Quick convergecast in ZigBee beacon-enabled tree-based
wireless sensor networks
Meng-Shiuan Pan
*, Yu-Chee Tseng
Department of Computer Science, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsin-Chu 30010, Taiwan Available online 25 December 2007
Abstract
Convergecast is a fundamental operation in wireless sensor networks. Existing convergecast solutions have focused on reducing latency and energy consumption. However, a good design should be compliant to standards, in addition to considering these factors. Based on this observation, this paper defines a minimum delay beacon scheduling problem for quick convergecast in ZigBee tree-based wireless sensor networks and proves that this problem is NP-complete. Our formulation is compliant with the low-power design of IEEE 802.15.4. We then propose optimal solutions for special cases and heuristic algorithms for general cases. Simulation results show that the proposed algorithms can indeed achieve quick convergecast.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Convergecast; IEEE 802.15.4; Scheduling; Wireless sensor network; ZigBee
1. Introduction
The rapid progress of wireless communication and embedded micro-sensing MEMS technologies has made wireless sensor networks (WSNs) possible. A WSN consists of many inexpensive wireless sensors capable of collecting, storing, processing environmental information, and com-municating with neighboring nodes. Applications of WSNs include wildlife monitoring [3,4], object tracking [16,18], and dynamic path finding[15,19].
Recently, several WSN platforms have been developed, such as MICA[6]and Dust Network[2]. For interoperabil-ity among different systems, standards such as ZigBee[24]
have been developed. In the ZigBee protocol stack, physi-cal and MAC layer protocols are adopted from the IEEE 802.15.4 standard[13]. ZigBee solves interoperability issues from the physical layer to the application layer.
ZigBee supports three kinds of networks, namely star, tree, and mesh networks. A ZigBee coordinator is
responsi-ble for initializing, maintaining, and controlling the net-work. A star network has a coordinator with devices directly connecting to the coordinator. For tree and mesh networks, devices can communicate with each other in a multihop fashion. The network is formed by one ZigBee coordinator and multiple ZigBee routers. A device can join a network as an end devices by the associating with the coordinator or a router. In a tree network, the coordinator and routers can announce beacons. However, in a mesh network, regular beacons are not allowed. Beacons are an important mechanism to support power management. Therefore, the tree topology is preferred, especially when energy saving is a desirable feature. To support ZigBee bea-con-enabled tree networks, the IEEE 802.15 WPAN Task Group 4 further defines a revision of the IEEE 802.15.4
[14] specification in 2006. One of the major changes is structure of superframes to support power management. On the contrary, to our understanding, power management is still impossible for mesh-based ZigBee networks in the current specification. Therefore, we will focus on tree-based, beacon-enabled ZigBee networks in this work.
Considering that data gathering is a major application of WSNs, convergecast has been investigated in several 0140-3664/$ - see front matterÓ 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.comcom.2007.12.015 *
Corresponding author. Tel.: +886 933243597.
E-mail addresses: [email protected] (M.-S. Pan), yctseng@cs.
nctu.edu.tw(Y.-C. Tseng).
www.elsevier.com/locate/comcom Computer Communications 31 (2008) 999–1011
works[8,9,11,17,20,23]. With the goals of low latency and low energy consumption, Ref. [20] shows how to connect sensors as a balanced reporting tree and how to assign CDMA codes to sensors to diminish interference among sensors, thus achieving energy efficiency. The work [23]
aims to minimize the overall energy consumption under the constraint that sensed data should be reported within specified time. Dynamic programming algorithms are pro-posed by assuming that sensors can receive multiple pack-ets at the same time. As can be seen, both[20]and[23]are based on quite strong assumptions on communication capability of sensor nodes and they do not fit into the Zig-Bee specification. In [17], the authors propose an energy-efficient and low-latency MAC, called DMAC. Sensors are connected by a tree and stay in sleep state for most of the time. When sensors change to active state, they are first set to the receive mode and then to the transmit mode. DMAC achieves low-latency by staggering wake-up sched-ules of sensors at the time instant when their children switch to the transmit mode. Similar to [17], Ref. [11]
arranges wake-up schedule of sensors by taking traffic loads into account. Each parent periodically broadcasts an advertisement containing a set of empty slots. Children nodes request empty slots according to their demands. In
[9], the authors propose a distributed convergecast schedul-ing algorithm. The basic concept is to connect nodes by a spanning tree. Then the algorithm reduces the tree to multi-ple lines. For each line, the algorithm schedules nodes’ transmission times in a bottom-up manner. Ref. [8] pre-sents a centralized solution to convergecast. The algorithm divides nodes into many segments such that the transmis-sion of a node in a segment does not cause interference to other transmissions in the same segment. The aim is to increase the degree of parallel transmissions to decrease latencies. Although these results [8,9,11,17] are designed for quick convergecast, the solutions are not compliant to the ZigBee standard for the following two reasons. Firstly, in these works, nodes’ wake/sleep times are dynamically changed according to their schedules. However, in a Zig-Bee beacon-enabled tree network, nodes’ wake/sleep times must be fixed in the way that each router wakes up twice in each cycle to receive its children’s packets and to transmit packets to its parent, respectively. The coordinator (resp., an end device) wakes up once to receive its children’s pack-ets (resp., to transmit packpack-ets to its parent). Secondly, the scheduling of [8,9,11,17]is transmission-based, while ours are receiving-based. The implication is that the former may cause a router to be active multiple times per cycle. This is incompatible with the ZigBee specification.
