# On Frequency Assignment in Cellular Networks

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1 On Frequency ssignment in ellular Networks Sanguthevar Rajasekaran Dept.ofISE,Univ. offlorida Gainesville, FL David Wei Dept. of S, Fordham University New York, NY K. Naik Dept. of S, Univ. of izu bstract In this paper we consider the problem of frequency assignment in cellular networks. The model we consider is general. We present an algorithm for frequency assignment and analyze its performance. 1 Introduction Frequency assignment is an important problem in the operation of mobile networks. The area available for the network is typically partitioned into hexagonal cells (see Figure 1for an example). base station is located at the center of each cell. This station is in-charge of handling calls arising from mobile hosts situated in its cell. The problem of frequency assignment is to allocate a frequency to each call for communication such that the interference among different calls is minimized and the total number of distinct frequencies used is minimized. ellular networks are usually represented as planar graphs where the nodes correspond to cell centers. There is an edge between two nodes if the corresponding hexagons have at least one common side. The graph G corresponding to the cells of Figure 1is shown with dotted lines. Mobile hosts in any cell communicate with the base station in the cell to serve their various needs. Each node of the graph G can be weighted with the number of hosts in the corresponding cell. Each host should be assigned a frequency for communication in such a way that interference with the other hosts in the network is minimized. The hosts in the same cell should be assigned different frequencies. lso, any two hosts that are at a distance of d or This work is supported in part by an NSF ward R and an EP Grant R

2 ... b g R r r b g b g r b g r b r b g r b g. Figure 1: onfiguration of cells less should be assigned different frequencies (where d is chosen appropriately to minimize the interference). fourth power attenuation law is typically assumed [1]. It is conceivable that a host can dynamically migrate from one cell to another. Thus the weighted graph G also changes dynamically. If one assumes that possible interferences are restricted to adjacent cells, then the frequency assignment problem reduces to weighted graph coloring. In this case, the graph G is refered to as the interference graph. The static frequency assignment problem refers to the problem of assigning frequencies to the hosts given a snapshot of the network at a specified time. The online (or dynamic) frequency assignment refers to the problem of continuous frequency assignment taking into account the changes that might occur in the graph G. In this case, the state of the network can be modeled with an ordered sequence of graphs G t,t 0. The problem corresponding to G t should be solved before considering G t+1. The static frequency assignment problem has been studied extensively. Even for the case when the interference is restricted to adjacent cells, it has been shown that the problem of optimal coloring of the interference graph G is NP-complete [6]. Several approximation algorithms have been devised for this version. For example, a simple algorithm called Fixed llocation (F) uses no more than three times the optimal number of frequencies (or colors). Janssen et. al. s algorithm [2] uses no more than 1.5 times the optimal number of colors. The number of colors used by Narayanan and Shende s algorithm [5] is no more than 4 3 times the optimal. Janssen et. al. [3] obtain the same performance for the dynamic version of the problem also. Several dynamic algorithms are known for the general version of the problem as well. Four of these algorithms are: the geometric strategy; orrowing with Directional hannel-locking (DL) [7]; Nanda-Goodman [4]; and the Two-Step Dynamic-Priority (TSDP) [1]. These algorithms have been demonstrated to do well in practice. However no bounds have been proven on their performances. 2

