Capacity of Dual-Radio Multi-Channel Wireless Sensor Networks for Continuous Data Collection

Transcription

1 This paper was presented as part of the main technical program at IEEE INFOCOM 2011 Capacity of Dual-Radio Multi-Channel ireless Sensor Networks for Continuous Data Collection Shouling Ji Department of Computer Science Georgia State University Atlanta, Georgia 30303, USA Yingshu Li Department of Computer Science Georgia State University Atlanta, Georgia 30303, USA Xiaohua Jia Department of Computer Science City University of ong Kong Kowloon, ong Kong Abstract Data collection is an important operation of wireless sensor networks (SNs). The performance of data collection can be measured by its achievable network capacity. Most existing works focus on the capacity of unicast, multicast or snapshot data collection in single-radio single-channel wireless networks, and no dedicated works consider the continuous data collection capacity for SNs in detail under the protocol interference model. In this paper, we first propose a multi-path scheduling algorithm for the snapshot data collection in single-radio multi-channel SNs and prove that its achievable network capacity is at least, 3.63 ρ2 +o(ρ) which is a tighter lower bound compared with the previously best result in [5] which is, where is the bandwidth over a 8ρ 2 channel, is the number of the available orthogonal channels, ρ is the ratio of the interference radius over the transmission radius of a sensor and o(ρ) is a linear equation of ρ. For the continuous data collection problem, although the authors in [5] claim that data collection can be pipelined with existing works, we find that such an idea cannot actually improve network capacity. e explain the reason for this and propose a novel continuous data collection method for dual-radio multi-channel SNs. This method significantly speeds up the data collection process, and achieves a capacity of when e 12, or when M e ( 3.63 e > 12, where n is the number of sensors, M is a constant value and usually M n, and e is the maximum number of leaf nodes having a same parent node in the routing tree (i.e. data collection tree). The simulation results also indicate that the proposed algorithms significantly improve network capacity compared with the existing works. I. INTRODUCTION ireless Sensor Networks (SNs) are mainly used for collecting data from the physical world. Data gathering can be categorized as data aggregation [1], [2], which obtains aggregated values from SNs, and data collection [3]-[5], which gathers all the data from a network without any data aggregation. For data collection, the union of all the sensing values from all the sensors at a particular time instance is called a snapshot [5]. The problem of collecting one snapshot is called snapshot data collection. The problem of collecting multiple continuous snapshots is called continuous data collection. Different from wired networks, SNs suffer from the interference problem, which degrades the network performance. Consequently, network capacity, which can reflect the achievable data transmission rate, is usually used as an important measurement to evaluate network performance. Particularly, for a data collection SN, we use the data receiving rate at the sink, referred as data collection capacity, to measure its achievable network capacity, i.e. data collection capacity reflects how fast data been collected to the sink. After the first work [8], extensive works emerged to study the network capacity issue for variety of network scenarios, e.g. multicast capacity [11], [12], unicast capacity [13], broadcast capacity [19], snapshot data collection capacity [3]-[5] and etc. Most of the previous studies on network capacity are for single-radio single-channel SNs [3]-[5], [8]-[16], where a network consists of a number of nodes with only one radio each and all the nodes communicate over a common single channel. Because of the inherent limitations of such networks, transmissions suffer from the radio confliction problem [6], [7] and the channel interference problem [17], [18], [7] seriously. This degrades network performance significantly. The radio confliction problem is caused by the fact that each node is equipped with only one radio, which means a node can only work on a half-duplex mode, i.e. this node cannot receive and transmit data simultaneously. The channel interference problem is caused by all the nodes working over a common channel. hen one node transmits data, all the other nodes within its interference radius cannot receive any other data and all the other transmissions interfering with this transmission cannot be carried out simultaneously. Fortunately, many current offthe-shelf sensor nodes are capable of working over multiple orthogonal channels, e.g. IEEE b/g standard supports 3 orthogonal channels and IEEE a standard supports 13 orthogonal channels [6] respectively, which can greatly mitigate the channel interference problem. Furthermore, with the development of hardware technologies and the decreasing of hardware cost, a sensor node can be equipped with multiple radios. This helps with solving the radio confliction problem. Different from the previous works which investigate the capacity issues for single-radio single-channel SNs, we study the network capacity problem for both continuous data collection and snapshot data collection in dual-radio multi-channel SNs under the protocol interference model. Meanwhile, the results obtained in this paper can be extended to SNs under the physical interference model [19]. The motivation of this work lies in the fact that dual-radio multi-channel SNs can make nodes work in a full-duplex manner without incurring /11/$ IEEE 1062