This paper aims at designing quick convergecast solu-tions for ZigBee tree-based, beacon-enabled WSNs. This work is motivated by the following observations. First, we see that most related works are not compliant to the ZigBee standard. Second, we believe that tree-based topol-ogy is more suitable if power management is a main con-cern in WSNs. The network scenario is shown in Fig. 1. The network contains one sink (ZigBee coordinator), some
ZigBee routers, and some ZigBee end devices. Each ZigBee router is responsible for collecting sensed data from end devices associated with it and relaying incoming data to the sink. According to specifications, a ZigBee router can announce a beacon to start a superframe. Each superframe consists of an active portion followed by an inactive portion. On receiving its parent router’s beacon, an end device has to wake up for an active portion to sense the environment and communicate with its coordinator. However, to avoid collision with its neighbors, a router should shift its active portion by a certain amount.Fig. 1shows a possible allo-cation of active portions for routers A, B, C, and D. The collected sensory data of A in the k-th superframe can be sent to C in the same superframe. However, because the active portion of B in the k-th superframe appears after that of C, the collected data of B in the k-th superframe can only be relayed to C in the (k + 1)-th superframe. The report delay from B to C is almost the length of one superframe. The delay can be eliminated if the active por-tion of B in the k-th superframe appears before that of C. The delay is not negligible because of the low duty cycle design of IEEE 802.15.4. For example, in 2.4 GHz PHY, with 1.56% duty cycle, a superframe can be as long as 251.658 s (with an active portion of 3.93 s). Clearly, for large-scale WSNs, the convergecast latency could be signif-icant if the problem is not carefully addressed. The quick convergecast problem is to schedule the beacons of routers to minimize the convergecast latency. We prove that this problem is NP-complete by reducing the 3-CNF-SAT problem to it. We show two special cases of this problem where optimal solutions can be found in polynomial time and propose two heuristic algorithms for general cases. To the best of our knowledge, this is the first result that provides convergecast solutions in ZigBee beacon-enabled tree networks.
The rest of this paper is organized as follows. Section 2
briefly introduces IEEE 802.15.4 and ZigBee. The quick convergecast problem is formally defined in Section3. Sec-tion4presents our scheduling solutions. Simulation results are given in Section 5. Finally, Section 6 concludes this paper.
2. Overview of IEEE 802.15.4 and ZigBee standards IEEE 802.15.4 [13] specifies the physical and data link protocols for low-rate wireless personal area networks (LR-WPAN). In the physical layer, there are three fre-quency bands with 27 radio channels. Channel 0 ranges from 868.0 to 868.6 MHz, which provides a data rate of 20 kbps. Channels 1 to 10 work from 902.0 to 928.0 MHz and each channel provides a data rate of 40 kbps. Channels 11 to 26 are located from 2.4 to 2.4835 GHz, each with a data rate of 250 kbps.
IEEE 802.15.4 devices are expected to have limited power, but need to operate for a longer period of time. Therefore, energy conservation is a critical issue. Devices are classified as full function devices (FFDs) and reduced
function devices (RFDs). IEEE 802.15.4 supports star and peer-to-peer topologies. In each PAN, one device is desig-nated as the coordinator, which is responsible for maintain-ing the network. A FFD has the capability of servmaintain-ing as a coordinator or associating with an existing coordinator/ router and becoming a router. A RFD can only associate with a coordinator/router and can not have children.
The ZigBee coordinator defines the superframe structure of a ZigBee network. As shown inFig. 2(a), the structure of superframes is controlled by two parameters: beacon order (BO) and superframe order (SO), which decide the lengths of a superframe and its active potion, respectively. For a beacon-enabled network, the setting of BO and SO should satisfy the relationship 0 6 SO 6 BO 6 14. (A non-bea-con-enabled network should set BO¼ SO ¼ 15 to indicate that superframes do not exist.) Each active portion consists of 16 equal-length slots, which can be further partitioned into a contention access period (CAP) and a contention free period (CFP). The CAP may contain the first i slots, and the CFP contains the rest of the 16 i slots, where 1 6 i 6 16. Slotted CSMA/CA is used in CAP. FFDs which require fixed transmission rates can ask for guaran-tee time slots (GTSs) from the coordinator. A CFP can support multiple GTSs, and each GTS may contain multi-ple slots. Note that only the coordinator can allocate GTSs. After the active portion, devices can go to sleep to save energy.
In a beacon-enabled star network, a device only needs to be active for 2ðBOSOÞ portion of the time. Changing the value of ðBO SOÞ allows us to adjust the on-duty time of devices. However, for a beacon-enabled tree net-work, routers have to choose different times to start their active portions to avoid collision. Once the value of ðBO SOÞ is decided, each router can choose from 2BOSO slots as its active portion. In the revised version of IEEE 802.15.4 [14], a router can select one active por-tion as its outgoing superframe, and based on the active portion selected by its parent, the active portion is called its incoming superframe (as shown inFig. 2(b)). In an out-going/incoming superframe, a router is expected to trans-mit/receive a beacon to/from its child routers/parent router. When choosing a slot, neighboring routers’ active portions (i.e., outgoing superframes) should be shifted away from each other to avoid interference. This work is motivated by the observation that the specification does not clearly define how to choose the locations of routers’ active portions such that the convergecast latency can be reduced. In our work, we consider two kinds of interfer-ence between routers. Two routers have direct interferinterfer-ence if they can hear each others’ beacons. Two routers have indirect interference if they have at least one common neighbor. Both interferences should be avoided when choosing routers’ active portions. Table 1 lists possible choices of ðBO SOÞ combinations.
A B
C
Sink
ZigBee router (FFD) ZigBee end device (RFD) Interference neighbor k-th superframe CAP CAP Schedule of A Schedule of C CAP CAP Schedule of B report report CAP Schedule of D report to sink CAP report report CAP (k+1)-th superframe data from end devices data from end devices data from end devices data from end devices data from end devices data from end devices data from end devices C A B D
3. The minimum delay beacon scheduling (MDBS) problem This section formally defines the convergecast problem in ZigBee networks. Given a ZigBee network, we model it by a graph G¼ ðV ; EÞ, where V contains all routers and the coordinator and E contains all symmetric commu-nication links between nodes in V. The coordinator also serves as the sink of the network. End devices can only associate with routers, but are not included in V. From G, we can construct an interference graph GI¼ ðV ; EIÞ, where edge ði; jÞ 2 EI if there are direct/indirect interfer-ences between i and j. There is a duty cycle requirement a for this network. From a and Table 1, we can determine
the most appropriate value of BO SO. We denote by
k¼ 2BOSOthe number of active portions (or slots) per bea-con interval.
