A Viterbi-like Algorithm with Adaptive Clustering for Channel Assignment in Cellular Radio Networks

—A new channel assignment algorithm, called the Viterbi-like algorithm (VLA), is proposed to solve the channel assignment problem in cellular radio networks. The basic idea of the proposed algorithm is step-by-step (sequential) channel assignment with the objectives of minimum bandwidth required at every step, subject to adjacent channel and cochannel separation constraints. The VLA provides the benefits of minimum required bandwidth, stability of solution, and fast execution time. The performance of the VLA is evaluated by computer simulation, applied first to 19 benchmark problems on channel assignment and then applied to study cellular radio network performance. Results from computer simulation studies show that bandwidth requirements by VLA closely match or are sometimes better than those of the existing channel assignment algorithms. Furthermore, it is found that execution of VLA is approximately two times faster than the local search algorithm—the existing channel assignment algorithm with the least bandwidth requirements. The combined advantages of minimum required bandwidth, stability of solution, and fast execution time make the VLA a useful candidate for cellular radio network planning.


I. INTRODUCTION
T HE need for efficient channel assignment algorithms in wireless networks is becoming increasingly important as the wireless subscriber base continues to grow rapidly while the radio-frequency spectrum remains a scarce resource.This paper presents a new channel assignment algorithm for solving the channel assignment problem (CAP) in cellular radio networks.Historically, the CAP has been formulated as a nonlinear optimization problem whose solution continues to be of active research interest in the literature.Examples of previously proposed solution techniques include the genetic algorithm [1], neural networks [2]- [4], simulated annealing [5], [6], local search approach [7], cochannel interference minimization [8], [9], heuristic approaches [10]- [12], and theoretical lower bounds [13], [14].The genetic algorithm proposed in [1] is a blind search procedure, where the initial population is semirandomly generated and the algorithm selects a new population with better fitness.
Several approaches based on neural networks have been proposed [2]- [4].For example, [4] proposes a modified Hopfield neural network, where the CAP is formulated as an energy-minimization problem.Solution of CAP using the simulated annealing approach starts with the selection of an arbitrary solution from which a minimum is then searched [5].During the optimization, a so-called Metropolis criterion is used to prevent the solution from converging to local minima.A local search approach (LSA), which is a special case of the simulated annealing approach, is proposed in [7].The LSA searches the neighborhood of the current solution point in the feasible region for a better solution.At the next solution point, if a solution is found such that the functional , then becomes the new solution point.The LSA, when applied to a set of benchmarking problems, offers the best results so far in the literature [7].Considering a pattern of hexagonal shaped cells, [8] and [9] present an approach that is based on selection of sets of channels such that cochannel interference is minimized by appropriately grouping the cells into compact patterns or clusters.Finally, heuristic-based solutions are presented in [10], [11], and [12].For example, in [10], the cells are first sorted in alphabetical order and the channels are sequentially assigned until a denial occurs.Subsequently, the cells are resorted in the order of increasing difficulty, and a reassignment is made until no further denial occurs.An excellent description of systematic sorting is given in [12], where most of the results of channel assignment problems considered are very close to the theoretical bounds derived in [13].
The major focus of this paper is development of a heuristic algorithm that achieves the objectives of 1) minimum bandwidth requirement that is comparable to or better than the best performance reported in the literature; 2) faster execution time compared to those of previous algorithms.
The channel assignment algorithm proposed in this paper is called the Viterbi-like algorithm (VLA) because of its similarity to the sequential trellis search algorithm of the original Viterbi algorithm used in the field of information decoding [15], [16].However, the VLA minimizes bandwidth instead of the Euclidean distance minimization performed in the original Viterbi algorithm.Minimum bandwidth is achieved by VLA through the removal of redundant channel assignments [a channel assignment is said to be redundant if its excess frequency factor (EFF) is negative] from further consideration and keeping only the survivors after each step of channel assignment.A benefit of retaining only the survivors is a reduction in the number of iterations performed in the subsequent steps of channel assignment; this translates to fast execution time for the proposed VLA algorithm.
Another reason for the minimum bandwidth achieved by VLA is the use of dynamic clustering.The dynamic clustering approach adaptively groups the cells for frequency reuse according to the instantaneous traffic demand while meeting acceptable cochannel and adjacent channel interference limits.In essence, dynamic clustering does the efficient cluster organization while the VLA manages the frequency planning within each cluster.
The remainder of this paper is organized as follows.Section II defines the channel assignment problem and also specifies the assumptions made along with justifications.In Section III, the concept of adaptive clustering of cells is introduced, taking into account the maximum allowable interference limits.A step-by-step description of the proposed Viterbi-like algorithm for solving channel assignment problem forms the main topic of Section IV.Other topics covered in Section IV include performance of VLA under insufficient available channels compared to traffic demand, as well as stability and limitations of the VLA.In Section V, we first assess the quality of the proposed VLA by using it to solve 19 benchmark channel assignment problems and then compare the results with those of the existing channel assignment algorithms.Next, we use the VLA to solve the channel assignment problem in a cellular radio network with emphasis on studying the effects of traffic demand and adjacent cell and co-cell separation on required bandwidth.Section VI concludes this paper.

