Optimal Backpressure Routing for Wireless Networks with Multi-Receiver Diversity

Transcription

1 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH Optimal Backpressure Routig for Wireless Networks with Multi-Receiver Diversity Michael J. Neely Uiversity of Souther Califoria mjeely Abstract We cosider the problem of optimal schedulig ad routig i a ad-hoc wireless etwork with multiple traffic streams ad time varyig chael reliability. Each packet trasmissio ca be overheard by a subset of receiver odes, with a trasmissio success probability that may vary from receiver to receiver ad may also vary with time. We develop a simple backpressure routig algorithm that maximizes etwork throughput ad expeds a average power that ca be pushed arbitrarily close to the miimum average power required for etwork stability, with a correspodig tradeoff i etwork delay. The algorithm ca be implemeted i a distributed maer usig oly local li error probability iformatio, ad supports a blid trasmissio mode (where error probabilities are ot required) i special cases whe the power metric is eglected ad whe there is oly a sigle destiatio for all traffic streams. Idex Terms Broadcast advatage, distributed algorithms, dyamic cotrol, mobility, queueig aalysis, schedulig I. INTRODUCTION I this paper, we cosider a multi-ode, multi-hop wireless etwork with ureliable chaels. Each trasmissio li has a associated error probability that may vary with time due to exteral factors such as eviromet chages or user mobility. May previous studies assume that accurate chael iformatio is available so that error probabilities are relatively small ad ca be eglected. However, i this work we cosider the opposite case where precise chael iformatio is difficult or impossible to obtai, but where simple estimates of chael quality ca be made based o limited chael feedback. A motivatig example is a uderwater sesor etwork that uses acoustic chaels with large propagatio delays. This is a particularly challegig eviromet due to time varyig wave ripple, complex sigal reflectios betwee surface ad groud, ad large delay spreads [1] [2]. While it may ot be practical to assume that a accurate chael quality ca be determied at the time of packet trasmissio, it is reasoable to estimate the error probability based o past sigal stregth values ad/or ACK/NACK history from previous trasmissios. The problem of ureliable chaels is also importat i other cotexts, such as mobile etworks where kowledge of which receivers are withi trasmissio rage may be ucertai, or i dese ad-hoc etworks where upredictable trasmissios of other odes ca act as radom iter-chael iterferece. It is imperative to develop flexible mathematical models of such etworks, ad to develop robust etworkig This material is based o work supported by the Natioal Sciece Foudatio grat OCE error X broadcastig Fig. 1. A multi-hop etwork with chael errors ad multi-receiver diversity. I this example there is a sigle destiatio idicated by the star ode. Note that a closest-to-the-destiatio heuristic might result i data beig routed from ode 1 to 2 to 3, resultig i a deadlock. strategies that exploit all system resources to operate efficietly i these extreme eviromets. I this paper, we desig robust algorithms by exploitig the broadcast advatage of wireless etworks. Specifically, our etwork model icludes the fact that a sigle packet trasmissio might be overheard by a subset of receiver odes withi rage of the trasmitter. This creates a multi-receiver diversity gai, where the probability of successful receptio by at least oe ode withi a subset of receivers ca be much larger tha the correspodig success probability of just oe receiver aloe. Hece, it is desirable to desig flexible routig algorithms that do ot require a sigle ext hop receiver to be specified i advace. Such algorithms ca dyamically adjust routig ad schedulig decisios i respose to the radom outcome of each trasmissio. The wireless broadcast advatage has bee used i various cotexts, for example, i [3] for the desig of wireless multicast algorithms, ad i [4] for the desig of miimum eergy disjoit paths. Our model ad problem formulatio is closest to the work by Zorzi ad Rao i [5], ad more recetly by Biswas ad Morris i [6], where efficiet methods of usig multi-receiver diversity for packet forwardig are explored. We ote that such formulatios ievitably ivolve situatios where the same packet is redudatly distributed over differet etwork odes. A fudametal decisio is whether to allow the differet versios of the packet to simultaeously propagate throughout the etwork, or to desigate oly a sigle copy that is allowed to proceed. The work i [5] cosiders the