The beacon scheduling problem is to find a slot assign-ment sðiÞ for each router i 2 V , where sðiÞ is an integer and sðiÞ 2 ½0; k 1, such that router i’s active portion is in slot sðiÞ and sðiÞ 6¼ sðjÞ if ði; jÞ 2 EI. Here, the slot assign-ment means the position of the outgoing superframe of each router (the position of the incoming superframe, as clarified earlier, is determined by the parent of the router). Motivated by Brook’s theorem [21], which proves that n colors are sufficient to color any graph with a maximum degree of n, we would assume that k P DI, where DI is the maximum degree of GI.
Given a slot assignment for G, the report latency from node i to node j, whereði; jÞ 2 E, is the number of slots, denoted by dij, that node i has to wait to relay its collected sensory data to node j, i.e.,
dij¼ ðsðjÞ sðiÞÞ mod k: ð1Þ
Note that the report latency from node i to node j ðdijÞ may not by equal to the report latency from node j to node i ðdjiÞ. Therefore, we can convert G into a weighted directed graph GD¼ ðV ; EDÞ such that each ði; jÞ 2 E is translated into two directed edges ði; jÞ and ðj; iÞ such that wðði; jÞÞ ¼ dij and wððj; iÞÞ ¼ dji. The report latency for each i2 V to the sink is the sum of report latencies of the links on the shortest path from i to the sink in GD. The latency of the con-vergecast, denoted as LðGÞ, is the maximum of all nodes’ report latencies.
Definition 1. Given G¼ ðV ; EÞ, G’s interference graph GI¼ ðV ; EIÞ, and k available slots, the minimum delay beacon scheduling (MDBS) problem is to find an interfer-ence-free slot assignment sðiÞ for each i 2 V such that the convergecast latency LðGÞ is minimized.
To prove that the MDBS problem is NP-complete, we define a decision problem as follows.
Definition 2. Given G¼ ðV ; EÞ, G’s interference graph GI¼ ðV ; EIÞ, k available slots, and a delay constraint d, 0 1 2 3 4 5 6 7 8 9 10111213 1415 Received Beacon Transmitted Beacon Inactive BI = aBaseSuperframeDuration×2BO symbols Inactive Received Beacon Start Time >SD 0 1 2 3 4 5 6 7 8 9 10111213 1415 SD = aBaseSuperframeDuration×2SO symbols (Incoming superframe) SD = aBaseSuperframeDuration×2SO symbols (Outgoing superframe) 10 9 7 8 6 5 4 3 2 1 0 1112 131415 GTS 1 GTS 2 Beacon Beacon Inactive CAP CFP BI = aBaseSuperframeDuration×2BO symbols GTS 0 SD = aBaseSuperframeDuration×2SO symbols (Active)
a
b
Fig. 2. IEEE 802.15.4 superframe structure. Table 1
Relationship of BO SO, duty cycle, and the number of active portions in a superframe
BO SO 0 1 2 3 4 5 6 7 8 P9
Duty cycle (%) 100 50 25 12.5 6.25 3.13 1.56 0.78 0.39 60.195
the bounded delay beacon scheduling (BDBS) problem is to decide if there exists an interference-free slot assignment sðiÞ for each i 2 V such that the convergecast latency LðGÞ 6 d.
Theorem 1. The BDBS problem is NP-complete.
Proof. First, given slot assignments for nodes in V, we can find the report latency of each i2 V by running a shortest path algorithm on GD. We can then check if LðGÞ 6 d. Clearly, this takes polynomial time.
We then prove that the BDBS problem is NP-hard by reducing the 3 conjunctive normal form satisfiability (3-CNF-SAT) problem to a special case of the BDBS problem in polynomial time. Given any 3-CNF formula C, we will construct the corresponding G and GI. Then we show that C is satisfiable if and only if there is a slot assignment for
each i2 V using no more than k ¼ 3 slots such that
LðGÞ 6 4 slots.
Let C¼ C1^ C2^ ^ Cm, where clause Cj¼ xj;1 _xj;2_ xj;3, 1 6 j 6 m, xj;i2 fX1; X2; . . . ; Xng, and Xi2 fxi; xig, where xi is a binary variable, 1 6 i 6 n. We first construct G from C as follows:
1. For each clause Cj, j¼ 1; 2; . . . ; m, add a vertex Cjin G. 2. For each literal Xi, i¼ 1; 2; . . . ; n, add four vertices xi1,
xi2, xi1, and xi2 in G.
3. Add a vertex t as the sink of G.
4. Add edgesðt; xi2Þ and ðt; xi2Þ to G, for i ¼ 1; 2; . . . ; n. 5. Add edgesðxi1; xi2Þ and ðxi1; xi2Þ to G, for i ¼ 1; 2; . . . ; n. 6. For each i¼ 1; 2; . . . ; n and each j ¼ 1; 2; . . . ; m, add an edgeðCj; xi1Þ (resp., ðCj; xi1Þ) to G if xi(resp., xi) appears in Cj.
Then we construct GI as follows. 1. Add all vertices and edges in G into GI.
2. Add edgesðxi1; xi1Þ and ðxi2; xi2Þ to GI, for i¼ 1; 2; . . . ; n. 3. Add edgesðCj; xi2Þ and ðCj; xi2Þ to GI, for i¼ 1; 2; . . . ; n
and j¼ 1; 2; . . . ; m.
Then we build a one-to-one mapping from each truth assignment of C to a slot assignment of G. We establish the following mapping:
1. Set sðtÞ ¼ 0.
2. Set sðCjÞ ¼ 0, j ¼ 1; 2; . . . ; m.
3. Set sðxi1Þ ¼ 1 and sðxi2Þ ¼ 1, i ¼ 1; 2; . . . ; n, if xi is true; otherwise, set sðxi1Þ ¼ 2 and sðxi2Þ ¼ 2.
4. Set sðxi2Þ ¼ 1 and sðxi1Þ ¼ 1, i ¼ 1; 2; . . . ; n, if xi is true; otherwise, set sðxi2Þ ¼ 2 and sðxi1Þ ¼ 2.
The above reduction can be computed in polynomial time. By the above reduction, vertices xi1 or xi1, i¼ 1; 2; . . . ; n, that are assigned to slot 1 (resp., slot 2) will have a report latency of 2 (resp., 4) and vertices xi2 or xi2, i¼ 1; 2; . . . ; n, that are assigned to slot 1 (resp., slot 2) will have a report latency of 2 (resp., 1). Hence, for those
vertices xi1, xi1, xi2, and xi2, i¼ 1; 2; . . . ; n, the longest report latency will be 4.