2 high hardware cost, while the channel interference problem can be mitigated significantly. To the best of our knowledge, the previous works only address the snapshot data collection capacity problem, and our work is the first dedicated one investigating the continuous data collection capacity problem in detail under the protocol interference model. The main contributions of our work are as follows. For the snapshot data collection problem in singleradio multi-channel SNs, we propose a new multi-path scheduling algorithm. e prove that this algorithm can achieve the order-optimal network capacity Θ( ) and has a tighter lower bound 3.63 ρ2 +o(ρ) compared with the previously best result in [5], which is 8ρ, where is the 2 channel bandwidth, is the number of the orthogonal channels, ρ is the ratio of the interference radius over the transmission radius of a node, and o(ρ) is a linear equation of ρ. Currently, there are no dedicated solutions for the continuous data collection capacity problem in SNs. An intuitive method is to combine a snapshot data collection method with the pipeline technology [5]. owever, this idea does not work for most of the existing works. e explain the reasons for the failure of this idea and show that pipeline with the methods in [4] and [5] cannot improve network capacity. e propose a novel pipeline scheduling algorithm that combines Compressive Data Gathering (CDG) [3] and pipeline together, which significantly improves the continuous data collection capacity for dual-radio multichannel SNs. e also prove that the achievable asymptotic network capacity of this algorithm in a long-run is when e 12, or M e( 3.63 when e > 12, where n is the number of the sensors, M is a constant value and usually M n, and e is the maximum number of the leaf nodes having a same parent in the routing tree (i.e. data collection tree). A straightforward upper bound of a dual-radio SN is 2, since a dual-radio sink can simultaneously receive two packets at most. hereas, thanks to the benefit brought by the pipeline technique, the analysis shows that our pipeline scheduling algorithm can even achieve a capacity higher than 2. The simulation results indicate that the proposed algorithms have a better snapshot data collection capacity compared with the previously best works. The rest of this paper is organized as follows: Section II summarizes the related works. Section III introduces the network model and preliminaries. The multi-channel scheduling algorithm for snapshot data collection in single-radio multi-channel SNs is proposed and analyzed in Section IV. Section V presents a novel multi-channel scheduling algorithm for continuous data collection and its theoretical achievable asymptotic network capacity. The simulations to validate the performance of the proposed algorithms are shown in Section VI and we conclude this paper in Section VII. II. RELATED ORK A. Capacity for Single-Radio Single-Channel ireless Networks The works in [9], [10] focus more on the MAC layer to improve network capacity. In [9], the authors studied the capacity of CSMA wireless networks. They formulated the models of a series of CSMA protocols and study the capacity of CSMA scheduling versus TDMA scheduling. In [10], the authors considered the scheduling problem where all the communication requests are single-hop and all the nodes transmit at a fixed power level. They proposed an algorithm to maximize the number of links in one time-slot. The works in [11]-[14] study the multicast and/or unicast capacity of wireless networks. The multicast capacity for wireless ad hoc networks under the protocol interference model and the Gaussian channel model are investigated in [11] and [12] respectively. In [11], the authors showed that the network multicast capacity is Θ( n log n n k ) when k = O( log n ) and is Θ( ) when k = Ω( n log n ). In [12], the authors showed n that when k θ 1 and n s θ 2 n 1/2+β, the capacity (log n) 2α+6 that each multicast session can achieve is at least c 8 n s k. Considering the problem of characterizing the unicast capacity scaling in arbitrary wireless networks, the authors proposed a general cooperative communication scheme in [13]. A more general casting capacity problem was investigated in [14] for random wireless networks. The snapshot data collection capacity of SNs is studied in [4], [5], and [3]. In [4], the authors considered the collisionfree delay-efficient data gathering problem. They proposed a family of path scheduling algorithms. The authors of [5] extended the work of [4]. They derived tighter upper and lower bounds of the capacity of snapshot data collection for arbitrary SNs. The work in [3] studies how to distribute a data collection task to the entire network to achieve load balancing. In this work, all the sensors transmit the same number of data packets during the data collection process. In [15], a method which decomposes a large network into many small zones is proposed, and then localized scheduling algorithms which can achieve the order optimal capacity are executed in each zone independently. A general framework to characterize the capacity of wireless ad hoc networks with arbitrary mobility patterns is studied in [16]. B. Capacity for Multi-Channel ireless Networks Since wireless nodes can work over multiple orthogonal channels, the issue of the capacity of multi-channel wireless networks also attracts a lot of attention [17], [18], and [7]. In [17], the authors studied the connectivity and capacity problem of multi-channel wireless networks. They considered a multichannel wireless network under the constraints on channel switching, proposed some routing and channel assignment strategies for multiple unicast communications and derived the per-flow capacity. In [18], the authors first proposed a multi-channel network architecture, called MC-MDA, and then obtained the capacity of multiple unicast communications n 1063