2 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH simple heuristic that shifts packet forwardig resposibilities to the receiver that is closest to the destiatio. While this scheme has may desirable properties, especially for large adhoc etworks, it is clear that for a give etwork of fixed size, the closest-to-destiatio heuristic either maximizes throughput or miimizes average power expediture. Further, this scheme ca lead to a udesirable deadlock mode if data is cosistetly forwarded to a particular ode for which there are o other ext-hop receivers that are closer to the destiatio (see Fig. 1). Thus, it is ofte better to route packets alog paths that temporarily take them further from the destiatio, especially if these paths evetually lead to lis that are more reliable ad/or that are ot as heavily utilized by other traffic streams. The work i [6] cosiders a routig heuristic based o a estimated delivery cost, computed by a estimate of the expected umber of hops required to reach the destiatio alog a traditioal shortest path. However, this method is ot ecessarily optimal i terms of eergy or throughput. There are several difficulties associated with developig a throughput optimal algorithm i this cotext. First, idividual odes might oly kow the error probabilities o their ow outgoig lis, ad may ot kow the error rates or traffic loads o other portios of the etwork. Secod, eve if cetralized etwork kowledge were fully available, a optimal algorithm would eed to specify a cotigecy pla for each possible radom trasmissio outcome. For example, suppose a give ode trasmits a packet for which there are k potetial receivers. There are 2 k possible outcomes of this sigle trasmissio (oe for each possible subset of successful receivers). A optimal algorithm would require a decisio for each possible outcome, perhaps also allowig for redudat packet forwardig. Hece, the desig of a optimal algorithm must overcome these geometric complexity issues. This is further complicated if there are multiple simultaeous packet trasmissios ad multiple traffic streams sharig the same etwork, ad if the etwork topology ad li error probabilities are chagig with time. I this paper, we overcome these challeges with a simple solutio that uses the cocept of backpressure routig ad Lyapuov drift. We first show that it is possible to restrict attetio to algorithms that do ot allow redudat forwardig, without loss of optimality. We the show that the optimal packet commodity to trasmit at each etwork ode ca be determied by a backpressure idex that compares the curret queue backlog of each commodity to the backlog i the potetial receivers. Oce a packet from this optimal commodity is trasmitted, the resposibility of forwardig the packet to its destiatio is shifted to the receiver ode that maximizes the differetial backlog. Resposibility is retaied by the origial trasmitter if o suitable receivers are foud o a give trasmissio attempt. Backpressure techiques of this type were first applied to multi-hop wireless etworks by Tassiulas ad Ephremides i [7], where throughput optimal algorithms were developed usig Lyapuov drift theory. Lyapuov theory has sice bee a powerful mathematical tool for the developmet of stable schedulig strategies for wireless etworks ad switchig systems [7]-[18], icludig our ow work i [15]-[18] that applies backpressure cocepts to solve problems of optimal power allocatio, routig, ad fair flow cotrol i wireless etworks with mobility. Related work o eergy efficiet wireless schedulig is developed i [19]-[22]. The work i [7]-[22] does ot cosider the broadcast advatage of wireless etworks, ad assumes that all trasmissios are fully reliable. Lyapuov schedulig for wireless MIMO dowlis with multiple trasmit ad receive ateas is cosidered i [23], ad related MIMO results are developed for chaels with errors i [24] [25]. Recet work i [26] cosiders backpressure techiques i combiatio with etwork codig, ad work i [27] cosiders backpressure strategies for cooperative trasmissio (where multiple odes ca trasmit redudat iformatio simultaeously for a power ehacemet at the receiver). Complexity issues of cooperative commuicatio uder the wireless broadcast advatage are discussed i [28]. We do ot cosider etwork codig or cooperative trasmissio i this paper, ad restrict attetio to the multi-user diversity problem for etworks with errors, as described above. It is likely that our formulatio ca be exteded to cosider more sophisticated cotrol actios by augmetig the set of decisio optios available to the etwork cotroller, i which case redudat packet forwardig may be required for optimality. I the ext sectio, we develop a simple etwork model i terms of (potetially time varyig) li error probabilities, ad specify the cotrol decisio optios for this model. I Sectio III, we specify the etwork capacity ad the miimum average power for stability associated with this model. I Sectio IV we develop the dyamic cotrol algorithm, ad i Sectio VI we exted the formulatio to iclude dyamic resource allocatio with variable rate ad power optios, where li error probabilities ca deped o trasmissio decisios. II. THE BASIC NETWORK MODEL We cosider a timeslotted system with slots ormalized to itegral uits t {0, 1, 2,.... There are N etwork odes ad L potetial trasmissio lis (possibly a sigle