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International Journal of Distributed Sensor Networks

Volume 2012 (2012), Article ID 891321, 12 pages

http://dx.doi.org/10.1155/2012/891321

## Optimal Convergecast Scheduling Limits for Clustered Industrial Wireless Sensor Networks

Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China

Received 27 July 2012; Revised 12 October 2012; Accepted 14 October 2012

Academic Editor: Hongwei Zhang

Copyright © 2012 Xiaoling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Increased mobility coupled with a possible reduction of cabling costs and deployment time makes wireless communication an attractive alternative for the industrial process monitoring and control. The major obstacles toward the utilization of wireless industrial networks are predominantly the timeliness and reliability requirements. In this paper, orienting to clustered industrial wireless sensor networks, we analyze the performance bounds of the convergecast scheduling, which is a typical many-to-one communication in wireless industrial networks. The analysis aims to the cluster-line and cluster-tree topologies. Each kind is future divided into three scenarios according to an application parameter, named data update rate. Firstly, we establish the lower bounds on the number of timeslots to finish the intracluster and the intercluster convergecast transmissions. Secondly, we establish the lower bounds on the number of channels based on the lower bounds on the number of timeslots and maximum available channels in a multichannel scenario. Lastly, we carry out the extensive analysis-taking packet retransmissions into consideration so as to meet the reliability requirement. Experiment results validate the correctness and tightness of our theory analysis.

#### 1. Introduction

With the success of wireless technologies in consumer electronics, standard wireless technologies are envisioned for the deployment in industrial environments to improve the functionality and the efficiency, which boost the formation of the industrial wireless sensor networks (IWSNs) [1]. Recently, some international organizations are actively promoting the standardized process of IWSNs and have achieved several productions, such as WirelessHART [2] and Wireless network for industrial automation-process automation (WIA-PA) [3]. Convergecast is a typical many-to-one communication pattern in IWSNs. In convergecast, many or all nodes in the network send data to a sink node during a relatively short time period [4]. In these applications all packets generated in the network have to reach a sink node either for record or for computationally intensive analysis.

A guarantee on packet delivery and a bound on convergecast latency are highly desirable in mission critical applications. Guarantee on packet delivery ensures availability of accurate information about the sensing field. An optimal bound on convergecast latency leads to timely detection of the events. Furthermore, a known time bound on convergecast latency can help the sink node to schedule convergecast requests and related computations [4].

As concluded in [4], collisions present a major challenge in convergecast when contention-based MAC protocol like CSMA are employed. Collisions will result in loss of packets. The recovery methods, such as retransmissions method, will increase the latency. Therefore, the convergecast latency incurred by radio interference is far from the optimal [4]. Contention-free MAC protocols like TDMA can be used to eliminate collisions and obtain a bound on the time required to complete convergecast. Hence, we are interested in determining a TDMA schedule such that the entire convergecast can be completed in minimal number of timeslots and in minimal number of channels.

Choi et al. [5] proved that convergecast scheduling problem is NP-hard. Florens et al. [6] consider that the convergecast is the reverse operation of the broadcast scheduling. If node sends packets to node by using timeslot during broadcast scheduling, then node send packets to node by using timeslot . is the minimal schedule length and is calculated according to the number of packets that sink node sends to node . Gandham et al. [4] proved that once convergecast scheduling requires at most timesltos, where represents the number of nodes in the network. They also proved that at lease timeslots are required to complete convergecast in a linear network with nodes; at least timeslots are required to complete convergecast in a multiline network; at least timeslots are required to complete convergecast in a tree network. Yu et al. [7] consider a network as a disc and proved that the upper bound on the number of timeslots is , where is the network diameter and is the maximum node degree. Soldti [8, 9] consider the jointly time- and channel-optimal convergecast scheduling, and proved the lower bound on the number of timeslots and channels for convergecast transmission in mesh network. Shang et al. [10] considered how to collect data from sensors deployed in the Euclidean plane in a time-efficient way. Zhang et al. [11] focused on bursty convergecast and proposed a windowless block acknowledgement scheme to achieve reliable and real-time error control and contention control. Liu et al. [12] estimated the probabilistic path delay bounds in an agile manner and designed a delay-sensitive routing protocol.

