Open Access

A Stochastic Multiobjective Optimization Framework for Wireless Sensor Networks

  • Shibo He1,
  • Jiming Chen1, 2,
  • Weiqiang Xu1,
  • Youxian Sun1,
  • Preetha Thulasiraman2 and
  • Xuemin(Sherman) Shen2Email author
EURASIP Journal on Wireless Communications and Networking20102010:430615

DOI: 10.1155/2010/430615

Received: 31 October 2009

Accepted: 17 February 2010

Published: 20 May 2010

Abstract

In wireless sensor networks (WSNs), there generally exist many different objective functions to be optimized. In this paper, we propose a stochastic multiobjective optimization approach to solve such kind of problem. We first formulate a general multiobjective optimization problem. We then decompose the optimization formulation through Lagrange dual decomposition and adopt the stochastic quasigradient algorithm to solve the primal-dual problem in a distributed way. We show theoretically that our algorithm converges to the optimal solution of the primal problem by using the knowledge of stochastic programming. Furthermore, the formulation provides a general stochastic multiobjective optimization framework for WSNs. We illustrate how the general framework works by considering an example of the optimal rate allocation problem in multipath WSNs with time-varying channel. Extensive simulation results are given to demonstrate the effectiveness of our algorithm.

1. Introduction

The layered architecture approach has achieved great success in traditional wired network design by dividing the whole architecture into several modules, called layers, each of which performs a separate functionality. As each layer design only needs some interface variables from the layer below, the complexity of other layers can be hidden. The layered architecture approach suggests that the network design can be scalable, evolvable, and implementable. However, it may have limitations in improvement of efficiency and fairness, and suffer potential risks of manageability [1], which motivates the optimization of network design. Chiang et al. [1, 2] propose an optimization decomposition technique to systematically understand the network architecture, known as "layering as optimization decomposition". They model the network as an optimization problem and decompose the problem into many subproblems. They classify the decompositions into vertical decomposition and horizontal decomposition. Vertical decomposition layers the network architecture into several modules and horizontal decomposition provides distributed algorithms to fulfill the functionality within the modules. According to the requirements of the applications, the decomposition may be different, yielding different layers and distributed algorithms. There are usually two steps in the process of layering as optimization decomposition: ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq1_HTML.gif ) modeling the network problem as a specific NUM problem, and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq2_HTML.gif ) exploring the alternative decompositions to design different modules and distributed algorithms. Most existing efforts have been put to the second step and simply assume that the network problems can be modeled by a unified utility function at the first step [36]. However, not all network problems can be modeled by a unified utility function in a tractable way since there may exist many objectives to be achieved, such as guaranteeing fairness, maximizing throughput, reducing packet dropping and delay, prolonging the network lifetime, and so forth. It may not be possible to integrate all these objectives into a single unified utility function, that is, network problems should be formulated as multiobjective optimization problems.

While the performance of the network can be greatly enhanced by adopting the NUM approach, the corresponding cost of algorithm implementation also increases. As we usually design and implement an algorithm for a specific application from scratch, the implementation can hardly be transplantable to other applications. This is especially aggravated in wireless sensor networks (WSNs) due to the application-oriented and infrastructureless nature of these networks. For example, if we design an efficient algorithm for events monitoring, the network lifetime is the main concern and the propagation delay can be tolerant, but it is difficult to apply such algorithm to online query applications, where the query delay is the primary objective.

In this paper, we utilize the concept of multiobjective optimization and provide a general framework for a specific class of applications in WSNs. It is well known that the TCP/IP reference model, one of the most popular layered architectures, divides the whole architecture into five layers (modules), and each layer only communicates with the layers next to it, while recent work on the NUM approach divides the architectures according to applications. We first list all the constraints and objectives which the applications may have. Then the network architecture is divided into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq3_HTML.gif modules, each of which has several interface variables with other modules. In this way, we can inherit the advantages of both the layered architectures (as we have fixed modules) and the NUM approach (as each module can communicate with other modules). We illustrate this in Figure 1, in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq5_HTML.gif are the interface variables (see Section 3 for detailed definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq7_HTML.gif ), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq8_HTML.gif is the objective vector function in each module https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq9_HTML.gif . We transform the objectives of the network into specific modules through the interface variables. Each sensor optimizes its own objective vector function to achieve the global optimal solution of the whole network. In this way, for different requirements from the network, we do not redesign the framework, that is, the modules and interfaces in Figure 1 can be kept unchanged. We only need to introduce multiobjective methods to optimize the vector function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq10_HTML.gif in each module. This will greatly simplify network design for WSNs.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Fig1_HTML.jpg
Figure 1

An illustration of our proposed framework.

