# How To Beat Traffic Mathematically

## Woe Is Traffic

Traffic: the commuter’s bane. It plagues major city drivers around the globe and shows no sign of letting up.1 In fact, the average U.S. commuter spends about 100 hours a year driving just to work – 20 hours more than a typical year’s supply of vacation.2 This personal “daily grind” uses more than 15,000 miles and 1,000 gallons of gas every year, which might not be so bad if much of it wasn’t waste: 1.6 million hours and 8 million gallons of gas are wasted every day in traffic jams across the nation. Traffic even affects your health, raising blood pressure, increasing stress, and producing more Type-A personalities.3

Of course, some places are much worse than others. New York tops the list, with Chicago, Newark and Riverside following, albeit at a distance. L.A. comes in at #6 and Houston, where I reside and commute, is #15.4 Other cities, such as Nashville, TN and Kansas City, MO, show up much further down the list, but something tells me that even commuters in those relative traffic havens dedicate significant effort and conversation to ‘beating traffic.’

Resources are sometimes available to help in this quest. Houston Transtar provides up to the minute traffic information for all major Houston highways.5 Average traveling speed, construction and accident information are all available at the click of the mouse, but how to avoid the perpetual web of red during the morning and evening rush hours is nowhere to be found. Obvious answers such as public transportation and carpooling are legitimate, but trends show that Americans are meeting the increase in traffic by using such transportation methods less, not more.6 Also, if the online traffic-reporting graphic warns of potential issues, there is no indication of how long they might persist, leaving the traffic conscientious commuter right where he started: guessing.

Tired of the typically inefficient and contradictory workplace chatter on the subject and feeling the pull of a slight worksheet obsession, I set out to statistically analyze my commute in order to determine how I might minimize my time behind the wheel. If there was a way to figure out how to give myself an advantage over the almost 900,000 other Houstonian workers out there (who average a 26.1 minute commute),7 math and a smidgeon of obsessive compulsive disorder had to be essential ingredients. At the very least, I would be able to ascertain just how much of my commute time was up to me – and how much depended on a “higher power” (e.g., weather, school districts, wrecks, etc.).

## Gathering Data

From March of 2005 to March of 2006, I recorded my departure and arrival times both to and from work, along with whether school was in or out.8 Other factors, although most likely important, were excluded to keep the scope of the experiment narrow and measurable.

### Driving Data

Every morning, I took note of the time on my car clock as I pulled out of my driveway at the Riata Ranch subdivision of northwest Houston9 and then again as I pulled into the parking garage at my office building close to the north-bound frontage road of Sam Houston Pky and Clay Rd.10 In the evening, I followed the same process in reverse. The morning route 11 and evening route12 differed slightly in length, but data was only recorded when the planned course was followed, allowing for only slight variations.13

### School District & Government Data

Being suspicious of the influence of the school session, I collected official 2004-2005 and 2005-2006 calendar data from Cypress Fairbanks Independent School District,14 which covers almost all of my commute route,15 and took note of all full student holidays (i.e., teacher in-service days, but not student early release days).16 I also collected official 2005 and 2006 government holiday information from the city of Houston17 and the US Federal Government,18 but this proved next to useless as I only commuted to work on one city and two federal government holidays.

## Analysis

To set up the gathered information, I first organized the variables into inputs and outputs as shown in Table 1.

 |borderTable 1: Input and Output Variables

To determine which variables had a statistically significant effect on my commute times, I ran one-way ANOVAs19 on the discrete variables and plotted smoothed graphs of means for the continuous variables.20