To prove the if part, we need to show that if C is satisfiable, there is a slot assignment such that k¼ 3 and LðGÞ 6 4. Since C satisfiable, there must exist an assign-ment such that each clause Cj, j¼ 1; 2; . . . ; m, is true. If a clause Cj is true, at least one variable in Cj is true. According to the reduction, Cj can always find an edge ðCj; xi1Þ or ðCj; xi1Þ with wððCj; xi1ÞÞ ¼ 1 or wððCj; xi1ÞÞ ¼ 1, where i¼ 1; 2; . . . ; n. Thus, when C is satisfiable, the reporting latency for each clause is 3. This achieves LðGÞ ¼ 4.
For the only if part, if each vertex Cj, j¼ 1; 2; . . . ; m, can find at least an edge with weight 1 to one of xi1and xi1, for i¼ 1; 2; . . . ; n, to achieve a report latency of 3, it must be that each clause has at least one variable to be true. So formula C is satisfiable. Otherwise, the report latency of Cj, j¼ 1; 2; . . . ; m, will be 6. h
For example, given C¼ ðx1_ x2_ x3Þ ^ ðx1_ x2_ x3Þ^ ðx1_ x2_ x3Þ,Fig. 3shows the corresponding G. The truth assignment ðx1; x2; x3Þ ¼ ðT ; F ; T Þ makes C satisfiable. According to the reduction and the mapping in the above proof, we can obtain the network G and its slot assignment as shown inFig. 3such that LðGÞ ¼ 4.
4. Algorithms for the MDBS problem 4.1. Optimal solutions for special cases
Optimal solutions can be found for the MDBS problem in polynomial time for regular linear networks and regular ring networks, as illustrated in Fig. 4. In such networks, each vertex is connected to one or two adjacent vertices and has an interference relation with each neighbor within h hops from it, where h P 2. In a regular linear network, we assume that the sink t is at one end of the network. Clearly, the maximum degree of GIis 2h. We will show that an optimal solution can be found if the number of slots kP hþ 1. The slot assignment can be done in a bottom-up manner. The bottom node is assigned to slot 0. Then,
1 2 0 0 2 1 0 1 2 2 1 2 1 1 2 0 C1 C2 C3 x11 x12 x11 x12 x22 x31 x21 x32 x21 x22 x31 x32 t
Fig. 3. An example of reduction from the 3-CNF-SAT to the BDBS problem.
for each vertex v, sðvÞ ¼ ðk0 þ 1Þ mod k, where k0 is the slot assigned to v’s child.
Theorem 2. For a regular linear network, if k P hþ 1, the above slot assignment achieves a report latency ofj V j 1, which is optimal.
Proof. Clearly, the slot assignment is interference-free. Also the report latency of j V j 1 is clearly the lower
bound. h
For a regular ring network, we first partition vertices excluding t into left and right groups as illustrated in
Fig. 4(b) such that the left group consists of the sink node t andbjV j12 c other nodes counting counter clockwise from t, and the right group consists of thosedjV j12 e nodes counting clockwise from t. Now we consider the ring as a spanning tree with t as the root and left and right groups as two lin-ear paths. Assuming thatbjV j1
2 c P 2h and k P 2h, the slot assignment works as follows:
1. The bottom node in the left group is assigned to slot 0. 2. All other nodes in the left group are assigned with slots in a bottom-up manner. For each node i in the left group, we let sðiÞ ¼ ðj þ 1Þ mod k, where j is the slot of i’s child. 3. Nodes in the right group are assigned with slots in a
top-down manner. For each node i in the right group, we let sðiÞ ¼ ðj cÞ mod k, where j is the slot assigned to i’s parent and c is the smallest constant (1 6 c 6 k) that ensures that sðiÞ is not used by any of its interference neighbors that have been assigned with slots.
It is not hard to prove the slot assignment is interfer-ence-free because nodes receives slots sequentially and we have avoided using the same slots among interfering neigh-bors. Although this is a greedy approach, we show that c is
equal to 1 in step 3 in most of the cases except when two special nodes are visited. This gives an asymptotically opti-mal algorithm, as proved in the following theorem. Theorem 3. For a regular ring network, assuming that kP2h andbjV j12 c P 2h, the above slot assignment achieves a report latency LðGÞ ¼ bjV j12 c þ h, which is optimal within a factor of 1.5.
Proof. We first identify three nodes on the ring (refer to
Fig. 4(b)):
l1: the bottom node in the left group. r1: the first node in the right group.
r2: the node that is h hops from l1 counting counter clockwise.
The report latency of each node can be analyzed as follows. The parent of node x is denoted by parðxÞ.
A1. For each node i in the left group except the sink t, the latency from i to parðiÞ is 1.
A2. The latency from r1 to t is h.
A3. For each node i next to r1 in the right group but before r2 (counting clockwise), the latency from i to parðiÞ is 1.
A4. The latency from r2 to parðr2Þ is 1 if the ring size is even; otherwise, the latency is 2.
A5. For each node i in the right group that is a descendant of r2, the report latency from i to parðiÞ is 1.
It is not hard to prove that A1, A2, and A3 are true. To see A4 and A5, we make the following observations. The function pariðxÞ is to apply i times the parðÞ function on node x. Note that par0ðxÞ means x itself.
0 1 2 0 t 1 2 0 1 2 0 1 0 2 3 1 3 2 1 2 0 3 size:11 t
left group right group l1 r1 r2 1 0 2 3 1 2 3 2 0 0 1 3 size:12 t left group right group l1 r1 r2
a
b
O1. When the ring size is even, the equality sðpari1ðl1ÞÞ ¼ sðpariðr2ÞÞ holds for i¼ 1; 2; . . . ; bjV j12 c h 1. More specifically, this means that (i) l1 and parðr2Þ will receive the same slot, (ii) parðl1Þ and par2ðr2Þ will receive the same slot, etc. This can be proved by induction by showing that the i-th descendant of t in the right group will be assigned the same slot as theðh þ i 1Þ-th descendant of t in the left group (the induction can go in a top-down manner). This property implies that when assigning a slot to r2 in step 3, c¼ 1 in case that the ring size is even. Further, r2 and its descendants will be sequentially assigned to slots k 1, k 2, . . ., k h, which implies that c¼ 1 when doing the assignments in step 3. So properties A4 and A5 hold for the case of an even ring.
