Bridging the gap between dynamic and static methods for cell planning

1224 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 5, SEPTEMBER 2001

Bridging the Gap Between Dynamic and StaticMethods for Cell Planning

Q. T. Zhang, Senior Member, IEEE

Abstract—Frequency reuse distance is a crucial parameter incellular system design that determines the service quality, which isusually measured in terms of the signal-to-interference ratio (SIR).The instantaneous SIR is a random variable; theoretically, its char-acterization is completely provided by the outage probability. Incellular engineering, however, frequency reuse planning is usuallybased on a simple static design, without taking into account the dy-namic propagation effects. The static method has the advantage ofsimplicity, but failing to indicate the system dynamic performance.In this paper, we determine a simple mathematical expression be-tween the static and dynamic SIRs whereby a new cell planningtechnique is developed. The new technique makes it possible toincorporate dynamic considerations into a static design therebybridging the gap between the two methods. We also use the newresults to study the system capacity, revealing the dependence ofthe system capacity on the minimum required outage probability.

Index Terms—Dynamic cell planning, geographical model,outage probability, static cell planning, system capacity.

I. INTRODUCTION

T HE basic idea of a cellular system is frequency reuse overdifferent cells to improve spectral efficiency. The increase

in system capacity is ultimately limited by the cochannelinterference. Therefore, reuse distance is a crucial parameterin the system design and should be chosen so as to meet thevoice quality objective which is usually measured in termsof signal-to-interference ratio (SIR). Theoretically, the SIRat a moving vehicle is a random variable, and its completedescription is provided by the outage probability. The designermust, therefore, ensure that the SIR be greater than a specifiedthreshold with a sufficiently large probability. The outageperformance of a cellular system under various operationalenvironments has been intensively studied in the literature.In cellular engineering practice, however, outage analysisis usuallynot adopted for frequency reuse design due to itscomplexity. The popular technique for this purpose is simplybased on the static formula instead, as shown by [1]–[4]

(1)

where denotes the number of cochannel interferers. Thisis often called the geographical model. Here, is thedistance from the desired transmitter with power and

Manuscript received January 14, 2000; revised December 19, 2000. Thiswork was supported by Strategic Research Grant 7001136 from the CityUniversity of Hong Kong.

The author is with the Department of Electronic Engineering, City Universityof Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(01)08253-6.

denotes the distance from theth interfererwith power . The constant stands for the propagationpathloss exponent. It is common practice in engineering designto assume that and that the interferingstations have equal distance from the receiver and the mobile islocated at the boundary of a cell with outer radius. Namely

(2)Under these conditions, the SIR reduces to

(3)

It is simple, easy to use, and requires no prior knowledge of thefading channels. As the expression of static SIR, the physicalsignificance of (3) is clearonlywhen a vehicle is fixed. Debatesbetween the use of a statistical model or the geographical modelhave been long existed among academic researchers and prac-ticing engineers. The major points of arguments are listed indetail in [5, pp. 68–73]. The significance of the static methodis unclear in a dynamic environment. Its advocates believe allthe statistical effects of a fading environment can be character-ized by a single control parameter. This can be done by carefullychoosing a protection ratio based on field measurement data.

It is the purpose of this paper is to reveal and establish the con-nection between the static and dynamic methods so that the dy-namic operational outage performance can be kept under controlwhile retaining the simplicity of the static method. We are alsointerested in the system capacity under varying requirements onthe minimum probability of outage.

II. DYNAMIC AND STATIC SIRS

In a real operational environment, the influence of multipathpropagation must be taken into account. Letdenote the re-ceived signal power and denote the interference from thethcochannel interferer where . Then the instanta-neous (dynamic) SIR is given by

(4)

All s are random variables depending on the transmitted powerand on the channel characteristics. For generality, we assumethat all channels suffer from independent Nakagami fading. Assuch, the received power follows the gamma distribution

(5)

0018–9545/01$10.00 © 2001 IEEE

ZHANG: BRIDGING THE GAP BETWEEN DYNAMIC AND STATIC METHODS 1225

where is the fading parameter. The average powerde-pends on the transmitted power , the distance from thethtransmitter , and the path-loss exponent, as shown by

