Efficient Scheduling for Periodic Aggregation Queries in Multihop Sensor Networks

Transcription

1 1 Efficient Schedling for Periodic Aggregation Qeries in Mltihop Sensor Networks XiaoHa X, Shaojie Tang, Member, IEEE, XiangYang Li, Senior Member, IEEE Abstract In this work, we stdy periodic qery schedling for data aggregation with minimm delay nder varios wireless interference models. Given a set Q of periodic aggregation qeries, each qery Q i Q has its own period p i, and the sbset of sorce nodes S i containing the data, we first propose a family of efficient and effective real-time schedling protocols that can answer every job of each qery task Q i Q within a relative delay O(p i ) nder resorce constraints by addressing the following tightly copled tasks: roting, transmission plan constrctions, node activity schedling, and packet schedling. Based on or protocol design, we frther propose schedlability test schemes to efficiently and effectively test whether, for a set of qeries, each qery job can be finished within a finite delay. Or theoretical analysis shows that or methods achieve at least a constant fraction of the maximm possible total tilization for qery tasks, where the constant depends on wireless interference models. We also condct extensive simlations to validate the proposed protocol and evalate its practical performance. The simlations corroborate or theoretical analysis. Index Terms Qery schedling, aggregation, delay, periodic, interference, tilization, schedlability. 1 INTRODUCTION Wireless sensor networks (WSNs) consists of different types of sensors collaborating to monitor physical or environmental conditions. WSNs are being widely sed in cyber-physical systems [24], [37] to provide qery services. In response to qery reqests of a control application, the corresponding sensory data need to be streamed to a control center. In contrast to the direct implementation of raw data collection, in-network aggregation [26], [8], [28] can redce the reqirements for both network bandwidth and power consmptions while garantee the validities of the aggregated data for answering qeries. Ths, for most qery services, data aggregation has been a promising approach to redce energy consmption. The majority work, e.g. [35], [39], [38], [34], [5], [9], [11], [14], [33], for data aggregation schedling focsed on the so-called single one-shot qery, where each node v i in the network contains only one data item, say x i, to be delivered to the control center, The research of athors are partially spported by NSF CNS , NSF CNS , program for Zhejiang Provincial Key Innovative Research Team, program for Zhejiang Provincial Overseas High- Level Talents (One-hndred Talents Program), National Basic Research Program of China (973 Program) nder grant No. 2010CB and 2010CB334707, and fnded by Tsingha National Laboratory for Information Science and Technology (TNList). Any opinions, findings, conclsions, or recommendations expressed in this paper are those of athor(s) and do not necessarily reflect the views of the fnding agencies (NSF, and NSFC). XiangYang Li is with Department of Compter Science, Illinois Institte of Technology, and holds a visiting position at Tsingha National Laboratory for Information Science and Technology (TNList), and HangZho DianZi University. xli@cs.iit.ed. Xiaoha X, Shaojie Tang are with Department of Compter Science, Illinois Institte of Technology, Chicago, USA. s: {xx23, stang7}@iit.ed. and the control center is only interested in getting the aggregated vale f(x 1, x 2,, x n ) for some aggregation fnction f() (e.g., min, average, or variance) with minimm delay. However, in many practical systems, qery reqests often come in a periodic fashion. For example, a qery in a strctral health monitoring system may reqest the sensory data of vibration periodically. For periodic qery tasks, sensor nodes are reqired to deliver their sampled data to the control center for each period. Moreover, mltiple qeries may be performed simltaneosly in the network, and qeries may differ in many aspects, e.g., some qery asks for the average temperatre while another asks for the monitored video in the same local area. On the other hand, for mission-critical real-time systems, the semantics and the validities of data often highly depend on the time of tilizing the data. For example, a srveillance system may reqire the positions of an intrder to be reported to a control center within a delay of seconds so that prsing actions can be initiated in time. For every period of a qery, the delay can be interpreted as the dration from the time when the job for this period is released, to the time when the control center receives all (possibly aggregated) data for this period. Then, qeries are often sbject to stringent delay constraints, in addition to their already complex appearances (i.e., periodic and mltiple qeries). We describe the main problem to be stdied as follows. Given a WSN consisting of a set of sensor nodes and a control center (or sink node), the sink node will isse to the network a set Q of periodic qeries for data aggregation. For each qery Q i Q, the sink node may be interested in data only from a certain region and therefore, only a sbset of sensor nodes

2 2 will generate data to satisfy the qery; We call these nodes as sorce nodes. For each period of a qery, the sink node expects to receive the corresponding (possibly aggregated) data from the sorce nodes of this qery. For a given wireless interference model (we do not restrict orself to a specific interference model), the problem Periodic Aggregation Qery Schedling (PAQS) seeks to jointly design a roting tree for each aggregation qery, and an interference-free schedle of activities for all nodes (i.e., when to transmit and which packet to transmit) sch that for each qery Q i, every job can be answered within a finite delay (typically a constant mltiple of its period). Nmeros milestone reslts have been developed for varios related schedling problems, e.g., periodic jobs at single processor [17], [15], [41], and packetlevel schedling for the Internet [42], [31]. All the existing reslts only consider job schedling at a single node, they did not toch in-network aggregation schedling for qeries, which reqires coordinations of data transmissions among all nodes in the WSNs. On the other hand, when considering aggregation schedling in the network, de to wireless interferences and resorce constraints in WSNs, even for the simple problem of Schedling for One-shot Qery (OQS) which aims to find a schedle to minimize the delay for answering this qery, it has been proved to be NP-hard [5]. Only a few previos literatres [32], [13], [6] have stdied the real-time periodic data aggregation/collection schedling in mlti-hop WSNs. Moreover, none of these reslts provides a theoretical performance assrance on the delay and maximm throghpt for WSNs. The main challenges may come from the fact that existing soltions for OQS can not serve directly as a basis for solving the problem PAQS after comparing these two problems. Comparing to One-shot Qery Schedling: While PAQS shares the notion of delay bonded in-network aggregation schedling with the problem OQS, these two problems differ in three aspects. First, the prereqisites for these two problems are different. For OQS, to the best of or knowledge, all methods assme that there