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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

By eliminating redundant data flows, data aggregation capabilities in wireless sensor networks could transmit less data to reduce the total energy consumption. However, additional data collisions incur extra data retransmissions. These data retransmissions not only increase the system energy consumption, but also increase link transmission delays. The decision of when and where to aggregate data depends on the trade-off between data aggregation and data retransmission. The challenges of this problem need to address the routing (layer 3) and the MAC layer retransmissions (layer 2) at the same time to identify energy-efficient data-aggregation routing assignments, and in the meantime to meet the delay QoS. In this paper, for the first time, we study this cross-layer design problem by using optimization-based heuristics. We first model this problem as a non-convex mathematical programming problem where the objective is to minimize the total energy consumption subject to the data aggregation tree and the delay QoS constraints. The objective function includes the energy in the transmission mode (data transmissions and data retransmissions) and the energy in the idle mode (to wait for data from downstream nodes in the data aggregation tree). The proposed solution approach is based on Lagrangean relaxation in conjunction with a number of optimization-based heuristics. From the computational experiments, it is shown that the proposed algorithm outperforms existing heuristics that do not take MAC layer retransmissions and the energy consumption in the idle mode into account.

Wireless sensor networks (WSNs) have received attention increasingly in recent years. First, the sensor nodes can probe and collect environmental information, such as temperature, atmospheric pressure and irradiation by providing ubiquitous sensing, computing and communication capabilities. Second, thanks to the development of sensor node hardware technologies, the cost of sensor nodes has declined rapidly. This makes it possible to deploy large scale WSNs [

Since each sensor node is powered by a battery and the exchange of batteries at the depleted sensor nodes is unlikely, data aggregation routing has been put forward as a particularly useful function for routing in terms of energy consumption in WSNs [

In WSNs, any sensor node that is within another's interference range trying to transmit simultaneously would result in

For example, in

Basically, there are two operating stages (active stage and sleep stage) for sensor nodes in WSNs. In the sleep stage, a sensor node will turn off its transceiver so that there is no power consumption. Whereas, in the active stage, a sensor node could either transmit data or listen from other sensor nodes. To facilitate data communication, the sensor nodes in the data aggregation tree should be in the active stage, and all the nodes that are not included in the data aggregation tree (e.g., node

When a sensor node in a data aggregation tree waits for data from the downstream nodes, this sensor node would operate in the idle mode during the waiting time (which is the maximum end-to-end delay from the farthest downstream node). For example, in

Intensive research has been conducted on data aggregation routing, but the important MAC layer retransmission issue as described above has been relatively seldom addressed. Krishnamachari [

Several works have addressed the MAC aware data aggregation routing problem in WSNs. In [

Several works have addressed the latency issue for data aggregation in WSNs. In [

How to minimize the energy consumption of MAC-aware data aggregation routing in WSNs under end-to-end delay QoS constraints (denoted as the

We propose an optimization-based heuristics to solve this E2EDAR problem. The problem is first formulated as an integer and non-convex mathematical programming problem where the objective function is to minimize total power consumption (includes data transmission power and the power consumption in idle time to wait data from downstream nodes) subject to data aggregation tree, transmission power coverage and end-to-end delay. Then Lagrangean relaxation scheme in conjunction with the optimization-based heuristics is proposed to solve this problem. From the computational experiments, the proposed solution approaches outperform the existing heuristics.

The remainder of this paper is organized as follows. In Section 2, a mathematical formulation of the E2EDAR is proposed. In Section 3, solution approaches based on Lagrangean relaxation are presented. In Section 4, heuristics are developed for calculating good primal feasible solution. In Section 5, computational results are reported. Finally, Section 6 concludes this paper.

We assume that each sensor node is equipped with a CSMA/CA compatible transceiver. In WSN, since there is no base station as the coordinator, the communication between sensor nodes is via ad-hoc mode. Hence, the sensor nodes will contend for the channel to transmit the data. We first examine the contention-based CSMA/CA protocol – Distributed Coordination Function (DCF), to derive the energy consumption function and delay function for the E2EDAR problem.

For the DCF in

Before we derive mathematical equations for the retransmission times and the end-to-end delay for data aggregation routing in WSNs, we first define notation in the following table.

The WSN is modeled as a graph in which sensors are represented as nodes and the arc connected two nodes indicates that one sensor is within the other's transmission radius. The definition of notation adopted in the formulation is listed below.

The notation used in the formulation is as follows.

