Distributed throughput maximization in wireless networks via random power allocation.bak
IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 577 Distributed Throughput Maximization in Wireless Networks via Random Power Allocation Hyang-Won Lee, Member, IEEE, Eytan Modiano, Fellow, IEEE, and Long Bao Le, Member, IEEE Abstract—We develop a distributed throughput-optimal power allocation algorithm in wireless networks. The study of this problem has been limited due to the nonconvexity of the underlying optimization problems that prohibits an efficient solution even in a centralized setting. By generalizing the randomization framework originally proposed for input queued switches to SINR rate-based interference model, we characterize the throughput-optimality conditions that enable efficient and distributed implementation. Using gossiping algorithm, we develop a distributed power allocation algorithm that satisfies the optimality conditions, thereby achieving (nearly) 100 percent throughput. We illustrate the performance of our power allocation solution through numerical simulation. Index Terms—Throughput-optimal power allocation, randomization framework, SINR-based interference model. Ç1 INTRODUCTIONR ESOURCE allocation in multihop wireless networks In the seminal work of , Tassiulas and Ephremides involves solving a joint link scheduling and power introduce the concept of stability region, defined as the set ofallocation problem which is very difficult in general , all arrival rate vectors that can be stably supported. They. Due to this difficulty, most of the existing works in also propose a joint routing and scheduling policy thatthe literature consider a simple setting where all nodes in achieves 100 percent throughput, meaning that it stabilizesthe network use fixed transmission power levels and the the network whenever the arrival rate vector is in theresource allocation problem degenerates into simply a link stability region. More recently, this throughput-optimal http://ieeexploreprojects.blogspot.comscheduling problem , , , . Furthermore, the link policy has been extended to wireless networks with powerscheduling problem has been mostly studied assuming a control ,  and for the scenario where arrival rates liesimplistic graph-based interference model. outside the capacity region , , . In fact, the resource allocation problem has been All these resource allocation algorithms, however, re-considered mainly in two different network settings. The quire repeatedly solving a global optimization problemfirst setting is a static one which does not take randomness which is NP-hard in general , . Hence, in multihopin the traffic arrival processes into consideration. Inparticular, it is usually assumed that users either have wireless networks, it may be impractical to find its solutionunlimited amount of traffic to transmit or have predeter- in every time slot due to limited computation capability, andmined traffic demands. Here, resource allocation aims at the need for distributed operation. As an alternative,achieving fair share of resource among competing traffic distributed greedy scheduling has been proposed andflows or developing resource allocation algorithms which analyzed , , , , . However, most of thehave nice performance properties (e.g., constructing mini- existing works in this context adopt the graph-based inter-mum length schedule to support a traffic demands) , ference models, where transmissions on any two links in the, , , . The second setting assumes random network are assumed to be either in conflict or conflict free.arrival traffic and one of the main objectives of the Moreover, the use of greedy scheduling typically results inresource allocation problem is to maximize the averagearrival rates which can be supported while maintaining throughput reduction by factor of up ð2Kþ1Þ2under the to 2network stability. primary interference model ,  and bK=2c in K-hop interference model for K ! 2 . It has been recognized that graph-based interference. H.-W. Lee is with Konkuk University, Seoul 143-701, Republic of Korea. models may be overly simplistic because they ignore the E-mail: firstname.lastname@example.org.. E. Modiano is with the Massachusetts Institute of Technology, 77 cumulative effect of wireless interference. However, going Massachusetts Ave., Cambridge, MA 02139. E-mail: email@example.com. beyond these simplistic interference models is challenging.. L.B. Le is with the INRS Energie, Materiaux et Telecommunications, In fact, the power allocation problem under the SINR rate- University of Quebec, Place Bonaventure 800, de la Gauchetiere Ouest, bureau 6900, Montreal, Quebec H5A 1K6, Canada. based interference model is nonconvex; therefore, obtaining a E-mail: firstname.lastname@example.org. global optimal power allocation even in a centralizedManuscript received 15 Jan. 2010; revised 10 Jan. 2011; accepted 21 Jan. 2011; manner is not practical. This nonconvexity issue in thepublished online 17 Mar. 2011. power allocation problem has been addressed by severalFor information on obtaining reprints of this article, please send e-mail to:email@example.com, and reference IEEECS Log Number TMC-2010-01-0025. papers ,  considering either the high or low SINRDigital Object Identifier no. 10.1109/TMC.2011.58. regimes. Recently, it has been shown that this problem is 1536-1233/12/$31.00 ß 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
578 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012NP-hard , , where the optimality conditions for sum hence link ða; bÞ exists whenever ðb; aÞ does. For simplicityrate maximization are extensively studied. of exposition, we start by assuming that there is only In this paper, we develop a distributed throughput- single-hop traffic and single channel available in theoptimal power allocation algorithm under the SINR rate- network. Extension to the case of multihop traffic andbased interference model. As mentioned above, the pre- multichannels can be found in . Node a maintains aviously known condition for throughput-optimal power data buffer for each outgoing link ða; bÞ, and its backlog atallocation under this model requires solving a nonconvex time t is denoted by qab ðtÞ.optimization problem for every time slot. Hence, its Denote by pab the transmit power allocated to linkdistributed implementation may be prohibitive in practice. ða; bÞ. Each node a has a limited power budget Pa , and maxWe take a randomization approach to develop the optimality the total transmit power constraint can be written as Pconditions that enable distributed power allocation algo- max Pa . We assume SINR rate-based interfer- b2V ðaÞ pabrithms. The randomization technique was originally devel- ence model. That is, under a power allocation vectoroped for input queued switches , and later extended for p ¼ ½p ; 8ða; bÞ 2 E, link ða; bÞ’s rate r ðpÞ is given by ab abmultihop wireless networks assuming the graph-based !primary and secondary interference models , . Its key gab pabfeature is that it does not seek to find an optimal schedule in rab ðpÞ ¼ log 1þ P P P ; nb þ i2V ðaÞnfbg gab pai þ i6¼a gib j2V ðiÞ pijevery time slot, and consequently, solving a difficultscheduling problem can be avoided. Motivated by this ð1Þobservation, our work attempts