In summary, there are three problems in previous researches. First, the network topology is ad hoc or mesh and the network deployment is plane or quadrate, which are not suitable for industrial applications. Second, the cyclic data feature is not taken into consideration. Third, previous research is based on one channel, which are inherently inefficient. To solve these three problems, a new result specific to clustered IWSNs, where in particular, cyclic data collection over the network is assumed. In addition, we consider data update rates (DURs) under various assumptions and derive corresponding time and channel performance. This work extends the results in [8], where only mesh topology is assumed and the DURs are not taken into consideration. It should be noted that most of industrial networks use hierarchical topology and faced to different applications with different data features. Therefore, when requirements are more stringent, the results in [8] may no longer be practical. In particular, this paper presents the following contributions.(i)For a cluster-line IWSN, we analyze and prove the performance limits for completing the convergecast transmission. We establish the lower bounds on the number of timeslots and channels based on three scenarios: all the data update rates (DURs) in a cluster are same, all the DURs in a cluster are different, and the DURs in a cluster are partly same.(ii)For a cluster-tree IWSN, the same problem as the first item is studied.

The rest of this paper is organized as follows. First, the system model and problem formulation of convergecast scheduling are described in Section 2. The analysis of the time- and channel-optimal convergecast scheduling for networks with cluster-line and cluster-tree topologies are studied in Sections 3 and 4, respectively. This paper is concluded in Section 5.

#### 2. Problem Formulation

Time is equally divided into timeslots. A collection of timeslots repeating constitute one superframe. Our superframe structure extends the IEEE STD 802.15.4-2006 superframe [13], and divides the inactive period into intracluster communication period, intercluster communication period, and sleep period. The extended superframe structure is shown in Figure 1. The superframe length of the gateway device or a routing device can be configured and the basic superframe length is also configurable.

One field device initially generates at most one packet destined to the gateway device every DUR time during each convergecast. Packets are routed along , where is the set of routing paths. For every device , and , respectively, denote the set of links that arrive at or leave device ; denotes the capability of buffer after timeslot , where we defined that . Let be the number of packets of a cluster with being the cluster head in one superframe cycle. is decided by the DURs of all field devices in a cluster. Let be the superframe length (in timeslots). Let be a binary variable indicating whether timeslot has been allocated to link (). If timeslot has been allocated to link (), ; otherwise, .

The time- and channel-optimal scheduling to complete convergecast with the objective to minimize both the number of timeslots and the number of channels can be informally formulated as [8].

*Problem 1. *Time-optimal convergecast scheduling:
s.t.

Objective (1) guarantees the minimization of the convergecast scheduling length. Constraint (2) is the half-duplex requirement, which restricts devices not to transmit and receive simultaneously. Constraint (3) guarantees that all packets generated by field devices have been sent to the gateway device. Constraint (4) is the memory status. And constraint (5) indicates the link status.

*Problem 2. *Time- and channel-optimal convergecast scheduling:
s.t.
where is the number of channels used for transmission.

#### 3. Time- and Channel-Optimal Convergecast Scheduling for Cluster-Line Topology

Cluster-line topology that is shown in Figure 2 is instrumental for more general routings, and is the preferred topology in certain applications such as pipeline monitoring and unmanned offshore gas production [8]. Without loss of generality, the GateWay device (GW) is placed at the right end of the line, and routing devices are placed from right to left. For routing device , , it acts as a cluster head and manages several field devices. denotes the number of field devices in the cluster with being cluster head; and denotes the th field device, ; denotes the DUR of . The following content studies three scenarios: all the DURs in a cluster are same, all the DURs in a cluster are different, and the DURs in a cluster are partly same, which are briefly denoted as ADS, ADD, and PDS.

