A Viterbi-like algorithm with adaptive clustering for channel assignment in cellular radio networks

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 1, JANUARY 2002 73

A Viterbi-Like Algorithm With Adaptive Clusteringfor Channel Assignment in Cellular Radio Networks

Xavier N. Fernando and Abraham O. Fapojuwo, Senior Member, IEEE

Abstract—A new channel assignment algorithm, called theViterbi-like algorithm (VLA), is proposed to solve the channelassignment problem in cellular radio networks. The basic ideaof the proposed algorithm is step-by-step (sequential) channelassignment with the objectives of minimum bandwidth required atevery step, subject to adjacent channel and cochannel separationconstraints. The VLA provides the benefits of minimum requiredbandwidth, stability of solution, and fast execution time. Theperformance of the VLA is evaluated by computer simulation,applied first to 19 benchmark problems on channel assignmentand then applied to study cellular radio network performance.Results from computer simulation studies show that bandwidthrequirements by VLA closely match or are sometimes better thanthose of the existing channel assignment algorithms. Furthermore,it is found that execution of VLA is approximately two times fasterthan the local search algorithm—the existing channel assignmentalgorithm with the least bandwidth requirements. The combinedadvantages of minimum required bandwidth, stability of solution,and fast execution time make the VLA a useful candidate forcellular radio network planning.

Index Terms—Cellular network planning, cellular radionetwork, channel assignment, Viterbi algorithm, wireless commu-nications.

I. INTRODUCTION

T HE need for efficient channel assignment algorithms inwireless networks is becoming increasingly important

as the wireless subscriber base continues to grow rapidlywhile the radio-frequency spectrum remains a scarce resource.This paper presents a new channel assignment algorithm forsolving the channel assignment problem (CAP) in cellularradio networks. Historically, the CAP has been formulated asa nonlinear optimization problem whose solution continuesto be of active research interest in the literature. Examples ofpreviously proposed solution techniques include the geneticalgorithm [1], neural networks [2]–[4], simulated annealing[5], [6], local search approach [7], cochannel interferenceminimization [8], [9], heuristic approaches [10]–[12], andtheoretical lower bounds [13], [14]. The genetic algorithmproposed in [1] is a blind search procedure, where the initialpopulation is semirandomly generated and the algorithm selectsa new population with better fitness.

Manuscript received March 23, 2000; revised August 17, 2001. The work ofA. O. Fapojuwo was supported by NSERC Canada under a Grant.

X. N. Fernando is with the Department of Electrical and Computer Engi-neering, Ryerson University, Toronto, ON, Canada.

A. O. Fapojuwo is with the Department of Electrical and Computer Engi-neering, University of Calgary, Calgary T2N 1N4 AB, Canada (e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(02)00438-3.

Several approaches based on neural networks have been pro-posed [2]–[4]. For example, [4] proposes a modified Hopfieldneural network, where the CAP is formulated as an energy-min-imization problem. Solution of CAP using the simulated an-nealing approach starts with the selection of an arbitrary solu-tion from which a minimum is then searched [5]. During theoptimization, a so-calledMetropoliscriterion is used to preventthe solution from converging to local minima. A local search ap-proach (LSA), which is a special case of the simulated annealingapproach, is proposed in [7]. The LSA searches the neighbor-hood of the current solution point in the feasible regionfor 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 ap-plied to a set of benchmarking problems, offers the best resultsso far in the literature [7]. Considering a pattern of hexagonalshaped cells, [8] and [9] present an approach that is based onselection of sets of channels such that cochannel interferenceis minimized by appropriately grouping the cells into compactpatterns or clusters. Finally, heuristic-based solutions are pre-sented in [10], [11], and [12]. For example, in [10], the cells arefirst sorted in alphabetical order and the channels are sequen-tially assigned until a denial occurs. Subsequently, the cells areresorted in the order of increasing difficulty, and a reassignmentis made until no further denial occurs. An excellent descriptionof systematic sorting is given in [12], where most of the resultsof channel assignment problems considered are very close to thetheoretical bounds derived in [13].

The major focus of this paper is development of a heuristicalgorithm that achieves the objectives of

1) minimum bandwidth requirement that is comparable to orbetter than the best performance reported in the literature;

2) faster execution time compared to those of previous algo-rithms.

The channel assignment algorithm proposed in this paper iscalled theViterbi-likealgorithm (VLA) because of its similarityto the sequential trellis search algorithm of the original Viterbialgorithm used in the field of information decoding [15], [16].However, the VLA minimizes bandwidth instead of the Eu-clidean distance minimization performed in the original Viterbialgorithm. Minimum bandwidth is achieved by VLA throughthe removal of redundant channel assignments [a channel as-signment is said to be redundant if its excess frequency factor(EFF) is negative] from further consideration and keeping onlythesurvivorsafter each step of channel assignment. A benefit ofretaining only the survivors is a reduction in the number of itera-tions performed in the subsequent steps of channel assignment;

0018-9545/02$17.00 © 2002 IEEE

74 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 1, JANUARY 2002

this translates to fast execution time for the proposed VLA al-gorithm.

