Quorum-based Localized Scheme for Duty Cycling in Asynchronous Sensor Networks
Jianhui Zhang, Shao-jie Tang, Xingfa Shen, Guojun Dai, Amiya Nayak
aa r X i v : . [ c s . N I] N ov Quorum-based Localized Scheme for DutyCycling in Asynchronous Sensor Networks
Jianhui Zhang ∗‡ , Shao-jie Tang † , Xingfa Shen ∗ , Guojun Dai ∗ , Amiya Nayak ‡∗ College of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou 310018 China. † Department of Computer Science, Illinois Institute of Technology, Chicago, IL60616, USA ‡ SITE, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Abstract — Many TDMA- and CSMA-based protocolstry to obtain fair channel access and to increase channelutilization. It is still challenging and crucial in Wire-less Sensor Networks (WSNs), especially when the timesynchronization cannot be well guaranteed and consumesmuch extra energy. This paper presents a localized and on-demand scheme ADC to adaptively adjust duty cycle basedon quorum systems . ADC takes advantages of TDMA andCSMA and guarantees that (1) each node can fairly accesschannel based on its demand; (2) channel utilization canbe increased by reducing competition for channel accessamong neighboring nodes; (3) every node has at least onerendezvous active time slot with each of its neighboringnodes even under asynchronization. The latency bound ofdata aggregation is analyzed under ADC to show thatADC can bound the latency under both synchronizationand asynchronization. We conduct extensive experimentsin TinyOS on a real test-bed with TelosB nodes to evaluatethe performance of ADC. Comparing with B-MAC, ADCsubstantially reduces the contention for channel access andenergy consumption, and improves network throughput.
Index Terms —Duty Cycle; Quorum Systems; MediaAccess Control; Wireless Sensor Networks
I. I
NTRODUCTION
WSNs have been applied in various environmentssuch as ecological surveillance [1]. Because of hard-ware limitation, sensor nodes have limited energy andunprecise clocks. Various approaches have been designedto save energy and improve some network performanceson throughput, delay and per-node fairness. In order toachieve good cooperation among nodes, synchronizationprotocols, e.g FTSP [2], were proposed but considerableenergy and time were consumed especially in large scalenetworks as well. How to design protocols to guaranteethe communication among nodes under the asynchronousnetwork becomes a very critical and challenging prob-lem. Media Access Control (MAC) protocols let nodesto know when and how to access common channels [3].Some popular MAC protocols, such as TDMA- andCSMA-based, were designed to share communicationmedium among nodes by assigning each node some fixedactive time slots in TDMA or by letting nodes locallycontest their channel access in CSMA. Both of two typesof protocols try to build a physically connected network
This paper was published on IEEE 8th International Conferenceon Mobile Ad hoc and Sensor Systems (MASS, 2011) while controlling nodes’ active time period in orderto reduce energy consumption and improve networkthroughput.TDMA has the advantage that time slots are previ-ously scheduled to each node. Therefore, a network canachieve high channel utilization under high media accesscontention and reduce collision among neighbors witha low cost when their clocks are well synchronized.But TDMA also has some disadvantages [3], some ofwhich are caused by clock asynchronization. B-MAC [4]adopts Low Power Listening (LPL) to solve communica-tion failure caused by clock asynchronization. AlthoughCSMA doesn’t austerely require the clock synchroniza-tion, it cannot achieve channel utilization as high asTDMA and costs additional time and energy on channelaccess contention. Thus some hybrid MAC protocols,such as B-MAC [4], S-MAC [3] and T-MAC [5], com-bining the strengths of both TDMA and CSMA, wereproposed. These MAC protocols essentially adopt theLPL technique or improved LPL to allievate localizedasynchronization problem. However, they cannot avoidchannel contention and obtain channel allocation fairnessin many scenarios [6]. Thus another challenging problemis to decrease the unfair contention for medium accesswithout synchronization while increasing the channelallocation fairness.This paper designs a localized scheme, named Adjust-ment of Duty Cycle (
ADC ), based on
Quorum Systems (QS) [7], to adaptively adjust the duty cycle of eachnode. A QS is a set of subsets of a universe set suchthat every pair of subsets intersect with no empty. Inrecent years, QS is applied to establish control chan-nels in dynamic spectrum access networks [8], to savepower [9], and to maximize throughput in limited infor-mation multiparty MAC [10]. By
ADC , each node canselect sufficient amount of active time slots, composinga set ς ⊂ T (called a quorum) according to the amountof its demand while it can sleep to save energy at itsrest time in a period. Therefore, its duty cycle | ς | / | T | isadaptively adjusted when the amount of active time slots | ς | is changed. Each node will inform its neighbors ofa quorum it selected thus the channel contention amongthem is decreased. The contributions of this paper are asfollows: • ADC can adaptively adjust duty cycle by demand, and increase the channel allocation fairness comparing withexisting contention-based MAC protocols. • ADC guarantees each pair of nodes having sufficientrendezvous active time to implement demand, and theworst case of channel utilization is lower bounded. • By the rotation closure property and intersectionproperty of QS, the successful connectivity of a wholenetwork is guaranteed even under asynchronization sono extra energy is consumed on synchronization. • This paper analyzes the performance of
ADC underdata aggregation, and derives the impact of QS load onnetwork delay, which is defined as a duration from onemoment some data is sampled to another that all data isreceived by the sink.The organization of this paper is as follows. We firstgive the network model, formulate our problem andintroduce the QS technology in Section II. In Section III,we design our protocol
ADC and address its preliminaryproperties. Under clock synchronization and asynchro-nization, the performances of
ADC are presented whencertain demand is implemented in Section IV and Vrespectively. In Section VI, we implement our scheme
ADC in a real test-bed consisting of TelosB nodes andevaluate its performance on real systems. Section VIItells the relative work in recent years. The work of paperis concluded in Section VIII.II. S
YSTEM M ODEL AND P RELIMINARY
A. Network Model and Quorum System
We formulate a network by a graph G ( V, E ) , where V (or E ) is a set of all nodes (or edges). Let n denote thenumber of total nodes and s denote an only sink in thenetwork. Each node is assigned a unique ID. A radius ofthe network G with respect to s , denote by R , is definedas the maximum distance (hops) between s and thosenodes in G . This paper studies the duty cycle adjustmentunder several popular interference models (denoted by I ): RTS/CTS, protocol model and physical model [11].QS, denoted by Ω , was used and introduced in pre-cious papers [12][13][8]. A QS Ω ⊂ T , containingquorums, denoted by Q , is a set of subsets ς of T , where T = { τ , · · · , τ m } is a period and composes of m timeslots. A rotation of a quorum Q is defined as S ( Q , i )= { ( τ j + i ) mod m | τ j ∈ Q } , where i is a non-negativeinteger. Some QSs satisfy the rotation closure property , i.e. ∀ i ∈ { , · · · , m − } : Q ∩ S ( Q , i ) = ∅ , where Q , Q ∈ Ω . Lemma 1:
Grid, torus and cyclic QS all satisfy the rotation closure property [13].