##### 3.1. Scenario I: ADS

Each field device in this cluster initially generates one packet at the begin of each superframe cycle. Considering the single-packet memory, the generated data must be sent out before the end of each superframe. Otherwise, the old data will be overlapped by newly generated data.

###### 3.1.1. Performance Limits on Intracluster Convergecast Scheduling

A routing devices manages one intracluster convergecast scheduling, and assigns timeslots and channels for communications between it and field devices. Because of the half-duplex radio transceiver, the length of intracluster convergecast scheduling should not be less than the number of field devices. For network scalability, we assume that the intracluster communication periods of all clusters are equal, which is defined as . Each field device is assigned one timeslot during the intracluster communication period.

The intracluster convergecast scheduling is simple and intuitive. In one cluster, different field devices using different timeslots, while using one channel can eliminate collisions completely. We only need to consider the interference among clusters. Neighboring clusters using different channels can avoid collisions, which will be an extension in our future work.

###### 3.1.2. Performance Limits on Intercluster Convergecast Scheduling

manages the intercluster convergecast scheduling, and allocates timeslots and channels for communications among routing devices and . The routing device nearest such as in Figure 2 is the “bottleneck" of all communications, so that the lower bound on the number of timeslots for intercluster convergecast scheduling is determined by .

Theorem 1. *If all the data update rates of field devices in a cluster are same, that is and , the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-line routing network is .*

*Proof. *At the beginning of each intercluster communication period, if (), the number of packets in satisfies . In Figure 2, is responsible for receiving and forwarding packets from to ; is also responsible for forwarding packets in its cluster to . Due to the half-duplex constraint, needs timeslots. Reminded that , then needs timeslots to forward all packets to . Therefore, the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-line network is .

Theorem 2. *Given any scheduling algorithm that can complete intercluster convergecast in timeslots in a cluster-line routing network, the lower bound on the number of channels used in is
*

*Proof. *Let be the number of channels that can be used in a network, and denotes the number of parallel transmissions scheduled in timeslot (). In order to complete convergecast in timeslots, the following conditions should be satisfied.

(i) When , there should be only one packet in the network and forwards it to . That is, only one device could be scheduled to transmitted if .

(ii) When , there exist two situations. First, sends its last packet to in the timeslot ; and forwards this packet to in timeslot . Second, there exist only two packets in ; and sends these two packets to , respectively, in timelsot and . In all, only one device could be scheduled to transmitted if .

(iii) Deduced as so forth, the maximum number of devices allowed to transmit in timeslot is .

Since , when and the upper bound on the number of channels is , we can obtain the following conclusion as in [8]:

By (9), the maximum number of transmissions that can be scheduled in using channels is

For intercluster convergecast, only routing devices and along the line routing transmit packets. Thus, the number of transmissions for completing convergecast is at least . Combined with (10), the number of transmissions satisfy the following inequality:

Replacing with , the lower bound on the number of channels used to complete convergecast in timeslots is

From Theorems 1 and 2, we can discover that the results deduced in [8, 9] is special cases of our work when all the data update rates of field devices in a cluster are same.

##### 3.2. Scenario II: ADD

If all the DURs of field devices in a cluster with () being cluster head are different, is satisfied the following.

###### 3.2.1. Performance Limits on Intracluster Convergecast Scheduling

Field devices with different DURs can realize the timeslot multiplex. For example, if the superframe cycle is 1 and the DURs of two field device and () are 2 and 4, respectively, then the timeslot of can be occupied by when has no data to be transmitted.

Theorem 3. *Given a cluster with more than one field device, if all the data update rates of field devices are different, that is , for all , , , the lower bound on the number of timeslots used for the intracluster convergecast in a cluster-line routing network is 2.*

*Proof. *Obviously, for a cluster with only one field device, the intracluster convergecast can be completed by using only one timeslot. We consider the cluster with more than one field device below. For a cluster with () being cluster head, the number of field devices is () and DUR of the th field device is (). We assume that the DUR of a field device is , where is a natural number. Without loss of generality, we suppose that ( is a natural number).