are no activities schedled in advance at any node before rnning the schedling protocol. However, for mltiple periodic qeries in PAQS, this prereqisite was not satisfied from the second qery: when we schedle the i-th qery (i 2), nodes already have lots of time-slots reserved to serve the first (i 1) qeries. In other words, for some nodes, some time-slots are not candidates any more to transmit data for some qeries. Note that, we may nify the prereqisites for these two problems by defining a hyper-period (the least common mltiple of the individal periods). We then merge mltiple qeries into one meta qery with a hyper-period. One main conseqence of the nification is the inevitable large delay for some job instances in PAQS as we wold have many job instances from one qery in one hyper-period. Second, the objectives of two problems are different. For OQS, since each node only needs to transmit once, we can always satisfy this qery. The matter here is to minimize the delay of answering the qery. However, for periodic qeries, we can not take for granted that we can answer them with finite delay. Even if there is only one node in the network (one-node network example is the widely stdied real-time schedling problem in real-time commnity), when the reqest rates of qeries/tasks become large, the delay for answering some task(s) will race towards infinity. That is becase in a system of periodic tasks, larger reqest rates imply larger total tilization; when the total tilization of all tasks exceeds the schedlable tilization of any algorithm ( capacity of the network), the system is not schedlable [18]. As an illstration, let s consider a one-node network instance with two periodic tasks {Q 1, Q 2 }. Assme each task has a period of one time-slot, and reqests exactly one time-slot to process for each period. Observe that both periodic tasks have the reqest rate of one; the total tilization is two which exceeds one. Then, the network instance is overloaded with processing for these two tasks {Q 1, Q 2 }; this fact will reslt in an infinite delay for at least one task. For periodic qeries in the problem PAQS, the objective is to satisfy as many qeries/tasks as possible instead. Third, the conflict constraints imposed on the soltions of these two problems are different. For oneshot qery, since each node only needs to transmit data once, the soltion (schedle) only needs to worry abot the interferences from nearby nodes (we call this the intra-qery spatial constraints). However, when answering mltiple qeries, we need to accont for additional constraints: (a) for any node, the timeslots schedled for a period of some qery cannot overlap with the time-slots schedled for another period of the same qery or for another qery (we call them inter-period node constraints and the interqery node constraints respectively), (b) the timeslots schedled for any node to answer one qery cannot overlap with the time-slots schedled for some nearby nodes to answer any qery (we call this the inter-qery spatial constraints and intra-qery spatial constraints), Or Main Contribtions: De to niqe challenges for PAQS, we need to propose a novel design of schedling protocols to orchestrate both the real-time job schedling and in-network aggregation for answering a given set of qeries. This is one of the two main contribtions of or work. For a set of periodic data aggregation qeries, we design a family of roting, node- and packet-level schedling protocols nder varios wireless interference models sch that each qery can be satisfied (the sink node can receive all the data for each qery), within a bonded end-to-end delay. The main idea

3 3 for or protocol design is to split the sensor network spatially and temporally and then to find a schedle that makes efficient and carefl se of resorces. We prove theoretically that or protocol can achieve a total load that is at least a constant fraction of the optimm load; and at the same time, for each qery, the delay is at most a small constant factor of the minimm delay by which any protocol can achieve. Or second main contribtion lies in schedlability test schemes that can test whether a given set of qeries can be satisfied sing any possible method. We propose necessary conditions for schedlability (smmarized in Theorem 6), sch that if a set of qeries does not satisfy the conditions, we can determine immediately that the WSN is overloaded with qery tasks (or the total reqest rate of all qeries exceeds the capacity of the network). We also propose sfficient conditions for schedlability of a set of periodic aggregation qeries (smmarized in Theorem 5) based on or protocol design. The gap between the proposed sfficient conditions and necessary conditions is proved to be a constant. This implies that the proposed sfficient conditions can achieve a tilization that is at least a constant fraction of the optimm tilization for schedlability. In addition to theoretical analysis, we condct extensive simlation stdies of or protocol design and schedlability test, the reslt of which corroborate or theoretical analysis. The rest of the paper is organized as follows. Section 2 formlates the qery schedling problem in WSNs. Section 3 reviews the related work, and introdces or preliminary reslts. In Section 4, we present or protocol design for schedling periodic aggregation qeries nder varios interference models. In Section 5, we propose schedlability test schemes for a set of periodic aggregation qeries. We present or simlation reslts in Section 6. Section 7 discsses the limitation of this work and possible ftre work. Section 8 concldes the paper. 2 SYSTEM MODEL, PROBLEM FORMULA- TION 2.1 System Model Let G = (V, E) represent a WSN consisting of a set V of n nodes and a set E of bi-directional commnication links. Let v s V be the sink node. There exists a commnication link between two nodes iff they are within the transmission range of each other. To transmit data, we assme TDMA schedling where the time domain is divided into time-slots of fixed length, and the transmission of each packet costs one time-slot. Note that in practice, the time granlarity is frame which consists of mltiple time-slots. A node will be continosly transmitting mltiple packets for a whole frame as an atomic action. For simplicity, we assme time granlarity is time-slot here. For a set of commnication links to transmit simltaneosly, we have to avoid wireless interferences. In the wireless network commnity, several interference models have been commonly adopted, e.g., Protocol Interference Model (PrIM), RTS/CTS Model, and Physical Interference Model (PhIM). In PrIM [10], each node has a fixed transmission range normalized to one, and a fixed interference range of ρ. Any node v V will be interfered by the signal from another node V, if v ρ and the node v is not the intended receiver of the transmission from. In the RTS/CTS model [1], for every pair of active transmitter and receiver, any other node that lies within the interference range of either the transmitter or the receiver cannot transmit simltaneosly. In PhIM [4], [36], there is a threshold vale β > 0, sch that a node v V can correctly receive the data from a sender iff the Signal to Interference pls Noise Ratio, SINR = P v κ ξ+ w I Pw wv κ β, where v is the Eclidean distance between the nodes and v, ξ > 0 is the backgrond Gassian noise, while I