Improving the capacity in wireless networks through integrated channel base station and power assignment

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 2, MAY 1998 417

Improving the Capacity in Wireless NetworksThrough Integrated Channel Base

Station and Power AssignmentSymeon Papavassiliou,Member, IEEE,and Leandros Tassiulas,Member, IEEE

Abstract—The limited availability of radio frequency spectrumwill require future wireless systems to use more efficient andsophisticated resource allocation methods to increase networkcapacity. In this work, we propose a joint resource allocationalgorithm (JRAA) that makes the channel base station and powerassignment in a wireless network with an arbitrary number ofbase stations and mobiles attempting to minimize the numberof channels needed to provide each user in the system with anacceptable radio connection. We compare the performance of theJRAA for both the forward (downstream) and reverse (upstream)directions, in terms of the achievable traffic capacity, with somebounds on the performance of the maximum packing (MP), cliquepacking (CP), and reuse partitioning (RP) techniques, whichare usually used as benchmarks on the capacity that can beachieved by any traffic-adaptive dynamic channel assignmentstrategy, where the quality is guaranteed by the reuse distance.Those performance results verify the improvement that can beachieved by the integration of the channel base station and powerassignment. Finally, several versions of the two-way channelassignment problem are studied and evaluated.

Index Terms—Capacity management, integrated resource allo-cation, wireless networks.

I. INTRODUCTION

W ITH THE current trend toward digital systems thatare flexible and have increased processing capability

in mobile and base stations, sophisticated resource allocationmethods are an attractive way to improve network capacityand performance. The goal of an efficient resource allocationmethod is to guarantee the quality of service of the ongo-ing connections, while at the same time the available radiospectrum is used efficiently. The radio connection comprisesthree dimensions: the allocated channel (channel assignmentprocedure), allocated power (power assignment procedure),and allocated location (base-station assignment procedure).

In the existing analog system, the frequency channels ofeach cell are fixed and preallocated off line during the fre-quency planning phase. A call can be served only if the cellwhere it arises has a free channel, otherwise, it is lost. Inthe digital system of the near future, there will be increasedflexibility in radio network control. All the channels will poten-tially be available to all base stations, allocated in a dynamic

Manuscript received June 28, 1996; revised December 27, 1996.S. Papavassiliou is with AT&T New and Emerging Services Organization,

Holmdel, NJ 07733 USA.L. Tassiulas is with the Department of Electrical Engineering, University

of Maryland, College Park, MD 20742 USA.Publisher Item Identifier S 0018-9545(98)02484-0.

fashion. Usually, many users share a common channel, andone user’s transmission causes interference to the others. As aresult, transmitted powers should be controlled both to provideeach user an acceptable connection and to limit the interferenceseen by the other users sharing the same channel. Moreover,the potential flexibility on the way that mobiles are assigned tobase stations to communicate with is another factor that helpstoward the alleviation of frequency congestion.

The problem of joint channel base station and power as-signment is illustrated in Fig. 1. Clearly, the three problemsare interrelated. For certain channel allocations and base-station assignments, there may be power vectors that satisfythe interference constraints while for others not. Therefore,these problems need to be considered jointly. In this paper, weprovide an algorithm that makes the joint resource allocation ina wireless network (in the following, we refer to that algorithmas JRAA). Specifically, for every available channel in thesystem, the algorithm tries to allocate as many mobiles aspossible, according to some rule, making simultaneously thebase-station assignment and the power selection.

The channel assignment problem in cellular networks (whenthe powers and base stations are preassigned and fixed) hasbeen shown to be equivalent to a generalized graph-coloringproblem [10], [22], which is known to be NP hard. If wehave only cochannel interference, then the graph is obtained byrepresenting each cell by a vertex with an edge connecting twovertices if the involved cells are forbidden from using commonchannels. The problem is to assign channels (or colors) to thevertices of this graph such that adjacent vertices are assigneddisjoint sets of color. The assignment should use as fewchannels (colors) as possible. Since no efficient algorithm thatsolves this problem exists, many heuristic channel assignmentalgorithms with varying complexities have been suggestedand evaluated in the literature. In fixed channel assignmentstrategy [4], [6], [13], a set of nominal channels is permanentlyallocated to each cell, and an arriving call can only be servedby the nominally allocated channels. In dynamic channelassignment [3], [7], [22], [26], all the channels are kept ina central pool, and any channel can be used by any userand base station. However, a channel in one cell can onlybe reassigned simultaneously to another cell in the systemif the separation distance between the two cells is greaterthan a prespecified minimum distance to avoid cochannelinterference. Moreover, different channel borrowing strategiesand hybrid channel assignment methods which try to combine

0018–9545/98$10.00 1998 IEEE

418 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 2, MAY 1998

(a) (b)

(c)

Fig. 1. (a) A base station is selected by each mobile for its forward and reverse links. The two base stations need not be the same. (b) A channelis selected by each link. The cochannel links are represented by the same type of line. (c) The transmission powers of the cochannel links should beselected such that the interference constraints at each mobile are satisfied.

the advantages of the fixed and dynamic channel assignmenthave been proposed and simulated [12], [22].