### Morning Commute ANOVAs

#### Day of the Work Week

The one-way ANOVA of the morning commute duration versus the day of work week (y1 vs. x1) showed a statistically significant effect.21 The table in the ANOVA output22 and the boxplot below confirm that this effect comes on Fridays, on which there is a significantly shorter commute time:

```Source        DF      SS     MS     F      P
Day of Week    4   544.4  136.1  3.87  0.005
Error        202  7103.2   35.2
Total        206  7647.6
S = 5.930   R-Sq = 7.12%   R-Sq(adj) = 5.28%```
```                          Individual 95% CIs For Mean Based on
Pooled StDev
Level   N    Mean  StDev  ----+---------+---------+---------+-----
1      43  22.209  5.726                  (------*------)
2      44  22.886  5.891                    (-------*------)
3      47  23.447  6.382                       (------*------)
4      39  22.462  7.014                  (-------*------)
5      34  18.559  3.855  (-------*-------)
----+---------+---------+---------+-----
17.5      20.0      22.5      25.0
Pooled StDev = 5.930```
 |borderFigure 1. A boxplot of the morning commute time versus the day of the work week.

#### Week of the Month

The results from an ANOVA of the week of the month versus the morning commute duration (y1 vs. x2) showed no statistically significant impact, although week 5 has the highest average commute time:

```Source          DF      SS    MS     F      P
Week of Month    4   226.5  56.6  1.54  0.192
Error          202  7421.1  36.7
Total          206  7647.6
S = 6.061   R-Sq = 2.96%   R-Sq(adj) = 1.04%```
```                          Individual 95% CIs For Mean Based on
Pooled StDev
Level   N    Mean  StDev  ---+---------+---------+---------+------
1      52  22.673  7.040         (------*-----)
2      44  21.636  4.760    (-------*------)
3      51  21.706  6.090     (------*------)
4      42  21.048  5.441  (------*-------)
5      18  24.944  7.075              (----------*----------)
---+---------+---------+---------+------
20.0      22.5      25.0      27.5
Pooled StDev = 6.061```
 |borderFigure 2. A boxplot of the morning commute time versus the week of the month.

#### Month of the Year

The month of the year versus morning commute time (y1 vs. x3) ANOVA results showed even less of an effect:

```Source          DF      SS    MS     F      P
Month of Year   11   496.5  45.1  1.23  0.269
Error          195  7151.1  36.7
Total          206  7647.6
S = 6.056   R-Sq = 6.49%   R-Sq(adj) = 1.22%```
```                          Individual 95% CIs For Mean Based on
Pooled StDev
Level   N    Mean  StDev  ------+---------+---------+---------+---
1     21  22.476  5.793              (--------*--------)
2      8  22.875  4.121          (-------------*-------------)
3     19  24.053  7.764                   (--------*--------)
4     19  22.737  4.053               (--------*--------)
5     19  23.842  5.650                  (--------*---------)
6     18  21.722  5.278           (--------*---------)
7     19  18.947  4.441  (--------*--------)
8     23  20.652  5.556         (-------*-------)
9     17  21.824  7.502           (---------*--------)
10     17  21.353  4.182          (--------*---------)
11     16  24.250  9.774                   (---------*---------)
12     11  20.545  5.336    (-----------*-----------)
------+---------+---------+---------+---
18.0      21.0      24.0      27.0
Pooled StDev = 6.056```
 |borderFigure 3. A boxplot of the morning commute time versus the month of the year.

#### Cypress-Fairbanks ISD

Whether or not the local school district was in session proved to be the greatest measured variable in explaining the morning commute time variation (y1 vs. x6):

```Source   DF      SS     MS      F      P
CyFair    1   774.0  774.0  23.08  0.000
Error   205  6873.6   33.5
Total   206  7647.6
S = 5.791   R-Sq = 10.12%   R-Sq(adj) = 9.68%```
```                           Individual 95% CIs For Mean Based on
Pooled StDev
Level    N    Mean  StDev    -+---------+---------+---------+--------
0       63  19.159  4.646    (------*------)
1      144  23.361  6.222                           (----*----)
-+---------+---------+---------+--------
18.0      20.0      22.0      24.0
Pooled StDev = 5.791```
 |borderFigure 4. A boxplot of the morning commute time versus Cypress-Fairbanks ISD Session.

