Tag Archives: Transportation

Federal van pools: a case of too many constraints

My husband works for the Federal government. He takes a van pool to and from his work on most days (he has a 120 mile commute round trip). The van pools are a great deal. If someone rides in a van pool, they commit to riding a certain number of days per quarter. They pay a membership fee and are reimbursed for saving gas.

In response to fraud, two significant changes were made to the van pool contracts:

  1. Van riders have to ride the van at least 50% of the working days each month (instead of each quarter).
  2. The van needs to be more than 50% full (not including the driver) at least 80% of the time.

First requirement

The first requirement is a problem. Vacations, has off-site training, goes to a conference, etc, interfere with ridership. Add two of these events together, and you end up with a severely inflexible policy. Employees who have to travel or who want to take a vacation may not be able to abide with the new requirements. Employees that travel a lot may not be the best candidates for the van pool, but employees who have one long trip ever should be able to ride the van.

Second requirement

The second change is even more restrictive. My husband’s van holds 14 people. They need to have at least 8 people ride (7 out of 13 + 1 driver) at least 4 out of 5 days in the week. It’s worth noting that small vans are going to have more of a problem with this because they typically have more variability in who shows up.

Looking at the long-term, a van needs to have at least 8 riders 80% of the time. That implies that the average number of days until the van is not “half” full needs to be 5. If we assume that each day is independent of the others, then we can model this as a geometric distribution. To get the desired average of 5, we need a “success” probability of 1/5. That is, the probability of a van being less than half full on any single day needs to be 20%. Is that realistic?

Let’s look at the 14 riders. Let’s say they each ride each day with probability p and that each day is independent. Then we can model the number of riders each day R~ Binomial(14,p). We need P(R<8) < 0.2 to meet requirement #2 (from the last paragraph). But if p=0.5 (as required), then P(R<8) = 0.605.  We need p > 0.642 to get P(R<8) < 0.2.

Thus far, I’ve assumed that each day is independent. Making that assumption will yield optimistic results. The “optimistic” results suggest that even under idealistic assumptions, the new van pool requirements will be extremely difficult to meet.

But this assumption is not valid. Riders ride less on Mondays and Fridays (40% of the work week). In practice, this means a van is likely to have at least 8 riders on three days of the week (Tue-Thu), but not four as required. For simplicity, let’s say that a van always has 8 riders on Tue-Thu (an idealistic assumption, but not too unreasonable). Let’s say that each rider takes the van with probability p = 1/3 on Mondays and Fridays (a modest change), and let’s assume that each rider and day are independent. Then the van has 8 riders with probability 0.0576 on these days. The probability that the van covers at least one Monday or Friday per week is 0.111. Therefore, under more realistic assumptions, it is even more unlikely that a van can meet assumption #2.

Vans could meet these requirements if they “overbook” and allow more riders than there are seats in the van. They will surely need to risk turning riders away to meet their requirements if more than 14 show up some day.  This may happen when gas prices change, since ridership increases when gas prices shoot up. However, my husband has noted problems meeting requirement #2 in the past couple of months when gas has been ~$4 per gallon.

Van pools and (un)intended consequences

Van pools are intended to improve the environment by encouraging car pooling. Van pools ultimately create an incentive for people to live very, very far from where they work, thus leading to more fuel usage. In the end, I do not think they save much (or any) fuel from being consumed. But this is open for debate.

On the other hand, van pools are great if your significant other wants a career. It has made our “two body problem” quite manageable. Accommodating one’s spouse’s career is rarely cited as a benefit to van pools. I am grateful that van pools exist so that both my husband and I can have careers. I hope it’s still possible for him to carpool with new rules.

Final thoughts

I applaud the Federal government for trying to crack down on fraud. But I encourage them to create rules that make it possible for riders to follow the letter of the law. And maybe someone who has taken a course on probability should look over the next set of rules and crunches some numbers.

Do you ride a van? Please correct anything I got wrong in the comments and add your two cents about how they can be improved.

Happy 12th birthday to my Honda Civic! Let’s compute how much money I spent on gas.

The first and only time I bought a car was in the year 2000 when gas cost about $1.40 per gallon. A year or two earlier, gas cost about $1 per gallon, so $1.40 seemed expensive at the time. Despite how trendy SUVs had become, I opted for a small, fuel efficient Honda Civic. Twelve years and three kids later, I am still driving that same Honda Civic. It has been a great investment for many reasons. This week when I filled up the tanked for $3.83/gallon, I calculated that a fuel economy of 37 mpg. this is much better than many newer, fuel efficient cars.

I wanted to compute how lucrative a fuel efficient car has been compared to the SUVs that I passed over. So I built a spreadsheet. If you want to check out my calculations, my spreadsheet is here.

Here is what I did. I have averaged 10,000 miles per year for the past 12 years at an average of 34 mpg. I assumed that model year 2000 SUVs get 19 mpg. I also assumed that the mpg was constant over 12 years, which seems reasonable from my back of the envelope mpg estimate every time I fill up the tank. This meant that I used 24.5 gallons per month for my Civic and 43.9 gallons per month for a hypothetical SUV.  Monthly averages for fuel prices were obtained from fueleconomy.gov, and I computed the net present fuel costs assuming a 2% rate of inflation.

