# Frequency Assignment in Mobile Phone Systems

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1 Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D Berlin-Dahlem Germany MARTIN GRÖTSCHEL Frequency Assignment in Mobile Phone Systems ZIB-Report (Dezember 2000)

5 50% of their area with low noise. What if interference is high but almost no traffic is expected? How can a single interference value reflect such a difference in the interference behaviour? There is no clear answer. The planners have to investigate such cases in detail and have to come up with a reasonable compromise. The result, in general, is a number, the interference value, which is usually normalized to be between 0 and 1. This number should to the best of the knowledge of the planners characterize the interference between two TRXs (in terms of the model, the technological assumptions, etc., used by the operator). Separation and Blocked Channels There are also hard constraints. If two or more TRXs are installed at the same location (or site), there are restrictions on how close their channels may be. For instance, if a TRX operates on channel, a TRX at the same site is not allowed to operate on channels. Such a restriction is called co-site separation. Separation requirements may even be tighter if two TRXs are not only co-site, but also serve the same cell. Separation requirements may apply also to TRXs that are in close proximity. The situation is even more complex. Due to government regulations, agreements with operators in neighbouring regions, requirements from military forces, etc., an operator may not be allowed to use its whole spectrum of channels at every location. This means that, for each TRX, there may be a set of so called blocked channels. Interference Graph A feasible assignment of channels to TRXs clearly has to satisfy all separation constraints. Blocked channels must not be used. What should one do about interference? On our way to an adequate mathematical representation of all technical constraints let us first introduce the interference graph. has a node for every TRX, two nodes are joined by an edge, if interference occurs when the associated TRXs operate on the same channel or on adjacent channels or if a separation constraint applies to the two TRXs. With each edge, two interference values, denoted by and, are associated; the number is the co-channel interference that occurs when TRXs and operate on the same channel while denotes the interference value coming up when and operate on adjacent channels. In general,. If a separation constraint applies to and then a suitable large number is allocated to and. Two Natural Approaches A first attempt to solve the frequency assignment problem is obvious. We try to find a spectrum, i.e., a number of channels such that the available channels can be assigned to TRXs so that no interference occurs. 4

6 No interference is, of course, a good aim, but this task is unrealistic for several reasons. A mobile phone network is a living system, i.e., new BTSs, antennas, etc., are installed regularly, old ones are replaced by new ones with different characteristics. It is impossible to change the spectrum each time the network changes. Moreover, the number of channels may be fixed or channels may only be available in bundles (i.e., one may buy 75, 100 or 125, but nothing else). Frequencies are expensive and cost reasons may require that some interference is tolerated. In fact, some interference may be unavoidable. We have data of mobile phone systems where the largest clique in the interference graph is about twice as large as the number of available channels and where the largest degree of a node is ten times as large as the number of available channels. Another classic choice for the solution of the frequency assignment problem is the following. We choose a threshold value and consider the graph where. Now we try to find the coloring number of, or try to color with colors. In other words, we consider interference below tolerable and try to find an assignment of channels to TRXs such that as few channels as possible are used (a color represents a channel, no two nodes with the same color are allowed to be adjacent) or we try to use the available channels so that no high interference occurs. Of course, if for a given threshold no feasible coloring can be found, one has to modify and try again. This approach is unable to handle separation constraints and ignores adjacent channel interference. It was the standard approach in the early days of the mobile phone era but did not prove efficient in the more complex environment we have today. Minimizing Interference There are several other ways of modelling the frequency assignment problem mathematically, see [5]. For reasons of brevity I will focus on an approach that was employed in a joint project of the Konrad-Zuse-Zentrum and E-PLUS, one of the four operators in Germany, and which has resulted in very satisfactory channel assignments. Let be the interference graph introduced before. Let be the set of available channels, and let, for each TRX, denote the subset of channels blocked at node. The values, denote, for each edge, the co-channel and adjacent-channel interference arising when TRXs and operate on the same or on adjacent channels. Moreover, let denote the separation necessary between the channels assigned to TRXs and. Thus, the input to a frequency assignment problem is a -tuple!!, briefly called network here. A frequency assignment for the network is a function. It is called feasible if it satisfies the following side constraints "# for all % for all The objective is to minimize the sum of co- and adjacent-channel interference, more 5

7 formally: feasible (FAP) This version of the frequency assignment problem is a generalization of list colorings in graph theory and it is related to the well known T-coloring problem. There are several ways to reformulate (FAP) in terms of other standard models of combinatorial optimization, e.g., there is a stable set model and a so called orientation model which is related to linear ordering. Several modifications of FAP have to be considered in practice. No operator wants to change all channel assignments whenever a new plan has to be computed. Some assignments have to stay fix (that is easy to achieve), sometimes one looks for the smallest number of channel adjustments within a certain range of interference, or one requires that, e.g., at most 100 of the assignments are changed. FAP is difficult in terms of complexity theory. Deciding whether a TRX network allows a feasible assignment is -complete; the optimization problem is strongly -hard. FAP is also difficult in practice. Nobody can solve realistic instances to optimality. Satisfactory lower bounds on the objective function value are very hard to obtain. All approaches based on polyhedral combinatorics and linear programming have failed so far. There is some hope to exploit semidefinite relaxations of FAP. The whole available zoo of heuristics has been tried for the solution of FAPs. Considerable improvements over previous approaches can be achieved. There are some spectacular successes, but at present, the gap between lower and upper bounds computable in practice is still very large. In my talk on this subject I will elaborate on the mathematical modelling of the FAP, on the development of heuristics and on the approaches with which lower bounds have been computed. I will present examples from practice that show what can be achieved today and how this mathematical approach compares to more traditional planning techniques. This lecture is based on joint work of the telecommunications group at the Konrad- Zuse-Zentrum, particular on the work of Andreas Eisenblätter [3]. Further references are [1], [2], [4]. The FAP website [5] is another excellent source of information. References [1] R. Borndörfer, A. Eisenblätter, M. Grötschel, and A. Martin, Frequency Assignment in Cellular Phone Networks, Annals of Operations Research, 76:73-93 (1998). [2] R. Borndörfer, A. Eisenblätter, M. Grötschel, and A. Martin, The orientation model for Frequency Assignment Problems, Technical Report TR , Konrad-Zuse-Zentrum für Informationstechnik Berlin, (1998a). 6

8 [3] A. Eisenblätter, Frequency Assignment in GSM Networks: Models, Heuristics, and Lower Bounds, Ph.D. Thesis, TU Berlin 2000, to appear. [4] A.M.C.A. Koster, Frequency Assignment Models and Algorithms, Ph.D. Thesis, Universiteit Maastricht, Maastricht, The Netherlands (1999). [5] FAP Web A Website about Frequency Assignment Problems, veröffentlicht in: Lecture Notes in Computer Science 1974, Kapur, Prasat 7

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