Optimal Distributed Scheduling in Wireless Networks under the SINR interference model
OOptimal Distributed Scheduling in Wireless Networksunder SINR Interference Model
P. Chaporkar (cid:63) , A. Proutiere †∗ ABSTRACT
Radio resource sharing mechanisms are key to ensuring good per-formance in wireless networks. In their seminal paper [1], Tassiu-las and Ephremides introduced the Maximum Weighted Schedulingalgorithm, and proved its throughput-optimality. Since then, therehave been extensive research efforts to devise distributed imple-mentations of this algorithm. Recently, distributed adaptive CSMAscheduling schemes [2] have been proposed and shown to be op-timal, without the need of message passing among transmitters.However their analysis relies on the assumption that interferencecan be accurately modelled by a simple interference graph. In thispaper, we consider the more realistic and challenging SINR inter-ference model. We present the first distributed scheduling algo-rithms that (i) are optimal under the SINR interference model, and(ii) that do not require any message passing . They are based ona combination of a simple and efficient power allocation strategyreferred to as
Power Packing and randomization techniques. Wefirst devise algorithms that are rate-optimal in the sense that theyperform as well as the best centralized scheduling schemes in sce-narios where each transmitter is aware of the rate at which it shouldsend packets to the corresponding receiver. We then extend thesealgorithms so that they reach throughput-optimality.
1. INTRODUCTION
The throughput experienced on a given link in wireless networksis affected by the interference generated by the transmitters of otherlinks. Interference management constitutes the main issue in thedesign of simple and efficient resource allocation (or Multiple Ac-cess Control) algorithms for such networks. Solving this issue be-comes even more challenging when links have to share radio re-sources in a distributed manner. Distributed power control [3, 4] isoften used (e.g. in cellular systems) to tackle this issue. How-ever, when links strongly interfere each other, power control isinefficient as the set of rates that can be simultaneously achievedon the competing links exhibits non-convexities. For such sce-narios, scheduling transmissions over time is much more efficient ∗ (cid:63) IIT Mumbai; † KTH, The Royal Institute of Technology.
Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00. and results in a much larger rate region. Most existing MAC algo-rithms for WLANs, Mesh, and AdHoc networks are scheduling al-gorithms: transmitters only decide when to be active, and when ac-tive, they use a single power level, often the maximum power level.In their seminal paper [1], Tassiulas and Ephremides proposed thequeue-length based
Maximum Weighted Scheduling (MWS) algo-rithm, and proved its throughput-optimality (meaning that it canstabilize the network whenever this is at all possible). Howeverthe MWS algorithm is centralized, and often requires to repeatedlysolve instances of NP-hard optimization problems.Over the last two decades, there have been important research ef-forts towards the design of low-complexity and distributed versionsof the MWS algorithm (refer to the related work section for refer-ences). Recently, in [2, 5–7], simple and throughput-optimal adap-tive versions of CSMA have been proposed. These algorithms en-joy the property of being fully distributed, in the sense that they donot require any kind of message passing among the various trans-mitters. However their analysis and performance guarantees rely onthe strong assumption that interference can be modelled as a sim-ple undirected graph (in the interference graph, vertices representlinks, and an edge between two links mean that these links cannotbe simultaneously activated). In particular, this simplistic interfer-ence model cannot account for the well-known hidden and exposedterminal problems, and more generally does not accurately capturethe very nature of interference. In this paper, we revisit the designof efficient and distributed MAC protocols under the more realis-tic SINR interference model. Specifically, we aim at answering thefollowing question:
Can we devise fully distributed and optimal scheduling algo-rithms for wireless networks under the SINR interference model? By fully distributed , we mean that transmitters are not allowed toexchange any signalling message, and the only feedback availableat a given transmitter is the level of interference measured at thecorresponding receiver (just as in classical distributed power con-trol mechanisms [3, 4]). Optimal may have several meanings. Todiscuss the different versions of optimality, let us first introduce thenotion of rate region defined as the set of rates that can be simul-taneously achieved on the various links using some (centralized)scheduling algorithms. (i) Rate-optimality: in this case, transmit-ters always have packets to send, i.e., they are fully backlogged. Analgorithm is rate-optimal, if it can achieve any rate vector within therate region. (ii) Throughput-optimality: in this case, each transmit-ter receives, in its (infinite) buffer, packets arriving according to astationary ergodic process with fixed average rate. An algorithm isthroughput-optimal if it stabilizes all buffers as long as the meanarrival rate vector belongs to the largest open set contained in the The assumptions made on the packet arrival processes and the no-tion of stability are described in Section 7. a r X i v : . [ c s . I T ] M a y ate region.In this paper, we show that surprisingly, the answer to the abovequestion is positive, and develop fully distributed and rate-optimalscheduling algorithms. We also demonstrate how these algorithmscan be used towards the design of throughput-optimal schedulingschemes. In the proposed framework, we first divide time intoframes consisting of a fixed number of slots. Each transmitter isthen allowed to adapt the power levels used in the various slots ofa frame to achieve the rate it is targeting. Our solution is basedon a simple power control mechanism, referred to as Power Pack-ing (PP). Under this mechanism, each transmitter aims at achiev-ing its target rate while minimizing the number of slots actuallyused, hence leaving as many radio resources as possible to the othertransmitters. PP algorithms are shown to be rate-optimal when twolinks compete for the use of resources. However, in more generalnetworks and in some rare scenarios, they may fail at achievingcertain rate vectors that could have been realized using centralizedscheduling. By just adding to the algorithms some level of random-ization in the power allocation, we overcome this issue and recoverrate-optimality. All the proposed algorithms are simple and do notrequire any message passing: each transmitter adapts its power lev-els in the various slots depending on the observed interference lev-els. To our knowledge, the proposed algorithms constitute the firstscheduling schemes that are fully distributed (no message passing)and optimal under the SINR interference model.The paper is organized as follows:(i) In Section 2, we present a brief overview of the existing liter-ature on distributed resource allocation algorithms in wireless net-works.(ii) In Sections 3 and 4, we present our generic framework, PowerPacking algorithms and explain their rationale.(iii) We establish the rate-optimality of Iterative Power Packing al-gorithms for 2-link networks in Section 5.(iv) For more general networks, we explain, in Section 6, why It-erative Power Packing algorithms may in some rare cases fail. Tosolve this issue, we introduce some Perturbed versions of IterativePP algorithms and show their rate-optimality.(v) In Section 7, we show how our rate-optimal algorithms can beadapted to achieve throughput-optimality.vi) Finally, in Section 8, we illustrate the efficiency of our algo-rithms using numerical experiments.
2. RELATED WORK
There have been, over the last two decades, a tremendous re-search effort towards the design of distributed resource sharing mech-anisms in wireless networks under various interference models (seee.g. surveys [8, 9]). For the simplistic interference graph model,researchers have developed scheduling algorithms that implementthe celebrated throughput-optimal MWS algorithm [1] in a dis-tributed manner. Some of these algorithms use message passing,see e.g. [10], some others do not require message passing, e.g. asthe adaptive versions of CSMA, see e.g. [2, 5–7].In this paper, we are interested in the more realistic SINR in-terference model. This model has also attracted a lot of attentionrecently, see e.g. [11–14]. For example in [12], the authors deriveutility-optimal power control schemes, but the achieved rate regionis restricted to that achieved by power control only. In [13, 14],the authors design schemes also enabling time sharing, and hencescheduling. These schemes implement the MWS algorithm, but re-quire message passing (basically, a transmitter need to know its im-pact on the throughputs on other links). In a series of papers [15–17], Bambos et al. design power control algorithms that ressem-ble Foschini-Miljanic algorithm [3, 4] in the sense that the power update at a transmitter only depends on the measured interferencelevel, and on some local queue size. These schemes are fully dis-tributed, and seem to realize time sharing when needed. However,their optimality has not been established, and there may be networkexamples where these schemes are not optimal.