3 under arbitrary and random network models. The impact of the number of the channels, the number of the interfaces and the interface switching delay on the capacity of multi-channel wireless networks is investigated in [7]. In this work, the authors derived the network capacity under different situations for arbitrary and random networks. C. Remarks Unlike the above mentioned works, our work has the following two main characteristics. First, most of the above mentioned works are specifically for single-radio single-channel wireless networks, while our work considers dual-radio multichannel SNs. Second, to the best of our knowledge, our work is the first dedicated one that investigates network capacity for continuous data collection in detail under the protocol interference model, whereas most of the previous works study network capacity for multicast or/and unicast and etc. For the works that study the data collection capacity of wireless networks, they only consider snapshot data collection which is just a special case of continuous data collection. III. NETORK MODEL AND PRELIMINARIES A. Network Model e consider a SN consisting of n sensors and one sink, represented by a connected undirected graph G = (V, E), where V is the set of all the nodes in the network and E is the set of all the possible links among the nodes in V. Every sensor in the SN produces one packet in a snapshot. Each sensor has two radios and each radio has a fixed transmission radius normalized to one and a fixed interference radius, denoted by ρ, ρ 1. Since we use the protocol interference model, for any receiving node v, v can successfully receive a packet from a transmitting node u if u v 1 and there is no other node s satisfying s v ρ and trying to transmit a packet simultaneously over the same channel with u. ere is the Euclidean distance. Furthermore, we say two links are interference links if at least one transmission over them will fail if they transmit data simultaneously over a common channel. Each radio can work over orthogonal channels, denoted by λ 1, λ 2,..., λ respectively. A fixed data-rate channel model [5] is adopted in this paper, which means each sensor can transmit at a rate of bits/second over a wireless channel. The size of all the packets transmitted in the network is set to be b bits. e also assume that all the transmissions are synchronized and the size of a time slot is t = b/ seconds. e further give the formal definition of the capacity of data collection. The capacity Υ of data collection is the ratio between the size of the data successfully received by the sink and the time τ used to collect these data, e.g. in a SN consisting of n sensors, the capacity to collect N snapshots of data is Υ = Nnb τ, which is actually the data rate at the sink. B. Routing Tree G is a unit-disk graph representing a SN. e define the sink s 0 as the center of G. The radius of G with respect to s 0 is the maximum depth of the Breadth-First-Search (BFS) tree rooted at s 0. For a subset U of V, U is a Dominating Set (DS) of G if every node in V is either an element of U or adjacent to at least one node in U. If the subgraph of G induced by U is connected, then U is called a Connected Dominating Set (CDS) of G. e build a CDS based on routing tree T using the method proposed in [1]. T is rooted at the sink and can be built according to the following steps. First, obtain a Maximal Independent Set (MIS) D according to the BFS search sequence. Note that D is also a DS of G and an element in D is called a dominator. Let G be a graph on D in which two nodes in D are linked by an edge if and only if these two nodes have a common neighbor in G. Obviously, sink s 0 is in G. Suppose that the radius of G with respect to s 0 is L and we denote the union of dominators at level l(0 l L ) as set D l. Second, we choose nodes, also called connectors, to connect all the nodes in D to form a CDS. Let S l (0 l L ) be the set of the nodes adjacent to at least one node in D l and at least one node in D l+1 and compute a minimal cover C l S l for D l+1. Let C = L 1 0 C l and therefore D C is a CDS of G. Finally, for any other node u, also called a dominatee, not belonging to D C, choose the nearest dominator as u s parent node. Thus, the routing tree T is obtained. For each link in T, the receiving (resp., transmitting) node, i.e. parent (resp., child) node, of this link is called a head (resp., tail). Suppose that A is a set of links of T. The corresponding conflicting graph of A is denoted by R(A) = (V A, E A ), where each link in A is abstracted to a node in V A and two nodes in V A form an edge in E A if the corresponding two links of these two nodes are interference links. Lemma 1 in [1] can be used to derive some useful results of the routing tree T. Lemma 1: [1] Suppose that O is a half-disk with radius r, and U is a set of points with mutual distances of at least one. Then the number of the points β r in a half-disk is β r = U O π 3 r 2 + ( π 2 + 1)r + 1. From Lemma 1, the authors in [1] derived the following properties of the routing tree T. First, for each 0 l L 1, each connector in C l is adjacent to at most 4 dominators in D l+1. Second, for each 1 l L 1, each dominator in D l is adjacent to at most 11 connectors in C l. Third, C C. Vertex Coloring Problem For a graph G = (V, E), the maximum degree (minimum degree) of G is denoted by (G) (δ(g)). A subgraph of G on U V is denoted by G(U). The inductivity of G is defined as δ (G) = max δ(g(u)). A vertex coloring of G is a scheme of U V coloring all the vertices in G such that no two adjacent vertices share the same color. The chromatic number χ(g) of G is the least number of colors used to color G. Deciding the lower bound of χ(g) is a well-known NPC problem. owever, the upper bound of χ(g) has been derived in graph theory [20], [1]. The following lemma was proved in [20] and [1]. Lemma 2: χ(g) 1+δ (G) and a vertex coloring scheme, called first-fit coloring, for G using at most 1 + δ (G) colors 1064