A. Problem Statement
The channel assignment problem in a cellular radio network is formulated as follows [7].
The cellular network comprises distinct geographic cells.These cells are grouped into small clusters, each of size cells, where .The offered traffic to a cluster is characterized by a demand vector , where is the number of traffic channels required in cell .The total number of traffic channels required in all the cells of a cluster (i.e., total demand) is denoted by , where .The channel separation constraint is modeled by an symmetric compatibility matrix , where denotes the minimum required channel separation between a channel assigned to a call in cell and a channel assigned to another call in cell .
When is called the co-cell separation (CCS), and when is called the adjacent cell separation (ACS) instead.Let denote the th call in the th cell where and .Assume that channels are available, where is initially made equal to to ensure that a channel can be assigned to each call in the cluster.Denote the channels by , where , is the th channel in the list of channels.After channel is assigned to call , the suffix is changed to (i.e., is changed to ) for consistency with the suffixes of call .A set of channel separation constraints is defined by the compatibility matrix , such that for all .Given a cellular radio network as described above, the CAP is to find a conflict-free assignment of channels to calls such that the total bandwidth required is a minimum.The CAP is well known to be a computational challenging nonlinear optimization problem.In this paper, we solve the CAP using the Viterbi-like algorithm.

B. Modeling Assumptions
The following assumptions are made in the analysis.Assumption 1) Sufficient channels are available to meet the total traffic demand.Assumption 2) The available channels are pooled and dynamically assigned to service the traffic demand.Assumption 3) All the channels have equal interference characteristics.Assumption 4) The total demand in a cluster is transmitted instantaneously to a call control server where the channel allocation is done.The implication/rationale behind Assumptions 1) to 4) is stated as follows.Assumption 1) is a starting assumption made to simplify the development of the Viterbi-like algorithm.Assumption 1) is relaxed in a later section to cover the case where the number of available channels is less than the total demand.Pooling of the available channels [Assumption 2)] is necessary for simplicity of channel management and flexibility to assign a channel to any cell in a cluster.The implication of Assumption 3) is that channel separation is independent of channel or channel .Assignment of the channels at a central location [Assumption 4)] provides a system-level view of used and available channels.In practice, the central call control server can be implemented at either the base station controller (BSC) or the mobile services switching center (MSC), depending on the network architecture.

III. CELL ORGANIZATION AND DYNAMIC CLUSTERING
The first step in the channel allocation procedure is to suitably arrange the cells in a cellular radio network into clusters (cell repeat patterns), each of size cells.For optimum frequency reuse with minimum interference, both the cluster size and the cell arrangement within a cluster are important.Conventionally, clusters have fixed size due to hardware and infrastructure constraints.Small cluster size results in high capacity but with a penalty of high interference.Conversely, if the cluster size is too large, there is an advantage of low interference but at the cost of low capacity.Also noting that, in a cellular radio network, traffic demand exhibits both spatial and temporal variability, it may not be desirable to use a fixed cluster size throughout the network.Hence, in this paper, we assume that cluster size is not fixed but is dynamically selected according to traffic demand.The preceding statement is referred to as dynamic clustering.By dynamically clustering the cells according to traffic demand, overall radio network capacity is improved.
As a preamble to dynamic clustering, we introduce two distance parameters: cochannel reuse distance and adjacent channel reuse distance.Cochannel reuse distance is defined as the geographic distance between the centers of any two cells using the same channel sets.Adjacent channel reuse distance is defined as the geographic distance between any two cells using adjacent channel sets.Fig. 1 illustrates the distance parameters and .Let the distances and be normalized with respect to cell radius so that these become independent of the actual cell size.Using the properties of hexagonal cell shape geometry listed in [9], cell dynamic clustering can be automated.That is, for a given demand vector and acceptable and , the cluster size can be automatically determined.Once the value of is decided, the next important step is to organize the cells within each cluster to achieve optimum values of and (i.e., optimum in the sense of meeting the specified objectives for and ).For a fixed value of , it turns out that different values of and are possible depending on cell arrangement within a cluster.The cell arrangement that gives optimum values of and is selected.The foregoing is best demonstrated by an example.

