Travel Time Equation and Route Efficiency

[adwerkema]

** MATH WARNING **

So, I was thinking about time efficiency when traveling. How do we determine whether a route is time-efficient or not? Sure, we can get a general feel based on traffic and speed limits. However, I couldn't think of a clean-cut way to determine the efficiency of a route. Thus, I decide to make an equation that gave an estimated travel time based on the length of a route. By simply comparing estimated travel time to actual travel time, the efficiency of a route could be determined.



How then, was this equation made? ** Skip to the end if you don't want to read about equation derivation **

First, I started with the fact the most interstates in America have a speed limit of 70 mph. Thus, it's rare that a trip would feature an average speed above 70 mph. Therefore, average speed in the equation needed converge to 70 mph as route length increased. (Most often, more time is spent of an interstate when a trip is longer. Thus would push average speed closer to 70 mph as route length increased.)

Converging to 70 mph, I obtain equation 1:

^Equation 1, where x is route length (in miles) and y is average speed (in mph)^

There are a couple problems with equation 1, though. First, the curve doesn't start at (0,0):

^Equation 1 graph^

This would mean that trips between 0 and 1 miles would have a negative average speed. To account for this, we shift the graph to the left by 1 mile. Thus, we obtain equation 2:

^Equation 2, where x is route length (in miles) and y is average speed (in mph)^

However, equation 2 by itself would be grossly inaccurate. It shows that a 6 mile drive would have an average speed of 60 mph.

To fix this error, another detail has to be added to the equation. I estimated that an average 1-hour drive in America has an average speed of 55 mph. Using this estimate, equation 2 needs to pass through the graph point (55,55). We can make this happen by putting the (x+1) term to an exponential power. The exact exponent needed is 0.38268.

Thus, our expression becomes equation 3:

^Equation 3, where x is route length (in miles) and y is average speed (in mph)^

Finally, it would be useful if equation 3 gave us travel time instead of average speed. This is possible if we take the mileage (x) and divide it by the average speed (y). Thus, we obtain equation 4:

^Equation 4, where x is route length (in miles) and z is travel time (in hours)^

The expression is almost complete; we just have to clean it up now. First, we run into the same problem as before: the curve doesn't start at (0,0):

^Equation 4 graph^

We shift the curve to the right and our expression becomes equation 5:

^Equation 5, where x is route length (in miles) and z is travel time (in hours)^

Finally, the last step 
The exponent needs slight adjusting due to the shifting and dividing that previously occurred. I previously estimated that an average 1-hour drive in America has an average speed of 55 mph. Thus, equation 5 needs to pass through the graph point (55,1). Adjusting the exponent gives us 0.3683.



The final travel time equation is:

^Where x is length of route (in miles) and z is travel time (in hours)^

Here are some graphs to visualize the equation:



To see how reliable this equation was, I performed a statistical analysis on it using a random sample of 30 routes.
Here is the website I found my random samples from.

After sampling and calculating the errors, I found this equation had an average error of 3.8%. Thus, I concluded that the travel time equation gives a fairly accurate estimate on the duration of a trip based on its mileage.

Now, back to determining efficiency. Since the travel time equation has a 3.8% average error, I determined that any route whose actual travel time is 3.8% more than the estimated should be deemed as inefficient.

For example, the route length between Philadelphia and Virginia Beach 275 miles. The travel time equation estimates a 275 mile trip to take 4.48 hours. However, the actual travel time is 5.23 hours. 5.23 hours is more than 4.48*1.038. Thus, I would consider the route between Philadelphia and Virginia Beach as inefficient.

To check out the equation yourself, here is a link to an interactive graph.

Thanks for your time and I hope this generates a conversation about other routes that are inefficient.

[kalvado]

That is perfectly true for car with N>150 wheels traveling in interstellar vacuum!
But does it have anything to do with the real road?

[kphoger]

First, I started with the fact the most interstates in America have a speed limit of 70 mph. Thus, it's rare that a trip would feature an average speed above 70 mph.