3. MODELS AND PRELIMINARIES3.1 Network model
We consider a network consisting of N interfering links (transmitter-receiver pairs). We are primarily interested in the design of rate-optimal algorithms, and so each link i has a target rate requirement R ti (corresponding to the QoS requirements of the underlying ap-plication). To achieve this target rate, link- i transmitter may adaptits transmission power p i . The transmission power at any trans-mitter cannot exceed P max . Links interfere, and we assume herethat each receiver treats interference as noise. Let g ji denote thechannel gain from link- j transmitter to link- i receiver. Thermalnoise is Gaussian, with power N . Under these assumptions, themaximum rate that link i can achieve can be written as: r i ( p ) = f (cid:16) g ii p i N + (cid:80) j (cid:54) = i g ji p j (cid:17) , where f ( · ) is an increasing positive concavefunction, typically f ( x ) = W log(1 + x ) , and p = ( p , . . . , p N ) . Notation.
Let U be a subset of R N + . We denote by conv( U ) theconvex hull of U , and by ∂ U the Pareto-boundary of U : x ∈ ∂ U iff x ∈ U and ∀ y ∈ U , y ≥ x coordinate-wise implies that x = y .We further define ¯ U = { r ∈ R N + : ∃ R ∈ U , ∀ i, r i ≤ R i } as thesmallest coordinate-convex set containing U . = (1 , . . . , . We define R pc1 = { r ( p ) : ∀ i, p i ∈ [0 , P max ] } as the set ofvectors representing rates that can be achieved on the various linksusing power control. This set is known to be non-convex, and mayexhibit different types of shapes, depending on the values of gains ( g ij , i, j ) , i.e., on the network geometry. Let S = { r ( p ) : ∀ i, p i ∈{ , P max }} be the set of vectors representing link rates achievedusing binary power control, i.e., for any i , link- i transmitter eitherremains silent or transmits at maximum power P max . S is referredto as the set of schedules . The set R sched of link rates that can beachieved by switching schedules over time is the convex hull of S : R sched = conv( S ) . Now we may allow transmitters to use bothpower control and time sharing. In this case, the set of achievablerate vectors is R = conv( R pc1 ) . In general, both power controland time sharing are required, in the sense that we may have forthe same network: R pc1 (cid:40) R and R sched (cid:40) R . We illustrate theseobservations in Figure 1, where we depict the Pareto-boundaries ∂ R pc1 and ∂ R sched of the set R pc1 and R sched , respectively, fordifferent interference scenarios. When links strongly interfere eachother, time sharing (scheduling) is enough, whereas when interfer-ence becomes weaker, power control may be necessary. In thispaper, our goal is to design fully distributed algorithms enablingthe various links to reach their target rates R t = ( R t , . . . , R tN ) ,provided that R t ∈ R sched . To share radio resources among links, we divide time into frames.Each frames consists of a fixed number M of time slots of equal du-rations. If each transmitter is allowed to use different power levelson the various slots, the rates R ( p ) achieved on the various linkscan be written as: R i ( p ) = 1 M M (cid:88) m =1 f ( g ii p im N + (cid:80) j (cid:54) = i g ji p jm ) , ∀ i, R sched r r ∂ R pc1 Figure 1: Rate regions under power control and scheduling:strong (left) and weak (middle and right) interference cases. where p = ( p im , i = 1 , . . . , N, m = 1 , . . . , M ) and p im is thepower level used by link- i transmitter on the m -th slot in eachframe. The set of achievable rates using such multi-slot power con-trol is then: R pc M = { R ( p ) : ∀ i, ∀ m, p im ∈ [0 , P max ] } . R pc M canalso be expressed as combinations of rate vectors in R pc1 : R pc M = { r : ∃ s m ∈ R pc1 , m = 1 , . . . , M : r = M (cid:80) Mm =1 s m } . Observethat we do not impose any constraint on the total power used by atransmitter per frame.Now consider scenarios where transmitters are allowed, in a givenslot, either to use maximum power P max or to remain silent. Asearlier, we may define a set S M of schedules : S M = { R ( p ) : ∀ i, ∀ m, p i,m ∈ { , P max }} . Sharing time among the various schedules in S M increases the setof achievable rates, i.e., S ⊂ S M . Observe that the set of achiev-able rates on the various links using a single schedule in S M is ¯ S M is the smallest coordinate convex set containing S M . The variousnotions of rate regions and their Pareto-boundaries are illustratedin Figure 2.Note that designing distributed radio resource allocation schemesachieving any R t ∈ R sched is difficult for this requires to identifythe various proportions of time schedules in S M are used. Design-ing schemes achieving any R t ∈ ¯ S M may seem easier because inthis case we only need to identify a single schedule in S M satisfy-ing the rate requirements.As stated in the following lemma, when the number of slots perframe is large, we can achieve the largest rate region R by justimplementing power control per slot, and R sched by choosing afixed schedule from S M . All proofs are presented in appendix.L EMMA lim M →∞ R pc M = R , lim M →∞ ¯ S M = R sched . Here lim M →∞ A M = B means that for every point R ∈ B ,there exists a sequence of points ( X M , M ≥ such that X M ∈ A M for all M , and lim M →∞ X M = R .In practice, we observe that the introduction of frames, even ofsmall sizes, considerably increases the rate region: in other words,the sequence of sets ¯ S M , M = 1 , , ... rapidly approaches R sched .Based on this observation and on previous lemma, we use the fol-lowing strategy to design distributed resource allocation schemesapproximately achieving rates in R sched : (i) We select a framesize M so that ¯ S M provides a good approximation of R sched , e.g. M = 16 ; (ii) we devise distributed resource allocation schemesachieving any rate vector in ¯ S M .
4. POWER PACKING
Figure 2: Rate regions under power control (top-right, top-left)and scheduling (bottom-right, bottom-left) in multi-slot sys-tems. For scheduling, the black dots correspond to the Pareto-boundary of S . The network for figures on the left (resp. right)corresponds to gains g ij = 1 , ∀ i, j (resp. g = 2000 , g = 0 . , g = 0 . , g = 0 . ), N = 0 . , P max = 1 . In this section, we present power packing algorithms for themulti-slot systems introduced in the previous section. When exe-cuting such algorithm, a transmitter aims at minimizing the numberof slots actually used (a slot is used on a link, if the correspondingtransmitter selects a strictly positive power level in this slot) whileachieving the target rate. To run power packing algorithms, trans-mitters just need to measure the interference generated by othertransmitters in the slots composing a frame.
Let I im ( p ) denote the interference perceived at link- i receiverduring the m -th slot of the frame, given the power allocation p =( p jm ) j,m : I im ( p ) = N + (cid:80) j (cid:54) = i g ji p jm . We also introduce h i :[0 , P max ] M × R M + → R + that gives the rate on link i as a functionof link- i transmitter power levels, and perceived interference levelsin the various slots: h i ( p i , I i ) = M (cid:80) Mm =1 f (cid:16) p im g ii I im (cid:17) . Power packing algorithm is executed by a transmitter in responseto the observed interference levels in the various slots of a frame.The principle of power packing is to sequentially fill with powerslots in increasing order of perceived interference and until the tar-get rate is reached. If the latter cannot be reached, the transmitterjust remains silent in all slots. The algorithm, whose pseudo-codeis presented below, is illustrated in Figure 3.
