The book, and we, start with a few definitions and a little historic context, to get a better sense of the field. There will be more basics as we go on, mixed in with the more complicated stuff.

• When were the U.S. highways built?
  • 1920–1930
• When were the interstate highways and urban freeways built?
  • 1950–1960
• What improvement has happened to the U.S. car transportation system since the 1960s?
  • Capacity has improved significantly through things like flow analysis and control, rather than building significantly more road.
• What is time headway?
  • The length of time between when the front of two consecutive cars pass an observation point along the road
• Which part of the vehicle is conventionally used for time headway measurements?
  • The front
• Which two components make up the time headway?
  • Occupancy time followed by time gap
• What is the microscopic version of flow rate?
  • Time headway
• What is the microscopic version of density rate?
  • Distance headway

I have long been confused by the terms used for different types of U.S. roads, so I clarified these for myself now that I figured out what is what.

• What is the Swedish concept corresponding to U.S. highways?
  • Landsvägar
• What is the Swedish concept corresponding to U.S. interstate highways?
  • Motorvägar
• What is the Swedish concept corresponding to U.S. urban freeways?
  • Motorvägar

I wouldn’t phrase this next one the same way given what I know now, but when I wrote it down, I had barely started reading the book so the wording used is a little clumsy. I’m okay with that. If I get confused down the line, I can either take this flashcard out or change the wording.

• Why is the minimum time headway an important variable in total capacity?
  • Because the inverse of time headway is literally vehicles per time, i.e. flow rate!

I find that understanding how variation behaves in different circumstances helps a lot in getting a good intuition for a concept, so I made some observations around that based on diagrams in the book.

• What would one expect about the standard deviation of time headways during low flow rates?
  • Roughly equal to the mean, because arrivals don’t interact and thus are Poisson
• As flow increases, does the standard deviation of time headways shrink or grow, absolutely speaking?
  • shrinks, because the distribution narrows toward smaller numbers
• As flow increases, does the standard deviation of time headways shrink or grow, relative to the mean?
  • shrinks, because the distribution narrows toward smaller numbers
• What is the human lower limit for time headway (not mean, but lowest value observed under most conditions)?
  • 0.5 seconds
• The human lower limit for time headway is 0.5 seconds – what’s another way to look at this?
  • Virtually all humans will require a time headway more than 0.5 seconds

We go on to modeling headway with theoretical distributions.

• Which distribution does the authors suggest for modeling time headways during low flow?
  • Exponential
• Which distribution does the authors suggest for modeling time headways during high flow?
  • Normal
• What is the drawback of using the exponential distribution for modeling time headways during low flow?
  • It suggests a mode at t=0 which cannot happen
• What is the drawback of using the normal distribution for modeling time headways during high flow?
  • When fitted based on mean, it suggests people drive less aggressively than they really do, because it’s fitting to a long tail

The author used theoretical distributions I’m not too familiar with, so i looked up some more information on those (mainly on Wikipedia) and added flashcards also for that.5⁵ This may be a faux pas for some, but when reading a book I often end up going on sidequests on Wikipedia or in scientific papers. What I learn on those side quests I include as flashcards ostensibly from the book, even though they were not. Oops.

• What is the value of Γ(4)?
  • 6 (because when k is integer, Γ(k) = (k-1)!)
• What is the value of Γ(19), in factorials?
  • 18!
• What is the relationship between the gamma and Erlang distributions?
  • When the gamma distribution is parametrised with a positive integer k and a rate λ, it becomes an Erlang-k distribution with rate λ
• What is the relationship between the Erlang and exponential distributions?
  • The Erlang-1 distribution is the exponential distribution.
• What is the relationship between the gamma and exponential distributions?
  • The gamma(1, λ) distribution is the exponential distribution.
• What is the relationship between the Chi-square distribution with k degrees of freedom and the gamma distribution?
  • The Chi-square distribution with k degrees of freedom is gamma(α=k/2, β=1/2)
• What are the two parametrisations of the gamma distribution?
  • With shape-scale (k, θ) or with shape-rate (α, β).
• What is the relationship between shape-scale (k, θ) and shape-rate (α, β) for the gamma distribution?
  • α=k and β=1/θ
• When is the shape-scale (k, θ) parametrisation of the gamma distribution more common?
  • Traditional industrial statistics, such as econometrics
• When is the shape-rate (α, β) parametrisation of the gamma distribution more common?
  • In Bayesian statistics, as it is trivially updated from Poisson and exponential draws.
• What is the relationship between the gamma and Pearson type III distributions?
  • Pearson type III is the gamma distribution except offset from the origin by a parameter a.