Another reason for the minimum bandwidth achieved byVLA is the use of dynamic clustering. The dynamic clusteringapproach adaptively groups the cells for frequency reuseaccording to the instantaneous traffic demand while meetingacceptable cochannel and adjacent channel interference limits.In essence, dynamic clustering does the efficient cluster organ-ization while the VLA manages the frequency planning withineach cluster.

The remainder of this paper is organized as follows. Sec-tion II defines the channel assignment problem and also spec-ifies the assumptions made along with justifications. In Sec-tion 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 algo-rithm for solving channel assignment problem forms the maintopic of Section IV. Other topics covered in Section IV includeperformance of VLA under insufficient available channels com-pared to traffic demand, as well as stability and limitations ofthe VLA. In Section V, we first assess the quality of the pro-posed VLA by using it to solve 19 benchmark channel assign-ment problems and then compare the results with those of theexisting channel assignment algorithms. Next, we use the VLAto solve the channel assignment problem in a cellular radio net-work with emphasis on studying the effects of traffic demandand adjacent cell and co-cell separation on required bandwidth.Section VI concludes this paper.

II. THE CHANNEL ASSIGNMENTPROBLEM

A. Problem Statement

The channel assignment problem in a cellular radio networkis formulated as follows [7].

The cellular network comprises distinct geographic cells.These cells are grouped into small clusters, each of sizecells, where . The offered traffic to a cluster is charac-terized by a demand vector , where

is the number of traffic channels required in cell. The totalnumber of traffic channels required in all the cells of a cluster(i.e., total demand) is denoted by, where . Thechannel separation constraint is modeled by an symmetriccompatibility matrix , where denotes the min-imum required channel separation between a channel assignedto 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 areavailable, where is initially made equal to to ensure that achannel can be assigned to each call in the cluster. Denote the

channels by , where ,is the th channel in the list of channels. After channelis assigned to call , the suffix is changed to (i.e., ischanged to ) for consistency with the suffixes of call .

A set of channel separation constraints is defined by the com-patibility 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 suchthat the total bandwidth required is a minimum. The CAP iswell known to be a computational challenging nonlinear opti-mization problem. In this paper, we solve the CAP using theViterbi-like algorithm.

B. Modeling Assumptions

The following assumptions are made in the analysis.

Assumption 1) Sufficient channels are available to meet thetotal traffic demand.

Assumption 2) The available channels are pooled and dy-namically assigned to service the traffic de-mand.

Assumption 3) All the channels have equal interferencecharacteristics.

Assumption 4) The total demand in a cluster is transmittedinstantaneously to a call control serverwhere the channel allocation is done.

The implication/rationale behind Assumptions 1) to 4) is statedas follows. Assumption 1) is a starting assumption made to sim-plify the development of the Viterbi-like algorithm. Assump-tion 1) is relaxed in a later section to cover the case wherethe number of available channels is less than the total demand.Pooling of the available channels [Assumption 2)] is necessaryfor simplicity of channel management and flexibility to assign achannel to any cell in a cluster. The implication of Assumption3) is that channel separation is independent of channelor channel . Assignment of the channels at a central location[Assumption 4)] provides a system-level view of used and avail-able channels. In practice, the central call control server can beimplemented at either the base station controller (BSC) or themobile services switching center (MSC), depending on the net-work architecture.

III. CELL ORGANIZATION AND DYNAMIC CLUSTERING

The first step in the channel allocation procedure is to suitablyarrange the cells in a cellular radio network into clusters (cellrepeat patterns), each of sizecells. For optimum frequencyreuse with minimum interference, both the cluster sizeandthe cell arrangement within a cluster are important. Convention-ally, clusters have fixed size due to hardware and infrastructureconstraints. Small cluster size results in high capacity but with apenalty of high interference. Conversely, if the cluster size is toolarge, there is an advantage of low interference but at the costof 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 thenetwork. Hence, in this paper, we assume that cluster size is notfixed but is dynamically selected according to traffic demand.The preceding statement is referred to asdynamic clustering.By dynamically clustering the cells according to traffic demand,overall radio network capacity is improved.

As a preamble to dynamic clustering, we introduce twodistance parameters: cochannel reuse distance and adjacentchannel reuse distance. Cochannel reuse distanceis definedas the geographic distance between the centers of any twocells using the same channel sets. Adjacent channel reuse

FERNANDO AND FAPOJUWO: VITERBI-LIKE ALGORITHM WITH ADAPTIVE CLUSTERING 75

Fig. 1. Seven-cell clustering withd � p7 andd � 1.

distance is defined as the geographic distance between anytwo cells using adjacent channel sets. Fig. 1 illustrates thedistance parameters and . Let the distances and benormalized with respect to cell radius so that these becomeindependent of the actual cell size. Using the properties ofhexagonal cell shape geometry listed in [9], cell dynamicclustering can be automated. That is, for a given demandvector and acceptable and , the cluster size can beautomatically determined. Once the value ofis decided, thenext important step is to organize the cells within each clusterto achieve optimum values of and (i.e., optimum in thesense of meeting the specified objectives forand ). Fora 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 ofand isselected. The foregoing is best demonstrated by an example.

