Hierarchical cellular network design with channel allocation

https://doi.org/10.1016/j.ejor.2003.06.017Get rights and content

Abstract

The design of a cellular network is a complex process that encompasses the selection and configuration of cell sites and the supporting network infrastructure. This investigation presents a net revenue maximizing model that can assist network designers in the design and configuration of a cellular system. The integer programming model takes as given a set of candidate cell locations with corresponding costs, the amount of available bandwidth, the maximum demand for service in each geographical area, and the revenue potential in each customer area. Based on these data, the model determines the size and location of cells, and the specific channels to be allocated to each cell. To solve problem instances, a maximal clique cut procedure is developed in order to efficiently generate tight upper bounds. A lower bound is constructed by solving the discrete optimization model with some of the discrete variables fixed. Computational experiments on 72 problem instances demonstrate the computational viability of our new procedure.

Introduction

Some 20 years after their commercial introduction, cellular mobile communication services are as popular as ever, with demand increasing at an exponential rate and network expansion following suit. Although not comparable to the wire-based communication infrastructure, investments in cellular systems are large and an economical use of resources is a necessity for companies operating in this competitive market.

All wireless systems are constrained in capacity by the available communication bandwidth. As demand for services has expanded in the cellular segment, several innovations have been made in order to increase the utilization of bandwidth. The cellular concept itself is one of the most prominent examples of such innovation. The cellular design increases bandwidth utilization by limiting the reach of a radio tower so that a frequency channel can be re-used by a tower that is located sufficiently far away from other towers utilizing the same channel. Given a service area and potential demand in this service area, an important decision is to determine the size of each cell in the service area.

Another proposed improvement is dynamic channel assignment. In traditional cellular systems designs, the allocation of frequency channels to cells is fixed. That is, once a frequency channel has been allocated to a cell, it cannot be used by other cells that may interfere with this. Realizing that most of the time, all channels will not be in use in all cells, schemes have been designed so that channels may be `borrowed' from other cells, given that the borrowing of the channel will not cause interference in other cells. While the advantages of channel borrowing (and other dynamic cell assignment schemes) are obvious, they also have some drawbacks. In order to borrow a channel from another cell, the cell that borrows the channel must have sufficient capacity to handle the extra channel. This has implications for tower equipment allocation, mobile telephone switching office capacity, cellular network backbone design, and traffic management.

A third proposed method to improve resource utilization is the concept of hierarchical (or multi-tier) network design. In a hierarchical system, various cell sizes may be deployed. In the initial proposal, there were two types of cells; macrocells and microcells. The microcells are similar to regular cells in traditional designs. Superimposed on this structure are the macrocells, each of which covers the same geographical area as several microcells. There are two principal benefits of a two-tier design. First, if a macrocell covers an area, it is not necessary to cover the same area with a microcell. In rural areas with low overall demand and scattered high demand points, this means that fewer resources need to be spent on providing adequate service. Instead of covering the entire service area with microcells, a macrocell can be used in combination with microcells at high demand points. The cost savings from this design can be substantial compared to a traditional design. Second, if there are mobile phone users that are traveling at high speeds, these users can be serviced by the macrocell and thereby reduce the need for handoff of the calls. When a handoff is avoided, the risk of call dropping due to channel shortage in the handoff service area is eliminated. The two-tier model can be extended into a three-tier model where a smaller cell size (picocell) is introduced. Picocells can be used in very high density areas, such as urban commercial centers. Thus, a multi-tiered design combines high frequency reuse in high demand areas with low system investment in low demand areas. In this investigation, a new model is proposed that incorporates multi-tiered design with channel allocation based on demand for service in different geographical locations in the service area.

Hierarchical cellular networks have been the topic of several studies, with focus in particular on accommodation of users that travel at different velocities. Characteristics of hierarchical cellular networks, such as call blocking and handoff failure probabilities have been investigated extensively by Hu and Rappaport [11] and Rappaport and Hu [18] for entirely ground based networks as well as for hybrid networks that combine services from a ground based network with those of satellite based overlay networks. Extensions of this work have been concerned with resource management in hierarchical networks, with an emphasis on channel allocation between network levels (both static and dynamic allocation schemes), cell size determination, and user mobility management [6], [14], [24].

Several hierarchical cellular network design models have been proposed in the literature. Ganz et al. [8] propose a model that minimizes total system deployment cost subject to quality of service constraints for customers assigned to each tier. The quality of service measure used is the probability of call loss. While this formulation takes into account user mobility, it assumes that demand is uniformly distributed across the network service area. Moreover, the formulation is based on the assumption that all demand has to be serviced at some given quality of service level. In a more recent paper, Wu and Lin [25] develop a model that minimizes the cost of system development subject to quality-of-service (QoS) constraints. In this model, the authors use cell radius as an independent variable and find a solution that satisfies demand subject to QoS constraints. In their work, QoS measures include signal-to-noise ratios for each cell location and a requirement that the system meets average demand for service. Neither of the papers consider economic factors other than cost.

