A column generation method for spatial TDMA scheduling in ad hoc networks

Abstract

An ad hoc network can be set up by a number of units without the need of any permanent infrastructure. Two units establish a communication link if the channel quality is sufficiently high. As not all pairs of units can establish direct links, traffic between two units may have to be relayed through other units. This is known as the multi-hop functionality. In military command and control systems, ad hoc networks are also referred to as multi-hop radio networks.

Spatial TDMA (STDMA) is a scheme for access control in ad hoc networks. STDMA improves TDMA by allowing simultaneous transmission of multiple units. In this paper, we study the optimization problem of STDMA scheduling, where the objective is to find minimum-length schedules. Previous work for this problem has focused on heuristics, whose performance is difficult to analyze when optimal solutions are not known. We develop novel mathematical programming formulations for this problem, and present a column generation solution method. Our numerical experiments show that the method generates a very tight bound to the optimal schedule length, and thereby enables optimal or near-optimal solutions. The column generation method can be used to provide benchmarks when evaluating STDMA scheduling algorithms. In particular, we use the bound obtained in the column generation method to evaluate a simple greedy algorithm that is suitable for distributed implementations.

Introduction

An ad hoc network consists of a collection of radio units with a wireless interface forming temporary connections. No fixed infrastructure is involved in the communication. Instead, two radio units establish a communication link, if the signal-to-noise ratio is high enough. Two radio units far away from each other may communicate, if the units between them are participating in the ad hoc network, and are willing to forward packets for them (called the multi-hop functionality). This type of network is often referred to as multi-hop radio networks in military command and control systems, for which it is often not feasible to install any permanent communication infrastructure. In recent years, there is a growing interest in other applications of ad hoc networks, such as peer-to-peer computer communications, and communications between mobile sensors (e.g., traffic safety systems).

As pointed out in [11], ad hoc networks pose many design challenges. In this paper, we address the issue of resource allocation when designing link access schemes. One access scheme for ad hoc networks is Time Division Multiple Access (TDMA), in which the transmission resource of a radio frequency is divided into time slots, and each unit (or each link) receives a dedicated slot. Although simple to implement, TDMA is very inefficient from the resource utilization point of view. One promising approach to increase the network capacity is Spatial TDMA, or STDMA [19], which takes into account the fact that radio units are usually spread out geographically, and hence units that are spatially separated can use the same time slot for transmission.

In STDMA, the efficiency of the spatial reuse depends highly on the algorithm used for generating the transmission schedule. Scheduling algorithms become thus very important for implementing STDMA. A number of scheduling problems, with different levels of complexity, can be identified in this context. From an algorithmic perspective, the most challenging problem is to derive distributed algorithms that can handle mobile scenarios, and can perform STDMA scheduling based on the traffic distribution in the network. However, to design such algorithms, solutions to simple scenarios are very useful. The most simple version of the problem is to compute a schedule centrally for a static network, without taking the traffic distribution into account. In this case, the objective is to find a schedule that is as short as possible, i.e., a minimum-length schedule in which each unit (or each link) receives at least one time slot. In this paper, we will focus on this problem. Even if this is the most simple scenario of STDMA scheduling, finding the optimal solution to the problem is difficult. In particular, it can be shown that, from a computational complexity point of view, the problem is NP-hard. Previous work for this problem (e.g., [5], [8], [13], [16], [20]), has focused on heuristics. However, in the absence of optimal solutions, it is hard to judge the performance of heuristics, and to assess the true potential of STDMA.

In this paper, we present novel linear integer formulations for minimum-length STDMA scheduling. Moreover, we present a column generation method that effectively exploits the structure of the formulations. The main strength of the proposed methodology is its scalability. The method can be used to find optimal or near-optimal schedules for networks with arbitrary topology and realistic size. The basic idea behind the solution approach was briefly presented in [2]. Using the solutions provided by the column generation method, we are able to evaluate a simple heuristic that is suitable for distributed implementation.

The remainder of this paper is organized as follows. We present our network model in Section 2. In Section 3, two assignment strategies for STDMA scheduling are discussed. The optimization problems are formalized in Section 4, and the computational complexity is studied in Section 5. We present mathematical formulations in Section 6, and the column generation method in Section 7. In Section 8 we discuss two approaches to obtain feasible schedules. Numerical results are presented in Section 9. In Section 10, we draw some conclusions and outline future work.