In [25], the optimum transmitter power control problem forthe case of preassigned base stations and considering only onechannel is studied. A power control scheme that is optimumin the sense that it maximizes the number of simultaneousconnections in the system is discussed. The algorithm removescombinations of cells and computes the eigenvalues of eachreduced system until the requirement is fulfilled. Such asubmatrix problem is computationally hard, and at the present,it is not known if an algorithm exists to solve the problemin polynomial time. The above procedure was found to beimpractical since the computational complexity of the opti-mum procedure proposed is exponentially increasing with thesize of the cochannel set. However, the satisfiability problemis solvable [25], that is, for a given set of cochannel linkswhether there exists a feasible power vector that guarantees

ratios at all mobiles above a prespecified threshold. Ifnot, then the previous step by step removal algorithm needs tobe applied. It has been shown that the largest achievablelevel in such systems is reached by balancing the ’sof all users. Similar results of applying power balancing toboth the forward and reverse links of a spread spectrumcellular mobile radio system have been reported in [15].Grandhi et al. [8] and Zander [24] proposed synchronousdistributed power control algorithms that converge to theoptimal solutions with varying convergence rates. Foschini andMiljanic [5] presented a distributed power control algorithmthat avoids some of the difficulties that may occur in theaforementioned distributed algorithms due to the absolute

power settings problem by the inclusion of the receiver noisein the definition of the interference. In [14], Mitra extendedFoschini’s synchronous algorithm in allowing asynchrony be-tween the various mobiles and base stations reducing theneed for coordination. All those distributed power controlschemes that follow the balancing approach have thenice property of fast, geometric convergence if a feasiblepower vector exists, satisfying the requirements of allusers. However, if no such power vector exists, they producefluctuations of the powers, diverging to infinity. Moreover,during the updating the ’s of operational links may dropbelow their lower thresholds rendering the links unreliableand “killing” on-going transmissions. In [1], a distributedpower control algorithm that protects the operational links andprevents the dropping of on-going calls at the expense of somereduction in the system capacity and slower convergence, ispresented.

In [11] and [23], the combined problem of regulating trans-mitter powers and assigning users to base stations, assumingthat all users share one channel, is considered. Specifically,Hanly [11] proposed a spread spectrum cellular system inwhich cells are allowed to expand and contract enablinga user to transmit to the optimal base station based oncurrent interference levels and on the user’s particular locationwith respect to the cell sites. In [23], an algorithm thatrepresents an implementation of Hanly’s concept of expandingand contracting cells and always finds a power vector andbase-station assignment that provides acceptable connectionsfor all users, as long as such a feasible solution exists, isprovided. When a feasible solution does not exist, the primary

PAPAVASSILIOU AND TASSIULAS: IMPROVING THE CAPACITY IN WIRELESS NETWORKS 419

issue becomes how to maximize the number of acceptableconnections, and in that case, the algorithm does not providegood results since it is possible even for all users to endup with unacceptable ’s. In [2], several algorithms forjoint power control and channel assignment with varyingcomplexities, assuming that the assignment of users to basestations is fixed or prespecified by outside means, are proposedand compared.

In [18] and [19], the joint optimal resource allocationproblem has been defined and formulated, and an algorithmthat solves it for two base stations and arbitrary numberof mobiles has been proposed. The algorithm identifies allthe possible pairs of mobiles that can share a channel, and,therefore, the problem is reduced to the solution of a maximummatching on the appropriate compatibility graph. In the generalcase of an arbitrary number of base stations, the same stepscould be followed, but instead of identifying pairs of mobilesthat can share a channel we should identify all the possiblesubsets of mobiles that can make use of the same channel.Then, we have to select those subsets that correspond to theuse of the minimum number of channels in the system, as wellas to assign base stations and select powers. As the numberof base stations and mobiles increases, such a procedurebecomes intractable. Therefore, some heuristic algorithms thatapproximate the optimal assignment should be devised as theycould provide useful bounds on the performance of the system.

In this work, we propose a heuristic algorithm (JRAA) thatmakes the channel, base station, and power assignment in awireless network with an arbitrary number of base stations andmobiles, trying to minimize the number of channels needed toaccommodate all the calls. The only quality of service criterionused here, in regard with the acceptance or rejection of a newincoming call, is to satisfy the carrier to interference ratiorequirements at all the active mobiles and base stations in thesystem by maintaining the possible cochannel interference atacceptable level. By minimizing the number of channels in useat the current state, the system is better prepared to accept newincoming calls in the future and therefore adapts itself betterto the new traffic requirements. We compare the performanceresults of the algorithm we propose with the correspondingresults of different traffic-adaptive channel allocation strate-gies, based on the techniques of maximum packing (MP),clique packing (CP) [21], and reuse partitioning (RP) [17].The performance evaluation of the different resource allocationstrategies is done in terms of the achievable traffic capacitiesappropriately defined. Let be the minimum numberof channels needed to establish a forward (reverse) connectionto mobiles, randomly and uniformly distributed. The trafficcapacity in the forward direction is defined as

(1)

and similarly for the reverse direction. We obtain the trafficcapacities in the forward and reverse directions, under theapplication of JRAA algorithm, by simulation for both thecases of fixed powers and controllable powers, and we studythe effect of power control on the system capacity. Moreover,we observe a considerable improvement on the traffic capacity

that is achieved by our algorithm over what can be achievedby many different fixed and dynamic channel assignmentstrategies, due to the joint consideration of the channel, basestation and power assignment problem. The problem of two-way channel assignment is also studied and similar results areobtained. It should be noted that the JRAA is a centralizedalgorithm, and, therefore, the amount of data that would haveto be managed by the central controller would be enormous,especially in a large network. The JRAA mainly aims toquantify the limits on the gains that can be achieved throughjoint channel base station and power assignment. Future re-search efforts should be devoted to developing other heuristicalgorithms that could be operated in a more distributed fashionand, therefore, could be more easily implemented in real time.