O2. When the ring size is odd, the equality
sðpariðl1ÞÞ ¼ sðpariðr2ÞÞ holds for i¼ 1; 2; . . . ; bjV j12 c h. This means that (i) parðl1Þ and parðr2Þ will receive the same slot, and (ii) par2ðl1Þ and par2ðr2Þ will receive the same slot, etc. Again, this can be proved by induction as in O1. This property implies that c¼ 2 when assigning a slot to r2 in step 3, and c¼ 1 when assigning slots to descendants of r2. So properties A4 and A5 hold for the case of an odd ring.
The equality of slot assignments pointed out in O1 and O2 is illustrated in Fig. 4(b) by those numbers in gray nodes. In summary, the report latency of the left group is bjV j12 c. When the ring size is even, the report latency of the right group is the number of nodes in this group, jV j2,
plus the extra latency h 1 incurred at r1. So
LðGÞ ¼jV j2 þ h 1 ¼ bjV j12 c þ h. When the ring size is odd, the report latency of right group is the number of nodes in this group, jV j12 , plus the extra latency h 1 incurred at r1 and the extra latency 1 incurred at r2. So LðGÞ ¼ bjV j12 c þ h.
A lower bound on the report latency of this problem is the maximum number of nodes in each group excluding t. Applying bjV j12 c as a lower bound and using the fact that bjV j12 c P 2h, LðGÞ will be smaller than 1:5 bjV j12 c, which implies the algorithm is opti-mal within a factor of 1.5. Note that the condition bjV j12 c P 2h is to guarantee that t will not locate within h hops from r2. Otherwise, the observation O2 will not
hold. h
4.2. A centralized tree-based assignment scheme
Given G¼ ðV ; EÞ, GI¼ ðV ; EIÞ, and k, we propose a centralized slot assignment heuristic algorithm. Our algo-rithm is composed of the following three phases:
phase 1. From G, we first construct a BFS tree T rooted at sink t.
phase 2. We traverse vertices of T in a bottom-up manner. For these vertices in depth d, we first sort them according to their degrees in GI in a descending order. Then we sequentially traverse these vertices in that order. For each vertex v in depth d visited, we compute a temporary slot number tðvÞ for v as follows.
1. If v is a leaf node, we set tðvÞ to the minimal non-neg-ative integer l such that for each vertex u that has been visited andðu; vÞ 2 EI, (tðuÞ mod k) 6¼ l.
2. If v is an in-tree node, let m be the maximum of the numbers that have been assigned to v’s children, i.e., m¼ maxftðchildðvÞÞg, where childðvÞ is the set of v’s children. We then set tðvÞ to the minimal non-negative integer l > m such that for each vertex u that has been visited andðu; vÞ 2 EI, (tðuÞ mod k) 6¼ ðl mod kÞ. After every vertex v is visited, we make the assignment sðvÞ ¼ tðvÞ mod k.
phase 3. In this phase, vertices are traversed sequentially from t in a top-down manner. When each vertex v is visited, we try to greedily find a new slot l such that (sðparðvÞÞ l) mod k < ðsðparðvÞÞ
sðvÞÞ mod k, such that l6¼ sðuÞ for each
ðu; vÞ 2 EI, if possible. Then we reassign sðvÞ ¼ l. Note that in phase 2, a node with a higher degree means that it has more interference neighbors, implying that it has less slots to use. Therefore, it has to be assigned to a slot earlier. Also note that, the number tðvÞ is not a modulus number. However, in step 2 of phase 2, we did check that if tðvÞ is converted to a slot number, no interference will occur. Intuitively, this is a temporary slot assignment that will incur the least latency to v’s chil-dren. At the end, tðvÞ is converted to a slot assignment sðvÞ. Phase 3 is a greedy approach to further reduce the report latency of routers. For example, Fig. 5(a) shows the slot assignment after phase 2. Fig. 5(b) indicates that B, C, and D can find another slots and their report laten-cies are decreased. This phase can reduce LðGÞ in some cases.
The computational complexity of this algorithm is analyzed below. In phase 1, the complexity of construct-ing a BFS tree is Oðj V j þ j E jÞ. In phase 2, the cost of sorting is at most Oðj V j2Þ and the computational cost to compute tðvÞ for each vertex v is bounded by OðkDIÞ, where DI is the degree of GI. So the time complexity of phase 2 is Oðj V j2þ kDIj V jÞ. Phase 3 performs a similar procedure as phase 2, so its time complexity is
also OðkDIj V jÞ. Overall, the time complexity is
Oðj V j2þ kDIj V jÞ.
4.3. A distributed assignment scheme
In this section, we propose a distributed slot assignment algorithm. Each node has to compute its direct as well as indirect interference neighbors in a distributed manner. To achieve this, we will refer to the heterogeneity approach
in [22], which adopts power control to achieve this goal. Assuming routers’ default transmission range is r, interfer-ence neighbors must locate within range 2r. From time-to-time, each router will boost its transmission power to dou-ble its default transmission range and send HELLO pack-ets to its neighbor routers. Each HELLO packet further contains sender’s (1) depth,1 (2) the location of outgoing superframe (i.e., slot), and (3) number of interference neighbors. Note that all other packets are transmitted by the default power level. When booting up, each router will broadcast HELLO packets claiming that its depth and slot are NULL. After joining the network and choos-ing a slot, the HELLO packets will carry the node’s depth and slot information. The algorithm is triggered by the sink t setting sðtÞ ¼ k 1 and then broadcasting its beacon. A router v6¼ t that receives a beacon will decide its slot as follows.
1. Node v sends an association request to the beacon sender.
2. If v fails to associate with the beacon sender, it stops the procedure and waits for other beacons.
3. If v successfully associates with a parent node parðvÞ, it computes the smallest positive integer l such that ðsðparðvÞÞ lÞ mod k 6¼ sðuÞ for all ðu; vÞ 2 EI and sðuÞ 6¼ NULL. Then v chooses sðvÞ ¼ ðsðparðvÞÞ lÞ mod k as its slot.