PP algorithm. (Executed at link- i transmitter)Input: target rate R ti , interference levels I i = ( I im ) m .1. Compute the rate ¯ R i = h i ( P max , I i ) achieved using maxi-mum power in each slot,2. If ¯ R i < R ti : select power allocation p i = (0 , . . . , ; igure 3: Example of power allocation obtained after powerpacking algorithm - frame size M = 4 .
3. If ¯ R i ≥ R ti : order slots in increasing interference levels:let σ be a permutation of { , . . . , M } such that I iσ (1) ≤ . . . , I iσ ( M ) . Define ˜ m = min { m : 1 M m (cid:88) k =1 f ( P max g ii I iσ ( k ) ) ≥ R ti } . Select the unique power allocation p i such that: ∀ m < ˜ m , p iσ ( m ) = P max , ∀ m > ˜ m, p iσ ( m ) = 0 , and R ti = h i ( p i , I i ) . The PP algorithm has a binary version, where the transmitter isallowed in a given slot to either use full power P max or remainsilent. BPP algorithm is identical to PP algorithm, except for step3 where the power allocation differs: If ¯ R i ≥ R ti , the transmitteruses the power allocation p iσ ( m ) = P max { m ≤ ˜ m } . We now provide a game theoretical perspective on PP and BPPalgorithms. Consider a noncooperative game played by the N trans-mitters. Each transmitter competes rationally against the others byselecting a power allocation across the M available slots. The setof strategies available to any transmitter consists of all possiblepower allocation across slots. In the case where transmitters canuse any power level between 0 and P max , the set of strategies is P = { p i : ∀ m, p im ∈ [0 , P max ] } , whereas in case of binary powercontrol, this set reduces to P B = { p i : ∀ m, p im ∈ { , P max }} .The utility function U i ( p ) of transmitter i is defined as follows: U i ( p ) = C × { R i ( p ) ≥ R ti } − (cid:80) Mm =1 p im g ii I im ( p ) , where C is a positiveconstant such that C > (cid:80) Mm =1 P max g ii /N , for any link i . Wedenote by G ( R t ) (resp. G B ( R t ) ) the game defined above when theset of strategies is P (resp. P B ). It can be easily shown that withour choice of utility functions, the PP and the BPP algorithms exe-cuted by link- i transmitter can be interpreted as the best response tothe power allocations p − i = ( p jm ) j (cid:54) = i,m used by the other trans-mitters. In other words, assume that the power allocation p j usedby link- j transmitter is fixed for all j (cid:54) = i . These allocations re-sult in interference levels ( I im ) m at link- i receiver. For example,the power allocation obtained when link- i transmitter executes PPalgorithm under these conditions solves the following optimizationproblem: maximize U i ( q i , p − i ) , over q i ∈ P .
5. TWO LINK CASE: ITERATIVE POWERPACKING
In this section, we restrict our attention to two-link networks. Wepropose and analyze the convergence of Iterative Power Packing(IPP) algorithms. The latter consist in letting transmitters sequen-tially update their power allocation using PP or BPP algorithms.
To define IPP and IBPP algorithms, we first introduce a sequence s = ( s [ t ]) t ≥ , s [ t ] ∈ { , } , defining the order in which transmit-ters update their power allocation. We assume that the sequencesatisfies the following property, stating that each transmitter gets toupdate its power allocation an infinite number of times:(P1) ∀ t ≥ , ∃ t , t ≥ t : s [ t ] = 1 and s [ t ] = 2 .This property is referred to as liveness property in game theory. Asequence of updates satisfying this property is in principle easy togenerate in a distributed manner, for example using independentPoisson clocks with identical rate at the various transmitters. Referto §6.1 for more details. We are now ready to define IPP algorithm: IPP algorithm.
Input: target rate vector R t , update sequence s , initial power allo-cation p [0] .For each step t ≥ : Let i = s [ t ] .1. Link- i transmitter measures interference levels I i ( p [ t − I im ( p [ t − m in the different slots;2. Link- i transmitter runs PP algorithms with inputs R ti and I i ( p [ t − .IPP algorithm has a binary version, IBPP algorithm, obtainedby just replacing PP algorithm by the BPP algorithm in the abovepseudo-code. IPP and IBPP algorithms correspond to the best re-sponse dynamics or Nash dynamics of the games G ( R t ) and G B ( R t ) ,respectively. They can easily be implemented in a fully distributedmanner: when a transmitter updates its power allocation, it onlyneeds to measure interference levels on the various slots and toknow its own target rate. To study the convergence of IPP and IBPP algorithms, we in-troduce the notion of repulsive power allocation. We say that p =( p , p ) ∈ [0 , P max ] M is repulsive if and only if there exist a per-mutation σ of { , . . . , M } and two integers m , m ∈ { , , . . . , M +1 } such that for all m ∈ { , . . . , M } (i) m ≤ m implies p σ ( m ) = P max , and m > m + 1 implies p σ ( m ) = 0 , (ii) m ≥ m implies p σ ( m ) = P max and m < m − implies p σ ( m ) = 0 .The set of rate vectors that can be achieved using repulsive powerallocation is then defined as: R IPP M = { R ∈ R : ∃ p repulsive: R = R ( p ) } In the case the power allocation is binary, we similarly define: R IBPP M = { R ∈ R : ∃ p ∈ { , P max } M repulsive: R ≤ R ( p ) } In what follows, we show that R IPP M (resp. R IBPP M ) is the rateregion achieved under IPP (resp. IBPP) algorithm.T HEOREM
Let R t ∈ R IPP M (resp. ∈ R IBPP M ). Then fromany initial power allocation, IPP (resp. IBPP) algorithm con-verges to a repulsive power allocation p ∈ [0 , P max ] M (resp. p ∈ { , P max } M ) such that R t = R ( p ) (resp. R t ≤ R ( p ) ). From a game theoretical perspective, this result states that if R t ∈ R IPP M , then the Nash dynamics converge to a pure Nash Equi-librium corresponding to a repulsive power allocation and achiev-ing the target rates R t . Figure 4 illustrates the rate regions R IPP M . igure 4: Rate regions obtained through IPP algorithm. We do not represent R IBPP M , because in fact, R IBPP M = ¯ S M .This can be shown by applying the following argument: Let p bea binary power allocation; modify this allocation such that (i) thenumber of slots used by each transmitter is not changed, and (ii)the new allocation is repulsive. It is easy to see that the new al-location provides greater rates to all transmitters. notice howeverthat R IPP M is smaller than R pc M : this is true when the initial rateregion R pc1 has concave parts – see Fig. 4 (right). In this case,some points of the Pareto-boundary of R pc M can only be achievedby non-repulsive power allocations of the type p = ( x , P max ) ,where x ∈ [0 , P max ] .To conclude this section, one can show (as in Lemma 1) that R IPP M and R IBPP M approximate R sched (when M is large).L EMMA lim M →∞ R IPP M = R sched , lim M →∞ R IBPP M =¯ R sched . As a consequence, any target rate vector R t inside R sched canbe achieved either using IPP or IBPP algorithm, provided that theframe size is large enough. In other words, IPP or IBPP algo-rithms are approximately rate-optimal in 2-link networks (we willgive a more precise definition of what we mean by "approximatelyrate=optimal" in the next section).