The paper is organized as follows. In Section II, we providethe model as well as the main assumptions that we are going touse in our simulations. In Section III-A, we describe the jointresource allocation algorithm (JRAA) we propose. Section III-B contains the power assignment procedure that we use in ouralgorithm, and in Section III-C, we discuss the complexity ofthe proposed algorithm, while in Section III-D we describehow the resource allocation algorithm can be applied in thedifferent versions of the two-way channel assignment problem.In Section IV, we compare the performance of the JRAAwith the performance of different traffic-adaptive dynamicchannel assignment strategies. Specifically, in Section IV-A,we provide some bounds on the performance of the MP,CP, and RP techniques, which are used in the literature astraditional benchmarks on the traffic capacity achieved byany traffic-adaptive dynamic channel assignment strategy incellular networks, where the quality is guaranteed by the reusedistance. In Section IV-B, we present some simulation resultsthat indicate the improvement on traffic capacity achieved bythe joint channel base station and power assignment.

II. SYSTEM MODEL

The formulation of the resource allocation problem is sim-ilar to the one described in detail in [18], but for the sake ofcompleteness we present it briefly here too. There arebasestations and mobiles at arbitrary locations with respect tothe base stations. The path-loss coefficients between anybase station and mobile are provided. They characterizecompletely the propagation properties of the system in thesense that when transmits power , receives power

. Throughout our simulations, we assume that the signalstrength decreases inversely proportional to, where bywe denote the distance between transmitterand receiver. Moreover, we prohibit the reuse of the same channel at

the same base station. The channels may be either frequencybands in a frequency-division multiple-access (FDMA) systemor carrier and a time slot in a time-division multiple-access(TDMA) system or different codes in a code-division multiple-access (CDMA) system. We denote by the transmittedpower from base stationin forward channel. Similarly,denotes the transmitted power from mobilein the reversechannel . Cochannel interference is the prevailing interferencetype—this is equal to in mobile receiving


from base station at channel . The carrier to interferenceratio at mobile in channel is equal to

(2)

The interference constraint at mobilethat receives on channelis satisfied if

(3)

where is a threshold imposed by physical layer’s constraints.Similarly, the carrier to interference ratio at basestation in channel is equal to

(4)

The reverse radio link from mobile to base station atchannel satisfies the interference constraints if

(5)

In this paper, we consider a snapshot during the dynamicoperation of the system. The same problem should be solvedcontinuously as the system state and parameters change withthe evolution of the system (call arrivals, departures, mobility,handoffs, fading, etc.). The static problem considered heremay be related to the dynamic situation as follows: considerthese resource allocation algorithms that permit reassignmentof all calls at every change in the configuration of users. Thesolution of the static problem gives the “best” assignment ateach step. Certainly such algorithms yield higher capacity thanalgorithms that do not reassign existing users or do limitedreassignments. The study of the static problem aims to quantifythe limits of the gains achievable through joint power control,channel, and base-station assignment.

III. JOINT RESOURCEALLOCATION ALGORITHM

In the following, we describe an algorithm (JRAA) forthe joint channel, base station, and power assignment in ageneral network with mobiles and base stations tryingto minimize the number of channels needed to accommodateall the calls. The main idea of the algorithm is the following:for every available channel in the system, allocate as manymobiles as possible making simultaneously the base-stationassignment and power selection, if the latter is allowed (whenthe powers are controllable parameters). First, we consider thecase of unidirectional channels, that is, each channel can beused in only one direction (either as forward or as reversechannel), and we describe the application of the resourceallocation algorithm in one direction (forward or reverse).Later on, we will consider several versions of the two-waychannel assignment problem including the possibility of usingbidirectional channels.

A. Description of the Algorithm

Before proceeding with the presentation of the various stepsof the algorithm, we describe in detail the major parame-ters and the heuristic rules that are involved in the JRAAimplementation.

Each mobile is usually associated with its closest basestation, but the algorithm gives the flexibility to a mobile to getconnected to another neighboring base station instead, if thisis preferable. Such a flexibility on the choice of base stationshelps the base stations with many mobiles close to them to getuncongested by assigning some mobiles to neighboring basestations with lower traffic density in their coverage area. Thisis achieved by keeping a list of base-station preferencefor every mobile for the channel under consideration, thatdictates the order of the base stations that a mobile shouldget associated with. Actually, is an ordered list where theordering is on the path gain between mobileand each one ofthe available base stations for the channel under consideration.With such an ordering, initially the first element in the list ofmobile is the base station for which is the maximumamong all , that is, the base stationclosest to mobile .