Buying a Civic instead of an SUV proved to be quite lucrative: My model suggests that I saved $5693 in gas in the past 12 years compared to an SUV. This is quite significant, as gas has been pretty cheap most of the 12 years and haven’t driven much.  Had I driven more, I would have saved more. If I drove 12,000 miles per year (about the national average), I would have saved $6569. If gas cost the same as it does now (a national average of $3.58) for all 12 years, I would have saved $9938 over 12 years.  I would save much more if you consider the typical lifetime of a Honda being about 250,000 miles–I’m not even halfway there yet.  The Civic cost about $15K back in 2000, which means that the gas money saved over the lifetime of the car is close to the purchase price of the car. So if you’re in the market for a car now, fuel economy is very important!

I can’t say that I really saved $5693. A report from the Texas Transportation Institute indicates that when people have more fuel efficient cars, they tend to drive more, thus negating some of their savings (I think it averages out to something like 15% more miles driven). The amount I drive is well below average, especially for someone who drives to and from work, so it’s doubtful that I would have driven much less with an SUV, but it’s something that could be added to the spreadsheet. It’s hard to overstate how much Americans love SUVs. Their market share has grown steadily since the 1970s (see the image below), and this has only recently started to erode. Gas prices have been pretty bad since 2005, and yet SUVs have only declined in popularity after the downturn in the economy. Cars have also been less popular lately–people are not buying cars or SUVs.

Here is another fuel economy fact:

Men drive way more than women. Men 35-54 drive 18,858 miles per year compared to 11,464 for women. That means that men in this age group drive 62% more than women. Similar trends occur for other age groups. There are several explanations: (a) adult women are more likely to stay at home with young children, (b) men work farther from home, (c) men like to do more joy riding.

If you’re curious, my Civic is pictured below. I hope it continues to serve me well.


My sweet ride!

The number of cards and light trucks (mostly SUVs) sold per year. SUVs have become very popular.

The monthly cost calculations are below.