### Evening Commute ANOVAs

#### Day of the Work Week

While the day of the week proved to have a significant impact on the morning commute, the evening commute showed no such relationship (y2 vs. x1):

```Source        DF      SS    MS     F      P
Day of Week    4    68.5  17.1  0.82  0.516
Error        158  3312.1  21.0
Total        162  3380.7
S = 4.579   R-Sq = 2.03%   R-Sq(adj) = 0.00%```
```                          Individual 95% CIs For Mean Based on Pooled
StDev
Level   N    Mean  StDev     +---------+---------+---------+---------
1      40  22.125  4.333              (--------*--------)
2      40  21.275  5.002         (--------*--------)
3      34  21.706  5.190           (---------*--------)
4      33  20.697  4.149     (--------*---------)
5      16  22.875  3.304              (-------------*-------------)
+---------+---------+---------+---------
19.2      20.8      22.4      24.0
Pooled StDev = 4.579```
 |borderFigure 5. A boxplot of the evening commute time versus the day of the work week.

#### Week of the Month

Again, the week of the month did not explain the commute time variation (y2 vs. x2):

```Source          DF      SS    MS     F      P
Week of Month    4    86.4  21.6  1.04  0.390
Error          158  3294.2  20.8
Total          162  3380.7
S = 4.566   R-Sq = 2.56%   R-Sq(adj) = 0.09%```
```                          Individual 95% CIs For Mean Based on
Pooled StDev
Level   N    Mean  StDev  ---+---------+---------+---------+------
1      34  21.176  4.496   (-------*-------)
2      39  20.769  4.782  (------*------)
3      42  21.857  4.176       (------*------)
4      35  22.000  4.583       (-------*-------)
5      13  23.462  5.238          (-----------*------------)
---+---------+---------+---------+------
20.0      22.0      24.0      26.0
Pooled StDev = 4.566```
 |borderFigure 6. A boxplot of the evening commute time versus the week of the month.

#### Month of the Year

Another change from the morning results, the month of the year proved to have a significant effect, with February, April and November showing the longest evening commute times (y2vs. x3):

```Source          DF      SS    MS     F      P
Month of Year   11   541.2  49.2  2.62  0.004
Error          151  2839.4  18.8
Total          162  3380.7
S = 4.336   R-Sq = 16.01%   R-Sq(adj) = 9.89%```
```                          Individual 95% CIs For Mean Based on
Pooled StDev
Level   N    Mean  StDev  ----+---------+---------+---------+-----
1     15  21.400  3.418          (------*-------)
2      9  24.222  3.833                 (---------*--------)
3     17  20.529  3.319        (-----*------)
4     10  23.700  6.325                (--------*--------)
5     14  20.143  3.416      (------*-------)
6     14  21.357  4.584          (------*-------)
7     14  19.143  4.400  (-------*------)
8     21  21.905  5.078             (-----*-----)
9     14  20.929  4.811        (-------*------)
10     11  20.091  3.590    (--------*--------)
11     16  25.625  4.731                        (------*-------)
12      8  20.625  3.021     (---------*---------)
----+---------+---------+---------+-----
18.0      21.0      24.0      27.0
Pooled StDev = 4.336```
 |borderFigure 7. A boxplot of the evening commute time versus the month of the year.

#### Cypress-Fairbanks ISD

The school session again showed signification influence, but it was not as strong in the evening as in the morning (y2 vs. x6):

```Source   DF      SS     MS     F      P
CyFair    1   106.2  106.2  5.22  0.024
Error   161  3274.4   20.3
Total   162  3380.7
S = 4.510   R-Sq = 3.14%   R-Sq(adj) = 2.54%```
```                           Individual 95% CIs For Mean Based on
Pooled StDev
Level    N    Mean  StDev  ---------+---------+---------+---------+
0       50  20.400  4.677  (------------*------------)
1      113  22.150  4.434                        (--------*-------)
---------+---------+---------+---------+
20.0      21.0      22.0      23.0
Pooled StDev = 4.510```
 |borderFigure 8. A boxplot of the evening commute time versus the Cypress-Fairbanks ISD Session.

### Departure Time Analysis

For the continuous variable of departure time, I plotted smoothed curves of the mean commute time at each minute.