is the set of other actively transmitting nodes when node is transmitting, and κ > 2 is the path loss exponent, P [P min, P max ] is the transmission power of the node. Here, we assme niform power assignment where each nodes is assigned with the same power, i.e., P = P, V. Note that nder PhIM, to ensre concrrent transmissions, we will constrct roting trees in a strong connected commnication graph which is a sbgraph of G (see [16] for details). In this work, we will extensively stdy qery schedling and the schedlability test in a WSN nder each of these interference models. Note that for PhIM, we define maximm transmission radis as L = κ P ξβ, which is the tight pper-bond on the possible commnication distances. A node cannot transmit data to another node v more than the distance L away even in the absence of other concrrent transmissions. We observe that if a commnication link has a length very close to L, the SINR at the corresponding receiver will flctate arond the threshold β in practice, then the transmission of this link is prone to fail. Ths, to avoid data transmission on the edge of commnication bondary, we need to se shorter links, say with length at most δ L with a parameter δ. We can derive a redced commnication graph, denoted as G δ L (V ), which consists of all links with length at most δ L. If G δ L (V ) is connected, we can perform data transmissions in this sbgraph of G instead. Therefore, nder PhIM, we will constrct roting trees in a redced commnication graph G δ L (V ). Assme the control application isses a set of qery tasks Q = {Q 1, Q 2,, Q m }. For qery Q i Q, let p i be its period, S i V be the set of sorce nodes which contain data for answering qery Q i, and each sorce node v S i will generate a data nit in every period to be gathered to the sink v s. We assme that

4 4 it will take χ i time to transmit one data nit for qery Q i over any commnication link. For different qeries, the size of data nits may be different; then, χ i may be different. For Q i, let a i represent the release time (or phase) let d i represent the end-to-end delay reqirement for receiving the answer; then, the j-th instance, denoted as q i,j, of this qery will be released at time a i + (j 1) p i, and the deadline for the sink to receive the answer is a i + (j 1) p i + d i. We will focs on data aggregation qeries. Data aggregation allows in-network fsion of data from different sensors when en-roting data towards the sink. We implicitly assme that the clocks of different nodes are synchronized sch that only data from the same period of the same qery are allowed to be aggregated. For simplicity, we assme that a node can aggregate mltiple incoming data nits into a single otgoing data nit of the same size. For some qery Q i Q, the data nit generated by each node may be large, ths we need to split the data nit into mltiple packets, then aggregate one packet at a time. For example, consider a qery where each data nit can be split into two packets: one is for the temperatre; the other is for the hmidity. Then a node can perform aggregation on the packets for temperatre first, when it received the corresponding packets, and then perform aggregation on the packets for hmidity later. A packet for temperatre cannot simply be aggregated with a packet for hmidity for this qery. We assme that χ i /t 0 is an integer, where χ i is the processing time for qery Q i, and t 0 is the time dration of a time-slot. When transmitting mltiple packets generated by a node in one period, we allow the perfect preemption which means that we can interleave transmissions of packets originated from different qeries or from different periods of the same qery. However, we assme that preemption only happens after a time-slot is finished, since we assme time-slot is an atomic, indivisible time nit. For simplicity, we assme that p i is an integer and the actal vale of a qery Q i s period is p i t Problem Formlation Given a set of preemptive, independent and periodic aggregation qeries, Here independent means that reqests for a certain task do not depend on the initiation or the completion of reqests for other tasks [17]. The first objective is to design a roting strctre and a transmission schedle to answer all qeries and meet the delay reqirement of each qery. Here the roting strctre consists of a set of roting trees {T i : 1 i m}, one tree T i for each qery Q i. A transmission schedle, denoted as S, consists of assigned time-slots for packets at each node to transmit. Let t q i,j,p be the time-slot assigned to the p-th packet of a job q i,j of qery Q i at a node. A transmission schedle is said to answer a set of v 1 v 7 Sink v s v 8 v 5 v 6 v 2 v 3 v 4 Node(s) Schedle Time-Slot v 1, v 3 3k + 1 v 2, v 4 3k + 2 v 5, v 6 3k + 3 v 7 3k + 4 v 8 3k + 5 Fig. 1: Example for Periodic Qery Schedling qeries (or check if there is a feasible schedle) iff the sink can receive all data for every period of each qery, when every packet is transmitted according to the schedle and roted via the roting tree. Given a feasible schedle S, the end-to-end delay of a job q i,j in a qery task Q i (denoted as D(S, Q i, j)) is defined as the lapse of time slots from the time-slot when this job q i,j is released to the time-slot when the sink node received the final aggregated vale for this job, i.e., D(S, Q i, j) = max max p {t q i,j,p ( ) a i + (j 1) p i }. The end-to-end delay of qery task Q i nder the schedle S is defined as max j D(S, Q i, j), which is reqired to be at most the delay reqirement d i for qery Q i. We assme implicitly that the delay reqirement d i for qery Q i meets the following constraints: (1) d i η 1 p i t 0 for some integer constant η 1 1, and (2) d i η 2 Rt 0, where η 2 > 0 is a constant; R is the radis of the network (i.e., the maximm hop distance from any node in the network to the sink node), which is at least half of the network diameter; and t 0 is the dration of a time-slot. The second condition comes from the fact that, for one-shot qery, the minimm delay of answering an aggregation qery is Ω(R t 0 ) (e.g., [38], [34]) de to the network delay for delivering data from within the network to the sink node. Given a qery, the qery is said to be satisfied if the qery is answered and the delay for the sink to get the answer is finite. In the example in Fig. 1, assme there is one periodic aggregation qery Q 1 with χ 1 = 1, p 1 = 3. Let V = {v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8 } {v s }. For each of the three sets {v 1, v 3, v 7 }, {v 2, v 4, v 8 }, {v 5, v 6 }, all membering nodes can transmit cocrrently. We give a feasible transmission schedle in Fig. 1 (k N}). Here for i {1, 2, 3, 4, 5}, {3k + i} means that the corresponding node(s) will transmit at all the time slots of 3k + i : k N. By sing schedle S after the arrival of the aggregated reslt for the first job, all other aggregated reslts for later jobs arrive in a periodic seqence time of two time-slots apart. Ths the delay for each period is exactly five time slots. The second objective is the schedlability test. Definition of Schedlability: Given a network, a set Q of independent, preemptive, and periodic qeries is schedlable iff there exists a roting strctre