to alleviate the difficulty in where nb is the noise power, and gab is the channel gainsolving the nonconvex optimization problem involved in from node a to b. It is assumed gab ¼ 1 if a ¼ b. Since theoptimal power allocation, using randomization. nodes are static, the channel gains are assumed to be fixed As mentioned above, the throughput-optimal schedulingproblem under the graph-based interference model has over time. Note that the second term in the denominator ofbeen relatively well understood. In particular, the rando- (1) is self-interference, and the third is mutual interference.mization has been successfully applied for developing Let Aab ðtÞ represent the amount of exogenous data thatefficient throughput-optimal scheduling algorithms , arrive to the buffer at the source of link ða; bÞ during slot t,. On the other hand, there are few results that deal with and pðtÞ the power allocation vector for slot t. Then, thethe throughput-optimal power control problem under the backlog qab ðtÞ evolves according to the following dynamics:SINR-based interference model in which the amount of qab ðt þ 1Þ ¼ max½0; qab ðtÞ À rab ðpðtÞÞ þ Aab ðtÞ: ð2Þinterference and noise is explicitly taken into account. In, , , , optimal scheduling problems are The arrival process Aab ðtÞ is assumed to be i.i.d. over time http://ieeexploreprojects.blogspot.comconsidered assuming that every transmitting node uses with average ab , i.e., E½Aab ðtÞ ¼ ab ; 8t. We assume that allfixed power levels and the success or failure of a arrival processes Aab ðtÞ have bounded second moments andtransmission is determined by certain SINR threshold. they are upper bounded by Amax (i.e., Aab ðtÞ Amax , In contrast, we assume a SINR rate-based interference 8ða; bÞ 2 E). Now, we define the network stability.model where the transmission rate of a link is given as acontinuous function of its SINR. In , the throughput- Definition 1. A queue qab ðtÞ is called strongly stable ifoptimal power control problem was considered under this 1X tÀ1model; however, the performance of the proposed power lim sup Efqab ðÞg 1: ð3Þ t!1 t ¼0allocation algorithm is not guaranteed. To the best of ourknowledge, there is no known work that assumes the SINR A network of queues is called strongly stable if all individualrate-based interference model and solves the throughput- queues are strongly stable.optimal power control problem in the stability framework of. As mentioned above, the problem needed to be solved in For convenience, we will instead use the term stable toeach time slot was shown to be NP-hard in . Hence, represent the term strongly stable.achieving throughput optimality under the SINR rate-based Let us drop the indices of a variable to denote its vectorinterference model is likely to be a hard problem. To form, for example, qðtÞ ¼ ½q ðtÞ; 8ða; bÞ 2 E. Define the abcircumvent this difficulty, we develop new tractable through- stability region, denoted by Ã, to be the union of arrivalput-optimality conditions by extending the randomization rate vectors ¼ ð ; ða; bÞ 2 EÞ such that there exists a abframework, and develop a distributed power allocation power allocation policy which stabilizes the networkalgorithm that satisfies the new optimality conditions. queues. In , the stability region for wireless networks with power control was characterized. Let F be the2 MODEL AND PROBLEM DESCRIPTION feasible region of transmit power vectors, i.e., F ¼ fp ! 0 : P maxWe consider a multihop wireless network modeled by a b2V ðaÞ pab Pa ; 8a 2 V g where p ! 0 is component-wisegraph G ¼ ðV ; EÞ, where V is the set of nodes and E is the inequality. The stability region Ã consists of all arrival rateset of links. Let N be the number of nodes, i.e., N ¼ jV j. It vectors ¼ ðab ; ða; bÞ 2 EÞ such thatis assumed that there is a link between two neighboring 2 Convex HullfrðpÞ : p 2 F g: ð4Þnodes if they want to communicate with each other. Weassume that time is slotted and a time slot interval is of Note that it is the convex hull of all the feasible link rateunit length. Let V ðaÞ be the set of node a’s neighbors, i.e., vectors. In , it was shown that if in each time slot t,V ðaÞ ¼ fb 2 V : ða; bÞ 2 Eg. We assume bidirectional links, power is allocated according to the following max-weight
LEE ET AL.: DISTRIBUTED THROUGHPUT MAXIMIZATION IN WIRELESS NETWORKS VIA RANDOM POWER ALLOCATION 579rule, then the network will be stable for all arrival rates let qðtÞT rðpÞ be the objective value in (5). A natural extensionwithin the stability region of the randomization framework to SINR rate-based inter- X ference model will be as follows: first, in each time slot t, the pÃ ðtÞ ¼ arg max qab ðtÞrab ðpÞ: ð5Þ nodes generate a new random power allocation vector, p2F ða;bÞ denoted by pðtÞ, in a distributed manner. Second, the current ~The optimal solution pÃ ðtÞ may not be unique, but in the power vector pðtÞ is selected by comparing the new powercase of multiple optimal solutions, our randomization vector pðtÞ and the previous one pðt À 1Þ; namely, pðtÞ ¼ pðtÞ ~ ~framework performs better. Hence, assuming unique qÃ ðtÞ if qðtÞT rð~ðtÞÞ qðtÞT rðpðt À 1ÞÞ and pðtÞ ¼ pðt À 1Þ other- pwill give a lower bound on the performance of our wise. These two steps are summarized in Algorithm 1. Therandomization framework. Note that in the graph-based key challenge in this setting is that it may not be possible tointerference model, link rates are fixed and the resource devise a power allocation policy RAND-POW that has aallocation problem degenerates into the link scheduling positive probability of being optimal since the optimalproblem, where the max-weight scheduling policy which power allocation takes on real values. Consequently, thereturns a feasible schedule achieving the maximum weight randomization approach to the power allocation problemin each time slot is throughput optimal. will not be able to achieve 100 percent throughput as in the The optimization problem (5) is nonconvex in p, and case of the graph-based interference model. We address thishence, it may not be possible to find an optimal power issue by generalizing the condition on RAND-SCH in thevector for every time slot t, even in a centralized manner. graph-based interference model; namely, the newly gener-We address this issue by using randomization, originally ated power vector is not required to be optimal, but isproposed for input queued switches  and wireless required to be within a small factor of optimal.networks under graph-based interference models , . Algorithm 1. Randomized Power Control Framework (for each time slot t)3 RANDOMIZATION FRAMEWORK 1. RAND-POW: Generate a new random power allocation3.1 Background on Randomization Framework vector pðtÞ in a distributed manner. ~The randomization approach was first developed for 2. DECIDE: Determine the current power allocation pðtÞscheduling in input queued switches , and extended by comparing the previous power allocation pðt À 1Þ andfor distributed operations in multihop wireless networks the new power allocation pðtÞ, and selecting the one with ~, . Recall that under these settings, a feasible schedule higher weight in (5).is to be found in each time slot. The key feature of the http://ieeexploreprojects.blogspot.com in the DECIDE part, as the