3 In this paper we consider a general version of the frequency assignment problem and present an algorithm that uses no more than three times the optimal number of colors. This is the first algorithm with a proven bound on the number of colors. We also illustrate how to use this algorithm to develop dynamic frequency assignment algorithms. The rest of the paper is organized as follows. In Section 2 we provide a summary of known techniques. In Section 3 we present a coloring algorithm and analyze its performance. In this section we also extend our coloring algorithm to obtain dynamic frequency assignment algorithms. Section 4 concludes the paper. 2 Literature Survey Static frequency assignment problem where interference is restricted to adjacent cells (call this version unit-distance frequency assignment problem) can be stated as follows. Input is a graph G(V,E,w) wherev is a set of nodes (cell centers), E is a set of edges (there is an edge from node u to node v if they are adjacent cells), and w is the weight function. In particular, w(u) stands for the number of hosts that the cell u has to handle. The problem is to multicolor the graph with as few colors as possible such that 1) for any u V, w(u) distinct colors are assigned to node u, and 2) for any edge (u, v) E, the colors assigned to the nodes u and v are disjoint. Let χ(g) be the minimum number of colors needed to color G. In the general frequency assignment problem, there cannot be a common frequency between two cells if they are separated by a distance of d (for some specified d). We can think of the space available for the mobile network as consisting of rows and columns of cells where the leftmost bottom cell is denoted as (0, 0) whose center is located at (0, 0). Let R be the radius of each cell (see Figure 1). The cell in row i and column j (denoted as (i, j)) ( has its center at 3Rj Ri, 3 2 ). Ri Thus in the case of unit-distance frequency assignment, the value of d is 3R. For the general case, d could take on any value. We can also define an interference graph corresponding to the general frequency assignment problem. This graph has the cell centers as its nodes. There is an edge from node u to node v if these nodes are at a distance of d. The performance of any coloring algorithm is measured using its competitive ratio which is the ratio of the number of colors used to χ(g). n algorithm is said to be c-competitive if its competitive ratio is c or better. 2.1 Unit-distance Frequency ssignment learly, the total weight on any maximal clique of the interference graph is a lower bound on χ(g). The only maximal cliques in the interference graph are edges and triangles. If D2 G and D3 G denote the maximum weight of any edge or triangle, respectively, then DG =max{d2 G,DG 3 } is a lower bound on χ(g). 3

4 simple coloring algorithm that is 3-competitive can be devised using the fact that the interference graph (where the weight of each node is 1) is 3-colorable. Figure 1 shows this coloring. The three colors used are red, blue, and green. Let the colors to be used be called 0, 1, 2,... To color new calls, the red, blue, and green nodes use the smallest integers available from the sets of integers that are mod 0, 1, and 2, respectively. For example, the sequence of colors used by any red node will be 0, 3, 6,... learly, there cannot be any interference between adjacent cells under this coloring. This algorithm is known as the Fixed llocation (F) algorithm. The competitive ratio of the above algorithm can be improved to 3 2 as has been shown by Janssen et. al. [2]. For any node v consider all the triangles in which v is a vertex. Let D 1 (v) denote the maximum total weight of any of these triangles. D 1 (v) is also known as 1-local maximum clique weight [3]. The colors are divided into three palettes. Red colors are integers that are mod 0. lue and green colors are integers that are mod 1and 2, respectively. Let v be any node. The node v employs three local spectra consisting of red, blue, and green colors each spectrum being of size D 1 (v)/2. If v s base color is red it uses the first w(v)/2 colors from its red spectrum and the last w(v)/2 colors from its blue spectrum. If v s color is blue it uses the first w(v)/2 colors from its blue spectrum and the last w(v)/2 colors from its green spectrum. Finally, if v s color is green it uses the first w(v)/2 colors from its green spectrum and the last w(v)/2 colors from its red spectrum. learly, the number of colors used by any node is no more than 3 2χ(G). It is easy to verify that no two adjacent cells will access the same color. Narayanan and Shende [5] have given an intricate algorithm for coloring that is 4 3 -competitive. ased on this algorithm, Janssen et. al. [3] have offered a dynamic frequency assignment algorithm that is 4 3-competitive. In this algorithm, at any given time, a node has to have a view of only a small neighborhood around it. 2.2 General Frequency ssignment Several works have been done that address the problem of dynamic general frequency assignment under both TDM and FDM multiplexing schemes. The problem of frequency assignment is also refered to as the carrier allocation problem (see e.g., [1]). The bandwidth available for communication is partitioned into carriers by frequency division. Each carrier is divided into certain number of channels by time division. channel is used to support a call. In this section we give a brief summary of four techniques for carrier allocation, namely, the geometric strategy, borrowing with directional channel-locking (DL), Nanda-Goodman strategy, and the two-step dynamic-priority (TSDP). Let D min be the minimum distance between two cells in order for them not to interfere with each other. If c is any cell, let IN(c) stand for the interference region of c, i.e., IN(c) is the set of cells that are at a distance of less than D min from c. The status of a carrier r in a cell c, denoted as status(r, c), is defined as 4