6. MULTIPLE LINK CASE: ITERATIVE PER-TURBED POWER PACKING
In this section, we consider general networks with more than twolinks. We first explain why IPP or Binary-IPP may fail at converg-ing for some specific target rates in R sched . We then present twobinary power control algorithms to overcome this issue.An example of networks and target rates where IPP does notwork is as follows. Consider a network consisting of 3 links shar-ing the same receiver (Access Point scenario), and let M = 3 slots. The two first transmitters are close to the receiver, whereasthe third one is further away. Assume that the target rates can beachieved by the unique following power allocation: p = p =( P max , P max , and p = (0 , , P max ) . This happens for exam-ple if R t = f ( P max g N ) , and R t > f ( P max g N + P max g i ) , i = 1 , (in words, the third link cannot accomodate any kind of interfer-ence). Now the problem stems from the fact that if transmitters 1and 2 select their allocation using PP first, then they would pick p = ( P max , P max , , p = (0 , , P max ) . The third transmitteron the other hand needs to be alone in a slot to be satisfied (i.e., toachieve its target rate), but it cannot, and hence remains silent. Theissue with IPP is actually common to all distributed power controlprotocols. A transmitter that causes low interference to others, butthat is strongly interfered by others, has difficulties indicating itsstate to others through power control. Similarly, a link that cannot suffer much from interference has difficulties in gauging the impactof its power allocation on other links.The proposed solution to this problem marries Power Packingprinciple and randomization. PP is used to (quickly) reach a fea-sible power allocation, when PP can indeed go there. We believethat in most cases PP actually finds a feasible allocation. Random-ization only helps IPP algorithm when the latter cannot converge tothe desired allocation. Thus, the proposed schemes can be thoughtof as the perturbed version of IPP algorithm. The key idea of the algorithm is to force transmitters that aresatisfied but whose power allocation is not compatible with anyglobally feasible allocation to explore other power allocations. Thisexploration is here triggered when unsatisfied transmitters createenough interference so that the target rate of satisfied links cannotbe achieved anymore. The algorithm works as follows. First wegenerate a sequence of transmitters selected to update their powerallocation over the frame. Then the power updates satisfy rules thatwe describe below.
Update sequences.
As for IPP and IBPP algorithms, transmittersupdate their power allocation sequentially. The sequence of updatesis driven by s = ( s [ t ]) t ≥ , assumed here to satisfy the followingproperty:(P2) ( s [ t ]) t ≥ is a stationary ergodic Markov chain with state space { , . . . , N } , such that P [ s [ t ] = i ] > for all transmitter i .A sequence satisfying (P2) may be generated when updates are trig-gered by independent Poisson clocks of identical rates, say ν , at thevarious transmitters. To be more specific, when the clock of a trans-mitter ticks, the latter starts a power update at the next frame. Whenthe common clock rate is relatively low (compared to the inverse ofthe frame duration), it is very unlikely that updates at two trans-mitters overlap (the convergence of our algorithms holds even incase of unfrequent update overlapping – in fact convergence takesa finite number of updates, and so we just need that such sequenceof updates occurs with positive probability). Under the above sce-nario, observe that for each new update, the selected transmitteris selected uniformly at random, so that (P2) is satisfied. Note alsothat the time between updates occur at instants of a Poisson processof mean rate Nν . Updating rules.
When an unsatisfied transmitter is picked for apossible update, it picks a power allocation as per the Binary-PPalgorithm with probability (w.p.) (1 − α ) , and picks a randomallocation w.p. α . A random power allocation can be obtained byusing power P max on each slot w.p. 1/2 independently of the powerlevels used in other slots. When a satisfied transmitter i is selected,it checks whether its target rate has been achieved because of itsown power allocation decision in the past ( β i = 1 ) or because ofchanges in the power allocation by other transmitters ( β i = 0 ). Itcan for example happen that an other transmitter decided to remainsilent (applying Binary-PP algorithm), which made i satisfied. Inthis case, the power allocation used by i might not be compatiblewith any globally feasible allocation, and transmitter i should ex-plore other allocations. Thus in the algorithm, when β i = 0 , i doesnot update its power allocation w.p. ( − α ) , and chooses a randompower allocation w.p. α . Parameters α and α characterize thelevel of randomization in the algorithm. In what follows, we alwaysassume that < α , α < . When they are small, the algorithm isclose to the initial IBPP algorithm, and converges very fast to a fea-sible allocation if IBPP can find one, but the algorithm would thentake more time to identify a feasible allocation that IBPP cannotreach. The pseudo-code of the algorithm is presented below. terative Perturbed Binary-PP (IPB-PP) algorithm. Input: target rate vector R t , update sequence s , power allocation p [0] , β [0] ∈ { , } K .For each step t ≥ : Let i = s [ t ] .1. Tx i measures interference levels I i ( p [ t − in the differentslots.2. Tx i updates its power allocation to p i [ t ] :(i) If R i ( p [ t − < R ti ) , then p i [ t ] is obtained as per BPPalgorithm with inputs R ti and I i ( p [ t − w.p. − α and is a random power allocation w.p. α ;(ii) Else If ( β i [ t −
1] = 0 ), p i [ t ] = p i [ t − w.p. (1 − α ) and p i [ t ] is random w.p. α ;Else p i [ t ] = p i [ t − .4. Tx i sets β i [ t ] = 1 if R i ( p i [ t ] , p − i [ t − ≥ R ti , and β i [ t ] =0 otherwise.We prove the convergence of IPB-PP under the following as-sumption. Let U ( p ) denote the set of unsatisfied links under binarypower allocation p .(A1) For any power allocation p such that U ( p ) (cid:54) = ∅ and U ( p ) (cid:54) = { , . . . , N } , either there exists i ∈ U ( p ) such that for R ( P max , p − i ) ≥ R ti , or for p (cid:48) such that p (cid:48) i = P max for all i ∈ U ( p ) and p (cid:48) i = p i for i / ∈ U ( p ) , U ( p ) (cid:40) U ( p (cid:48) ) .The assumption states that for any given power allocation p , ei-ther there exists a unilateral change in the power allocation of anunsatisfied transmitter that makes it satisfied, or when unsatisfiedtransmitters all select P max , at least one other link becomes un-satisfied.T HEOREM
If there exists a binary power allocation p (cid:63) suchthat R ( p (cid:63) ) ≥ R t , if (A1) holds, then from any initial condition,IPB-PP algorithm converges almost surely to a power allocation p such that R ( p ) ≥ R t . The previous theorem does not lead to the rate-optimality of IPB-PP algorithm. Although the algorithm performs well in practice,there are still some target rate vectors that it cannot reach. Thisis typically the case where one link has very low target rate, inwhich case, assumption (A1) may not be satisfied (the correspond-ing transmitter can be hardly affected by interference). Next wepropose a rate-optimal algorithm whose principles are similar tothose of IPB-PP algorithm.
The next algorithm follows the same design principles as IPB-PP algorithm. However, the way satisfied transmitters are forced toexplore other power allocations is different: they explore new allo-cations if they perceive significant changes in interference. Moreprecisely, exploration is triggered when the change in the sum ofthe interference measured in the various slots exceeds a threshold δ . The pseudo-code of this new algorithm is presented below. Interference-Triggered IPB-PP (IT-IPB-PP) algorithm.