In WSNs, some parameters (e.g., the topology of the networks or channel condition) are time-varying. In [7], Lee et al. demonstrated that the state of the network can be more efficiently utilized to improve the performance of the network (e.g., increasing the throughput and reducing packet delay), by appropriately exploiting the variability of the time-varying channels. Also there are measurement errors in the implementation of distributed algorithms, such as the noisy feedback [8] or lossy links [9]. Therefore, we also characterize these random factors in our model. Our contributions in this paper are summarized as follows.

  1. (1)

    We formulate a general multiobjective stochastic optimization problem for WSNs. We decompose the optimization problem through Lagrange dual decomposition and adopt the stochastic quasigradient method to solve the primal-dual problem. In other words, we transform the multiple objectives of WSNs into the multiple objectives of each individual sensor node. The global optimal solution can be obtained when each sensor node maximizes its own objective vector function. Therefore, our approach provides a general framework for multiobjective optimization for WSNs.

     
  2. (2)

    We study the stability of the algorithm by using the knowledge of stochastic programming, and show that our algorithm for stochastic multiobjective optimization problem (ASMOP) can converge to the optimal solution of the primal problem.

     
  3. (3)

    We demonstrate how the general framework can be applied to different applications, by considering the rate allocation problem as an example. We introduce three multiobjective optimization methods: ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq11_HTML.gif ) constraint method, ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq12_HTML.gif ) linear weighted method, and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq13_HTML.gif ) hierarchical sequence method. The three paradigms show that although different requirements may lead to different models [6, 10], we can solve them in the general framework.

     

The remainder of the paper is organized as follows: in Section 2, we discuss related work regarding the NUM problem and stochastic network utility maximization (SNUM). We formulate a general mathematical model and design a distributed algorithm to solve the problem in Section 3, and the stability of the algorithm is also discussed. We provide three paradigms in Section 4 to demonstrate the general framework for different applications. Simulation results are given in Section 5. We conclude the paper in Section 6.

2. Related Work

There are several research works in the literature studying the NUM problem [4, 1114]. Kelly et al. were the first to propose the optimization approach, which provides a mathematical foundation for NUM problem [11]. In [12], Chiang adopted the NUM approach to obtain a cross-layer design including the physical layer and the transport layer. Zhu et al. [13] considered the energy model in the cross-layer design. In [1], Chiang et al. provided a mathematical theory of network architectures. Wang et al. studied joint interference-aware routing and TDMA link scheduling to improve the throughput in multihop wireless networks [15]. Zhang et al. [8] elaborated on the impact of the feedback in the implementation of distributed NUM algorithms. Since feedback is often collected using error-prone measurement mechanisms, for example, biased estimator or unbiased estimator, they adopted the knowledge of stochastic approximation and proved stability of the algorithms of single-time scale and two-time scale. Lee et al. utilized the variation of channels to guide power and rate control in cross-layer design [7]. In this paper, we formulate a more general mathematical model by considering stochastic multiple objectives in objective functions. We apply our approach to rate allocation problem in multipath WSNs with time-varying channels. Rate allocation is a fundamental problem and has been extensively investigated [1619]. Low and Lapsley [16] first introduced the Lagrange dual method to decompose the problem and proposed two algorithms under synchronous and asynchronous scenarios. A multipath formulation for rate control in multi-cast networks was proposed in [20], and three distributed algorithms were proposed to solve the problem. The goal is to maximize the aggregate utility. In [6], Srinivasan et al. considered two objectives: utility maximization and guaranteeing prespecified network lifetime for multipath wireless ad hoc networks. In [10], Zhu et al. also focused on the network lifetime and application performance (utility), and employed the linear weighted method from the multiobjective optimization to transform these two objective functions into a single one which was named to the utility-lifetime tradeoff function.