(6)

It is easy to recognize that is related to a chi-square variableby

(7)

The symbol signifies the chi-square variable havingdegrees of freedom. We use the subscriptto indicate

that they are independent variables. The notation “” means“distributed as.” Inserting these notations into (4) yields

(8)

In dynamic reuse distance planning, it is usually assumed thatall the channels have the same characteristics such that

(9)

in addition to the equal-power and equal-distance conditionsgiven in (2). As such, we can rewrite

(10)

which, when inserted into (8) gives

(11)

Using the closure property of chi-square distribution [7] and re-calling that the ratio of two independent chi-square variables,when normalized by their degrees of freedom follows the-dis-tribution [7], [8], we can rewrite (11) as

(12)

The probability density function (pdf) of is explicitlygiven by [7], [9]

(13)

We note that the instantaneous SIR,, consists of two factors:a deterministic and a random factor. The former

(14)

is exactly the same as the static SIR characterizing the influencedue to the relative location of the mobile unit and the random

factor accounts for the random fluctuation of thereceived signals caused by multipath propagation. In the linearscale, is given by

(15)

which, when expressed in the decibel scale gives

(16)

The second term represents the random gain caused by the mo-bility during propagation.

III. STATISTICAL MEANS AND CONFIDENCEINTERVAL

When cell planning is based on a static design, one may ex-pect that the received SIR should equal if the vehicle doesnot move. The natural question is, “What doesrepresent in adynamic operational environment?” Let us study some charac-teristic parameters on the basis of (15), aiming to gain a deeperunderstanding of the physical significance of the static SIR.

We first relate to the mean value of. To this end, recallthat the th moment of an variable is given by [9]

(17)

which, when applied to (15), produces

(18)

Clearly, differs from the mean value of the instantaneous SIRby a bias term . The relative difference becomesnegligible as or increases. When the fading parameterincreases, fading becomes less severe and the mean SIR getscloser to the static one. That is consistent with intuition.

We next examine the median value of. It represents thevalue below which the instantaneous SIR falls with probabilityof . Let denote the pdf of and let

denote the upper-point of . It followsthat

(19)

which along with (15) allows us to obtain the median value ofas shown by

Median (20)

Since is usually not equal to unity, doesnot stand for the median of. From the previous discussion,it is clear that the static SIR represents neither the mean valuenor the median of the instantaneous SIR. Therefore, a naturalquestion is, “What does the static SIR really represent?”

To reveal its physical significance, it is necessary to determinethe variation behavior of around the static SIR. Given the con-fidence level , we would like to find and such that

(21)


where the short-form means the probability of event. Following the convention adopted in statistical theory, we

evenly allocate the upper and lower points such that

(22)

The first expression is not easy to determine from a lookup tablesince is very close to one. However, we can invokethe basic relation (26.6.9) of [9]

(23)

to rewrite

(24)

The methods for determining the upper-point of an -distri-bution are briefly described in Appendix A. If we rewrite (21)in the decibel scale such that

(25)

then, clearly represents the accuracy of using tocharacterize the dynamic SIR. Thus, one significance of thestatic SIR is that the instantaneousfalls into the neighborhoodof , with probability .

Example 1: As an illustration, consider the well-known de-sign example described in many textbooks [1], [4]. Suppose thedesired signal is corrupted by six cochannel interferers on thefirst tier. Given and dB, the reuse distancebased on the static SIR is given by

(26)

Suppose that the designed system is to operate in a Rayleigh-fading environment (i.e., ). Determine the following char-acteristic parameters of the dynamic SIR: 1) the average SIR;2) the median SIR; and 3) the 80% confidence interval of SIR.

Repeat the above calculation for a Nakagami fading environ-ment where the fading parameter .