Given parameters:
_{sq}_{p(n,k)}_{nk}_{data}_{n}_{n}(r_{n})_{idle}

Decision variables:
_{sp}_{(n,k)}_{n}_{(n,k)}_{n}_{nk}_{nk}

Basically, the size of _{sq}_{sq}_{sp}_{sq}

In order to better explain the objective function to be shown in the proposed mathematical formulation, we next provide a more detailed description on the network operation as follows. First, it is assumed that the network operates in a synchronous fashion, where in each “data collection cycle” all the nodes in the network cooperatively collect, aggregate and transmit a final result to the sink node. It is also assumed that a separate communication channel exists for, e.g., the sink node to initiate and control each data collection cycle. More precisely, during each data collection cycle, every data source node collects a measurement and sends it back to the sink node via the selected data aggregation tree. Once an intermediate node alone the data aggregation tree successfully receives data packets from all of its downstream nodes, the corresponding data aggregation process is performed and the intermediate result is transmitted upwards.

Based on the results in [_{nk}

The meaning of the above retransmission function is the mean value of a geometrically distributed random variable where the successful transmission probability _{success}_{(n,k)} is that for a transmission from node _{success}_{(n,k)} is in general an underestimation, where the maximum degree of interference during the entire data collection cycle is assumed. As a result, _{nk}

The link transmission delay from node _{(n,k)}

By substituting (3) into (2), we get:

The denominator in

Objective function:

The first term in the objective function indicates the energy consumption from data packet transmission and RTS packet retransmissions. The second term in the objective function indicates the energy consumption in the idle period. Hence, the objective function of (IP) is to minimize data transmission power and the power consumption in idle time.

_{(n,k)}

_{n} ≥ d_{kn}, _{nk}_{nk}_{nk}_{(n,k)}_{nk}_{(n,k)}_{nk}

_{(n,k)}_{k}_{(}_{k,n}_{)}) − _{n}_{(n,k)}_{k}_{(k,n)}) ≤ _{n}_{A}_{B}_{(A, C)}_{C}_{A}_{(A, C)}_{C}_{(C, E)}_{E}_{C}_{(C, E)}_{S}_{G}_{(G, S)}

In

In addition, the approximation function exceeds the original function. This guarantees that if the approximation function could satisfy the maximum end-to-end delay function from the data source node back to the sink, then the original function will also satisfy. This kind of overestimation is valid from the engineering perspective. We then take natural logarithm on both sides in order to make this function solvable:

Problem (P) is a non-convex programming problem because a number of decision variables are coupled in product forms in

The algorithm development is based upon Lagrangean relaxation. In (IP), by introducing Lagrangean multiplier vectors ^{1}, u^{2}, u^{3}, u^{4}, u^{5}, u^{6}, u^{7}^{8}

subject to

(LR) is then decomposed into the following 6 independent subproblems.

Subproblem 1: for _{n}

Subproblem 2: for _{(n,k)}

Subproblem 3: for _{sp}

Subproblem 4: for _{n}_{nk}

Subproblem 5: for _{nk}

Subproblem 6: for _{(n,k)}

(SUB1) can be further decomposed into |

_{n}

When the coefficient of _{n}_{n}_{l}_{n}

Subproblem 2 is to determine decision variable _{(n,k)}.

The proposed algorithm to optimally solve (SUB2) is shown as follows.

_{(n,k)}

_{(n,k)}_{(n,k)}_{(n,k)}

_{g}_{g}_{(n,k)}_{g}_{g}_{(n,k)}_{(n,k)}

The computational complexity of the above algorithm is O(|^{2}).

(SUB3) can be further decomposed into |^{2}) for each data source node.

(SUB4) can be optimally solved by exhaustively searching all combinations of radius _{n}_{nk}_{n}

In (SUB5), if the corresponding coefficient
_{nk}

We can further decompose (SUB6) into |^{2} independent subproblems. For each link (

_{(n,k)} ^{0.115}·(

If
_{(n,k)} to be ^{0.115}·(_{(n,k)} by the following procedure. Apply the first derivative with respect to _{(n,k)} on the objective function of (SUB6.1) and let it be 0,

Then calculate the second derivative with respect to _{(n,k)} on the objective function of (SUB6.1).

Since the second derivative is larger than or equal to zero, the objective function of (SUB6.1) is a convex function. Then the optimal value of _{(n,k)} is either
^{0.115} (

According to the algorithms proposed above, we could effectively solve the Lagrangean relaxation problem optimally. Based on the weak Lagrangean duality theorem, _{D}^{1},u^{2},u^{3},u^{4},u^{5},u^{6},u^{7},u^{8}_{IP}

The basic idea of getting primal feasible solution (denoted as _{sp}_{sp}_{n}_{nk}

We perform rerouting algorithm to decrease the maximum end-to-end delay of the routing path for each data source node. The steps of the rerouting heuristic are as follows:

Identify the path (denoted as

Investigate nodes located on _{(n,k)}, is smaller than the maximum end-to-end delay of node

Update the decision variable _{(n,k)} and recalculate the maximum end-to-end delay of the new routing path.