3. General Multiobjective Formulation and Solutions

Throughout the paper, we will denote sets by capital letters, variables by lowercase letters, vectors by bold lowercase letters, and matrices by bold capital letters. For a vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq14_HTML.gif , we denote its https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq15_HTML.gif th component by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq16_HTML.gif and its transpose by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq17_HTML.gif . We use capital letters for both the sets and the cardinality of sets.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq18_HTML.gif sensing nodes and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq19_HTML.gif sink nodes in the region of interest. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq20_HTML.gif be a probability space with a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq21_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq22_HTML.gif of random events, and have a finite set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq23_HTML.gif with the corresponding probability https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq24_HTML.gif . There are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq25_HTML.gif objective functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq26_HTML.gif , defined on the subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq27_HTML.gif of a Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq29_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq30_HTML.gif is the objective vector function of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq31_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq32_HTML.gif be the column vector of variables of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq34_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq35_HTML.gif a column vector function. We can formulate the primal problem (PP) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ1_HTML.gif
(1)
The objective function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq36_HTML.gif , may be a coupled one. In order to design a distributed algorithm, we introduce auxiliary variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq37_HTML.gif to decouple it. Assume that the node set associated with coupled variables of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq38_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq39_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq40_HTML.gif , which denotes the node set associated with coupled variables of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq41_HTML.gif of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq42_HTML.gif , then the decoupled primal problem (DPP) can be given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ3_HTML.gif
(3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq43_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq44_HTML.gif th decoupled objective function, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq45_HTML.gif is the corresponding vector of auxiliary variables.

In our formulation, the objective functions are deterministic, taking the advantage that each sensing node can be obtained from the network. The constraint set contains the random factors of the networks, such as message exchange, and environmental effect. If we know the distribution, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq47_HTML.gif , of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq48_HTML.gif , we can transform the problem into a deterministic one, by calculating the expectation. However, in WSNs, there is often no prior knowledge about the randomness from the networks themselves and the environmental effect. Therefore, we develop an algorithm without this prior knowledge, which can be achieved by the stochastic quasigradient method [7].

To decompose the problem, we take Lagrange dual approach. The Lagrange function [21] of (3) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ4_HTML.gif
(4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq49_HTML.gif is the objective vector function of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq50_HTML.gif . It is a formal expression which can be transformed into different objective functions for different applications.

We call https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq52_HTML.gif decoupled prices ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq53_HTML.gif is used to decouple the coupling of variables and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq54_HTML.gif is used to decouple the coupling of objective functions). Since (4) is separable, we exploit the decomposable structure of Lagrangian function and decompose the problem into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq55_HTML.gif subproblems. Maximization is achieved in each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq57_HTML.gif , with the knowledge of local variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq58_HTML.gif ) and the current state https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq59_HTML.gif , by solving the following optimization problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq60_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ6_HTML.gif
(6)
At iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq61_HTML.gif , each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq62_HTML.gif updates its resource variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq63_HTML.gif and auxiliary variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq64_HTML.gif according to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ7_HTML.gif
(7)
We proceed to solve the dual problem. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq65_HTML.gif . Then the dual problem ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq66_HTML.gif ) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ8_HTML.gif
(8)
At iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq67_HTML.gif , each sensing node can acquire the state of random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq68_HTML.gif . The stochastic quasigradient method only needs this current state information of the system and utilizes it to form the stochastic subgradients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq69_HTML.gif at iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq70_HTML.gif . For the dual problem, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq71_HTML.gif , prices are updated according to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ10_HTML.gif
(10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq73_HTML.gif are the stochastic quasigradients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq74_HTML.gif .

In our algorithm, we set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ13_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq75_HTML.gif is the state of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq76_HTML.gif at iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq77_HTML.gif .

We summarize our algorithm for the general formulation of stochastic multiobjective optimization problem (ASMOP) in the Algorithm 1.

Algorithm 1: ASMOP.

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq78_HTML.gif ) Price update algorithm: At times https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq79_HTML.gif , decoupled prices are updated according to

            https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq80_HTML.gif

          https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq81_HTML.gif

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq82_HTML.gif ) Sensor node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq83_HTML.gif 's Algorithm: At time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq84_HTML.gif , each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq85_HTML.gif updates its variables

according to

            https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq86_HTML.gif

To prove that the algorithm can converge to the optimal solution of the primal problem, we make the following assumptions.