We continue learning about more advanced modeling of time headway.

• What value is typically chosen for the shift parameter a of the Pearson type III distirbution in time headway modeling?
  • 0.5, since that is the practical lower bound of time headway for humans
• When is the Pearson type III distribution used in time headway modeling?
  • It fits the skewed distribution of time headway better than the pure exponential or pure normal distribution.
• What is an alternative to the Pearson type III distribution for time headway modeling?
  • A composite distribution, where a fraction p of vehicles are assumed to follow the normal, high-flow regime, while (1-p) follow the shifted exponential low-flow regime.

At this point, the book had mentioned a difference between interrupted and uninterrupted traffic situations a lot, but never explained them. However, due to the frequency of the mentions I figured they must be important concepts, so I made a note.

• What is the difference between interrupted and uninterrupted traffic situations?
  • Interrupted situations are those in which external control measures (traffic signals, stop signs) may force vehicles to stop, whereas in uninterrupted situations only the interactions between vehicles force vehicles to stop.

There was a little information on human reaction times, which I found useful for personal reasons, to become a better driver myself!

• Humans react quicker when they are expecting something to happen. What consequence does this have for highway infrastructure?
  • It is useful to have systems that alert the driver to potential vehicle-closing situations.
• What is the relationship between time headway, safety, and capacity?
  • Increased time headway increases safety margins but decreases capacity.
• What is the median driver’s reaction time in optimal conditions?
  • 0.7 seconds
• What two conditions are required for optimal reaction times?
  • (a) The driver already expects something to happen, and (b) The situation requires no decision-making, i.e. reacting is the only option on the table.
• How much worse are reaction times when there’s a binary choice to be made first?
  • Increase by about half a second
• How much worse are reaction times when the driver does not expect something to happen?
  • Increase by about half a second
• What is the reaction time of the median driver when there is a choice to make among four alternatives, and the event is unexpected?
  • About two seconds (0.7 base reaction + 2×0.5 seconds for each bit of information + 0.5 to account for not expecting it)

Some data on vehicle and driver performance. I made notes here mainly because it might be useful when driving to know what to expect of others.

• What are the sizes of accelerations typically encountered in passenger cars?
  • 0.9–1.5 m/s²
• What is the size of acceleration a passenger car is typically capable of at speeds under 60 km/h?
  • Just over 2 m/s²
• What is the size of acceleration a truck is typically capable of at speeds under 60 km/h?
  • 0.5 m/s²
• What fraction of vehicles exceed the speed limit, (a) when it matches the design speed, (b) when it is lower than the design speed?
  • (a) 40 % and (b) 60 %

We now come into a section on intersections and interrupted situations.

• What is the rough size of headway that the average driver finds gives an acceptable gap in intersections?
  • 6 seconds
• What is the definition of discharge headway?
  • The time between successive vehicles passing the stop line in a signalised intersection
• What numbers does one practically encounter for discharge headway?
  • Four seconds for the first vehicle in the queue, then going down to just over two seconds for later vehicles
• Why is the discharge headway smaller for later vehicled behind a traffic signal? (Two reasons)
  • They don’t need a reaction time, nor do they need to build a distance buffer to the vehicle in front
• What is the discharge headway a measure of?
  • Intersection capacity
• Even if vehicles start out bunched up when a signal turns green, they will spread out after a while. What is this called?
  • Platoon diffusion

And back to uninterrupted situations, but now circling around the concept of capacity. The difference between capacity and service volume was particularly enlightening – it’s something I discuss frequently in terms of software also, but I never before had the words for it. Now I do!

