Team 11774 1

“Can you hear me now?”: A Practical Strategy for VHF Repeater

Coordination

Elizabeth Liang∗ Philip Pham† Robert Won†

February 14, 2011

1 Introduction

Amateur, or ham, radio makes use of designated radio bands for communication. A radio user can use acompact transceiver to both transmit and receive radio communications. In North America, the 2-meterband comprises frequencies between 144MHz and 148MHz. This band is part of the very high frequency(VHF) radio spectrum. For user-to-user communications, VHF radio requires line-of-sight transmission andreception. In a flat area, this line-of-sight transmission can range up to 15 miles [3]. To improve this range,devices known as “repeaters” receive, amplify, and retransmit weak signals. Under good conditions, repeaterscan improve range up to 50 miles [3]. Thus, the use of repeaters can make direct user-to-user communicationspossible over much further distances than simple line-of-sight transmission.

In order to avoid interference, repeaters receive and transmit at different frequencies. Thus, a repeatermight receive all signals at a frequency of 145MHz, and retransmit these signals at 145.6MHz. To preventinterference between different repeaters, repeaters must either be placed far enough away from one anotheror transmit on sufficiently separated frequencies. In addition to these two tactics, the “continuous tone-codedsquelch system” (CTCSS) allows for even more efficient use of available frequencies by associating to eachrepeater a subaudible tone that must be transmitted by any user wishing to use the repeater. Using differentCTCSS tones, two repeaters at the same location can have the same frequency pair without any interference.

In this paper, we consider the problem of 1,000 simultaneous radio users located within a flat, circular areaof radius 40 miles. Repeaters are required for communications over sufficiently long distances. However, thereis a cost associated with setting up and operating a repeater. Therefore, we seek to minimize the number ofrepeaters by optimizing repeater location, range, and frequency pair (transmission and reception) over thiscircular area.

1.1 Assumptions

First, we state our assumptions about repeaters and radio communications:

� We are able to use the spectrum between 145 to 148MHz.

� Transmitter frequency in a repeater is either 600kHz above or below the receiver frequency.

� There are 54 CTCSS tones available (in addition to no tone).

� Line-of-sight transmission is possible up to a range of 7 miles, so users closer than 7 miles apart do notneed to use a repeater.

� Repeaters can have a range of anywhere between 7 miles and 40 miles.

∗Duke University Department of Computer Science†Duke University Department of Mathematics

Team 11774 2

� If a repeater has a range of r miles, then all communications within a circle of radius r have perfectreception and transmission through the repeater.

� Any communications outside of the circle of a repeater’s range are unaffected by interference from therepeater.

� A separation of 600kHz is a sufficient frequency difference to avoid repeater interference. That is, arepeater transmitting at 145.6MHz does not interfere with a repeater receiving at 146.2MHz.

� Repeaters are placed only within the circular area of radius 40 miles.

� Once a repeater is placed, its location, frequency, and CTCSS tone never change.

Next, we state our assumptions about the population of users:

� The population consists of 1,000 users.

� There is a uniform distribution of the 1,000 users over the circular area.

� Each user is engaged in two-way communication with some subset of other users. Thus, the worst caseis 500 pairs of users attempting to communicate pair-wise.

� Any user may attempt to communicate with any other user.

� Users remain stationary while communicating.

1.2 Frequency and tone assignments

Repeaters can use any frequency between 145 and 148MHz. We assumed sufficient separation for non-interference to be 600kHz. To make maximum use of this range, we transmit or receive on one of sixdifferent frequencies: 145, 145.6, 146.2, 146.8, 147.4, and 148 MHz. Any repeater will occupy two adjacentfrequencies, and any other repeater using either of these two frequencies causes interference. Therefore, forconvenience, we adopted the convention that all repeaters receive at 145, 146.2 or 147.4 MHz. Each repeaterretransmits at a frequency 600kHz above its reception frequency.

This means that with no CTCSS tones, we would be able to place three repeaters in the same locationwithout the risk of interference. Using CTCSS tones, we could place 3 × 55 = 165 repeaters in the samelocation – one for every frequency and tone pair (including no tone).

We assumed that line-of-sight transmission was possible up to a range of 7 miles. We therefore approxi-mated the average number of user pairs that would not need to use a repeater. Because we assumed uniformpopulation distribution over the entire circle, the probability that a user is within any particular region ofthe circle is proportional to the area of the region.