4 can be found in polynomial time. Given a link set A of T, the channel assignment problem for A can be abstracted to the vertex coloring problem for its corresponding conflicting graph R(A). If the tail (resp., head) of every link in A is a dominator, then Lemma 3 in [1] gives the upper bound of δ (A). Lemma 3: [1] δ (A) β ρ+1 1. Lemma 3 implies that in the worst case, at most β ρ+1 channels may be assigned to all the links in A without channel interference by a first-fit coloring method. Fig s 0 1 s 1 0 s 0 1 s 1 1rd 1 s 0 0 s 1 2rd 1 s 0 1 s 1 3rd 2 s 0 0 s 1 4rd 2 s 0 1 s 1 5rd 3 s 0 0 s 1 1 s 2 1 s 3 1 s 2 1 s 3 2 s 2 1 s 2 1 s 2 0 s 2 0 s 2 (a) (b) (a) A single path and (b) its scheduling (rd = round). IV. CAPACITY OF SNAPSOT DATA COLLECTION In this section, we first investigate the traditional snapshot data collection problem, and propose a novel scheduling algorithm with a tighter lower capacity bound for this problem in single-radio multi-channel SNs. Subsequently, we discuss the proposed algorithm further, and point out that the proposed algorithm and most existing works cannot improve network capacity with the pipeline technology. Since at any time slot, the sink can receive data from at most one sensor, the upper bound of the snapshot data collection capacity is [5]. Aiming at this upper bound, we design a scheduling algorithm for snapshot data collection which is order-optimal and has a tighter lower bound than the previously best result [5]. A. Scheduling Algorithm for Snapshot Data Collection In this subsection, we design a new multi-path scheduling algorithm based on the routing tree T built in Section III, which is proven to have a better performance. e first study how to schedule a single path and then extend it to the scheduling of multi-path in T. For simplicity, we introduce the concept of round. A round is a period of time consisting of multiple continuous time slots. e take the path shown in Figure 1(a) as an example to explain the idea of the single path scheduling scheme. In Figure 1(a), the path, denoted by P, consists of one sink s 0 and three sensors s 1, s 2, and s 3, where s 0 and s 2 are dominators, s 1 is a connector, and s 3 is a dominatee. The value marked in each node is the number of the packets at this node to be transmitted. Initially, every sensor on P has one packet and there is no packet at s 0. P o (resp., P e ) denotes the set of links on P whose heads (resp., tails) are dominators and whose tails have at least one packet to be transmitted. e schedule P according to the following two steps and repeat them until all the packets have been collected by s 0. Step 1: In an odd round, schedule every link in P o once, i.e. assign a dedicated channel and one dedicated time slot to each link in P o. Step 2: In an even round, schedule every link in P e once, i.e. assign a dedicated channel and one dedicated time slot to each link in P e. The detailed scheduling in Step 1 can be done in the following way: first, calculate the interference degree i.e. the number of the links interfering with another link, for each link in P o ; second, sort the links in P o according to their interference degrees in a non-decreasing order and denote the resulting link sequence as {l 1, l 2,..., l Po }; finally, during the i-th (1 i ) time slot of a round, let the j- Po th ((i 1) < j i) link in P o work on channel λ j%+1. The detailed scheduling in Step 2 is similar to that of Step 1. The scheduling process of P is shown in Figure 1(b). During the first (odd) round, links (s 3, s 2 ) and (s 1, s 0 ) are scheduled and the packets in s 3 and s 1 are transmitted to their parent nodes. After the first round, s 3 has no packet to transmit. During the second (even) schedule, link (s 2, s 1 ) is scheduled and s 2 transmits one packet to its parent node. This schedule process continues until all the packets on path P has been transmitted to s 0. e now consider the scheduling of the routing tree T built in Section III. Suppose that there are m leaf nodes in T, and the path from leaf node s i (1 i m) to the sink s 0 is denoted by P i. Two paths P i and P j are said intersecting if they have at least one common node besides the sink node. The common node of two intersected paths having the largest number of hops from the sink is called an intersecting point. If path P i intersects with other paths, the route from s i to the nearest intersecting point of s i is called a sub-path, denoted by F i. Otherwise, F i is actually P i. The basic idea of our multi-path scheduling algorithm is as follows: For the non-intersecting paths without wireless interference, schedule them by the single path scheduling algorithm concurrently. For the non-intersecting paths with wireless interference, schedule them according to a certain order. The path with the smallest subscript has the highest priority. For the intersecting paths, schedule them according to a certain order, e.g. if P i is the one of these intersecting paths with the smallest subscript, schedule P i until there is no packet having to be transmitted on the sub-path F i and then continue to schedule the next path. e further explain the multi-path scheduling algorithm by the routing tree shown in Figure 2(a). In Figure 2(a), there are three leaf nodes s 1, s 2, and s 3 and their corresponding paths are P 1, P 2 and P 3, respectively. Path P 1 and path P 3 are not intersecting, therefore they are scheduled concurrently. Since path P 1 and path P 2 are intersecting paths, path P 1 is scheduled first. After no packets on the sub-path F 1 of P 1 have to be transmitted, path P 2 is scheduled as shown in Figure 2(b). 1065