• What is a synonym of flow rate?
  • Volume
• What is a synonym of volume?
  • Flow rate
• What is the definition of service volume?
  • The traffic flow that can be supported without breaking a service level requirement
• What is the first example of a service level requirement given in the book?
  • The maximum fraction of traffic that travels in platoon (specifically, has a time headway of less than five seconds)
• What is the difference between service volume and capacity?
  • Capacity is maximum flow rate, service volume is flow rate while maintaining service level requirements.
• What is the difference between capacity and saturation flow rate?
  • Saturation flow rate one gets when ignoring red lights etc. Capacity is measured under a specific set of control regimes (red light periods, ramp metering, etc.)
• Why does the flow rate become constant during high flow?
  • At that point the flow rate experienced is the capacity, which is constant for a given configuration.

The following notes on highway flow might be unnecessary, but I thought them as fun facts at the time. If they turn out to be difficult to memorise, I will not hesitate to throw them out as they are not useful for anything.

• How much of total traffic does a highway shoulder lane carry? (Low, medium, and high flow)
  • 40 % at low flow, 20 % at medium and high flow
• How much of total traffic does a highway middle lane carry? (Low, medium, and high flow)
  • Around 40 % at all flow rates
• How much of total traffic does a highway median lane carry? (Low, medium, and high flow)
  • 20 % at low flow, 40 % at medium and high flows
• At low flows, how much of total traffic do the (median, middle, shoulder) lanes carry?
  • (20, 40, 40)
• At medium flows, how much of total traffic do the (median, middle, shoulder) lanes carry?
  • (40, 40, 20)
• At high flows, how much of total traffic do the (median, middle, shoulder) lanes carry?
  • (40, 40, 20)
• Why does the shoulder lane carry less of the traffic during high flow rates?
  • It is home to slower vehicles
• What is the name of the outermost (overtaking) lane on a highway?
  • Median lane
• What is the name of the innermost (slow) lane on a highway?
  • Shoulder lane

Anyone who has ever driven on a highway know that trucks go slowly and appear to impede flow, but by how much? A traffic engineer can tell you, and now so can I! Fun to know.

But also useful for software engineering: it shows us that if some requests are more expensive than others, we might not have to account for them entirely separately: we might be able to estimate a “heavy vehicle adjustment factor” for these requests instead.

• What is the heavy vehicle adjustment factor?
  • The number of passenger cars that would be needed to reduce capacity by as much as one heavy vehicle (e.g. truck or bus.)
• What three things does the heavy vehicle adjustment factor depend on?
  • Heavy vehicle type (higher for slower, e.g. trucks), road type (higher for narrower, e.g. rural highway), and hilliness (higher for more hills.)
• What is the range of heavy vehicle adjustment factors commonly encountered?
  • 1.5–5 for trucks, depending on hilliness and road type. Slightly less for buses.

Here comes some good stuff. When capacity planning for software services I have never quite found a good way to relate peak flows to longer-term averages. It didn’t strike me before, but of course traffic engineers deal with the exact same problem. They work around it using very primitive means, but maybe the same techniques could work for software? I’m looking forward to try!

The reason I made so many flashcards on this part is specifically that I expect it to be one of the more useful bits for software engineering.