A. Example Problem 1

Consider a cellular network whose coverage area comprises49 hexagonal-shaped cells (i.e., ). The constraints forand are given as and , respectively. Denotethe cluster size by. Due to spatial traffic variability, a distribu-tion is assumed for traffic demand in each cell of a cluster. In thisproblem, it is assumed that the traffic demand vectoris gen-erated from a uniform distribution ll ul where “ll” and “ul”are the lower and upper traffic demand limits, respectively. Theassumption of uniform traffic distribution is based on the initialassumption that subscribers are uniformly distributed over theservice area. For purpose of illustration, ll and ul are set to 10and 15, respectively, in this example problem. The traffic de-mand of all the cells in a cluster can be arranged in a vectorof length as shown in

(1)

where is the traffic demand (i.e., number of traffic channelsrequired) in cell . The problem is to determine the minimumcluster size and the optimum cell arrangement within eachcluster so that the specified and constraints are satisfied.

B. Solution to Example Problem 1

Using the geometry properties of hexagonal-shaped cells, thefollowing constraint must be satisfied [9]:

(2)

where is the cluster size, is a rhombic number,1 and is anonnegative integer. The values for and are selected tosatisfy the following constraints [9]:

(3)

(4)

(5)

For a given value of cluster size, Table I presents calculatedvalues for and using (2)–(5). It is seen from Table Ithat a cluster size of at least seven is required to achieve thespecified constraints and . Notice also that thereare two solutions for the minimum cluster size in Table I.The first solution satisfies the given

and constraints but the second solutiondoes not because of the second solution is less

than the specified constraint. Other solutions with largerthan the minimum value of seven (in the example problem) canbe found, but these are not optimal. For example, the secondsolution for , that is, , givesacceptable values for both cochannel and adjacent channel reusedistance constraints, but this solution is not optimal because

is much larger than the minimum cluster size of seven in theexample problem.

C. Further Discussions on Dynamic Clustering Approach

Suppose that for a cluster size, there are multiple solutionssatisfying the specified and constraints. There arises a

1Any number that can be expressed as(p +q +pq) for some integerp andq is a rhombic number. All the distances in hexagonal cell patterns are usuallyrhombic.


TABLE ICALCULATED COCHANNEL AND ADJACENT CHANNEL REUSEDISTANCES

problem of how to select the optimal solution among all thefeasible solutions. As an illustration, suppose that is theminimum cluster size in the previous example problem. FromTable I, it is observed that the two solutions for satisfythe specified constraints and . Note that the twosolutions have the same value of but different valuesof . We select the second solution as the optimum becauseit has a larger value of , the rationale being that cochannelinterference is inversely proportional to , for , where

is the path loss exponent. We present the following rules forselecting the optimum solution when there are several possiblesolutions satisfying the given set of distance constraints.

Step 1) Determine the minimum for which at least oneof its solution(s) satisfies the specified distance con-straints.

Step 2) If there is only one such solution, this solution isoptimum. Stop.

Step 3) Otherwise, if there exists more than one solution sat-isfying the specified distance constraints, then selectone of the following four cases.

Case 1) If the multiple solutions have the samebut different , select the optimum

solution as that having the higher.Case 2) If the multiple solutions have the same

but different , select the optimumsolution as that having the higher.

Case 3) If the multiple solutions have the sameand also the same , pick any one of thesolutions as the optimum.

Case 4) If the multiple solutions all have differentand also all have different , the op-

timum solution is determined by an op-

timality criterion, for example, total dis-tance separation (TDS). Compute totaldistance separation for solutionTDS ,by

TDS

(6)

where TDS is the net separation of so-lution , denoted by , from thegiven constraints . TheTDS metric is introduced to compare thequality of all the feasible solutions.Select the optimum solution as thathaving the largest TDS; such a solu-tion exhibits the least interference. Notethat the TDSs are nonnegative becausefor each solution satisfying the dis-tance constraints, and

.

IV. THE PROPOSEDVITERBI-LIKE ALGORITHM (VLA) FOR

SOLVING CAP

As a prelude to presenting the proposed Viterbi-like algo-rithm for solving the CAP in cellular radio networks, it is in-structive to briefly describe the original Viterbi algorithm usedfor information decoding in digital communication systems. Ba-sically, the Viterbi algorithm uses the Hamming distance metricsto find the optimum path (i.e., minimum distance path) throughthe decoder trellis. At each time instant during the decodingprocess, the Viterbi algorithm consists of computing the dis-tance metrics for the paths entering each state and retaining only


TABLE IICOMPARISONBETWEEN THEORIGINAL VITERBI ALGORITHM (VA) AND THE PROPOSEDVITERBI-LIKE ALGORITHM (VLA) FOR CAP

the path with the smallest distance metric (called the survivingpath). If more than one path has the smallest distance metric,then one of the paths is selected randomly. The path computa-tion proceeds as a function of time, advancing deeper into thetrellis until the first transmitted bit is decoded unambiguously(i.e., only one branch of the minimum distance path exists be-tween two states of the first decoding interval). The sequentialtrellis search process continues until all the transmitted bits aredecoded.