A. Example Problem 1
Consider a cellular network whose coverage area comprises 49 hexagonal-shaped cells (i.e., ).The constraints for and are given as and , respectively.Denote the cluster size by .Due to spatial traffic variability, a distribution is assumed for traffic demand in each cell of a cluster.In this problem, it is assumed that the traffic demand vector is generated from a uniform distribution ll ul where "ll" and "ul" are the lower and upper traffic demand limits, respectively.The assumption of uniform traffic distribution is based on the initial assumption that subscribers are uniformly distributed over the service area.For purpose of illustration, ll and ul are set to 10 and 15, respectively, in this example problem.The traffic demand of all the cells in a cluster can be arranged in a vector of length as shown in (1) where is the traffic demand (i.e., number of traffic channels required) in cell .The problem is to determine the minimum cluster size and the optimum cell arrangement within each cluster so that the specified and constraints are satisfied.

B. Solution to Example Problem 1
Using the geometry properties of hexagonal-shaped cells, the following constraint must be satisfied [9]: (2) where is the cluster size, is a rhombic number, 1 and is a nonnegative integer.The values for and are selected to satisfy the following constraints [9]: For a given value of cluster size , Table I presents calculated values for and using ( 2)-( 5).It is seen from Table I that a cluster size of at least seven is required to achieve the specified constraints and .Notice also that there are two solutions for the minimum cluster size in Table I.The first solution satisfies the given and constraints but the second solution does not because of the second solution is less than the specified constraint.Other solutions with larger than the minimum value of seven (in the example problem) can be found, but these are not optimal.For example, the second solution for , that is, , gives acceptable values for both cochannel and adjacent channel reuse distance constraints, but this solution is not optimal because is much larger than the minimum cluster size of seven in the example problem.

C. Further Discussions on Dynamic Clustering Approach
Suppose that for a cluster size , there are multiple solutions satisfying the specified and constraints.There arises a problem of how to select the optimal solution among all the feasible solutions.As an illustration, suppose that is the minimum cluster size in the previous example problem.From Table I, it is observed that the two solutions for satisfy the specified constraints and .Note that the two solutions have the same value of but different values of .We select the second solution as the optimum because it has a larger value of , the rationale being that cochannel interference is inversely proportional to , for , where is the path loss exponent.We present the following rules for selecting the optimum solution when there are several possible solutions satisfying the given set of distance constraints.
Step 1) Determine the minimum for which at least one of its solution(s) satisfies the specified distance constraints.
Step 2) If there is only one such solution, this solution is optimum.Stop.
Step 3) Otherwise, if there exists more than one solution satisfying the specified distance constraints, then select one of the following four cases.
Case 1) If the multiple solutions have the same but different , select the optimum solution as that having the higher .Case 2) If the multiple solutions have the same but different , select the optimum solution as that having the higher .Case 3) If the multiple solutions have the same and also the same , pick any one of the solutions as the optimum.Case 4) If the multiple solutions all have different and also all have different , the optimum solution is determined by an op-timality criterion, for example, total distance separation (TDS).Compute total distance separation for solution TDS , by TDS (6) where TDS is the net separation of solution , denoted by , from the given constraints .The TDS metric is introduced to compare the quality of all the feasible solutions.Select the optimum solution as that having the largest TDS; such a solution exhibits the least interference.Note that the TDS s are nonnegative because for each solution satisfying the distance constraints, and .

IV. THE PROPOSED VITERBI-LIKE ALGORITHM (VLA) FOR SOLVING CAP
As a prelude to presenting the proposed Viterbi-like algorithm for solving the CAP in cellular radio networks, it is instructive to briefly describe the original Viterbi algorithm used for information decoding in digital communication systems.Basically, the Viterbi algorithm uses the Hamming distance metrics to find the optimum path (i.e., minimum distance path) through the decoder trellis.At each time instant during the decoding process, the Viterbi algorithm consists of computing the distance metrics for the paths entering each state and retaining only the path with the smallest distance metric (called the surviving path).If more than one path has the smallest distance metric, then one of the paths is selected randomly.The path computation proceeds as a function of time, advancing deeper into the trellis until the first transmitted bit is decoded unambiguously (i.e., only one branch of the minimum distance path exists between two states of the first decoding interval).The sequential trellis search process continues until all the transmitted bits are decoded.
The channel assignment problem is solved in this paper using an approach similar to the original Viterbi algorithm; hence the name "Viterbi-like algorithm."The VLA uses the sequential trellis search to minimize a path metric called excess frequency factor, and hence to minimize the bandwidth required to support specified traffic demand across the cells of a cluster while satisfying the channel separation constraints.Table II summarizes the similarities and differences between the original Viterbi algorithm and the VLA proposed in this paper.