Considering that something like 18 states have speed limits of 75 mph or faster, it's not exactly rare for a trip to feature an average speed above 70 mph.  Furthermore, my usual cruising speed on Interstate highways in Kansas is 82 mph.  In states where the speed limit is 70 mph, I usually do around 78 or 79 mph.  So converging to 70 mph is pointless to me.

I estimated that an average 1-hour drive in America has an average speed of 55 mph.

Huh?  My typical one-hour drive is generally done on an Interstate or other multi-lane highway.  70 to 75 mph speed limit.  Even on the two-lane roads, the limit is 65 mph, and I typically drive about 72 mph on those roads.



The way I figure drive time of a route is thus:

[1] Plot the route on Google Maps.
[2] Multiply the result by 0.95 to account for normal speeding.
[3] If the route has a lot of stoplights, add an estimated 1 minute per stoplight.
[4] Plan 15-20 minutes per stop (gas station or potty break), at a frequency to be determined by your driving habits.

Even for trips of up to 700 miles, I usually come within 15 minutes of the correct duration this way.

[jeffandnicole]

It depends if you consider just the driving time of your trip, or the entire trip from start to finish. 25 mph side roads, 45 mph local roads, stop signs, traffic lights and even highway ramps can significantly slow down the overall average speed.

[tradephoric]

In the past i tried to consider the "efficiency" of a metro's arterial road network be comparing the "as the crow flies" distance of major surface arterial streets that branch out from downtown to the actual route distance.  The spoke/wheel design of Detroit had the most efficient arterial network while San Francisco (with the roads winding around the bays) had the least efficient network:







Basically all it's saying is that if you live in metro Detroit, there's a good chance you live near a street that will take you downtown in a straight shot.  Is this relevant or important?  Who knows.  But i do know it can be faster to take the surface streets to get downtown Detroit than taking the freeway.

[kphoger]

It depends if you consider just the driving time of your trip, or the entire trip from start to finish. 25 mph side roads, 45 mph local roads, stop signs, traffic lights and even highway ramps can significantly slow down the overall average speed.

Indeed.  How you do road trips makes a big difference.

For example, I posit a hypothetical 660-miles trip with an average speed limit of 70 mph.  You're in your car, and I'm in mine.

You drive exactly the speed limit, and you stop twice along the way for 15 minutes each time.  Maybe gas the first time and a burger at the drive-through the second time.  It ends up taking you 9 hours 56 minutes to complete your trip.

Code Select Expand


TRIP     660     MILES
AVERAGE   70     MPH
DURATION   9.43  HOURS
BREAKS     0.5   HOURS
TOTAL      9.93  HOURS


But I don't drive the speed limit.  In this scenario, I go 8 mph over the speed limit the whole time.  However I take three 15-minute breaks along the way, plus a 45-minute lunch stop.  Because of the additional stops, even though I drive substantially faster than you, I arrive two minutes after you.

Code Select Expand


TRIP     660     MILES
AVERAGE   78     MPH
DURATION   8.46  HOURS
BREAKS     1.5   HOURS
TOTAL      9.96  HOURS


[adwerkema]

First, I started with the fact the most interstates in America have a speed limit of 70 mph. Thus, it's rare that a trip would feature an average speed above 70 mph.

Considering that something like 18 states have speed limits of 75 mph or faster, it's not exactly rare for a trip to feature an average speed above 70 mph.  Furthermore, my usual cruising speed on Interstate highways in Kansas is 82 mph.  In states where the speed limit is 70 mph, I usually do around 78 or 79 mph.  So converging to 70 mph is pointless to me.

I estimated that an average 1-hour drive in America has an average speed of 55 mph.

Huh?  My typical one-hour drive is generally done on an Interstate or other multi-lane highway.  70 to 75 mph speed limit.  Even on the two-lane roads, the limit is 65 mph, and I typically drive about 72 mph on those roads.



The way I figure drive time of a route is thus:

[1] Plot the route on Google Maps.
[2] Multiply the result by 0.95 to account for normal speeding.
[3] If the route has a lot of stoplights, add an estimated 1 minute per stoplight.
[4] Plan 15-20 minutes per stop (gas station or potty break), at a frequency to be determined by your driving habits.