  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq87_HTML.gif as well as the objective vector function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq88_HTML.gif (which can be transformed into a single function in applications), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq89_HTML.gif , are twice continuous differentiable concave functions.

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq91_HTML.gif , are convex and twice continuous differentiable functions in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq92_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq93_HTML.gif .

     

Theorem 1.

If (1) hold, then from an arbitrary point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq94_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq96_HTML.gif , the sequence generated by (7), (9), and (10) converges. Every limit point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq97_HTML.gif of the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq98_HTML.gif is primal-dual optimal.

Proof.

Let the sequences of iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq100_HTML.gif be generated by (9) and (10), respectively. Then to guarantee the convergence of the algorithm, according to [7, 22], the current stepsize and quasigradients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq101_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq103_HTML.gif should be chosen such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ14_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ15_HTML.gif
(15)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ16_HTML.gif
(16)

It can be seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq105_HTML.gif , satisfy (14). From [22]; we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq107_HTML.gif from (12) and (13) also satisfy (15) and (16).

From assumptions (1) and (2), the primal function is concave and the dual function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq108_HTML.gif is convex in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq110_HTML.gif for a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq111_HTML.gif . From (7), (9), (10), (11), (12), and (13), we can conclude that the sequence converges to the optimal solution by solving the dual problem [22]. As the primal problem is a convex optimization problem, there is no gap between the primal and dual problems. So the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq112_HTML.gif generated by the algorithm is primal-dual optimal.

Remarks 3.

Because of multipath routing, the problem, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq113_HTML.gif , may not be strictly concave even if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq114_HTML.gif is strictly concave. This may lead to oscillation of the sequences generated by the algorithms. There are several ways to cope with this problem. For example, we can first add some augmented variables to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq115_HTML.gif and adopt the first-order Lagrangian method to solve it [23].

The main difference of our proposed approach is that we adopt the knowledge of multiobjective optimization and provide some potential interfaces for each layer. In this way, we can take the advantages of both the layered architectures and cross-layer design. In other words, we can implement different algorithms in each module according to specific applications. In Figure 1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq117_HTML.gif act as the interface variables between different modules and sensor nodes. Through https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq119_HTML.gif , the network architectures can be decomposed into different modules and each module fulfills corresponding functionality distributively. From (7), we can transform the multiple objectives of the whole network into the multiple objectives of each sensor node. Optimizing the objective vector function of each sensor node can achieve the global optimal solution. Therefore, it is very convenient to implement algorithms in each module https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq120_HTML.gif to solve the objective vector function independently according to different requirements.

4. Paradigms of Objective Optimization in WSNs

In the proposed general framework, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq121_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq122_HTML.gif is a vector function and can be transformed into a single function according to different requirements. Therefore, solving https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq123_HTML.gif is application-dependent, which provides the flexibility of solving a class of applications by the general framework.

In this section, we consider the rate allocation problem as an example and show how the general framework works. Rate allocation problem is a well-investigated problem [24], and has different requirements for different applications. There are usually three methods to cope with the requirements: ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq124_HTML.gif ) Constraint Method [6], ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq125_HTML.gif ) Linear Weighted Method [10], and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq126_HTML.gif ) Hierarchical Sequence Method. While these methods are extensively studied in existing works, we can integrate these methods together into the general framework. Hence, our approach can be applied to a class of applications with different background, which will offer significant convenience to the designers.

4.1. Preliminary Knowledge

In this section, we give a brief introduction to the three multiobjective methods.

  1. (1)
    The constraint method tries to solve the multiobjective problems by placing the most important one in the objective function, while other objective functions are constrained within the constraint set. In other words, constraint method can solve the following https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq127_HTML.gif of each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq128_HTML.gif . (without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq129_HTML.gif is the most important objective function)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ17_HTML.gif
    (17)
     
  2. where

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq131_HTML.gif , are constant constraints imposed by applications.

     
  3. (2)
    The linear weighted method focuses on solving the multiobjective optimization problems by first associating each objective function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq132_HTML.gif with a weight https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq133_HTML.gif and then taking the weighted sum as a new objective function. Using linear weighted method to solve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq134_HTML.gif , the variables in each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq135_HTML.gif are updated according to
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ18_HTML.gif
    (18)
     
  4. where

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq137_HTML.gif , are weight coefficients.

