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GAP ACCEPTANCE THEORY

GAP ACCEPTANCE THEORY

 The Negative Exponential Distribution and other headway distributions are very important for traffic analysis.  Unsignalised intersections, roundabouts, and pedestrian crossing movements all function through a gap acceptance mechanism – the capacity of a minor traffic stream to cross or merge depends on the presence of gaps (headways) of an acceptable size being present in the major traffic stream.

 We will cover unsignalised intersections and roundabouts over the next few weeks, but we need to have this basic probability theory beforehand.

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TYPICAL GAP ACCEPTANCE BEHAVIOUR Not all drivers require (or desire) the same size gap in traffic to merge or to cross We commonly use the Critical Gap, as the gap which 50% of drivers will use

T is referred to as the Critical Gap Austroads (2015). “Guide to Traffic Management”, Part 2: Traffic Theory. Section 5: Gap Acceptance. Austroads.

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PRINCIPAL GAP ACCEPTANCE FORMULAS

Assuming a Negative Exponential headway distribution:

The proportion of vehicles that will be delayed is

Proportion delayed 1 e  qT Average delay to all vehicles (including those that experience no delay)

d av  d t 0 =

1 1   T s/veh qe  qT q

Average delay to all vehicles that experience delay

d av  d ! 0  =

1 T s/veh  qe  qT 1  e  qT

where q is the volume of the conflicting major traffic stream, in veh/s, and T is the size of the critical gap, in s/veh Austroads (2015). “Guide to Traffic Management”, Part 2: Traffic Theory. Section 5: Gap Acceptance. Austroads.

Don’t bother to memorise these formulas, just know they exist !

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GENERATING RANDOM NUMBERS THAT FOLLOW A DISTRIBUTION

Want more information on this? Search for “Inverse transform sampling”

RANDOM NUMBERS FOLLOWING A DISTRIBUTION

 We may need to create a sequence of random numbers that follows a particular distribution – For example, we may be developing a simulation model of a roundabout or an unsignalised intersection – That model would require accurate modelling of the headways (gaps between vehicles) into which minor stream vehicles may merge

 We know the headway distribution (e.g. negative exponential), but how can we generate a set of random numbers having that distribution?

h

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RANDOM NUMBERS FOLLOWING A DISTRIBUTION

How can we generate a set of vehicle headways following a negative exponential distribution? OR, more generally… How can we generate a set of random numbers {x} having a particular numerical distribution?  The Probability Density Function, f(x) is given by f ( x) O exp  O x 

The Cumulative Density Function, F(x) is the probability of x a value less than x occurring. It ranges from 0 to 1, and is given by F  x  ³ f  x  dx 1  exp  O x  0

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RANDOM NUMBERS FOLLOWING A DISTRIBUTION



The Cumulative Density Function, F(x) is the probability ofx a value less than x occurring. It ranges from 0 to 1, and is given by F  x  ³ f  x  dx 1  exp O x  0



Compare this to a set of uniform random numbers {u} between 0 and 1, for which the probability of a value less than u is u.

We need to compute the values of x such that F(x) = u, i.e. x = F-1(u) – x will then be a random number drawn from the probability distribution f(x) 1 x F 1 u   ln 1  u  u F  x  1  exp  O x  O



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RANDOM NUMBERS FOLLOWING A DISTRIBUTION

 This is an example of a “Monte-Carlo method”  We can easily generate this set of uniform random numbers, {u} (e.g. using the RAND() function in Excel)  The transformation can also be done in Excel, giving our negative exponentially distributed headways, {x}, and the vehicle arrival times

^x `

F 1 ^Rand ` 

1

O

ln 1  ^Rand ` 

Random Headway, x Arrival Numbers (0-1) (s) time, t (s) 0.198 2.209 2.209 0.760 14.276 16.485 0.607 9.336 25.820 0.907 23.791 49.611 0.050 0.513 50.125 0.167 1.831 51.956 0.552 8.032 59.988 0.760 14.284 74.273

Random Numbers

This is how I generated the vehicle arrival times and speeds for Assignment 1 Question 1! {X X}

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0.310 0.852 0.683 0.971 0.132 0.449 0.101

3.706 19.099 11.490 35.489 1.421 5.968 1.064

97.700 116.799 128.289 163.778 165.200 171.167 172.231

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RANDOM NUMBERS FOLLOWING A DISCRETE DISTRIBUTION

This Monte Carlo method also works for discrete distributions.  

For example, we need to generate a series of time intervals in which a number of vehicles arrive, those arrivals follow the Poisson Distribution We use a similar process, generating uniform random numbers, and finding the cumulative probability that will match the random number Rand Number, X

Pr (x=X) 0 1 2 3 4 5 6

0.135335 0.270671 0.270671 0.180447 0.090224 0.036089 0.01203

Pr(x≤X)

0.198882 0.355364 0.969214 0.615992 0.245274 0.426208 0.712621 0.225492 0.063876 0.43653 0.182847

0.135335 0.406006 0.676676 0.857123 0.947347 0.983436 0.995466

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X 1 1 5 2 1 2 3 1 0 2 1

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SUMMARY OF TRAFFIC HEADWAY DISTRIBUTIONS

SUMMARY – TRAFFIC HEADWAY DISTRIBUTIONS

 Traffic is random, deal with it! – flows and headways can’t be predicted exactly – this is due to differences in drivers, vehicles and conditions

 We can model this randomness in arrivals by looking at the probabilities and frequencies  The Poisson Distribution models arrivals in a time interval x e  qt  qt  p( x) x!  The Negative Exponential Distribution models headways that are greater than a given time

p  h t t  e qt

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