Denoted that the superframe length of is . denotes the basis superframe length. that has the minimal DUR needs one timeslot per superframe cycle to transmit its packet. Other field devices cannot occupy its timeslot and only one timeslot can not finish the intracluster convergecast.

During superframe cycles, the first timeslot in every intracluster communication period is allocated to and is denoted as , where is the index of the timeslot during the first superframe cycle and .

For , the available timeslot is the second timeslot in each intracluster communication period. We choose one timeslot numbered as 2 in the intracluster communication period from the first superframe cycles and denoted as , which is to guarantee the packet in can be sent out before DUR. Since updates its data every timeslots, will occupied the timeslot indexed by during superframe cycles, where . During superframe cycles, the occupied number of timeslots number as 2 in the intracluster communication period is and the remainder number of the second timeslots in the intracluster communication period is .

The timeslot assignment for the rest field devices can be done in the same manner as . In all, the available timeslot for () is the second free timeslot in the intracluster communication period during the first superframe cycles and occupied number is during superframe cycles.

For the last field device , the number of idle timeslot numbered as 2 in the intracluster communication period during superframe cycles is:
which illustrates that the number of free timeslot number as 2 in the intracluster communication period during superframe cycles is enough for all field devices completing their intracluster convergecast.

###### 3.2.2. Performance Limits on Intercluster Convergecast Scheduling

The following Corollaries 4 and 5 can be deduced from Theorems 1 and 2.

Corollary 4. *If all the data update rates of field devices in a cluster are different, that is , for all , , the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-line routing network is , where is the number of routing devices in a network.*

*Proof. *Let be the number of packets in () at the beginning of each superframe cycle. The proving approach is similar to Theorem 1. The only difference is that the lower bound of is 2 because of Theorem 3. needs timeslots for forwarding packets from and needs timeslots for forwarding own packets to . In all, the lower bound on the number of timeslots used for intercluster convergecast scheduling is the sum of these two parts, which can be formulated as .

Corollary 5. *Given any scheduling algorithm that can complete intercluster convergecast in timeslots in a cluster-line network, the lower bound on the number of channels used in is , where is the number of routing devices in a network.*

*Proof. *The proving approach is similar to Theorem 2. The only difference is to replace in Theorem 2 with . Then, the lower bound on the number of channels satisfy the following inequality:

##### 3.3. Scenario III: PDS

If the DURs of field devices in a cluster with () being cluster head are partly same, the following is satisfied:

###### 3.3.1. Performance Limits on Intracluster Convergecast Scheduling

In this scenario, field devices with different DURs can also realize the timeslot multiplex.

Theorem 6. *Given a cluster with more than one field device and the data update rates of field devices in a cluster with being cluster head are partly same, that is , , , , the lower bound on the number of timeslots used for the intracluster convergecast in a cluster-line network is . is the number of field devices with DUR being and satisfies (); is the number of DUR kinds in this cluster; satisfies:*

where .

*Proof. *The basic idea of intracluster scheduling is to search free timeslots without increasing the number of total occupied timeslots during each superframe cycle. The scheduling cycle is designated as superframe cycles. We assume that .

The number of field devices that have the minimum DURs being is . These field devices can not occupy each other’s timeslots. Therefore, the number of timeslots needed by field devices with DURs being is .

The field devices with DURs being need to be allocated the free second timeslot in the intracluster communication period during the first superframe cycles. If these timeslots have already been occupied, the total number of timeslots should be increased. And the increased number of timeslots is .