• When a new facility is planned, one often gets to know only the average annual daily flow rate. How is this converted to get a more useful guide to necessary capacity? (Which two steps?)
  • First convert to peak hour flow, then convert from two-way flow to flow in major direction.
• To convert annual average daily flow to peak hour flow, one needs to do what?
  • Multiply with a factor K which relates the two for the given type of road.
• When multiplying by K to convert annual average daily flow to peak hour flow, how is K determined?
  • Empirically, by measuring at similar roads.
• When multiplying by K to convert annual average daily flow to peak hour flow, what values are typically encountered?
  • 0.1–0.2
• When multiplying by K to convert annual average daily flow to peak hour flow, what does it mean to use a K value of 0.14?
  • The peak hour of the year is expected to have 14 % of the annual average daily flow.
• When multiplying by K to convert annual average daily flow to peak hour flow, is K selected from the highest flow hour over the year?
  • Typically not, because that would be overengineering. Often the value for the 30th highest flow hour is used.
• When multiplying by K to convert annual average daily flow to peak hour flow, what sorts of roads typically have low K, and what roads have high K?
  • Urban roads have low K, rural roads have high K. Recreational access routes have very high K.
• When multiplying by K to convert annual average daily flow to peak hour flow, what is the intuitive interpretation of a low K value?
  • Low variation in flow; uniform hourly flow distribution over thr year.
• When multiplying by K to convert annual average daily flow to peak hour flow, what would be the K value of a completely uniform hourly flow distribution, and what is the lowest value encountered in practise?
  • Uniform would be K=1/24=0.04, but lowest in practise is K=0.1
• To give a quantitative idea of the within-hour variation of flow rates, what number is often used?
  • The ratio of the full hour flow to the hourly flow during the most busy 15 minutes.
• To give a quantitative idea of the within-hour variation of flow rates, the ratio of the full hour flow to the hourly flow during the most busy 15 minutes is often used. What is this called?
  • Peak hour factor
• What would a peak hour factor value of 0.25 mean?
  • That all traffic during the hour passed during 15 minutes.
• What would a peak hour factor value of 1.0 mean?
  • That each 15-minute block of the hour, an equal amount of traffic passed.
• Under what conditions would the peak hour factor be something like 0.96?
  • During high flow rates, when flow is close to capacity and therefore constant.
• Under what conditions would the peak hour factor be something like 0.82?
  • During low flow rates, with nearly Poisson arrivals.
• What values does the peak hour factor typically take?
  • 0.8–0.98
• When converting bidirectional flow rate to flow rate in the major flow direction, one multiplies by a factor D. What is the lowest possible value for D?
  • 0.5 (because the major direction must have at least half the flow or it wouldn’t be the major direction!)
• When converting bidirectional flow rate to flow rate in the major flow direction, one multiplies by a factor D. What values does D typically take?
  • 0.5–0.7
• When converting bidirectional flow rate to flow rate in the major flow direction, one multiplies by a factor D. On which types of roads is D low, and on which is it high?
  • Urban roads low, intercity highways high
• When converting bidirectional flow rate to flow rate in the major flow direction, one multiplies by a factor D. What is thw intuitive interpretation of D being high?
  • Traffic flows mainly in one direction at during some of the day, and then in reverse later

We move on to flows again, in relation to capacity and level of service.