4. Then, v broadcasts HELLOs including its slot assign-ment sðvÞ for a time period twait. If it finds that sðvÞ ¼ sðuÞ for any ðu; vÞ 2 EI, v has to change to a new slot if one of the following rules is satisfied and goes back to step 3.
(a) Node u has more interference neighbors than v. (b) Node u and v have the same number of interference
neighbors but the depth of u is lower than v, i.e., u is closer to the sink than v.
(c) Node u and v have the same number of interference neighbors and they are at the same depth but the u’s ID is smaller than v’s.
5. After twait, v can finalize its slot selection and broadcast its beacons.
In this distributed algorithm, slots are assigned to rou-ters, ideally, in a top-down manner. However, due to trans-mission latency, some routers at lower levels may find slots earlier than those at higher levels. Also note that the time twait is to avoid possible collision on slot assignments due to packet loss.
5. Simulation results
This section presents our simulation results. We first assume that the size of sensory data is negligible and that all routers generate reports at the same time, and com-pare the performances of different convergecast algo-rithms. Then we simulate more realistic scenarios where the size of sensory data is not negligible and routers need to generate reports periodically or passively driven by events randomly appearing in certain regions in the sens-ing field. More specifically, sensors generate reports according to certain application specifications. Devices all run ZigBee and IEEE 802.15.4 protocols to communi-cate with each other. Routers can aggregate child sen-sors’ reports and report to their parents directly. Each router has a fix-size buffer. When a router’s buffer over-flows, this router will not accept further incoming frames. We also measure the goodput of the network, which is defined as the ratio of sensors’ reports successfully received by the sink. Some parameters used in our simu-lation are listed in Table 2.
5.1. Comparison of different convergecast algorithms We compare the proposed slot assignment algorithms against a random slot assignment (denoted by RAN) scheme and a greedy slot assignment (denoted by GDY) scheme. In RAN, the slot assignment starts from the sink and each router, after associating with a parent router, sim-ply chooses any slot which has not been used by any of its interference neighbors. In GDY, routers are given a sequence number in a top-down manner. The sink sets its slot to k 1. Then the slot assignment continues in sequence. For a node i, it will try to find a slot sðiÞ ¼ sðjÞ l mod k, where j is the predecessor of i and l is the smallest integer letting sðiÞ is the slot which does not assign to any of i’s interference neighbors. In the sim-ulations, routers are randomly distributed in a circular region of a radius r and a sink is placed in the center. Our centralized tree-based scheme and distributed slot assignment scheme are denoted as CTB and DSA, respec-tively. We compare the report latency LðGÞ (in terms of slots).
Fig. 6 shows some slot assignment results of CTB and DSA when r = 35 m and k¼ 64. Devices are randomly dis-tributed. The transmission range of routers is set to 20 m. In this case, CTB performs better than DSA.
1 The depth of a node is the length of the tree path from the root to the node. The root node is at depth zero.
a
b
5 1 0 0 6 5 4 3 3 6 E EI t A B C D t A B C DNext, we observe the impact of different r, CR (number of routers), and TR(transmission distance).Fig. 7(a) shows
the impact of r when k¼ 64, TR¼ 25 m, and
CR¼ 3 ðr=10Þ 2
. CTB performs the best. DSA performs slightly worse than CTB, but still significantly outperforms RAN and GDY. It can be seen that RAN and GRY could result in very long convergecast latency. Both CTB and DSA are quite insensitive to the network size. But this is
not the case for RAN and GDY. Fig. 7(b) shows the
impact of TRwhen CR¼ 300, r ¼ 100 m, and k ¼ 64. Since a larger transmission range implies higher interference among routers, the report latencies of CTB and DSA will increase linearly as TR increases. The report latency of RAN also increases when TR¼ 17–21 m because of the increased interference. After TRP22 m, the latency of RAN decreases because that the network diameter is reduced. Basically, GDY behaves the same as CTB and DSA. But when the transmission range is larger, the report latency slightly becomes small.
Fig. 7(c) shows the impact of CR when r¼ 100 m, TR¼ 20 m, and k ¼ 128. As a larger CRmeans a higher net-work density and thus more interference, the report laten-cies of CTB and DSA increase as CR increases. Since the network diameter is bounded, the report latency of RAN is also bounded. GDY is sensitive to the number of routers when there are less routers. This is because that each router can own a slot and the report latency increases proportion-ally to the number of routers. With r¼ 100 m, CR¼ 300, and TR¼ 20 m,Fig. 7(d) shows the impact of routers’ duty cycle. Note that a lower duty cycle means a larger number of available slots. Interestingly, we see that the report laten-cies of CTB, DSA, and GDY are independent of the num-ber of slots. Contrarily, with a random assignment, RAN even incurs a higher report latency as there are more free-dom in slot selection.
5.2. Periodical reporting scenarios
Next, we assume that sensors are instructed to report their data in a periodically manner. We set r¼ 100 m, TR¼ 20 m, and CR¼ 300 with 6000 randomly placed
sen-sors associated to these routers, and we further restrict a router can accept at most 30 sensors. BO SO is fixed to six, so k¼ 2BOSO¼ 64. Since the earlier simulations show that CTB and DSA perform quite close, we will use only CTB to assign routers’ slots. Sensors are required to gener-ate a report every 251.66 s (the length of one beacon inter-val when BO¼ 14). We set the buffer size of each router is 10 KB.2We allocate two mini-slots for each child router of the sink as the GTS slot.3
SinceðBO SOÞ is fixed, a small BO implies a smaller slot size (and thus a smaller unit size of LðGÞ). So, a smaller slot size seemingly implies higher contention among sen-sors if they all intend to report to their parents simulta-neously. In fact, a smaller BO does not hurt the overall reporting times of sensors if we can properly divide sensors into groups. For example, inFig. 8, when BO¼ 14, all sen-sors of a router can report in every superframe. When BO¼ 13, if we divide sensors into two groups, then they can report alternately in odd and even superframes. Simi-larly, when BO¼ 12, four groups of sensors can report alternately. Since the length of superframes are reduced proportionally, the report intervals of sensors actually remain the same in these cases. In the following experi-ments, we groups sensors according to their parents’ IDs. A sensor belongs to group m if the modulus of its parent’s ID is m.