Fix a user at some point in the circle. If the user’s partner is located within a circle of radius 7 aroundthe user, then communication is possible without the use of a repeater. If we ignore cases that are close to

the boundary of the circle, this probability is P =π(7)2

π(40)2. Therefore, for every 1,000 users, the expected

number of conversations that do not need a repeater is 500 × P ≈ 15 pairs. A computer simulation verifiesthis approximation.

We thus reserved a small number of frequencies for line-of-sight transmissions only. We chose to reserveall no-tone frequencies (145, 145.6, 146.2, 146.8, 147.4, and 148 MHz) for non-repeater communications.There are, on average, about 15 pairs of users (distributed approximately uniformly over the entire area)attempting to communicate on these frequencies. Therefore, with six frequencies reserved, interference isunlikely to occur. A computer simulation verifies that with six frequencies, on average, less than one pair ofusers is unable to communicate due to line-of-sight transmission interference.

Team 11774 3

1.3 Defining the problem

In an ideal world, we might have a distribution of repeaters that guarantees clear communications to 500user pairs, regardless of the locations of those 1,000 users. However, it is clear that there exist certaindistributions of user pairs for which no solution is possible. For example, consider the case in which 500users are in a single location and their 500 partners are in a single location 20 miles away. Each of these500 conversations requires the use of a unique repeater with a unique frequency-tone combination. However,there are only 165 combinations available. Therefore, this distribution of users can never be supported usingonly the available frequencies and tones.

We therefore sought to place repeaters and assign their frequencies and tones such that we supported1,000 users conversing pair-wise as efficiently as possible, regardless of their distribution within the circle.The “efficiency” of a set of repeaters was measured using an objective function. Specifically, we wanted adistribution of repeaters that maximized the objective function:

Fk = kµ− ρ,

where µ was the mean number of pairs supported, ρ was the number of repeaters, and k was a weightingparameter.

2 Methods

The problem of placing repeaters is similar to the problem of placing cell phone base stations. Thus,the problem is related to the Minimum Dominating Set Problem, which is NP-hard [2]. In such cases,computation of even suboptimal solutions is difficult. Therefore, we chose to use a statistical model. Wedeveloped the model as follows.

� Generate training data. In order to develop a statistical model, we first generated data about thelocation of radio users and their partners.

� Fit models. To fit each model, we placed repeaters in order to best serve the user pairs in each setof training data.

� Test models. Each model was tested by generating new datasets of 1,000 users (500 pairs) andattempting to serve the user pairs with the set of repeaters in our model.

� Evaluate models. A best model was chosen based on an objective function that took into accountthe users served and the number of repeaters used.

2.1 Generating training data

We assumed that each user had an equal probability of being located anywhere in the entire circular area.We also assumed that each user’s partner had an equal probability of being located anywhere. Thus, wegenerate pairs of users with a uniform distribution.

2.2 Fitting models

When fitting the model to the pairs of users in the training data, repeaters were placed to maximize thenumber of pairs served in the training data. The number of repeaters in the fitted model was bounded bythe number of pairs in the training data since each pair could only make use of one repeater. For example,using 600 training pairs would yield a model with at most 600 repeaters.

There were 162 frequency-tone pairs available for repeater use. The main idea of the strategy used wasto maximize the utility of each frequency-tone pair in order to serve as many users as possible. Repeaterswith larger range are able to serve user-pairs that are further apart. However, large repeater ranges alsolead to more interference, and thus a less efficient use of the frequency-tone pairs.

Team 11774 4

Routine 1 Process for creating a model that places repeaters down based on training data and attempts tooptimize the placement of repeaters of the same frequency.

def Model createModel (d):

sort d from longest to shortest pair

new List<Repeater> UR # Repeaters with unassigned frequencies

new List<Repeater> AR # Repeaters with assigned frequencies

for each p in d:

if p.distance < 7:

p.supported = true

else:

Repeater r = createRepeater(p.midpoint, p.distance/2)

UR.append(r)

for each r in UR:

if !AR.contains(r):

# try to assign r a frequency so it does not conflict with any repeater

# in AR that intersects r

boolean assignmentSuccess = tryAssign(r,AR)

if assignmentSuccess = try:

AR.append(r)

for p in d:

if p.repeater = r:

p.supported = true

doOutsidePairs(AR,r,d)