For field devices with DUR being , the available timeslots can be searched from the free second timeslot in the intracluster communication period during the first superframe cycles, which is numbered from 2 to . If these timeslots are not enough for the field devices, the number of timeslots should be increased. The number of free timeslots indexed from 2 to is . The increased number of timeslots is determined byGenerally speaking, for field devices with DUR being (), the available timeslots are from the free second timeslot to in the intracluster communication period during the first superframe cycles. If these timeslots are not enough for these field devices, the number of timeslots should be increased. The number of free timeslots indexed from 2 to is . The increased number of timeslots is determined byIn all, the total number of timeslots used for intracluster convergecast is the sum of the increased number of timeslots, which is .

###### 3.3.2. Performance Limits on Intercluster Convergecast Scheduling

The following Corollaries 7 and 8 can be deduced from Theorems 1 and 2.

Corollary 7. *If the data update rates of field devices in a cluster are partly same, that is , , , , the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-line routing network is , where is the number of routing devices in a network and is the number of DUR kinds in a cluster with being the cluster head.*

*Proof. *Let be the number of packets in routing device () at the beginning of each superframe cycle. The proving approach is similar to Theorem 1. The only difference is that the lower bound of is according to Theorem 6.

needs timeslots for forwarding packets from and needs timeslots for forwarding own packets to . In all, the lower bound on the number of timeslots used for intercluster convergecast scheduling is the sum of these two parts, which can be formulated as .

Corollary 8. *Given any scheduling algorithm that can complete intercluster convergecast in timeslots in a cluster-line routing network, the lower bound on the number of channels used in is*

where is the number of routing devices in a network and is the number of DUR kinds in a cluster with being the cluster head.

*Proof. *The proof is similar to Theorem 2. The only difference is to replace in Theorem 2 with . Then, the lower bound on the number of channels used for intercluster convergecast scheduling is

#### 4. Time- and Channel-Optimal Convergecast for Cluster-Tree Topology

Figure 3 is a cluster-tree network. There exist routing devices in the network, which manage several field devices and constitute clusters. Let denote the depth of the routing tree. The is located at level 0 and has subtrees rooted at routing devices , respectively. The number of routing devices in subtrees , respectively, are . Without loss of generality, it is assumed that . Therefore, is the largest subtree. Let denote the th routing device in the th subtree and be the number of field devices in the cluster with being the cluster head (, ). Next, we will investigate the convergecast scheduling problems in cluster-tree networks from three scenarios: ADS, ADD, and PDS.

##### 4.1. Scenario I: ADS

For a cluster-tree network, if all the DURs in a cluster are same, the number of timeslots for intracluster convergecast with being cluster head is at least equal to the number of field devices in this cluster, that is, the number of timeslots for intracluster convergecast is (, ). For network scalability, we specify that the number of timeslots for intracluster convergecast is .

The lower bound on the number of timeslots used for intercluster convergecast scheduling is decided by both the largest subtree and the number of packets in network. For ADS, the number of packets in routing devices are equal to the number of timeslots needed for intracluster convergecast. That is to say, the number of packets in routing device is (, ).

Theorem 9. *If all the data update rates of field devices in a cluster are same, the lower bound on the number of timeslots required to complete the intercluster convergecast in a cluster-tree network and routing devices is , where is the number of field devices in the cluster with being cluster head (, ).*

*Proof. *For (), all packets in subtree must go through to . Hence, needs at least timeslots to forward all packets in . Since the largest subtree is , at least timeslots are needed to complete the convergecast. Meanwhile, since the can receive only one packet per timeslot, at least timeslots are needed to complete the intercluster convergecast in the network having packets. In all, the lower bound on the number of timeslots needed to complete the intercluster convergecast is .

Theorem 9 shows that the structure of the routing tree plays a fundamental role in minimizing the convergecast time and dominates the minimum network latency. If (i.e., ), then the lower bound on the number of timeslots to complete the intercluster convergecast is , which is equal to the number of packets in the whole network. Otherwise, the lower bound on the number of timeslots to complete the intercluster convergecast is , which is equal to the number of packets in the largest subtree . Finding a minimum spanning tree subject to cardinality constraints on the number of routing devices in any subtree, the so called capacitated minimum spanning tree problem, is known to be NP-hard [8], and the effective heuristics algorithm should be carefully designed.