At each step of the algorithm, for every unassigned mobile, we construct a tradeoff for connecting mobile with the

“nearest” available (first element in list ) base station (callit ). In this tradeoff, we take into consideration a measureof the interference caused to the receiver of this tentativeconnection by the other transmitters already using the samechannel as well as a measure of the interference that thiscandidate connection (mobile with base station ) wouldcause to the other already active links. Therefore, we consider

where

where by we denote the set of base stations alreadytransmitting on channel and denotes the set of mobilesalready using channel. Among all the unassigned mobiles,we want to choose as the next candidate connection thatone which will cause the smallest interference in the system.That connection is indicated by the maximum among all theavailable ’s.

Then, we check if the new configuration, old active linksplus the new proposed connection, satisfies the thresholdconstraints. If we assume that all base stations and mobilesuse the same transmission powers, equal to, then the checkcan be based on (2) and (3) for every mobilereceivingon forward channel and on (4) and (5) for every basestation receiving on reverse channel. If power control isallowed, then we find those powers, if there exist, that satisfythe threshold constraints based on some results described in[24], which will be discussed in detail in next subsection.If the constraints are satisfied, then we accept the proposedconnection and try to find the next candidate connection that


possibly can make use of the same channel, repeating thesame procedure. If not, then we reject the proposed connectionand try to find another possible candidate connection amongthe remaining unassigned mobiles. In the latter case, wecan either repeat this procedure until no candidate mobilethat could probably be assigned to this channel exists orlimit the repetition of the aforementioned procedure to aprespecified number ( ). At this point, by limitingthe number of candidate mobiles for assignment on a chan-nel, we decrease the running time of the algorithm withoutaffecting substantially the capacity of the system since if theaddition of the first candidate mobile in the system violatesthe threshold constraints, then in accordance to the way thatthe candidate connections are chosen, it is unlikely that thereexists another possible connection to be accommodated onthat channel.

Then, we choose a new free channel and repeat the aboveprocedure for the remaining unassigned mobiles until all ofthem are accommodated. In the following, we outline the mainsteps of the algorithm.

1) Find the first available free channel.2) For each unassigned mobile, create its preference list

.3) For each unassigned location, construct the tradeoff .4) Find the maximum among all the tradeoffs’s. The

connection indicated by this maximum (mobile, basestation ) is the next candidate connection.

5) Check if the new configuration (old active links and newproposed connection) satisfies the threshold constraints.

a) If all constraints are satisfied, mark locationas “assigned” and delete base station fromthe preference lists of all the remaining unas-signed mobiles. Go to Step 3). If power controlis allowed, then specify the powers that satisfythe constraints of all the mobiles assigned tochannel according to the method described inthe following section.

b) If some constraint is violated, then reject thatcandidate connection and go to Step 4). Repeatthis procedure either until you exhaust all thepossible candidate connections or until you checkthe first candidates.

6) Choose a new free channel and repeat from Step 2) untilall the mobiles are assigned.

Note that in the end of the algorithm’s operation, all mobilesare assigned. The performance criterion is how many channelshave been used by this assignment.

B. Power Assignment Procedure

In this section, we describe how the power assignmentis done when the power is a controllable parameter. In thecontext of the algorithm described in Section III-A the powerassignment procedure is invoked every time that we have a setof mobiles associated with base stations, and we want to findif there exist transmitted powers such that all those links canbe active simultaneously. If this is possible, we want to find a

transmission power vector, denoted by, such that in a givenchannel the ratios at all receivers using that channel arelarger than a prespecified threshold. The power assignmentscheme is described here for the forward link and is based onsome results presented in more detail in [25]. For the reverselink, a similar procedure can be followed.

In the following, we consider a configuration ofmobiles,already associated with base stations. For simplicity (withoutloss of generality), we assume that themobiles are numberedfrom one to , while we number the base stations such thatmobile is associated with base station. Note that at everystep of the algorithm, every mobile, , is associatedwith a base station , , where for ,and, therefore, the above assumption is always valid.

First, we define the link gain matrix . At somegiven instant with , mobiles will be a square matrix.Now rewriting relation (3) for in a matrix form,we get

(6)

where

and (7)

A carrier to interference ratio is defined to be achievable inthe cochannel set of themobiles if there exists a power vector

such that for all mobiles using this channel.An important result [25] is that the largest achievableratio is related to the spectral properties of matrixas

(8)

where is the largest real eigenvalue of matrix. Thepower vector achieving this maximum was found to bethe eigenvector of corresponding to the eigenvalue.

Therefore, the problem of checking if there exist transmittedpowers such that the mobiles under consideration can use thesame channel satisfying the threshold constraints is reducedto finding the largest eigenvalue of the correspondingmatrix. The complexity of such a procedure applied to aconfiguration with mobiles associated with the correspondingbase stations is [20]. Then, we can calculate themaximum achievable according to (8). If , thenthe powers of the mobiles trying to use the same channelare assigned according to . If , then there is notpositive power vector such that themobiles belonging to thecochannel set of channelcan use the same channel.