localizedrandomization approach is that it does not seek to find an Another challenge lies comparison in the graph-based interference model cannotoptimal schedule in every slot, and hence, it can signifi- work in our setting. With the SINR rate-based interferencecantly reduce the computation overhead. In every time slot, model, the interference level experienced at a node is affectedthe randomization framework does the following: by all the other nodes in the network. Hence, the localized 1. RAND-SCH: generate a new random schedule, comparison may lead to a wrong decision, and a network- 2. DECIDE: decide on the current schedule by compar- wide comparison will be inevitable. To resolve this problem, we will use randomized gossiping . ing and selecting the better of the new and old We first present new conditions for RAND-POW and schedules (i.e., the one with higher weight in (5)). DECIDE that will be used to characterize the performance of randomization framework.Lemma 1 (). Under the condition that the newly generated Condition 1 (C1). For every time slot t, schedule in RAND-SCH is optimal with positive probability, the randomization framework achieves 100 percent throughput. Pr½qðtÞT rð~ðtÞÞ ! ð1 À 1 ÞqðtÞT rðpÃ ðtÞÞ ! 1 0; p Note that in an input queued switch the number of where 1 and 1 are some positive constants, and pðtÞ, pÃ ðtÞ are ~possible activations is finite. Hence, it is trivial to develop a the new random power vector and optimal power vector,random algorithm to satisfy the condition in Lemma 1. respectively.Moreover, the comparison in a switch can be done in acentralized manner. However, in multihop wireless net- Condition C1 allows for the possibility that the newworks, the DECIDE step is challenging because each node random power allocation is within a factor of the optimal.must compare the network-wide weighted sum rates Notice that when 1 ¼ 0, C1 becomes the condition onachieved by the two schedules in a distributed manner. In RAND-SCH in ,  which requires the new scheduling to, this comparison is localized over connected subgraphs be optimal with positive probability. This generalization isconsisting of old and new link activations, where the the key to dealing with the power control problem (5) usingdecisions in one subgraph do not affect the decisions at the randomization approach, and the optimality loss underother subgraphs. The communication overhead can be this condition will be characterized in Theorem 1.substantially reduced using this localization. The following is the condition on DECIDE adopted from :3.2 Extension to SINR Rate-Based ModelOur work is motivated by the intuition that the difficulty Condition 2 (C2, ). For every time slot t,due to the nonconvexity in (5) can also be alleviated using qðtÞT rðpðtÞÞ ! ð1 À 2 ÞmaxfqðtÞT rðpðt À 1ÞÞ; qðtÞT rð~ðtÞÞg pthis randomization technique. For notational convenience,
580 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 X with probability (WP) at least 1 À 2 , where 2 and 2 ð( 1 Þ ¼ i rðpi Þ; ð12Þ are some positive constants. i P where i i ¼ 1 and i ! 0; 8i, and pi is a feasible power Condition C2 requires that the weight attained by the vector. Multiplying both side of (12) by qT = yieldschosen power vector pðtÞ should not be less than some X Xfactor of the maximum of the weights obtained by pðtÞ and ~ qT ¼ i qT rðpi Þ i qT rÃ ¼ qT rÃ ; ð13Þpðt À 1Þ. This condition was considered in  to account for i iimperfect comparison in multihop networks. In Section 5,we discuss a distributed implementation of the DECIDE step where rÃ is the optimal rate which achieves thethat satisfies C2. maximum weight (i.e., qT rÃ ¼ maxp2F qT rðpÞ). The above The achievable stability region under our randomization inequality also impliesframework can be characterized as follows: qT qT rÃ ¼ W Ã ðtÞ; ð14ÞTheorem 1. If RAND-POW and DECIDE in Algorithm 1 satisfy C1 and C2, then it stabilizes the network for any arrival rate for any lying inside Ã. qﬃﬃﬃ Using (13), the bound in (11) can be rewritten as vector in Ã where 1 À ð1 þ ð1 À 1 Þ2 Þ À 2 2 . 1Proof. Here, we briefly prove the theorem. More detailed X T À1 version of the proof can be found in . Consider the fqðt þ ÞT À qðt þ ÞT rÃ ðt þ Þg ¼0 ð15Þ following Lyapunov function Ã X ÀT ð1 À ÞW ðtÞ þ B3 : LðqðtÞÞ :¼ qab ðtÞ2 : ð6Þ Using (10) and (15), the conditional expectation of T -step ða;bÞ2E Lyapunov drift in (7)-(8) can be bounded as Then, the expected conditional T -step Lyapunov drift is bounded as ÁT ðtÞ À2T 1 Â 1 À À 1 À ð1 À 1 Þ2 À À 2 T W Ã ðtÞ þ B4 ; ÁT ðtÞ ¼ EfLðqðt þ T ÞÞ À LðqðtÞjqðtÞÞg 1 T X T À1 ð7Þ where B4 ¼ B1 þ 2B3 þ 2B2 is a finite number. Now, the 2 Efqðt þ ÞT À qðt þ ÞT rÃ ðt þ ÞjqðtÞg pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼0 proof can be completed by choosing T ¼ 1=ð1 2 Þ, and http://ieeexploreprojects.blogspot.com ﬃﬃﬃ (14) and the condition 1 À applying the inequality q X T À1 1 À ð1 À 1 Þ2 À 2 2 . 1 u t þ B1 þ 2 EfÉðt þ ÞjqðtÞg; ð8Þ When 1 is 0, i.e., when a new power vector is optimal ¼0 with probability 1 , the obtained throughput mainly depends on the comparison performance (2 ). However, where B1 is a finite constant the throughput loss increases as 1 increases. In case of perfect comparison (i.e., 2 ¼ 0 and 2 ¼ 0), the throughput ÉðtÞ :¼ qðtÞT rÃ ðtÞ À qðtÞT rðtÞ; ð9Þ loss depends only on the optimality loss in the random and rÃ ðtÞ is the optimal rate which corresponds to the power allocation. In brief, our randomized power control optimal power allocation given the queue length vector framework can achieve nearly 100 percent throughput if we qðtÞ at time t (i.e., it achieves the maximum weight). can develop a power allocation policy (RAND-POW) and a Ã T Ã Let W ðtÞ ¼ qðtÞ r ðtÞ. Then, using Conditions C1 and comparison algorithm (DECIDE) satisfying conditions C1 C2, we can show and C2 with small 1 , 2 , and 2 . In the rest of the paper, we focus on developing such algorithms. In particular, in X T À1 Section 4, we develop a random power allocation policy EfÉðt þ ÞjqðtÞg that satisfies C1 and in Section 5 we develop a comparison ¼0 ð10Þ algorithm that satisfies C2. 1 T 1 þ ð1 À 1 Þ2 þ þ 2 T W Ã ðtÞ þ B2 ; 1 T 3.3 Frame-Based Implementation In this section, we discuss some issues arising in the where B2 is a finite constant. The first term in (7) is implementation of our randomization framework. First, the bounded as RAND-POW step can be easily implemented as it is easy to X T À1 generate a random power vector in a distributed manner, fqðt þ ÞT À qðt þ ÞT rÃ ðt þ Þg as we demonstrate in Section 4. For the DECIDE phase, each ¼0 ð11Þ node has to estimate the global weights qT r in order to T ½qðtÞT À qðtÞT rÃ ðtÞ þ B3 ; make the same decision on the selection of the current power allocation. In small networks, a centralized entity where B3 is a finite number. Now, consider any arrival may exist (e.g., base station in cellular networks); hence, rate vector which lies inside the -scaled stability comparison and decision can be implemented in a region (i.e., inside Ã) for 0 1. Then, we have 2 Ã centralized manner. In large networks, however, such a and hence, the scaled rate vector can be represented as centralized comparison is prohibitive and we adopt a convex combination of feasible rate vectors gossiping for distributed comparison.