Theorem 10. *Given any scheduling algorithm that can complete intercluster convergecast in timeslots in a cluster-tree network, the lower bound on the number of channels used in is
**
where denotes the total number of packets in routing devices at level .*

*Proof. *Let be the number of channels that can be used in a network, and denotes the number of parallel transmissions scheduled in timeslot ().

To complete convergecast in timeslots, the following conditions should be satisfied [8].(i)For , there should be only one packet in the network and is required to send it to . That is, only one device could be scheduled to be transmitted if .(ii)For , two devices could be scheduled to be transmitted in the same timeslot.(iii)Deduced as so forth, the maximum number of devices allowed to transmit in timeslot is .

Since when and the upper bound on the number of channels is , we can obtain the following conclusion [8]:

By (23), the maximum number of transmissions that can be scheduled in using channels is
For the intercluster convergecast, only routing devices and along the routing tree transmit packets. Thus the number of transmissions for completing convergecast is at least . Combined with (24), the number of transmissions satisfy the following inequality:

The inequality (25) can rewritten as
where .

##### 4.2. Scenario II: ADD

In Scenario II, the DURs of field devices in the cluster with (, ) being cluster head satisfy the following relationship:

Theorem 11. *Given a cluster with more than one field device, if all the data update rates of field devices in a cluster are different, that is , for all , , , the lower bound on the number of timeslots used for the intracluster convergecast in a cluster-tree network is 2.*

*Proof. *Similar to Theorem 3.

For ADD, the number of packets in a routing device is equal to the number of timeslots for intracluster convergecast. According to Theorem 11, at the beginning of the intercluster convergecast, the number of packets in a routing device is 2.

Corollary 12. *If all the data update rates of field devices in a cluster are different, that is , for all , , , the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-tree network is , where is the number of routing devices in subtree .*

*Proof. *Let be the number of packets in during one superframe cycle. Reminded that the number of packets in a routing device is 2 at the beginning of the intercluster communication period, we can conclude that (, ). Similar to the proof of Theorem 9, the lower bound on the number of timeslots used for the intercluster convergecast in a network with cluster-tree routing is .

Corollary 13. *Given any scheduling algorithm in a network with cluster-tree routing that can complete intercluster convergecast in timeslots, the lower bound on the number of channels used in is
**
where denotes the total number of packets in routing devices at level .*

*Proof. *The proof of Corollary 13 is similar to Theorem 10. The only difference is that according to Corollary 13.

##### 4.3. Scenario III: PDS

In Scenario III, the DURs of field devices in the cluster with (, ) being cluster head satisfy the following relationship:

Corollary 14. *Given a cluster with () being cluster head, the data update rates of field devices in a cluster are partly same, that is , , . Supposing that the number of field devices with DUR being is and (), the lower bound on the number of timeslots used for the intracluster convergecast in a cluster-tree network is , where , , . is the number of DUR kinds in the cluster with being cluster head, and is defined as:*

where .

*Proof. *similar to Theorem 6.

The performance limits on the intercluster convergecast scheduling are determined by the performance limits on the intracluster convergecast scheduling, which have been established in Corollary 14.

Corollary 15. *If the data update rates of field devices in a cluster are partly same, that is , , , the lower bound on the number of timeslots used for the intercluster convergecast in a cluster-tree network is , where is the number of routing devices in subtree , , .*

*Proof. *Let be the number of packets in routing device during one superframe cycle. Reminded that the number of packets in the routing device is at the beginning of the intercluster convergecast, we can conclude that (, , ). Similar to the proof of Theorem 9, the lower bound on the number of timeslots used for the intercluster convergecast in a network with cluster-tree routing is .