     
  5. (3)
    The hierarchical sequence method is concerned with solving the multiobjective optimization problems sequentially, that is, solving the most important problem first and then the less important problems. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq138_HTML.gif and divide (5) into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq139_HTML.gif functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq141_HTML.gif . Maximization is achieved by solving the following https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq142_HTML.gif subproblems sequentially.
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ19_HTML.gif
    (19)
     

4.2. Rate Allocation Problem under ASMOP Formulation

In this section, we consider the rate allocation problem with two objectives: ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq143_HTML.gif ) maximizing aggregate utility and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq144_HTML.gif ) prolonging the network lifetime.

Assume the sensing nodes can transmit their rates to the sink nodes over a set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq145_HTML.gif of links, each of which has capacity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq146_HTML.gif . Each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq147_HTML.gif can transmit its rate through https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq148_HTML.gif of the routes. Each route https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq149_HTML.gif traverses over a set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq150_HTML.gif of links with a rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq151_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq152_HTML.gif be the rate vector of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq154_HTML.gif the set of sensing nodes using link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq156_HTML.gif the subset of routes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq157_HTML.gif used by sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq158_HTML.gif to traverse over link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq159_HTML.gif . We denote the set of sensing nodes that use sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq160_HTML.gif as an interim relay node by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq161_HTML.gif (not including the sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq162_HTML.gif itself). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq163_HTML.gif be the subset of routes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq164_HTML.gif which use sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq165_HTML.gif as a relay node and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq166_HTML.gif the relay nodes used by route https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq167_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq168_HTML.gif be finite number of state that the channels have and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq169_HTML.gif the probability of the state https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq171_HTML.gif . Each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq172_HTML.gif is characterized by three parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq173_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq174_HTML.gif is a strictly concave utility function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq176_HTML.gif which are the required minimum and maximum transmission rates for each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq177_HTML.gif , respectively.

From [11, 16], we know utility maximization can be formulated as follows.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ20_HTML.gif
(20)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ21_HTML.gif
(21)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq178_HTML.gif describes the distribution of the states of link channel condition and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq179_HTML.gif is the capacity of link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq180_HTML.gif under state https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq181_HTML.gif . We will establish algorithms that can guarantee convergence without prior knowledge of the underlying probability distribution of the system channel state.

The network lifetime is often defined as the time interval between initialization of the network and the exhaustion of the battery of the first sensing node. The total power dissipation, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq182_HTML.gif , at sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq183_HTML.gif is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ22_HTML.gif
(22)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq185_HTML.gif are the energy consumptions at sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq186_HTML.gif for transmitting or receiving unit data flow over route https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq187_HTML.gif , respectively.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq188_HTML.gif denote the initial energy of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq189_HTML.gif . Its lifetime https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq190_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq191_HTML.gif . Following [10], we have the energy model for the network lifetime:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ23_HTML.gif
(23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ24_HTML.gif
(24)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq192_HTML.gif .

We can have the multiobjective model for rate allocation problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ25_HTML.gif
(25)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ26_HTML.gif
(26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ27_HTML.gif
(27)

Here, parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq193_HTML.gif scales the values of the two objective functions into the same order of magnitude.

4.3. Algorithm Design

Similar to (3), the decoupled form of (25) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ28_HTML.gif
(28)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ29_HTML.gif
(29)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq195_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ30_HTML.gif
(30)
Then we have the Lagrange function:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ31_HTML.gif
(31)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq197_HTML.gif .

Notice that for given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq201_HTML.gif , the update for each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq202_HTML.gif is deterministic, no matter what the channel condition is. So at iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq203_HTML.gif , each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq204_HTML.gif updates its rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq205_HTML.gif and auxiliary variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq206_HTML.gif by solving the optimization below
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ32_HTML.gif
(32)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq207_HTML.gif is the objective vector function for sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq208_HTML.gif .

At iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq210_HTML.gif , sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq211_HTML.gif first receives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq214_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq215_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq216_HTML.gif , from the network, and then updates its rate and auxiliary variables according to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ33_HTML.gif
(33)
Notice that the update of decoupled prices, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq217_HTML.gif , does not depend on the channel condition. At each iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq218_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq222_HTML.gif , being collected, the decoupled price https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq223_HTML.gif is updated according to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ34_HTML.gif
(34)
To update price https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq224_HTML.gif , without prior knowledge about the distribution of the channel state, each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq225_HTML.gif can measure channel condition and get the link capacity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq226_HTML.gif for each link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq227_HTML.gif at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq228_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq229_HTML.gif can be updated according to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ35_HTML.gif
(35)

So far, we have introduced interface variables ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq230_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq231_HTML.gif ) for a fully distributed implementation and provided a framework for rate allocation in WSNs with time-varying channels. Next, we show how to apply the framework to applications under different methods.

4.4. Paradigms: Applications under Different Methods

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq232_HTML.gif ) Constraint Method. To maximize the utility of a network under the condition that the network lifetime exceeds a prespecified threshold time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq233_HTML.gif , the constraint method can be used to solve the optimization problem below of each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq234_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ36_HTML.gif
(36)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ37_HTML.gif
(37)
( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq235_HTML.gif ) Linear Weighted Method. In [10, 13], Zhu et al. considered the tradeoff between lifetime and rate allocation. By introducing the weight parameter, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq236_HTML.gif , to evaluate the importance of the two objectives, they can be combined into a single one. For (29), the desired tradeoff between network utility and lifetime can be achieved by solving the optimization problem below.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ38_HTML.gif
(38)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ39_HTML.gif
(39)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq237_HTML.gif , is the weight coefficient.

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq238_HTML.gif ) Hierarchical Sequence Method. In our rate allocation paradigm, we have two objectives: ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq239_HTML.gif ) find a rate allocation strategy to maximize the total utility and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq240_HTML.gif ) prolong the networks lifetime. To the best of our knowledge, this method has not been applied to the rate allocation problem before. We can achieve the two objectives by solving the two subproblems below sequentially.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ40_HTML.gif
(40)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Equ41_HTML.gif
(41)

It is sufficient to employ optimization methods to solve (37), (39), or (40), and (41) for different applications, while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq242_HTML.gif updates are kept unchanged (according to (34) and (35)).

5. Simulation Results

5.1. Simulation Setting

We use 9 sensing nodes and 1 sink node in our simulations. These sensor nodes are randomly deployed in an area of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq243_HTML.gif to perform a sensing task. The randomly generated topology of the sensor nodes is shown in Figure 2, in which sensing nodes are marked by triangle icons and the sink node is marked by a square icon. In our simulations, we only focus on the rate allocation problem. The study of the routing in the network layer is beyond the scope of our paper. We assume that there are 15 routes available for data transmission. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq244_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq247_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq248_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq249_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq251_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq252_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq253_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq257_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq258_HTML.gif . Every sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq259_HTML.gif can transmit its sensing data to the sink node from a set of routes: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq263_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq264_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq266_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq267_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq268_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Fig2_HTML.jpg
Figure 2

Topology of the simulated WSN.

We have two objectives: maximizing the aggregate utility and the network lifetime. For the utility objective, we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq269_HTML.gif for each sensor node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq270_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq271_HTML.gif . From [10], the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq272_HTML.gif can have a ratio higher than 0.95 to approximate the original lifetime problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq273_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq274_HTML.gif . In our simulations, we use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq275_HTML.gif . The link capacities vary from time to time according to a uniform distribution with the expected capacities of links 1–13 to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq276_HTML.gif 2000, 2000, 2200, 2000, 2000, 2500, 2800, 3500, 3500,  2000, 3000, 2800 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq277_HTML.gif (bit/s). For the energy consumption model, from [10], https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq278_HTML.gif is a constant and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq279_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq280_HTML.gif is the length of the outgoing link of sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq281_HTML.gif for transmitting rate of route https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq282_HTML.gif . We set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq283_HTML.gif  nJ/bit, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq284_HTML.gif  pJ/bit/ https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq285_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq286_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq287_HTML.gif  nJ/bit. The initial energy of the sensing nodes 1–9 is set to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq288_HTML.gif (J) and the sink node (node 10) is assumed to have enough energy. The minimum and maximum rates of each sensing node are set to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq289_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq290_HTML.gif , respectively.