B. Problem Statement
Two neighboring nodes u and v can communicate witheach other in WSNs iff they have at least one rendezvous active time slot. When a network is asynchronous, i.e. theclock of each node u has a shift t uδ ≥ from real time,the set of u ’s active time slots ς u accordingly has a shift, i.e. ς ′ u = ς u + { t uδ } . The following equation should besatisfied if a pair of neighboring nodes can communicatewith each other under asynchronization. ∀ u, v ∈ V and v ∈ ϑ u : ς ′ u ∩ ς ′ v = ∅ , ς ′ u , ς ′ v ⊂ T (1)where ϑ u is a communication set centered at a node u and a set containing u and those nodes in its communica-tion range. Here, we call a pair of nodes as neighboringwhen they respectively belong to the communication setof each other. Equation (1) means a pair of neighboringnodes can be physically connected even under asyn-chronization if their active time slot sets are properlydesigned.We define a parameter demand D to indicate theamount of data needed to transmit or receive in unit time.Notice that Equation (1) indicates a pair of neighboringnodes should have not only common active time butalso enough time to finish all of its demand. This paperaims to deriving a demand condition so that each node u can implement its demand by locally choosing a subset ς ′ u ⊂ T to guarantee each pair of nodes in ϑ u satisfyingEquation (1). To obtain the purpose, this paper designsthe localized duty cycle adjustment scheme ADC .III. Q
UORUM S YSTEM BASED T IME S LOT A SSIGNMENT
This section presents our designing of
ADC andanalyzes its properties. These properties indicate
ADC can achieve better solution than existing protocols on fairmedium access even under asynchronization as describedin Section I.
A. Designing of
ADC
ADC lets each node u obtain a time slot set ς u so that three following conditions can be satisfied: ① Equation (1); ② Demand condition; ③ The active timeis minimized to save energy.
ADC consists of two steps. The first step is to designquorums in a grid QS, denote by Ω g . In the second step,each node locally selects its quorum base on one-hopinformation about selected quorums.At the first step, we construct a grid QS Ω g with thesize ⌈√ m ⌉ × ⌈√ m ⌉ based on a period T as shown inFigure 1. The time slots from T are allocated into thegrids of Ω g from right to left in a row-major manner asshown in Figure 1. In each period, node u requires a setof time slots ς u to afford of its demand D u . The relationbetween the cardinality of ς u , denoted by κ u , and D u is κ u = D u | T | ̺ , where ̺ is the data rate. Thus we design Fig. 1. A grid QS Ω g contains T . There are ⌈√ m ⌉ rows andcolumns in Ω g .
123 1 2 3 41 2 3 4 5 67 8 9 10 11 1213 14 15 16 17 1819 20 21 22 23 2425 26 27 28 29 3031 32 33 34 35 365 6456 Q u Q v Fig. 2. Q u and Q v overlap at th line and column. a quorum Q u for u and Q u contains ⌈ D u √ m ̺ ⌉ rows andcolumns. Rule 1 [Quorum Design] : The quorums Q i ∈ Ω g areorganized by from i th to i + ⌈ D u √ m ̺ ⌉ th rows and from i th to i + ⌈ D u √ m ̺ ⌉ th columns, where the row and columnnumbers are labeled as shown in Figure 1 and i ∈ Z + and i ≤ √ m + 1 − ⌈ D u √ m ̺ ⌉ .At the second step, we design a quorum selectionmethod for each node as described in Algorithm 1, inwhich “a quorum Q i is occupied” means that | Q i ∩ Q j | > ⌈ D i D j m ̺ ⌉ , i = j when Q i is selected earlier than Q j .Figure 2 shows that Q u is occupied by Q v if Q u isselected earlier than Q v . Notice that the parameter K inAlgorithm 1 will be discussed in Lemma 7 and 8. Itsvalue is determined in advance. Algorithm 1
Quorum Selection
Input : Ω g , which is allocated time slots from T . Output : Each node v ∈ ϑ u is allocated a quorum. v ∈ ϑ u sets a positive natural k = 1 ; v sets a list L o storing the quorums occupied byothers. while ϑ u = ∅ do v randomly selects a quorum Q i with equal prob-ability, where i = 1 , · · · , ⌈√ m + 1 ⌉ − ⌈ D u √ m ̺ ⌉ ; if Q i was occupied and k ≤ K then Add Q i into L o ; v selects another quorum Q j , i = j and Q j / ∈ L o ; k + = 1 ; end if Set k = 1 and delete v from ϑ u . end while v informs its neighbors of its quorum in a message. B. Properties of
ADC
The properties of
ADC includes physical connectivity , maximum demand and maximum load . We analyze theeffect of ADC on network connectivity when the clocksare either synchronous or asynchronous.