M.repeaterList = AR

# Once the first repeater of a frequency is placed down, try to optimally set down

# repeaters around it with the same frequency.

def doOutsidePairs(AR,r):

List<Pair> O #list of pairs outside of r

for p in d:

if p.supported = false and !p.intersect(r):

O.append(p)

sort O from longest to shortest pair

for each pair q in O:

# try to assign r.frequency to q.repeater, assignment will be successful

# if q does not intersect any repeater in AR with r.frequency

boolean assignmentSuccess = tryAssign(q,AR,r.frequency)

if assignmentSuccess = true:

AR.append(q.repeater)

q.supported = true

List<Pair> B #all repeaters with frequency of r

for all b in AR:

if b.frequency is equal to r.frequency:

B.append(b)

optimize(AR, B)

Team 11774 5

The first step of fitting the model was to choose the user-pair that was separated by the greatest distance,r miles. A repeater with a range of r

2 was placed at the midpoint of the two users, and the repeater wasassigned one of the 162 frequency-tone pairs (call it frequency 0). We then identified every pair of users thatfell outside the range of this repeater (and would thus be unaffected by interference from the repeater). Fromthis collection of pairs, the pair separated by the furthest distance was chosen, and a repeater of frequency0 was placed at the midpoint of this pair in the same way. This was repeated until it was no longer possibleto use frequency 0 on any remaining pairs. For each new repeater added, we checked that the repeater didnot interfere with any existing conversations.

Next, the next-furthest remaining pair was assigned a repeater with a different frequency-tone pair (sayfrequency 1). The process above was repeated with frequency 1. The algorithm stopped when no frequencyexisted for which there would not be interference between the two users of the pair. By choosing pairsseparated by the furthest distance first, overfitting to the training data was avoided since the probabilityof a future pair being in the range of a short-range repeater is small. The described frequency assignmentalgorithm is depicted in Routine 1.

We performed two minor optimizations to this model fit. After fitting the mode, if there was anyfrequency that was used by only one repeater, we relocated that repeater to the center of the circle and setits range to 40 miles. Additionally, we always checked the smallest repeater for any given frequency. If thissmallest repeater’s range was entirely outside of all other repeaters of the frequency, we increased the rangeof the repeater so that it was tangent to the next closest repeater range. This allowed us to support moreconversations in the test data with zero additional cost.

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−40 −20 0 20 40

−40

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020

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−40

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−40 −20 0 20 40

−40

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Model (650 Training Pairs) Frequency: 28

Figure 1: A graphical view of repeaters in four of ourfitted models. Each image corresponds to a particularfrequency-tone pair (in this case frequency “28”) in oneof the models. Sometimes, we have one single repeaterplaced in the center with a radius of 40. At other times,the circular area is filled efficiently with repeaters of thegiven frequency-tone pair.

Figure 1 is a graphical representation of several frequencies. In each image, the green circle representsthe circular area of radius 40. Each image corresponds to one particular frequency-tone pair (like frequency0 above). Each repeater is represented by a colored dot, with hotter colors corresponding to longer ranges.The range of each repeater is also depicted by a blue circle.

User pairs that were separated by fewer than 7 miles were ignored, with the assumption that these userswould be served by line-of-sight communication over the six reserved frequencies.

2.3 Testing models

To test the flexibility of each model, we tested each model by generating new data sets. Each data setcorresponded to a different distribution of 500 pairs of users. This might represent a snapshot of our 1,000users at a different point in time. For example, our users may have moved, attempted conversations with

Team 11774 6

different users, or both. These data sets were all generated with 500 pairs of users in the same manner as thetraining data generation. We applied each of our fitted models to each new random data set and countedthe number of pairs of users that each placement of repeaters could serve.

In testing a model, we required an algorithm that would count the number of user pairs that could beserved by the model. This count relies upon efficient use of the available repeaters. Thus, repeaters wereassigned to pairs of users in the following way. First the user pair that was furthest apart was chosen.This pair was assigned the repeater of minimum range that allowed the pair to communicate. Next, thesame was done for the pair with the second largest distance and so on. Pairs that were separated by fewerthan 7 miles were counted as served, under the assumption that these users would be served by line-of-sightcommunication. This algorithm is depicted in Routine 2.

