ths s a restrctve assumpton because the channel fadng dstrbuton s usually unknown and can only be estmated based on channel observatons. Overcomng ths lmtaton motvates the development of adaptve algorthms to learn optmal operatng ponts based on local CSI 4, 37. The work n 4 proposes a heurstc adaptve algorthm for thresholdbased schedulers n whch the thresholds are tuned based on local channel realzatons n a tme wndow. The work n 37 develops an onlne learnng algorthm for transmsson probablty and power control under rate constrants usng game-theoretc approaches. However, nether 4 nor 37 guarantees global optmalty. The contrbuton of ths chapter s the development of an optmal dstrbuted adaptve algorthm for schedulng and power control gven that termnals only have access to local CSI and operate ndependently of each other. At each tme slot, termnals observe ther channel states 11

28 and decde whether to transmt or not. If they decde to transmt, they choose a power for ther communcaton attempt. As tme progresses, power budgets are satsfed almost surely, whle the network almost surely maxmzes a weghted proportonal far utlty. We remark that termnals operate ndependently wthout access to the channel state of other termnals and that the channel pdf s unknown. The proposed algorthm can handle general non-convex, even dscontnuous, rate functons wth manageable computatonal complexty. It s worth notng that under the frame work of network utlty maxmzaton (NUM) algorthms for computng optmal channel access probabltes n random access networks are developed (see e.g. 21). However, nether fadng nor power adaptaton s consdered n these work. The presentaton begns by formulatng optmal adaptve random access as a utlty maxmzaton problem whose objectve s to maxmze a weghted sum of throughput logarthms (Secton 2.1). The varables to be determned as a soluton of ths optmzaton problem are a schedulng functon that determnes f a termnal should transmt or not based on ts CSI, and a power allocaton functon that maps a termnal CSI to ts transmt power. It s mportant to remark that: () because fadng takes on a contnuum of values, ths optmzaton problem s nfnte-dmensonal; () the constrants modelng random access are non-convex; () despte the exstence of these non-convex constrants optmzaton problems of ths form are known to have null dualty gap 33; and (v) snce the number of constrants turns out to be fnte the optmzaton problem s fnte-dmensonal n the dual doman. A further complcaton s that the orgnal problem formulaton yelds solutons that requre access to global CSI. We start by overcomng the dependence on global CSI by ntroducng an equvalent decomposton n per-termnal subproblems whereby nodes maxmze local utltes (Secton 2.2.A). Whle ths reformulaton yelds solutons that depend on local CSI only, attemptng a soluton n the prmal doman s dffcult because the per-termnal subproblems nhert nfnte dmensonalty and lack of convexty from the orgnal problem formulaton, as well as the need to have access to the channel pdf. We therefore explot the lack of dualty gap to approach ther soluton through 12

29 a stochastc subgradent descent algorthm n the dual doman (Secton 2.2.B). Based on channel realzatons n each tme slot, the algorthm computes nstantaneous values for the schedulng and power allocaton functons and updates Lagrangan multplers n a drecton that can be proven to pont towards the set of optmal dual varables n an average sense (Proposton 1). Explotng ths fact we prove that the throughput utlty acheved by the algorthm almost surely converges to a value close to the optmal utlty. The gap between the optmal and the acheved utlty can be made arbtrarly small by reducng a fxed step sze (Theorem 1). The chapter closes wth a numercal evaluaton of the proposed algorthm for a randomly generated heterogeneous network (Secton 2.3). To llustrate generalty of the proposed approach we consder a system wth termnals employng capacty achevng codes (Secton 2.3.1) and a more practcal scenaro wth nodes employng adaptve modulaton and codng (Secton 2.3.2). Concludng remarks are presented n Secton Problem formulaton Consder a random access channel as ntroduced n Secton Wth rates r gven as n (1.6), our objectve s to maxmze a weghted proportonal far (WPF) utlty defned as n U(r) = w log(r ), (2.1) =1 where r = r 1,, r n T s the vector of rates and w R + s the weght coeffcent for termnal. Settng w = w j for all j n a homogenous system wth all channels havng the same pdf, the WPF utlty s equvalent to maxmzng the sum of throughputs. In a heterogeneous network where channel pdfs vary among users, maxmzng U(r) yelds solutons that are far snce t prevents users from havng very low transmsson rates. Groupng the objectve n (2.1) wth the constrants n (1.6) and (1.7), optmal adaptve random 13