The physical connectivity.
Physical connectivity is thepreliminary condition under which nodes can commu-nicate with each other and implement their demand.In this paper, physical connectivity means the time-to-rendezvous , which is the amount of rendezvous slotsbetween arbitrary nodes u and v . If a node v ∈ ϑ u and u are physically connected, the time-to-rendezvousbetween them should be at least one slot.By ADC , the physical connectivity under bothclock synchronization and asynchronization are different.When the clock is synchronous, we can easily obtainLemma 2. For example, two quorums Q u and Q v respectively contain one row and one column in Figure 1.They rendezvous at the time slot 11 and 26. Thus theyare physically connected and have two rendezvous activetime slots. If one node (for example v ) chooses a usedquorum Q u as the instance in Line 5 of Algorithm 1,then two quorums (for example, Q v and Q u ) have morethan two rendezvous time slots. Lemma 2:
If two arbitrary nodes u and v ∈ ϑ u arerespectively allocated quorums Q u and Q v according toAlgorithm 1, they have at least ⌈ D u D v m ̺ ⌉ rendezvousactive time slots when clocks are synchronous.When clocks are asynchronous, a node u is prone tohave clock shift t uδ , which is the difference between thelocal time of u ’s clock and the exact time. Therefore, therelative clock shift between u and v is t δ ( u, v ) = t uδ − t vδ .This paper always assumes t uδ , t δ ( u, v ) < + ∞ . Lemma 3:
Any pair of quorums in the same QS Ω g must have at least ⌈ D u D v m ̺ ⌉ rendezvous active time slotseven when the relative clock shift between any pair ofnodes is an arbitrary value. Proof:
A grid QS Ω g satisfies the rotation closureproperty according to Lemma 1. Thus any two quorums Q u and Q v satisfy Q u ∩ S ( Q v , i ) = ∅ , where i =0 , · · · , m − , when Q u , Q v ∈ Ω g . For any pair of nodes u and v , there is relative clock shift t δ ( u, v ) because ofthe clock asynchronization. Without loss of generality,let the rotation of Q v be S ( Q v , t δ ( u, v )) . It means thatthe slots in Q v shift because of the relative clock shift.Thus S ( Q v , t δ ( u, v )) ∩ Q u = ∅ for i = 0 , · · · , m − if v ∈ ϑ u since t δ ( u, v ) mod m is positive and not biggerthan m − . It means u and v are physically connectedbecause a period totally contains m time slots.Next we look for a lower bound of the cardinality of S ( Q v , t δ ( u, v )) ∩ Q u . In ADC , any two quorums in thesame QS has at least ⌈ D u D v m ̺ ⌉ rendezvous active timeslots according to Theorem 3 of [8] and Lemma 2, i.e. | S ( Q v , t δ ( u, v )) ∩ Q u | ≥ ⌈ D u D v m ̺ ⌉ .Notice that Lemma 3 is obtained without consideringthe interference since our scheme ADC is applied toMAC. Thus if | t δ ( u, v ) | mod m =0 for any pair of nodes u and v , then the physical connectivity of ADC under theclock asynchronization is same with that under the clock synchronization, and the same result can be obtainedon the parameter load. In the subsection IV-A, we willanalyze the effect of the interference.Notice that other kind of QSs are also applicable in
ADC according to Lemma 1 in spite that clocks aresynchronous or asynchronous. If a QS should satisfy therotate closure property, the quorums in the QS shouldsatisfy the condition in Lemma 4.
Lemma 4:
If a QS Ω satisfies the rotate closure prop-erty, then the cardinality of any quorum in Ω must bemore than √ m [13]. The maximum demand.