Routine 2 Run each model a number of times with new random data each time.def runModels(M,r):

for r in range(0,r):

for m in M:

List<Pair> d = createData(500)

m.reset() # set all repeaters in m to off

sort d from long to short

countSupported = 0

for each p in d:

assignFrequencies(p,m)

if p.isSupported

countSupported++

output countSupported

# Assign the shortest possible available repeater that satisfies the pair.

def assignFrequencies(p,M):

if p.distance < 7:

p.supported = true

else:

List<Repeater> A #Available repeaters in m for given p

for every repeater r in M.repeaterList:

if r.isOn == false and p.inrange(r):

A.append(r)

# if a frequency appears more than once in the list

# remove all repeaters with frequency from list

A.removeDuplicateFrequencies()

if A.isNotEmpty:

p.repeater = a.get(0)

a.get(0).turnOn

p.suppored = true

2.4 Evaluating models

In evaluating which model was best, we required an objective function. Since we were trying to serve asmany users as possible with as few repeaters as possible, this objective function depended upon the meannumber of conversations supported and the number of repeaters used. As described earlier, we used theobjective function:

Fk(µ, ρ) = kµ− ρ,

Team 11774 7

where µ is the mean number of pairs supported across all test data sets and ρ is the number of repeatersused in the model. k > 0 is the weighting parameter. This parameter can be interpreted as the “value”of one repeater in terms of conversations. A gain of one user pair served on average is worth k repeaters.The model that maximizes the objective function was considered the best model. For our analyses, we chosek = 3. However, in the real world, one could choose their own k depending upon the cost of repeaters andthe marginal value of additional conversations supported.

3 Results

3.1 Determining the optimal number of training pairs

An important question at the data-generation phase was: how much training data should we input to fitour model? Since the number of repeaters placed is bounded above by the number of training pairs, usingtoo few training pairs yields fewer repeaters than optimal. In contrast, using too many training pairs leadsto two possible pitfalls. First, since placing a repeater incurs a cost, placing too many repeaters decreasesthe value of our objective function. Second, because our model-fitting algorithm places repeaters betweenthe user pairs with greatest separation first, we run the risk of setting up too many “long” repeaters. Forexample, if we double the number of training pairs, then we also roughly double the number of pairs thatare separated by more than 70 miles.

We therefore used a statistical approach to determine the optimal number of training pairs. We generatedmodels using between 500 and 1,000 training pairs in increments of 10. For each of these training pair values,we generated 100 models, and tested each model with 100 test data sets. We chose k = 3 as our objectivefunction parameter. Thus, the average value of the objective function F3 for each of the 100 runs of the 100models was then calculated. These values are plotted against number of training pairs in Figure 2.

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500 600 700 800 900 1000

600

650

700

750

Training Pairs

Obj

ectiv

e F

unct

ion

((F3))

Objective Function vs. Number of Training Pairs

Figure 2: A value of k = 3 was chosen for the objective function. In order to determine how many training pairs togenerate, we fitted models using between 500 and 1,000 training pairs. For each model, the objective function, F3,was evaluated for 100 test models. This data was fitted with a degree 3 polynomial.

We performed a regression on this data using a degree 3 polynomial. We then maximized the value ofthe best-fit polynomial over the interval. For k = 3, the polynomial was maximized at 650 training pairs.Therefore, we chose to use between 600 and 700 pairs to train our models.

Team 11774 8

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ectiv

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500 600 700 800 900 1000

4600

5000

5400

Training Pairs

Obj

ectiv

e F

unct

ion

((F15

))

Varying Objective Functions vs. Number of Training Pairs

Figure 3: Varying the k parameter in the objective function. Between 500 and 1,000 training pairs were used to fitmodels. Each model was evaluated with four objective functions (F1, F2, F7, and F15).

We also could have chosen some other value of k for our objective function parameter. If the cost ofinstalling repeaters suddenly increased, then we could decrease k. If, in another context, we were willing tosupport each conversation at higher cost, we could increase the value of k. We plotted the same regressionas above for different k values in Figure 3. The maximum value of the polynomials for data F1, F2, F7, andF15 was 590, 640, 660, and 670, respectively.

Intuitively, these data make sense. The more repeaters we are willing to use per gained conversation,the more training pairs we should use. This also shows one of the strengths of our model. A change inthe objective function easily feeds back into our model generation phase. Thus, we are able to improve ouralgorithm regardless of the choice of parameter.