C. Complexity

For large networks with a large number of base stations, the most time-consuming and complex part of the al-

gorithm is the one that checks if, after the selection of thenext candidate connection, the new configuration satisfies thethreshold constraints in the system [Step 5) of the algorithm]and does the power assignment to the base stations and mobilethat are active on the channel under consideration. By takinginto consideration the power control procedure that involves


the identification of the largest eigenvalue of the matrix(defined in the previous subsection) and the correspondingeigenvector, the complexity of the algorithm turns out to be ofthe order . In extreme cases, where we may have asmall number of base stations and a large number of mobilesand/or a very nonuniform distribution of traffic to the differentbase stations, the most time-consuming part of the algorithmwill be the one of identifying the measuress and the nextcandidate connection [Steps 3) and 4) of the algorithm] andthe complexity of the algorithm may turn out to be in worstcase of the order .

D. Two-Way Channel Assignment Procedure

There are several versions of the two-way channel assign-ment problem depending on the constraints we pose. All theanalog cellular systems, which are based on FDMA, employfrequency-division duplexing (FDD), and, therefore, they usedifferent bands for the establishment of the forward and reverseconnections. Hence, in those systems a channel is prespecifiedas forward or reverse and it can be used only in one direction.Similarly, all the existing digital cellular systems [i.e., globalsystem for mobile communications (GSM) in Europe, IS-54and IS-95 in North America, and personal digital cellular(PDC) in Japan], independently of the multiple-access methodthat they use (FDMA/TDMA/CDMA), employ the FDD, and,therefore, they use unidirectional channels too [16]. This ismainly due to the fact that most of the existing digital cellularsystems operate in the frequency bands originally allocated foranalog systems, and in order to allow smooth transition fromthe analog systems to the digital ones, the FDD technique isemployed for the digital cellular systems.

This is certainly not a physical constraint since in principalthe same channel can be used in both directions simultaneouslyin sufficiently spatially separated locations. In the future digitalsystems, this constraint can be eliminated, that is, the samechannel may be used for a forward connection by somemobiles and for a reverse connection by some others. Byallowing a channel to be used in both directions (bidirectionalchannels) rather arbitrary frequency bands, it can be allocated(frequency-duplexed systems need symmetric pairs of bandsseparated by several megahertz), and the system is moreflexible to adjust to traffic asymmetries that may occur. Intelephony, the traffic requirements are usually symmetrical inboth directions of communication, while in data communi-cations a high level of asymmetry may occur in the trafficgenerated in the forward and reverse connections (i.e., duringdata base retrieval, file transfer). The flexibility of usinga channel for a forward or reverse connection on demandresults in a dynamic allocation of the available radio frequencyspectrum to the forward and reverse connections, therefore,allowing the systems to adapt themselves to various trafficsituations. Most of the digital cordless systems used today(CT2 common air interface, DECT, and PHS) employ thetime-division duplexing (TDD) [9], [16], thus, simplifyingfrequency planning.

Moreover, another constraint, usually imposed, that affectsthe system capacity is that a mobile should communicate with

the same base station in both directions. If we ignore thepractical constraints imposed in specific systems, each mobilemay use any channel and base station for each one of thereverse and forward connections.

In the following, we are going to describe how the afore-mentioned resource allocation algorithm can be applied in thedifferent versions of the two-way channel assignment problem.Initially, we assume that each channel can be used in onedirection only (either as forward or as reverse channel) andthat each mobile may use any base station for the forwardand reverse connection. Then, the two-way channel assignmentproblem is reduced to the solution of the forward and reversechannel assignment problems separately, and, therefore, thetotal number of channels needed to accommodate all themobiles in the system is equal to the sum of the forward andreverse channels.

If we impose the constraint that each mobile should com-municate with the same base station in both directions, thenthe same algorithm can be applied, but each time we considera pair of channel, one forward channel and one reversechannel . A new candidate connection is accepted if thenew configuration (old active links and the new proposedconnection) satisfy the threshold constraints in both directions.Then, mobile communicates with base station onin the forward direction and on channel in the reversedirection.

In the following, let us consider the case where a channelcan be used in both directions and each mobile may usedifferent base stations for the forward and reverse connection.In this case, for every physical mobile we consider two virtualmobiles: one for the forward link and one for the reverselink. Therefore, in a system with physical mobiles weconsider different virtual mobiles. The modifications inthe algorithm proposed in Section III-A are the following.For every unassigned virtual mobile, we construct thetradeoff for making the connection of virtual mobilewiththe “nearest” available base station. These tradeoffs aredifferent for the two different virtual mobiles that correspondto the same physical mobile. In those tradeoffs, we take intoconsideration a measure of the interference caused to thereceiver of the candidate connection by the other transmitterson this channel (those transmitters can be both base stationsand mobiles) as well as a measure of the interference causedby the transmitter of the new candidate connection to theother receivers already receiving on this channel. For a virtualmobile that corresponds to a forward connection, we have

where

where denotes the path-loss coefficients between anytwo mobiles (base stations)and , denotes the set of base


stations already using channelin the forward direction,denotes the set of mobiles already using the channelin thereverse direction, denotes the set of mobiles already usingthe channel in the forward direction, and denotes the setof base stations already using channelin reverse direction.