The channel assignment problem is solved in this paper usingan approach similar to the original Viterbi algorithm; hence thename “Viterbi-like algorithm.” The VLA uses the sequentialtrellis search to minimize a path metric called excess frequencyfactor, and hence to minimize the bandwidth required to supportspecified traffic demand across the cells of a cluster while sat-isfying the channel separation constraints. Table II summarizesthe similarities and differences between the original Viterbi al-gorithm and the VLA proposed in this paper.

A. Step-by-Step Channel Assignment Using the VLA

Step 0) Initialization step—determine the order of as-signing channels to the cells in a cluster, accom-plished in two stages. The first stage is to calculatethe degree of each cell using the formula [12]

(7)

where is the degree of cell .The notation is a heuristic measure of the diffi-culty of assigning a channel to cell[12]. The dif-ficulty is finding the minimum channel to assign toa cell without violating the channel separation con-straints. The second stage is to arrange the cells in acluster in a descending order based on the calculateddegrees. This ordered list is then used for channelassignment.

Once the cells are sorted, the next task is to assignthe available channels to calls starting from the cellat the top to the bottom of the ordered list. Channelassignment is done on a step-by-step basis where,during each step, a channel is assigned to supporta call (if any) in each cell in the ordered list, asillustrated in Fig. 2. Denote the total demand (i.e.,total number of channels required) across thecellsin a cluster by ; then

(8)

where is the demand of the cell atposition in the ordered list (note that the index ofthe cell at position in the ordered list is not neces-sarily equal to ). Initially, we evoke Assumption 1)so that the total channels availableis the same asthe total traffic demand, i.e., . The avail-able channels (denoted by )


Fig. 2. Step-by-step channel assignment. (ACS: Adjacent cell separation; CCS: Co-cell separation.)

are to be assigned to thecells starting with thecell at the top of the ordered list. Without loss ofgenerality, denote the index of the cell at position 1in 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 inthe cluster, whereprovided the channel separation distance betweencells and is zero, i.e., for all . De-fine a set whose elements are cells to whichchannel can be assigned. Mathematically,cell cells .

Step 2) Perform the second step channel assignment. Thenext channel to be assigned issince it is the nextadjacent channel to , the last channel assigned.Two constraints must be satisfied for channeltobe assigned to a cell. The first constraint is the ad-jacent channel constraint: channelis assigned toa call in cell provided

for all cell to which channel has been as-signed. Recall that (specified in the compati-bility matrix) is the minimum channel separationdistance between a channel assigned to a call in cell

and a channel assigned to another call in cell.The second constraint is that channelis assignedto other cells pro-vided there is zero separation constraint betweencells and . Denote by the set of cells towhich can be assigned. Mathematically,cell cells .

If none of the cells with nonzero traffic demand sat-isfies the above constraints (i.e., is an empty set),

then channel cannot be assigned and the algo-rithm considers the next available channel. Alter-natively, if set contains more than one element,some of the elements (i.e., cells) are redundant andwill be systematically removed at the next step(s) ofchannel assignment.

Step 3) The third step channel assignment consists of as-signing channel to a call in cellprovided adjacent channel and zero separation con-straints are satisfied. Denote by the set of cellsto which can be assigned. Mathematically, theconstraints cell cells

and mustbe satisfied where is used here generically to de-note the index of cells belonging to set . Thatis, each cell in has a corresponding cellin

. If set contains more than one element, someof these elements are redundant. The redundant ele-ments in are removed by comparing the requiredminimum channel separation between andassigned to every cell in sets and , respec-tively, with the actual channel separation ,where the subscriptsand are used generically todenote the index of cells in sets and , respec-tively. The rationale is that since the elements in set

already satisfy the channel adjacency constraintwith the elements in set , the only other preas-signed channel (nonadjacent to ) is then usedto remove any redundant elements in set. To thisend, we introduce a metric called the excess fre-quency factor between the assignments in setsand , denoted by EFF, where the subscriptsand are used generically to denote the index of


cells in sets and , respectively. Mathemati-cally, EFF for all in setand all in set . Now, EFF can have three pos-sible values and the decisions taken are described asfollows.

Case 1) EFF . This means that channel is assignedto a cell that is too close to a cellalready assignedwith channel ; hence the minimum channel sep-aration constraint is not met. All cells in forwhich EFF are redundant and have to be re-moved from set .