3.2 Solving the posed problem

The defined problem was to find the optimal placement of repeaters to serve 1,000 users or 500 pairs ofusers distributed uniformly throughout a flat circular area of radius 40. The best fitted model found (asmeasured by maximizing the objective function) is depicted in Figure 4. We called this fitted model our“best uniform” model.

This fitted model served users efficiently. Throughout all 200 test runs, this best uniform model served411 pairs or 822 users using only 477 repeaters. In 95.5% of the test runs, this fitted model managed toserve over 80% of the population (Table 1).

The arrangement of repeaters in the best uniform model conform to our expectations. We would expect,for example, that there should be relatively few long-range repeaters near the edge of the circle. In Fig-ure 4(a), we see that longer range repeaters are placed in the middle where they can best serve users thatare far apart, whereas pairs close to the edges need only short-range repeaters.

Team 11774 9

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−40

−20

020

40

Model (650 Training Pairs)

X (miles)

Y (

mile

s)

(a) Heatmap of best fitted model to the uniform case.

Repeater Ranges

Range (miles)F

requ

ency

0 10 20 30 40

020

4060

8010

012

0(b) Histogram of corresponding repeater ranges.

Figure 4: Repeater placement for best fitted model to the uniform case is shown. Colors indicate the range of therepeater, where hotter colors represent longer-range repeaters.

Model Mean No. of Rep. Min Max % above 400

Best uniform 411.2 477 395 432 95.5%

Worst uniform 336.5 404 320 353 0.0%

Best centered 410.5 466 384 429 92.5%

Best edge 415.1 477 396 433 99.5%

Table 1: The table displays various results for 500 pairs and with varying population distributions. All were runwith 650 training pairs.

Moreover, the histogram in Figure 4(b) shows that medium-range repeaters are optimal. This is becausewhile long-range repeaters are more likely to be usable by user pairs, they also cause more interference,limiting the number of repeaters that can be placed at that frequency. On the other hand, while small-rangerepeaters create little interference, they are less often usable by a test pairs. For instance, if a repeater hasa range of 8 miles or fewer, the expected number of pairs that can make use of it (as approximated by area)is less than 1:

E(# of pairs that can use the repeater) = 500 ×(

82

402

)2

= 0.8 < 1.

Thus, medium range repeaters in the 20-25 mile range are found to provide the best balance betweenminimizing interference and having the flexibility to serve a large number of pairs.

We can perform the above calculation and predict the expected value of user pairs that are m milesapart. We would expect our optimal repeater range distribution to be similar. These expected values are

Team 11774 10

plotted in Figure 5. This distribution is comparable to Figure 4(b); thus, the fitted model places repeatersefficiently. In an average case, there are just enough long-range repeaters to serve pairs that are far apart,which minimizes interference while serving these pairs. Placing the rest as shorter-range repeaters allows forefficient use of frequencies.

Pairs

Distance (miles)

Fre

quen

cy

0 20 40 60 80

020

4060

8010

0

Figure 5: Histogram of pair distances in a populationgenerated with a uniform distribution.

We contrast our “best uniform” model to our “worst uniform” model – the model that minimized ourobjective function. The histogram of repeater ranges (Figure 6(b)) differs significantly from the averagedistribution of pair distances in Figure 5. The peak of this distribution is higher than in the “best uniform”case. This creates interference which allows for the placement of only 404 repeaters. Further, there arerelatively more short-range repeaters. These short-range repeaters are often unable to be used and onaverage, out of 500 pairs, this fitted model only serves 336 conversations between pairs (Table 1).

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020

40

Model (650 Training Pairs)

X (miles)

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mile

s)

(a) Heatmap of worst fitted model to the uniform case.

Repeater Ranges

Range (miles)

Fre

quen

cy

0 10 20 30 40

020

4060

80

(b) Histogram of corresponding repeater ranges.

Figure 6: Repeater placement for worst fitted model to the uniform case is shown. Each point represents a repeater.Longer ranges are mapped with hotter colors.

Team 11774 11

3.3 Varying the distribution of users

After finding a solution to the uniform case, we performed an investigation on two different distributions ofusers. We altered our pair generation to generate distributions with more users clustered toward the centerof the circle, then toward the edges of the circle. We then tested each model with test data sets having thesame distributions. As shown in Table 1, our model performs nearly as well when the population of users isnot uniformly distributed.

In our best model for uniform, centered, and edge distributions, the mean number of conversationssupported ranged between 411.2 and 415.1. The number of repeaters also remained consistent. We can lookat the repeater placements and ranges in these two cases. The heatmaps and corresponding histograms forthese cases are shown in Figure 7 and Figure 8.