5 Fig. 2. F 1 s 1,P 1 s 0 (sink) s 2,P 2 (a) B. Capacity Analysis s 3,P 3 F 1 s 1,P 1 s 0 (sink) s 2,P 2 s 3,P 3 (b) s 0 (sink) s 2,P 2 (a) A routing tree and (b) its scheduling. The upper bound of the snapshot data collection capacity is which has been explained. Consequently, we focus on the lower bound of the snapshot data collection capacity. In the worst case, all the paths in the routing tree T are intersecting, which means only one path can be scheduled at any time. In order to derive the lower bound of the multi-path scheduling algorithm, we first investigate the number of the rounds needed to finish the scheduling of one single path and then study the number of the time slots in each round. Lemma 4 gives the maximum number of the rounds used for scheduling of one single path. Lemma 4: For a path P of length L, it takes at most 2L 1 rounds to collect all the packets on P at the sink node. Proof: Please refer to [21] of this paper. From Lemma 4, it is straightforward to obtain the number of the rounds used to collect all the data on the sub-path F of P as shown in Corollary 1. Note that the intersecting point also needs one round to transmit its packet. Corollary 1: For the sub-path F of length L s in P, it takes at most 2L s rounds to collect all the packets on F. The maximum number of the time slots in a round is as follows. Lemma 5: A round has at most time slots. Proof: During every odd (resp., even) round, the scheduled links are links in P o (resp., P e ). Since the heads (resp., tails) of links in P o (resp., P e ) are dominators, we can schedule all the links in P o (resp., P e ) in one time slot with at most β ρ+1 channels by Lemma 3 and Lemma 2. Now, we have available channels, which means we can finish the scheduling ρ2 +o(ρ) within time slots. Therefore, the lemma holds. Now we can obtain the lower bound of the achievable capacity of the multi-path scheduling algorithm as shown in Theorem 1. Theorem 1: The capacity Υ at the sink of T of the multipath scheduling algorithm is at least 3.63, which is order-optimal. Proof: Suppose that T has m paths and the length of each path is L i (1 i m). In the worst case, all the m paths cannot be scheduled concurrently. Then by Lemma 4, Corollary 1 and Lemma 5, the total time τ used to collect 2L i i=1 According to the multi-path scheduling algorithm, for any path all the packets of T at the sink is at most t m. P i, the time used to collect packets on P i is equal to the time used to collect packets on the corresponding sub-path F i of P i. Therefore, τ t m m 2L i = 2t F i. i=1 i=1 Since the number of the links in T is equal to the number of the sensors in T, m F i = n. Then, τ 2nt i=1 Therefore, the capacity Υ = nb 3.63 ρ2 +o(ρ) nb τ = 2nt. 2, where o(ρ) is a linear equation of ρ. Since the upper bound of Υ is, Υ is order-optimal. From Theorem 1, the multi-path scheduling algorithm has a tighter lower bond compared with the previously best result in [5], which has a lower bound of 8ρ. 2 C. Discussion hen we address the continuous data collection problem, an intuitive idea is to pipeline the existing snapshot data collection operations [5]. Nevertheless, such an idea cannot achieve a better performance. This is because the sink can receive at most one packet in a time slot. By pipeline, data transmissions at the nodes far from the sink are really accelerated. owever, the fact that a sink can receive at most one packet in each time slot makes the data accumulated at the nodes near the sink. Finally, the network capacity cannot be improved even with pipeline. This motivates us to investigate new methods for continuous data collection. V. CAPACITY OF CONTINUOUS DATA COLLECTION e propose a novel pipeline scheduling algorithm based on compressive data gathering (CDG) [3] for dual-radio multichannel SNs in this section, which augments the continuous data collection capacity significantly. ere we consider dualradio multi-channel SNs because dual radios can make a half-duplex single-radio node work in a full-duplex mode. Note that our pipeline scheduling algorithm can also be used in the single-radio scenario just with a little modification. A. Compressive Data Gathering (CDG) CDG is first proposed in [3] for snapshot data gathering in single-radio single-channel SNs. The basic idea of CDG is to distribute the data collection load uniformly to all the nodes in the entire network. e take the data collection on a path consisting of L sensors s 1, s 2,..., s L and one sink s 0 as shown in Figure 3 [3] as an example to explain CDG. In Figure 3, the packet produced at sensor s j (1 j L) is d j. In the basic data collection shown in Figure 3(a), s 1 transmits one packet d 1 to s 2, s 2 transmits two packets d 1 and d 2 to s 3, and finally all the packets on the path are transmitted to s 0 by s L. To balance the transmission load, the authors in [3] proposed the CDG method as shown in Figure 3(b). Instead of transmitting the original data directly, s 1 multiplies its data with a random coefficient ϕ i1 (1 i M), and sends the M results ϕ i1 d 1 to s 2. Upon receiving ϕ i1 d 1 (1 i M) from s 1, s 2 multiplies its data d 2 with a random coefficient ϕ i2 (1 i M), adds it to ϕ i1 d 1, and then sends ϕ i1 d 1 +ϕ i2 d 2 as one data packet to s 3. Finally, s L does the similar multiplication 1066