li for each ode pair (a, b)). All data arrives radomly to the etwork i (t) represet the umber of packets that exogeously arrive to etwork ode durig slot t that are iteded for delivery to etwork ode c. All packets destied for a particular ode c are defied as commodity c packets. Arrivals are assumed to be i.i.d. over timeslots, ad we packetized uits, ad we let A let λ = E{A (t) represet the arrival rate of commodity c data ito source ode (i uits of packets/slot). Iteral etwork queues store packets accordig to their commodities. Each packet is assumed to have a appropriate header field with commodity ad packet umber idetifiers. We assume that at most oe packet ca be trasmitted from ay give ode durig a sigle timeslot, ad let µ (t) represet the umber of packets trasmitted by ode durig slot t (where µ (t) {0, 1). Trasmissio opportuities are determied by a uderlyig radom access or time divisio multiple access (TDMA) structure, ad we let χ (t) represet a 0/1 process which is 1 if ad oly if ode is allowed to trasmit durig slot t. Each packet trasmissio is assumed to exped a costat amout of power P tra, ad is successfully

3 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH received by the other odes of the etwork accordig to receptio probabilities q (t) (for, k {1,..., N). For coveiece, we defie the etwork topology state process S(t) as the collective process of all ode trasmissio capabilities ad li coditios at time t, so that trasmissio opportuities ad li probabilities ca be determied as fuctioals of S(t) as follows: seder ode receiver ode k Cotrol Iformatio Cotrol Iformatio Packet Trasmissio ACK/NACK Fial Istructios t t+1 χ (t) = ˆχ (S(t)) q (t) = ˆq (S(t)) Let K (t) represet the set cosistig of all potetial receivers for ode durig slot t (which ca potetially chage from slot to slot if the etwork is mobile). The set K (t) ca geerally cotai all N 1 other etwork odes, although it typically has a much smaller size ad cosists oly of those odes withi realistic trasmissio rage of ode. Error evets for a sigle packet trasmissio ca be correlated over various lis, ad hece a more complete characterizatio of each trasmitter is give by probabilities q,ω (t), where Ω is a subset of odes withi the receiver set K (t), ad q,ω (t) represets the probability that the set of all odes that successfully receive the packet trasmitted by ode is exactly give by the subset Ω. This probability is also determied as a fuctioal of the topology state process: q,ω (t) = ˆq,Ω (S(t)) The error evets of differet packet trasmissios from differet odes may also be correlated, ad these correlatios i priciple are also determied by the topology state process S(t). However, we shall fid that these additioal correlatios are irrelevat to etwork capacity ad optimal cotrol. For aalytical purposes, the etwork topology state S(t) is assumed to take values i a fiite (but arbitrarily large) state space S. We ote that the success probabilities of a give li or set of lis are completely determied by the etwork topology state process S(t). That is, give S(t), these probabilities are ot affected by the trasmissio decisios µ (t) for {1,..., N. This assumptio is reasoable if all trasmittig odes use orthogoal sigals, or if iterchael iterferece ca be approximated as radomly ad idepedetly ifluecig the chael probabilities. A more geeral model where chael probabilities ca deped o trasmissio decisios is cosidered i Sectio VI. A. A Timig Diagram for Oe Timeslot The timig diagram of Fig. 2 illustrates our model of iformatio exchage betwee odes. The evets that take place betwee a trasmittig ode ad a potetial receiver ode k durig a sigle timeslot are outlied i the diagram. At the begiig of the timeslot, chael probability iformatio ad ay ecessary cotrol iformatio is passed betwee the two odes. This ca possibly take place over a dedicated cotrol chael, or might be implemeted by appedig header iformatio to packets trasmitted o previous slots. Next, the trasmitter ode observes the trasmissio opportuity process ˆχ (S(t)). If ˆχ (S(t)) = 0 the ode does ot trasmit, while if ˆχ (S(t)) = 1 the ode ca decide whether Fig. 2. A timig diagram illustratig the evets withi a sigle timeslot. or ot it desires to trasmit a packet. If it decides to trasmit, it chooses a particular packet ad trasmits it with power P tra, for a fixed amout of time as idicated i the timig diagram. Every potetial receiver ode the provides immediate ACK/NACK feedback to the trasmitter, iformig the trasmitter if the packet was successfully received. The absece of a ACK sigal is cosidered to be equivalet to a NACK (this treats the case whe the receiver ode did ot detect ay trasmissio). The trasmitter ode accumulates all of the ACK resposes, ad the trasmits a fial message that iforms the successful receivers of all other successful receivers. This fial trasmissio possibly also provides istructios for future packet forwardig. The 3-part hadshake of the timig diagram (trasmissio, ACK/NACK, ad fial message) is