5 5 and a transmission schedle S that can answer all qeries and the delay of every job q i,j of each qery Q i Q is finite. Given a set of periodic data aggregation qeries Q = {Q 1, Q 2,, Q m }, the objective is to test whether the set of qeries is schedlable. Note that, if we relax the delay reqirement, Q seems to be more likely to be satisfied. However, when the network is overloaded with qeries with large reqest rates, the qery set is not schedlable, irrespective of the delay reqirement. We will captre the schedlability of a set of qeries by sfficient conditions and necessary conditions. In addition, we want to minimize the gap between sfficient conditions and necessary conditions for schedlability of a set of qeries. It is easy to verify that the qery given in Fig. 1 is schedlable. Given a set of periodic tasks, determining the schedlability has been extensively stdied in the real-time literatre. Several pioneering reslts (see [18] and references therein) have been developed for the schedlability test of periodic tasks in single processor. For example, Earliest Deadline First (EDF) can always find a feasible schedle for a set of schedlable preemptive, independent and periodic qeries where the delay reqirement of each qery is at least its period [17]. 3 RELATED WORK, OUR PRELIMINARIES In each sbsection, in addition to the review, we will also present or preliminary reslt which serves as a basis for or protocol design. 3.1 Real-time Schedling Two representative classes of well-stdied real-time schedling algorithms are rate-monotonic (RM) and EDF schedling. The class of RM algorithms assigned static-priorities to qeries on the basis of the cycle dration of the jobs. Li and Layland [17] presented a RM algorithm in a single processor, and the first sfficient condition for schedlability of a set of qeries. This is then frther extended in [15], [27]. On the other hand, EDF is a dynamic schedling algorithm. EDF and its several extensions were proposed to garantee the end-to-end delay of packets, e.g., EDF with traffic shaper [30], [29], [31] that can reglate the distorted traffic from the EDF schedler to deal with the brsty traffic. Unfortnately, sing optimal traffic shaper is in general infeasible and introdces additional packet delays. Another approach, sch as deadline-crve based EDF (DC-EDF) [42], or similar one [3], is to jdiciosly adjst the local deadlines of packets at a node, based on the traffic load and/or the end-to-end deadlines. DC-EDF can garantee end-toend delay performances and provide a schedlable region as large as that of RC-EDF [42]. Recently, Chipara et al. [7], [6] stdied the real-time qery schedling in WSNs by assming a pre-given roting tree. Their reslts often sffer from the fact that different roting strctres have vast impact on the delay performances and flow data rates spported by a WSN. Next, we present or preliminary reslt of packet labeling for single-hop qeries. Here a single-hop qery differs from the qery defined in Section 2.1 in only one aspect: each packet reqests only a singlehop transmission (instead of mlti-hop transmissions across the network). We define the reqest rate of a single-hop qery as the reciprocal of its period. Let the tilization of a set of qeries be the smmation of their reqest rates. Definition of Packet Labeling: Given a set Q of preemptive and periodic single-hop qeries, the objective is to assign a different integer label for each packet, sch that if each packet transmits at the timeslot eqal to its label, the delay for each qery is at most its period. Observe that, a packet labeling scheme corresponds to a single processor periodic job schedling. Ths, we can label packets based on RM or EDF schedling. RM Schedling: Then, RM prioritizes packets simply based on reqest rates of qeries. When the nmber of qeries is large, RM schedling can achieve a tilization of 69% (all packets can make their deadlines). Lemma 1: Given a set Q of single-hop qeries with tilization at most 0.69 N, where N 1 is an integer constant, there exists a packet labeling scheme sch that each label can be divided by N (Lemma 1 has fond applications in Section 4 with the vale of N assigned to be either 2c 1 c 2 or 2c 1 ). Proof: If each packet wold cost N consective time-slots instead of one (we assme the period is greater than N here), then the qery set Q wold have a virtal tilization of at most By sing RM schedling, we can answer all qeries and each packet is assigned with N consective time-slots. We then select one time-slot among the N time-slots that can be divided by N, and assign it as the packet s label. This finishes or proof. EDF Schedling: We prioritize packets strictly according to their deadlines. EDF can achieve tilization of exactly one instead of 0.69 [17]. By replacing RM with EDF, we can achieve a reslt similar to Lemma Min-Delay Aggregation Schedling Minimm delay data aggregation problem has been proven to be NP-hard [5], even for the case of simple one-shot qery. Athors in [11], [39], [34] [38] proposed a seqence of constant-ratio approximation algorithms for OQS nder PrIM. Next, we present or preliminary reslt of node ranking in a Connected Dominating Set (CDS) (see definition in [25]). Let s review the concept of CDS first. In a graph G = (V, E), a sbset V V is a

6 6 dominating set if every node is either in V or adjacent to some node in V. Nodes in V are called dominators; nodes not in V are called dominatees. A sbset V V is a connected dominating set, if V is a dominating set and V indces a connected sbgraph. Definition of Node Ranking: Given a CDS T CDS, sink v s, interference model, the objective is to assign a rank r() for each node in CDS, sch that if all nodes transmit towards the sink v s at the time-slot eqal to its rank, the aggregated data from the CDS can be received by v s withot interferences. Clearly, given a CDS, a transmission schedle for OQS with the CDS as inpt graph corresponds exactly to a node ranking scheme; moreover, the delay of the schedle corresponds to the maximm rank among all nodes in the CDS. We will focs on a CDS whose maximm node degree is bonded by a constant 12 (Lemma 4.1 of [34]). Then, we can compte the ranks of nodes based on existing soltions for OQS. Corollary 1: Given a CDS, there exists a node ranking scheme with the maximm rank at most β ρ (2R O(log R)) nder PrIM [34] β 2 (2R O(log R)) nder RTS/CTS (12K 2 + 1)R + 72K 2 + O(log R) nder PhIM [16] Here ρ is the interference range nder PrIM, β ρ is a parameter with its vale given as β ρ = π 3 ρ 2 + ( π 2 + 1)ρ + 1, R is the graph radis of the CDS, and K is a constant depending on the parameters of PhIM. The vale of K is given in [16]. Note that nder RTS/CTS, a schedle for OQS with delay β 2 (2R O(log R)) exists de to or following observation: a schedle nder PrIM when ρ = 2 corresponds to a conflict-free schedle nder RTS/CTS for OQS. We then qantitatively captre the relevance between two nodes spatial distance and the temporal difference of their ranks. Let λ(m) be the conflict range of an interference model M sch that for a set of nodes, if the mtal distance between any pair of nodes is at least λ(m), then all nodes in this set can transmit concrrently. Lemma 2: (Spatial-Temporal Relevance) In any node ranking scheme in Corollary 1, for any pair of nodes, v of mtal distance at most λ(m), r() r(v) < c 1 (M) with the vales of c 1 (M) and λ(m) given as: c 1 (M) = 2(ρ + 1) 2 + O(log ρ) nder PrIM 30 nder RTS/CTS ((K 1)L) 2 + O(log KL) nder PhIM ρ + 1 nder PrIM λ(m) = 2 nder RTS/CTS (K 1)L nder PhIM 4 SCHEDULING PROTOCOL DESIGN The general framework of or protocol design is niversal for varios wireless interference models. 