Even for trips of up to 700 miles, I usually come within 15 minutes of the correct duration this way.

I'm not sure you understand the main idea. The purpose of the equation is to estimate travel times throughout America as a whole (including areas like rural Kansas and the cluttered BosWash corridor). If your actual travel time is faster than the equation's estimate (yours would be from the details you provided), then the roads in your area would be deemed as efficient.

[adwerkema]

In the past i tried to consider the "efficiency" of a metro's arterial road network be comparing the "as the crow flies" distance of major surface arterial streets that branch out from downtown to the actual route distance.  The spoke/wheel design of Detroit had the most efficient arterial network while San Francisco (with the roads winding around the bays) had the least efficient network:

Basically all it's saying is that if you live in metro Detroit, there's a good chance you live near a street that will take you downtown in a straight shot.  Is this relevant or important?  Who knows.  But i do know it can be faster to take the surface streets to get downtown Detroit than taking the freeway.

Excellent information! I'm not surprised to see other cities on the list that feature spoke/wheel or grid systems.

[Thing 342]

Worth noting that if you parameterize the model's assumptions, you get this more general equation:


Where s_c is the upper bound for expected average speed (set to 70mph in the OP), and s_d is the lower bound for expected average speed (set to 55mph in the OP). Essentially, the slope of the line (the average speed) converges towards the upper bound at a rate governed by the difference between it and the lower bound.

I'm not really sure that relying on averages to model an effect with a large amount of inherent, nonparametric variability is such a great idea. I'd suggest the use of calculus here, but having to factor in differing speed limits means that such an integral would be no fun to attempt to calculate.

[Rothman]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

[tradephoric]

Excellent information! I'm not surprised to see other cities on the list that feature spoke/wheel or grid systems.

Thanks adwerkema.  The city that surprised me when i did this little analysis was Phoenix.  It has a very structured grid network yet its "as the crow flies" efficiency isn't very high (even the spaghetti streets of Boston and Atlanta have a higher efficiency than Phoenix).  The problem with Phoenix is if you live NE or SE of downtown there is no direct route to get downtown.

[adwerkema]

Worth noting that if you parameterize the model's assumptions, you get this more general equation:


Where s_c is the upper bound for expected average speed (set to 70mph in the OP), and s_d is the lower bound for expected average speed (set to 55mph in the OP). Essentially, the slope of the line (the average speed) converges towards the upper bound at a rate governed by the difference between it and the lower bound.

I'm not really sure that relying on averages to model an effect with a large amount of inherent, nonparametric variability is such a great idea. I'd suggest the use of calculus here, but having to factor in differing speed limits means that such an integral would be no fun to attempt to calculate.

Thanks for the generalization. Using that, users would be able to fine-tune the equation based on regional speed limits, traffic conditions, and driving habits. And I agree, there is a large amount of variability that is not accounted for in the equation. My goal for the equation was to purposely not account for this variability. The equation is supposed to produce a general result that when compared to the actual, reveals the effects of variability (either a slower or faster travel time).

[adwerkema]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

Perhaps "efficiency" was not my best choice of words. The equation, when compared to reality, shows if a route has a slower or faster travel time compared to America's average travel time for that given distance.

[jeffandnicole]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

Perhaps "efficiency" was not my best choice of words. The equation, when compared to reality, shows if a route has a slower or faster travel time compared to America's average travel time for that given distance.

Well, then what is America's average road trip? You have seemed to based it on what you feel is the average top speed limit in each state. However, traffic volumes would dictate that a highway with a speed limit of 80 has very few people in those 80 zones. In the Northeast, most highways top out at 65 mph, but will have upwards of 10 times more traffic than a very rural 80 mph highway.

This is typical to threads on here when people post a news story with some vague criteria. Usually they take some piece of info and try to base their story on that piece of info, regardless if the two are even compariable.

[adwerkema]

Well, then what is America's average road trip? You have seemed to based it on what you feel is the average top speed limit in each state. However, traffic volumes would dictate that a highway with a speed limit of 80 has very few people in those 80 zones. In the Northeast, most highways top out at 65 mph, but will have upwards of 10 times more traffic than a very rural 80 mph highway.