Here the demand conditionis given so each node u can afford its demand D u .It is easy to know D u ≤ ̺ if the demand D u is im-plementable. When the interference models I presentedin the subsection II-A are considered, the demand D u cannot necessarily be close to ̺ . The summation of allnodes in the same communication set ϑ u should satisfythe condition in Lemma 5 if all of the demand of nodesin ϑ u can be implemented. Before we give out Lemma 5,we introduce a constant c ( I ) which is determined by theinterference model I . We calculate c ( I ) by the techniqueof vertex coloring. The vertex coloring means to colorall nodes with minimal number of colors under I . Thusnodes with same color are interference-free under I . Lemma 5:
When demands of all nodes are imple-mentable, the demands of nodes belonging to the same ϑ u should satisfy P v ∈ ϑ u D v ≤ ̺c ( I ) , where c ( I ) is aconstant related with the interference model I . Proof:
When no interference is involved, the nodesin ϑ u share a period. So P v ∈ ϑ u κ v ≤ T ⇒ P v ∈ ϑ u D v T̺ ≤ T ⇒ P v ∈ ϑ u D v ≤ ̺ . Under the interference model I , eachcommunicate set can transmit or receive in every c ( I ) periods in order to be interference-free. The average datarate is ̺c ( I ) . So P v ∈ ϑ u D v ≤ ̺c ( I ) .We find it is not always suitable to decrease themaximum demand of each node since a node shouldkeep active in at least √ m time slot to satisfy the rotateclosure property according to Lemma 4. Hence demandof a node u should have a lower bound. Since the Q u contains ⌈ D u √ m ̺ ⌉ rows and a column, the lower boundcan be obtained by Lemma 6. Lemma 6:
When the Ω g satisfies the rotation closureproperty, the demand of each node should satisfy D u ≥ c ̺ , where c = 1 − p − / √ m , when m > . Proof:
For an arbitrary node u , the cardinality of Q u is √ m ⌈ D u √ m ̺ ⌉ − ⌈ D u √ m ̺ ⌉ × ⌈ D u √ m ̺ ⌉ . By Lemma 4,we have √ m ⌈ D u √ m ̺ ⌉ − ⌈ D u √ m ̺ ⌉ × ⌈ D u √ m ̺ ⌉ ≥ √ m . ⇒ ⌈ D u √ m ̺ ⌉ − ⌈ D u ̺ ⌉ × ⌈ D u √ m ̺ ⌉ ≥ . When m = 1 ,the above inequality can be always satisfied because D u /̺ ≤ . When m > , D u ≥ c ̺ , where c = 1 − p − / √ m .Let D k = min v ∈ ϑ u D v so D k ≥ c ̺ according toLemma 6. Thus we have P v ∈ ϑ u D v ≥ c | ϑ u | ̺ . The load bound.
When the condition in Lemma 5 issatisfied, there still exists competition between a pair ofnodes u and v in the same ϑ , where the competitionbetween them because | Q u ∩ Q v | > ⌈ D u D v m ̺ ⌉ , u = v ,as described in Algorithm 1. Notice that the parameter K in Algorithm 1 is used to decrease the competition,where K is the number of times in which the same nodeselects different quorums. It is important to take fulladvantage of the time diversity of medium access, i.e. todecrease the competition between different quorums, inorder to control the channel congestion. Some previouswork designed protocol to minimize the load [8]. But itis not suitable under a more practical case in this paper.That is each node has demand different from others’because of their different network tasks. Furthermore,we show that it cannot fully use the time slots whenthe demand is very low, under which it will degrade thechannel utilization to decrease the load. In ADC , wepresent the upper- and lower-bound of the load underthe demand constraint given in Lemma 5 and 6.We first give out a lower bound of the quorum load.Two propositions 4.1 and 4.2 in [14] gave a result that L (Ω) ≥ max { c (Ω) , c (Ω) m } , where c (Ω) is the size ofsmallest quorum in Ω , and c (Ω) ≥ √ m according toLemma 6. We then have c (Ω) ≤ √ m and c (Ω) m ≥ √ m .Therefore, L (Ω) ≥ c (Ω) m , which is different from theresult in Theorem in [8], because the rotation closureproperty is considered.Next we discuss c (Ω) . Notice that the cardinality of Q u should be not less than √ m when the demand of eachnode D u ≥ c ̺ . Otherwise, the rotation closure propertycannot be satisfied. Thereinafter, we analyze the boundof the load in two case: K = 1 and K ≥ .In Lemma 5, an obvious upper bound of the QSload appears when a node is required to afford the fulldemand. i.e. , the demand D v of a node v ∈ ϑ u is not lessthan the maximal data rate, D v ≥ ̺ . Under this case, thecardinality of each quorum is m . Because each node ran-domly selects a quorum with equal probability and theprobability that each time slot is included in a quorumis | ϑ | , L S ( i ) = P Q ∈ Ω: τ i ∈ Q P S ( Q ) = P Q ∈ Ω: τ i ∈ Q | ϑ | = 1 , L S (Ω) = 1 . Under ADC , the cardinality of each quorumis not bigger than m and the probability that eachtime slot is included in a quorum is less than | ϑ | . So L S (Ω) ≤ . Lemma 7: In ADC , the load of Ω g is less than c | ϑ u | , i.e. , L S (Ω g ) ≤ c | ϑ u | , when the QS Ω g satisfies therotation closure properties and K = 1 , where c = ⌈ c ⌉ − ⌈ c ⌉ . Proof:
Since the cardinality of a quorum Q v ( v ∈ ϑ u ) is | Q v | under ADC , the probability that a time slotis included in Q v is | Q v | m . Suppose there are γ quorumsin each Ω . | ϑ u | ≤ γ ≤ √ m when the summation of allnodes’ demand in the same ϑ u is implementable. So theprobability that a quorum is chosen by a node is √ m ≤ γ ≤ | ϑ u | . Therefore, the probability that each time slot isincluded in the quorum Q v when there are | ϑ u | quorumsis | ϑ u | × | Q u | m . The load induced by the strategy S ona time slot τ i is L S ( τ i ) = P Q u ∈ Ω g : τ i ∈ Q u | ϑ u | × | Q u | m .Notice there are | ϑ u | nodes to select quorums from Ω g , i.e. Ω g = { Q v : Q v ∈ ϑ u } . So L S ( τ i ) = 1 γm X Q u ∈ Ω g : τ i ∈ Q u | Q u |≤ | ϑ u | m X τ i ∈ Q u ∈ Ω g { √ m ⌈ D u √ m ̺ ⌉ − ⌈ D u √ m ̺ ⌉ } = 1 | ϑ u | X τ i ∈ Q u ∈ Ω g {⌈ D u ̺ ⌉ − ⌈ D u ̺ ⌉ } ≤ | ϑ u | ( ⌈ c ⌉ − ⌈ c ⌉ ) (2)Thus the load induced by the strategy S on the quorumsystem Ω g is L S (Ω g ) = max τ i ∈ T L S ( τ i ) ≤ | ϑ u | ( ⌈ c ⌉ −⌈ c ⌉ ) = c | ϑ u | , where c = ⌈ c ⌉ − ⌈ c ⌉ .Now we analyze the upper bound of the QS loadwhen K ≥ . Here we treat a quorum containing a rowand a column and call the quorum as a bin, so eachnode actually selects several such bins according to the Rule 1 . Each node has K times to select its quorumin Algorithm 1. Thus the quorum selection problem inAlgorithm 1 is equivalent to the k -round ball placementproblem [15]. When m i = 1 , we can obtain that themaximum load achieved by Algorithm 1 is less than log log √ m log K w.h.p according to Theorem 6 of [16]. When m i ≥ in each round, i.e. several bins are selectedtogether in each round, it is equivalent to combiningseveral bins into one. Therefore, the total number of binsis correspondingly reduced. We can obtain Lemma 8. Lemma 8:
The maximum load achieved by Algo-rithm 1 is less than log log √ m log K w.h.p .IV. S YNCHRONOUS D EMAND I MPLEMENTATION
This section evaluates the performance of
ADC whenthe data aggregation is implemented and clocks aresynchronous, which we call as synchronous demandimplementation. In order to implement the demands, weconstruct a tree and design the specific demand imple-mentation methods. The performance of
ADC under theasynchronization will be analyzed in the next section byapplying the results of this section.