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Figure 7: Repeater placement for a population concentrated at the center.

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Figure 8: Repeater placement for a population concentrated around the edges.

The results are largely as expected. In the center case, repeaters are more densely concentrated in the

Team 11774 12

center (Figure 7(a)). Moreover, since pairs are generally closer together, in Figure 7(b), a higher numberof medium range repeaters are seen and fewer long-range repeaters are seen. On the other hand, when thepopulation is concentrated around the edges, (Figure 8(a)), the distribution of repeaters is pulled slightly tothe edges. However, in the center, there are many long-range repeaters and more of them overall due to thefact a pair (Figure 8(b)) is now more likely to consist of users on two separate ends of the circle.

Overall, our model appears capable of handling a wide variety of population distributions with a negligibleimpact on performance. This represents a major strength in our modeling approach. As long as we inputappropriate training data, our model offers consistently good performance over a range of distributions.

4 Discussion

4.1 Strengths of our model

The primary strength of our model is that it offers a practical, implementable, and flexible approach torepeater placement.

� The fitted models adapt to the training data. In this paper, we usd three different distributions ofusers in the area. However, if any information about the distribution of users is known, the model canbe improved by altering the training data. We achieved comparable results with all of our populationdistributions.

� The model does not depend upon any particular region shape or area. In the real world, our modelcould directly be used to place repeaters in a flat region of any shape and of any area.

� The model is easily implemented and outputs useful results. We were able to generate and test over10,000 models in only a few hours. Thus, the model could be used to actually determine repeaterlocations and placements in the real world.

� The evaluation phase of each fitted models is easy to understand and adapt. Our evaluation dependson a simple one-parameter objective function. The parameter k is easily interpretable. Potentialadaptations for the objective function are discussed below.

4.2 Weaknesses of our model

Although our model has many strengths, there is certainly room for improvement in our model.

� We are never actually able to serve 1,000 users conversing pair-wise. In fact, when we use 500 pairsof users to train the data, the resulting fitted model serves all 500 conversations less than 1% of thetime. A better repeater placement algorithm might be able to improve upon this success rate.

� Our model never considers non-pair-wise conversations. In the real world, groups of three or more peo-ple might attempt to communicate over the same channel. Our model does not address this situation.

� In the model-testing phase, we rely upon users to pick repeaters in an algorithmic fashion. We assignrepeaters to the furthest-separated users first. In an actual application, it is unlikely that we couldenforce this repeater assignment order. The model-testing is not completely unrealistic, however, aseach pair of users could conceivably choose the shortest range repeater that was available to them.

� We model the interference pattern of a user-pair simply as a circle centered at the repeater. The actualinterference pattern, however, is more similar to the one shown in Figure 9, where each user causesadditional interference.

� Our assumptions about interference in general are somewhat unrealistic. We assumed that interferencepatterns from repeaters and users to be perfect circles of exact radius. In actuality, signal quality isprobably a function of distance, and interference extends far beyond the usable range of a repeater.

Team 11774 13

Figure 9: A more realistic interfer-ence pattern than the one that wemodeled. In addition to interferencecaused by the repeater, there is alsointerference surrounding each user.

Further, repeaters are not manufactured on a continuous spectrum of ranges, so we cannot choose anyreal number for the range of our repeaters.

4.3 Adapting the model

As mentioned above, our model lends itself well to adaptations and tweaks. As illustrated in our brief investi-gation into non-uniform distributions, our model is stable with respect to changing population distributions.Thus, if any information about the distribution of users is known, this information should be used in thetraining pair-generation phase. Further, the objective function parameter, k, can also be used to tweakmodel performance.

Currently, we consider k > 0 to be a constant. A possibly more realistic approach would be to considerk(x) as a function number of conversations supported. We would try to choose k(x) such that it accuratelyreflected the marginal value of an extra conversation. For example, at low x, each additional conversationmight be worth more than when x approached 500. Thus, we could choose k to be a monotonically decreasing

logistic function, k(x) =a

1 + ex, where a is a scaling factor.

Since k is easily interpreted as the value of one conversation in terms of repeaters, we could use it as athreshold value in our model-training phase. Specifically, if any repeater were predicted to add fewer than1/k conversations, we should never add that repeater. The probability of a user pair being located within anyparticular repeater’s range is a function of the area of the repeater’s range. Therefore, for any repeater, wecan calculate an expected value of conversations occurring within that repeater range, and decide whetheror not to place the repeater.