Fig. 9 shows the theoretical and actual report latencies under different BOs. Note that a report may be delayed due to buffer constraint. As can be seen, the actual latency does not always favor a smaller BO. Our results show that BO¼ 10–12 performs better.Fig. 9(b) shows the goodput of sensory reports, channel utilization at the sink, and the
number of dropped frames at the sink. When BO¼ 14,
although there is no frames being dropped at the sink, the goodput is still low. This is because a lot of collisions happen inside the network, causing many sensory reports being dropped at intermediate levels (a frame is dropped after exceeding its retransmission limit). Fig. 10 shows a log of the numbers of frames received by a sink’s child rou-ter when BO¼ 14. We can see that more than half of the active portion is wasted. Overall, BO¼ 10 produces the best goodput and a shorter report latency.
Some previous works can be also integrated in this peri-odical reporting scenario, such as the adaptive GTS alloca-tion mechanism in[12]and the aggregation algorithms for WSNs in[7,10].Fig. 11shows an experiment that routers can compress reports from sensors with a rate cr when BO¼ 10. If a router receives n reports and each report’s size is 16 Bytes (as in Table 2), it can compress the size to 16 n ð1 crÞ. The report latencies decrease when the cr becomes larger. By compressing the report data, Table 2
Simulation parameters
Parameter Value
Length of a frame’s header and tail 18 Bytes
Length of a sensor’s report 16 Bytes
Beacon length 18 Bytes
Maximum length of a frame 127 Bytes
Bit rate 250 kbps
Symbol rate 62.5 k symbols/s
aBaseSuperframeDuration 960 symbols aUnitBackoffPeriod 20 symbols aCCATime 8 symbols MacMinBE 3 aMaxBE 5 MacMaxCSMABackoffs 4
Maximum number of retransmissions 3
2 Currently, there are some platforms which are equipped with larger RAMs. For example, Jennic JN5121 [5] has a 96 KB RAM and CC2420DBK[1]has a 32 KB RAM.
the goodput can up to 98% and the report can arrive to the sink more quickly.
5.3. Event-driven reporting scenarios
In the following, we assume that sensors’ reporting activities are triggered by events occurred at random loca-tions in the network with a rate k. The sensing range of
each sensors is 3 m and each event is a disk of a radius of 5 m. A sensor can detect an event if its sensing range overlaps with the disk of that event. Each router has an 1 KB buffer. When a sensor detects an event, it only tries to report that event once. All other settings are the same as those in Section5.2.
Fig. 12 shows the simulation results when k = 1/5s, 1/ 15s, and 1/30s. FromFig. 12(a), we can observe that when 0 50 100 150 200 250 300 110 100 90 80 70 60 50 40 30 Average L(G) Network radius (m) CTB DSA RAN GDY 0 50 100 150 200 250 300 17 18 19 20 21 22 23 24 25 26 Average L(G) Transmission range (m) CTB DSA RAN GDY
a
b
0 100 200 300 400 500 600 700 200 300 400 500 600 700 800 900 Average L(G)Number of ZigBee routers CTB DSA RAN GDY 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.098 0.195 0.39 0.78 1.56 Average L(G)
Network duty cycle (%) CTB
DSA RAN GDY
c
d
Fig. 7. Comparison of report latencies under different configurations.
L(G)=22
k = 64 k = 64
L(G)=19
BO is small, the report latency can not achieve to the the-oretical value. This is because that an active portion is too small to accommodate all reports from sensors, thus lengthening the report latency. When BO becomes larger, the theoretical and actual curves would meet. However, the good put will degrade, as shown inFig. 12(b). This is because reports are likely to be dropped due to buffer over-flow. How to determine a proper BO, which can contain most of the reports and guarantee low latency, is an impor-tant design issue for such scenarios.
6. Conclusions
In this paper, we have defined a new minimum delay beacon scheduling (MDBS) problem for convergecast with the restrictions that the beacon scheduling must be compli-ant to the ZigBee standard. We prove the MDBS problem is NP-complete and propose optimal solutions for special cases and two heuristic algorithms for general cases. Simu-lation results indicate the performance of our heuristic algorithms decrease only when the number of interference 0 20 40 60 80 100 120 140 160 14 13 12 11 10 9 8
L(G) x slot-size (in seconds)
BO Theoretical Actual 0 10 20 30 40 50 60 70 80 90 100 14 13 12 11 10 9 8 0 1 2 3 4 5 6 7 8 9 10
Goodput or channel utilization (%)
Number of dropped frames
BO Goodput Channel utilization The number of dropped frames
a
b
Fig. 9. Simulations considering buffer limitation and contention effects: (a) Theoretical vs. actual report latencies and (b) goodput, channel utilization, and number of dropped frames.
Report (Beacon) interval: 251.66 s Report (Beacon) interval: 251.66 s Active portion:
3.932 s
Active portion:
3.932 s Active portion: 3.932 s
time (s)
Number of frames received
0 5 10 15 20 25 30 134 135 136 137 138 139 385 386 387 388 389 390 391 636 637 638 639 640 641 642 643
Fig. 10. A log of the number of frames received by a sink’s child router when BO¼ 14.
BO=13 # of groups = 2 beacon beacon group 0 report beacon ... n th superframe (n+4)th superframe beacon (n+1)th superframe (n+2)th superframe beacon (n+3)th superframe
group 1 report group 0 report group 1 report
beacon beacon group 0 report beacon ... n th superframe beacon (n+1)th superframe beacon group 1 report (n+2)th superframe (n+3)th superframe (n+4)th superframe (n+5)th superframe (n+6)th superframe (n+7)th superframe (n+8)th superframe beacon
beacon beacon beacon
group 2 report group 3 report group 0 report group 1 report group 2 report group 3 report BO=12 # of groups = 4 BO=14 # of groups = 1 beacon beacon
all sensors report
beacon
...
n th superframe (n+1)th superframe (n+2)th superframe
all sensors report
neighbors is increased. Compared to the random slot assignment and greedy slot assignment scheme, our heuris-tic algorithms can effectively schedule the ZigBee routers’ beacon times to achieve quick convergecast. In the future, it deserves to consider extending this work to an asynchro-nous sleep scheduling to support energy-efficient converge-cast in ZigBee mesh networks.