Sarnecki et al. [21] compare the characteristics of macrocells and microcells and conclude that the cost per subscriber of using microcells may be as much as 60–70% lower than the cost of using macrocells. However, this cost reduction is due to the use of low-power transmitters in microcells and the figures do not include the possible incremental cost of connecting the cell sites to switching offices. Using an engineering cost approach, Reed [19] estimates the costs of PCS networks and studies the economies of scope in PCS service provision. He finds that there is some minimum bandwidth that allows PCS operators to reach a point where all economies of scope are exhausted and concludes that this finding may have important regulatory implications. Finally, Gavish and Sridhar [9] use a cost and revenue model to find optimal configurations (in terms of cell size and channel allocation) of cellular systems. Their model assumes that demand is uniformly distributed and that each cell in the system will be configured exactly the same. Due to the symmetry arguments made, the channels available to the system can be divided a priori in equal numbers between cells without causing interference. The remaining channel allocation problem is to decide how many of the channels available to a cell should actually be used. Thus, interference considerations do not play a role in their model.

The problem considered in this work includes as part the assignment of frequency channels to cells in the network. The frequency assignment problem is closely related to the graph coloring problem and has been subject to intensive study in the literature (see [17] for a comprehensive survey). The frequency assignment problem has been examined for a number of objective functions, mostly related to minimizing the number of channels used to satisfy demand or to minimizing the total system interference while satisfying demand for communication. In this context, methods for deriving strong lower bounds on the number of frequency channels utilized have been developed (see, e.g., [7], [23]). In common for most of the work on finding feasible solutions to the frequency assignment problem is that the solution methodology relies on specific geometric cell structures [3] or on the use of randomized local search procedures without known deviation from the optimal solution [2], [4], [10], [12], [15], [20], while integer programming based approaches have successfully solved only small problem instances [16].

The model presented in the next section differs from previous research in that it simultaneously incorporates base station location, cell size choice, and channel allocation in a multi-tier cellular network. Both cost and revenues are considered in the design of the cellular network. The model does not, however, directly address queueing or mobility aspects of cellular network operations. It is assumed that the service demand parameters are adjusted for mobility and queueing effects and, thus, the proposed model is suited for planning rather than operational purposes.

We claim four contributions from the investigation described in this manuscript. First, we present a new optimization model for the hierarchical network design problem. This is the first three-tier model that we have seen and it includes macrocell, microcell and picocell selection. It is a profit maximization model, as opposed to the usual cost minimization strategy, which better reflects the requirements of potential clients. Channels are assigned to cells in blocks, which reduces the problem size and improves computational tractability.

Our basic model is a large mixed-integer linear program whose problem instances are not solvable using standard commercial optimization software. However, there exists an underlying graph (called the interference graph) that can be used to create an almost unlimited number of valid inequalities for any large problem instance. While this class of valid inequalities is well known for the frequency assignment problem [1], our second contribution is a set of algorithms to successively and selectively generate valid inequalities from the interference graph by repeatedly solving maximal clique problems to produce a problem with a much stronger continuous relaxation. Our third contribution is a computational study that provides convincing evidence that good solutions to realistically sized problems can be obtained with our procedures. The feasible solutions obtained, combined with upper bounds on the optimal solution value generated, provide a solution quality measure in terms of a known maximal deviation from the optimal solution value. Finally, we make our AMPL models and algorithms available on the World Wide Web for downloading free of charge for immediate use by industrial design groups who need solutions to this type of problem. Users only need to supply their specific data to have a working design tool. Other research groups also have complete access to our models, algorithms, and test data for independent verification and comparison with their design tools.

Section snippets

The model

In this section, we present a model for the hierarchical cellular network design problem. The model proposed in this investigation is applicable for cell planning for analog cellular systems (in particular, AMPS using FDMA), digital cellular (standard IS-54 using TDMA), and GSM. The minimum service requirements in the model conform to the guidelines for markets in the United States established by the Federal Communications Commission (FCC). In the remainder of this section, we present the

Solution procedure

The linear-programming (LP) relaxation of the ILP model presented in Section 2.4 is weak; i.e., the gap between the LP upper bound and the ILP optimum can be relatively large. One difficulty is that the LP relaxation allows for channels to be shared among tower-cell combinations that are close enough to interfere with each other. Our experience with this model indicates that for even a toy problem, an ILP solver may have to branch an enormous number of times in order to resolve these conflicts.

Empirical analysis

The upper and lower bound algorithms have been implemented using A Mathematical Programming Language (AMPL) and CPLEX (http://www.ampl.com and http://www.ilog.com/products.cplex). Each iteration of the upper bound procedure requires the solution of three problems, UB, C1, and C2. Problem UB is , , , , , , , , , , , , , , , , , , , , , problem C1 is , , , , and C2 is , , , (28). Exact solutions are obtained for C1 and C2 and an optimality gap of 1% is used for UB. That is, CPLEX terminates as

Conclusion

This manuscript presents a formulation and solution procedure for the hierarchical network design problem with channel allocation. The proposed model includes the cost of system investments and operation, as well as revenues generated by the system based on customer demand for service. In contrast to previous work in this area, the model specifically includes heterogeneous demand for service. Simultaneously, the model also solves the channel allocation problem taking into consideration

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    This research was supported in part by the Office of Naval Research Award Number N00014-96-1-0315.

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