LEE ET AL.: DISTRIBUTED THROUGHPUT MAXIMIZATION IN WIRELESS NETWORKS VIA RANDOM POWER ALLOCATION 581 as close to pÃ as possible, and hence, identifying the optimality properties of (16) would be helpful for generat- ing such p. The following lemma characterizes some useful ~ properties of pÃ : Lemma 2. Under the optimal power allocation pÃ , 1. a node does not transmit while receiving, and vice versa, 2. a node transmits to at most one of its neighbors. Proof. Recall the assumption gaa ¼ 1; 8a. Under this assumption, if a node transmits while trying to receive, it will achieve zero rate due to infinite interference. Hence, at optimal pÃ , case 1 does not happen. P To prove case 2, let pÃ ¼ b2V ðaÞ pÃ , i.e., pÃ is the a ab a total power transmitted by node a at the optimal allocation. It is obvious that solving the problem (16) P Ã with the additional constraints b2V ðaÞ pab ¼ pa ; 8a will result in the same optimal solution. Hence, the objective function in (16) can be written asFig. 1. Frame-based and control-channel-based implementations. ! X X gab pab In the proof of Theorem 1, we assumed for simplicity qab log 1 þ P : a b2V ðaÞ nb þ gab ðpÃ À pab Þ þ i6¼a gib pÃ a ithat the power allocation is updated for each datatransmission slot by running the RAND-POW and DECIDE Clearly, changing transmit power pab ; 8b 2 V ðaÞ understeps. However, it may not be practical to run these two fixed total power does not affect the mutual inter-steps on a slot-by-slot basis, because the DECIDE step may ference, but only changes the self-interference. Hence,require a significant amount of communications. In fact, this the new optimization problem can be solved separatelyassumption can be easily relaxed by running the RAND- with respect to each node, i.e., for each a, we only needPOW and DECIDE on a frame basis as shown in Fig. 1a, to maximizewhere they are performed for every multiple data transmis- http://ieeexploreprojects.blogspot.com !sion slots so that the same power allocation is kept for X gab pabmultiple data slots. By doing so, the control overhead can be qab log 1 þ P nb þ gab ðpÃ À pab Þ þ i6¼a gib pÃ a isignificantly reduced. Moreover, it was shown in  that b2V ðaÞthis frame-based scheduling still achieves throughput P subject to p ! 0 and b2V ðaÞ pab ¼ pÃ . Since this function aoptimality as long as the RAND-POW and DECIDE steps is strictly convex in ½pab ; 8b 2 V ðaÞ, it is maximized at aare performed at regular intervals. Alternatively, the power corner point, i.e., pab ¼ pÃ for some b 2 V ðaÞ and 0 for all acontrol algorithm can be done on a separate low-bandwidth others. This shows that it is optimal for each node tocontrol channel, in parallel with data transmission, asshown in Fig. 1b. Again, throughput optimality can be transmit to at most one neighbor. u tachieved as long as a new power allocation is generated at According to Lemma 2, at an optimal allocation, a noderegular (finite duration) intervals. The advantage of this is not allowed to transmit to multiple neighbors, and to be aimplementation over the frame-based is clear, that is, the transmitter and receiver simultaneously. Note, however,data transmission does not need to wait until the update of that it is possible for a node to receive from multiplepower allocation is finished, and consequently it will transmitters. This is in contrast to a matching in which aachieve better performance. node cannot be shared by multiple edges. For ease of exposition, we define the notion of a pairing as follows:4 RANDOMIZED POWER ALLOCATION Definition 2. Assume that the tail and the head of a directed edgeWe present a power allocation policy RAND-POW that denote a transmitter and a receiver, respectively. A directed subgraph of G is called a pairing if it satisfies cases 1 and 2 insatisfies C1, i.e., finds with positive probability a power Lemma 2.vector within a small factor of the optimal value in (5). Theproblem (5) is to maximize Note that a pairing is different from a matching because X X pÃ ¼ arg max qab it allows a node to be shared by multiple edges. Fig. 2 p2F a2V b2V ðaÞ shows an example of matching and pairing ! gab pab 4.1 Transmitter-Receiver Pairing Â log 1 þ P P P ; nb þ gab i2V ðaÞnb pai þ i6¼a gib j2V ðiÞ pij From Lemma 2, it is clear that finding a power allocation can be decomposed into two steps. First, find a pairing, and then ð16Þ P select the transmit power levels for the given pairing. Since maxwhere F ¼ fp ! 0 : b2V ðaÞ pab Pa ; 8a 2 V g. Clearly, there is a finite number of pairings, and at least one of them isthe new power vector p in RAND-POW is desired to be ~ optimal, it is easy to generate an optimal pairing with
582 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 generates a pairing I ¼ ½Iab ; b 2 V ðaÞ; a 2 V . Given this pairing, the problem (16) is rewritten by pÃ ðIÞ ¼ ! X X g p arg max qab log 1 þ P ab ab P :Fig. 2. Matching: either (single) transmission or reception is allowed for p2F a2V b:Iab ¼1 nb þ 6¼a j:Iij ¼1 gib pijeach node. Pairing: node can receive from multiple neighbors buttransmitting to multiple neighbors are not allowed. ð17Þ Notice that the self-interference has been removed and thepositive probability. One such algorithm is given by RAND- mutual interference has been simplified due to thePAIR (see Algorithm 2). Let Iab ¼ 1 if node a transmits to its constraints 1 and 2 in Lemma 2. Since the pairing I foundneighbor b, and 0 otherwise. The goal of RAND-PAIR is to by RAND-PAIR has a positive probability of being optimal,generate a vector I ¼ ½Iab ; b 2 V ðaÞ; a 2 V satisfying the the condition C1 can be satisfied if a power level is selectedpairing constraints. To do this, each node a decides to be a such that it is within a factor of the objective in (17) withtransmitter with probability 1 and a receiver WP 1 . Then, 2 2 positive probability. To meet this requirement, Algorithmeach transmitting node a sends a pair-request message to one RAND-PSEL simply chooses power levels uniformly atof its neighbors b. If b decided to be a receiver, it accepts the random. In particular, each transmitting node a randomlyrequest and sends an acceptance message. Otherwise, it is selects its transmit power from the feasible region, i.e.,ignored and nothing happens. Once node a receives the max ½0; Pa . This random power selection meets the require-acceptance message, it updates Iab to Iab ¼ 1. This algorithm ment as