Solution: Given and , the mean SIR isobtained by using (18)

dB (27)

To determine the median SIR, we find the value ofto be 0.735 from the lookup table in [9, p. 986]. Hence,

Median

dB

We next determine the confidence interval. From the lookuptable, we have

(28)

After inserting into (25), the 80% confidence interval in decibelscale is obtained

dB (29)

More explicitly, we write

dB dB (30)

Since dB, it is clear that the static SIR is close tothe mean SIR; yet, it well deviates from the median SIR. ForRayleigh fading, the 80% confidence interval is very large,clearly indicating that we cannot use the static SIR alone todescribe the dynamic behavior of a cellular system in a fadingenvironment.

For Nakagami fading channels with , we can proceedin a similar way yielding

dB

Median dB

dB dB

In general, as fading parameter increases, the static SIR getscloser to the mean SIR and the confidence interval becomesnarrower. This is not difficult to understand since a larger valueof corresponds to a less severe fading environment.

IV. DYNAMIC AND STATIC PLANNING

In most cellular systems, a power protection ratiois presetto ensure the minimum quality of service, the value of whichdepends on the modulation/codec to be used. Onceis speci-fied, we can express the outage probability as

(31)

which, when written explicitly, leads to

(32)

The two formulas are equivalent and their lookup tables are welldocumented (see for example, [9, pp. 986–989]). If one choosesto use a lookup table, (31) is more convenient. Equation (32),on the other hand, is suitable for numerical calculation by usinga computer.

We can use the above formulas to determine the frequencyreuse distance by assuming that a mobile is located at theboundary of the home cell. This represents the worst-casescenario. Let us show how to achieve this goal by using alookup table. Suppose we want to control outage probabilityat the boundary to meet the requirement of . To thisend, we rewrite (31) as

(33)


By definition of the upper point of the distribution,we have

(34)

Comparing (34) with (33) yields

(35)

which, when invoking (23) again, gives

(36)

Equation (36) clearly reveals the relationship among the staticSIR , dynamic power protection ratio and the specifiedoutage probability . It takes into consideration both servicequality and influence of the fading channels, which are repre-sented by the first and second factors on the right-hand side of(36), respectively. Remember that a smaller outage probability

will result in a larger value of . Accordingly,we conclude that increase in the power protection ratio or reduc-tion in the outage probability requires an increase in frequencyreuse distance.

At this point, it is interesting to take a close look at the phi-losophy of the advocates of geographic models and the alter-native statistical models. The relation (36) allows us to gain adeep insight into the difference and connection behind the twomodels. The advocates of statistical models maintain that op-erational environments are complicated and are random in na-ture. Statistical models thus can provide a better description andthe reuse distance design on its basis will ensure the dynamicSIR required by a cellular mobile system. The advocates of geo-graphical models, on the other hand, consider statistical modelsto be too difficult to use in practice. They feel that the impactof the randomness of a fading channel can be absorbed into thepower protection ratio [5]. When put into the framework of (36)we obtained above, the connection between the two conflictingmethodologies becomes clear immediately. The advocates ofgeographical models usually set a dynamic protection ratio first,say 18 dB, and then add a margin, say 6 dB, on the top to accountfor the effect of the fading channel. In the other words, they use24 dB as the threshold for the static design. In comparison with(36), this simply implies that what they set are

dB

dB

It is clear that in reuse distance design, the advocates of thetwo models are essentially consistent except for different view-points. Another difference is that the geographic model advo-cates determine by resorting to measurementdata.

Example 2: Let us illustrate how to conduct a dynamic de-sign in the static framework. Suppose the system is to operate ina Rayleigh-fading environment. Design the system to meet SIRrequirement at the boundary of 18 dB with outage probabilityequal to 0.25.

Solution: For Rayleigh fading, we have . SincedB and , it follows from

(36) that

(37)

which, when inserted into (3), produces

(38)

This shows that dynamic considerations result in a much largerreuse distance, as opposed to that based on the static design,which is 4.41. We would like to see the influence of fading pa-rameter on reuse distance. Supposeis increased to be

. Using the same procedure as above, we can determine that

(39)

which is smaller compared to the case of . Generallyspeaking, the larger the fading parameter, the less severe thefading and, thus, the smaller the reuse distance.