6 and addition and sends the result L j=1 ϕ ijd j (1 i M) to s 0. After s 0 receives all the M packets, s 0 can restore the original packets based on the compressive sampling theory [3]. The number of the transmitted packets is O(n 2 ) in Figure 3 (a) and is O(nM) in Figure 3 (b), and usually M n for large scale SNs. Therefore, CDG reduces the number of the transmitted packets. (a) (b) Fig. 3. Comparing of (a) basic data collection and (b) CDG [3]. B. Pipeline Scheduling Thanks to the benefit brought by CDG, we can address the continuous data collection problem with the pipeline technique. From the building process of the routing tree T, we know that the nodes in T can be divided into sets by levels D e, D L, C L 1, D L 1, C L 2,..., D 1, C 0, D 0 = {s 0 } in a bottom-up way, where D e is the set of all the dominatees, D i (0 i L ) is the set of the dominators at the i-th level, and C i (0 i L 1) is the set of the connectors at the i-th level. Since every node has two radios, one radio can be dedicated to receive data and the other is dedicated to transmit data. Therefore, the nodes at every level can receive and transmit data simultaneously over different channels. Consequently, for a continuous data collection task consisting of N snapshots, we propose a pipeline scheduling algorithm as follows. Step 1: All the nodes in D e transmit the packets of the j-th (1 j N 1) snapshot to their parent nodes in the CDG way, i.e. for every node s D e, s multiplies its data with M random coefficients respectively, and sends the M products to its parent node. After all the packets of the j-th snapshot have been transmitted successfully, the nodes in D e immediately transmit the packets of the (j + 1)-th snapshot in the CDG way. Step 2: After all the nodes in D l (1 l L ) receive all the packets of the j-th snapshot from their child-level, they send the packets of the j-th snapshot to their parent nodes in the CDG way, i.e. every node s D l combines its packet of the j-th snapshot with the received packets of the j-th snapshot, and sends the M new packets to its parent node. After all the packets of the j-th snapshot have been transmitted successfully, the nodes in D l immediately transmit the packets of the (j + 1)-th snapshot to their parent nodes in the CDG way, if they have received all the packets of the (j + 1)-th snapshot from their child-level. Step 3: After all the nodes in C l (0 l L 1) receive all the packets of the j-th snapshot from their child-level, they send the packets of the j-th snapshot to their parent nodes in the CDG way, i.e. every node s C l combines its packet of the j-th snapshot with the received packets of the j-th snapshot, and sends the M new packets to its parent node. After all the packets of the j-th snapshot have been transmitted successfully, the nodes in C l immediately transmit the packets of the (j+1)-th snapshot in the CDG way if they have received all the packets of the (j +1)-th snapshot from their child-level. Step 4: The sink restores the data of a snapshot in the CDG way after it receives all the packets of this snapshot. Steps 1-4 provide the general frame of our pipeline scheduling scheme. Now, we discuss how to prevent radio confliction and channel interference in Steps 1-3. If two or more nodes have the same parent node, we call them sibling nodes. In Steps 1-3, radio confliction may arise if two or more sibling nodes send data to their parent node simultaneously even over different orthogonal channels. This is because every sensor only has one radio dedicated to receiving data. Suppose that there are at most e (resp., d and c ) nodes in D e (resp., D l (1 l L ) and C l (1 l L 1)) which have the same parent node. Usually, e < (T ) except in onehop SNs, where any sensor is just one hop away from the sink, e = (T ). Then, d 4 and c 11 (Note that C 0 12.) (see Section III-B). To avoid confliction, we divide the nodes in D e (resp., D l and C l ) into e (resp., d and c ) subsets to guarantee that each node belongs to one subset and no sibling nodes belong to the same subset. Then, when we schedule the nodes of each level, we schedule these subsets in a certain order. For the nodes in C 0, we schedule them in a certain order. Different from the multi-path scheduling algorithm, in which a sensor sends one packet over a link in one time slot, we employ CDG, where a sensor sends M packets for a snapshot. e now introduce the concept of a Super Time Slot (STS) which consists of M time slots. In a STS, a sensor can send M packets over a channel for a snapshot. For the links working simultaneously, we assign channels and STSs in the similar way of the multi-path scheduling algorithm. C. Capacity Analysis In this subsection, we analyze the achievable network capacity of the proposed pipeline scheduling algorithm. Lemma 6 indicates the inductivity (Section III-C) of the corresponding conflicting graph of the links scheduled simultaneously in the pipeline scheduling algorithm. Lemma 6: Suppose that A is the set of the links in T scheduled simultaneously in the pipeline scheduling algorithm, and R(A) is the corresponding conflicting graph of A, then, δ (R(A)) 2β ρ+2 1, where δ (R(A)) is the inductivity of R(A). Proof: Please refer to [21]. Based on the result of Lemma 6, we can determine the number of the STSs used to schedule all the links in A as follows. 1067