shown in the following lemma. Assumehas Oð1Þ computation and communication complexity, and max Pa ¼ 1; 8a.will find an optimal pairing with positive probability, asstated in the following lemma. Algorithm 3. RAND-PSEL (for given pairing I) 1: Each node a initializes pab ¼ 0; 8b 2 V ðaÞ. ~Algorithm 2. RAND-PAIR 2: Every paired transmitting node a does the following: 1: Each node a decides to be a transmitter w.p. 1=2 and a max (i) Select a number, say u, from ½0; Pa uniformly at receiver w.p. 1=2, and initializes Iab ¼ 0; 8b 2 V ðaÞ. random, and set pab ¼ u for b such that Iab ¼ 1. ~ 2: Each transmitting node a sends a pair-request message (PQM) to one of its neighbors in V ðaÞ uniformly at Lemma 4. For any 2 ð0; 1Þ, Algorithm RAND-PSEL generates Ã random. http://ieeexploreprojects.blogspot.com that p 2 Bðp ðIÞ; Þ with probability at a power vector p such N ~ ~ 3: If node b receives a PQM, one of the following happens: least ðpﬃﬃﬃÞ , where Bðp ðIÞ; Þ ¼ fp 2 F : kp À pÃ ðIÞk2 g. N Ã (i) If node b is a receiver, then it accepts the request and Proof. See . u t sends a pair-request-accepted message (PAM) to node a. Note that this lemma can be easily extended to the case (ii) Otherwise, ignore the PQM and nothing happens max of general Pa . Combining Lemmas 3 and 4, we can show for node a. that Condition C1 can be satisfied by RAND-PAIR and 4: If node a receives a PAM from node b, set Iab ¼ 1, RAND-PSEL. meaning that node b is a receiver of node a. Theorem 2. Choosing a power allocation according to RAND-Lemma 3. Algorithm RAND-PAIR finds an optimal pairing with PAIR and RAND-PSEL satisfies C1 with arbitrarily small probability at least ð4NÞÀN . 1 0 and positive 1 which is a function of 1 .Proof. See . u t Proof. Let fðpÞ be the objective function in (17), and consider an arbitrary 1 2 ð0; 1Þ. Due to the continuity of fðpÞ, Note that in the interference graph model, a new there exists 0 such that fðpÞ ! ð1 À 1 ÞfðpÃ ðIÞÞ for anyscheduling should be a max-weight matching (or indepen- feasible p such that kp À pÃ ðIÞk . Let I be a pairingdent set in general) with positive probability. Because the generated by RAND-PAIR and p be a power vector ~max-weight matching is one of maximal matchings, and obtained through RAND-PSEL, given pairing I. Let I Ã besuch a probability can be increased by performing multiple an optimal pairing. Then, it follows thatiterations until the obtained matching becomes maximal. Pr½qT rð~Þ ! ð1 À 1 ÞqT rðpÃ Þ pHowever, in our case, maximal pairing1 may not be alwaysoptimal. Hence, performing multiple iterations does not ¼ Pr½fð~Þ ! ð1 À 1 ÞfðpÃ ðIÞÞjI ¼ I Ã Pr½I ¼ I Ã pnecessarily enhance the probability of being optimal, and ! Pr½k~ À pÃ ðIÞk2 jI ¼ I Ã Pr½I ¼ I Ã pfurther it may not guarantee that the obtained pairing has a N Npositive probability of being optimal. ! pﬃﬃﬃﬃﬃ Áð4NÞÀN ¼ ; N 4N 3=24.2 Power Level Selection where the last inequality is due to Lemmas 3 and 4.Now, what remains is to select a power level which together Therefore, the power allocation obtained by RAND-PAIRwith RAND-PAIR satisfies C1. Recall that RAND-PAIR and RAND-PSEL achieves at least ð1 À 1 Þ fraction of optimal value of problem (16) with probability at least 1. A pairing I is maximal if adding a link (not in I) to I makes it nolonger a pairing. 1 ¼ ð4N3=2 ÞN 0, satisfying Condition C1. u t
LEE ET AL.: DISTRIBUTED THROUGHPUT MAXIMIZATION IN WIRELESS NETWORKS VIA RANDOM POWER ALLOCATION 583 Therefore, the optimality loss 1 can be arbitrarily small that gossiping has been used extensively for computing(with small enough and thus small probability 1 ). averages (See ,  and references therein).According to Theorem 1, the throughput loss due to this Typically, gossiping generates a matching for eachoptimality loss (1 ) under our power allocation is negligible, iteration. Let xa ðkÞ be the value at node a after iteration k.as long as 2 ( 1 . The initial value is thus xa ð0Þ and the global average is PRemark. Theorem 2 implies that the random power a xa ð0Þ=N. If any two nodes a and b share the same link allocation hits a near optimal solution in every ð4N ÞN 3=2 under the current matching, then they update their values to slots (in average sense). As a consequence, it can their average, i.e., xa ðk þ 1Þ ¼ xb ðk þ 1Þ ¼ xa ðkÞþxb ðkÞ . Gossip- 2 experience large delay or network backlog, although ing keeps generating a random matching for this averaging our work in this paper focuses on the long-term operation, and every node eventually obtains an estimate of P throughput performance. In fact, recent results in  the global average a xa ð0Þ=N. In this paper, we use a show that there may not exist a polynomial time random matching policy in  that works as follows: let (deterministic or randomized) throughput-optimal pol- dðaÞ be the degree of node a, i.e., dðaÞ ¼ jV ðaÞj and dÃ be the icy for NP-hard scheduling problem such that it achieves maximum node degree, i.e., dÃ ¼ maxa2V dðaÞ. Each node a polynomial network backlog. The power allocation decides to be active with probability 1 and inactive WP 1 . 2 2 problem (5) contains a maximum weight independent An active node a does nothing WP 1 À dðaÞ , and randomly dÃ set problem . Hence, any polynomial time power contacts one of its neighbors WP dðaÞ .2 Consider an inactive dÃ allocation policy that takes on the problem (5) will node b. If b is contacted by node a while it has not been experience large delay, as our random power allocation contacted by any other, then nodes a and b average their Ã algorithm will. This implies that our random power values WP ð1 À 2dÃ Þd ÀdðbÞ . Otherwise, nothing happens for a. 1 allocation algorithm may not scale very well as the network size grows. As mentioned above, in this paper, 5.1 Compare and Agree we focus on the throughput performance, and we leave The COMPARE procedure estimates the averages Xnew and this delay and scalability issue as future research. X old using the gossiping described above, and is shown in Algorithm COMPARE. Note that at each iteration, a matching is generated and any two nodes sharing a link5 COMPARISON AND AGREEMENT in that matching average their values. Note also that theThe goal of the DECIDE step is to choose a power same matching is used for new and old values. After Kallocation pðtÞ by selecting one of the two power iterations, every node will obtain the estimates of new andallocations pðt À 1Þ and pðtÞ, so