Cost Calculations Miles driven Price per gallon Gallons (Civic) Gallons (SUV) Gas cost (Civic) Gas cost (SUV) NPV (Civic gas) NPV (SUV gas)
Mar-00 833 $1.52 24.5 43.9 $37.16 $66.49 $29.29 $52.42
Apr-00 833 $1.47 24.5 43.9 $35.91 $64.25 $28.36 $50.74
May-00 833 $1.49 24.5 43.9 $36.45 $65.22 $28.83 $51.59
Jun-00 833 $1.63 24.5 43.9 $40.02 $71.62 $31.71 $56.75
Jul-00 833 $1.55 24.5 43.9 $38.01 $68.03 $30.17 $53.99
Aug-00 833 $1.47 24.5 43.9 $35.91 $64.25 $28.54 $51.08
Sep-00 833 $1.55 24.5 43.9 $37.99 $67.98 $30.25 $54.13
Oct-00 833 $1.53 24.5 43.9 $37.55 $67.19 $29.95 $53.59
Nov-00 833 $1.52 24.5 43.9 $37.18 $66.54 $29.70 $53.16
Dec-00 833 $1.44 24.5 43.9 $35.37 $63.29 $28.30 $50.65
Jan-01 833 $1.45 24.5 43.9 $35.47 $63.46 $28.43 $50.87
Feb-01 833 $1.45 24.5 43.9 $35.54 $63.60 $28.53 $51.06
Mar-01 833 $1.41 24.5 43.9 $34.53 $61.80 $27.77 $49.69
Apr-01 833 $1.55 24.5 43.9 $38.04 $68.07 $30.64 $54.83
May-01 833 $1.70 24.5 43.9 $41.72 $74.65 $33.66 $60.23
Jun-01 833 $1.62 24.5 43.9 $39.61 $70.88 $32.01 $57.28
Jul-01 833 $1.42 24.5 43.9 $34.83 $62.32 $28.19 $50.45
Aug-01 833 $1.42 24.5 43.9 $34.83 $62.32 $28.24 $50.53
Sep-01 833 $1.52 24.5 43.9 $37.30 $66.75 $30.30 $54.22
Oct-01 833 $1.32 24.5 43.9 $32.23 $57.68 $26.22 $46.92
Nov-01 833 $1.17 24.5 43.9 $28.70 $51.36 $23.39 $41.85
Dec-01 833 $1.09 24.5 43.9 $26.62 $47.63 $21.73 $38.88
Jan-02 833 $1.11 24.5 43.9 $27.13 $48.55 $22.18 $39.70
Feb-02 833 $1.11 24.5 43.9 $27.30 $48.86 $22.36 $40.01
Mar-02 833 $1.25 24.5 43.9 $30.61 $54.78 $25.11 $44.93
Apr-02 833 $1.40 24.5 43.9 $34.24 $61.27 $28.13 $50.34
May-02 833 $1.39 24.5 43.9 $34.12 $61.05 $28.08 $50.24
Jun-02 833 $1.38 24.5 43.9 $33.87 $60.61 $27.92 $49.97
Jul-02 833 $1.40 24.5 43.9 $34.24 $61.27 $28.27 $50.59
Aug-02 833 $1.40 24.5 43.9 $34.22 $61.23 $28.30 $50.64
Sep-02 833 $1.40 24.5 43.9 $34.31 $61.40 $28.43 $50.87
Oct-02 833 $1.45 24.5 43.9 $35.42 $63.38 $29.39 $52.59
Nov-02 833 $1.42 24.5 43.9 $34.78 $62.24 $28.91 $51.73
Dec-02 833 $1.39 24.5 43.9 $33.97 $60.79 $28.28 $50.61
Jan-03 833 $1.46 24.5 43.9 $35.74 $63.95 $29.80 $53.33
Feb-03 833 $1.61 24.5 43.9 $39.53 $70.75 $33.02 $59.10
Mar-03 833 $1.69 24.5 43.9 $41.50 $74.25 $34.72 $62.12
Apr-03 833 $1.59 24.5 43.9 $38.95 $69.69 $32.64 $58.40
May-03 833 $1.50 24.5 43.9 $36.69 $65.66 $30.80 $55.11
Jun-03 833 $1.49 24.5 43.9 $36.59 $65.48 $30.77 $55.06
Jul-03 833 $1.51 24.5 43.9 $37.08 $66.36 $31.23 $55.89
Aug-03 833 $1.62 24.5 43.9 $39.71 $71.05 $33.50 $59.94
Sep-03 833 $1.68 24.5 43.9 $41.15 $73.64 $34.77 $62.23
Oct-03 833 $1.56 24.5 43.9 $38.33 $68.60 $32.44 $58.06
Nov-03 833 $1.51 24.5 43.9 $37.06 $66.32 $31.42 $56.22
Dec-03 833 $1.48 24.5 43.9 $36.25 $64.87 $30.78 $55.09
Jan-04 833 $1.57 24.5 43.9 $38.53 $68.95 $32.77 $58.65
Feb-04 833 $1.65 24.5 43.9 $40.39 $72.28 $34.42 $61.59
Mar-04 833 $1.74 24.5 43.9 $42.55 $76.14 $36.31 $64.98
Apr-04 833 $1.80 24.5 43.9 $44.07 $78.86 $37.67 $67.41
May-04 833 $1.98 24.5 43.9 $48.60 $86.97 $41.62 $74.47
Jun-04 833 $1.97 24.5 43.9 $48.26 $86.36 $41.39 $74.07
Jul-04 833 $1.91 24.5 43.9 $46.84 $83.82 $40.24 $72.00
Aug-04 833 $1.88 24.5 43.9 $46.03 $82.37 $39.61 $70.88
Sep-04 833 $1.87 24.5 43.9 $45.83 $82.02 $39.51 $70.70
Oct-04 833 $2.00 24.5 43.9 $49.02 $87.72 $42.32 $75.73
Nov-04 833 $1.98 24.5 43.9 $48.50 $86.80 $41.95 $75.06
Dec-04 833 $1.84 24.5 43.9 $45.12 $80.75 $39.09 $69.94
Jan-05 833 $1.83 24.5 43.9 $44.88 $80.31 $38.94 $69.68
Feb-05 833 $1.91 24.5 43.9 $46.81 $83.77 $40.69 $72.81
Mar-05 833 $2.08 24.5 43.9 $50.96 $91.18 $44.36 $79.37
Apr-05 833 $2.24 24.5 43.9 $54.98 $98.38 $47.93 $85.78
May-05 833 $2.16 24.5 43.9 $52.97 $94.78 $46.26 $82.78
Jun-05 833 $2.16 24.5 43.9 $52.84 $94.56 $46.23 $82.72
Jul-05 833 $2.29 24.5 43.9 $56.13 $100.44 $49.18 $88.01
Aug-05 833 $2.49 24.5 43.9 $60.93 $109.04 $53.48 $95.70
Sep-05 833 $2.90 24.5 43.9 $71.15 $127.32 $62.56 $111.94
Oct-05 833 $2.72 24.5 43.9 $66.59 $119.17 $58.64 $104.94
Nov-05 833 $2.26 24.5 43.9 $55.32 $98.99 $48.80 $87.32
Dec-05 833 $2.19 24.5 43.9 $53.55 $95.83 $47.32 $84.67
Jan-06 833 $2.32 24.5 43.9 $56.76 $101.58 $50.24 $89.90
Feb-06 833 $2.28 24.5 43.9 $55.88 $100.00 $49.54 $88.65