5 follows [1]. 1) We say r is a used carrier for c if at least one channel in r is used by some user in c. Inthiscasestatus(r, c) =U.2)Wesayr is an interfered carrier for c if there is a cell c IN(c) such that status(r, c )=U.Inthiscase,status(r, c) =I. 3)Otherwise,wesay r is available for c. Inthiscasestatus(r, c) =. Geometric Strategy. Here the cells are partitioned into k sets S 0,S 1,...,S k 1 as evenly as possible. These sets are such that no two cells in the same set are D min or less apart, i.e., no two cells interfere with each other. The set of available carriers is also divided into k sets P 0,P 1,...,P k 1 as evenly as possible. arriers in each P i are ordered. When a cell in S i needs a carrier, it chooses the first available carrier in the sequence P i,p i+1,...,p k 1,P 0,P 1,...,P i 1. Thus the cells in S i prefer the carriers in P i the most and hence P i are called the first-choice or nominal carriers for S i. orrowing with Directional hannel-locking (DL). This technique [7] is closely related to the geometric strategy. Here also the cells are partitioned into S 0,S 1,..., S k 1 and the carriers are partitioned into P 0,P 1,...,P k 1. arriers in P i are the nominal carriers for cells in S i. When a cell c in S i needs a carrier, it chooses the first available carrier in the sequence P i. (Note that P i is a list ordered according to priority, with the first carrier having the highest priority.) If no nominal carrier is available, the cell borrows the carrier of lowest priority from the richest cell in IN(c), i.e., the cell with the largest number of unassigned nominal carriers. Nanda-Goodman Technique. In this technique [4] each carrier is assigned an acquisition priority and a release priority. These priorities are assigned dynamically. For any cell c and a carrier r available for c, c s acquisition priority for r is defined as the number of cells in IN(c) in which r is an interfered carrier. In other words, acquisition priority(r, c) = {c IN(c) :status(r, c )=I}. Let r be used in c. Then the release priority of r is defined as the number of cells in IN(c) in which r will become available if r is released by c. Inotherwords, release priority(r, c) = {c IN(c) :ζ(r, c )=1} where ζ(r, c ) is the number of cells in IN(c ) that are currently using r. Two-Step Dynamic Priority (TSDP) Technique. This algorithm [1] is based on an optimal reuse pattern of the carriers. Let the reuse pattern S(r) of a carrier r be defined as S(r) ={c status(r, c) = U}. yoptimal reuse pattern of a carrier we mean the largest reuse pattern possible for that carrier. ny carrier that is used in a cell c canbereusedinanother cell c provided that these two cells are at least D min apart. Dong and Lai [1] show that an 5