A. Step-by-Step Channel Assignment Using the VLA
Step 0) Initialization step-determine the order of assigning channels to the cells in a cluster, accomplished in two stages.The first stage is to calculate the degree of each cell using the formula [12] (7 where is the degree of cell .The notation is a heuristic measure of the difficulty of assigning a channel to cell [12].The difficulty is finding the minimum channel to assign to a cell without violating the channel separation constraints.The second stage is to arrange the cells in a cluster in a descending order based on the calculated degrees.This ordered list is then used for channel assignment. Once the cells are sorted, the next task is to assign the available channels to calls starting from the cell at the top to the bottom of the ordered list.Channel assignment is done on a step-by-step basis where, during each step, a channel is assigned to support a call (if any) in each cell in the ordered list, as illustrated in Fig. 2. Denote the total demand (i.e., total number of channels required) across the cells in a cluster by ; then (8) where is the demand of the cell at position in the ordered list (note that the index of the cell at position in the ordered list is not necessarily equal to ).Initially, we evoke Assumption 1) so that the total channels available is the same as the total traffic demand, i.e., . The available channels (denoted by ) are to be assigned to the cells starting with the cell at the top of the ordered list.Without loss of generality, denote the index of the cell at position 1 in the ordered list by , where .
Step 1) Perform the first step channel assignment.First, channel is assigned to a call in cell .Next, channel also is assigned to other cells in the cluster, where provided the channel separation distance between cells and is zero, i.e., for all .Define a set whose elements are cells to which channel can be assigned.Mathematically, cell cells .
Step 2) Perform the second step channel assignment.The next channel to be assigned is since it is the next adjacent channel to , the last channel assigned.Two constraints must be satisfied for channel to be assigned to a cell.The first constraint is the adjacent channel constraint: channel is assigned to a call in cell provided for all cell to which channel has been assigned.Recall that (specified in the compatibility matrix) is the minimum channel separation distance between a channel assigned to a call in cell and a channel assigned to another call in cell .The second constraint is that channel is assigned to other cells provided there is zero separation constraint between cells and .Denote by the set of cells to which can be assigned.Mathematically, cell cells .If none of the cells with nonzero traffic demand satisfies the above constraints (i.e., is an empty set), then channel cannot be assigned and the algorithm considers the next available channel.Alternatively, if set contains more than one element, some of the elements (i.e., cells) are redundant and will be systematically removed at the next step(s) of channel assignment.
Step 3) The third step channel assignment consists of assigning channel to a call in cell provided adjacent channel and zero separation constraints are satisfied.Denote by the set of cells to which can be assigned.Mathematically, the constraints cell cells and must be satisfied where is used here generically to denote the index of cells belonging to set .That is, each cell in has a corresponding cell in .If set contains more than one element, some of these elements are redundant.The redundant elements in are removed by comparing the required minimum channel separation between and assigned to every cell in sets and , respectively, with the actual channel separation , where the subscripts and are used generically to denote the index of cells in sets and , respectively.The rationale is that since the elements in set already satisfy the channel adjacency constraint with the elements in set , the only other preassigned channel (nonadjacent to ) is then used to remove any redundant elements in set .To this end, we introduce a metric called the excess frequency factor between the assignments in sets and , denoted by EFF , where the subscripts and are used generically to denote the index of cells in sets and , respectively.Mathematically, EFF for all in set and all in set .Now, EFF can have three possible values and the decisions taken are described as follows.Case 1) EFF .This means that channel is assigned to a cell that is too close to a cell already assigned with channel ; hence the minimum channel separation constraint is not met.All cell s in for which EFF are redundant and have to be removed from set .Case 2) EFF .For each cells in with EFF , such a cell represents an optimum solution and the cell is called a survivor at the third step of channel assignment.Recall that each cell in has a cell in with which it satisfies the channel adjacency constraint .Such a cell also represents a survivor in at the third step of channel assignment.Case 3) EFF .For all cells in with EFF , determine the minimum value of the EFF s and denote it by EFF .Retain only the cells in whose EFF s is equal to EFF .Retain also the cells in corresponding to cells in with EFF .Finally, when different combinations of the above three cases occur, the following decisions are taken.