1) For each qery Q i Q, constrct a roting tree T i for data aggregation; 2) For each node, constrct a transmission plan, which specifies the data to transmit at the crrent moment. For each qery Q i, based on roting tree T i, each node in T i ( may not be a sorce node) needs to add data for each period to its plan. 3) For each node, for each packet from s transmission plan, assign concrete time to transmit. The packet schedling will be based on or preliminaries of Packet Labeling (Sbsection 3.1) and Node Ranking (Sbsection 3.2). This phase is the key part. We then describe the details of each phase separately. The first phase is roting. The constrctions of roting trees are the same nder varios wireless interference models. Observe that, for each data aggregation qery Q i Q, the roting tree T i shold be a Steiner Tree inter-connecting the terminals of S i {v s }. Let the commnication graph be G = (V, E), we select a CDS T CDS of G by sing an existing approach [25]. We then constrct a spanning tree T G by connecting each node not in the CDS to one of s neighboring dominators. Based on the spanning tree T G, for each qery Q i Q, we prne each node V and an incident commnication link v (the link from to its parent node v) in T G if the intersection of two node sets S i and the node set from the sbtree of T G rooted at (noted as T G ) is empty: S i V (T G ) =. The prning operation reslts in a roting tree T i for the qery Q i. The second phase is constrcting transmission plans, based on roting trees for data aggregation qeries. For each qery Q i Q with a roting tree T i, dring each period, first each leaf node in T i adds the sorce data to its transmission plan; then every internal node in T i (noted as a relay node for qery Q i ) only generates one nit of data by aggregating all received data with its own data (if it has), while it may receive mltiple data nits from its children; Note that, before adds the data nit to its transmission plan, it needs to wait ntil receiving the corresponding data from all its children in T i (the roting tree for qery Q i ). Ths, for a qery Q i, the data nit at node can be either (1) original or (2) aggregated one, depending on whether this data nit comes from one node. The third phase is packet schedling at each node which contains data nits in its transmission plan. Here a data nit may consist of several packets. We divide all nodes into two complementary grops: nodes not in the CDS T CDS (noted as leaf nodes) and nodes in the CDS T CDS (noted as intermediate nodes). We then describe or packet schedling for each grop sepa-

7 7 (a) 4-Coloring for PrIM, RTS/CTS (b) c 2 (M)-coloring for PhIM Fig. 2: Grid Partition and Coloring: if at most one node from every grid with a monotone color transmits simltaneosly, the transmissions are interferencefree. For PhIM, the constant c 2 (M) depends on the parameters. rately. We will ensre that all leaf nodes transmit at even time-slots only, and all intermediate nodes transmit at odd time-slots only; the time-disjoint property can avoid interferences between nodes from different grops. Packet Schedling at Leaf Nodes: We employ a grid partition of the deployment plane. The vertical lines x = i λ for i Z and horizontal lines y = j λ for j Z partition the planes into half-open and halfclosed grids of side λ (here λ represents λ(m), and Z represents the integer set): {[i λ, (i + 1) λ [j λ, (j + 1) λ : i, j Z}. We then color the grids sch that if at most one node from every grid with a monotone color transmits simltaneosly, the transmissions are interferencefree. Note that, the nmber of colors sed here (noted as c 2 (M)) depends on the interference model M. Under PrIM, RTS/CTS model, no neighboring grids share a common color is enogh to avoid interferences, ths, c 2 (M) = 4; while nder PhIM, c 2 (M) is a larger constant (see Fig. 2 and [16]). We index the colors sed and denote σ g as the color of grid g (σ g {0, 1,, c 2 (M) 1}). For each grid g that contains leaf nodes, we find the sbset (noted as V leaf (g)) of all leaf nodes lying in g; and for each node V leaf (g), we find the sbset (noted as Q ) of qeries at (a qery Q i Q iff its roting tree contains the node, i.e., T i ). Let Q g = V leaf(g) Q be the collection of qery sbsets at leaf nodes from V leaf (g). Note that here, in Q g, a qery Q i at different nodes is perceived as different qeries. Then, we create an instance of Packet Labeling with the inpt single-hop qery set as Q g. For this instance, when the tilization is larger than the constant c 3 (M), the grid g wold be overloaded with data transmissions for answering qeries in the long rn (The complete argments are presented in Sbsection 5.3). Ths, no matter what tilizations are for other grids, the qery set Q wold be not schedlable, de to a bottleneck of grid g. On the other hand, when the tilization is smaller than c 1c 2 (or 1 2c 1c 2 ), by Lemma 1, we can obtain a packet labeling scheme by sing RM (or EDF) schedling where each packet (say the p-th packet for j-th job of qery Q i at some leaf-node V leaf (g)) is assigned with a label ; and at the same time, we have (2c 1 c 2 ) ). Here c 1 (M), c 2 (M) are (i.e., 2c 1 c 2 divides abbreviated to c 1 and c 2 respectively, we will keep this convention from now on. Note that, for different nodes and v, the label shold not be eqal to v, since a packet at two different nodes are perceived as two different packets for labeling. For every leaf node, let σ g be the color index of the grid g where lies. We assign the p-th packet for qery Q i s j-th job at node with a transmission time: t q i,j,p = + 2σ g. This finishes packet schedling at leaf nodes. Packet Schedling at Intermediate Nodes: For each intermediate node, we find the set of qeries Q for which participates in roting. Then, we map each qery Q i Q to a new one Q i with modification of only the release time: a i = a i + p i + 2c 1 c 2. Let Q = Q i Q Q i. Finally, we create an instance of Packet Labeling with the single-hop qery set Q. Note that, for sch an instance of Packet Labeling, the tilization is at most c 1 (M) Q χ i i Q p. i If χ i Q i Q p > 1, the sink node wold be overloaded i with data receptions, i.e., the sink cold not receive all data completely in the long rn, for answering the given aggregation qery set Q (The complete argments are presented in Sbsection 5.3). Ths, the qery set Q wold be not schedlable, de to a bottleneck of the sink node. On the other hand, when χ i Q i Q p 0.69 i 2c 1, the tilization of this Packet Labeling instance is at most c 1, by Lemma 1, we can obtain a packet labeling scheme by sing RM (or EDF) schedling where each packet (say the p-th packet for qery Q i s j-th job at the intermediate node ) TABLE 1: Parameters sed in or protocol design T i roting tree for qery Q i ρ interference range for PrIM M interference model i.e.,prim, CTS/RTS, PhIM λ(m) conflict range σ g color index of the grid g label of p-th packet for Q i s j-th job at node in Packet Labeling r() rank of node in Node Ranking U G,Q (g v,h ) initial load of grid g v,h c 1 (M) or c 1 refer to Lemma 2 c 2 (M) or c 2 nmber of colors for grid coloring c 3 (M) or c 3 maximm nmber of nodes that can transmit in any grid t q i,j,p final schedled time-slot for p-th packet for Q i s j-th job