Similarly, for a virtual mobile that corresponds to a reverseconnection, we have

where

After the new candidate connection is selected, we check if thenew configuration, old active links plus the new proposed con-nection, satisfies the threshold constraints at all the receivers.In the case of fixed powers, the check can be based on thecorresponding expressions that give the ratios at themobiles and the base stations when we consider bidirectionallinks. In the case of power control, the power assignmentprocedure described in Section III-B can also be applied here.In fact, at any given instant of virtual mobiles, we have

transmitters (these transmitters could be both mobiles andbase stations) associated withreceivers, and we want tofind a power vector for the transmitters that satisfies the

ratios at all the receivers. Following the same steps asin Section III-B, we conclude that the power vector (if thereexist such a vector) which satisfies all the ratios at thereceivers should be such that

(9)

where and is the path-loss coefficient betweentransmitter and receiver normalized by the path-losscoefficient between transmitter and receiver . The sameresults presented in Section III-B hold here too, and, therefore,the problem is reduced again to finding the largest eigenvalueof matrix .

IV. PERFORMANCE EVALUATION

In this section, we compare the performance of JRAA, interms of traffic capacity, with the corresponding performanceof a conventional cellular system with fixed channel assign-ment (FCA), where all the channels are allocated followingthe same reuse pattern as well as with the performanceof traffic-adaptive dynamic channel assignment algorithms,which adapt the channels to the number of active users ineach cell in the system. We obtain the traffic capacities inthe forward and reverse directions, under the application ofJRAA algorithm, by simulation for both the cases of fixedpowers and controllable powers, and we study the effect ofpower control on the system capacity. Moreover, we observea considerable improvement on the traffic capacity that isachieved by our algorithm over what can be achieved by many

different fixed and dynamic channel assignment strategies, dueto the joint consideration of the channel, base station, andpower assignment problem. The problem of two-way channelassignment is also studied and similar results are obtained.

In Section IV-A, we discuss some bounds on the per-formance of the MP and CP techniques, which are usedin the literature as traditional benchmarks on the minimumnumber of channels required to carry all calls, given thenumber of calls of all cells in a conventional cellular systemwith fixed powers, under any traffic-adaptive dynamic channelassignment scheme, where the quality is guaranteed by thereuse distance. We also combine the MP and CP strategieswith the RP concept in order to achieve better performancebounds. In Section IV-B, we compare by simulation JRAAwith different traffic-adaptive dynamic channel assignmentschemes.

A. Maximum Packing, Clique Packing, and Reuse Partitioning

1) Maximum Packing:We consider that the whole cover-age area in the system is divided into cells, where acell defines the area covered by the corresponding basestation . Therefore, we can assume a cellularcover . The occupancy of the systemis described by a vector , where isthe number of active calls in cell . Assume that in order tosatisfy the constraint (3) for all mobiles in the system, we mustuse a reuse factor at least equal to. MP is the method thataccepts every call to which a channel can be assigned withoutviolating the reuse constraints, regardless of the number ofreassignments involved [21]. A particular occupancy vector

is said to be admissible under MP if and only if thereis an allocation of channels to cell , ,which respects the reuse constraints. Denote by the setof admissible vectors.

A set of cells that all interfere with each other is called aclique. Denote by the set of all cliques in the network.Obviously, for calls generated in cells that belong to the sameclique, we must assign distinct channels. Let us denote by

the minimum number of channels needed to satisfyall the reuse constraints for a given occupancy vectorunderthe MP policy. Then, satisfies the following:

(10)

In a linear network with a reuse factor equal to , achannel can be used simultaneously by two cells if theyare separated by cells such that ( -cellseparation rule). For a linear network, it is known that

,and .

2) Clique Packing: Consider again the cellular networkand assume that there areavailable channels in the system.Define . In general,

. The CP strategy accepts a new call in cellifand only if the network state at the time of arrival issuch that , where denotes the unit coordinatevector of length containing a “one” at the th position


Fig. 2. One-way channel assignment for a linear system with 12 base stations,(C=I) � 18 dB, and reuse factorsN1 = 1 andN2 = 2.

and zeros elsewhere [21]. For a given occupancy vector, theminimum necessary number of channels under the CP strategyis given by the expression on the right-hand side of (10). Inlinear networks, every state achieved by CP is also achievedby MP [21]. That is, for linear networks . But,in general, CP violates the reuse constraints and thereforehas no practical application. However, if the probability ofstates is sufficiently small, it can serve as anapproximation to MP.

3) Reuse Partitioning and Maximum Packing (RPMP):Inthis section, we combine the concepts of MP and RP in orderto provide a more efficient spectrum reuse in cellular systems.

In the following, we assume that we usedifferent reusefactors, denoted by . Then, a cellular coveris denoted by , whereby we denote the zoneof cell , , and

. The occupancy vector of the system is describedby a vector , where bywe denote the number of active calls in zoneof cell . Aparticular vector is said to be admissible if and only if there isan allocation of channels to zone of cell , ,and , which respects the reuse constraints in allzones. We denote by the set of all admissible vectorsin network under the MP policy. Let us also denote bythe set of all cliques in the network . Obviously, for callsgenerated in zones (subcells) belonging to the same clique, wemust assign distinct channels. Let us denote by theminimum number of channels needed to satisfy all the reuseconstraints for a given occupancy vectorunder the MP policyin a network with different reuse factors. Then,

satisfies the following: .4) Reuse Partitioning and Clique Packing (RPCP):In this

section, we apply the CP strategy in a reuse partitioned systemwith zones. Consider again the cellular network andassume that there are available channels in the system.Define . In general,

. The RPCP strategy accepts a new callin subcell if and only if the network stateat the time of arrival is such that ,where denotes the unit coordinate vector of length ( )containing a “one” at theth position and zero’s elsewhere. In

general, RPCP violates the reuse constraints. However, if theprobability of states is sufficiently small, itcan serve as an approximation to RPMP.