desiged to clealy describe a system where trasmissio outcomes are kow to all relevat odes at the ed of a sigle timeslot. This facilitates mathematical aalysis. However, i practice the last two steps of the hadshake may take place by appedig this iformatio to the packet header of future packet trasmissios. This creates a system with delayed feedback iformatio, which i priciple does ot affect throughput optimality (provided some regularity assumptios hold cocerig the timeliess of the feedback) but may affect ed-to-ed etwork delay, as discussed i more detail i Sectio VII. Throughout this paper, we make the idealistic assumptio of perfect cotrol iformatio, so that the cotrol sigals themselves are ot subject to errors. I particular, for the timig diagram of Fig. 2, it is assumed that if a packet trasmitted at ode was successfully received at ode k, the the chael from k to ad from to k is good eough for the remaiig parts of the hadshake to be successful. This is a reasoable assumptio if forward ad backward chaels are relatively similar for the duratio of a timeslot, or if the dedicated cotrol chael is reliable. The possibility of cotrol chael errors ca create aother situatio of delayed feedback iformatio, ad this is also briefly discussed i more detail i Sectio VII. B. Network Objective ad Cotrol Decisio Variables The goal is to desig a cotrol algorithm that stabilizes the etwork wheever possible. Further, the average power cost should be as small as possible. Specifically, for a power vector P = (P 1,..., P N ), we defie the separable cost fuctio h(p ) = h 1 (P 1 ) h N (P N ), where each compoet h (P ) is o-egative, cotiuous, ad has the property that h (0) = 0. The power expeded o each timeslot t is give by the vector P (t) =P tra (µ 1 (t),..., µ N (t)), ad the time

4 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH average power cost h is defied: h = lim t 1 t t 1 τ=0 E {h(p (τ)) Note that choosig h(p ) = N =1 P coicides with the objective of miimizig the time average expected power expediture. Uder our simple etwork model, we have P (t) {0, P tra for all t, so that h (P (t)) {0, h (P tra ). I this case, the h ( ) fuctio plays oly a limited role i geeralizig the miimum average power objective, although it shall be more meaigful i the exteded formulatio of Sectio VI that cosiders a cotiuum of power optios. I Sectio III we show that throughput ad eergy optimality ca be achieved without usig redudat packet forwardig. This allows the followig more detailed etwork queueig variables ad cotrol decisio variables to be defied. At each timeslot t, every etwork ode makes a trasmissio decisio µ (t) subject to µ (t) {0, 1 ad µ (t) = 0 wheever χ (t) = 0. It the chooses a packet commodity to trasmit by selectig cotrol variables µ (t) subject to: µ (t) {0, 1, N c=1 µ (t) µ (t), µ c (t) = 0 (1) That is, µ (t) represets a opportuity for commodity c packet trasmissio by ode durig slot t. This ca be either 0 or 1, but ca be 1 for at most oe commodity c. We set (t) = 0 as it does ot make sese to retrasmit a packet that has already reached its destiatio. We say that µ (t) is a trasmissio opportuity because it is useful to imagie the possibility of choosig these decisio variables idepedet of queue backlog. I cases whe a trasmissio opportuity arises but there is o commodity c packet available, the o packet is actually trasmitted. We let H (t) represet the radom variable that is 1 if a packet trasmitted from ode was successfully received by receiver k, ad zero otherwise. After receivig ACK/NACK feedback, ode selects a ew ode to take resposibility for the packet (possibly choosig itself), ad iforms its receivers of the choice. This is doe accordig to cotrol decisio µ c variables β (t), represetig the umber of commodity c packets whose resposibility is shifted from ode to ode k durig slot t, where: β (t) {0, 1, β (t) µ (t)h (t) β (t) = 0, N k=1 β (t) 1 (2) That is, the β (t) variables are either 0 or 1, ca be 1 oly if a commodity c trasmissio opportuity occurs o slot t ad H (t) = 1, ad ca be 1 for at most oe receiver ode k (where such a ode k is ecessarily i the set of potetial receivers K (t)). If β (t) = 0 for all k K (t), the ode retais resposibility for the packet. It shall be coveiet to also allow these decisio variables to be idepedet of queue backlog, ad so both β (t) ad µ (t) ca potetially equal 1, regardless of whether or ot ode was holdig a commodity c packet that it actually trasmitted. I this case, the H (t) value is viewed as a radom variable that is distributed the same as if a packet had actually bee trasmitted. The actual cotrol decisios β (t) i the case of o packet trasmissio are irrelevat as they do ot affect the system. However, it is useful to formally allow choosig o-zero β (t) values i this case. Specifically, we fid it useful for mathematical proofs to imagie the existece of a statioary radomized cotrol policy that chooses decisio variables idepedetly of queue backlog, but where o packets are