7 Lemma 7: For the links of set A in Lemma 6, we can use STSs to schedule them without channel interference. Proof: Please refer to [21] of this paper. From the pipeline scheduling algorithm we know that the transport of subsequent snapshots has some time overlap with the transport of preceding snapshots. Therefore, we first analyze the time used to collect the packets of the first snapshot since it is the base of the pipeline, and then analyze the achievable capacity of the entire pipeline. Theorem 2: The number of the time slots used to collect the packets of the first snapshot by the pipeline scheduling algorithm is at most M ( e + 15L + 1). Proof: In Step 1 of the pipeline scheduling algorithm, we divide the nodes in D e into e subsets and schedule them in a certain order. By Lemma 7, each scheduling uses at most STSs. Consequently, Step 1 needs at most e STSs to finish the scheduling for the first snapshot. Similarly, Step 2 (resp., Step 3) needs at most 4L (resp., 11(L 1) ) STSs to finish the scheduling for the first snapshot. Furthermore, it needs at most 12 STSs for C 0 to transmit the packets for the first snapshot to the sink. In summary, the total number of the STSs used for the first snapshot is at most e + 4L + 11(L 1) +12 = ( e + 15L + 1). Since every STS has M time slots, then the number of the time slots usedfor the collection of the first snapshot is at most M ( e + 15L + 1). On the basis of the result in Theorem 2, we obtain the number of the time slots used to collect all the packets of N continuous snapshots for the pipeline scheduling algorithm as shown in Theorem 3. Theorem 3: The time slots used for the pipeline scheduling algorithm to collect N continuous snapshots are at most M ( e + 15L + 12N 11) when e 12 or M (N e + 15L + 1) when e > 12. Proof: From the proof of Theorem 2 we know, it takes the nodes in D e (resp., D l (1 l L ), C l (1 l L 1) and C 0 ) at most e (resp., 4, 11 and 12 ) STSs to transmit packets for a snapshot. In order to obtain the upper bound of the number of the time slots used, we assume the STSs used by nodes in D e (resp., D l (1 l L ), C l (1 l L 1) and C 0 ) are e (resp., 4, 11 and 12 ) in the following proof. Then, we prove Theorem 3 by cases. Case 1: e 4. For clearness, we use the transmission of two snapshots S-1 and S-2 shown in Figure 4(a) as an example for explanation. In Figure 4(a), the vertical axis denotes the levels in the routing tree T and the horizontal axisdenotes time slots. t e = e, t d = 4, t c = 11, and t 0 = 12. From Figure 4(a) we know, the nodes at D e - level begin to send packets of S-2 immediately after they send out the packets of S-1. Since e 4, after the nodes at D L - level receive all the packets of S-2, they may still be busy with the transmission of packets of S-1. Nevertheless, from the C L 1-level to the D 1 -level, the pipeline can be utilized in a maximum degree, which means whatever the packets of S-1 or the packets of S-2, they can be sent immediately. After the packets of S-2 are sent from the nodes at D 0 -level to the nodes in C 0, they may have to wait for a while at the nodes of the C 0 -level, since the transmission for the packets of S-1 may last for as long as 12 STSs. This implies the sink will receive all the packets of S-2 in 12 STSs after it receives all the packets of S-1. According to the description of the pipeline scheduling algorithm, the subsequent snapshots will be transmitted in the same way, which means the sink will receive all the packets of a snapshot at most every 12 STSs after it receives the packets of the first snapshot which takes at most M ( e + 15L + 1) time slots by Theorem 2. As a result, the number of time slots used to collect the packets of N continuous snapshots by the pipeline scheduling algorithm is at most M = M = M ( e + 15L + 1) + (N 1) 12M ( e + 15L N 12) ( e + 15L + 12N 11). Case 2: 4 < e 11 and Case 3: e = 12. These two cases can be proved by the similar method used in Case 1. The number of the time slots used to collect the packets of N continuous snapshots is also at most M ( e +15L + 12N 11). Case 4: e > 12. e use the data transmission of two snapshots shown in Figure 4(b) as an example to show the proof process. Since e > 12, the pipeline can be utilized in a maximum degree at D L -level and continue to C 0 -level. Then, the sink can receive all the packets of a subsequent snapshot every e STSs after it receives the packets of the first snapshot. Therefore, the number of the time slots used to collect the packets of N continuous snapshots is at most M = M = M ( e + 15L + 1) + (N 1) e M ( e + 15L (N 1) e ) (N e + 15L + 1). As a conclusion, Theorem 3 is true. Theorem 3 shows the number of the time slots used to collect N continuous snapshots. This prepares us to derive the achievable capacity of the pipeline scheduling algorithm. 1068

8 Levels Levels t e t d t e t d t c S-1 t 0 S t c (a) S-1 t 0 S (b) Time Slots Fig. 4. Data transport in (a) Case 1 and (b) Case 4. Time Slots Theorem 4: For a long-run continuous data collection, the lower bound of the achievable asymptotic network capacity of the pipeline scheduling algorithm is when e 12 or when M e( 3.63 e > 12. Proof: e prove Theorem 4 in two cases. Case 1: e 12. In this case the number of the time slots used to collect N continuous snapshots is at most M ( e + 15L + 12N 11) as proved in Theorem 3. Therefore, the lower bound of the capacity of the pipeline scheduling algorithm is N nb tm ( e+15l +12N 11) nb = tm ( e N + 15L 11 N +12 N ) = M ( e N + 15L 11 N +12 ). N hen N, the above equation approaches to, which means the asymptotic 12M network capacity in this case is. Case 2: e > 12. The asymptotic network capacity in this case is, which can be proved by the similar M e ( 3.63 method as in Case 1. In a dual-radio multi-channel SN, since every node has two radios, the upper bound of the network capacity is 2. This is because the sink can receive at most two packets in one time slot. From Theorem 4, it is possible for both and to be greater than 2. M e ( 3.63 There are two main reasons for this result. The most important one is the pipeline scheduling, which accelerates the data collection process directly and significantly, and as a result the network capacity is increased obviously. Another reason is the use of the CDG method. As indicated in [3], CDG reduces the total number of transmission times of packets in some degree. These two reasons are also validated by the simulation results in Section VI. Furthermore, we find that the pipeline scheduling