V. AVERAGE OUTAGE PROBABILITY

The relation obtained above provides a relatively simple ap-proach to determining the outage probability at the boundaryand averaged over the entire cell. Consider the worst case for amobile located at the cell’s boundary. The outage probability isgiven by

(40)

where was defined in (14).Suppose, instead, the mobile is located inside the cell, say

separated from the base, then the outage probability is equalto

(41)

where is defined

(42)

Clearly, the outage probability is a function of the location of themobile. Suppose the mobile is uniformly located over the entirecell and thus, in the polar coordinate system, we can express theprobability that the mobile is situated in an area element as

(43)


Fig. 1. Average outage probability with protection ratio= 18 dB.

This expression enables us to determine the average outageprobability over the entire cell yielding

(44)

The last line was obtained by setting . After inserting(32) into the above expression and simplifying, we obtain

(45)

Change of variable leads to

(46)

This is the outage probability averaged over the entire cell.The average outage probability versus reuse distance for the

protection ratio equal to 18 dB is shown in Fig. 1. The remainingparameters used are: and . For comparison, theoutage probability at the cell’s boundary is also included. Twoobservations can be made. First, both average and boundaryoutage probabilities decrease as the reuse distance increases.

Fig. 2. Average outage probability with protection ratio= 12 dB.

Second, the average outage probability is lower than the corre-sponding one at the boundary. The case for the protection ratioof 12 dB is shown in Fig. 2.

VI. SYSTEM CAPACITY

The system capacity in terms of user population is an im-portant design requirement for a cellular network. Consider aTDMA cellular system using a hexagonal cell pattern. Letand denote the total system and RF channel bandwidth, re-spectively, and let denote the cluster size. Then its systemcapacity is given by

(47)

Clearly, given bandwidths of the total system and each channel,the capacity depends on the clutter size which, in turn, dependson the reuse distance. Specifically, for a hexagonal cell pattern

.In a dynamic operational environment, the reuse distance will

influence the outage probability of the system. We therefore ex-pect that the capacity is a function of the acceptable systemoutage probability. Let us determine their relationship. Com-bining (14) and (36) allows us to write

(48)

Hence, we obtain

(49)

It follows that

(50)

In this expression, and reflect the impact ofthe quality of service on the system capacity. The former repre-sents the protection ratio and latter stands for the requirement ofoutage performance at the boundary. From this expression, the


Fig. 3. System capacity with protection ratio= 18 dB.

variation of the system capacitywith the outage probabilitycan be visualized. The smaller the outage probability, the largerthe value of and, thus, the lower the system ca-pacity. This reflects the conflicting design requirements on theoutage probability and system capacity. Accordingly, a systemdesign is a tradeoff between these parameters.

Thesystemcapacityagainstoutageprobability isshowninFig.3 for the protection ratio equal to 18 dB where and .As expected, the number of users that can be simultaneouslysupported in a given cell depends on the acceptable outageprobability at the cell’s boundary. A more restricted requirementon the outage probability will reduce the system capacity.In this sense, the outage performance can be used to tradefor the system capacity and vice versa, in much the same asthe relationship between the data rate and error performancein digital communication over an additive white Gaussianchannel.

According to the conventional theory as shown in (47), thesystem capacity is independent of operational environment. Fig.3 reveals that this is not the case for the capacity of a cellularsystem operating in a fading environment. It is observed that fora specified outage probability, the system capacity varies withthe fading parameter. The larger the fading parameter, theless severe the fading and the larger the system capacity. Fig.4 shows the case for protection ratio equal to 12 dB. Clearly,the lower the required outage probability, the higher the systemcapacity.

VII. CONCLUSION

In this paper, we established a simple relationship betweenthe dynamic and static methods for cell planning. The staticmethod is based on the geographic model alone whereas the dy-namic method takes both geometrical and statistical influencesinto account. The static SIR is related to its dynamic counterpartby a factor, which represents the influence of a fading environ-ment.