Another implementable modification to our model is the ability to make the objective function a functionof the cost of a repeater rather than the number of repeaters. Smaller range repeaters might be cheaper thanlarger range ones. For each fitted model, we could calculate the total cost of repeaters, and then minimizea objective function that depended upon number of conversations and total cost. This would be a usefulmodification in a real application.

4.4 Real-world considerations

Although our model performs reasonably well in our simulations, there are practical considerations thatmust be taken into account before deployment in a real-world application. First, 1,000 users over a 5,000square mile area is a very sparse distribution. However, when we consider the case of 10,000 users (5,000pairs), our model performs extremely poorly. We fitted 50 models to 6,000 training pairs and tested themon 50 sets of 5,000 test pairs generated by a uniform distribution.

Unfortunately, while we were able to apply the model in the same manner, performance dropped dras-tically. In the best fitted model, on average only 623 conversations could be supported with 538 repeaters.There are more conversations than repeaters because of the high number of pairs closer than 7 miles apart,as we modeled these pairs as being able to communicate without repeaters. We very likely overcount becausewith 5,000 pairs, there will be approximately 5, 000 × (72/402) ≈ 150 such pairs, which means that the sixfrequencies allocated to such short-range conversations would likely not be sufficient to avoid interference.

Team 11774 14

In the case of 10,000 users, space and frequency limited the number of repeaters that could be placedwithout creating interference. In our 50 modes, we placed more than 600 repeaters. Thus, we were boundedabove by 600 conversations using repeaters.

Another real-world concern is non-flat terrain. Hills, mountains, and even the curvature of the Earthaffects signal propagation. Our model never considers terrain and also never considers repeater height. Wecould integrate our model with existing irregular-terrain radio propagation models like the Longley-Ricemodel [1]. A naive solution might be to choose the highest height between any user-pair as the repeaterlocation before setting the range of the repeater. We might also use a metric that assigns a value to eachregion of the area based on its height. We could then take the “value” of a repeater location into accountwhen placing repeaters. However, we never incorporated variable terrain into our model.

Before deployment, we would also need to consider communications with more than one user per repeater.In this case, we might “weight” each user equally and choose repeater location based on the center of massof the users who are trying to communicate with one another. This would likely lead to an higher averagerange for the repeaters. This would lead to an increase in interference. However, the number of conversations(and thus repeaters needed) would also decrease, thus leading to a decrease in interference.

Finally, a strength of our model in terms of real-world applications is its ability to integrate with existingnetwork pieces. Most locations already have existing communications infrastructure in place. Thus, we needto consider interference with existing parts of the infrastructure. Our model handles this situation well.When we place a repeaters in the model-fitting phase, the location is based on all other repeaters that arecurrently placed. Thus, we can simply input data as to existing network pieces, and our model will generaterepeaters locations that could smoothly integrate with the existing infrastructure.

4.5 Conclusion

We developed a statistical model to solve the problem of coordinating VHF repeaters for a populationof 1,000 users of a circular area of radius 40. Our algorithm prioritized long-range communications, andattempted to efficiently use each frequency with repeaters of as long a range as possible. Our model is quick,understandable, and easy to implement.

In our best model, we were able to serve more than 80% of users over 95% of the time. We were able,on average, to serve over 822 users attempting to communicate pair-wise. We achieved this using only 477repeaters. Moreover, our model has stable performance with respect to population distributions. Whenthe distribution of our users was varied, a corresponding change in training data led to consistently goodperformance. Overall, we believe that we have developed a novel way to allocate radio frequencies to servemany different distributions of users. We believe that the techniques employed here can be implemented inreal-world applications.

References

[1] K. Chamberlin and R. Luebbers. “An evaluation of Longley-Rice and GTD propagation models.” An-tennas and Propagation, IEEE Transactions 30, 6 (1982) 1093- 1098.

[2] R.M. Mathar and T. Niessen. “Optimum positioning of base stations for cellular radio networks”. WirelessNetworks 6 (2000) 421-428.

[3] M.J. Wilson and M. Weinberg. The ARRL Operating Manual For Radio Amateurs. Amer Radio RelayLeague. 2008, pp. 2-3, 2-8.