Acknowledgements
Y.-C. Tseng’s research is co-sponsored by Taiwan MoE ATU Program, by NSC Grants 93-2752-E-007-001-PAE, 96-2623-7-009-002-ET, E-009-058-MY3, 95-2221-E-009-060-MY3, 95-2219-E-009-007, 95-2218-E-009-209, and 94-2219-E-007-009, by Realtek Semiconductor Corp., by MOEA under Grant No. 94-EC-17-A-04-S1-044, by ITRI, Taiwan, by Microsoft Corp., and by Intel Corp.
References
[1] Chipcon CC2420DBK. Available from:<http://www.chipcon.com/>. [2] Dust network Inc. Available from:
<http://dust-inc.com/flash-index.shtml>.
[3] Design and construction of a wildfire instrumentation system using networked sensors. Available from:<http://firebug.sourceforge.net/>. [4] Habitat monitoring on great duck island. Available from:<http://
www.greatduckisland.net/technology.php>.
[5] Jennic JN5121. Available from:<http://www.jennic.com/>. [6] Motes, smart dust sensors, wireless sensor networks. Available from:
<http://www.xbow.com/>.
[7] S.-J. Baek, G. de Veciana, X. Su, Minimizing energy consumption in large-scale sensor networks through distributed data compression and hierarchical aggregation, IEEE Journal on Selected Areas in Com-munications 22 (6) (2004) 1130–1140.
[8] H. Choi, J. Wang, E.A. Hughes, Scheduling on sensor hybrid network, in: Proceedings of IEEE International Conference on Computer Communications and Networks (ICCCN), San Diego, USA, 2005.
[9] S. Gandham, Y. Zhang, Q. Huang, Distributed minimal time convergecast scheduling in wireless sensor networks, in: Proceedings of IEEE International Conference on Distributed Computing Systems (ICDCS), Lisboa, Portugal, 2006.
[10] D. Ganesan, B. Greenstein, D. Perelyubskiy, D. Estrin, J. Heidemann. An evaluation of multiresolution storage for sensor networks, in: Proceedings of ACM International Conference on Embedded Networked Sensor Systems (SenSys), Los Angeles, USA, 2003.
[11] B. Hohlt, L. Doherty, E. Brewer, Flexible power scheduling for sensor networks, in: Proceedings of ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN), Berkeley, USA, 2004. 0 20 40 60 80 100 120 140 160 90 70 50 30 10
L(G) x slot-size (in seconds)
Compression rate (%) Theoretical Actual 0 10 20 30 40 50 60 70 80 90 100 90 70 50 30 10 0 0.5 1 1.5 2
Goodput or channel utilization (%)
Number of dropped frames
Compression rate (%) Goodput Channel utilization Number of dropped frames
a
b
Fig. 11. Simulations considering data compression: (a) Theoretical vs. actual report latencies and (b) goodput, channel utilization, and number of dropped frames. 0 5 10 15 20 25 30 35 40 7 8 9 10 11 12
L(G) x slot-size (in second)
BO Theoretical Actual(λ=1/5s) Actual(λ=1/15s) Actual(λ=1/30s) 0 10 20 30 40 50 60 70 80 90 100 7 8 9 10 11 12 Goodput (%) BO λ=1/5s λ=1/15s λ=1/30s
a
b
[12] Y.-K. Huang, A.-C. Pang, T.-W. Kuo, AGA: adaptive GTS allocation with low latency and fairness considerations for IEEE 802.15.4, in: Proceedings of IEEE International Conference on Communications (ICC), Istanbul, Turkey, 2006.
[13] IEEE standard for information technology – telecommunications and information exchange between systems – local and metropolitan area networks specific requirements part 15.4: wireless medium access control (MAC) and physical layer (PHY) specifications for low-rate wireless personal area networks (LR-WPANs), 2003.
[14] IEEE standard for information technology – telecommunications and information exchange between systems – local and metropolitan area networks specific requirements part 15.4: wireless medium access control (MAC) and physical layer (PHY) specifications for low-rate wireless personal area networks (LR-WPANs) (revision of IEEE Std 802.15.4-2003), 2006.
[15] Q. Li, M. DeRosa, D. Rus, Distributed algorithm for guiding navigation across a sensor network, in: Proceedings of ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), Maryland, USA, 2003.
[16] C.-Y. Lin, W.-C. Peng, Y.-C. Tseng, Efficient in-network moving object tracking in wireless sensor networks, IEEE Transactions on Mobile Computing 5 (8) (2006) 1044–1056.
[17] G. Lu, B. Krishnamachari, C.S. Raghavendra, An adaptive energy-efficient and low-latency MAC for data gathering in wireless sensor networks, in: Proceedings of IEEE International Parallel and Distributed Processing Symposium (IPDPS), New Mexico, USA, 2004.
[18] Y.-C. Tseng, S.-P. Kuo, H.-W. Lee, C.-F. Huang, Location tracking in a wireless sensor network by mobile agents and its data fusion strategies, The Computer Journal 47 (4) (2004) 448–460.
[19] Y.-C. Tseng, M.-S. Pan, Y.-Y. Tsai, Wireless sensor networks for emergency navigation, IEEE Computer 39 (7) (2006) 55–62. [20] S. Upadhyayula, V. Annamalai, S.K.S. Gupta, A low-latency and
energy-efficient algorithm for convergecast in wireless sensor net-works, in: Proceedings of IEEE Global Telecommunications Confer-ence (Globecom), San Francisco, USA, 2003.
[21] D.B. West, Introduction to Graph Theory, Prentice Hall, 2001. [22] M. Yarvis, N. Kushalnagar, H. Singh, A. Rangarajan, Y. Liu, S.
Singh, Exploiting heterogeneity in sensor networks, in: Proceedings of IEEE INFOCOM, Miami, USA, 2005.
[23] Y. Yu, B. Krishnamachari, V.K. Prasanna, Energy-latency tradeoffs for data gathering in wireless sensor networks, in: Proceedings of IEEE INFOCOM, Hong Kong, 2004.