8 8 Algorithm 1: Schedling Protocol for PAQS Inpt : A set of periodic aggregation qeries Q, an interference model M. 1 for each qery Q i Q do 2 constrct a data aggregation roting tree T i ; 3 for each j-th instance of each qery Q i Q do 4 for each node do 5 if is a leaf node in T i then 6 adds the data to s transmission plan; 7 if is an internal node in T i then 8 if has only one child v in T i then 9 when received the data from v; 10 adds data to s transmission plan; 11 else 12 when received data from all children 13 in T i, generates aggregated data; 14 adds the data to s transmission plan; 15 for each leaf node (i.e., / T CDS ) do 16 σ g color index of the grid g where lies; 17 for packet in (p-th packet for Q i s j-th job ) do 18 assign time: t q i,j,p + 2σ g ; 19 for each intermediate node (i.e., T CDS ) do 20 for packet at (p-th packet for Q i s j-th job ) do 21 assign time: t q i,j,p + 2r() retrn Time to transmit for each packet at each node. is assigned with a label ; and at the same time, we have (2c 1 ) (i.e., 2c 1 divides ). For each intermediate node, for each packet (assme it is the p-th packet for Q i s j-th job), we assign it with a time-slot: +2r()+1. Here r() is the rank of for Node Ranking with the CDS T CDS as the inpt. This finishes packet schedling at intermediate nodes. To sm p, we present Algorithm 1 for or protocol design and Table 1 for the notations. We will analyze the performance of Algorithm 1 in Section SCHEDULABILITY ANALYSIS In this section, we will propose both sfficient conditions and necessary conditions for schedlability of a set of periodic qeries. The validness of proposed sfficient conditions will rely on the correctness of Algorithm Performance of Algorithm 1 We first prove the correctness of Algorithm 1. This means that Algorithm 1 otpts an interference-free schedle (Lemma 3) and at the same time, every node will transmit strictly after it receives the packets from all its children (Lemma 4), for every period of each qery. We then prove the bonded delay of or schedle (Theorem 1). All the reslts hold whenever is obtained based on either RM or EDF schedling in Section 3.1. Lemma 3: Algorithm 1 can avoid interferences. the packet label Proof: If two packets p q i,j,p and p q i,j,p v at node are assigned with the same time-slot, we have either + 2σ g = l q i,j,p + 2σ g ( is leaf) or + 2r() + 1 = l q i,j,p + 2r() + 1 ( is intermediate node). Then, = l q i,j,p, this constradicts the fact that labels are different for any single packet labeling instance. Else, there exist conflicting transmissions. Let, v be the pair of conflicting nodes with minimm mtal distance and the corresponding transmitting packets be p q i,j,p and p q i,j,p v respectively. As leaf nodes transmit at even time-slots and intermediate nodes transmit at odd time-slots,, v mst be both leaf nodes or both intermediate nodes. If, v are leaf nodes, then packets p q i,j,p and p q i,j,p v are assigned with + 2σ g and l q i,j,p v +2σ g respectively. If, v lie in the same grid, σ g = σ g, then = l q i,j,p v, contradiction. If, v lie in different grids g and g, we have (2c 1 c 2 ) (2c 1 c 2 divides ) and (2c 1 c 2 ) l q i,j,p v. Ths, we have +2σ g l q i,j,p v +2σ g mod 2c 1 c 2 2σ g 2σ g mod 2c 1 c 2, then σ g = σ g. Ths, g and g share a common color, i.e.,, v are not conflicting, this leads contradiction. If, v are intermediate nodes, then packets p q i,j,p and p q i,j,p v are assigned with +2r()+1 and l q i,j,p v +2r(v)+1. We have 2c 1 (2c 1 divides ) and 2c 1 l q i,j,p v. Then, 2r() + 2r(v) + l q i,j,p v mod 2c 1 2r() 2r(v) mod 2c 1. If v l(m), by Lemma 2, r() r(v) < c 1 r() = r(v); this cases contradiction as {r() : V (T CDS )} is a conflict-free schedle. If v > l(m), then, v are not conflicting pair, contradiction. Lemma 4: In the schedle otpt by Algorithm 1, for every job of each qery Q i, for each packet, each leaf node transmits after the job is released; each intermediate node transmits after its children in T i transmit. Proof: Given a packet (say the p-th packet for qery Q i s j-th job), its release time is a i + (j 1) p i. Clearly, a i + (j 1) p i + 2σ g = t q i,j,p. For any node v, assme node is a child of v. If is a leaf node, we have t q i,j,p = + 2σ g a i + jp i + 2σ g < a i + jp i + 2c 1 c 2. Here the ineqality

9 9 a i + j p i holds becase any packet labeling scheme can ensre that each packet is delayed by at most its corresponding period. At the same time, v a i +(j 1)p i = (a i + p i +2c 1 c 2 )+(j 1)p i = a i + jp i + 2c 1 c 2. Ths, we have t q i,j,p v + 2r(v) + 1. < t q i,j,p v = If is an intermediate node, we want to prove that, for each qery, for every packet, node transmits strictly before its parent v transmits. To begin with, let s constrct a packet labeling scheme for all intermediate nodes { : i, j, p, T CDS } sch that for any intermediate node, assme v is s parent in T G, we have i, j, p : v. We constrct labeling of packets at intermediate nodes level by level in tree T CDS, starting from the root : i, j, p} for packets at node v. For each child of v, let Q, Q v be the qery sets for which, v participates in roting. We observe that Q Q v. Then, for node, for each packet (say p-th packet for j-th job of qery Q i ), we v s. After constrcting labeling { v have = v. Based on the constrcted property of packet labeling, we have v. At the same time, we have r() r(v) where r() r(v) are the ranks of node and v respectively; otherwise the parent v will miss node s reported data when aggregating data. Then, we have +2r() r(v)+1. p() By Lemma 3, or schedle can avoid interferences, ths the transmission time of and v are different for any packet. Ths, we have + 2r() + 1 < +2r(v)+1. To sm p, for each qery, for every packet, node transmits strictly before its parent v transmits. This completes the proof. p() Theorem 1: For each qery Q i Q, Algorithm 1 can achieve the following end-to-end delays nder varios wireless interference models: 2β ρ (2R + O(log R)) + 2c 1 c 2 + 2p i nder PrIM 2β 2 (2R + O(log R)) + 2c 1 c 2 + 2p i RTS/CTS (12K 2 + 1)R + O(log R) + 2c 1 c 2 + 2p i nder PhIM where the parameters ρ, β ρ, R, K are defined in Corollary 1; p i is the period of the qery Q i, c 1 is given in Lemma 2, and c 2 is the nmber of colors for grid coloring. Proof: For the j-th period of Q i, the end-to-end delay is max{t q i,j,p ( ) a i + (j 1) p i : p, }. By Lemma 4, for any packet, intermediate nodes will transmit after leaf nodes transmit, i.e., max{t q i,j,p p, / T CDS } max{t q i,j,p : : p, T CDS }. We have: max{t q i,j,p : p, } = max{t q i,j,p : p, T CDS } = max{ + 2r() + 1 : p, T CDS } a i + j p i + 2 max {r() : T CDS} + 1 = ( a i + p i + 2c 1 c 2 ) + j pi + 2 max {r()} + 1 = a i + (j + 1) p i + 2c 1 c max {r()} + 1 Here, the ineqality a i +j p i holds becase any packet labeling scheme can ensre that each packet is delayed by at most its corresponding period. Ths, the delay is at most 2p i + 2c 1 c max {r()} + 1. By Corollary 1, max {r()} β ρ (2R O(log R)) nder PrIM β 2 (2R O(log R)) nder RTS/CTS (12K 2 + 1)R + O(log R) nder PhIM Therefore, the theorem follows. As it costs at least R time-slots for sink node to receive the data from the farthest node, combining with Theorem 1, Theorem 2 follows. Theorem 2: (Approximation Ratio) For each qery, assme the end-to-end delay reqirement is at least its period (note that this assmption is tre for nearly all qeries), Algorithm 1 can achieve constant approximation in terms of delay; the approximation ratio is as follows. { } min 4β ρ + 2p i +O(log R) 4βρ+O(log R) R, 2 + p nder PrIM { i } min 4β 2 + 2p i +O(log R) 4β2+O(log R) R, 2 + p RTS/CTS { i } min c 4 + 2p i +O(log R) c4r+o(log R) R, 2 + p nder PhIM i where the parameters ρ, β ρ, R, K are defined in Corollary 1; p i is the period of the qery Q i, and c 4 = 12K Sfficient Conditions for Schedlable Qeries χ j p j, To characterize a sfficient condition for schedlability, let s first introdce the concept of initial tilization. Given a WSN G = (V, E) and a set of qeries Q, assme each qery Q j Q is associated with its time χ j needed by transmitting over a commnication link, its period p j, and a set of sorce nodes S j V that contains the needed data. We define the initial tilization of a node V as U G,Q () = ( S i) (Q i Q) and the initial tilization of a grid g as the smmation of the initial tilizations of all nodes from this grid, i.e. U G,Q (g) = V (g) U G,Q(). Note that in the first phase of or protocol design (Section 4), for the instance of Packet Labeling created for each grid g, to generate a packet labeling scheme based on RM schedling, the tilization (which is at most U G,Q (g)) has to be pper-bonded by a constant

10 c 1(M) c 2(M). At the same time, for the instance of Packet Labeling created for each intermediate node, to generate a packet labeling scheme based on RM schedling, the tilization (which is at most χ i Q i Q c 1(M) p ) has to be pper-bonded by a constant i. To sm p, we propose a sfficient condition for the correctness of Algorithm 1 based on RM schedling. Theorem 3: There exists a RM-based conflict-free schedle if the following conditions are satisfied: { 0.69 UG,Q (g) 2c, g 1(M) c 2(M) (1) χ i Q i Q p 0.69 i 2c 1(M) where U G,Q (g) is the initial tilization of a grid g, c 1 (M) is given in Lemma 2, c 2 (M) = 4 nder PrIM and RTS/CTS model, and c 2 (M) = K 2 where the constant K (depending on the parameters of PhIM) is given in [16]. Similar to Theorem 3, we give a sfficient condition for the correctness of Algorithm 1 based on EDF schedling. Theorem 4: There exists a EDF-based schedle if the following conditions are satisfied: { 1 UG,Q (g) 2c, g 1(M) c 2(M) χ i 1 (2) Q i Q p i 2c 1(M) To sm p, we propose a sfficient schedlability test scheme as follows. Theorem 5: (Sfficient Schedlability Test) Ineqality (1) (and Ineqality (2)) is a sfficient condition for schedlability of a set of periodic aggregation qeries sing RM (and EDF respectively) schedling. 5.3 Necessary Conditions for Schedlable Qeries Given a set of periodic aggregation qeries, to make it possible to schedle nodes transmissions, observe that, for a sbset of nodes where any pair of nodes conflict with each other (the indced conflict graph is a cliqe), the smmation for the initial tilizations of all nodes in this sbset can not exceed one. Generally, given a grid g, where the maximm nmber of nodes in g that can transmit concrrently is a constant c 3 (M), the initial tilization of this grid can not exceed c 3 (M). On the other hand, for a single periodic data aggregation qery Q i Q, it costs the sink v s of time χ i to receive data dring every period p i. Obviosly, for a set of qeries Q, the initial tilization at sink v s, given by χ i Q i Q p, can not exceed one if Q can be i satisfied. To sm p, we propose a necessary schedlability test scheme as follows. Theorem 6: (Necessary Schedlability Test) If a set of periodic aggregation qeries Q is schedlable nder an interference model M, then the following conditions mst be satisfied: { UG,Q (g) c 3 (M), g χ i (3) Q i Q p 1 i The constant c 3 (M) 1 is maximm nmber of nodes that can transmit concrrently in a grid, nder interference model M. If we se RM-based schedling method instead of EDF-based schedling, the pper-bond of initial tilization for each grid (or node) has to be decreased correspondingly by a factor of 0.69 for schedlable qeries. We can obtain the vale of c 3 (M) withot mch effort nder each wireless interference model as follows: 16ρ 2 (ρ 1) nder PrIM 2 c 3 (M) = 36 nder RTS/CTS 2κ P nder PhIM N 0β 2 To conclde, we have the following main theorem: Theorem 7: The gap between proposed sfficient conditions (Theorem 5) and necessary condition (Theorem 6) for schedlability is a constant 2c 1 (M) c 2 (M) c 3 (M). 6 SIMULATION RESULTS In this section, we present the simlation reslts which evalate the proposed algorithm (Algorithm 1). We implemented Algorithm 1 on TOSSIM of TinyOS In or simlation, we randomly deploy a nmber of wireless sensor nodes in a 2-D sqare region, all nodes have the same transmission range. Each node will generate a random 4-byte non-negative nmber as its own datm. The objective of the sink node is to collect the aggregation (max, min, sm or average) reslt from all sorce nodes periodically. We assme that each node is eqipped with 10 types of sensors. Then there are totally 10 types of data in the networks. Two data packets can be aggregated into a new data packet iff they are generated at the same period from the same qery. The sink node will generate p to 20 qeries with random qery parameters. A typical qery message Q i has the following components: the data type, the reqired starting time at which each node will start its dty, the period, and IDs of the nodes (sorce nodes) who shold respond to the qery Q i. To evalate the performance of Algorithm 1, we also implemented another data aggregation method by combining Breath First Search(BFS) tree and CTP (Collection Tree Protocol, which is provided by TinyOS 2.0.2) sing TOSSIM. The main idea of BFS/CTP method is to constrct a BFS tree rooted at the sink node based on the link qality. In other words, dring the procedre of constrcting BFS, the

11 11 link qality compted by CTP will be considered as the link weight. Notice that, the original CTP method (components) provided in TinyOS is designed for data collection. To spport data aggregation, we modified CTP in the pper layer sch that each node will not send data to its parent (in the BFS tree) ntil it aggregates all necessary data from all its children (in the BFS tree). The main flow of or evalation system is as follows. Firstly, the sink node will generate and broadcast a set of data aggregation qeries. Secondly, the sink node initiates to constrct a CDS (sing the algorithms in [25]). Then, a roting tree (based on the CDS) which covers all nodes (may need other relay nodes) will be constrcted for data aggregation. When receiving the parameters of start time and time period, each sorce node generates a datm in each period. All leaf nodes will send data to the corresponding dominators. The dominator will relay data for each period to the sink node based on the roting tree. Basically, each dominator in CDS will locally broadcast information for transmission schedling along each link to all its dominatees throgh beacon message. Once the qery procedre begins, the sink node will contine to analyze all received data packets for each period of each qery. When all crrently existing qeries are satisfied, i.e., for each qery, the sink node can collect data packets from all sorce nodes completely and correctly for each period, the sink node will release next 20 qeries. Otherwise, the algorithm will terminate. A qery