Mathematical Problems in Engineering

Volume 2015, Article ID 390457, 9 pages

http://dx.doi.org/10.1155/2015/390457

## Minimum-Energy Wireless Real-Time Multicast by Joint Network Coding and Scheduling Optimization

^{1}College of Computer and Information Engineering, Hohai University, Nanjing 210098, China^{2}Shanghai Microsystem and Information Technology Research Institute, Chinese Academy of Sciences, Shanghai 200050, China

Received 21 January 2015; Revised 31 March 2015; Accepted 31 March 2015

Academic Editor: Joaquim Joao Judice

Copyright © 2015 Guoping Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For real-time multicast services over wireless multihop networks, to minimize the energy of transmissions with satisfying the requirements of a fixed data rate and high reliabilities, we construct a conflict graph based framework by joint optimizing network coding and scheduling. Then, we propose a primal-dual subgradient optimization algorithm by random sampling *K* maximal stable sets in a given conflict graph. This method transforms the NP-hard scheduling subproblem into a normal linear programming problem to obtain an approximate solution. The proposed algorithm only needs to adopt centralized technique for solving the linear programming problem while all of the other computations can be distributed. The simulation results show that, comparing with the existing algorithm, this algorithm can not only achieve about 20% performance gain, but also have better performance in terms of convergence and robustness.

#### 1. Introduction

Since Ahlswede et al. proposed network coding (NC) [1], many studies have shown that NC can not only increase the throughput significantly but also achieve better robustness. Actually, it can achieve network multicast capacity by using random network coding (RNC) over a multihop wireless network [2]. This has promoted many studies on RNC based distributed optimization algorithms. For example, Lun et al. proposed to decompose the multicast optimization problem into two subproblems [3]: one is to search for a NC subgraph with minimum cost by modelling as a linear or convex program problem, which can be solved by a distributed primal-dual subgradient optimization algorithm; the other is to design a network coding scheme for the optimal subgraph obtained from the first subproblem; then a simple RNC solution can be employed for this subproblem. Similarly, Wu et al. proposed a distributed NC optimization algorithm over mobile ad hoc networks to minimize the energy for multicast services [4]. Lee and Vishwanath recently proposed a distributed algorithm for rate allocations to achieve the network capacity with minimizing the operation cost [5].

Another research direction is to introduce scheduling techniques in medium access control (MAC) layer into an optimization framework. The research in [3, 6] has shown that network performance can be improved significantly by optimizing a scheduling technique in MAC layer. Recently, using interference graph model, Jaramillo et al. have studied an optimization problem on resource scheduling when real-time and non-real-time services coexist in wireless multihop networks [7]. In addition, they have also studied the optimal rate allocation problem under heterogeneous delay constraints [8]. Although these studies cover the resource allocation and scheduling optimization problem for real-time services, they do not introduce NC into the optimization framework.

Apparently, we must integrate the two problems mentioned above together for achieving the best overall performance. By taking both NC and scheduling into account jointly, Rajawat and Giannakis proposed a joint optimization technique to improve the wireless multicast throughput performance under strict delay constraints [9]. Using hyperarcs to model the natural properties of wireless multicasts, Traskov et al. proposed a conflict graph model to identify effective network settings for studying joint NC and scheduling optimization algorithms [10]. In fact, this conflict graph based framework can be used to build an interference model for those active nodes in wireless networks. In order to avoid interference, we can select an efficient scheduling policy by sampling stable sets in a conflict graph. The studies have shown that, comparing with the scheduling technique with simple orthogonal models, the joint scheduling and NC graph optimization algorithm can improve the multicast throughput significantly [10].

Unfortunately, the joint optimization framework proposed in [10] cannot be used for wireless real-time multicast services without changes, because real-time multicast services usually need networks to support a predefined fixed and qualified data rate. By addressing this issue, Lun et al. [3] and Wu et al. [4] proposed a NC subgraph based optimization framework with minimum cost for supporting fixed multicast rates. However, they do not integrate scheduling techniques into their optimization frameworks. Moreover, since the conflict graph based scheduling optimization problem built in [10] is a NP-hard problem, they proposed a greedy algorithm with sampling the maximum weight stable sets. Nevertheless, there is a major drawback in this algorithm: it is very sensitive to the parameters of iterations such as the step size. In other words, the results often fall far short of the global optimum value in case that the parameters of iterations cannot be set properly. Since those parameters of iterations can only be chosen through trial and error methods, it is thus difficult to meet real-time requirements.