The morning departure time plot shows relatively long commute times until about 7:40AM, at which time a gradual decrease starts that continues in an overall linear fashion for the next hour. After 8:40AM, traffic appears to have only minimal impact. (y1 vs. x4):

 |borderFigure 9. A smoothed plot of the mean of the recorded morning commute durations versus the home departure time.

The evening departure time plot shows a peak commute time at about 5:10PM, tapering off linearly through the next two or so hours. Departure times prior to 5:00PM showed erratic results, but it is obvious that traffic played a decreasing role in evening commute time duration moving back through 4:00PM, before which it’s influence is noticeable, but slight. (y2 vs. x5):

 |borderFigure 10. A smoothed plot of the mean of the recorded evening commute durations versus the work departure time.
 |borderFigure 11. A close-up of Figure 10.

I usually leave home at 8:00AM and work at 5:30PM, but a 30 minute delay of each looks like it would shave five minutes off the morning commute and about 2.5 minutes off the evening. Additional half-hour delays bring 2.5 minutes of commute time savings in the evening, but little to no savings in the morning. Slightly earlier departure times appear to result in commute time increases for both trips. Moving back past 4:30 in the evening brings slight improvement in the evening commute, but savings in the morning would most likely require leaving before 6:30AM.

## Conclusions

Given the above data and analysis, what can be done to improve my commute times? Changing my morning or evening departure time looks promising. The best bet appears to be moving my schedule out a half-hour to 8:30AM and 6:00PM, bringing significant savings (about 7.5 minutes of commute time per day) without getting too far from normal business hours. Spread out over 50 work weeks, that results in a total savings of over 30 hours a year – the equivalent of about a 38% boost to my existing 80 hours of vacation.

Departure time isn’t the say-all, however, and making this shift won’t always result in a smooth and fast commute. The day of the week in the morning and the month of the year in the evening both have significant impacts, and whether or not school is in session affects both. I could possibly squeeze out a few more minutes of savings by scheduling my vacation days to align with the potentially longest commutes (e.g., non-Friday school days in the months of November, February and April), but the data shows significant variation up and above that described by the measured variables – much likely due to factors outside of the control of the commuter (e.g., weather, wrecks, breakdowns, response to traffic predictions, etc.).23

The commuter may have more control than it appears, however. Adjusting your commute times and rearranging your vacation schedule will probably help in the meantime, but getting cars off the road is the only sure solution – one that is within commuters’ sphere of influence.24 It might require punching your “free reign” in the gut, but getting involved in your community by writing your Congressperson or attending city council meetings in promotion/defense of improved mass transit could be the most effective way to “curb” your drive times in the long run.25