The new relation bridges the gap between the geographicaland statistical methods for cell planning. It takes into account

Fig. 4. System capacity for protection ratio= 12 dB.

the influence of both geographical factors and fading channelswhile retaining the simplicity of the static method. We used thenew relation to study several aspects of a cellular system withinteresting findings. First, we revealed the physical implicationof the static SIR in a dynamic operational environment. Second,we provided a simple method for evaluating the average outageprobability. Finally, we determined the relationship between theconflicting requirements on the system capacity and on the min-imum outage performance.

Finally, we would like to mention that the influence of codingand interleaving on cell planning is not considered in this paper.If coding and interleaving are used, our analysis is still appli-cable. The only thing we need to do is to appropriately adjust thepower protection ratio, , to account for the gain from codingand interleaving.

APPENDIX

CALCULATION OF

There are three ways to determine the upperpoint of thevariable. The most straightforward one is to find its

value from a lookup table. This type of table is well documentedin many textbooks. If the given parameters are not given in thetable, one can use the method of interpolation to determine thedesired value.

For numerical calculation using a computer, it is more con-venient to determine using an analytical method. Tothis end, we may use a second method. The idea is to relate theupper point of an variable with its counterpart of a Gau-sian variable. The reason is that the latter can be expressed interms of a function and, thus, much easier to determine. Oncewe obtain the upper point of the Gaussian distribution, it isstraightforward to use (26.5.22) and (26.5.1) of [9] to determinethe corresponding one for the distribution.

The third method is simple for our purpose if the fading pa-rameter is the multiples of 0.5. Denote and

(A.1)


Then, for of even value, we have [9]

(A.2)

If is of even value, a similar formula is available. For details,see (26.6.5) of [9]. For a fading parameter which assumes valueother than specified above, one can use the method of interpo-lation.

REFERENCES

[1] W. C. Y. Lee,Mobile Cellular Telecommunications, 2nd ed. New York:McGraw-Hill, 1995.

[2] , Mobile Communications Design FundamentalsNew York,1993.

[3] A. Mehrotra, Cellular Radio: Analog and Digital Systems. Boston,MA: Artech House, 1991.

[4] T. S. Rappaport,Wireless Communications: Principles and Prac-tice. Upper Saddle River, NJ: Prentice-Hall, 1996.

[5] H. Hammuda,Cellular Mobile Radio Systems. Chichester, U.K.:Wiley, 1997, pp. 68–72.

[6] Q. T. Zhang, “Probability of error for equal-gain combiners overRayleigh channels: Some closed form solutions,”IEEE Trans. Commu-nications, vol. 45, pp. 70–73, Mar. 1997.

[7] M. H. DeGroot,Probability and Statistics. Menlo Park, CA: Addison-Wesley, 1975, pp. 423–428.

[8] A. M. Mood, F. A. Graybill, and D. C. Boes,Introduction to the Theoryof Statistics, 3rd ed. New YorK: McGraw-Hill, 1963, p. 246.

[9] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Func-tions. New York: Dover, 1972.

Q. T. Zhang (S’84–M’85–SM’95) received theB.Eng. degree from Tsinghua University, Beijing,and M.Eng. degree from South China Universityof Technology, Guangzhou, China, both in wirelesscommunications, and the Ph.D. degree in electricalengineering from McMaster University, Hamilton,ON, Canada, in 1970, 1981, and 1986, respectively.

After graduation from McMaster Universityin 1986, he held a research position and AdjunctAssistant Professorship at the same institution. InJanuary 1992, he joined the Spar Aerospace Ltd.,

Satellite and Communication Systems Division, Montreal, QC, Canada, as aSenior Member of Technical Staff. At Spar Aerospace, he participated in thedevelopment and manufacturing of the Radar Satellite (Radarsat). He was sub-sequently engaged in the development of the advanced satellite communicationsystems for the next generation. He joined Ryerson Polytechnic University,Toronto, ON, Canada, in 1993 and became a Full Professor in 1999. In 1999,he took one-year sabbatical leave at the National University of Singapore.He is now with the City University of Hong Kong. His research is primarilyfocused on various technical issues in the physical and data-link control layersof wireless communications.

Dr. Zhang is currently an Associate Editor for IEEE COMMUNICATIONS

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