Mar-06 833 $2.43 24.5 43.9 $59.44 $106.36 $52.77 $94.43
Apr-06 833 $2.74 24.5 43.9 $67.21 $120.26 $59.77 $106.96
May-06 833 $2.91 24.5 43.9 $71.25 $127.50 $63.47 $113.58
Jun-06 833 $2.89 24.5 43.9 $70.71 $126.54 $63.10 $112.91
Jul-06 833 $2.98 24.5 43.9 $73.06 $130.75 $65.30 $116.86
Aug-06 833 $2.95 24.5 43.9 $72.35 $129.47 $64.78 $115.91
Sep-06 833 $2.56 24.5 43.9 $62.62 $112.06 $56.16 $100.49
Oct-06 833 $2.25 24.5 43.9 $55.02 $98.46 $49.43 $88.45
Nov-06 833 $2.23 24.5 43.9 $54.63 $97.76 $49.16 $87.96
Dec-06 833 $2.31 24.5 43.9 $56.69 $101.45 $51.09 $91.43
Jan-07 833 $2.24 24.5 43.9 $54.90 $98.25 $49.56 $88.69
Feb-07 833 $2.28 24.5 43.9 $55.83 $99.91 $50.49 $90.35
Mar-07 833 $2.56 24.5 43.9 $62.82 $112.41 $56.89 $101.80
Apr-07 833 $2.85 24.5 43.9 $69.73 $124.78 $63.26 $113.20
May-07 833 $3.15 24.5 43.9 $77.11 $137.98 $70.06 $125.38
Jun-07 833 $3.06 24.5 43.9 $74.90 $134.04 $68.17 $121.99
Jul-07 833 $2.97 24.5 43.9 $72.67 $130.04 $66.25 $118.55
Aug-07 833 $2.79 24.5 43.9 $68.28 $122.19 $62.36 $111.58
Sep-07 833 $2.80 24.5 43.9 $68.70 $122.94 $62.84 $112.45
Oct-07 833 $2.80 24.5 43.9 $68.70 $122.94 $62.94 $112.64
Nov-07 833 $3.08 24.5 43.9 $75.49 $135.09 $69.28 $123.98
Dec-07 833 $3.02 24.5 43.9 $73.97 $132.37 $68.00 $121.68
Jan-08 833 $3.04 24.5 43.9 $74.58 $133.46 $68.68 $122.89
Feb-08 833 $3.03 24.5 43.9 $74.22 $132.81 $68.45 $122.49
Mar-08 833 $3.24 24.5 43.9 $79.51 $142.28 $73.45 $131.44
Apr-08 833 $3.46 24.5 43.9 $84.75 $151.67 $78.43 $140.34
May-08 833 $3.77 24.5 43.9 $92.30 $165.18 $85.55 $153.09
Jun-08 833 $4.05 24.5 43.9 $99.36 $177.81 $92.25 $165.08
Jul-08 833 $4.06 24.5 43.9 $99.56 $178.16 $92.58 $165.67
Aug-08 833 $3.78 24.5 43.9 $92.62 $165.75 $86.28 $154.39
Sep-08 833 $3.70 24.5 43.9 $90.76 $162.41 $84.68 $151.54
Oct-08 833 $3.05 24.5 43.9 $74.78 $133.82 $69.89 $125.06
Nov-08 833 $2.15 24.5 43.9 $52.62 $94.17 $49.26 $88.15
Dec-08 833 $1.69 24.5 43.9 $41.35 $73.99 $38.77 $69.38
Jan-09 833 $1.79 24.5 43.9 $43.82 $78.42 $41.16 $73.66
Feb-09 833 $1.92 24.5 43.9 $47.13 $84.34 $44.34 $79.35
Mar-09 833 $1.96 24.5 43.9 $48.01 $85.92 $45.24 $80.96
Apr-09 833 $2.05 24.5 43.9 $50.22 $89.87 $47.40 $84.82
May-09 833 $2.27 24.5 43.9 $55.54 $99.39 $52.51 $93.96
Jun-09 833 $2.63 24.5 43.9 $64.49 $115.39 $61.07 $109.28
Jul-09 833 $2.53 24.5 43.9 $61.94 $110.83 $58.75 $105.13
Aug-09 833 $2.62 24.5 43.9 $64.12 $114.74 $60.92 $109.01
Sep-09 833 $2.55 24.5 43.9 $62.60 $112.02 $59.58 $106.61
Oct-09 833 $2.55 24.5 43.9 $62.52 $111.89 $59.60 $106.66
Nov-09 833 $2.65 24.5 43.9 $64.98 $116.27 $62.04 $111.03
Dec-09 833 $2.61 24.5 43.9 $63.90 $114.34 $61.11 $109.36
Jan-10 833 $2.72 24.5 43.9 $66.54 $119.08 $63.75 $114.08
Feb-10 833 $2.64 24.5 43.9 $64.80 $115.96 $62.19 $111.29
Mar-10 833 $2.77 24.5 43.9 $67.94 $121.58 $65.30 $116.85
Apr-10 833 $2.85 24.5 43.9 $69.80 $124.91 $67.20 $120.26
May-10 833 $2.84 24.5 43.9 $69.51 $124.39 $67.03 $119.95
Jun-10 833 $2.73 24.5 43.9 $66.96 $119.82 $64.68 $115.74
Jul-10 833 $2.73 24.5 43.9 $66.89 $119.69 $64.71 $115.80
Aug-10 833 $2.73 24.5 43.9 $66.91 $119.74 $64.85 $116.04
Sep-10 833 $2.71 24.5 43.9 $66.30 $118.64 $64.36 $115.17
Oct-10 833 $2.80 24.5 43.9 $68.65 $122.85 $66.75 $119.45
Nov-10 833 $2.86 24.5 43.9 $70.07 $125.39 $68.25 $122.13
Dec-10 833 $2.99 24.5 43.9 $73.36 $131.27 $71.57 $128.06
Jan-11 833 $3.10 24.5 43.9 $75.86 $135.75 $74.13 $132.65
Feb-11 833 $3.21 24.5 43.9 $78.70 $140.83 $77.04 $137.85
Mar-11 833 $3.56 24.5 43.9 $87.28 $156.18 $85.56 $153.11
Apr-11 833 $3.80 24.5 43.9 $93.14 $166.67 $91.46 $163.66
May-11 833 $3.91 24.5 43.9 $95.74 $171.32 $94.16 $168.50
Jun-11 833 $3.68 24.5 43.9 $90.20 $161.40 $88.87 $159.02
Jul-11 833 $3.65 24.5 43.9 $89.46 $160.09 $88.28 $157.98
Aug-11 833 $3.64 24.5 43.9 $89.19 $159.61 $88.17 $157.77
Sep-11 833 $3.61 24.5 43.9 $88.50 $158.38 $87.64 $156.82
Oct-11 833 $3.45 24.5 43.9 $84.51 $151.23 $83.82 $149.99
Nov-11 833 $3.38 24.5 43.9 $82.94 $148.42 $82.40 $147.45
Dec-11 833 $3.27 24.5 43.9 $80.05 $143.25 $79.65 $142.54
Jan-12 833 $3.38 24.5 43.9 $82.84 $148.25 $82.57 $147.76
Feb-12 833 $3.58 24.5 43.9 $87.72 $156.97 $87.58 $156.73