## Notes

1. “Beating Traffic.” Mathematical Moments. American Mathematical Society. 2005. Accessed April 2006 from http://www.ams.org/ams/mm31-traffic.pdf. According to the publication, “In the last 30 years while the number of vehicle-miles traveled has more than doubled, physical road space has increased only six percent.”
2. “Americans Spend More Than 100 Hours Commuting to Work Each Year, Census Bureau Reports.” US Census Press Release. March 20, 2005. Accessed April 2006 from http://www.census.gov/Press-Release/www/releases/archives/ american_community_survey_acs/004489.html.
3. “Understanding Traffic.” Discovery Channel Features. January 30, 2006. Accessed April 2006 from http://www.odeo.com/audio/674920/view.
4. “Average Travel Time to Work of Workers 16 Years and Over Who Did Not Work at Home.” U.S. Census Bureau: American Community Survey 2003. Accessed April 2006 from http://www.census.gov/acs/www/Products /Ranking/2003/pdf/R04T160.pdf.
5. Houston Real-Time Traffic Map. HoustonTranstar.org. Accessed April 2006 from http://traffic.houstontranstar.org/layers/.
6. Reschovsky, Clara. “Journey to Work 2000.” US Census Bureau. Accessed April 2006 from http://www.census.gov/prod /2004pubs/c2kbr-33.pdf. According to Table 1: Means of Transportation to Work: 1990 and 2000, 2.5% more Americans drove to work alone in 2000 when compared with ten years earlier. All public transportation used saw at least a minor decline.
7. “Houston city, Texas: Selected Economic Characteristics: 2004.” U.S. Census Bureau: American Fact Finder. Accessed April 2006 from http://factfinder.census.gov/servlet/ADPTable?_bm= y&-geo_id=16000US4835000&-qr_name=ACS_2004_EST_G00 _DP3&-ds_name=ACS_2004_EST_G00_&-_lang=en&-_sse=on.
8. My data can be viewed in the linked Excel file: http://www.omninerd.com/articles/attachments/Brandon/a50/commute_data.xls
9. Google Local – Cypress N Houston Rd & Riata Ranch Blvd, Houston, TX 77095. Google Maps. Accessed April 2006 from http://maps.google.com/maps?f=q&hl=en&q=Cypress+N+ Houston+Rd+%26+Riata+Ranch+Blvd,+Houston,+TX+77095&om=1. My exact home address is withheld purposely.
10. Google Local – W Sam Houston Pky N & Clay Rd, Houston, TX 77041. Google Maps. Accessed April 2006 from http://maps.google.com/maps?f=q&hl=en&q= W+Sam+Houston+Pky+N+%26+Clay+Rd,+Houston,+TX+77041&om=1. Again, the exact details of my office location are purposely omitted.
11. My 12.7 mile route to work consists of the following:
a. Proceed .1 miles from home to Riata Ranch Blvd & Cypress N Houston Rd.
b. Proceed west .2 miles on Cypress N Houston Rd.
c. Turn right on Barker Cypress Rd. Proceed .8 miles.
d. Turn right on US-290 E. Proceed 1 mile.
e. Take US-290 ramp. Proceed 6.7 miles.
f. Take Frontage Road Exit to Beltway 8 / FM-529 / Senate Ave. Proceed .7 miles. (I exit here instead of taking the shorter – and most likely faster – freeway to avoid the toll. Yes, I’m cheap and I like spreadsheets.)
g. Turn right on Senate Ave. Proceed 3.1 miles to Clay Rd.
h. Proceed .1 miles to office.
12. My 13.0 mile route home consists of the following:
a. From the office, proceed north on Sam Houston Parkway frontage road for 3.1 miles.
b. Turn left on US-290 frontage road. Proceed 1.0 mile.
c. Take US-290 ramp. Proceed 6.8 miles.
d. Take Barker Cypress Rd Exit. Proceed .9 miles, veering right at split.
e. Turn left on Barker Cypress Rd. Proceed .9 miles.
f. Turn left on Cypress N Houston Rd. Proceed .2 miles to Riata Ranch Blvd.
g. Proceed .1 miles to home.
13. I occasionally took two variations, one on the way to work and one on the way home. In the morning, I sometimes drove around the south side of the fast food restaurants on the southwest bound frontage road of US-290 to avoid the backup at the light at Senate Ave. In the evening, heading north on Senate Ave, I occasionally continued straight under US-290 to avoid the backup in the left-hand turn lanes. Although the road is not shown on the map, the first left after crossing the US-290 frontage road proceeds about .2 miles, then makes a left turn and dead-ends back into the frontage road. A detail of the US-290 and Senate Ave intersection, which contains both variations, is available from Google Maps: http://maps.google.com/maps?f=q&hl=en&q=US-290 +W+%26+Senate+Ave,+Houston,+TX+77040&ll=29.877 341,-95.564607&spn=0.006549,0.013561&t=h&om=1