Input: target rate vector R t , update sequence s , power allocation p [0] , previous interference levels I last [0] ∈ R K + .For each step t ≥ : Let i = s [ t ] . 1. Tx i measures interference levels I i ( p [ t − in the differentslots.2. Tx i updates its power allocation to p i [ t ] :(i) If R i ( p [ t − < R ti ) , then p i [ t ] is obtained as per BPPalgorithm with inputs R ti and I i ( p [ t − w.p. − α and is a random power allocation w.p. α ;(ii) Else If | (cid:80) Mm =1 I im ( p [ t − − I last i [ t − | > δ , p i [ t ] = p i [ t − w.p. (1 − α ) and p i [ t ] is random w.p. α ;Else p i [ t ] = p i [ t − .4. Tx i sets I last i [ t ] = (cid:80) Mm =1 I im ( p [ t − .We prove the convergence of the algorithm under the followingassumption on δ .(A2) For every set U (cid:54) = ∅ or { , . . . , N } , there exists a set U (cid:48) (cid:54) = ∅ satisfying (1) U ∩ U (cid:48) = ∅ and (2) for every j ∈ U (cid:48) , MP max (cid:80) i ∈ U g ij > δ .Assumption (A2) states that any set U of transmitters can be"heard" by at least one link in U c = { , . . . , N } \ U . Note that aslong as g ij > for every i, j , for any δ > , one can find a framesize M (large enough) such that (A2) is satisfied. In this sense, theassumption is not restrictive: one may choose δ depending on thesensitivity of receivers, and then tune M so that (A2) holds.T HEOREM
If there exists a binary power allocation p (cid:63) suchthat R ( p (cid:63) ) ≥ R t , and if with our choices of δ and M , (A2) holds,then from any initial condition, IT-PIB-PP algorithm converges al-most surely to a power allocation p such that R ( p ) ≥ R t . The above theorem states that if R t ∈ ¯ S M and if (A2) holds,then IT-IPB-PP algorithm converges to a feasible power allocation.Now combining, this result with that of Lemma 3.1, we deduce thatIT-PIB-PP is approximately rate-optimal. To be more precise, for (cid:15) ∈ (0 , , we say here that an algorithm is (cid:15) -rate-optimal if it canachieve any rate vector R t such that R t + (cid:15) ∈ R sched .C OROLLARY
For any (cid:15) > , and any threshold δ > ,there exsists a frame size M ( (cid:15), δ ) such that if M ≥ M ( (cid:15), δ ) , IT-IPB-PP algorithm is (cid:15) -rate-optimal.
7. THROUGHPUT-OPTIMALITY
In the previous section, we developed an approximately rate-optimal and fully distributed scheduling scheme. We now turn ourattention to scenarios where each transmitter is equipped with aninfinite buffer where it stores packets before sending them, and weaddress the design of throughput-optimal and distributed schedul-ing algorithms. We first describe our assumptions on the arrivalprocesses, and on the notion of system stability.
Arrival processes.
We assume that packets arrive in transmitter- i ’s buffer according to an i.i.d. process. Let A i [ t ] denote thenumber of bits arriving in transmitter- i ’s buffer during frame t . ( A i [ t ]) t ≥ forms an sequence of i.i.d. random variables such that A i [ t ] ≤ A < ∞ for all i and t . The mean arrival rate (per frame) attransmitter i is denoted by λ i = E [ A i [ t ]] . Let λ = ( λ , . . . , λ N ) .Finally, we assume that arrival processes are independent acrosstransmitters. Stability.
Let Q i [ t ] denote the number of bits in transmitter- i ’sbuffer at the beginning of frame t . It evolves as: Q i [ t + 1] =max(0 , Q i [ t ] + A i [ t ] − S i [ t ]) , where S i [ t ] is the number of bitssent during frame t . Let B denote the time required to empty allueues, i.e., B = inf { u : u ≥ , Q i [ u ] = 0 , ∀ i } . We say thatthe system is stable if E [ B | Q [0]] < ∞ for all initial queue vector Q [0] = ( Q [0] , . . . , Q N [0]) such that Q i [0] < ∞ , for all i . We saythat an algorithm is (cid:15) -throughput optimal if it stabilizes the systemwhenever λ + (cid:15) ∈ R sched .We use IT-IPB-PP algorithm to design approximately throughput-optimal and fully distributed scheduling schemes. If each transmitter i is aware of its arrival rate λ i , this design isstraightforward: each transmitter i selects a target rate R ti slightlybigger than λ i , and we then run the IT-IPB-PP algorithm with thesetarget rates, even when its queue is empty (using dummy packets).Under this strategy, after convergence of the IT-IPB-PP algorithm(which occurs after a finite time with finite mean), queues behavesindependently and each of them has an arrival rates strictly less thanits fixed service rate, which ensures stability. Next we make thesestatements precise.L EMMA
Let M be a frame size such that IT-IPB-PP is ( (cid:15)/ -rate-optimal. Assume that λ − (cid:15) ∈ R sched . Then underIT-IPB-PP algorithm with target rate vector R t = λ + ( (cid:15)/ , thesystem is stable. The above lemma simply states that IT-IPB-PP algorithm pro-vides an (cid:15) -throughput-optimal algorithm, if each transmitter knowsits arrival rate. (cid:15) can be made as small as desired by increasing theframe size M . When the arrival rate λ i is not known, transmitter i estimates it.When its estimate is precise enough, it selects a target rate appropri-ately (again slightly bigger than its estimated arrival rate) and thenruns the IT-IPB-PP algorithm with this target rate. More precisely,for any i , let λ i [ t ] = t (cid:80) ts =1 A i [ s ] and let µ = (cid:15)/ . Further de-fine the interval e k = [2( k − µ, kµ ) . The target rate vector iscontinuously updated as follows: for any i ,if λ i [ t ] ∈ e k , then R ti [ t ] = (4 k + 1) µ/ . When λ i lies in the interior for some e k , since λ i [ u ] → λ i a.s.as u → ∞ , after a finite time T i , R ti [ t ] does not change any-more. In appendix we briefly explain how the case λ i = 2 kµ can be handled. The following lemma then relies on the facts that R ti [ T i ] ∈ ( λ i , λ i + (cid:15)/ and E [ T i ] < ∞ (proved in appendix):L EMMA
Let M be a frame size such that IT-IPB-PP is ( (cid:15)/ -rate-optimal. Assume that λ − (cid:15) ∈ R sched , and that IT-IPB-PP algorithm is executed jointly with the above target rate updatealgorithm. Then the system is stable. According to the above lemma, the proposed joint target rate up-date and scheduling algorithm is (cid:15) -throughput-optimal. It is worthremarking that this algorithm proceeds in three phases: in the firstphase, each transmitter aims at identifying a target rate that is juststrictly greater than the arrival rate of bits in its buffer; in the sec-ond phase, IT-IPB-PP algorithm finds a power allocation compat-ible with the target rate vector; and finally, transmitters apply thispower allocation, and queues empty. Also note that our algorithmis not designed so as to adapt to changing traffic conditions (i.e.,changes in the arrival rates). A way to devise adaptive algorithmswould be to let each transmitter continuousloy updates its targetrate, depending on its observed queue length. To study such queuelength based algorithm, one would need to understand the interac-tion between dynamics of the queues and of our IT-IPB-PP algo-rithm, which would require a significantly more involved analysis.