the license plate game: the raw numbers

My last post discussed how one might estimate how many state license plates one would expect to see on a road trip. I made a spreadsheet to compute the probability of seeing each state license plate.


  1. The probability of seeing a state license plate A in another state B depends on the distance between their state capitals. It is scaled by the  number of licensed drivers in state A. (This indirectly means that the probability does not depend on how long we are in a state).
  2. Seeing state license plates A, B, etc. are independent from other license plates in a given state D.
  3. Seeing given state license plate A is independent when driving across states B, C,…
  4. We do not adjust for round trips.

The distance between state capitals was found here. The number of licensed drivers per state is here. I estimated the odds of seeing a license plate from state A in state B is captured by this formula:

P = exp(-K * (Distance from A to B in miles) / # of licensed drivers)

with K = 7000 – 2000*Summer01 – 1000*ExpensiveGas01. Summer01 is 1 if it is summer break and 0 otherwise. ExpensiveGas01 is 1 if it gas is “expensive” and AAA predicts that road trips will be down and 0 otherwise. I didn’t have time to properly identify a meaningful formula or calibrate the parameters. Suggestions here are welcome!


  • We predicted 28.3 states for our summer trip from Richmond to Chicago. We saw ~35. Here, the discrepancy seemed to be the amount of time we spent in each state. We went through fewer states, but was in each state (especially Kentucky and Indiana) a relatively long time.
  • We predicted 26.8 license plates for our winter trip from Richmond to Vermont. We saw 26. Not bad!

The results make me conclude that the first assumption is probably not true: the probabilities do depend on how long we are in a state. When driving to Vermont, we went through many (8) little states. When driving to Chicago, we went through fewer (5) states but were in each state for longer.  Moreover, many of the Midwest states are not “destination” states. Take Indiana for instance. I love Hoosiers as much as the next person, but Indiana truly is the “Crossroads of America”–it’s a state that many people from other states drive through. It’s a better place to spot license plates than, say, Delaware. I didn’t take that into account.

Below is a detailed review of our winter trip numbers. It indicates the predicted probability of seeing each state license plate and whether we actually saw it. As asterisk (*) indicates whether the model is “off”–whether we (1) did not see a state with probability greater than 0.5 or (2) did not see a state with a probability of 0.5 or lower.

A copy of my spreadsheet is here if you want to see how I computed the numbers.

State Cumulative probability of seeing each state      States we saw
Alabama 0.671 *
Alaska 0 Yes *
Arizona 0.065
Arkansas 0.060
California 0.961 Yes
Colorado 0.083 Yes *
Connecticut 1 Yes
Delaware 1 Yes
District of Columbia 1 Yes
Florida 0.999 Yes
Georgia 0.971 Yes
Hawaii 0
Idaho 0
Illinois 0.973 Yes
Indiana 0.950 *
Iowa 0.056
Kansas 0.028
Kentucky 0.710 *
Louisiana 0.236
Maine 0.565 Yes
Maryland 1 Yes
Massachusetts 1 Yes
Michigan 0.990 *
Minnesota 0.269
Mississippi 0.060 Yes *
Missouri 0.563 Yes
Montana 0
Nebraska 0.001
Nevada 3.53E-06
New Hampshire 0.911 Yes
New Jersey 1 Yes
New Mexico 3.14E-05
New York 1 Yes
North Carolina 0.999 Yes
North Dakota 0
Ohio 0.998 Yes
Oklahoma 0.032 Yes *
Oregon 0.0006
Pennsylvania 0.999 Yes
Rhode Island 0.863 *
South Carolina 0.878 Yes
South Dakota 0
Tennessee 0.841 *
Texas 0.983 Yes
Utah 5.61E-05
Vermont 1 Yes
Virginia 1 Yes
Washington 0.037
West Virginia 0.416
Wisconsin 0.671 Yes
Wyoming 0

the license plate game

My family took a lot of road trips when I grew up. To combat boredom, we tried to see how many state license plates we would see on our trip. On a trip to see Mount Rushmore, we found almost all of the states.