30 access s formulated as the followng optmzaton problem P = max U(r) s.t. r = E h Q (h )C (h P (h )) E h Q (h )P (h ) p avg, n j=1,j 1 Ehj Q j (h j ), Q (h ) Q, P (h ) P, (2.2) where Q s the set of functons R + {0, 1} takng values on {0, 1} and P represents the set of functons R + 0, p nst takng values on 0, p nst. Notce that the jont optmzaton across users requred to solve (2.2) ntroduces functonal dependence between the actons of dfferent termnals. Ths s not ncongruent wth the requrement of statstcally ndependent schedules n each tme slot. In fact, the notatons Q (h ) and P (h ) n (2.2) stpulates that termnals are requred to make channel access and power allocaton decsons based on local CSI only. Consequently, although problem (9) requres jont optmzaton across users, t restrcts optmzaton to polces that result n statstcally ndependent operatons. The goal of ths chapter s to develop an onlne algorthm to determne schedules q (t) and power assgnments p (t) havng statstcs that solve the optmzaton problem n (2.2). The algorthm s requred to: () operate wthout knowledge of the channel dstrbuton; and () yeld functons q (t) and p (t) that depend on the current and past values of the local channel h (t) but are ndependent of other termnal s channels h j (t) for j. Remark 1. In order to allow termnals to know f ther transmssons are successful or not, the AP provdes feedback on whether the transmsson attempt was successful or a collson detected. If a termnal transmts a packet but detects a collson, t can reschedule the packet for retransmsson n a subsequent tme slot. We remark that feedback does not ntroduce correlaton between the transmsson decsons of dfferent termnals. The provded feedback only tells termnals f they should retransmt prevous packets or not, but does not enforce them to make channel access or power allocaton decsons. 14

31 2.2 Adaptve algorthms for optmal random access channels The stated goal s to devse schedulng and power control polces based on local CSI that are globally optmal as per (2.2). These two objectves,.e., global optmalty whle relyng on local CSI, seem to contradct each other. Because r depends not only on Q (h ) and P (h ) but on Q j (h j ) for all j, t seems that optmal Q (h ) and P (h ) solvng (2.2) mght also be functons of other termnals CSI. To see that ths s not the case, we wll show that t s possble to decompose (2.2) n per termnal subproblems. After ntroducng ths decomposton the complcatng fact that the channel pdf f h (h ) s unknown remans. To overcome ths complcaton, we wll ntroduce a stochastc subgradent descent algorthm n the dual doman that s optmal n an ergodc sense Problem decomposton and ts dual Begn then by separatng (2.2) n per termnal subproblems. To do so, we substtute (1.6) nto (2.1) and express the logarthm of a product as a sum of logarthms. As a result, the global utlty n (2.1) can be rewrtten as n U(r) = log E h Q (h )C (h P (h )) + =1 w n j=1,j log 1 E hj Q j (h j ). (2.3) Note that each summand n (2.3) only nvolves varables related to a partcular node. Thus, we can reorder summands n (2.3) to group all of the terms pertanng to node. Further defnng w := n j=1,j w, we can rewrte (2.3) as U(r) = n =1 w log E h Q (h )C (h P (h )) + w log 1 E h Q (h ) := n U, (2.4) where we have defned the local utltes U. Snce U only nvolves varables that are related to termnal, t can be regarded as a utlty functon for termnal. To maxmze U(r) for the whole system t suffces to separately maxmze U for each termnal. Introducng auxlary varables x = E h Q (h )C (h P (h )) and y = E h Q (h ), t follows that (2.2) s equvalent to =1 15