B. Numerical Results

We are going to apply these allocation strategies on a linearsystem with a threshold of 18 dB and two different reusefactors of one and two as well as on a planar system with a

threshold of 18 dB and two reuse factors of one andseven. In the linear system, the base stations are uniformlyplaced on the line and the distance between two successivebase stations is assumed to be equal to one unit, while for theplanar system, we consider 49 base stations placed in suchpositions that the whole structure resembles a 77 cellularlay out. We approximate the performance of MP and RPMPby the CP and RPCP strategies, respectively, since they give,in general, an upper bound of the traffic capacities that can beachieved by MP and RPMP, as we explained in the previoussections.

In Fig. 2, we present the capacities (number of mobilesper channel) of the forward and reverse channels versus thenumber of mobiles that are randomly and uniformly distributedon a linear system with 12 base stations.

For the FCA, CP, and RPCP strategies, the forward andreverse channel assignment are exactly equivalent, and, there-fore, they are represented by the same lines in the graph.However, for the JRAA we simulated separately the forwardand reverse channel assignment for both the cases of powercontrol and nonpower control. As we see from Fig. 2, in anycase, the forward and reverse channel capacities are very closeto each other. This does not mean that the forward and reversechannel assignments are necessarily the same, but it means thaton the average we need about the same number of channelsfor the forward and reverse connections in the system.

As we see from Fig. 2, CP strategy behaves much betterthan FCA resulting to higher channel capacities. This wasexpected since CP is a dynamic channel assignment schemethat accepts every call to which a channel can be assignedwithout violating the reuse constraints regardless of the num-ber of reassignments involved. Moreover, we confirm the largeimprovements on the channel capacity that can be achievedby the use of the RP concept. The increase on the channel


Fig. 3. One-way channel assignment for a planar system with 49 base stations,(C=I) � 18 dB, and reuse factorsN1 = 1 andN2 = 7.

capacity achieved by RPCP over the one that correspondsto the CP strategy is about 30%. The JRAA outperformsall these strategies resulting, for the case of fixed powers,in an increase on the channel capacity of about 10% overthe maximum channel capacity that may be achieved by theRPCP strategy. Furthermore, the expected number of channelswhen the powers are controllable parameters is lower than thecorresponding value for the case of fixed powers resulting toan extra increase of 5%–10% on the corresponding channelcapacity. Similar qualitative results can be observed in Fig. 3,where we plot the capacities of the forward and reversechannels versus the number of mobiles that are randomlyand uniformly distributed on a planar system with 49 basestations.

In this case, the traffic that corresponds to the inner zoneof the cell, and, therefore, the assignment of those mobilesis done according to reuse factor of one, represents only asmall portion of the total traffic (actually, only 5% of the totaltraffic as calculated in [17]). This explains the fact that theimprovement on the channel capacity achieved by RPCP overthat achieved by CP is only 5%. In [17], we have seen thatby choosing different reuse factors we may achieve highertraffic capacities. However, as we see from Fig. 3 the JRAAimproves the channel capacity by 100% over the one thatcan be achieved by RPCP with reuse factors of one andseven.

The way that JRAA works resembles, in some sense, theRP concept. By assigning each time to the channel underconsideration the unassigned mobile with the maximum,which is a function of the received desired signal as well asof the interference from the other transmitters to the candidatereceiver and to the other receivers by the candidate transmitter,we result to an allocation such that: the mobiles close tothe base stations are assigned a channel which is reusedvery densely, while the mobiles that are far from the basestations use channels that are reused more sparsely. Thisallocation is very similar to the one achieved by strategiesbased on the RP concept (i.e., RPCP). However, the JRAAoutperforms the RPCP because the JRAA gives the flexibilityto a mobile to get connected to another neighboring basestation instead of getting associated with the closest basestation. Therefore, we allow base stations with many mobiles

TABLE I

close to them to get uncongested by assigning some of themobiles to neighboring base stations with lower traffic densityaround them. The feature of “base-station selection” combinedwith the power control feature which is an integral part ofthe JRAA further improves the performance of the systemespecially in the cases of nonuniform traffic patterns. Next,we apply the JRAA on a linear system with 25 base stationsand a threshold of 14 dB in order to gain someinsight on how the “base-station selection” feature combinedwith the “power control” feature of the JRAA results in adecrease on the necessary number of channels. We consider

mobiles distributed as follows: /5 mobiles are randomlyand uniformly distributed within a unit length around eachone of the base stations 3, 8, 13, 18, and 23. In Table I,we present the number of forward channels under the RPCPstrategy (denoted by ) and under the JRAA (denotedby ) as well as the number of mobiles that are notassociated with the closest base station during the operationof the JRAA. The following results are averages over tenexperiment repetitions.

In the following graph, we present the channel capacity forseveral versions of the two-way channel assignment mentionedin Section III-D versus the number of mobiles distributedrandomly in a linear system with 12 base stations. Thecorresponding graph for a planar system with 49 base stationsis depicted in Fig. 5.