actually trasferred if these decisios attempt trasmissio from a empty queue. Packets are stored at every ode accordig to their commodity, ad we defie U (t) as the curret umber of commodity c packets i ode at the begiig of slot t. The U (t) process takes values i the set of o-egative itegers, ad evolves accordig to the followig queueig dyamics: [ U (t + 1) max U (t) ] N k=1 β (t), 0 + N a=1 β a (t) + A (t) (3) The expressio above is a iequality rather tha a equality because the actual edogeous arrivals to ode may be less tha N a=1 β a (t) if there are little or o actual commodity c packets trasmitted from the other odes a. We formally defie U () (t) to be zero for all ad all t. III. NETWORK CAPACITY AND MINIMUM POWER Here we defie the optimal throughput ad average power cost operatig poits. The etwork layer capacity regio Λ is defied as the closure of all iput rate matrices (λ ) that ca be stabilized by the etwork accordig to some cotrol algorithm, perhaps a algorithm that uses redudat packet forwardig. We ote that this otio of capacity assumes that etwork cotrol actios are withi the scope of the system model described i Sectio II, ad i particular this model does ot iclude the possibility of cooperative trasmissio or etwork codig, which ca potetially improve performace. Suppose that the etwork topology state process S(t) takes values o a fiite state space S, ad has well defied time average probabilities π s for each s S. For each ode, let H deote the set of all subsets Ω of {1,..., N {. For each subset Ω, recall that ˆq,Ω (s) is the probability that Ω is exactly the set of all successful receivers of a packet trasmitted by ode, give such a packet is trasmitted whe the topology state is give by S(t) = s. Theorem 1: (Network Capacity ad Miimum Cost) The etwork capacity regio Λ cosists of all rate matrices (λ ) for which there exist multi-commodity flow variables {f together with probabilities α (s), θ (Ω ) for all, k, c, all topology states s S, ad all subsets Ω H, such that: c a f c f ab 0, f cb f a + λ b s S π s α (s) = 0, f aa = 0 (4) f b for all c (5) [ ˆq,Ω (s)θ (Ω ) Ω H where (4) holds for all a, b, c {1,..., N, (6) holds for all lis (, k), ad where the probabilities θ (Ω ) satisfy for all Ω H : θ (Ω N ) = 0 if k / {Ω {, k=1 θ (Ω ) = 1 ] (6)

5 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH ad for all s S the α (s) probabilities satisfy: N c=1 α (s) 1, α (s) = 0 if ˆχ (s) = 0 Furthermore, the miimum average power cost required for etwork stability is give by the value h that miimizes the followig metric: h = [ s S π N ] N s c=1 α (s)h (P tra ) over all {f, α =1 (s), θ (Ω ) variables that satisfy (4)-(6). The above theorem is similar to the capacity theorem of [16] [15], where the costraits (4) represet o-egativity ad flow efficiecy costraits for the flow variables {f ab, the costraits (5) represet flow coservatio costraits, ad the costraits (6) represet li costraits for each li (, k). Each α (s) value ca be iterpreted as the coditioal probability that ode trasmits a commodity c packet give that S(t) = s. Each θ (Ω ) value ca be iterpreted as the coditioal probability that ode shifts packet forwardig resposibilities to ode k, give that ode trasmits a commodity c packet that is heard exactly by the subset Ω of receivers. With this iterpretatio, the theorem ca be simplified accordig to the followig corollary. For each iput rate matrix λ = (λ ) Λ, we defie Φ(λ) as the miimum power cost h required to stabilize the system. Suppose that the iput rate matrix is iterior to the capacity regio, so that there exists a positive value ɛ such that (λ + ɛ1 ) Λ, where 1 is a idicator fuctio equal to 1 if ad oly if c, ad zero else. Corollary 1: If the topology state S(t) is i.i.d. over timeslots, the a rate matrix (λ + ɛ1 ) is i the capacity regio Λ if ad oly if there exists a statioary radomized algorithm that chooses cotrol decisio variables µ (t) ad β (t) (accordig to the costraits specified i Sectio II-B) based oly o the curret topology state S(t) (ad hece idepedet of curret queue backlog), to yield: E a { β a (t) + λ + ɛ { E β b (t) c b (7) E {h(p (t)) = Φ(λ + ɛ) (8) where ɛ = (ɛ1 ) ad P (t) = P tra (µ 1 (t),..., µ N (t)). The expectatios i (7) ad (8) are take with respect to the radom topology state S(t) ad the radom cotrol decisios based o this topology state, ad do ot deped o queue backlog. The above theorem ad its corollary demostrate that for ay rate matrix (λ ) Λ, there exists a statioary radomized algorithm (with probabilities precisely matched to the etwork traffic rates ad topology state probabilities) that ca achieve a multi-commodity flow that supports the iput rate matrix by routig all data to its proper destiatio, ad that icurs a average power cost exactly give by h. However, eve if all topology state probabilities π s were fully kow, the geometric complexity of the optimizatio problem i Theorem 1 demostrates the extreme difficulty of directly solvig for the parameters required to