1) If all the computed EFF s are less than zero, then all the cells in are redundant.Hence cannot be assigned and the next channel is tried.
2) If all the computed EFF s are equal to zero, then all the cells in are optimum solutions and all are retained (i.e., all are survivors).
3) If all the EFF s are greater than zero, then retain only the cells having EFF .4) If some of the EFF s are less than zero, some equal to zero, and some greater than zero, then retain only the cells whose EFF s are equal to zero (i.e., the optimum solutions).
The procedure outlined in Step 3) is repeated for every subsequent step of channel assignment until all the available channels are assigned.In general, the procedure to be used for assigning channel to a cell with index , given that the channel is already assigned to a cell with index , is outlined as follows.
Step 1) Generate a set whose elements satisfy the adjacent channel constraint .
Step 2) If the condition specified in Step 1) is not satisfied, then pick the next channel.That is, replace by and go back to Step 1).
Step 3) Calculate the relevant excess frequency factors between channel and each of the previously assigned channels nonadjacent to EFF for all .Here we have assumed that the channel (for ) is assigned to a cell with index .Note that the index need not be the same for all .
Step 4) For each in Step 3), do the following.
Step 4.1) Remove all the elements in set with negative excess frequency factor.Also, remove the elements in set that corresponds to the deleted elements in set .Using the new Viterbi-like approach, the comparison of the required channel separation with the actual separation is reduced to just one or two at every step of channel assignment because only the optimal solutions are retained from previous assignments.The preceding statements serve as the intuitive explanation for the quality of solution (i.e., minimum bandwidth) and the savings in execution time achieved by the Viterbi-like algorithm.
Example Problem 2: We apply the VLA to solve the channel assignment problem for the cellular network considered in Example Problem 1.It is assumed that adjacent cell separation (ACS) and cocell separation (CCS) constraints are 1 and 3, respectively [7].Traffic demand in each cell is selected from a uniform distribution , as assumed in Example Problem 1.The generated traffic demand is listed according to a decreasing order in table heading row #2 of columns 2 to 8 in Table III.Clearly, the total traffic demand is 96.Starting from row #3, the channels assigned to cells during each step of channel assignment are shown.For example, channel is assigned to cell #1 during the first step, channel is assigned to cell #1 during the second step, and so on.It is observed that during the first 12 steps of channel assignment, the actual channel separation distance between two successive channel assignments in each cell is a fixed number equal to seven units.Constant channel separation of seven is due to cluster size and the fact that each cell with nonzero demand is assigned a maximum of one channel during each step of channel assignment, according to VLA operation.Hence, prior to servicing all 12 traffic demands in each cell of the seven-cell cluster, adjacent cell separation is the limiting factor with a consequence of a constant channel separation distance between two successive channel assignments in a cell.As channel assignments to cells 7, 6, 5, and 4 are complete, the channel separation distance between two successive assignments in cells 1, 2, and 3 is no longer a fixed number.For example, for cell 1, the channel separation distance between steps 13 and 14 assignments is five, while the corresponding separation distance between steps 14 and 15 assignments is four (both CCS values of five and four satisfy the CCS constraint of three units).We conclude from Table III that the solution obtained using the Viterbi-like algorithm (assuming a cluster size  of seven) yields a bandwidth of 96 units.This result agrees with the previously reported result in [7] under the same assumption.By assuming other values of cluster size, the corresponding required bandwidth (normalized with respect to the bandwidth of one channel) determined using the Viterbi-like algorithm is summarized in Table IV.It is seen that required bandwidth increases with cluster size and, for the range of cluster sizes considered in Table IV, a cluster size of seven provides the minimum bandwidth.However, implication of small cluster size is high interference.This implication leads us to quantify the relationship between interference and cluster size as depicted by Fig. 3.It is observed from Fig. 3 that adjacent channel interference decreases as cluster size increases, but the rate of decrease is less at large values of .Beyond a certain value of cluster size (e.g., ), adjacent channel interference becomes relatively insensitive to further increases in .It is interesting to note that the preceding observation on the behavior of ACI with respect to is contrary to that observed for required bandwidth, which increases monotonically with .Hence, there exists a tradeoff between required bandwidth and tolerable level of adjacent channel interference.