8. NUMERICAL EXPERIMENTS
In this section, we present simulation results to illustrate therate-optimality of IPB-PP and IT-IPB-PP algorithms. For all ex-periments, the sensitivity parameter δ in IT-IPB-PP algorithm isfixed. We first experiment with a 3-link network. The networkgeometry is such that transmitters 1 and 2 strongly interfere link3, whereas transmitter 3 does not produce much interference, i.e., g = g = 60 , g = g < , and the other gains are equal to 1.The target rate vector is chosen so that it cannot be reached by sim-ple iterative Power Packing. It corresponds to a power allocationclose to p = p = ( P max , P max , , p = (0 , , P max ) . Fig. 5shows the convergence time (in number of updates) of IPB-PP andIT-IPB-PP algorithms as a function of the exploration rate α (wechoose α = α ). The convergence time is averaged over 10,000simulations starting from random power allocations. The conver-gence time rapidly grows either when the exploration rate is closeto 0, or when it becomes too large. In the former, the algorithmsbehave like Binary-IPP, and cannot find a feasible allocation. Inthe latter, the algorithms get closer to a random search algorithm,and the convergence time explodes. Hence, in IPB-PP and IT-IPB-PP algorithms, it is clear that both Power Packing and randomiza-tion components are crucial: PP accelerates the convergence andrandomization helps where PP fails at identifying a feasible allo-cation. It is worth noting that when the target rate vector can beachieved through simple Power Packing (without randomization),the convergence of the algorithm is very fast. C on v e r gen c e t i m e Figure 5: Convergence time vs Exploration rate α – N = 3 = M Next we consider randomly generated networks by placing 10links on a 2D square (gains are computed using a path loss expo-nent equal to 3). For each generated network topology, we furthergenerate target rate vectors in S M . For each vector we analyzethe convergence time if the latter remains less than updates.We use two metrics for comparison: (1) the average (over topolo-gies and rate vectors) number of updates required for convergence,given that it remains less than , and (2) the proportion of ratevectors for which the algorithm does not converge in less up-dates.We first investigate the performance of our algorithms when theframe size M varies. Here we fix α = α = 0 . . Figure 6(a)shows that as M increases, the proportion of rate vectors not reachedby the algorithms rapidly decreases. For IT-IPB-PP, all vectors areachieved when M = 16 , illustrating the rate-optimality of the al-gorithm. Note that this is not the case of IPB-PP, as we predicted.Figure 6(b) shows how the convergence time varies with M . IT-IPB-PP seems to conevrge faster, and for both algorithms the con-vergence speed is increased when M grows large.We now challenge our algorithms, and evaluate their performancewhen the frame size is not sufficient to guarantee the rate-optimalityof IT-IPB-PP: we fix M = 8 , and vary the exploration rate α . In N on - c onv e r g e d fr ac ti on Number of Slots(M)IPB-PPIT-IPB-PP (a) Prop. of non-achieved ratevectors vs. frame size U pd a t e s f o r C onv s e r g e n ce Number of Slots(M)IPB-PPIT-IPB-PP (b) Avg. number of updates forconvergence vs. frame size
Figure 6: Performance of IPB-PP and IT-IPB-PP - α = 0 . U pd a t e s f o r C onv e r g e n ce Exploration rate (alpha1)IPB-PPIT-IPB-PP (a) Avg. number of updates forconvergence vs. exploration rate N on - c onv e r g e d fr ac ti on Exploration rate (alpha1)IPB-PPIT-IPB-PP (b) Prop. of non-achieved ratevectors vs. exploration rate
Figure 7: Performance of IPB-PP and IT-IPB-PP - M = 8 Figure 7(a), we observe that in this case, the convergence time in-creases when α decreases, which again ilustrates the importanceof the Power Packing component in the algorithms. In Figure 7(b),the proportion of rate vectors not reached by the algorithms within updates seem to increase as α increases, which indicates thenegative effect of an aggressive random exploration.
9. CONCLUSION
This paper presents the first distributed scheduling algorithmsthat are optimal under the realistic SINR interference model, anddo not require message passing among transmitters. The fact thatalgorithms combining such properties exist in surprising. Our solu-tion is based on combining a simple power allocation strategy, andrandomization techniques. Without randomization, the power allo-cation scheme could not, alone, achieve all parts of the throughputregion (although numerical experiments show that it reaches a vastmajority of it), and hence randomization is needed. We actuallybelieve that randomization is always needed, i.e., no deterministicresource allocation scheme can be optimal. It would be interestingto formally establish this result. We are also interested in studyingthe convergence time of our iterative power allocation scheme, andits impact on actual queueing delays.
10. REFERENCES [1] L. Tassiulas and A. Ephremides, “Stability properties ofconstrained queueing systems and scheduling policies formaximum throughput in multihop radio networks,”
IEEETrans. on Automatic Control , vol. 37, no. 12, pp. 1936–1948, dec 1992.[2] L. Jiang and J. Walrand, “A distributed csma algorithm forthroughput and utility maximization in wireless networks,”in
Proc. of 46th Allerton Conference on Communication,Control, and Computing , 2008.[3] J. Zander, “Distributed power control in cellular radiosystems,”
IEEE Trans. on Vehicular Technology , vol. 12,no. 3, August 1992. [4] G. Foschini and Z. Miljanic, “A simple distributedautonomous power control algorithm and its convergence,”
IEEE Trans. on Vehicular Technology , vol. 42, no. 4, pp. 641–646, nov 1993.[5] L. Jiang, D. Shah, J. Shin, and J. Walrand, “Distributedrandom access algorithm: scheduling and congestioncontrol,”
IEEE Trans. on Information Theory , vol. 56, no. 12,pp. 6182–6207, Dec. 2010.[6] L. Jiang and J. Walrand, “A distributed CSMA algorithm forthroughput and utility maximization in wireless networks,”
IEEE/ACM Trans. on Networking , vol. 18, no. 3, pp.960–972, 2010.[7] J. Ni, B. Tan, and R. Srikant, “Q-CSMA: Queue-lengthbased CSMA/CA algorithms for achieving maximumthroughput and low delay in wireless networks,” in
Proc.IEEE Infocom , 2010.[8] L. Georgiadis, M. Neely, and L. Tassiulas, “Resourceallocation and cross-layer control in wireless networks,”
Foundations and Trends in Networking , vol. 1, no. 1, pp.1–144, 2006.[9] L. Jiang and J. Walrand, “Scheduling and congestion controlfor wireless and processing networks,”
Synthesis Lectures onCommunication Networks, Morgan-Claypool Publishers ,2010.[10] E. Modiano, D. Shah, and G. Zussman, “Maximizingthroughput in wireless networks via gossiping,” in
Proc. ofACM Sigmetrics , 2006, pp. 27–38.[11] M. Neely, E. Modiano, and C. Rohrs, “Dynamic powerallocation and routing for time varying wireless networks,”in
Proc. of IEEE Infocom , vol. 1, 2003, pp. 745 – 755 vol.1.[12] J. Papandriopoulos, S. Dey, and J. Evans, “Optimal anddistributed protocols for cross-layer design of physical andtransport layers in manets,”
IEEE/ACM Trans. onNetworking , vol. 16, no. 6, pp. 1392–1405, Dec. 2008.[13] Y. Xi and E. M. Yeh, “Throughput optimal distributed powercontrol of stochastic wireless networks,”
IEEE/ACM Trans.on Networking , vol. 18, no. 4, pp. 1054–1066, Aug. 2010.[14] H.-W. Lee, E. Modiano, and L. B. Le, “Distributedthroughput maximization in wireless networks via randompower allocation,”
IEEE Trans. on Mobile Computing ,vol. 11, no. 4, pp. 577 –590, april 2012.[15] N. Bambos and S. Kandukuri, “Power controlled multipleaccess (pcma) in wireless communication networks,” in
Proc. of IEEE Infocom , vol. 2, 2000, pp. 386 –395 vol.2.[16] S. Kandukuri and N. Bambos, “Multimodal dynamicmultiple access (mdma) in wireless packet networks,” in
Proc. of IEEE Infocom , vol. 1, 2001, pp. 199 –208 vol.1.[17] D. Vengerov, N. Bambos, and H. Berenji, “A fuzzyreinforcement learning approach to power control in wirelesstransmitters,”
IEEE Trans. on Systems, Man, andCybernetics , vol. 35, no. 4, pp. 768 –778, aug. 2005.