5.2. Simulation Results for Paradigms

First, we show the performance of the ASMOP algorithm by using the constraint method. Let the threshold of the network lifetime https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq291_HTML.gif  h. We collect the rate of each route https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq292_HTML.gif at each iteration. For each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq293_HTML.gif , the rates are updated according to (37) and the decoupled price https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq294_HTML.gif is updated according to (34). Each link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq295_HTML.gif first collects information about the channel condition and computes its corresponding capacity, then updates its link price according to (35). The results are shown in Figure 3. Since there are 9 sensing nodes and 15 routes in our simulations, due to space limitation, we only show the aggregate rate of each sensing node. In Figure 3, we can be see that the rates first change sharply and then converge to the optimal one, which indicates the effectiveness of our ASMOP algorithm.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Fig3_HTML.jpg
Figure 3

Convergence of the ASMOP algorithm by using constraint method.

Next, we show the simulation results of the ASMOP algorithm using the linear weighted method. Here we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq297_HTML.gif . At each iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq298_HTML.gif , each sensing node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq299_HTML.gif updates its rates according to (39), and the decoupled price https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq301_HTML.gif are updated as the same in paradigm I. The results are shown in Figure 4, and similar conclusions can be made as in the paradigm I.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Fig4_HTML.jpg
Figure 4

Convergence of the ASMOP algorithm by using linear weighted method.

A similar simulation is performed for the hierarchical sequence method and the corresponding results are shown in Figure 5. We can see that the rates change sharply at the beginning of each iteration, and then converge to the optimal one in Figure 5.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_Fig5_HTML.jpg
Figure 5

Convergence of the ASMOP algorithm by using hierarchical sequence method.

We further set the threshold of the network lifetime to be 800 h in the simulation for the constraint method and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F430615/MediaObjects/13638_2009_Article_1907_IEq302_HTML.gif for linear weighted method. These two methods mainly target the energy-constraint in WSNs. For the hierarchical sequence method, we focus on the utility of the network. From Figures 3, 4, and 5, it can seen that the rates in Figure 5 are much larger than those in Figures 3 and 4. On the other hand, as the rates in each sensor node become large, the energy consumption increases. So the network lifetime under the hierarchical sequence method is less than that under the constraint method or linear weighted method. In addition, different multiobjective methods obtain different network performances. The results of the three simulations also demonstrate the efficiency and convenience of our proposed framework.

6. Conclusions

In this paper, we have proposed a general stochastic multiobjective optimization framework for WSNs. Our approach inherits advantages of both layered architectures and cross-layer design. Therefore, even the requirements and objectives are changed, it is not necessary to redesign the optimization framework but to have minor modifications of specific modules to meet the corresponding requirements. Although there may be uncertainty in WSNs, our approach can still achieve desired performance. In our future work, we will focus on investigating the general multiobjective optimization problem, instead of transforming the multiple objectives into a single one. We will study the distributed algorithms to optimize the objective vector function under some criteria, for example, Pareto optimality.

Declarations

Acknowledgment

The research was supported in parts by NSFC Guangdong joint Project grant U0735003, NSFC grants 60736021 and 60974122, China 863 High-Tech Project 2007AA041201, the Fundamental Research Funds for the Central Universities grant 2009QNA5007.

Authors’ Affiliations

(1)
State Key Lab of Industrial Control Technology, Zhejiang University
(2)
Department of Electrical and Computer Engineering, University of Waterloo