Corollary 16. *Given any scheduling algorithm that can complete intercluster convergecast in timeslots in a cluster-tree network, the lower bound on the number of channels used in is
**
where denotes the total number of packets in routing devices at level .*

*Proof. *The proof of Corollary 16 is similar to Theorem 10. The only difference is that according to Corollary 16.

#### 5. Extensive Analysis

The theoretical analysis of the lower bounds on the number of timeslots and the number of channels gives deep insights into the network planning and reliable scheduling algorithm designing. In order to increase the reliability for IWSNs, a retransmission scheme is introduced.

We consider the general cluster-tree network. According to the theoretical analysis in Section 4, the lower bound on the number of timeslots for intracluster convergecast in Scenario I is and the lower bound on the number of timeslots for intercluster convergecast is . If we consider retransmissions for each packet (default value of *mac Max Frame Retries* in IEEE STD 802.15.4-2006 [10]), the lower bounds on the number of timeslots for intracluster and intercluster convergecast in Scenario I are times of and , respectively. Meanwhile, the number of timeslots during the beacon, CAP and CFP periods are 16 according to the IEEE STD 802.15.4-2006 [11]. In all, the superframe length can be calculated as × × *duration of timeslot*. For a network having 1000 field devices, that is , and the duration of the timeslot being 10 ms, if and , the maximum superframe length is 80.16 s, which is allowable in most industrial process applications. In Scenario II and Scenario III, the superframe lengths are less than 80.16 ms because of smaller number of timeslots.

The analysis of superframe length indicates that the network with hierarchical topology can accommodate more devices and can guarantee better timeliness. For applications required large network and tight timeliness, if we assign field devices with different DURs in same cluster, the requirements can be well satisfied.

#### 6. Experiment

We validate the tightness from a real implementation on the SIA2420 modules [14]. The SIA 2420 module is completely compliant to the IEEE STD 802.15.4-2006 physical layer and built in multiple specifications, such as WIA-PA, WirelessHART and Zigbee. It is a high reliable, low energy-consumption and long-distance wireless product of original equipment manufacturer (OEM), which is controlled by a 16-bit microcontroller and is installed by a MultiMedia Communication eXchange (MMCX) antenna connector. The transmission radius of SIA 2420 is 25 meters. The user interface is built on the Microsoft Visual C++ 6.0. Figures 4 and 5 exhibits the experiment platform, network topology and user operation interface, respectively.

Based on our experiment system, we investigate the bounds of the number of timeslots and channels with cluster-line and cluster-tree routing topologies, respectively. We use three parameters to represent the network with cluster-tree topology, which are , , . denotes the number of children of ; denotes the depth of the tree; and denotes the number of field devices in the cluster. Each simulation setting is represented as . Different combination of these three parameters will generate different topology. Figure 6 compares the experiment convergecast time with the optimal results for three scenarios in the network with the cluster-line routing topology. The experiment results show that there exist the optimal strategies for convergecast in the network with cluster-line topology, the conclusion of which is same as the proof results. Figure 7 compares the experiment convergecast time with the optimal results for three scenarios in the network with the cluster-tree routing topology. The experiment results of the network with cluster-tree routing topology show that our two-stage scheduling algorithms are near-optimal and the maximum degree of approximation is 99.9%.

#### 7. Conclusion

In this paper, orienting to clustered networks including cluster-line and cluster-tree, we analyze the performance limits of the convergecast scheduling. Our analysis is divided according to the network topology (cluster-line or cluster-tree) and the application scenarios (ADS, ADD, and PDS). We establish the lower bounds on the number of timeslots to finish the intracluster and the intercluster convergecast transmissions. In addition, we establish the lower bounds on the number of channels based on the lower bounds on the number of timeslots and available channels in multichannel scenario. The analysis results are future extended to take packet retransmissions into consideration so as to meet the reliability requirement.