If no node on path

However, the union of the routing paths of all data source node might not be a data aggregation tree because the tree constraints [i.e.,

The algorithm for the drop heuristic is as follows:

Based on the solutions of (SUB3) we can get the set of decision variables, _{sp}_{nk}_{nk}

According to the arc weight calculated in Step 1, we sort the links from small to large.

We sequentially remove the links from the largest arc weight to the smallest one, but we ignore the links with infinity costs. At the time when link, say link (_{sp}

After executing the drop heuristic we get a data aggregation tree without any cycles. The computational complexity of this getting primal feasible heuristic (include rerouting and drop heuristic) is ^{3}). Note that the reasons that we choose
^{1}^{2}^{1}^{2}^{6}^{7}^{8}

In the following, we show the complete algorithm (denoted as LGR) to solve Problem (P). The computational complexity of the above LGR algorithm is ^{4}) for each iteration.

^{i}

_{LR}

_{LR}

_{LR}

The proposed algorithms for solving E2EDAR problem are coded in C and run on a PC with an INTEL™ PIV-2G CPU.

We assume that a sensor network operates in periodic mode where, the sensor nodes periodically report information to the sink node. The network topology comprises _{n}_{n}_{idle}_{n}

In

We summarize the improvement ratio of the LGR algorithm over the other 4 heuristics in

Data aggregation could decrease redundant data transmissions so as to minimize the total transmission energy. However, data aggregation also increases the collision probabilities so as to increase the system energy consumption and link delays from data retransmissions. Optimizing the trade-off between data aggregation and retransmission in terms of energy consumption and delay is an interesting and challenging issue in WSNs. In this paper, for the first time, we model the E2EDAR problem as an optimization problem, where the objective function is to minimize the total (including transmission time and idle time) energy consumption subject to delay QoS, retransmission and data aggregation tree constraints. The proposed solution approach is based on Lagrangean relaxation for calculating an energy-efficient data aggregation tree that considers routing assignment, transmission radius assignment, data retransmission, and maximum end-to-end delay constraints. Note that the values of Lagrangean multipliers reflect the violation cost for the corresponding relaxed constraints. Hence they could provide useful information to get primal feasible solutions. According to the computational experiments, the LGR algorithm is superior to the other heuristics under all tested cases. More precisely, the LGR algorithm outperforms the LGRMAC, CCA, CNS, and GIT heuristics by 17%, 123%, 30% and 49%, respectively. In addition, the LGR algorithm could not only obtain feasible data aggregation trees than the other heuristics under stringent delay QoS constraints but also identify more energy efficient data aggregation trees under loose ones.

Besides the objective considered in this paper to construct an energy-efficient data-aggregation tree so as to meet the delay QoS requirements, it is also an important design issue for an MSN to maximize its life time. Although the proposed model and algorithm may inherently tend to prolong the lifetime of a WSN, nevertheless, to specifically address the aforementioned issue of MSN lifetime, particularly taking into consideration of the effects of sensors' residual energy, the following mechanisms may be adopted directly based upon the proposed algorithm:

The proposed algorithm may be re-executed periodically or on an event-driven basis. When the residual energy of a node is below a certain level, this node is reserved for future use unless it is absolutely necessary.

Also execute the proposed algorithm periodically or on an event-driven basis, where the energy consumption function _{n}_{n}

This work is supported in part (i) by the Shih-Hsin University, Taiwan, under Grant No. P 9711, (ii) by the National Science Council, Taiwan, under Grant No. NSC 98-2622-H-128-002-CC3, and (iii) by the National Taiwan University and the Ministry of Education, Taiwan, under the Top University Project.

^{2}RP: The Multi-Rate and Multi-Range Routing Protocol for IEEE 802.11 Wireless Ad Hoc Networks

Data aggregation in MAX.

DCF mode in CSMA/CA protocol.

Link delay approximation function versus the original function.

Performance comparison with respect to traffic loads.

Performance comparison with respect to maximum end-to-end delays (i.e.,

Performance comparison with respect to network sizes.

Performance comparison between LGR and the other three heuristics.

| |||
---|---|---|---|

LGRMAC | 17% | 11% | 10% |

CCA | 123% | 33% | 50% |

CNS | 30% | 12% | 28% |

GIT | 49% | NA |
44% |

The GIT algorithm does not identify feasible solutions.