A. Tree Construction
Firstly, a tree T is constructed based on G byconstructing a connected dominator set (CDS). we thendefine a new conception region in order to obtain aconflict-free quorum assignment for each node.We construct a CDS by the breadth-first-search (BFS)based on G . Each dominatee connects with the domi-nator closest to it. In this way, T , rooting at s , can beconstructed and is ranked into L levels from s . Thelevel of s is labeled l . The parent and the children of anode u are denoted by p ( u ) and c ( u ) respectively.We assign QSs for each region in two phases. At thefirst phase, a new conception region is defined to obtainthe conflict-free partition by the vertex coloring. At thesecond phase, we assign each region with a period sonodes can be active without confliction. Phase I:
After T is constructed, each node can knowits own level and its one-hop neighbors’ IDs and levels.The one-hop neighborhood of each node u is denotedby N ( u ) and notice that u ∈ N ( u ) . We call a one-hop neighborhood of a non-dominatee node as a region (denoted by σ ) in the tree T . Notice that any dominateedoes not form a region. Because of interference within anetwork, the QSs of some neighboring regions cannot beassigned a same time slots set. Here, we say two regions σ and σ are neighboring (or over-lap ) if there are twonodes u ∈ σ and v ∈ σ and u (or v ) locates in theinterference range of v (or u ). xu v (a) u vx y (b)Fig. 3. (a) σ u and σ v are two over-lap regions. (b) σ u and σ v aretwo neighboring regions. If two regions are conflict-free, i.e. they are notneighboring and over-lap, we color them with a samecolor. So it is vertex coloring problem to find the minimalnumber of colors. The least number of colors, denotedby c , necessary to color all regions, is affected by theinterference model I , i.e. we need at least c ( I ) to colorall regions and the regions with same color are conflict-free. We label each region σ u with a color index θ σ u , θ σ u ∈ C = { , · · · , c ( I ) } . In order to determine theleast number of colors, we define a parameter (denotedby ϕ ) to denote the interference range under differentinterference models. Thus any pair of regions can havesame color if they are more than η hops apart. Phase II:
Allocate each region with a QS. Because thequorum is designed in the subsection III-A, each nodebelongs to at least one region and some belong to several regions. For example, x belongs to two regions σ u and σ v in Figure 3(a). Thus the number of quorums eachnode occupies is same with the number of regions itbelongs to. We assign each region with a period, i.e. aQS, according to the color so each neighboring or over-lap regions can be conflict-free. For example, σ u and σ v are respectively assigned two periods T u and T v . Let T u = { , , } and T v = { , , } so T u ∩ T v = ∅ .We use a natural number i ( i ∈ Z ) to label the IDof a region in order to assign the time slot set conve-niently. Firstly, we color all the regions by Algorithm 2.Secondly, each region i is assigned a slot set L i byAlgorithm 3. Algorithm 2
Region Coloring
Input : The labels of all regions, σ i , i = 1 , · · · , | CDS | and the color set C . Output : The colored regions. for i = 1 , · · · , | CDS | do if There is no region colored within η hopscentered at a region σ i then σ i labels itself with a color θ σ i ( ∈ C ). else σ i selects a color θ σ i from C and θ σ i isdifferent from the colors of other nodes within η hops centered at σ i ’s. end if end forAlgorithm 3 Slot Set Assignment
Input : All the colored regions, θ σ i , i = 1 , · · · , | CDS | and θ σ i ∈ CDS . Output : Each region σ i obtains a slot set L σ i . for j = L , · · · , and σ i ⊂ l j do for i = 1 , · · · , | CDS | do A region σ ji colored with a color index θ σ ji isassigned the slot set L σ ji = θ σ ji mod ϕ × κ . if L σ ji > max all c ( i ) L σ ( j +1) c ( i ) then S ( L σ ji , m ) . end if end for end for After each region is assigned a period, it can obtainits quorum according to Algorithm 1. We can designa determinate quorum selection method rather than therandom one in Algorithm 1. The determinate quorumselection method is given in Algorithm 4. The differencebetween Algorithm 1 and Algorithm 4 is that each nodedoesn’t select quorums randomly. By Algorithm 4, wecan obtain some properties. Here we give a notion: logical connection . Nodes u and v are logically connected if they locate in eachother’s transmission range. If u and v cannot be active incommon time slots, they cannot communicate with eachother. A graph can be logically connected by topologycontrol algorithms. Thus T is logically connected if G is. According to Algorithm 4, it is easy to know thateach pair of nodes are physically connected in T if G is logically connected. Algorithm 4
Determinate Quorum Selection Each node u collects the level label and quorums ofits neighbors in N ( u ) ; u classifies its neighbors into three sets: S , S and S . S contains the node in l k − and S containsthe nodes in l k and S contains the nodes in l k +1 ; if u is the sink then S = ∅ ; end if for i = 1 , · · · , w do if Q i is not chose then According to the order S , S and S , eachnode in S , S and S chooses Q i . end if end for Lemma 9:
Each pair of neighboring or over-lap re-gions, and each pair of links in a same region areconflict-free according to Algorithm 4.
