Ecole Polytechnique Fédérale de Lausanne
School of Architecture, Civil and
Environmental Engineering
N. Geroliminis
Fall 2011
Transportation Systems Engineering (CIVIL‐351) Notes for weeks 3-4 – Properties of traffic streams, Basic assessment tools
1. Time-Space Diagrams (TS)
2. Puzzle
3. Transit-Stop Delay
4. Constructing TS Diagram
5. Traffic stream definitions
6. Fundamental Diagram
7. Moving Observer
8. Input-Output Curves
9. Little’s Formula
10. Variations to the Queuing Diagrams
11. Kinematic wave theory
The notes and slides that are posted on the web do not substitute for class
attendance. Students are highly encouraged to attend class and keep personal notes.
1. Time – Space Diagrams


Trajectories are curves in the time-space diagram that define a single position for
every moment of time  x(t)
Recall from basic physics 1. dx/dt velocity 2. d2x/dt2  acceleration
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Example 1
Example 2
Example 3
Example 1: 1 goes faster than 2
Example 2: 1 is not moving and 2 is moving
backwards
Example 3: 1 is accelerating and 2 is
decelerating
2. Example – Stations of a transit vehicle
Note that if the distance between stations is not long enough the vehicle can’t reach its
cruising speed.
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
From the above figure we see that the total delay due to a stop is:
v
v v

 ,
2a 2a a
where a is both, the acceleration and deceleration rate.

If the acceleration and deceleration rates, a+ and a- , differ the result is
v
v
 

2a
2a
If there are N stops, distance between stops is S, free flow speed of the bus is v, P
passengers per stop are boarding and alighting and time needed for a passenger to board
or alight is t seconds, then the average speed of a bus is (a+ =a-=α):
vav 
 N  1 S
 N  1 S 
v
 N  1
v
 NPt
a
We described in class why we use N-1 instead of N.
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3. Puzzle
• Assume there are three people going from EPFL to Lausanne
• However, they only have ONE tandem bike (only two people can ride it at a time)
• Riders (solo or tandem) travel at 20 km/hr
• Any person who doesn’t ride the bike, can jog at 4 km/hr
• In order to get to Lausanne as fast as possible, two of the three people start riding
the bike, and the third starts jogging in the same direction. After a while, one of
the riders gets off and starts jogging. The other one rides back to pick up the
original jogger. Then they jointly ride in the San Jose direction until catching up
with the one that got off some time before. At that point they are together again
(with the bike) and they repeat the process.
• How fast do they go? (average speed?)
The solution to this problem can be computed two ways: graphically and analytically.
Here we present the graphical procedure, but this graph can also be used to understand
the analytical solution.
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4. Development of time-space diagrams
Time-space diagrams are “complete”. They offer a lot of valuable information in a
condensed manner.
There are different ways of constructing time-space diagrams:


Aerial surveys

Take photographs to the same road segment (between two given points)

Place them next to each other, separated according to the time interval
between shots

Draw lines across the different pictures following the location of the
individual vehicles (these are the trajectories)
Traffic detectors
 This is similar to having stationary observers at specific locations along the
road
 Observers (or loop detectors) measure the time at which every vehicle passes
them
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
Driver logs

Drivers record the time at which they pass by certain locations
5. Traffic stream Characteristics (part II)
We say that traffic on a long stretch of road represents steady-state conditions during a
period of observation if you cannot get any clues as to what time it is or where you are by
inspecting the time-space diagram through a small window in a template (stationary
conditions=steady-state conditions for our class).
Next table summarizes how one can estimate various traffic characteristics using two
observation methods. Underlined expressions correspond to the original definitions
introduced in class. These formulas are correct for stationary conditions.
Density
Method of Observation
Aerial Photograph
Stationary Observer
1 m 1
n/L

T i 1 vi
1 n
 ui
L i 1
1 n
 ui
n i 1
Flow
Space-mean speed
n
Time-mean speed
m/T
n
u u
i 1
2
i
i 1
6
i
1 m 1
  
 m i 1 vi 
1 m
 vi
m i 1
1
6. Fundamental Diagram
• Up to now we have seen 5 descriptor of the traffic stream (v, q, k, s, h) and three
relations (q=kv, q=1/h, k=1/s)
• Therefore we only have 2 degrees of freedom (meaning we only need to keep track of
two variables)
• Greenshield in the 1930’s conjectured that there was a linear relationship between speed
and density
o He observed that when there were just a few vehicles on the road the speed was
very high (around vf = free flow speed) and when the road was full (density was closed to
kj = jam density) the speed was almost zero o From experimental data he then got this
graph.
• From the previous graph it is easy then to construct a flow-density diagram. This
diagram is called the “fundamental diagram”.
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o Where qmax is the maximum flow in the highway (capacity) and is obtained with
an optimal density kop
• Similarly, we can construct a speed vs. flow diagram
• However, these diagrams are not very realistic. Researchers now know that the
flow-density relation is better described by a triangle than by a parabola.
• The following graph shows the Fundamental Diagram as we use it today. It
contains enough information to find any of the 5 descriptors, if one is given k.
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o As shown by the picture, for every flow we have two densities. We call the
states on the left side uncongested (or “free flow”, or “unqueued”), and those on
the right congested (or “queued”).
o The diagram is a property of the road.
o Points on the diagram describe possible traffic conditions (or “steady states”).
7. Moving Observer
Recall as we showed in class that the rate, q0 , at which cars pass an observer that moves
with speed v0 when traffic is in a steady flow-density state (q,k) is given by the flow
conservation formula:
q0= q – k v0 (1)
If v0 and (q,k) are given then q0 is the vertical separation between the corresponding
steady-state point on the (k,q)-plane and the dotted ray shown in Fig. 1.
flow
q00
v0
density
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8. Input-Output Curves
• We can use two observers
• Observer A looks at all the arrivals to the system
• Observer D looks at all the departures from the system
• The following diagram depicts the cumulative number of arrivals and departures as seen
by the two observers
• Q(0) = number of customers in queue at time
• Q(t) = number of customers in queue at time
th
• An = time of arrival of the n customer
th
• Dn = time of departure of the n customer
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