Case 2) EFF . For each cells in with EFF ,such a cell represents an optimum solution and thecell is called asurvivorat the third step of channelassignment. Recall that each cellin has a cor-responding cell in with which it satisfies thechannel adjacency constraint . Sucha cell also represents a survivor in at the thirdstep of channel assignment.

Case 3) EFF . For all cells in with EFF ,determine the minimum value of the EFFs anddenote it by EFF . Retain only the cells in

whose EFF s is equal to EFF . Retain alsothe cells in corresponding to cells in withEFF .

Finally, when different combinations of theabove three cases occur, the following decisionsare taken.

1) If all the computed EFFs are less than zero,then all the cells in are redundant. Hence

cannot be assigned and the next channelis tried.

2) If all the computed EFFs are equal to zero,then all the cells in are optimum solutionsand all are retained (i.e., all are survivors).

3) If all the EFF s are greater than zero, thenretain only the cells having EFF .

4) If some of the EFF s are less than zero, someequal to zero, and some greater than zero,then retain only the cells whose EFFs areequal to zero (i.e., the optimum solutions).

The procedure outlined in Step 3) is repeated for every subse-quent step of channel assignment until all the available channelsare assigned. In general, the procedure to be used for assigningchannel to a cell with index , given that thechannel is already assigned to a cell with index, is out-lined as follows.

Step 1) Generate a set whose elements satisfy the adja-cent channel constraint .

Step 2) If the condition specified in Step 1) is not satisfied,then pick the next channel. That is, replaceby

and go back to Step 1).Step 3) Calculate the relevant excess frequency factors be-

tween channel and each of the previously as-signed channels nonadjacent to EFF

for all . Here we have assumed

that the channel (for ) is as-signed to a cell with index . Note thatthe 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 nega-

tive excess frequency factor. Also, remove the el-ements in set that corresponds to the deletedelements in set .

Step 4.2) Retain all the elements in having zero excessfrequency factor.

Step 4.3) For the elements whose excess frequency factorsexceed zero, determine the minimum excess fre-quency factor between the channel assigned to acell with index and channel assigned to a cellwith index , denoted by EFF .

Step 4.4) Retain only the elements corresponding toEFF . Remove all other entries.

Using the new Viterbi-like approach, the comparison of the re-quired channel separation with the actual separation is reducedto just one or two at every step of channel assignment becauseonly the optimal solutions are retained from previous assign-ments. The preceding statements serve as the intuitive explana-tion for the quality of solution (i.e., minimum bandwidth) andthe savings in execution time achieved by the Viterbi-like algo-rithm.

Example Problem 2:We apply the VLA to solve the channelassignment problem for the cellular network considered in Ex-ample Problem 1. It is assumed that adjacent cell separation(ACS) and cocell separation (CCS) constraints are 1 and 3, re-spectively [7]. Traffic demand in each cell is selected from a uni-form distribution , as assumed in Example Problem 1.The generated traffic demand is listed according to a decreasingorder in table heading row #2 of columns 2 to 8 in Table III.Clearly, the total traffic demand is 96. Starting from row #3, thechannels assigned to cells during each step of channel assign-ment are shown. For example, channelis assigned to cell #1during the first step, channel is assigned to cell #1 during thesecond step, and so on. It is observed that during the first 12steps of channel assignment, the actual channel separation dis-tance between two successive channel assignments in each cellis a fixed number equal to seven units. Constant channel sep-aration of seven is due to cluster size and the fact thateach cell with nonzero demand is assigned a maximum of onechannel during each step of channel assignment, according toVLA operation. Hence, prior to servicing all 12 traffic demandsin each cell of the seven-cell cluster, adjacent cell separation isthe limiting factor with a consequence of a constant channel sep-aration distance between two successive channel assignments ina cell. As channel assignments to cells 7, 6, 5, and 4 are com-plete, the channel separation distance between two successiveassignments in cells 1, 2, and 3 is no longer a fixed number.For example, for cell 1, the channel separation distance betweensteps 13 and 14 assignments is five, while the correspondingseparation distance between steps 14 and 15 assignments is four(both CCS values of five and four satisfy the CCS constraint ofthree units). We conclude from Table III that the solution ob-tained using the Viterbi-like algorithm (assuming a cluster size


TABLE IIICHANNEL ASSIGNMENTSOLUTIONS FORn = 7

TABLE IVNORMALIZED BANDWIDTH REQUIRED AS A FUNCTION OF CLUSTER SIZE n

of seven) yields a bandwidth of 96 units. This result agrees withthe previously reported result in [7] under the same assump-tion. By assuming other values of cluster size, the correspondingrequired bandwidth (normalized with respect to the bandwidthof one channel) determined using the Viterbi-like algorithm issummarized in Table IV. It is seen that required bandwidth in-creases with cluster size and, for the range of cluster sizes con-sidered in Table IV, a cluster size of seven provides the min-imum bandwidth. However, one implication of small cluster sizeis high interference. This implication leads us to quantify therelationship between interference and cluster size as depictedby Fig. 3. It is observed from Fig. 3 that adjacent channel in-terference decreases as cluster sizeincreases, but the rate ofdecrease is less at large values of. Beyond a certain value ofcluster size (e.g., ), adjacent channel interference be-comes relatively insensitive to further increases in. It is inter-esting to note that the preceding observation on the behavior ofACI with respect to is contrary to that observed for requiredbandwidth, which increases monotonically with. Hence, there

exists a tradeoff between required bandwidth and tolerable levelof adjacent channel interference.