Proof:
According to Algorithm 2, any neighboring(or overlap) regions are colored different colors. In Algo-rithm 3, the regions σ i and σ j colored different colors areassigned different slot set L σ i and L σ j . L σ i ∩ L σ j = ∅ according the line 3 in Algorithm 3. In Algorithm 4, eachnode is assigned a quorum different from that of othersin the same region. According to the definition of QS,there are only two nodes to be active simultaneously. Lemma 10:
If a node u (expect s ) and its parent p ( u ) are respectively assigned the time slots τ u and τ p ( u ) , then τ u < τ p ( u ) according to Algorithm 4.In Algorithm 3, the active period of each node isearlier than that of its parent. According to Algorithm 4,the node in the level l i is active earlier than that in l i − in the same region. Thus we obtain Lemma 10. It is easyto obtain that each parent will transmit after it receivesall packets from all of its children in the same period. B. Data Aggregation
This section discusses performance of
ADC underdata aggregation. Under data aggregation, the demand ofa parent is the summation of its children, i.e. D p ( u ) = P u ∈ σ p ( u ) D u . We can obtain that the maximal size of a QS among all QSs is determined by the maximal degree of T so we have Lemma 11. Lemma 11: max | Ω | = max u ∈ T d u .The delay of data aggregation is given in Theorem 12. Theorem 12:
The maximal delay of data aggregationis O ( ϕmR + ∆ ) by Algorithm 2 and 4. Proof:
Each region σ i is assigned a QS Ω i so it costs | Ω i | to finish all the transmission in σ i . Since | Ω i | ≤ max | Ω | according to Lemma 11, max | Ω | = max u ∈ T d u =∆ . There exists an assignment method such that thenumber of different color is at most ϕ in each level ofthe tree T . Let each region σ i be assigned a slot set L σ i .Thus it needs at most ϕ L σ i in each level. Because thereare totally L levels and the lower level l i is active earlierthan the higher level l j ( i > j ), the sink costs ϕ L σ i L time to collect all data. Because L σ i ≤ m , L ≤ R and L σ i ≤ m , ϕ L σ i L ≤ ϕmR , the maximal delay of thedata aggregation by our method is ϕmR + ∆ .Any schedule has delay at least R (or D ), where D is the radius of a network [17]. When s locates at thecenter of a topology, the delay lower boundary can bereduced to be ( ϕm + 1) D + ∆ , where D = R/ [17].V. A SYNCHRONOUS D EMAND I MPLEMENTATION
ADC does not require the global clock synchroniza-tion. This section aims to analyze the delay of dataaggregation under asynchronization. Existing algorithmsare designed to bound the delay, such as time slot as-signment algorithm in [18]. However, neighboring nodesmay not be physically connected under asynchronization.Thus an additional method is given to ensure each pairof neighboring nodes are physically connected in thesubsection V-B.
A. Asynchronous Delay of Data Aggregation
We assume that the clock shift t uδ ( t ) of a node u randomly and uniformly distributes in the interval [ −∞ , ∞ ] , where t u and t are local time and exact time.When u selects a quorum Q u , u is actually active in S ( Q u , t δ ( t )) because of the clock shift t δ . Accordingto Algorithm 3, every region is assigned a period in every ϕ periods. Without loss of generality, u ∈ σ u is active ina period T u + iϕ , where i = 1 , , · · · . For a pair of neigh-boring nodes u and v , the relative clock shift betweenthem is t uvδ ( t ) = t u − t v . Thus u and v have commonactive time slots only if S ( Q v , t uvδ ( t )) ∩ Q u = ∅ , where Q u ∈ T u + iϕ . We have the following lemma. Lemma 13: If u and v locate in the same region and i − < t uvδ ( t ) T ϕ < i + 1 , where i = 1 , , · · · , u and v arephysically connected.According to Lemma 13, nodes may not be physi-cally connected because the data aggregation scheme is adopted under the clock asynchronization. Notice that itis not caused by ADC .Asynchronous clock causes additional delay on thedelay of data aggregation in order to ensure each pairof u and v could have common active time slots tocommunicate with each other in T u under asynchro-nization. That means u and v have to postpone theircommunication because of the clock shift t uvδ . But theycan communicate with each other within at most ϕ − additional periods delay if they are physically connectedaccording to Lemma 13. Therefore, we can obtain thedelay of data aggregation based on Theorem 12 asillustrated in the following lemma. Lemma 14:
The delay of data aggregation is O ((2 ϕ − T R + ∆ ) under asynchronization if each pair ofneighboring nodes are physically connected. B. Quorum Share