B. Performance of VLA Under Insufficient Available Channels
In this section, we study the performance of VLA under the assumption that the number of available channels is less than the total demand , in other words, Assumption 1) of Section II-B is removed.First note that the VLA works in the same way as described previously until all the available channels have been assigned.Clearly, if no free channel is available, the VLA cannot be applied and the remaining calls are delayed prior to being assigned channels.Due to the fact that VLA distributes channel assignment across the cells with nonzero traffic demand, we conclude that the delayed calls are also distributed evenly across the cells with nonzero traffic demand when the available channels become exhausted.As an illustration, suppose the total demand (sum of the entries in table heading row #2 of columns 2 to 8 in Table V) but the available number of channels .This means that four calls in total  V.The first call is the fourteenth call in cell 4 during step 14 assignment (shown as A in Table V).The second, third, and fourth calls delayed are the fifteenth calls in cells 1, 2, and 3, respectively (shown as B, C, and D in Table V), occurring during step 15 assignment.The delay in assigning a channel to a call that cannot be serviced due to insufficient channels is the minimum of all the channel holding times.Assuming that the channel holding times of all the active calls are indepen- dent exponentially distributed random variables with parameter , the delay in assigning a channel to a delayed call is also exponential with parameter [17].

C. Stability of VLA
We observe that channel assignment algorithms whose solutions usually start at an arbitrary point (e.g., the LSA) runs the risk of instability because a solution point may fall in a local minimum, thereby rendering the algorithm unstable (i.e., the algorithm converges to a solution that is not optimal), as illustrated in Fig. 4(a).Unlike the algorithms whose solutions always start from an arbitrary point, the solutions obtained using the VLA do not run the risk of falling into a local minimum because the starting solution (obtained at the first step channel assignment) is always the best solution.Based on VLA operation, the initial best solution is maintained at all subsequent steps of channel assignments.From the VLA operation described in Section IV-A, the solution of the first step channel assignment (i.e., the cells within a cluster that are assigned the first channel) are all selected if and only if the specified channel separation constraints at the first step assignment are satisfied.Such a solution is considered to be optimal with respect to first step channel assignment constraints.VLA offers an advantage that during subsequent steps of channel assignment, only the cells meeting the required constraints of the current step and those of the previous steps are retained and redundant assignments are removed from further consideration.In this way, the best solution (i.e., minimum bandwidth) is maintained at subsequent steps of channel assignment [as illustrated in Fig. 4(b)], leading us to conclude that the VLA provides a stable solution (i.e., best solution at all the steps of channel assignment).

D. Limitations of VLA
The first limitation of the proposed VLA is the storage requirement for storing the channels that have been assigned to each cell during successive steps of channel assignment.For a cluster with cells, the VLA requires no more than storage locations.Assuming that the word size for each channel is bits, the maximum amount of storage required is given by ( 9) Equation ( 9) shows that storage requirement increases linearly with cluster size .Recall from Table IV that bandwidth requirement is minimum at low value of cluster size.Hence, it is concluded that VLA performs better (in terms of bandwidth and memory requirements) at low cluster size compared to large cluster size.A second limitation of the VLA is computational complexity.The computations performed at step of channel assignment consists of subtractions and comparisons, where .The first term of unity on the right-hand side represents the single subtraction and comparison required to check the adjacent channel separation constraint; the second term represents the EFF calculations performed to check the nonadjacent channel separation constraint.Each EFF calculation consists of a subtraction and a comparison.Fig. 5 illustrates the types of computations (i.e., adjacency versus nonadjacency constraint checking) performed at each step of channel assignment.Note that even with the preceding computation requirements, the VLA still has better real-time performance than the existing algorithms, as will be discussed in a later section.

V. FURTHER NUMERICAL EXAMPLES
Two additional numerical examples of the application of the VLA are presented in this section.The first example is the solution of 19 benchmark problems on channel assignment proposed in the literature [7].Our objective in the first example is to assess the quality of VLA solution against those of existing channel assignment algorithms in the literature.The second example is solution of the channel assignment problem in a cellular radio network, with an objective of determining the bandwidth required to support a given traffic demand, without violating adjacent and cocell separation constraints.First, the 19 benchmark problems are classified into two groups depending on whether the traffic demand is a fixed number (Group I problems) or selected from a given probability distribution (Group II problems).A description of the key features of Groups I and II problems is provided in Tables VI and   VII, respectively.The first nine problems in Table VI assume a cluster size of 21, but problem 10 assumes a cluster size of 25.The values of the fixed traffic demand and compatibility matrix are as specified in [3], [7], and [12].Table VI also presents the solutions obtained from two existing algorithms in the literature: 1) lower bounds (i.e., the minimum bandwidth required [2], [3], [12], [13]) Fig. 6.Demand vector assumed for the seven-cell cluster of a cellular radio network example.and 2) LSA [7].For purpose of comparison with the previous results, results obtained using the VLA are presented in the last column of Table VI.It is observed from Table VI that the VLA meets the lower bound solution in 70% of the problem cases but provides slightly higher bandwidth ( 6% higher) than the lower bound in problems 4, 6, and 8.The quality of the solution provided by the VLA is due to the removal of redundant assignments from further consideration and keeping only the survivors at each step of channel assignment.
Next, the VLA is applied to solve the Group II problems whose key features are specified in Table VII.Here, all nine problems assume a seven-cell cluster structure, and the traffic demand is assumed to be uniform over a specified interval (ll, ul), where ll and ul are the lower (minimum) and upper (maximum) traffic demand, respectively [7].Each problem listed in Table VII is unique in terms of the specified values for ll and ul and the interference constraints and .Table VII presents the results obtained using two of the existing algorithms-the two-phase algorithm [8] and the LSA [7].Also included for comparison are the results obtained using the VLA.It is observed from Table VII that the solutions obtained by the VLA are always better than or match those of the two-phase algorithm.The solutions obtained using the VLA are comparable to those achieved by the LSA, the difference between the LSA and VLA results being less than 8% and the LSA exhibiting a lower bandwidth requirement.Note, however, that other factors, such as computation complexity and execution time, must be considered in reaching a conclusion on the better algorithm between the LSA and VLA.As stated previously, VLA achieves its performance through sequential assignment of channels subject to meeting adjacent and nonadjacent channel constraints.
Solution obtained by LSA begins by assuming an arbitrary initial solution followed by exhaustive search for a better solution through interchange of already assigned channels.Of course, this channel interchange operation of LSA requires intensive computation, which is much higher than that for VLA.For example, using a DEC workstation, a straightforward implementation of the local search algorithm without any run-time optimization takes, on the average, between 2-3 min, whereas the proposed Viterbi-like algorithm takes around 1-2 min for all 19 benchmarking problems.