B. Performance of VLA Under Insufficient Available Channels

In this section, we study the performance of VLA under theassumption that the number of available channelsis lessthan the total demand, in other words, Assumption 1) of Sec-tion II-B is removed. First note that the VLA works in the sameway as described previously until all the available channels havebeen assigned. Clearly, if no free channel is available, the VLAcannot be applied and the remaining calls are delayedprior to being assigned channels. Due to the fact that VLA dis-tributes channel assignment across the cells with nonzero trafficdemand, we conclude that the delayed calls are also distributedevenly across the cells with nonzero traffic demand when theavailable channels become exhausted. As an illustration, sup-pose the total demand (sum of the entries in tableheading row #2 of columns 2 to 8 in Table V) but the availablenumber of channels . This means that four calls in total


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

TABLE VCHANNEL ASSIGNMENTWITH BLOCKING

are delayed and these calls are distributed evenly across the fourcells with nonzero demand as shown by entries A, B, C, and Din Table V. The first call delayed is the fourteenth call in cell 4during 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), occurringduring step 15 assignment. The delay in assigning a channel toa call that cannot be serviced due to insufficient channels is theminimum of all the channel holding times. Assuming thatthe channel holding times of all the active calls are indepen-


(a)

(b)

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-likealgorithm.

dent exponentially distributed random variables with parameter, the delay in assigning a channel to a delayed call is also ex-

ponential with parameter [17].

C. Stability of VLA

We observe that channel assignment algorithms whose solu-tions usually start at an arbitrary point (e.g., the LSA) runs therisk of instability because a solution point may fall in a localminimum, thereby rendering the algorithm unstable (i.e., the al-gorithm converges to a solution that is not optimal), as illustratedin Fig. 4(a). Unlike the algorithms whose solutions always startfrom an arbitrary point, the solutions obtained using the VLAdo not run the risk of falling into a local minimum because thestarting solution (obtained at the first step channel assignment)is always the best solution. Based on VLA operation, the initialbest solution is maintained at all subsequent steps of channel as-signments. From the VLA operation described in Section IV-A,

the solution of the first step channel assignment (i.e., the cellswithin a cluster that are assigned the first channel) are all se-lected if and only if the specified channel separation constraintsat the first step assignment are satisfied. Such a solution is con-sidered to be optimal with respect to first step channel assign-ment constraints. VLA offers an advantage that during subse-quent steps of channel assignment, only the cells meeting therequired constraints of the current step and those of the previoussteps are retained and redundant assignments are removed fromfurther consideration. In this way, the best solution (i.e., min-imum bandwidth) is maintained at subsequent steps of channelassignment [as illustrated in Fig. 4(b)], leading us to concludethat the VLA provides a stable solution (i.e., best solution at allthe steps of channel assignment).

D. Limitations of VLA

The first limitation of the proposed VLA is the storage re-quirement for storing the channels that have been assigned toeach cell during successive steps of channel assignment. For acluster with cells, the VLA requires no more thanstoragelocations. Assuming that the word size for each channel isbits, the maximum amount of storage requiredis given by

(9)

Equation (9) shows that storage requirement increases linearlywith cluster size . Recall from Table IV that bandwidth re-quirement is minimum at low value of cluster size. Hence,it is concluded that VLA performs better (in terms of band-width and memory requirements) at low cluster size comparedto large cluster size. A second limitation of the VLA iscomputational complexity. The computations performed at step

of channel assignment consists of subtractionsand comparisons, where . The firstterm of unity on the right-hand side represents the single sub-traction and comparison required to check the adjacent channelseparation constraint; the second term represents the EFFcalculations performed to check the nonadjacent channel sepa-ration constraint. Each EFF calculation consists of a subtrac-tion and a comparison. Fig. 5 illustrates the types of computa-tions (i.e., adjacency versus nonadjacency constraint checking)performed at each step of channel assignment. Note that evenwith the preceding computation requirements, the VLA still hasbetter real-time performance than the existing algorithms, aswill be discussed in a later section.

V. FURTHER NUMERICAL EXAMPLES

Two additional numerical examples of the application of theVLA are presented in this section. The first example is the so-lution of 19 benchmark problems on channel assignment pro-posed in the literature [7]. Our objective in the first example isto assess the quality of VLA solution against those of existingchannel assignment algorithms in the literature. The second ex-ample is solution of the channel assignment problem in a cel-lular radio network, with an objective of determining the band-width required to support a given traffic demand, without vio-lating adjacent and cocell separation constraints.