Q(t’) = accumulation (number of customers in the system) at time t’
A(t) = arrivals as seen by observer A
D(t) = departures as seen by observer D
Shaded area = “total” wait time (time in the system)
Average time in system = w = total area / N, where N is 4 for the previous
example
• In order to simplify the process, we can also use a piecewise linear approximation
instead of the step function shown in the previous graph.
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• This queuing diagram can also be obtained from the time-space diagram (see below)
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9. Little’s Formula
• This queuing diagram shows the beginning and end of a typical bottleneck
o t0 = time when congestion starts (queue begins)
o t1 = time when congestion ends (queue disappears)
o Note that Area   n1  n0   w , where w is the average horizontal distance
between A(t) and D(t) (average waiting time)
o Note also that Area   t1  t0   Q , where Q is the average vertical distance
between A(t) and D(t) (average accumulation)
o Therefore,  n1  n0   w   t1  t0   Q
o Rearranging the equation we get
flow, so Q   w
o Recall that q=kv
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 n1  n0   Q
 t1  t0  w
where
 n1  n0   
 t1  t0 
i.e.
o From the previous figure we can see that, w 
o Therefore,   Q 
w
L
, and Q  kL
v
kL
 kv which satisfies our original equation
L/v
10. Variations to the Queuing Diagrams
• In many queuing systems L ~ small (e.g., doctor’s office).
• In others, L ~ large (e.g., highway). In these cases it takes us some time to go from
one observer to the other one even if the system is uncongested
• Take for example a piece of highway where the free flow travel time between two
observers if fftt. If there is congestion the travel time increases, but the actual
delay is just the travel time with congestion minus the free flow travel time.
o V(t) represents virtual departures: the time you would have departed the
downstream end if there was no queue.
o If L ~ small  A  t   V  t  
o If L ~ large  A  t   V  t  , so we use V(t)
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Incomplete Information
• In many cases we have incomplete information
e.g., we know V(t) or A(t) and the operating features of the server (e.g.,
constant service rate, µ), but we need to find D  t 
• There are certain rules we can use to construct the departure curve
o When V initially is > µ  D  t    
o D  t   min   , V  t   when V(t)=D(t)
On-off Service
• In some cases the service rate is not constant, but on-and-off (e.g., traffic signal)
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11. Kinematic wave theory

Fundamental Diagram

Flow Conservation

Shockwave Analysis
Fundamental Diagram (Continuation)
• Recall that:
o As shown by the picture, for every flow we have two densities. We call the
states on the left side uncongested (or “free flow”, or “unqueued”), and those on the
right congested (or “queued”).
o The diagram is a property of the road.
o Points on the diagram describe possible traffic conditions (or “steady states”).
Example 1
• The following figure represents a queue directly upstream of a bottleneck
We assume that locations 0 and 1 correspond to identical roads:
o because vehicles are conserved between observers q0=q1=QB
o because vehicles at location 0 are in a queue while vehicles at location 1 are
freely flowing v0<v1
o The following fundamental diagram shows these traffic conditions
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• If the previous diagram corresponds to a two-lane highway, and the
bottleneck only has one lane, what is the diagram like for the bottleneck
section? Answer below:
Capacity
• Capacity of a link: maximum sustained flow that can pass through the link when
there is a queue upstream feeding it and no restriction downstream blocking its
exit
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• In the previous example the capacity of the two-lane links was qmax and the capacity
of the middle link QB.
Flow conservation
n3
x
A
C
n4
n2
B
D
n1
t
As we described in class we have that n1+n4=n2+n3
Using density and flow definitions we get:
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qBD   t2  t1   k AB  x2  x1   q AC   t2  t1   kCD  x2  x1  
 k BD  k AC  x2  x1     qBD  q AC  t2  t1  
 k BD  k AC    qBD  qAC   0 
 x2  x1 
 t2  t1 
k q

0
t x
This is the flow continuity equation.
Shockwave analysis
Example 1
• Vehicles travel at speed vf = vA (state A)
• First vehicle decides to stop
• Some time later traffic will be in the jam state J
• From the fundamental diagram we know the density for state J (jam density)
• So we know the spacing when all the vehicles are stopped
• Now we can draw the trajectories in the time-space diagram that satisfy both state A
and state J conditions
• There is a clear line in the time-space diagram separating the states A and J, the shock
wave
• This line has a clearly defined slope. Geometrical considerations would show that the
q  qJ
slope of this line is u AJ  A
k A  kJ
• Conveniently for us, this is also the slope of the line going through points A and J of
the fundamental diagram. (Note the parallel dashed lines in both diagrams.)
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Example 2
• With a similar approach we can determine the effects of a recurrent bottleneck
(i.e., a bottleneck that is always there)
• Data: trajectories for an isolated group of vehicles in state “A” approaching the
bottleneck, and the capacity of the bottleneck QB.
• From this, draw trajectories for state “B” and the shock trajectory.
• When would the queue disappear?
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