• Six service levels are standardised, A–F. Which is the least dense?
  • A
• Roughly what average time headway gives service levels (A, B, C, D, E)?
  • (5.2, 3.2, 2.2, 2, 1.8) seconds
• What are typical flow rates of free flowing traffic?
  • < 800 vehicles per hour per lane
• What are typical maximum flow rates when there are multiple lanes in each direction?
  • 1800 vehicles per hour per lane
• What is the per-lane capacity of single-lane highways with an opposing adjacent lane, in ideal conditions?
  • 1400 vehicles/hour
• When vehicle speeds match the design speed, what relative size are flow rates?
  • Very low, because this only happens when there are sporadic, few vehicles on the road.
• When vehicle speeds are very low, what relative size are flow rates?
  • Also very low – duh!
• What is the typical maximum density of a lane?
  • 125 passenger cars per kilometre
• Roughly at what density is optimal flow achieved?
  • 30 passenger cars per kilometre
• When the density is 30 cars/km and flow is at its maximum, what is the occupancy/utilisation?
  • 25 %
• At an utilisation of 25 % when flow is maximised, what is stability like?
  • Extremely unstable
• At what levels of utilisation are operations stable?
  • 0–10 %
• At what levels of utilisation are vehicles free-flowing?
  • 0–5 %
• What does congestion behave like around a fixed bottleneck?
  • Once demand exceeds bottleneck capacity a linearly backward-moving forming shockwave marks the transition between free flow and congestion. When demand decreases, a forward recovery shock wave resolves congestion from the rear.
• What is the difference in the recovery shockwaves between a fixed bottleneck and a temporary one?
  • With a temporary bottleneck, the recovery shockwave travels backwards rather than forwards.
• What example is given in the book of a situation in which the frontal shockwave of a bottleneck can be non-stationary (i.e. forward forming)?
  • A bottleneck in front of an incline, where trucks have trouble getting up to desired speed.
• How is congestion elegantly described?
  • As the location where excess demand is stored.
• Under what conditions is there a rear stationary shockwave?
  • When demand exactly matches capacity. (If it does not, the rear shockwave will be backwards forming, or recovering, depending on sign.)
• In the simplest case, which are the steps to calculate demand from density?
  • Convert excess density to hypothetical excess flow rate (through excess vehicles in a time period), then add hypothetical excess flow rate to observed flow rate!
• When estimating demand of a bottleneck via stored demand in the upstream congestion, there is a possibility not all stored demand intends to exercise the bottleneck (might exit at an earlier off-ramp). How does one account for this?
  • Estimating a destination factor for each section and multiply with that.
• When is estimated demand typically lower than observed flow?
  • After a period of higher demand than capacity, and the queue is resolving. The bottleneck is now serving – at capacity still – old demand.

Then we end with some car-following theory, which might be useful if-or-when I try writing my own highway traffic simulation.

• Why do highway simulations require a burn-in portion of road before the characteristics of something can be analysed?
  • Because we cannot generate vehicles with a realistic enough distribution without actually simulating them for a bit
• What does Pipes’ car following theory suggest?
  • That the distance headway increases linearly with speed (at approximately one car length per 10 miles per hour).
• How well does Pipes’ car following theory correspond to real measurements?
  • Surprisingly well for such a simple theory!
• Where does Pipes’ car following theory disagree with real measurements?
  • At very low and relatively high speeds – under those conditions real drivers have more distance headway than the theory suggests.
• What does Pipes’ car following theory suggest about flow rate as speeds increase?
  • That flow rate increases without limit
• Pipes’ car following theory suggests that flow rate increases linearly with speed. What happens in practice?
  • At higher speeds, vehicles have a greater average distance headway than suggsted by the theory. This maximises flow rates at about 50–60 km/h.
• What is the difference between Pipes’ and Forbes’ car-following theories?
  • Pipes’ is based on distance headway, whereas Forbes’ is based on time headway.
• What is the similarify between Pipes’ and Forbes’ car-following theories?
  • Both (contrary to the real world) result in a distance headway that is linear in speed.
• What three variables go into the fourth GM car following throry?
  • My speed, the distance to the vehicle ahead, and the relative speed between us.
• Other than three variables, what goes into the fourth GM car-following theory?
  • A normalising constant that indicates individual sensitivity to closing distances – or optimum speed, from a macroscopic view.

I don’t expect anyone to actually read this dump down here, but if you did, congrats! You have now read everything on traffic flow I didn’t already know but now do.

I’m not sure I’ll pick up the second half of Traffic Flow Fundamentals anytime soon – I have learned a lot already and might need to digest this for a bit before moving on. I have found that when taking a break and coming back after a iterations of working with the flashcards (or even after a year!), I more easily grasp the more advanced content that builds on the basics – all thanks to spaced repetition, of course.