The lowest curve in the graph (labeled by jointnopower)corresponds to the case that each channel can be used in oneonly direction, and a mobile must use the same base stationfor the forward and reverse connections under the assumptionof fixed powers, while the curve labeled by jointpowerrepresents the corresponding case when the powers are con-trollable parameters. The curves labeled by jointfree nopowerand joint free power reflect the channel capacities for the


Fig. 4. Linear system with 12 base stations,(C=I) � 18 dB, and reuse factorsN1 = 1 andN2 = 2.

Fig. 5. Planar system with 49 base stations,(C=I) � 18 dB, and reuse factorsN1 = 1 andN2 = 7.

nonpower and power control cases, respectively, under theassumption that each mobile uses any base station for itsreverse and forward connection. The remaining two curves,denoted by jointbinopower and jointbipower, correspond tothe cases that we use bidirectional channels. The differenceson the performance of the systems represented in the lattergraph are due to the different constraints that we have posedin each of these systems. From Figs. 4 and 5, we confirmthat the power control increases the channel capacity by 10%(20%) when we use unidirectional links and by 5% (15%) if weconsider bidirectional links. Moreover, the use of bidirectionallinks by itself results in an extra increase of up to 10% on thechannel capacity compared to the one that can be achieved byusing unidirectional links.

V. DISCUSSION

The problem of joint channel base station and power as-signment in a wireless network with an arbitrary number ofmobiles and base stations has been considered here. In general,the optimal resource allocation in such a network is a hardoptimization problem. A procedure similar to the one describedin [18] that achieves the optimal assignment for a system withtwo base stations, in the sense that minimizes the number ofchannels needed to establish communication for all mobiles,becomes intractable as the number of mobiles and base stationsincreases.

In this paper, we have proposed a JRAA that makes thechannel base station and power assignment in a generalwireless network. The algorithm allocates as many mobiles aspossible to every available channel in the system, according tosome heuristic rules, making simultaneously the base stationassignment and power selection. The traffic capacities for theforward and reverse channel, as well as for the several ver-sions of the two-way channel assignment, have been obtainedthrough simulation. The corresponding numerical results forthe performance of the JRAA indicate that with the integrationof channel base station and power assignment, significantimprovements on the traffic capacity are obtained over whatcan be achieved by dynamic channel allocation strategies.

The JRAA considers every time the static problem anddepends on the full knowledge of the gain (loss) in all prop-agation paths, both intended base-mobile paths and unwantedinterference paths. Estimating these gains would require a sig-nificant measurement effort, and the amount of data that wouldhave to be managed by the central controller would be enor-mous in a large network. Therefore, the JRAA mainly aims toquantify the limits on the gains that can be achieved throughjoint power control and channel and base-station assignment.Further studies should be devoted to developing algorithmsthat use only limited path gain information and could beoperated in a distributed fashion based on information thatcan be derived from local measurements in every base station.


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Symeon Papavassiliou(S’92–M’96) was born inAthens, Greece, in December 1967. He received thediploma in electrical engineering from the NationalTechnical University of Athens, Athens, Greece, in1990 and the M.S. and Ph.D. degrees in electricalengineering from Polytechnic University, Brooklyn,NY, in 1992 and 1995, respectively.

While at Polytechnic University, he was aResearch Fellow in the Center for AdvancedTechnology in Telecommunications (CATT). InAugust 1995, he joined the AT&T Bell Laboratories,

Holmdel, NJ, where he is currently a Senior Technical Staff Member in theAT&T New and Emerging Services Organization. Since June 1996, he hasalso been an Industry Professor at the Electrical Engineering Department,Polytechnic University. His main research interests lie in the mobile radiocommunication system design, network operations and capacity management,and optimization of stochastic systems.

Dr. Papavassiliou received the Best Paper Award in INFOCOM’94. He isa Member of the Technical Chamber of Greece.

Leandros Tassiulas(S’89–M’91) was born in 1965in Katerini, Greece. He received the diploma in elec-trical engineering from the Aristotelian Universityof Thessaloniki, Thessaloniki, Greece, in 1987 andthe M.S. and Ph.D. degrees in electrical engineeringfrom the University of Maryland, College Park, in1989 and 1991, respectively.

From 1991 to 1995, he was an Assistant Pro-fessor in the Department of Electrical Engineering,Polytechnic University, Brooklyn, NY. In 1995, hejoined the Department of Electrical Engineering,

University of Maryland, where he is now an Associate Professor. He holds ajoint appointment with the Institute for Systems Research and is a Member ofthe Center for Satellite and Hybrid Communication Networks, established byNASA. His research interests are in the fields of computer and communicationnetworks with emphasis on wireless communications (terrestrial and satellitesystems) and high-speed network architectures and management, control andoptimization of stochastic systems, and parallel and distributed processing.

Dr. Tassiulas received a National Science Foundation (NSF) ResearchInitiation Award in 1992, an NSF Faculty Early Career Development Awardin 1995, and an Office of Naval Research Young Investigator Award in 1997.He received the INFOCOM’94 Best Paper Award for coauthoring a paper.

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Improving the capacity in wireless networks through integrated channel base station and power assignment