As an adult and geek, the license plate game has (subtlety?) changed. Now, I combat boredom by talking with my husband about how to come up with a probability distribution for how many state license plates we would expect to see on a road trip from point A to point B.

We took two road trips this year: one from Richmond, VA to Chicago, IL over the summer, the second from Richmond, VA to Burlington, VT over the winter break. We saw ~35 states in our first trip and ~25 states in our second trip.  My husband and I immediately noticed that we accrued license plates at a slower rate on our winter trip, which we suspect was from fewer people making road trips over the winter as compared to summer.

We wondered if one could estimate how many license plates you would expect to see in a road trip based on

  • the states you drive through,
  • the time of year (more people take road trips in the summer)

The state that you are in determines how likely you are to see other state license plates based on their relative distances as well as the number of licensed drivers in other states.

We simplified the problem to avoid looking at how long you drove through a state as well as interstate connectivity issues. That is, there is no difference between driving through West Virginia on I-70 and driving through Pennsylvania on I-80. Additionally, if you are in I-80 in Illinois, you are connected to neighbor states Iowa and Indiana but not neighbor states Missouri and Wisconsin, and therefore, one might expect to see Iowa and Indiana plates. We ignored this and just noted that you would be in Illinois, which gives the likelihood of seeing license plates from other states regardless of “route distance.”

My next post summarizes the model, the assumptions, and the results.

Have you tallied license plates on road trips? What do you think are the salient aspects of this problem to include in a probability model?

amazing facts about the origins of the U.S. highway system

Americans love their cars.  Our love of cars goes back more than 100 years. Americans fell in love with cars when they were first introduced. They embraced the automobile like nothing else and demanded that a transportation system be built to accommodate their wanderlust.

I learned this in a podcast with Earl Swift, author of The Big Roads: The Untold Story of the Engineers, Visionaries, and Trailblazers Who Created the American Superhighways. In the podcast, Earl Swift discussed the origins about the U.S. highway system. I learned quite a few things about our highways.

The interstate system did not begin in the 1950s with Eisenhower, as I had always believed. It began in 1913, when the inevitable dominance of the automobile over other forms of transportation was certain. Early adopters of the cars drove on dirt roads that were incredibly muddy.  The roads resembled the tracks that farm equipment leave behind next to fields of corn and soybeans. Driving from coast to coast was impossible. A US FHWA cite describes the roads at the time:

Railroads dominated interstate transportation of people and goods. Roads were primarily of local interest. Outside cities, “market roads” were maintained, for better or worse, by counties or townships. Many States were prohibited by their constitution from paying for “internal improvements,” such as road projects. The Federal-aid highway program would not begin until 1916 and, because of structural problems and the advent of World War I in 1917, would not accomplish much until 1921. The country had approximately 2,199,600 miles of rural roads and only 190,476 miles (8.66 percent of the total) had improved surfaces of gravel, stone, sand-clay, brick, shells, oiled earth, bituminous or, as a U.S. Bureau of Public Roads (BPR) bulletin put it, “etc.” Many people thought of interstate roads as “peacock alleys” intended for the enjoyment of wealthy travelers who had time to spend weeks riding around the country in their automobiles.

In 1913, private entrepreneurs led by Carl Fisher (of the Indianpolis 500 fame) established the Lincoln Highway Association to build the first paved highway in the United States: The Lincoln Highway (pictured below). The 3,398 mile Lincoln Highway eventually linked New York to San Francisco via Chicago (it follows the path near what is now I-80). This  “Coast-to-Coast Rock Highway” simply transformed the transportation system in the U.S. I included two pictures from 1920 and 1922 below to show how radically different the road looked from the dirt roads it replaced. I drove by the Lincoln Highway (now Route 30) every time I came home from college. I had no idea that it had such an interesting birth story.

The next major highway was the Dixie Highway (pictured below), which linked Chicago and Michigan to Miami. Not coincidentally, this is when Miami Beach was established (Carl Fisher was one of the developers). The Dixie Highway crossed the Appalachian Mountains, which was a big deal at the time. The highway looks like a ladder (see the picture below), because every city and municipality clamored to be along the route. These highways were a game-changer. The Dixie Highway still goes through Chicago. I knew of it was growing up, and as a result, I never associated the word “Dixie” with the South, even though it was originally intended to link Chicago to the South.

The later expressways (Eisenhower’s highways) were not originally designed to move around troops and missiles (at least Congress never seriously used that for justification). They were built and designed to support the many drivers in the U.S. Our economy depended on our transportation system. It still does.

It took some time to build the interstate system as we now know it. My grandfather grew up in Chicago in the 1920s and 1930s, where horse drawn carts delivered milk and other commodities. This was after the highway system began, but before cars completely replaced other modes of transportation. My grandfather remembers all of the horse manure that collected in the streets and alleyways. Most of the games he and the other kids played in the streets hard to be designed to work around the manure and the infectious diseases that could be transmitted from horse to human. This is yet another advantage of cars (Freakonomics covered the environmental trade-offs between car and horse transportation. Cars win).