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

TABLE VICOMPARISON OFMINIMUM BANDWIDTH OBTAINED USING THE VITERBI-LIKE ALGORITHM AND THOSE OFPREVIOUS ALGORITHMS FOR THE TEN

BENCHMARK (GROUPI) PROBLEMS WITH FIXED DEMAND IN EACH CELL

TABLE VIICOMPARISON OFMINIMUM BANDWIDTH OBTAINED USING THEVITERBI-LIKE ALGORITHM AND THOSE OFPREVIOUSALGORITHMS FOR THENINE BENCHMARK

(GROUPII) PROBLEMSASSUMING UNIFORM DEMAND DISTRIBUTION IN EACH CELL

A. Solution of 19 Benchmark Problems Using the VLA

First, the 19 benchmark problems are classified into twogroups depending on whether the traffic demand is a fixednumber (Group I problems) or selected from a given probabilitydistribution (Group II problems). A description of the key fea-tures of Groups I and II problems is provided in Tables VI and

VII, respectively. The first nine problems in Table VI assumea cluster size of 21, but problem 10 assumes a cluster size of25. The values of the fixed traffic demandand compatibility matrix are as specified in[3], [7], and [12]. Table VI also presents the solutions obtainedfrom 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 previousresults, results obtained using the VLA are presented in thelast column of Table VI. It is observed from Table VI that theVLA meets the lower bound solution in 70% of the problemcases but provides slightly higher bandwidth (6% higher)than the lower bound in problems 4, 6, and 8. The quality ofthe solution provided by the VLA is due to the removal ofredundant assignments from further consideration and keepingonly the survivors at each step of channel assignment.

Next, the VLA is applied to solve the Group II problemswhose key features are specified in Table VII. Here, all nineproblems assume a seven-cell cluster structure, and the trafficdemand is assumed to be uniform over a specified interval (ll,ul), where ll and ul are the lower (minimum) and upper (max-imum) traffic demand, respectively [7]. Each problem listed inTable VII is unique in terms of the specified values for ll and uland the interference constraints and . Table VII presentsthe results obtained using two of the existing algorithms—thetwo-phase algorithm [8] and the LSA [7]. Also included forcomparison are the results obtained using the VLA. It is ob-served from Table VII that the solutions obtained by the VLAare always better than or match those of the two-phase algo-rithm. The solutions obtained using the VLA are comparable tothose achieved by the LSA, the difference between the LSA andVLA results being less than 8% and the LSA exhibiting a lowerbandwidth requirement. Note, however, that other factors, suchas computation complexity and execution time, must be consid-ered in reaching a conclusion on the better algorithm betweenthe LSA and VLA. As stated previously, VLA achieves its per-formance through sequential assignment of channels subject tomeeting adjacent and nonadjacent channel constraints.

Solution obtained by LSA begins by assuming an arbitraryinitial solution followed by exhaustive search for a bettersolution through interchange of already assigned channels. Of

Fig. 7. Normalized bandwidth required.

course, this channel interchange operation of LSA requiresintensive computation, which is much higher than that forVLA. For example, using a DEC workstation, a straightforwardimplementation of the local search algorithm without anyrun-time optimization takes, on the average, between 2–3 min,whereas the proposed Viterbi-like algorithm takes around 1–2min 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 thechannel assignment problem in a cellular radio network, takinginto account the effect of traffic demand ACS and CCS. A cel-lular network with a cluster size of seven is assumed. First, weassume a fixed traffic demand in each cell as shown in Fig. 6,


Fig. 8. Normalized bandwidth versus normalized demand range.

where the minimum and maximum demands are five and 18,respectively. The cells are ordered according to decreasing de-mand using node degree ordering, so that channel assignmentbegins with cell 1, using the VLA. In this example network, theperformance metric is the bandwidth required to support the as-sumed traffic demand, normalized by the bandwidth requiredto support the minimum demand of five calls. That is, normal-ized bandwidth is defined as the ratio of bandwidth required tosupport a given demand (selected in the range [5, 18] calls) tobandwidth required to support the minimum traffic demand offive calls.

Fig.7 shows a three-dimensional plot of normalized band-width (determined using the VLA) versus the cell index in thecluster versus traffic demand. Two operating regions are ob-served for the required bandwidth shown in Fig. 7. In the firstregion (traffic demand varying from one to five calls), the re-quired bandwidth in each cell increases linearly with traffic de-mand. Under high traffic demand range (from nine to 18 calls),the bandwidth required increases nonlinearly with demand andtends to a limiting value. An explanation for the linear and non-linear behavior of the bandwidth required versus demand is as

follows. First, when all the cells in a cluster have nonzero de-mand, the ACS constraint is the limiting factor causing a linearincrease in bandwidth at a rate equal to the product of clustersize and ACS constraint. When there are only few cells in thecluster with nonzero demand, ACS constraint exerts less influ-ence, resulting in a reduction in the rate of bandwidth increase.In the extreme case when only one single cell in the cluster hasdemand (i.e., cell 1 in Fig. 7), the CCS becomes the limitingfactor causing a nonlinear increase in bandwidth, a consequenceof CCS assuming a nonconstant value at subsequent steps ofchannel assignment.