implemet such a policy. Theorem 1 is prove by first showig that the costraits (4)-(6) are ecessary for etwork stability. The sufficiecy part of the theorem is prove by costructig a stabilizig algorithm for ay rate matrix (λ ) that is iterior to the capacity regio (so that (λ + ɛ1 ) Λ, for some positive value ɛ). Such stabilizig policies ca be costructed with resultig average power costs that are arbitrarily close to h (by choosig ɛ arbitrarily small), with a correspodig tradeoff i ed-to-ed etwork delay. The proof of ecessity uses the fiite state space assumptio for the topology state variable S(t), ad is related to similar proofs of capacity ad miimum eergy i [16] [17] [15] (proof omitted for brevity). Sufficiecy does ot require the fiite state space property, ad is prove i the ext sectio, where a simple dyamic cotrol algorithm is costructed that ca be implemeted i real time. IV. THE DYNAMIC CONTROL ALGORITHM We have the followig dyamic cotrol algorithm, defied i terms of a o-egative cotrol parameter V that determies the degree to which we emphasize power cost miimizatio. Diversity Backpressure Routig (DIVBAR): Every timeslot t, each etwork ode observes the queue backlogs i each of its potetial receiver odes k K (t), ad observes the curret li chael probabilities associated with its receivers. Each ode determies if χ (t) = 1 (i.e., it determies if a trasmissio opportuity is available o the curret slot). If so, it performs the followig operatios: 1) For each commodity c ad each receiver k K (t), the differetial backlog weights W (t) are computed as follows: W (t) = max[u (t) U k (t), 0] (9) That is, the weight W (t) is equal to the differece betwee the commodity c backlog i ode ad the commodity c backlog i ode k (maxed with zero). 2) The receivers k K (t) are priority raed accordig to the W (t) weights, so that receivers with larger weights are ordered with higher priority (breakig ties arbitrarily). We defie k(, c, t, b) as the ode k K (t) with the bth largest weight W (t) for commodity c. Thus, by defiitio we have: W,k(,c,t,1) (t) W,k(,c,t,2) (t) W,k(,c,t,3) (t)... 3) Defie φ (t) as the probability that a packet trasmissio from ode is correctly recieved by ode k, but is ot received by ay other odes k K (t) that are raed with higher priority tha ode k accordig to the commodity c ra orderig of the previous step. 4) Defie the optimal commodity c (t) as the commodity c {1,..., N that maximizes (breakig ties arbitrarily): K (t) b=1 W,k(,c,t,b) (t)φ,k(,c,t,b) (t)

6 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH where K (t) deotes the umber of odes i the set K (t). Defie W (t) as the resultig maximum value: W (t) = K (t) b=1 W (c ),k(,c,t,b)(t)φ(c ),k(,c,t,b)(t) 5) If W(t) V h (P tra ) > 0, ode trasmits a packet of commodity c (t). Else, ode remais idle for slot t. 6) After receivig ACK/NACK feedback about the successful recipiets of the trasmissio, ode shifts resposibility of packet forwardig to the successful receiver k with the largest positive differetial backlog W (c (t)) (t). If o successful receivers have positive differetial backlog, ode retais resposibility of the packet. The above algorithm is fully distributed, i that each ode oly requires queue backlog ad li probability values for each of its eighborig odes (i.e., each ode withi K (t)). The queue backlogs ca be passed durig the cotrol iformatio phase of the timeslot, or ca be based o backlog updates received i the headers of previous packets. We ote that, as i the Dyamic Routig ad Power Cotrol (DRPC) policy of [15] [16], the algorithm ca be implemeted without loss of throughput optimality by usig out of date backlog iformatio, provided that some regularity coditios hold (see Chapter of [15]). The li error probabilities ca be obtaied based o cotrol iformatio exchage at the begiig of the timeslot (such as a pilot sigal ad a correspodig SINR measuremet, as i [16]), or ca be estimated based o previous ACK/NACK history. The above algorithm cosiders the geeral case where li error evets ca be correlated. However, computatio of the φ (t) probabilities ca be greatly simplified uder the assumptio that error evets are idepedet over each li. I this case, φ (t) is obtaied from a simple multiplicatio of the appropriate success or error probabilities of the correspodig lis. A. Algorithm Performace To facilitate mathematical aalysis, we assume the etwork topology state S(t) is i.i.d. over timeslots. 