Although Lemma 3 ensures each pair of nodes(including non-neighboring) are physically connected,Lemma 13 indicates each pair of neighboring nodesmust be unable to communicate with each other in someperiods since Algorithm 3 assigned each region withdiscontinuous periods. This section designs a schemeto solve this problem. Suppose σ u is assigned periods T i , i = 0 , ϕ, ϕ, · · · , and the clock shift of a node v ∈ σ u is t vδ and Q v contains the time slots set ς v .When ς ′ v = ς v + { t vδ } locates in the periods whichdoes not satisfy the inequality in Lemma 13, v conflictswith some nodes, i.e. the quorum Q v shifts into theQS of some other nodes x i , i = 1 , , · · · , ≤ | ϑ x | , inanother region σ x , where v locates in the interferencerange of x i . Notice that there are two kinds of quorumshifting: (1) a quorum Q v only shifts between periodswhen t δ mod | T | = 0 ; (2) a quorum Q v shifts among aQS when t δ mod | T | 6 = 0 .In our scheme described in Algorithm 5, v and x i , i = 1 , , · · · , ≤ | ϑ x | share the quorum Q v with equalprobability. Our scheme can deal with two kinds ofquorum shifting. Algorithm 5
Quorum Share Each node v sets a list L t storing the nodes’ ID,which occupy v ’s quorum Q v . while v detects that Q v is occupied by the nodes notin its one-hop neighborhood. do v sets a positive natural number k to be k + = 1 ; v sets itself to be active in periods T i , where i =0 , kϕ, kϕ, · · · ; end while By Algorithm 5, each pair of neighboring nodes arephysically connected even when the time slot allocation algorithm is implemented in Algorithm 3. Accordingto Algorithm 5, u would be active in T i , where i =0 , kϕ, kϕ, · · · . It means that u is active in part ofperiods. Since the grid QS satisfies the rotation closureproperty, u is still physically connected with its neigh-boring nodes. Therefore, we can obtain Lemma 15. Lemma 15:
A pair of neighboring nodes can be phys-ically connected by Algorithm 5 under asynchronization.
Proof:
Suppose a pair of neighboring nodes u and v respectively select quorums Q u and Q v . Thus Q u ∩ Q v = ∅ . Q u and Q v respectively contain the time slot set ς u and ς v . Denote the relative clock shift between u and v is t uvδ . Thus ς ′ u = ς u + { t uvδ } . Since the grid QS, ς ′ u ∩ ς v = ∅ .In Algorithm 5, each node u would be active inpart of periods under asynchronization. We find thatthe whole network delay is prolonged while each pairof neighboring nodes are guaranteed to be physicallyconnected. The demand of u would be implementedlingeringly because the quorum Q u of u moves to Q ′ u in another period as shown in Figure 4. We suppose Q ′ u locates in the period T and T is originally assignedto the nodes in the region σ . So u would share thesame quorum with some nodes in σ . At the worst case,all nodes in σ shunt one turn, i.e. they are active in , ϕ, ϕ, · · · . The delay caused by the clock shift is atmost t uδ periods when there is only u which has clockshift under the interference model I . When the clocksof every nodes shift, their quorum may also shift. Forexample, Q i and Q j respectively move to new places,such as Q ′ i and Q ′ j , which locate in different periodsin Figure 4. The worst case is that ϕ − regions shiftinto one regions, thus the additional delay is at most ϕ ( ϕ − m . According to Theorem 12, we have thefollowing lemma. Lemma 16:
By Algorithm 5, the delay on the dataaggregation is at most O ( ϕ mR + ∆ ) when the clocksare asynchronous. T T T Q u Q v T (cid:77) u Q (cid:99) v Q (cid:99) t (cid:71) t (cid:71) t (cid:71) t (cid:71) Fig. 4. Quorums shift because of the clock shift.
VI. P
ERFORMANCE E VALUATION
This paper evaluated
ADC and B-MAC in a realtestbed running TinyOS on TelosB motes. The testbedcomposes of one hundred nodes. We compare the per-formance
ADC against B-MAC on the throughput andpacket receiving ratio (PRR).
A. Experiment Setup
We randomly deployed 100 nodes on an in-door test-bed. Each sensor node works with its modified internal antenna, the transmission range of which can be assmall as 10cm. Thus nodes in the original network canstill communicate with each other by multi-hop. Afterdeployment, we start our experiment, composing of twophases. At first phase, all the nodes are initially set with100% duty cycle. At the second phase, nodes in a sameregion selected their quorums according to their locations(parent or leaf node) in the tree and the number of theleaf nodes under
ADC . The duty cycle is set 20% underB-MAC.Under
ADC , each Ω contains 100 time slots, i.e. m =100 . Each time slot is respectively set as ms , s , s and s . Each node samples data in every ms , ms , ms , ms , ms , ms , ms , s , . s and s , which are called as the data generation period inFigure 5 and 6. When the experiment starts, the sinkbroadcasts a message to synchronize the clocks of allnodes. B. Performance Comparison
In this section, we compare the performance of B-MAC and
ADC on the network throughput and PRR.Although we care about the channel utility, fairnessand energy consumption, the experiment results on twoparameters, throughput and PRR, synthetically reflect thechannel utility.