B. Solution of CAP in Cellular Radio Network Using the VLA
In this section, we use the Viterbi-like algorithm to solve the channel assignment problem in a cellular radio network, taking into account the effect of traffic demand ACS and CCS.A cellular network with a cluster size of seven is assumed.First, we assume a fixed traffic demand in each cell shown in Fig. 6, where the minimum and maximum demands are five and 18, respectively.The cells are ordered according to decreasing demand using node degree ordering, so that channel assignment begins with cell 1, using the VLA.In this example network, the performance metric is the bandwidth required to support the assumed traffic demand, normalized by the bandwidth required to support the minimum demand of five calls.That is, normalized bandwidth is defined as the ratio of bandwidth required to support a given demand (selected in the range [5,18] calls) to bandwidth required to support the minimum traffic demand of five calls.Fig. 7 shows a three-dimensional plot of normalized bandwidth (determined using the VLA) versus the cell index in the cluster versus traffic demand.Two operating regions are observed for the required bandwidth shown in Fig. 7.In the first region (traffic demand varying from one to five calls), the required bandwidth in each cell increases linearly with traffic demand.Under high traffic demand range (from nine to 18 calls), the bandwidth required increases nonlinearly with demand and tends to a limiting value.An explanation for the linear and nonlinear behavior of the bandwidth required versus demand is as follows.First, when all the cells in a cluster have nonzero demand, the ACS constraint is the limiting factor causing a linear increase in bandwidth at a rate equal to the product of cluster size and ACS constraint.When there are only few cells in the cluster with nonzero demand, ACS constraint exerts less influence, resulting in a reduction in the rate of bandwidth increase.In the extreme case when only one single cell in the cluster has demand (i.e., cell 1 in Fig. 7), the CCS becomes the limiting factor causing a nonlinear increase in bandwidth, a consequence of CCS assuming a nonconstant value at subsequent steps of channel assignment.
Next, we investigate the effect of normalized demand range on the required bandwidth.is defined as demand range divided by average demand or, mathematically, ul ll ul ll .For this investigation, we assume that traffic demand in each cell is selected from a uniform distribution ll ul , where the minimum traffic demand is set to five calls and the value of the maximum demand is calculated by ul ll (10) where is assumed to be known and lies in the interval [0, 2) to calculate the corresponding value of ul.
Fig. 8 shows the normalized bandwidth plotted against the normalized demand range .Although the scatter plot in Fig. 8 is quite dispersed, the general trend is that the normalized bandwidth increases with normalized demand range.When the normalized demand range is small, a low bandwidth is often enough because the channels can be evenly distributed among the cells.At high values of normalized demand range (i.e., ), there is an uneven distribution of channels among the cells, resulting in an increase in bandwidth required.Fig. 9 shows a plot of the normalized bandwidth versus the average traffic demand ( ul ll for the assumed uniform traffic distribution).It is seen that the normalized required bandwidth increases linearly with the average traffic demand, as expected.