1 Note that this also icludes the case whe the topology state does ot chage over time. Defie the followig costat µ i max to be the largest umber of edogeous packet arrivals that ay sigle ode ca receive durig a timeslot. Further, defie A 2 max as a upper boud o the secod momet of the total exogeous arrivals to ay ode durig a timeslot, so that: { (N max E c=1 A (t) ) 2 A 2 max We assume the iput rate matrix is iterior to the capacity regio Λ (so that stability is possible), ad defie ɛ max as the largest scalar such that (λ + ɛ max 1 ) Λ. Theorem 2: (Algorithm Performace) If topology state variatios S(t) are i.i.d. over timeslots, ad if the iput rate matrix is strictly iterior to the capacity regio Λ, the the 1 The same algorithm ca be show to be throughput optimal for o-i.i.d. topology state variatios usig a similar T -slot Lyapuov drift argumet, see [16][15] for such a aalysis for a related algorithm. DIVBAR algorithm stabilizes all queues of the system (ad hece provides maximum throughput). Furthermore, average etwork cogestio ad average power cost satisfies: lim sup t lim sup t 1 t 1 t t 1 τ=0,c { E U (τ) NB + V h max ɛ max t 1 E {h (µ (τ)p tra ) h + NB/V τ=0 where h max = h (P tra ), ad where B is defied: B= (µi max + A max ) (10) 2 Note that choosig the cotrol parameter V to be zero leads to the best cogestio boud but does ot lead to ay power efficiecy guaratees. The parameter V ca be icreased to drive average power cost arbitrarily close to the miimum cost h required for etwork stability, with a correspodig liear icrease i average etwork cogestio (ad hece, by Little s Theorem, average delay). B. Chael Blid Packet Trasmissio I the special case whe power optimizatio is eglected (so that V = 0) ad there is a sigle destiatio for all packets, the DIVBAR algorithm ca be sigificatly simplified to allow for blid packet trasmissios. Specifically, because there is just a sigle commodity, the steps (1)-(5) ca be avoided ad the algorithm reduces to havig ode trasmit a packet wheever possible (i.e., wheever χ (t) = 1). It the receives ACK/NACK feedback from the various receivers, ad chooses the receiver k with the largest positive differetial backlog U (t) U k (t), breakig ties arbitrarily ad retaiig the packet if o receiver has a positive differetial backlog. Note that the backlog of each receiver ca simply be icluded i the ACK/NACK sigal. The algorithm thus achieves throughput optimality without requirig chael probability iformatio. This is a remarkable property, ad eables perfect throughput optimality to be achieved eve whe chael probabilities are rapidly chagig due to dramatic ode mobility. No effort is eeded to estimate error rates, or to track them if they vary with time. V. PERFORMANCE ANALYSIS Here we prove Theorem 2. The proof uses the followig result from [15] [17] [18] cocerig performace optimal Lyapuov schedulig, which is a simple but importat extesio of classical Lyapuov stability results of [7]-[16]. Let U(t) = (U (t)) represet the matrix of queue backlog values, ad assume these backlogs evolve accordig to a give probability law ad are affected by a cotrol process P (t) = (P 1 (t),..., P N (t)). Let h(p ) be ay o-egative fuctio of P, ad let h represet a target value for the time,c (U average of h(p (t)). Let L(U) = 1 2 ) 2 represet a quadratic Lyapuov fuctio, ad defie the oe step Lyapuov drift (U(t)) as follows: (U(t)) =E {L(U(t + 1)) L(U(t)) U(t)

7 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH Theorem 3: (Lyapuov Optimizatio [15] [17][18]) If there exist positive costats B, V, ɛ such that for all timeslots t ad for all queue backlogs U(t), the Lyapuov drift satisfies: (U(t)) + V E {h(p (t)) U(t) B ɛ,c U (t) + V h the all queues are stable, ad time average cogestio ad etwork cost satisfies:,c U = 1 lim sup t t t 1 τ=0,c { E U (τ) B + V h ɛ h= 1 t 1 lim sup E {h(p (τ)) h + B/V t t τ=0 The above theorem suggests the strategy of miimizig the metric (U(t)) + V E {h(p (t)) U(t) every timeslot t, which is the motivatio behid DIVBAR. A. Proof of the DIVBAR Performace Theorem (Theorem 2) The coditioal Lyapuov drift ca be computed from the queue backlog expressio (3) accordig to stadard drift techiques (see [7][15][16]), ad is give by: (U(t)) NB {,c U (t)e k β (t) a β a (t) λ U(t) where B is defied i (10). Addig the cost metric to both sides (where P (t) = P tra (µ 1 (t),..., µ N (t))), we have: (U(t)) + V E {h(p (t)) U(t) NB + V E {h(p (t)) U(t) { U (t)e β (t),c a k β a (t) λ U(t) (11) The DIVBAR algorithm is desiged to choose cotrol actios that greedily miimize the right had side of the above iequality over all possible choices of the cotrol variables µ (t), µ (t), ad β (t) that satisfy the costraits (1) ad (2). This ca be see by switchig the sums to ote that: { U (t)e β (t) β a (t) U(t) =,c k a { [ ] E β ab (t) U(t) U a (t) U b (t) ab c which reveals the differetial backlog metric (details omitted for brevity). It follows that the right had side of (11) uder the DIVBAR algorithm is less tha or equal to the correspodig expressio whe the cotrol variables are replaced with ay others, ad i particular those of the statioary radomized algorithm from (7) ad (8) of Corollary 1, ad so: (U(t)) + V E {h(p (t)) U(t) NB + V Φ(λ + ɛ),c U (t)ɛ The above iequality is i the exact form for applicatio of the Lyapuov Optimizatio Theorem (Theorem 3), ad we thus have (otig that Φ(λ + ɛ) h max ): U (NB + V h max )/ɛ (12),c h Φ(λ + ɛ) + NB/V (13) The above performace bouds hold for ay value ɛ > 0 such that (λ +ɛ1 ) Λ, ad hece the bouds ca be optimized separately