thathttp://ieeexploreprojects.blogspot.com ðKÞ and xold ðKÞ. ~ Condition C2 can be old average values xnew a asatisfied. Such a selection is easy in a centralized setting;namely, a central entity can compare qðtÞT rðpðt À 1ÞÞ and Algorithm 4. COMPAREqðtÞT rð~ðtÞÞ, pick the one having larger value, and p 1: For iteration k ¼ 1; . . . ; K, do the following:disseminate the selection to every node. In small net- (i) Each node a updates xnew ðkÞ ¼ xnew ðk À 1Þ and a aworks, such a centralized comparison might be possible, xold ðkÞ ¼ xold ðk À 1Þ. a aor a spanning tree could be computed in a distributed (ii) Each node decides to be active w.p. 1=2 andmanner and used for the comparison . However, in inactive w.p. 1=2. An active node a doeslarge networks, such a centralized computation is prohi- nothing w.p. 1 À dðaÞ , and contacts one of its dÃbitive. For this reason, we develop a distributed DECIDE neighbors uniformly at random (i.e., with 1policy by using randomized gossiping . It consists of equal probability dÃ ).two procedures: COMPARE and AGREE. The COMPARE (iii) If node b is contacted, one of the followingprocedure estimates the objective values achieved under happens:the new and old power allocations, and the AGREE (b) If b is inactive and has not been contacted,procedure uses these estimates to make a unanimous they average as xnew ðkÀ1Þþxnew ðkÀ1Þdecision on the selection of current power allocation. xnew ðkÞ ¼ xnew ðkÞ ¼ a a b 2 b and new old Let xb ð0Þ and xb ð0Þ be the weighted (receiving) rates xold ðkÀ1Þþxold ðkÀ1Þat node b under the new power pðtÞ and old power pðt À 1Þ, ~ xold ðkÞ ¼ xold ðkÞ ¼ a a b Ã 2 b w.p. new 1 d ÀdðbÞrespectively. Then, they can be expressed as xb ð0Þ ¼P P ð1 À 2dÃ Þ . old a2V qab ðtÞrab ð~ðtÞÞ and xb ð0Þ ¼ P P p new a2V qab ðtÞrab ðpðt À 1ÞÞ. (c) Otherwise, b ignores the contact and nothingLet Xnew ¼ a2V xa ð0Þ and Xold ¼ a2V xold ð0Þ, i.e., Xnew a happens for a.and Xold are the objective function values under the newly If the estimates are exact, a unanimous decision satisfy-generated power vector and the old power vector, respec- ing C2 can be easily made since every node a will havetively. The DECIDE step must choose the new power xnew ðKÞ xold ðKÞ (or xnew ðKÞ xold ðKÞ). Such a unani-allocation if Xnew Xold , and the old one if Xnew Xold . a a a aThis can also be accomplished using the average values mous decision can also be guaranteed if the estimates are Xnew and Xold instead of Xnew and Xold , where Xnew ¼ 2. Under the Algorithm COMPARE, each active node a has 1 dðaÞ inactiveXnew =N and X old ¼ Xold =N. Therefore, if every node can neighbors in average. Hence, for better chance of matching, 2 is desirable it compute an accurate estimate of Xnew and Xold , they will be for an active node with high degree to make an attempt to match with high probability, while a node with low degree is desired to attempt with lowable to make a decision leading to C2. A randomized probability. This is why the contact probability is proportional to the node gossiping algorithm is used to estimate Xnew and Xold . Note degree.
584 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 Lemma 6. For every k ! 0, the sum is conserved as X X xa ðkÞ ¼ xa ð0Þ: a2V a2V Proof. Let xa ðkÞ and xðkÞ be the value of node a after iteration k and its vector, respectively. Denote by MðkÞ the matching found in iteration k. The update of node values can be expressed as a linear equation by xðk þ 1Þ ¼ W ðkÞxðkÞ; ð18Þ where W ðkÞ is an N Â N matrix given by X ðea À eb Þðea À eb ÞT W ðkÞ ¼ I À ; ð19Þ ða;bÞ2MðkÞ 2 where I is the identity matrix and ei is the ith element vector whose ith coordinate is 1 and all others are 0. The first term in W ðkÞ corresponds to the original value of each node and the second term describes the change Fig. 3. Impact of difference jXnew À Xold j on unanimous decisions. from the original value. For example, when node a averages with b, its new value becomes 1 xa ðkÞ þ 1 xb ðkÞ ¼ 2 2 highly accurate, provided that the difference jXnew À Xold j is xa ðkÞ À 1 xa ðkÞ þ 1 xb ðkÞ, where the first term corresponds 2 2sufficiently large. However, in the case of small difference, to I and the last two terms correspond to the seconddecisions can be mixed even under highly accurate term in (19). Note that the matrix W ðkÞ is doublyestimation (See Fig. 3), which can lead to the violation of stochastic, and as a consequence, the following holds:C2. An additional procedure is thus needed to ensure that Xevery node makes the same and right decision. xa ðk þ 1Þ ¼ ~T Á xðk þ 1Þ 1 The AGREE procedure keeps the decision made by a2V http://ieeexploreprojects.blogspot.com ¼ ~T W ðkÞxðkÞCOMPARE if it is unanimous. Otherwise, it keeps the old 1 Xpower allocation. Note that in the case of small difference, ¼1~T Á xðkÞ ¼ xa ðkÞ:this selection policy will not incur big losses in throughput, a2Vbecause there is only a small difference between selectingeither the new or the old power allocation. To do this, the The third equality is due to the fact that W ðkÞ is doubly newAGREE procedure uses the estimates xa ðKÞ and xa ðKÞ as old stochastic. This proves the lemma.follows: each node a initiates a variable za ð0Þ as follows: We now prove Lemma 5. Under the assumption of unanimous decisions after COMPARE, there can be 1 if xnew ðKÞ xold ðKÞ a a only two cases including 1) xnew ðKÞ xold ðKÞ; 8a or a a za ð0Þ ¼ 2) xnew ðKÞ xold ðKÞ; 8a. Suppose case 1, in which 0 if xnew ðKÞ xold ðKÞ: a a a a every node selects the new power. Then, it follows thatNamely, za ð0Þ is equal to 1 if node a prefers the new power X X ~allocation and 0 otherwise. It runs gossiping for K iterations xnew ðKÞ xold ðKÞ P a a ¼(as in COMPARE) to estimate the average Z a2V za ð0Þ=N. a a X XAfter that, each node a decides to use the new power if ) xnew ð0Þ a xold ð0Þ; a ~ ¼ 1 and the old one otherwise, where za ðkÞ is theza ðKÞ a avalue at node a after iteration k. Note that if za ð0Þs are all 0 or where the second line is due to Lemma 6. Therefore,all 1, then the convergence and unanimous decision are selecting the new power is the right decision. Case 2 canguaranteed immediately. We will show that this is the right be proved similarly. u tdecision. If there is a mixture of decisions at the end of Therefore, it is desirable to keep any unanimous decisionCOMPARE, the AGREE procedure tries to keep the old power made after COMPARE, because the