VI. CONCLUSION
The major focus of this paper is development of a heuristic algorithm, called the Viterbi-like algorithm, that achieves the objectives of minimum bandwidth requirement comparable to or better than the best performance reported in the literature and faster execution time compared to those of previous algorithms.The basic idea of the VLA is sequential assignment of channels to the cells in a cluster subject to meeting cocell and adjacent cell separation constraints.The VLA provides advantages of minimum bandwidth requirement, stability of solution, and fast speed of execution.However, the VLA has a limitation of high bandwidth and memory requirement at large value (e.g., 13) of cluster size for which adjacent channel interference is minimum.Hence, a tradeoff exists between bandwidth/memory requirement and tolerable level of adjacent channel interference.The VLA, like the other existing channel assignment algorithms, also suffers from high computation requirement.According to our experimental results, we found that the computation requirement of the VLA is less (by up to a factor of two) than those of the existing algorithms.
Performance of the VLA is assessed first by using the algorithm to solve 19 benchmark problems and then by studying the important characteristics of cellular radio networks.Results from the performance assessment show that the VLA provides bandwidth requirements that closely match or are sometimes better than those obtained with the existing algorithms in the literature.Minimum bandwidth is achieved by the VLA through the removal of redundant channel assignments (measured by a metric called excess frequency factor) from further consideration and keeping only the survivors after each step of channel assignment.A byproduct of retaining only the survivors is a reduction in the number of comparisons performed at subsequent steps of channel assignment; this translates to fast speed of execution for the VLA.Furthermore, the solutions obtained by the VLA are stable because they always start from the best solution that is maintained at all subsequent steps of channel assignment.The combined advantages of minimum bandwidth requirement, stability of the solutions, and fast speed of execution make the VLA a viable candidate for implementation in cellular radio network planning.

A
Viterbi-Like Algorithm With Adaptive Clustering for Channel Assignment in Cellular Radio Networks Xavier N. Fernando and Abraham O. Fapojuwo, Senior Member, IEEE Abstract-A new channel assignment algorithm, called the Viterbi-like algorithm (VLA), is proposed to solve the channel assignment problem in cellular radio networks.The basic idea of the proposed algorithm is step-by-step (sequential) channel assignment with the objectives of minimum bandwidth required at every step, subject to adjacent channel and cochannel separation constraints.The VLA provides the benefits of minimum required bandwidth, stability of solution, and fast execution time.The performance of the VLA is evaluated by computer simulation, applied first to 19 benchmark problems on channel assignment and then applied to study cellular radio network performance.Results from computer simulation studies show that bandwidth requirements by VLA closely match or are sometimes better than those of the existing channel assignment algorithms.Furthermore, it is found that execution of VLA is approximately two times faster than the local search algorithm-the existing channel assignment algorithm with the least bandwidth requirements.The combined advantages of minimum required bandwidth, stability of solution, and fast execution time make the VLA a useful candidate for cellular radio network planning.Index Terms-Cellular network planning, cellular radio network, channel assignment, Viterbi algorithm, wireless communications.

Step 4 . 2 )
Retain all the elements in having zero excess frequency factor.Step 4.3) For the elements whose excess frequency factors exceed zero, determine the minimum excess frequency factor between the channel assigned to a cell with index and channel assigned to a cell with index , denoted by EFF .Step 4.4) Retain only the elements corresponding to EFF .Remove all other entries.

Fig. 3 .
Fig. 3. Effect of cluster size on normalized bandwidth and adjacent channel interference.

Fig. 4 .
Fig. 4. Stability of VLA solution compared to that of other CAP algorithms, e.g., LSA.(a) Solution trajectory for LSA.(b) Solution trajectory for Viterbi-like algorithm.

Fig. 5 .
Fig. 5. Types of computations performed at each step of channel assignment.

TABLE I CALCULATED
COCHANNEL AND ADJACENT CHANNEL REUSE DISTANCES

TABLE II COMPARISON
BETWEEN THE ORIGINAL VITERBI ALGORITHM (VA) AND THE PROPOSED VITERBI-LIKE ALGORITHM (VLA) FOR CAP

TABLE III CHANNEL
ASSIGNMENT SOLUTIONS FOR n = 7

TABLE IV NORMALIZED
BANDWIDTH REQUIRED AS A FUNCTION OF CLUSTER SIZE n

TABLE VI COMPARISON
OF MINIMUM BANDWIDTH OBTAINED USING THE VITERBI-LIKE ALGORITHM AND THOSE OF PREVIOUS ALGORITHMS FOR THE TEN BENCHMARK (GROUP I) PROBLEMS WITH FIXED DEMAND IN EACH CELL

TABLE VII COMPARISON
OF MINIMUM BANDWIDTH OBTAINED USING THE VITERBI-LIKE ALGORITHM AND THOSE OF PREVIOUS ALGORITHMS FOR THE NINE BENCHMARK (GROUP II) PROBLEMS ASSUMING UNIFORM DEMAND DISTRIBUTION IN EACH CELL A. Solution of 19 Benchmark Problems Using the VLA