References

  1. Chiang M, Low S, Calderbank A, Doyle J: Layering as optimization decomposition: a mathematical theory of network architectures. Proceedings of the IEEE 2007, 95(1):255-312.View Article
  2. Yi Y, Chiang M: Stochastic network utility maximisation. European Transactions on Telecommunications 2008, 19(4):421-442. 10.1002/ett.1285View Article
  3. Chen L, Low S, Chiangs M, Doyle J: Cross-layer congestion control, routing and scheduling design in ad hoc wireless networks. Proceedings of the 25th IEEE International Conference on Computer Communications (INFOCOM '06), April 2006, Barcelona, Spain 1-13.
  4. Long C, Li B, Zhang Q, Zhao B, Yang B, Guan X: The end-to-end rate control in multiple-hop wireless networks: cross-layer formulation and optimal allocation. IEEE Journal on Selected Areas in Communications 2008, 26(4):719-731.View Article
  5. Jin J, Wang W, Palaniswami M: Application-oriented flow control forwireless sensor networks. Proceedings of the 3rd International Conference on Networking and Services (ICNS '07), June 2007, Athens, Greece 71.View Article
  6. Srinivasan V, Chiasserini C, Nuggehalli P, Rao R: Optimal rate allocation for energy-efficient multipath routing in wireless ad hoc networks. IEEE Transactions on Wireless Communications 2004, 3(3):891-899. 10.1109/TWC.2004.826343View Article
  7. Lee J, Mazumdar R, Shroff N: Joint opportunistic power scheduling and end-to-end rate control for wireless ad hoc networks. IEEE Transactions on Vehicular Technology 2007, 56(2):801-809.View Article
  8. Zhang J, Zheng D, Chiang M: The impact of stochastic noisy feedback on distributed network utility maximization. IEEE Transactions on Information Theory 2008, 54(2):645-665.MATHMathSciNetView Article
  9. Gao Q, Zhang J, Hanly S: Cross-layer rate control in wireless networks with lossy links: leaky-pipe flow, effective network utility maximization and hop-by-hop algorithms. IEEE Transactions on Wireless Communications 2009, 8(6):3068-3076.View Article
  10. Zhu J, Hung K, Bensaou B, Nait-Abdesselam F: Rate-lifetime tradeoff for reliable communication in wireless sensor networks. Computer Networks 2008, 52(1):25-43. 10.1016/j.comnet.2007.09.011MATHView Article
  11. Kelly F, Maulloo A, Tan D: Rate control for communication networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society 1998, 49(3):237-252.MATHView Article
  12. Chiang M: Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control. IEEE Journal on Selected Areas in Communications 2005, 23(1):104-116.View Article
  13. Zhu J, Chen S, Bensaou B, Hung K: Tradeoff between lifetime and rate allocation in wireless sensor networks: a cross layer approach. Proceedings of the 26th IEEE International Conference on Computer Communications (INFOCOM '07), May 2007, Anchorage, Alaska, USA 267-275.
  14. Chen J, He S, Sun Y, Thulasiraman P, Shen X(Sherman): Optimal flow control for utility-lifetime tradeoff in wireless sensor networks. Computer Networks 2009, 53(18):3031-3041. 10.1016/j.comnet.2009.07.014MATHView Article
  15. Wang Y, Wang W, Li X, Song W: Interference-aware joint routing and TDMA link scheduling for static wireless networks. IEEE Transactions on Parallel and Distributed Systems 2008, 19(12):1709-1726.View Article
  16. Low S, Lapsley D: Optimization flow control—I: basic algorithm and convergence. IEEE/ACM Transactions on Networking 1999, 7(6):861-874. 10.1109/90.811451View Article
  17. Cai L, Shen X(Sherman), Pan J, Mark J: Performance analysis of TCP-friendly AIMD algorithms for multimedia applications. IEEE Transactions on Multimedia 2005, 7(2):339-355.View Article
  18. Wang L, Cai L, Liu X, Shen X(Sherman): Stability and TCP-friendliness and delay performance of AIMD/RED system. Computer Networks 2007, 51(15):4475-4491. 10.1016/j.comnet.2007.06.022View Article
  19. Zhang J, Gursoy M: Achievable rates and resource allocation strategies for imperfectly-known fading relay channels. EURASIP Journal on Wireless Conmunications and Networking 2009, 2009:-16.
  20. Cui T, Chen L, Ho T: Optimization based rate control for multicast with network coding: a multipath formulation. Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), December 2007, New Orleans, La, USA 6041-6046.
  21. Bertsekas D: Parallel and Distributed Computation: Numerical Methods. 2nd edition. Athena Scientific, Belmont, Mass, USA; 1999.
  22. Kall P, Wallace S: Stochastic Programming. John Wiley & Sons, Chichester, UK; 2003.
  23. Wang W, Palaniswami M, Low S: Optimal flow control and routing in multi-path networks. Performance Evaluation 2003, 52(2-3):119-132. 10.1016/S0166-5316(02)00176-1View Article
  24. Chen J, Xu W, He S, Sun Y, Thulasiraman P, Shen X(Sherman): Utility-based asynchronous flow control algorithm for wireless sensor networks. IEEE Journal on Selected Areas in Communications. In press

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© Shibo He et al. 2010

This article is published under license to BioMed Central Ltd. 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.