better power allocationallocation. The following lemmas show a unanimous is always selected under such a decision. This justifies thedecision is the right decision, hence justifying the AGREE. AGREE procedure that always keeps unanimous decisionsLemma 5. Suppose that there was an agreement after COMPARE, made after COMPARE. i.e., za ð0Þs are all 0 or all 1. Then, it is the right decision We now analyze and prove that the combination of regardless of the values of Xnew and Xold in that the power COMPARE and AGREE can satisfy Condition C2. For the allocation selected based on za ð0Þs achieves the objective value proof, we need to define some parameters. Let xðkÞ be the P of maxfXnew ; Xold g. As a consequence, unanimous wrong vector of xa ðkÞs and X ¼ a xa ð0Þ=N, where xa can be xnew , a decisions cannot happen after COMPARE. xold , or za . aProof. To prove Lemma 5, we need the following lemma. Definition 3 (-convergence time, ). For given 0, the Again, let xa be xnew , xold , or za . a a u t -convergence time Kð; Þ is defined by
LEE ET AL.: DISTRIBUTED THROUGHPUT MAXIMIZATION IN WIRELESS NETWORKS VIA RANDOM POWER ALLOCATION 585 ( # ) 1k kxðkÞ À X~ and RAND-PSEL. The DECIDE step runs two rounds of Kð; Þ ¼ sup inf k : Pr ! 1 À ; ð20Þ xð0Þ kxð0Þk gossip algorithm which requires OðN 3 Þ computations in the worst case . Therefore, our algorithm requires where k Á k is l2 -norm. OðN 3 Þ computations in total. Briefly, the -convergence time is the time until the 5.2 Proof of Theorem 3estimation vector xðkÞ falls into the -neighborhood (in The theorem is proved in three steps. First, in Section 5.2.1, we 1relative sense) of the average vector X~ with high analyze the case of large difference jXnew À Xold j. In parti-probability. cular, we show that a unanimous decision can be easily made,Assumption 1. Fix arbitrary 2 ; 2 2 ð0; 1Þ. Consider positive given that each has obtained a good estimate (-convergence) constants ; ; ; ; and assume the following: ^ ~ of averages. Since any unanimous decision made after COMPARE is the right decision, the probability of right 2 ^ 2 decision is the probability of -convergence in COMPARE. We ^ 0 ; ¼ ;0 2 À 2 N 2 show that this probability is high. Second, in Section 5.2.2, we 1 ~ deal with the case of small difference, where even a good ~ 0 ; ¼ : estimate can possibly result in mixed decisions. The key to N À1 N ~ dealing with this case is that selecting either the new or old Assume further that K ¼ Kð; Þ in COMPARE and K ¼ power allocation is not a bad choice due to jXnew % Xold j. We Kð; Þ in AGREE. show that the AGREE procedure attains a unanimous decision with probability, which in this case implies a fairly Let Xagr be the average objective value achieved by the good choice. Finally, Section 5.2.3 combines these two resultsabove described DECIDE algorithm that runs COMPARE and to show that COMPARE and AGREE will select the powerthen AGREE. It can be proved that this policy satisfies C2 as allocation that achieves almost the maximum of new and oldshown in Theorem 3. objective values with high probability.Theorem 3. Consider any 2 ; 2 2 ð0; 1Þ. Under Assumption 1, the DECIDE algorithm (COMPARE and AGREE) achieves 5.2.1 The Case of Large Difference Let us first delineate between large and small differences. Pr½Xagr ! ð1 À 2 ÞmaxfXnew ; Xold g ! 1 À 2 : Consider an arbitrary 2 ð0; 1Þ, and let ¼ =N. Recall the ^ ^ definition of Kð; Þ in (20). It can be easily shown that under Algorithm COMPARE, for any k ! Kð; Þ, http://ieeexploreprojects.blogspot.comProof. See Section 5.2. u tRemark. As seen above, the -convergence time Kð; Þ is a ^ jxnew ðkÞ À Xnew j Xnew ; 8a 2 V a ð21Þ critical parameter because Condition C2 can be guaran- old ^ jxa ðkÞ À Xold j Xold ; 8a 2 V ; teed after -convergence time in COMPARE and AGREE. It is known that in a line or ring topology, it is given by with probability at least 1 À . Define E 1 as the event that ÂðÀN 2 logðÞÞ . Moreover, in a complete graph, it is (21) is satisfied, under the assumption that K ¼ Kð; Þ in given by Kð; Þ ¼ ÂðÀ logðÞÞ . In wireless networks, Algorithm COMPARE. Then, it 4is obvious that Pr½E 1 ! 1þ^ the topology can be controlled by adjusting the coding 1 À . Define E 2 as the event that Xnew 1À^ Xold or new 1À^ Xold . Then, its complement E C is the event that and transmission rate. That is, if a strong coding is used X 1þ^ 2 1À^ 1þ^ with low transmission rate, then the communication 1þ^ Xold Xnew 1À^ Xold . Note that the event E 2 basically range can be increased (for the purpose of control indicates that the difference between the old and new C signaling only). This will make the topology closer to a average values is relatively large, whereas E 2 indicates C complete graph. In particular, a small network could be that they are fairly close. These two events E 2 and E 2 , made a complete graph. Hence, if this is used for respectively, define large and small differences. In the gossiping, the -convergence time will be substantially following, we will see how these two events affect the improved.3 The convergence time can be further im- performance of our decision policy. proved by exploiting the geographic information. In , Consider a naive policy Å such that each node a decides , geographic gossip algorithms were developed such its power based on its own estimates obtained by running OMPARE, that is, it switches to the new power if xnew ðKÞ that their convergence time is OðNÞ. Clearly, this is order Cold a optimal for network-wide averaging, and therefore, the xa ðKÞ and keeps the old one otherwise. gossiping-based comparison can be a practical solution in Lemma 7. Assume K ¼ Kð; Þ in Algorithm COMPARE. Then, real wireless networks. the policy ÅRemark. We briefly discuss the total overhead of our Pr½XÅ ! maxfXnew ; Xold gjE 1 ; E 2 ¼ 1; algorithm. Recall that our algorithm consists of RAND- POW and DECIDE. In RAND-POW, OðNÞ and Oð1Þ where XÅ is the average objective value of the power vector computations are needed, respectively, for RAND-PAIR, selected under the policy Å. Proof. Given E 1 , (21) holds, and consequently it follows that 3. In fact, the techniques used in ,  for analyzing convergence timeshow that as the number of disjoint paths increases, the convergence speed for all a,increases. Hence, such a topology control will enhance the convergencespeed. More details can be found in . 4. Note that Xnew is a random variable.