APPENDIXProof of Lemma 3.1
We prove lim M →∞ R pc M = R . lim M →∞ ¯ S M = R sched can beproved analogously. Let R ∈ R . Since R = conv ( R pc1 ) ⊂ R N ,by Caratheodory’s theorem, there exist a finite set { r , . . . , r N +1 } of ( N +1) points in R pc1 and positive real number λ , . . . , λ N +1 such that: R = (cid:80) N +1 j =1 λ j r j , and (cid:80) N +1 j =1 λ j = 1 .or any j = 1 , . . . , N + 1 , let p j ∈ [0 , P max ] N denote thevector representing power levels used by the various transmittersto achieve rate vector r j . Now for M ≥ , we propose the fol-lowing power allocation across the M slots of a frame: for all j = 1 , . . . , N + 1 , power levels p j are used for (cid:98) Mλ j (cid:99) slots(where (cid:98) x (cid:99) is the largest integer smaller than or equal to x ). Thepower allocation is arbitrary for the remaining slots. Using thispower allocation, the achieved rate vector U M satisfies the follow-ing component-wise inequality: U M ≥ (cid:80) N +1 j =1 (cid:98) Mλ j (cid:99) M × r j .Note that lim M →∞ (cid:98) Mλ j (cid:99) M = λ j for all j , and thus lim M →∞ U M ≥ R . (cid:50) Proof of Theorem 5.1
We prove the convergence of IPP algorithm (the proof of the con-vergence of Binary-IPP is similar and easier). Let s be a sequenceof updates satisfying property (P1). Without loss of generality, wecan assume that s [2 t ] = 1 and s [2 t + 1] = 2 , for all t ≥ (i.e.,transmitters alternatively apply PP algorithm). We denote by p [ t ] the power allocation after the t -th update, and abuse the notationby writing R [ t ] = R ( p [ t ]) . It can be readily seen that (i) afterboth transmitters update once, i.e., for t ≥ , the power alloca-tion p [ t ] is σ -repulsive (repulsive under the slot permutation σ ), fora fixed given permutation σ of slots; (ii) R [2 t ] ∈ { , R t } and R [2 t − ∈ { , R t } for any t ≥ . Observation (i) can beeasily proved by induction on t . Given the permutation σ , we in-troduce the following notation: let p , p (cid:48) ∈ [0 , P max ] M , we write p ≤ σ p (cid:48) if for all m , p σ ( m ) ≤ p (cid:48) σ ( m ) .Now let R t ∈ R IPP M . Let p (cid:63) = ( p (cid:63) , p (cid:63) ) be a σ -repulsive powerallocation such that R ( p (cid:63) ) = R t . We establish the convergence ofIPP algorithm to R t by investigating various possible initial condi-tions. Case 1: At time t , p [2 t ] = 0 . This means that after an update,link-1 transmitter actually chooses to remain silent. Without loss ofgenerality, we assume that t = 0 . We show by induction property ( N t ) , stating that the sequence of power allocations is monotoni-cally increasing, and that the target rates are alternatively achievedon links 1 and 2:Property ( N t ) : p [2 t − ≤ σ p [2 t ] ≤ σ p (cid:63) , R [2 t ] = R t , p [2 t − ≤ σ p [2 t − ≤ σ p (cid:63) , R [2 t −
1] = R t (with theconvention that p [ −
1] = 0 ).Let us prove ( N ) . We have p [0] = 0 . Then at time 1, link-2transmitter applies PP algorithm, and selects allocation p [1] suchthat R [1] = R t . Observe that link 2 has no interference, so that p [1] ≤ σ p (cid:63) . Then link-1 transmitter updates its power allocation.Since p [1] ≤ σ p (cid:63) , it can choose an allocation p [2] such that R [2] = R t and p [1] ≤ σ p (cid:63) . Thus ( N ) holds. Now assume that ( N t − ) holds, and let us prove ( N t ) . At time t , link-1 transmitterupdates its power, and since p [2 t − ≤ σ p (cid:63) , it can choose anallocation p [2 t ] such that R [2 t ] = R t and p [2 t ] ≤ σ p (cid:63) . Thesame argument applied for link-2 transmitter allows to finish theproof of ( N t ) .Now p [ t ] is monotonically increasing (w.r.t. ≤ σ ), and henceit converges. Remark that because of monotonicity, after a finitenumber of updates, the number of slots used by transmitter 1 or2 is fixed. Hence after these numbers are fixed, the transmittersjust update power on a single slot (always the same). The updatescorrespond to the synchornous version of Foschini-Miljanic algo-rithm, and hence converge to a feasible solution. In other words, if lim t →∞ p [ t ] = p (cid:48) , then R ( p (cid:48) ) = R t .The case where at time t + 1 , p [2 t + 1] = 0 , is similar to Case1. Case 2: At time t + 1 , R [2 t + 1] = R t and R [2 t + 1] ≤ R t . Without loss of generality, asume that t = 0 . Link-1 transmitterupdates its power at time 2. There are two cases:(i) R ( P max , p [1]) < R t , in which case, p [2] = 0 , and we re-turn to Case 1;(ii) R ( P max , p [1]) ≥ R t , in which case, p [2] is such that R [2] = R t and p [2] ≥ σ p [0] (because R [1] < R [2] ). Nowwe have R [2] ≤ R t because interference increased for link 2.We can show using induction arguments just as those used in Case1 that the power allocation is monotonically increasing until onetransmitter saturates and becomes silent. In the latter case, we areback to Case 1. If transmitters never reset their power, we haveconvergence towards the target rates (using the same argument asin Case 1).The case where at time t , R [2 t ] = R t and R [2 t ] ≤ R t , issimilar to Case 1. Case 3: At time t + 1 , R [2 t + 1] = R t and R [2 t + 1] > R t . In this case again, we can show convergence using monotonicty ar-guments exactly as in previous cases. Note that in this case, powerallocations are monotonically decreasing. The case where at time t , R [2 t ] = R t and R [2 t ] > R t is of course similar. (cid:50) Proof of Theorem 6.1
Since the update sequence satisfies (P2), x [ t ] = ( s [ t ] , p [ t ] , β [ t ]) t ≥ is an homogenous Markov chain withfinite state space. Observe that a set { ( s, p, ) , s ∈ { , . . . , N }} constitutes a communication class of this Markov chain if R ( p ) = R t (in such states, all links are satisfied, and do not update their al-locations anymore). To prove the theorem, we just need to showthat from any initial state, at least one of these communicationclasses are accessible, i.e., we construct a finite sequence of statetransitions occuring with positive probability and leading to one ofthe aforementioned communication classes. To construct such apath, we use the fact that from any state, all transmitters are pickedfor possible a power update with positive probability. We also usethe fact that if tx i is chosen for an update, and either its target rateis not satisfied or its β i is equal to 0, then tx i can pick any powerallocation with positive probability.W.l.o.g. we may assume that U ( p [0]) (cid:54) = ∅ and that there is notx i ∈ U ( p [0]) that can update its allocation and become satisfied.Indeed if this is not the case, we pick this tx. With positive probail-ity, it updates its power allocation to P max and becomes satisfied.We repeat this procedure: pick an unsatisifed tx that can becomesatisfied, and let it use P max . From allocation p [0] , power levelshave been only increased, and so we end up at a state where thereis no unsatisifed tx that can become satisified by unilateral powerupdate.Our constructed path consists of phases, indexed by k = 0 , , , ... .At the beginning of phase k , the set of unsatisfied links whose trans-mitters is not using allocation P max is denoted by V k . Phase k consists in letting links from V k select allocation P max (thisoccurs with positive probability because these links are not statis-fied). Note first that V (cid:54) = ∅ , for by assumption (A1), when tx from U ( p [0]) use P max , one