Of course, the highway system has its problems. Despite the popular support for a highway system, there were many early critics who were ahead of their time in identifying the downsides to highways. For example, Lewis Mumford preached that congestion could not be cured by capacity. In 1955, Mumford wrote, “People, it seems, find it hard to believe that the cure for congestion is not more facilities for congestion.” (see this Atlantic Cities article by Eric Jaffe for more on this topic).

Amazing transportation systems in the United States and elsewhere provide a foundation for operations research. The highway system allows us to transport goods from one end of the country to another in mere days. The highway system inspired many OR problems that use the Traveling Salesman Problem, truck routing problems, facility location models, shortest path problems, and many other classic operations research problems. The Lincoln Highway is where much of it began.

Map of the Lincoln Highway

The Lincoln Highway, 1920. It looks nothing like the dirt roads it replaced.

The Lincoln Highway, 1922

Map of the Dixie Highway

my first sleeper car in a train

I took Amtrak to and from the Society for Risk Analysis Annual Meeting. I took an overnight train home and reserved a sleeper car (a “roomette”). It was fantastic! The room was about two seats wide and long enough for me to lie down. In the tiny room was  retractable sink and toilet. A second bed could be lowered from the ceiling for a roommate. During the day, the bed could be converted to two seats. My trip was entirely overnight, so I only experienced the room in its nighttime mode. The roomette was a bit pricey ($194 plus the purchase of a regular train ticket), but it is comparable to a hotel stay and includes meals and excellent service. It is a real treat if you only fly/ride coach like me.

I took a few pictures of my roomette because I was so happy with it. Sorry the quality was so bad–I had to use my phone camera and tablet camera, neither of which take good pictures.


A picture of a roomette (from Amtrak). The room has a door that closes and locks for privacy.

A picture of a roomette (from Amtrak). The switches above the bed control several lights in the room. This was one of the many thoughtful details in the room.

A picture of my roomette bed.

Roomette sink and toilet

Roomette sink and toilet (folded up)

Roomette amenities included drinks, soap, towels, and dixie cups.

The toilet--seen with the lid on here--is literally in front of the door and window (open to the aisle). If you close the door and blinds, you have privacy.

Roomette sink and toilet

Roomette sink and toilet (open!).

Roomette sink

A closer look at the roomette sink. As you can see, there are outlets here for charging my laptop and using other electronics.

A closer look at the folded up roomette toilet. This was the only place to keep my luggage while sleeping.

Amtrak gave me my own little roll of toilet paper. It is the cutest toilet paper I have ever seen.

Those of you who have spent time in Europe may used to nice train accommodations. I have enjoyed the Amtrak service here in the Eastern United States, but it’s not equivalent to the trains in Europe. The sleeper car experience was a pleasant surprise that I will certainly try again.

What is your best train experience?

pumpkin patches and queuing theory

This weekend, my family and I went to a pumpkin patch. Everyone else had the same idea. The line stretched out of the pumpkin patch gates and through the parking lot.  We waited in line for ten minutes and then balked. When we left, about 90% of those that were leaving did not have pumpkins. We arrived in the morning on Sunday. It was only going to get busier. I cannot imagine the amount of revenue that was lost. We found out later that it took nearly two hours to get through the line.

During our short wait and on our drive to another orchard, we discussed queuing and pumpkin patches.

First, the pumpkin patch could make money by moving the long line inside  of the gates. Quite a few people left only because they could see how long the line was from their cars as they drove in. If they committed to at least getting out of their cars, they might have stayed long enough to buy pumpkins. Queuing in the parking lot was also a safety issue–nearly everyone in line had small children.

The long line was caused by the wait for the hayrides to the pumpkin fields. We couldn’t see the pumpkin fields from the front of the line.  It is a long walk, which is hard with small children and large pumpkins. The only bottleneck was for the hayrides. The traffic outside of the pumpkin patch was not too congested and the parking lot was not nearly full. The pumpkin patch hired people to make sure there was no gridlock in the parking lot (there wasn’t), but overlooked the bottleneck for the hayrides.

There was plenty to do at the pumpkin patch aside from picking pumpkins. There was a store, haunted house, and a restaurant. If people had been given tickets for a scheduled FCFS hayride (with a time of when the hayride would be leaving), they could spend money while they waited for their turn. Waiting two hours for a pumpkin isn’t so bad if you are enjoying donuts and cider.

The queue leveled off and seemed to reach steady state. The rate out only equaled the rate in because so many balked at the long line. For those of you who study queues, is this typical?

Picking pumpkins should be modeled as an infinite server queue, and in practice, this assumption should be a good approximation. (An infinite server queue can model self-service). Maybe this could happen if pumpkins are planted closer to the entrance, so some people could bypass the hayrides and walk instead. This should be somewhat achievable: when we pick apples, many more people can pick at the same time.

There are real constraints that would ultimately limit the pumpkin patch throughput, even if they had a continuous stream of tractors to and from the pumpkin fields. This can be contrasted with, say, an amusement park or stadium, where a large number of people can be served in a short amount of time (even stadiums have their limits).