Throughput . Figure 5 shows the network throughputrespectively under
ADC and B-MAC. Each node exceptthe sink generates data at different rate. Because thenodes should compete the channel access when transmit-ting each packet, much time is wasted. When the timeslot size is big, for example, 1s, 2s and 5s, the nodesshould cost time on the channel access competition andany pair of neighboring nodes have much continual timeto transmit packets. As shown in Figure 5(b), 5(c) and5(d), the throughput under B-MAC is much lower thanthat under
ADC when the time slot size is bigger, suchas s , s and s . Although the network is synchronizedat the right beginning of the experiment, the clocks ofall nodes shift off after a period of time. Some of nodesscheduled to wake up at common time may mismatchespecially when time slots are set to be very short. Thethroughput under B-MAC is litter higher than that under ADC when the time slots size is small, such as ms .Notice that the network throughput under ADC doesnot change much when the time slot size changes. Butthe time slot size has much effect on the throughputunder B-MAC. The throughput under both B-MAC and
ADC decrease with the increasing of the data generationperiod when the period is higher than ms . PRR . PRR reflects the channel utility within a net-work. The PRRs under both B-MAC and
ADC increasewith the increasing of the data generation period. Whenthe time slot size is big, such as, s , s and s , the T h r oughpu t ( K bp s ) Data generation period (ms) B-MAC ADC (a) 50ms T h r oughpu t ( K bp s ) Data generation period (ms) B-MAC ADC (b) 1000ms T houghpu t ( K bp s ) Data generation period (ms) B-MAC ADC (c) 2000ms T h r oughpu t ( K bp s ) Data generation period (ms) B-MAC ADC (d) 5000msFig. 5. The network throughput respectively under
ADC and B-MAC with different data generation periods. P RR Data generation period (ms) B-MAC ADC (a) 50ms P RR Data generation period (ms) B-MAC ADC (b) 1000ms P RR Data generation period (ms) B-MAC ADC (c) 2000ms P RR Data generation period B-MAC ADC (d) 5000msFig. 6. The network PRR respectively under
ADC and B-MAC with different data generation periods.
PRR under
ADC is much higher than that under B-MAC. The results are similar to those on the networkthroughput. The time slots size has much effect on thethroughput under B-MAC instead of that under
ADC .The PRR under B-MAC increases with the decreasingof the time slot size.VII. R
ELATED W ORK
A. Duty Cycle
In WSNs, many works were put on duty-cyclednetworks as following. [19] designed a data forwardingtechnique to optimize the data delivery ration, end-to-end delay or energy consumption under low-duty-cycleby synchronized mode. [20] designed an opportunisticflooding scheme for low-duty-cycle networks with unre-liable wireless links and predetermined wording sched-ules by locally synchronization. [21] provided a bench-mark for assessing diverse duty-cycle-aware broadcaststrategies and extend it to distributed implementation.[22] minimized broadcast transmission delay by a set-cover-based approximation scheme with both centralizedand distributed algorithms. Using the β -synchronizer, afast distributed algorithm built all-to-one shortest pathswith polynomial message and time complexity [23].[24] designed an asynchronous duty-cycle broadcastingto let a node be active very long time when it needbroadcast the data to a large number of neighbors. [25]analyzed the performance of geographic routing overduty-cycled nodes and presented a sleeping scheduling algorithm that can be tuned to achieve a target routinglatency. [26] presented an alternative frame-let basedLPL implementation to improve the network perfor-mance by opportunistically aggregating packets over theradio channel. B. MAC protocol
Some protocols were designed to combine the advan-tages of TDMA and CSMA. [27] proposed a hybridMAC protocol, called Z-MAC, in which, a node alwaysperforms carrier-sensing before transmission. Z-MACconsumes much energy on the carrier-sensing and alsoneeds local synchronization among senders in two-hopneighborhoods. S-MAC [3] and T-MAC [5] employRTS/CTS mechanism to solve the the synchronizationfailure. Since these protocols use RTS/CTS, their over-head is quite high [27]. B-MAC [4] is the default MACin the operate system of Mica2 and adopts Low PowerListening (LPL) to solve the asynchronization. SinceLPL consumes much energy, X-MAC reduces the energyconsumption and latency by employing short preambleand embedding address information of the target in thepreamble [28]. So the non-target receivers can quicklygo back to sleep and the energy is saved. LPL basedpreamble transmission may occupy the medium formuch longer time than actual data transmission. So [29]designed an asynchronous duty cycle MAC: RI-MAC.It wastes energy especially under low traffic load andthe interference is increased because of the periodical broadcasting of beacons.MAC protocols are also designed to reduce energyconsumption, such as S-MAC [3] and T-MAC [5]. [30]considered LPL approaches, such as WiseMAC and B-MAC, are limited to duty cycles of 1-2% and designeda new MAC protocol called scheduled channel polling(SCP) to ensure that duty cycles of 0.1% and beloware possible. [30] dynamically adjusts duty cycles in theface of busy networks and streaming traffic in orderto reduce the latency. [31] presented a new receiver-initiated link layer A-MAC to support multiple servicesunder a unified architecture more efficiently and scalablythan prior designs. [32] designed a TDMA-based MACprimitive module PIP to achieve high throughput forreliable bulk data transfer.VIII. C ONCLUSION
Energy conservation is a fundamental issue in WSNs,which usually relies on wise designs of duty cyclingmechanisms. In this paper, we propose a localizedscheme,
ADC , to adaptively adjust the duty cycles ofall nodes in WSNs.
ADC leverages the technique of QSand adjust the duty cycles of sensor nodes according totheir demand, so that all nodes can fairly access theircommon channels. We address both synchronous andasynchronous cases with
ADC and implement it on atest-bed with 100 TelosB nodes. The results demonstratethat ADC significantly improves the WSN performancesuch as network throughput and PRR. In our future work,we plan to design protocols of duty cycle adjustment,which has more high utilization of active time and lowerduty cycle, so the energy consumption efficiency can beincreased. R
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