This is typical to threads on here when people post a news story with some vague criteria. Usually they take some piece of info and try to base their story on that piece of info, regardless if the two are even compariable.

The equation created is a model for America's average road trip. To create such a general equation, assumptions were made about average speeds in America. The assumptions about averages may be inaccurate - I assumed to the best of my knowledge.

[froggie]

Considering that something like 18 states have speed limits of 75 mph or faster, it's not exactly rare for a trip to feature an average speed above 70 mph

May not be rare.  But given the bulk of those states and segments are in areas with light population and traffic compared to the national average, I wouldn't exactly call it common either.  He's right in that the vast bulk of American trips have an average speed of 70 or less.

I'd also argue that assuming an average speed closer to (or even under) the speed limit of a given highway is an effective counter to the time taken for gas/bathroom/food stops.

[Life in Paradise]

One other slight factor is that your speedometer is most likely off at 70 MPH.  It could be 68 or 69.  They are more likely a fraction lower than the showing speed.  (this is with factory supplied tires/wheels, etc).  I was curious when my GPS in several different vehicles would always show that I was traveling slower than my speedometer, then I did some research into why.

[kphoger]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

Perhaps "efficiency" was not my best choice of words. The equation, when compared to reality, shows if a route has a slower or faster travel time compared to America's average travel time for that given distance.

Isn't there a simpler solution, then?

(1)  Let n = what you think the average speed limit is.
(2)  Let s = the speed limit of your route.
(3)  If n<s, then your route is efficient.  If n>s, then it isn't.

[hotdogPi]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

Perhaps "efficiency" was not my best choice of words. The equation, when compared to reality, shows if a route has a slower or faster travel time compared to America's average travel time for that given distance.

Isn't there a simpler solution, then?

(1)  Let n = what you think the average speed limit is.
(2)  Let s = the speed limit of your route.
(3)  If n<s, then your route is efficient.  If n>s, then it isn't.

1. Short commutes are more likely to use surface roads.
2. Straight lines are faster, which is why Memphis to Kansas City is inefficient.

[kphoger]

I don't see how an equation based upon a bunch of averages would be a measure of efficiency, whatever that means.  Not sure what you're supposed to do with an inefficient route (either from a DOT or traveler's perspective), either.

Perhaps "efficiency" was not my best choice of words. The equation, when compared to reality, shows if a route has a slower or faster travel time compared to America's average travel time for that given distance.

Isn't there a simpler solution, then?

(1)  Let n = what you think the average speed limit is.
(2)  Let s = the speed limit of your route.
(3)  If n<s, then your route is efficient.  If n>s, then it isn't.

1. Short commutes are more likely to use surface roads.
2. Straight lines are faster, which is why Memphis to Kansas City is inefficient.

1.  If short commutes are more likely to use surface roads, then the routes in question are less efficient than average.  My morning commute has an average speed limit of 56.9 mph.  That's less efficient than, say, 70 mph.  But urban roads are bound to be less efficient than rural roads, because the speed limits tend to be lower.  But isn't that the point of the OP?  It's to find out if my morning commute (or any trip) is more or less efficient than the average trip in America.

2.  Straight lines are indeed faster, but the OP mentioned nothing about measuring straight-line distances.  From Memphis to KC, you could reasonably go US-63 to Willow Springs, US-60 to Springfield, MO-13 and MO-7 to Harrisonville, then I-49 to KC.  But you could also reasonably go I-55 to Saint Louis, then I-70 to KC.  By the OP criteria, the more efficient route has nothing to do with the straight-line distance between Memphis and KC.  Unless I'm reading it wrong.

[webny99]

I read the thread title and think of traffic flow and congestion, not speed limits. Not "how fast should I go?" but more along the lines of "how fast can I go?". And what role do other drivers play in efficiency, by keeping right, not braking excessively, and so on.

Either way, I'm pretty sure my commute is more efficient than average, especially if I take the highway route (which is longer - but I'm skeptical that that's relevant here).