32 the followng per termnal subproblems P = max w log x + w log(1 y ) s.t. x E h Q (h )C (h P (h )), y E h Q (h ), E h Q (h )P (h ) p avg, x 0, 0 y 1, Q (h ) Q, P (h ) P, (2.5) where we relaxed the equalty constrants to nequalty ones whch can be done wthout loss of optmalty. Fndng optmal solutons of (2.5) for all termnals s equvalent to solvng (2.2). Dfferent from (2.2), however, there s no couplng between varables of dfferent termnals n (2.5). Ths property leads naturally to optmal Q (h ) and P (h ) that are ndependent of other termnals CSI as requred by problem defnton. Alas, (2.5) nherts the complex structure of (2.2). As s the case wth (2.2), solvng (2.5) s dffcult because: () The optmzaton space n (2.5) ncludes functons Q (h ) and P (h ) that are defned on R +, mplyng that the dmenson of the problem s nfnte. () The rate functon C (h P (h )) s n general non-concave wth respect to h P (h ), and may be even dscontnuous as n (1.2). () The constrants nvolve expected values over random varables h whose pdfs are unknown. An mportant observaton s that the number of constrants n (2.5) s fnte. Ths mples that whle there are nfnte varables n the prmal doman, there are a fnte number of varables n the dual doman. Ths observaton suggests that (2.5) s more tractable n the dual space. Introduce then Lagrange multplers λ = λ 1, λ 2, λ 3 T assocated wth the frst three nequalty constrants n (2.5); defne vectors x := x, y T and P (h ) := Q (h ), P (h ) T ; and wrte the Lagragan of the optmzaton problem n (2.5) as L (x, P (h ), λ ) =w log x + w log(1 y ) + λ 1 E h Q (h )C (h P (h )) x 16

33 + λ 2 y E h Q (h ) + λ 3 p avg E h Q (h )P (h ) =λ 3 p avg + w log x λ 1 x + w log(1 y ) + λ 2 y + E h Q (h ) λ 1 C (h P (h )) λ 2 λ 3 P (h ). (2.6) where the second equalty follows after reorderng terms n the frst equaton. Notce that the frst term n the second equalty n (2.6) depends on x only, the second term on y and the thrd term on P (h ) and Q (h ). Ths property s exploted later on. The dual functon s then defned as the maxmum of the Lagrangan over the set of feasble x and P (h ),.e., g (λ ) := max L (x, P (h ), λ ) s.t. x 0, 0 y 1, Q (h ) Q, P (h ) P. (2.7) We now can wrte the dual problem as the mnmum of g (λ ) over postve dual varables,.e., D = mn λ 0 g (λ ). (2.8) In general, the optmal dual value D of (2.8) provdes an upper bound for the optmal prmal value P of (2.5),.e., D P. Whle the nequalty s typcally strct for non-convex problems, for the problem n (2.5) P = D as long as the fadng dstrbuton has no realzaton wth postve probablty 33. Notce that ths s true despte the non-convex constrants present n (2.5). Ths lack of dualty gap mples that the fnte dmensonal convex dual problem s equvalent to the nfnte dmensonal nonconvex prmal problem. Whle ths affords a substantal mprovement n computatonal tractablty, t does not necessarly mean that solvng the dual problem s easy because evaluaton of the dual functon s value requres maxmzaton of the Lagrangan. In partcular, ths maxmzaton ncludes an expected value over the unknown channel dstrbuton f h (h ). Stll, convexty of the dual functon allows the use of descent algorthms n the dual doman because any local optmal soluton s a global optmal soluton λ = λ 1, λ 2, λ 3 T. Ths property s exploted next to develop a stochastc subgradent descent algorthm that solves (2.8) usng observatons of nstantaneous channel realzatons h (t). 17