15. Harris County Appraisal District: Index Map: By School District. HCAD: I-Map Publication Service. Accessed April 2006 from http://www.hcad.org/maps/default.asp.
16. As an interesting aside, information was also gathered for surrounding school districts:
a. Houston (http://www.houstonisd.org)
b. Katy (http://www.katyisd.org)
c. Klein (http://www.kleinisd.net)
d. Spring Branch (http://www.springbranchisd.com)
e. Tomball (http://www.tomballisd.net)
f. Waller (http://www.waller.isd.esc4.net)
Analysis indicated that these schedules had no statistically significant impact on my commute, confirming that the effect of the school district schedule is limited to within its own boundaries.
17. “Official City Holidays.” HoustonTX.gov. 2006. Accessed April 2006 from http://www.houstontx.gov/abouthouston/cityholidays.html. 2005 city holidays confirmed via Mrs. Wilkerson of Houston City’s 3-1-1 Helpline, accessible per: “Contact Us.” HoustonTX.gov. 2006. Accessed April 2006 from http://www.houstontx.gov/contactus/index.html.
18. “2005 Federal Holidays.” OPM.gov. Accessed April 2006 from http://www.opm.gov/Fedhol/2005.asp. & 2006 Federal Holidays. OPM.gov. Accessed April 2006 from http://www.opm.gov/Fedhol/2006.asp.
19. “ANOVA” stands for ANalysis Of VAriance. For more details on ANOVAs and how/when they are used: “Chapter 12: Introduction to ANOVA.” HyperStat Online Textbook. Accessed April 2006 from http://davidmlane.com/hyperstat/intro_ANOVA.html.
20. Discrete variables are those whose values are represented in a limited set. For example, the “day of the work week” variable consists of five values (“Monday” through “Friday”) and a one-way ANOVA analyzes each to determine if it has a significant impact on the result variation. On the other hand, the “departure time” variable is practically continuous, with as many “categories” as there are minutes, and doesn’t lend itself well to ANOVA analysis.
21. For each of the ANOVA analyses, the significance level (α) is .05 and the null hypothesis (H0) is that the input variable has no statistically significant influence on the output. When the Pvalue < α, H0 is thrown out. For example, in the case of the day of the work week vs. the morning commute, the Pvalue is .005, which is less than .05. Thus, it is statistically improbable that the results could have occurred at random and, therefore, the day of the week is shown to exert a significant effect on the morning commute duration.
22. I used Minitab to run the ANOVAs. The top table of the output lists the output variable (Source), the degrees of freedom (DF), the sum of the squares (SS), the mean of the squares (MS), the Fvalue (F) and the Pvalue (P). The lower table lists the input variables (Level), the number of inputs for each (N), the mean of the inputs (Mean), the standard deviation of the inputs (StDev), and then these mean values graphed with a 95% confidence interval (CI) based on the pooled standard deviation. For more information on interpreting the output of one-way ANOVAs: “How to Read the Output From One Way Analysis of Variance.” Jerry Dallal’s Tufts Home Page. Accessed April 2006 from http://www.tufts.edu/~gdallal/aov1out.htm.
23. Some have even suggested chaos theory and driver psychology as ways to best model traffic behavior. More information on chaos theory and traffic: “Chaos and your everyday Traffic Jam.” FailedSuccess.com. Accessed April 2006 from http://www.failedsuccess.com/index.php?/ weblog/comments/traffic_jam_causes/. More information on driver psychology: Groegera, J. A. and Rothengatter, J. A. “Traffic psychology and behaviour.” Transportation Research Part F: Traffic Psychology and Behaviour. Volume 1, Issue 1, August 1998, Pages 1-9. Accessed April 2006 from http://dx.doi.org/10.1016/S1369-8478(98)00007-2.
24. “Understanding Traffic.” Discovery Channel Features. January 30, 2006. Accessed April 2006 from http://www.odeo.com/audio/674920/view. Every subway train takes 1,000 cars off the road. Every bus, 40 cars.
25. “Critical Relief for Traffic Congestion.” PublicTransportation.org. Accessed April 2006 from http://www.publictransportation.org/pdf/reports/congestion.pdf. Public transportation stands to improve commute times more than departure time adjustment. “The Benefits of Public Transportation: An Overview.” PublicTransportation.org. Accessed April 2006 from http://www.publictransportation.org/reports/asp/pub_benefits.asp. Public transportation brings unparalleled reliability and consistency.