satisfied link becomes unsatisfied. Suchan update requires that at least a tx from U ( p [0]) is able to increaseinterference, and hence is not already using allocation P max . Weprove similarly that at the beginning of phase k ≥ , either V k (cid:54) = ∅ or every transmitter uses P max . Assume that V k = ∅ , whichmeans that all unsatisfied transmitters use P max . Hence unsat-isfied transmitters cannot change their power allocation either tobecome satisfied or to disatisfy one link. From (A1), we deducethat all links are unsatisfied, and hence all use P max . In summaryafter at most N phases, all links are unsatisfied and use allocation max .After all links have become unsatisfied, we add the followingphase. We pick tx one after the other once. When tx i is picked,either it is unsatisfied, or due to power updates of previous trans-mitters, it has become satisfied, but the value of its parameter β i is0 (because it was not picked earlier in this phase). Hence when tx i is picked, it will update its power allocation. With positive prob-ability, it selects p (cid:63)i . After the last tx is picked, each tx is satisfied,but the β i ’s may not be all equal to 1. Finally we add a last phase:only the tx’s i such that β i = 0 are picked, and they again selectpower allocation p (cid:63)i . Thus we constructed a positive probabilitypath from any state to a state where every tx is satisfied and willnot update its power again. (cid:50) Proof of Theorem 6.2
The proof is similar to that of Theorem 6.1. y [ t ] = ( s [ t ] , p [ t ] , I last [ t ]) t ≥ is an homogeneous Markov chainwith finite state space. A set { ( s, p, I ) , s ∈ { , . . . , N }} consti-tutes a communication class of this Markov chain if R ( p ) ≥ R t (in such states, all links are satisfied, and do not update their allo-cations anymore). We show that these classes are accessible, andfrom any state, we build a positive probability path towards one ofthese classes.Let y [0] be any initial state of the Markov chain. As in the proofof Theorem 6.1, w.l.o.g. we may assume that U ( p [0]) (cid:54) = ∅ andthat there is no tx i ∈ U ( p [0]) that can update its allocation andbecome satisfied. Let U = U ( p [0]) . By (A2), there exists a max-imal set U (cid:54) = ∅ such that U ∩ U = ∅ and for every j ∈ U (cid:48) , MP max (cid:80) i ∈ U g ij > δ . Set U is maximal in a sense that no set U (cid:48) ⊃ U satisfies (A2) for the set U . Similarly, we recursivelydefine U w as the maximal set satisfying (A2) for the set ∪ w(cid:96) =1 U (cid:96) if ∪ w(cid:96) =1 U (cid:96) (cid:54) = { , . . . , N } . Let W : ∪ W(cid:96) =1 U (cid:96) = { , . . . , N } . Notethat such W exists and is less than or equal to N . Also note thatthe sets U (cid:96) ’s define a partition of { , . . . , N } .Our constructed path consists of phases, indexed by k = 0 , , , ... .We show that we can build these phases with positive probabilitysuch that:(i) In each phase, all tx’s are selected once; tx’s from U are se-lected first, then tx’s from U , and so on. (ii) In phase k , the tx’snot in U k +1 ∪ . . . ∪ U W do not update their power allocation. (iii)In phase k , each tx i updating its allocation selects p i = 0 . Inphase k + 1 , they select P max .(iv) there is a phase that ends with all tx having power allocation .If this construction is valid, then from the state where all tx remainsilent, we conclude as in the proof of Thoerem 6.1: we let each txpick p (cid:63)i , and run two phases to align the variables I last i .We now justify (i)-(ii)-(iii)-(iv). (i)-(ii) are immediate ( s satis-fies (P2), and a tx may always pick the same allocation as beforewith positive probability). Note that because of (i), in each phase,each tx i updates its value of I last i . In phase 0, all tx’s in U areunsatisfied, they update their power, and all choose 0 with positiveprobability. At the beginning of phase 1, tx’s in U are unsatisfied,and pick P max ; after that, from (A2), any tx i in U noticed theincreased interference in its parameter I last , and hence update itspower allocation with positive probability - it selects P max . Inphase 2, tx’s in U are still not satisfied because from their perspec-tive, interference has increased compared to that perceived initially;they can then update their allocations again and this time select 0power. This will be noticed by tx’s in U , that again will updatetheir allocations and select 0 power. In phase 3, tx’s in U and U will select allocation P max , which will be noticed by tx’s U . Thelatter will then select allocation P max . Repeating this argument, we justify (iii). (iv) is readily deduced from (iii). (cid:50) Proof of Lemma 7.1
Let T be the time at which the IT-IPB-PP has converged. Since T is the absorbing time of a finite state Markov chain, we have E [ T ] < ∞ . Now at T , a worst case (sample-path wise) is obtainedby assuming that in each queue i , there are AT + Q i [0] bits tobe served. From T , queues behave independently, and are alsoindependent of the r.v. T . Thus the system is stable if and onlyif each queue is stable. It remains to prove that each queue i isstable. W.l.o.g., assume that at time 0, queue i has AT + Q i [0] bits to be served, and let B i = inf { u : Q i ( u ) = 0 } . Define λ i [ u ] = u (cid:80) us =1 A i [ s ] . Let δ = R ti − λ i > . We have: P [ B i ≥ u ] ≤ P [ AT + Q i [0] + uλ i [ u ] ≥ uR ti ] ≤ P [ AT + Q i [0] u + λ i [ u ] − λ i ≥ δ ] ≤ P [ AT + Q i [0] u ≥ δ P [ λ i [ u ] − λ i ≥ δ ≤ P [ AT + Q i [0] u ≥ δ c e − c u , where the last inequality is obtained using Hoeffding’s inequality( c , c > ). We deduce that E [ B i ] = (cid:80) ∞ u =1 P [ B i ≥ u ] < ∞ ,and queue i is stable. (cid:50) Proof of Lemma 7.2
We just need to prove here that E [ T i ] < ∞ and R ti [ T i ] ∈ ( λ i , λ i + (cid:15)/ . After establishing these results, we can apply the same proofas that of Lemma 7.1. Indeed, note that after max i T i , at eachtransmitter, the target rate is fixed and greater than the arrival rate;also observe that E [max i T i ] ≤ (cid:80) i E [ T i ] .We only consider that λ i lies in the interior of e k for some k . Thus, there exists δ > such that δ -neighborhood of λ i lies in e k .Let T δ = inf { t : sup u ≥ t | λ i [ u ] − λ i | < δ } . Note that for every t ≥ T δ , λ i [ t ] ∈ e k and thus R ti [ t ] = (4 k + 1) µ/ . Also observethat T i ≤ T δ . We show that E [ T δ ] < ∞ . Consider P { T δ > t } andnote that P { T δ > t } = P {∪ ∞ u = t +1 {| λ i [ u ] − λ i | ≥ δ }}≤ ∞ (cid:88) u = t +1 P {| λ i [ u ] − λ i | ≥ δ }≤ ∞ (cid:88) u = t +1 c e − c u ≤ c c e − c t . The last inequality follows from Hoeffding’s inequality ( c , c > ). Now, E [ T δ ] = (cid:80) ∞ t =1 P { T δ ≥ t } ≤ (cid:80) ∞ t =1 c c e − c t < ∞ .Hence, E [ T i ] < ∞ . Finally, from the fact that λ i ∈ e k and R ti [ T i ] = (4 k + 1) µ/ , we simply deduce that R ti [ T i ] ∈ ( λ i , λ i + (cid:15)/ . (cid:50) The case where λ i may lie on the boundary of some e k can behandled similarly by choosing a slightly more complex target rateupdate algorithm: we consider two partitions of R + , ( e k ) k ≥ and ( f k ) k ≥ where f k = e k + µ/ for k ≥ and f = [0 , µ/ . Weconsider the same rate update, but switch partition when λ i [ t ] fallsinto a different interval than that of λ i [ t − . Using this, after λ i [ t ] concentrates around λ i , we do not switch partition anymore, and λ ii