algorithm is more effective for large scale SNs, since large scale SNs are more suitable for pipeline. The pipeline scheduling algorithm is also more effective for a long time of continuous data collection, which can also be seen from Theorem 4. VI. SIMULATIONS AND RESULT ANALYSIS e conducted simulations to verify the performances of the proposed algorithms through implementing them with the C language. For all the simulations, we assume every SN has one sink, and all the sensor nodes of each SN are randomly distributed in a square area and the communication radius of each node is normalized to one. Suppose the network MAC layer works with TDMA, i.e. the network time can be slotted. Every node produces one data packet in a snapshot and the size of a packet is normalized to one. Every available channel has the same bandwidth normalized to one. For any two different channels, we suppose they are orthogonal, i.e. the communications initialized over any two channels have no wireless interference. Furthermore, we assume a packet can be transmitted over a channel within a time slot. The compared algorithms are BFS [5], SLR [17] and CDG [3]. BFS is a snapshot data collection algorithm based on a breadth first search tree and scheduling is carried out path by path. BFS is specifically proposed for single-radio single-channel SNs. SLR is a straight-line routing method for multi-unicast communication in multi-channel wireless networks. e extend BFS and SLR to the general dual-radio multi-channel scenario in our simulations for fairness. The proposed multi-path scheduling algorithm is referred to as MPS and the pipeline scheduling algorithm is referred to as PS. In the remainder of this section, AR refers to the square area where a SN is deployed. ρ, n,, M, and N have the same meanings as mentioned before, respectively. A. Performance of MPS The snapshot data collection capacities of MPS, BFS, and SLR are shown in Figure 5. In Figure 5(a), the capacity of every algorithm increases when the number of the available channels increases, since more available channels enable more concurrent transmissions. MPS achieves a higher capacity compared with BFS and SLR. This is because MPS simultaneously schedules all the paths without radio confliction (except at the sink). hereas, BFS just schedules links without radio confliction on one path every time and SLR schedules all the transmission links simultaneously, which leads to serious radio confliction. On average, MPS achieves 77.49% and 41.95% more capacity than BFS and SLR, respectively. The effect of interference radius on capacity is shown in Figure 5(b). ith the increase of interference radius, more transmission interference occurs, which leads to the decrease of the capacities of all the algorithms. Nevertheless, MPS still achieves the largest capacity since it simultaneously schedules multiple paths without radio confliction, which suggests a nice tradeoff between BFS and SLR. On average, MPS achieves 67.45% and 37.37% more capacity than BFS and SLR, respectively. B. Performance of PS Since CDG, BFS, and SLR do not consider pipeline, they collect data snapshot by snapshot in continuous data collection. The continuous data collection capacities of PS, CDG, BFS, 1069

9 Capacity MPS BFS SLR (a) Capacity vs. (ρ=2) Capacity MPS BFS SLR (b) Capacity vs. ρ (=3) Fig. 5. Snapshot Data Collection Capacity (n = 4000, AR = 30 30). and SLR in different network scenarios are shown in Figure 6. Figure 6(a) reflects the effect of the number of the available channels on capacity. As explained before, the capacities of all the algorithms increase as more and more channels are available. PS has a much higher capacity compared with the other three algorithms. This is because: first and the most important, PS utilizes the pipeline technology; second, for the data of a single snapshot, PS collects them in the CDG way, which reduces the overall data transmission times. This also explains why CDG has a higher capacity compared with BFS and CDG. On average, PS achieves a capacity of 8.22 times of that of CDG, times of that of BFS and times of that of SLR, respectively. The effect of interference radius on capacity is shown in Figure 6(b). ith the increase of interference radius, a transmission link will interfere with more and more other transmission links, which leads to the decrease of capacity whatever algorithm it is. PS has the highest capacity among all the algorithms since it works with pipeline and CDG. On average, PS achieves a capacity of 7.31 times of that of CDG, times of that of BFS and times of that of SLR, respectively. Capacity PS CDG BFS SLR (a) Capacity vs. (ρ=2) Capacity (b) Capacity vs. ρ (=3) Fig. 6. Continuous Data Collection Capacity in Different Scenarios (n=10000, AR=50 50, N=1000, M=100). VII. CONCLUSION Motivated by the fact that there exist no works studying the capacity of continuous data collection, we investigate this problem in dual-radio multi-channel SNs. e first propose a multi-path scheduling algorithm for the snapshot data collection problem and prove that its achievable network capacity is at least 3.63 ρ2 +o(ρ) PS CDG BFS SLR. For the continuous data collection problem, although the authors in [5] claim that it can be solved by combining the pipeline technology with the existing works, we however find that pipeline with the existing works cannot actually improve the network capacity. e explain the reason of this, and then propose a novel continuous data collection method for dual-radio multi-channel SNs. This method speeds up the data collection process significantly. Theoretical analysis of this method shows that the achievable asymptotic network capacity is when e 12, or when M e ( 3.63 e > 12. 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