6 Figure 2: Equivalence classes when D min =3 3R optimal way of using a carrier is to employ it in cells that are regularly placed in the plane such that the distance between adjacent cells in any column (or row) is exactly equal to D min. This notion can be formalized as follows. Define an equivalence relation Γ on the collection of cells. For any c 1,c 2, (c 1,c 2 ) Γ if and only if one of the following is true: 1) c 1 = c 2 ;2)dist(c 1,c 2 )=D min ;or3)thereexists a c such that (c 1,c) Γand(c, c 2 ) Γ. Dong and Lai [1] have shown that an optimal reuse pattern for any carrier is an equivalence class of Γ. When D min =3 3R, there arise nine equivalence classes as shown in Figure 2. The TSDP algorithm partitions the cells into optimal reuse patterns G 0,G 1,...,G k 1 using the equivalence relation Γ. onsider only the case D min =3 3R. For any cell c G i, define LOP (c) ={c : dist(c, c )=D min }. When D min =3 3R, there are six cells in LOP (c). The algorithm also has an adjustable parameter λ which is an integer in the range [0, 7]. ny carrier r is called a primary carrier for cell c if r is currently used by at least λ cells in LOP (c). Otherwise, r is said to be a secondary carrier for c. y appropriately choosing λ, one can make sure that the carriers are used according to maximal reuse patterns as much as possible. The algorithm description will be complete if we specify the acquisition and release priorities for the various carriers. For any cell c and a primary carrier r for c, the acquisition priority is defined as follows. acquisition priority(r, c) = weight(r, c ) c LOP (c) where weight(r, c )isn if status(r, c )=U;1ifstatus(r, c )=; and0ifstatus(r, c )= I. HereN is a positive number greater than the number of cells in IN(c). lso, release priority(r, c) = acquisition priority(r, c) 6

7 Dong and Lai [1] show that this priority scheme results in the primary carriers being used in close approximation to their optimal reuse patterns. For the secondary carriers they employ the priority schemes of Nanda-Goodman in order to use these carriers in a compact manner. If r is a secondary carrier for c, then acquisition priority(r, c) = {c IN(c) :status(r, c )=I}. lso, release priority(r, c) = {c IN(c) :ζ(c,r)=1} where ζ(r, c ) is the number of cells in IN(c ) in which the carrier r is currently used. The experimental results of [1] indicate that the TSDP algorithm performs better than the other three techniques namely, geometric, DL, and Nanda-Goodman. In [1] they also present some variants of the algorithm TSDP. 3 The New lgorithm ll the carrier allocation algorithms we have seen thus far have the following common theme: Partition the cells as well as carriers into groups. For each group of cells and each groups of carriers assign acquisition as well as release priorities. The TSDP algorithm performs well because of the way the cells are partitioned and also because of the priority schemes. ell partition corresponds to optimal reuse patterns of the carriers and the priority scheme for primary carriers is such that they are used according to their optimal reuse patterns as much as possible. onsider the case D min = k 3R, for some positive integer k. For this case, there will be k 2 equivalence classes in the relation Γ. all the intersection of k successive columns and k successive rows of cells a k k-rectangle. How many colors are needed to multicolor the interference graph? The optimal reuse pattern suggests that a different palette be used for each cell in a k k-rectangle. lower bound on the number of colors needed is the maximum of total weight on any circle of diameter (k 1) 3R. Figure 3 shows the case when k =3. The total weight of all the cells whose centers lie within a circle of diameter 2 3R is a lower bound. Thus the TSDP algorithm uses no more than k 2 times the optimal number of colors. Example 1. onsider the following instance. Similar to the cells, name the rectangles also as (0, 0), (0, 1), etc. Rectangle (0, 0) has q requests in its cell (0, 0). The rest of the cells in the rectangle do not have any calls. Rectangle (0, 1) has q calls in its cell (0, 1)andtherest of the cells do not have any calls, and so on. lso assume that this pattern persists for a long time. For this instance, the TSDP algorithm will use k 2 colors. Thus the TSDP algorithm is k 2 -competitive. 7

8 E E h t D Figure 3: n efficient coloring of the graph Our algorithm is based on the fact that the interference graph can be colored with no more than 3 colors. We begin with a 5-competitive scheme. Figure 3 shows this coloring for the case k = 3. ut the result holds for any value of k. Lemma 3.1 There exists a coloring scheme that is 5-competitive. Proof. The idea is to partition the cells as illustrated in Figure 3. In particular we draw circles of diameter (k 1) 3R. ll the cells whose centers lie on or inside a circle constitute a class. We label these classes as follows. The classes that are in the same rows will be labeled,,,... The classes immediately below will be labeled,d,,... There are cells that are not covered by these circles. These are grouped such that all the cells in between two successive circles belong to the same class. Label these classes E. We use five palettes. Four palettes are used to color the classes labeled,,, and D. The fifth palette is used to color the class E. Let each of the five spectra be of size. learly, there won t be any conflicts among the classes,,, and D. Two classes that are labeled E might have some conflicts, since there are nodes from two adjacent classes labeled E that are at a distance of <D min. In order to avoid these conflicts, for any class labeled E, the cells in the top half (E h ) will use the spectrum from one end and the cells in the bottom half (E t ) will use the spectrum from the other end. This scheme reverses for the E classes that are in rows below. 8