By addressing the problems mentioned above, to minimize the energy with satisfying the requirements of real-time multicast services in wireless multihop networks, we will construct a conflict graph based framework for designing joint NC subgraph and scheduling optimization algorithms. Here we would like to point out that this framework is mainly inspired from [10], but there are two important differences: it focuses on real-time services with a fixed and qualified data rate rather than non-real-time services; the optimization target is to minimize the energy of transmissions rather than to maximize the network capacity. Afterwards, using Lagrangian relaxation, we will propose a joint optimization algorithm by sampling the maximum stable sets randomly in a conflict graph, which includes two steps: first, it randomly samples a certain number of maximum stable sets of the conflict graph at each iteration; then, it solves a scheduling optimization problem by searching for those random sets. This method transforms a NP-hard scheduling subproblem into a plain linear programming problem so that it can be solved efficiently. More importantly, the accuracy can be adjusted not only through the parameters of iterations, but also by the number of random samplings. It thus can have good convergence and robustness.

The remainder of this paper is organized as follows. The network model with a directed hypergraph and its corresponding conflict graph is introduced in Section 2. A conflict graph based real-time multicast optimization framework is presented in Section 3. Using Lagrangian relaxation, a joint optimization algorithm is proposed in Section 4. The numerical simulation results are presented in Section 5. Finally, we conclude the paper in Section 6.

#### 2. Network Model

Considering a wireless multihop network, we use a directed hypergraph (where denotes the set of nodes and is a collection of hyperarcs) to represent the model. We define as a hyperarc and as a set of neighbors of node . When node sends data, all nodes in are assumed to be within the receiving range. For any hyperarc , we have and . For each node, there are at most hyperarcs.

The scheduling problem studied here is a scheduling for all of the hyperarcs defined above. When scheduling multiple hyperarcs for transmissions, we must avoid the interference caused by those conflict nodes. The specific hyperarc conflict situation depends on the network interference model. We consider the following two commonly used protocol interference models: the primary interference model and the secondary interference model. It is assumed that each node can only receive data from one node every time in the primary interference model, while in the secondary interference model, besides the above constraints, it is also assumed that any receiving node can only receive data correctly when all other neighbors are in a dormant state. They are strictly defined as follows.

*Interference Models*. For any two simultaneous scheduling hyperarcs and , the necessary and sufficient conditions for no conflict are as follows:(1) and , ;(2)there is for the primary interference model; there is and , where is an empty set for the secondary interference model.

Note that in both the primary interference model and the secondary interference model parameters are defined symmetrically. Therefore, we can use an undirected graph to express the conflict among hyperarcs. In this paper, we use the method presented in [10] to formulate an undirected graph based conflict graph. It is defined as follows.

*Conflict Graph *. Given a directed hypergraph defined above, we can formulate an undirected graph , which represents conflicts among all hyperarcs under an interference model. The vertex set in is the collection of all the hyperarcs in . Each edge in represents the conflict between two connected vertices according to an interference model. That is, the vertices in represent the hyperarcs in ; each edge in represents a conflict between the two connected vertices. The two corresponding hyperarcs then cannot be scheduled at the same time.

Clearly, based on the above definitions and a specific interference model, we can easily formulate the corresponding conflict graph according to any hypergraph . For a conflict graph , we now define any subset that there is no edge connected between any two nodes as a stable set . A stable set can be indicated by a column vector of length , which is defined as

A maximal stable set is one that is not contained in any other stable set. A maximum stable set is a stable set of largest cardinality. The stability numbers of are the cardinality of the maximum stable set. The stable set polytope (denoted by CH_{SS}) is the convex hull of the incidence vectors of all stable sets of . For example, assuming that there are stable sets in and is the incidence vector of the stable set , then we have

Finally, we take an example to illustrate the notations. A directed hypergraph with five nodes is shown in Figure 1 and its corresponding conflict graph in Figure 2.