586 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 ^ ^ ð1 À ÞXnew À ð1 þ ÞXold Pr½Xagr ! ð1 À 2~=ð1 þ ÞÞmaxfXnew ; Xold gjE C ; E 3 ¼ 1: ~ xnew ðKÞ À xold ðKÞ a a ^ ^ ð1 þ ÞXnew À ð1 À ÞXold : 2 Further, given E 2 , we have Xnew 1þ^ Xold or Xnew 1À^ 1À^ 1þ^ Xold . In the first case, the above sandwich inequality Proof. Note first that given E 3 , the AGREE policy will result implies xnew ðKÞ xold ðKÞ; 8a. Consequently, every a a in agreed decisions. If za ð0Þs were all 0 or 1, then it follows node will select the new power under the policy Å so from Lemma 5 that Xagr ¼ maxfXnew ; Xold g. If this was that XÅ ¼ Xnew . Note that this is the right decision not the case, then every node will select the old power because Xnew Xold in this case. The second case can be given E 3 ; so that Xagr ¼ Xold . Further, given E C , we have 2 proved similarly. u t 1À^ new 1þ^ Xold . Consequently, it follows that 1þ^ Xold X 1À^ The following is a consequence of Lemma 7. 1À~ Xagr ¼ Xold ! 1þ~ maxfXnew ; Xold g. Therefore, the agreed 1þ^ 2~ Corollary 1. Given E C , i.e., if 1À^ Xold Xnew 1À^ Xold , the 2 1þ^ decisions are made achieving at least 1 À 1þ~ of the policy Å can result in mixed decisions even after -convergence. maximum of old and new values. u t Lemma 8 implies that the case of small difference can be Lemma 7 shows that when the difference between Xnew addressed by the AGREE procedure. That is, if the decisions and Xold is sufficiently large, the desired selection can be were unanimous after COMPARE, they are right decisionsmade easily based solely on the COMPARE procedure. On and kept by AGREE. Even if the decisions are mixed, thethe other hand, according to the above corollary, if they are AGREE procedure can guarantee almost the maximum oftoo close, the decisions can be mixed that can possibly lead new and old objective values with high probability.to the violation of C2. Note also that as the accuracy of 5.2.3 Combining All the Resultsestimation increases (i.e., decreases), the region of mixed ^decisions diminishes. We now combine all the above results to complete the proof. For any trivial event A (i.e., event having zero5.2.2 The Case of Small Difference probability measure), assume the convention P ðÁjAÞ ¼ 0So far, we have seen that when the difference is large (i.e., where P ðÁÞ is probability measure. We will use thegiven E 2 ), a unanimous decision can be easily made right following relationship for any events A; B; Cafter COMPARE. Moreover, any unanimous decision is kept P ðAjBÞ ¼ P ðAjB; CÞP ðCjBÞ þ P ðAjB; C C ÞP ðC C jBÞ: ð23Þby AGREE, and hence, Lemma 7 also hold for AGREE. Thissection studies the case where the decisions are mixed after Recall E 1 , E 2 , E 3 are, respectively, the events of - http://ieeexploreprojects.blogspot.comCOMPARE, in particular when the difference is small. convergence in COMPARE, relatively large difference Recall that if za ð0Þs are all 0 or all 1, then the convergence between Xnew and Xold , and -convergence in AGREE. and the right decision are guaranteed immediately under Let Xmax ¼ maxfXnew ; Xold g. First, note thatthe decision Algorithm AGREE (See Lemma 5). Hence, Pr½Xagr ! ð1 À 2 ÞXmax assume that such a case has not happened, so that there is a ¼ Pr½X agr ! ð1 À 2 ÞXmax jE 1 Á Pr½E 1 mixture of nodes with za ð0Þ ¼ 0 and za ð0Þ ¼ 1. Considerany 2 ð0; 1=ðN À 1ÞÞ and let ¼ =N. Then, as argued in ~ ~ þ Pr½Xagr ! ð1 À 2 ÞXmax jE C Á Pr½E C 1 1(21), after K ~ ¼ Kð; Þ iterations in AGREE, every node a ! ð1 À Þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ;will obtain where the inequality follows from the facts that the second ~ ~ ð1 À ÞZ za ðKÞ ð1 þ ÞZ; ~ ð22Þ term is nonnegative and Pr½E 1 ! 1 À . Using the relation- ship (23), the last line can be rewritten aswith probability at least 1 À . Consequently, the above ~ Èinequality implies 0 za ðKÞ 1; 8a since 1=N Z ¼ ð1 À Þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ; E 2 Á Pr½E 2 jE 1 ðN À 1Þ=N and 1=ðN À 1Þ. Hence, every node will ~ É þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ; E C Á Pr½E C jE 1 acknowledge that there are mixed decisions, and hence 2 2 Èthey will decide to use the old power. ! ð1 À Þ Pr½E 2 jE 1 É þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ; E C Á Pr½E C jE 1 Algorithm 5. AGREE 2 2 ~ 1: Run gossiping for K iterations to estimate the average ! ð1 À Þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ; E C : 2 P a2V za ð0Þ=N. The second inequality follows from Lemma 7, and the last ~ 2: Each node a selects the new power if za ðKÞ ! 1, and the inequality follows from the fact Pr½E 2 jE 1 þ Pr½E C jE 1 ¼ 1. 2 old one otherwise. Similarly to the above (where (23) was applied and the Let E 3 denote this event, i.e., E 3 is the event that every second term was removed for proceeding the inequality), ~ node obtains approximation of Z (as in (22)) under the the last line can be rewritten asassumption that K ~ ¼ Kð; Þ. Note that Pr½E 3 ! 1 À . The following lemma shows the performance of AGREE under ! ð1 À Þ Pr½Xagr ! ð1 À 2 ÞXmax jE 1 ; E C ; E 3 Á Pr½E 3 jE 1 ; E C : 2 2some conditions. Recall Pr½E 3 ! 1 À , and this is true regardless of the initial Lemma 8. Let Xagr be the average objective value achieved by value zð0Þ. The events E 1 and E C only affect the initial value, 2 AGREE. Then, hence, the conditional probability Pr½E 3 jE 1 ; E C is also no 2