  • The country road that leads up to the pumpkin patch is a two lane highway. It can only accommodate so many cars per unit of time.  Having someone direct traffic on busy weekends could help if it gets bad (the apple orchard we went to does this).
  • The parking lot was on grass. People drive slow and awkwardly on grass. The pumpkin patch isn’t profitable enough to warrant paving a huge lot to improve throughput. Amusement parks have many lanes so people can park in a short amount of time. That wouldn’t work in agricultural settings. Some congestion in the parking lot seems inevitable.
  • Ultimately, there is a choke point at these types of orchards, where everyone goes through some central area to pay. Even with many makeshift registers set up outside the barn, paying for pumpkins would get pretty chaotic if it got too busy, even if the registers are well-staged. The places I have seen that efficiently coordinate a huge number of cash registers are well-designed and inside.

So now that I figured out how to efficiently run a pumpkin patch, should I open my own?

pumpkin patch

a Christmas brain teaser

When finding some math worksheets to occupy my six year old daughter on her day off of school, I discovered a Christmas brain teaser for elementary students (in pdf format):

Find a route [between twelve cities] for Santa to follow that is as short as possible. After you have found a route, compare it to others to see if they found a shorter route.

Of course, you will immediately recognize this as the TSP.  The “solution” is amusing:

Try to view this question as open-ended, or your student(s) might be working on it for days. …And if someone figures out the best answer to this question, please let me know and I’ll add it here.

Finding the optimal solution to a twelve city TSP is indeed pretty hard.  So hard, in fact, that the folks at math-drills.com couldn’t find the solution.  I have confidence that my readers can find the optimal solution in less than a day.  But please spend Christmas with your families enjoying holiday cheer.

Happy holidays!

shortest paths and Thanksgiving travel

My annual Thanksgiving post is going to be a short one, mainly since I suspect that many of my readers are already traveling.  This past week, my family decided to visit family in Connecticut and New York City.  We are excited to celebrate with family and to show the kids the NYC skyline, Central Park, and the Statue of Liberty.  The biggest challenge is how to find the shortest path from point Richmond, VA to Connecticut, where the shortest path is measured in hours, not miles.  The routes take us past five metro regions: DC, Baltimore, Philadelphia, Newark, and New York City, with opportunities to be stuck in traffic throughout our trip.

We initially planned to take the train.  Amtrak (train) tickets are usually a bargain, but they were much more expensive for this holiday weekend, so we opted to drive.  Given that we are driving, we have some leeway about when we leave and the routes we take.  The biggest problem is uncertainty: we’ve never driven in this neck of the woods during normal traffic conditions, let alone holiday traffic conditions.  We’ve had some input from friends, family, coworkers, and the news.  For example, AAA predicts a 12% increase in driving this year, with today (Wednesday) being the most traveled day of the VDOT recommends avoiding Interstate 95 and traveling on Wednesday. As for the specifics of the route, we can’t simply rely on GPS, since our GPS (like most GPSs) recommend a single path based on the shortest route in miles, which follows Interstate 95 the entire path in this case.  Based on the feedback we’ve received, I printed out google map directions based on our a priori belief about what the shortest path will be.  An atlas will help us to be flexible and adjust our route in the moment.  I’ll keep my fingers crossed.

Have a great Thanksgiving!


carpooling using social networking

My university just launched a social networking site (hosted by Zimride–see a short video here) for students, staff, and faculty to rideshare and carpool.  Users can offer a ride by posting time, location information, and price, or they can search for existing rides.  Zimride seems to work using Google Maps, databases, social networking practices, and decent search options. Zimride was inspired by Zimbabwe’s low tech ridesharing system, which works well without the aid of google.  My university’s Zimride system been operational for about a week, but a quick search already indicates several rides in my neck of the woods.

There are not many economic incentives for carpooling in the metro Richmond area, which is why the bus system isn’t as popular as it could it. But I wonder if the wide use of social networking tools combined with a heavy dose of eco-guilt will help it take off. The benefit of Zimride is that it creates a flexible transportation system that can supplement public transportation routes.  It can be extremely helpful to those who cannot drive due to disabilities or health problems.  I am anxious to see how users create a set of ridesharing paths and hubs where carpoolers can meet up, since users will be indirectly recommending bus stops and routes for a better public transportation system.  The routes will reflect seasonal demand more so than a subway or bus system would.

My campus is a good candidate for ridesharing tools, since the options with the public bus system are limited.  Most bus routes go through the downtown, but most students–and I–are on the uptown campus.  Since I need to work around my kids’ schedule, I’m not sure if I will be able to take advantage of Zimride except in am energency, but I am glad to know that getting to campus will be a lot easier if something comes up.

On a side note, I tested out Zimride’s commute calendar, which  allows users to organize their trips (whether they rideshare, walk, bicycle or drive to work) and then generate a report about miles saved, CO2 saved, and dollars saved.  The report has some naive assumptions (e.g., it doesn’t take the miles driven to a rideshare location into account, it assumes a running cost of $0.55 per mile and 25 miles per gallon), but I suppose the report gets you into the ballpark.  Accuracy probably is not really the point.

Have you used Zimride?

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