will be considered as nsatisfied if the sink node cannot collect complete data packets from all sorce nodes of this qery more than 3 periods. We se qery delay and schedlability as the two metrics to compare the performance of different protocols. The schedlability is measred by sccess ratio, which is the ratio of the nmber of sccessfl ronds to total ronds for existing qeries. In the following we compare the performance of Algorithm 1 with BFS/CTP nder two different network settings. Based on the properties of a wireless TelosB node, each node has a data bffer with size 512 bytes in or simlation. A data packet is 4 bytes inclding all control and data information. In other words, every node only can temporarily save at most 128 data packets at the same time. When the data bffer is fll, the data packet with lowest priority will be dropped. Here, the priority of a data packet is inversely proportional to its period length. If two packets have the same period length, the one with early deadline has higher priority. If the deadlines for two packets are still same, the one received (or generated) firstly has higher priority. It will drop the packet if it already passes deadline. In the first case, we randomly generated the network topology (connected) with different network size (increasing from 50 to 250 ) with same network density, i.e., the maximm degree is fixed to 20 in or simlation. Average Delay (s) Alg. 1 BFS/CTP Nmber of Deployed Nodes (a) Qery Delay Sccess Ratio Alg. 1 BFS/CTP Nmber of Deployed Nodes (b) Schedlability Fig. 3: Increased network size with fixed maximm degree. As illstrated in Fig. 3(a), we can see Algorithm 1 otperforms BFS/CTP. Fig. 3(b) describes the average sccess ratio per node with the increment of network size for both methods. Clearly, for both algorithms, the average sccess ratio decrease slightly as the more nodes deployed and or method is better than the other in most cases. For the second case, we fix the deployment area as ( ) and contine to increase the network size from 50 to 200 with step 30 while keeping the network is connected. By doing this, we can fix the radis R and test the performance of both algorithms with the increment of network density (maximm degree ). Average Delay (s) Alg. 1 BFS/CTP Nmber of Deployed Nodes (a) Qery Delay Sccess Ratio Nmber of Deployed Nodes (b) Schedlability Fig. 4: Increased network size with fixed deployment region. As we can see from Fig. 4(a), both average delay and sccess ratio have big gaps between these two methods when network density contines increasing. This is mainly de the fact that interferences greatly decrease after all the data have been gathered to dominators. For the BFS/CTP method, we contine increasing the nmber of relay nodes with the increment of network size sch that the average delay increases significantly de to the interference. Fig. 4(b) shows the reslts of sccess ratio as the network size increases. For Algorithm 1, when the network size increases over 130, the sccess ratio qickly drops from arond 0.9 to 0.6. The BFS/CTP method follows a similar pattern. That is becase the new packets from newly increased nodes lead both methods satrated sch that many packets are dropped de to the bffer limit. As indicated in the simlation reslts,

12 12 Algorithm 1 has better performance than BFS/CTP method. 7 DISCUSSIONS We have proved that or methods can achieve a total rate that is at least a constant factor of the optimm load. There are still a few limitations in or work which will be smmarized as follows. This will serve as some ftre research challenges. First, we assmed that a node can aggregate any nmber of packets into a single packet while in practice, a different aggregation degree may be sed, i.e., the size of the aggregated data depends on the nmber of inpt data items. One challenge of extending the algorithm to different aggregation degree is to prove its performance. With different aggregation degree, we do not know if the aggregation tree constrcted in this paper will still be almost optimal or not. If the tree is still almost optimal, then the link and packet schedling can be readily extended to address this more challenging case. Second, we omitted the extra aggregation delay. To address this practical challenge, one possible approach is sending partially aggregated packet withot waiting for data packets from all its children nodes. However, this approach may not improve the delay performance; it may hrt it actally. Note that the delay here is defined as the last time the sink collected the data. Althogh sending partially aggregated packet allows the sink node to know some partial reslts earlier, it still did not help the sink to get the data from other children nodes earlier. The sink still has to wait. A potential disadvantage of sending partially aggregated packet is the increasing of the nmber of packets to be sent by the node (ths the traffic of the network), this in trn will case the delay onwards becase of the additional packets to be sent. A potential advantage of this approach is that, depending on the aggregation fnction, it may redce the nmber of temporarily stored packets at a node (redcing the memory sage, and ths redcing the risk of packets dropping). Therefore, this interesting approach is worthy of extensive ftre investigation. In addition, there are some other challenges sch as (1) the impact of nreliable network: dring data transmissions, sensor nodes and links may sffer from packet losses, which will often trigger re-roting and retransmissions of data. This will incr additional delay and overhead to the network; (2) the impact of the time synchronization errors on the performance of the proposed methods. 8 CONCLUSIONS Real-time qeries appear in many sensor network applications. For answering periodic qeries, we proposed a set of efficient schedling schemes for data commnications. Essentially, we jointly designed the roting strategy as well as packet schedling protocols nder varios interference models. 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His research interests and experience span a wide range of topics from theoretical analysis to practical design in wireless networks. He is a stdent member of the IEEE. Shaojie Tang is a Compter Science PhD stdent at Illinois Institte of Technology. He received B.S. at Radio Engineering, Sotheast University, P.R.China, His research field is on algorithm design, optimization, secrity of wireless networks, electronic commerce as well as online social network. He is a stdent member of IEEE. Dr. Xiang-Yang Li has been an Associate Professor since 2006 and Assistant Professor of Compter Science at the Illinois Institte of Technology from 2000 to He hold MS (2000) and PhD (2001) degree at Compter Science from University of Illinois at Urbana-Champaign. He received B.Eng. at Compter Science and Bachelor degree at Bsiness Management from Tsingha University, P.R. China in His research interests span wireless sensor networks, comptational geometry, and algorithms, and has pblished over 200 papers on these fields. He is an editor of IEEE TPDS, Networks: An International Jornal, and was a gest editor of several jornals, sch as ACM MONET, IEEE JSAC. In 2008, he pblished a monograph Wireless Ad Hoc and Sensor Networks: Theory and Applications. He is a senior member of the IEEE and a member of ACM.