over all such ɛ. Lettig ɛ ɛ max i (12) yields the cogestio boud of Theorem 2, ad lettig ɛ 0 i (13) yields the power cost boud of Theorem 2. VI. VARIABLE RATE AND POWER CONTROL Cosider ow a system with variable rate ad power cotrol optios, so that every timeslot the trasmissio rates µ(t) = (µ 1 (t),..., µ N (t)) ca be chose such that µ (t) {0, 1,..., µ out max for all t (for some pre-specified iteger µ out max), ad trasmissio power to support these rates is chose accordig a power vector P (t) = (P 1 (t),..., P N (t)), where 0 P (t) P peak for all t ad all (for some peak trasmissio power P peak ). Note that the µ (t) variable is still iteger valued, but there is o loger ay multiple access process χ (t) that places further restrictios o µ (t). Defie I(t) =(µ(t); P (t)) as the collective trasmissio cotrol decisios of all etwork odes durig slot t, ad defie I as the set of all possible optios for I(t). We assume that error probabilities are fuctios of I(t) ad the curret topology state S(t), so that: q,ω (t) = ˆq,Ω (I(t), S(t)) If m packets are trasmitted by ode, the each of them is assumed to have the same q,ω (t) probability. Correlatios i the error evets of differet packets withi the batch of m are arbitrary ad do ot affect capacity or optimal cotrol decisios. The cotrol objective of stabilizig the etwork ad miimizig h is the same as before. Usig similar reasoig, it ca agai be show that it is possible to restrict to algorithms that do ot allow redudat forwardig, without loss of optimality. A similar Lyapuov argumet the leads to the followig optimal policy: 1) Compute W (t) = max[u (t) U k (t), 0] as before. For each ode ad each commodity c, we agai ra order the receivers k K (t) with priority give by the largest values of W (t), ad defie k(, c, t, b) as ˆφ before. We defie (I(t), S(t)) as the probability that a packet trasmissio from ode durig slot t is correctly received by ode k, but ot received by ay other odes k K (t) that are raed with higher priority tha ode k accordig to the commodity c orderig. 2) Defie: G,c,t,b (I(t), S(t)) = W,k(,c,t,b) (t) ˆφ,k(,c,t,b) (I(t), S(t)) Choose a etwork-collaborative cotrol actio I (t) = (µ (t), P (t)) I ad a collectio of optimal commodities c (t) {1,..., N (for all odes ) that joitly

8 CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH maximizes the metric: K (t) G,c (t),t,b(i (t), S(t)) V h (P(t)) b=1 3) If K (t) [ b=1 G,c (t),t,b (I (t), S(t)) ] > V h (P(t)), ode trasmits µ (t) commodity c (t) packets (usig idle fill if there are ot eough such packets). 4) After receivig ACK/NACK feedback from each receiver about each of the µ (t) trasmitted packets, ode shifts resposibility of each packet to the successful receiver with the largest positive differetial backlog W (c (t)) (t). If o receivers of a give packet have positive differetial backlog, ode retais resposibility of the packet. Choosig the appropriate cotrol actio I(t) = (µ(t); P (t)) effectively optimizes over all multiple access decisios, but yields a optimizatio problem i step 2 that ca be quite difficult to solve ad may require full cetralized coordiatio. However, distributed implemetatio is possible if all odes trasmit with orthogoal sigals, ad costat factor throughput optimality results ca be achieved i a decetralized maer for some etworks models (such as the ode exclusive spectrum sharig model of [29]), if power optimizatio is eglected ad simple radom access methods are employed. We further ote that if simple radom access methods are used as i [16][15], ad if trasmissios are idepedet of queue backlog, the radom trasmissios themselves ca be viewed as part of the chael state, ad the algorithm thus achieves efficiet performace with respect to this (suboptimal) radom access. VII. DISCUSSION OF EXTENSIONS We ote that delay ca be improved by modifyig the differetial backlog W (t) by addig a estimated hop cout differetial to the destiatio, as i the EDRPC policy of [16] [15]. Optimizatio of geeral utility ad fairess metrics ca also be achieved i cases whe the iput rate matrix is either iside or outside of the capacity regio Λ by usig the simple ad optimal flow cotrol techiques of [15] [18] together with DIVBAR. Fially, we ote that Lyapuov drift theorems hold if drift expressios are off by a additive costat, so that queue backlog estimates ca be used i replacemet of actual queue backlogs [15]. This does ot effect throughput optimality, but may icrease delay by a costat proportioal to the differece betwee the true ad estimated backlog values. This eables DIVBAR to be implemeted with delayed feedback, provided the feedback delay is bouded. A queueig implemetatio of this would provide separate storage for data that has bee correctly received but has ot yet bee ackowledged by a fial istructio message from the trasmitter. REFERENCES [1] M. Stojaovic, J. A. Catipovic, ad J. G. Proakis. Phase-coheret digital commuicatios for uderwater acoustic chaels. 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