9 It is easy to verify now that there won t be any conflicts among the E classes also. Our algorithm will partition the cells according to the coloring imposed by Lemma 3.1. There are many possibilities. If we know in advance the value of, then we can use five spectra each of size and there won t be any interferences or failures. nother possibility is that the number of colors (i.e., carriers) available may be fixed. In this case we partition the available carriers into five: P 0,P 1,P 2,P 3, and P 4. all the cell partitions S 0,S 1,S 2,S 3, and S 4 (corresponding to,,, D, and E, respectively). We say P i are the primary carriers for S i, 0 i 4. Depending on the priority schemes to be used, we can derive many dynamic carrier allocation strategies. We list below some of them. lgorithm1 The priority scheme is similar to the geometric strategy. If a cell c S i requires a channel it picks the first available channel from the list P i,p i+1,...,p 4,P 0,..., P i 1,for 0 i 4. lgorithm2 Here the priority scheme corresponds to that of DL. lgorithm3 For each group S i, primary carriers are P i,0 i 4. For the secondary carriers, use the priority scheme of Nanda-Goodman. Useful Heuristic for lgorithm1. lgorithm1is based on the geometric strategy. Here we mention a heuristic that has the potential of improving the performance. ccording to the geometric strategy, a cell m labelled will look for a carrier in the ordered list,,. If there is no available carrier in, it will look for one from the palettes and. Say a carrier q is available in. Ifq is assigned to m, q cannot then be assigned to any cell in the interference region of m. This will affect the reusability of the carrier. Instead we can do the following. When the cell m does not find a carrier in, it first checks the carriers in and that are currently used by cells that are outside m s interference region. This way we can make sure that we are not using any new carriers from either or. Lemma 3.2 There exists a coloring scheme that is 3-competitive. Proof. The cells are partitioned into three classes as shown in Figure 4. The labeling of these classes correspond to the palettes that will be used to color them. It is easy to see that there won t be any conflicts among the three classes. 4 onclusions In this paper we have provided a summary of known algorithms for the frequency assignment problem. We have considered both unit-distance and general versions. coloring algorithm that is three-competitive for the general interference graph has been given. ased on this coloring scheme we have developed many dynamic carrier allocation algorithms. 9

10 References Figure 4: 3-competitive coloring of the graph [1] X. Dong and T. H. Lai, Dynamic arrier llocation Strategies for Mobile ellular Networks, Technical Report OSU-ISR-10/96-TR48, Dept. of omputer and Information Science, The Ohio State University, [2] J. Janssen, K. Kilakos, and O. Marcotte, Fixed Preference Frequency llocation for ellular Telephone Systems, unpublished manuscript, pril [3] J. Janssen, D. Krizanc, L. Narayanan, and S. Shende, Distributed Online Frequency ssignment in ellular Networks, Manuscript, [4] S. Nanda and D. Goodman, Dynamic Resource cquisition: Distributed arrier llocation for TDM ellular Systems, IEEE Globecom, 1991, pp [5] L. Narayanan and S. Shende, Static Frequency ssignment in ellular Networks, Technical Report, Dept. of omputer Science, oncordia University, [6]. McDiarmid and. Reed, hannel ssignment and Weighted olouring, manuscript,

11 [7] M. Zhang and T.-S. Yum, omparisons of hannel-ssignment Strategies in ellular Mobile Telephone Systems, IEEE Transactions on Vehicular Technology 38(4), 1989, pp

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