Working with this framework has opened up several possibilities of future research. Work currently in progress is under assumption of reliable link-level transmissions. In the future, an extension to the framework is planned to analyze the performance of the networks in presence of intermittent connectivity and channel errors. The channel-constrained convergecast scheduling when the number of available channels is less than the lower bound required for optimal convergecast is a great challenge for research. As comparison with the convergecast transmission in this paper, another interesting issue to be investigated in the future work is the dissemination communications from the gateway device to other devices in the network.

#### Acknowledgments

This work was supported by the Natural Science Foundation of China under Contract 61202418 and 61174026, the National High Technology Research and Development Program of China under Contract 2011AA040101, the Special Program for Key Basic Research Founded by MOST under Contract 2010CB334705, and the Foundation of Chinese Academy of Sciences under Contract (KGCX2-EW-104).

#### References

- A. Willig, “Recent and emerging topics in wireless industrial communications: a selection,”
*IEEE Transactions on Industrial Informatics*, vol. 4, no. 2, pp. 102–122, 2008. View at Publisher · View at Google Scholar · View at Scopus - HART Communication Foundation, “Wire- lessHARTTM technical data sheet,” May 2009.
- Industrial communication network—Field-bus specifications—WIA-PA communication network and communication profile, http://www.iec.ch/.
- S. Gandham, Y. Zhang, and Q. Huang, “Distributed minimal time convergecast scheduling in wireless sensor networks,” in
*Proceedings of the 26th IEEE International Conference on Distributed Computing Systems (ICDCS '06)*, pp. 462–466, Washington, DC, USA, 2006. - H. Choi, J. Wang, and E. A. Hughes, “Scheduling on sensor hybrid network,” in
*Proceedings of the 14th International Conference Computer Communications and Networks (ICCCN '05)*, pp. 503–508, San Diego, Calif, USA, October 2005. - C. Florens, M. Franceschetti, and R. J. McEliece, “Lower bounds on data collection time in sensory networks,”
*IEEE Journal on Selected Areas in Communications*, vol. 22, no. 6, pp. 1110–1120, 2004. View at Publisher · View at Google Scholar · View at Scopus - B. Yu, J. Li, and Y. Li, “Distributed data aggregation scheduling in wireless sensor networks,” in
*Proceedings of the 28th IEEE Conference on Computer Communications (INFOCOM '09)*, pp. 2159–2167, Orlando, Fla, USA, April 2009. View at Publisher · View at Google Scholar · View at Scopus - P. Soldti,
*On cross-layer design and resource scheduling in wireless networks [Ph.D. thesis]*, KTH, School of Electrical Engineering, Stockholm, Sweden, 2009. - P. Soldati, H. Zhang, and M. Johansson, “Deadline-constrained transmission scheduling and data evacuation in wirelesshart networks,” in
*Proceedings of the European Control Conference (ECC '09)*, Budapest, Hungary, September 2009. - W. Shang, P. Wan, and X. Hu, “Approximation algorithm for minimal convergecast time problem in wireless sensor networks,”
*Wireless Networks*, vol. 16, no. 5, pp. 1345–1353, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Zhang, A. Arora, Y. R. Choi, and M. G. Gouda, “Reliable bursty convergecast in wireless sensor networks,”
*Computer Communications*, vol. 30, no. 13, pp. 2560–2576, 2007. View at Publisher · View at Google Scholar · View at Scopus - X. Liu, H. Zhang, Q. Xiang, X. Che, and X. Ju, “Taming uncertainties in real-time routing for wireless networked sensing and control,” in
*Proceedings of the 13th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc '12)*, Hilton Head, SC, USA, 2012. - IEEE SA Standards Board, “Part 15.4: wireless medium access control (MAC) and physical layer (PHY) specification for low- rate wireless personal area networks (LR- WPANs),” Nmuber 802.15.4, 2006.
- SIA 2420, http://www.industrialwireless.cn/03.asp?pd=dl&nclas-sid=628&anclassid=14.