Next, we investigate the effect of normalized demandrange on the required bandwidth. is defined as de-mand range divided by average demand or, mathematically,

ul ll ul ll . For this investigation, we assumethat traffic demand in each cell is selected from a uniformdistribution ll ul , where the minimum traffic demand is setto five calls and the value of the maximum demand is calculatedby

ulll

(10)


Fig. 9. Normalized bandwidth versus average demand.

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 thenormalized demand range . Although the scatter plot in Fig. 8is quite dispersed, the general trend is that the normalized band-width increases with normalized demand range. When the nor-malized demand range is small, a low bandwidth is often enoughbecause 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, re-sulting in an increase in bandwidth required. Fig. 9 shows a plotof the normalized bandwidth versus the average traffic demand( ul ll for the assumed uniform traffic distribution). It isseen that the normalized required bandwidth increases linearlywith the average traffic demand, as expected.

VI. CONCLUSION

The major focus of this paper is development of a heuristicalgorithm, called the Viterbi-like algorithm, that achieves theobjectives of minimum bandwidth requirement comparable toor better than the best performance reported in the literature andfaster execution time compared to those of previous algorithms.The basic idea of the VLA is sequential assignment of channelsto the cells in a cluster subject to meeting cocell and adjacentcell separation constraints. The VLA provides advantages ofminimum bandwidth requirement, stability of solution, and fastspeed of execution. However, the VLA has a limitation of highbandwidth and memory requirement at large value (e.g.,13) ofcluster size for which adjacent channel interference is minimum.Hence, a tradeoff exists between bandwidth/memory require-ment and tolerable level of adjacent channel interference. TheVLA, like the other existing channel assignment algorithms,also suffers from high computation requirement. According toour experimental results, we found that the computation require-ment of the VLA is less (by up to a factor of two) than those ofthe existing algorithms.

Performance of the VLA is assessed first by using the algo-rithm to solve 19 benchmark problems and then by studyingthe important characteristics of cellular radio networks. Resultsfrom the performance assessment show that the VLA provides

bandwidth requirements that closely match or are sometimesbetter than those obtained with the existing algorithms in the lit-erature. Minimum bandwidth is achieved by the VLA throughthe removal of redundant channel assignments (measured by ametric called excess frequency factor) from further considera-tion and keeping only the survivors after each step of channelassignment. A byproduct of retaining only the survivors is a re-duction in the number of comparisons performed at subsequentsteps of channel assignment; this translates to fast speed of exe-cution for the VLA. Furthermore, the solutions obtained by theVLA are stable because they always start from the best solutionthat 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 theVLA a viable candidate for implementation in cellular radio net-work planning.

ACKNOWLEDGMENT

The authors wish to thank the anonymous reviewers for theirdetailed comments and very useful suggestions that have led toimprovements in the content and presentation.

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Xavier N. Fernando received the B.Sc.Eng. degreein electrical and electronic engineering (first classhonors) from the University of Peradeniya, SriLanka, in 1992 and the master’s degree from theAsian Institute of Technology, Bangkok, Thailand, in1993. He received the Ph.D. degree at the Universityof Calgary, Calgary, AB, Canada.

He was an R&D Engineer for AT&T, Thailand,until 1997 and from 1997 to 2001, he was with TR-Labs and the University of Calgary. He is now an As-sistant Professor at Ryerson University, Toronto, ON,

Canada. His research interests include wireless communications, adaptive signalprocessing, nonlinear channels, and infrared communications.

Dr. Fernando has received scholarships from the Natural Science and Engi-neering Research Council (NSERC) of Canada, Alberta Informatics Circle ofResearch Excellence, and Finnish International Development Agency.

Abraham O. Fapojuwo (S’85–M’89–SM’92)received the B.Eng. degree (first class honors) fromthe University of Nigeria, Nsukka, in 1980 and theM.Sc. and Ph.D. degrees in electrical engineeringfrom the University of Calgary, AB, Canada, in 1986and 1989, respectively.

From 1990 to 1992, he was a Research Engineerwith NovAtel Communications, Ltd., where he per-formed numerous exploratory studies on the architec-tural definition and performance modeling of digitalcellular systems and personal communications sys-

tems. From 1992 to 2002, he was with Nortel Networks, where he conducted,led, and directed system-level performance modeling and analysis of wirelesscommunication networks and systems. In January 2002, he joined the Depart-ment of Electrical and Computer Engineering, University of Calgary, as an As-sociate Professor. He is also a Research Scientist at TRLabs, Calgary. His cur-rent research interests include resource management in wireless IP networks,quality-of-service in mobilead hocnetworks, and software reliability engi-neering.

Dr. Fapojuwo is a registered Professional Engineer in the Province of Alberta.

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A Viterbi-like algorithm with adaptive clustering for channel assignment in cellular radio networks