### 7.3.2 Traffic Bunches

The formation of traffic bunches has been addressed briefly in Section 3.3.4, which deals with composite models of headway distributions in a traffic stream. At that point, the above distinction between bunches and platoons was not made, as composite headway distribution models may be applied to both situations, but it is now appropriate to focus attention on bunches as defined above.

Section 3.3.4 noted that a composite headway distribution model may be defined by specifying a sufficient number of items from a list of characteristics of the traffic stream. One such sufficient set of characteristics is:

- the distribution of bunch sizes
- the distribution of headways for restrained vehicles
- the distribution of inter-bunch headways (i.e. the difference between the times at which the lead vehicles of two consecutive bunches pass a given point).

#### Bunch sizes

If the distribution of bunch sizes is known, the mean bunch size, **m **, also is known and it is possible to calculate \(\theta \), the proportion of vehicles in the traffic stream which are restrained (in the sense of being following vehicles in a bunch of size two or more) as:

\[\theta = \frac{m - 1}{m} \] | 7.11 |

This is readily seen by considering a time period during which the number of vehicles passing a given point in the direction of interest is N. Given an average bunch size of m, these vehicles would be in N/m bunches, each consisting of exactly one free flowing vehicle, either alone or closely followed by other vehicles in a multi-vehicle bunch. Thus, there would be N/m free flowing vehicles and, hence, N – N/m vehicles following in bunches. The proportion of vehicles following in bunches would therefore be.

\[\Theta =\frac{N-N/m}{N}=1-\frac{1}{m}=\frac{m-1}{m}\]

A number of discrete probability distributions have been employed to represent the distribution of bunch sizes. Among these are the geometric distribution (Walpole et al. 2011), the Borel‑Tanner distribution (Tanner 1961) and the two-parameter and one-parameter Miller distributions (Miller 1961).

The geometric distribution was introduced as a discrete distribution relevant to traffic theory in Section 3.2.3. In the context of its application to the distribution of bunch sizes in a traffic stream, it predicts the probability of observing a bunch of size n as:

\[\Pr (n) = \theta ^{n-1} (1-\theta )\] | 7.12 |

Where θ is the proportion of vehicles in the traffic stream that are following in bunches, as previously defined. Given that θ can also be considered as the probability that any one vehicle in the traffic stream is a ‘restrained’ or ‘following’ vehicle, the right hand side of Equation 7.12 is seen to be the product of the probability that the lead vehicle of the observed bunch will be followed, first, by n–1 restrained vehicles (making up a bunch of size n), then by a free flowing vehicle (the lead vehicle of the next bunch).

The mean of the geometric distribution gives the average or expected bunch size as:

\[m = \frac{1}{1-\theta } \] | 7.13 |

which is seen to be consistent with Equation 7.11.

The Borel-Tanner distribution (Tanner 1961) has already been discussed in Section 3.3.4 in the context of its application to bunch size distributions and is not considered further here.

Miller (1961) originally proposed a two-parameter model of bunch size distributions in which the probability of observing a bunch of exactly n vehicles is:

\[\Pr (n) = \frac{(a+1) (a+b+1)! (b+n-1)!}{b! (a+b+n+1)!} \] | 7.14 |

where a and b are the two parameters, having values that provide the best fit of Equation 7.14 to observed data, while being related to the mean bunch size, m, by:

\[m = \frac{a+b+1}{a} \] | 7.15 |

The probability of observing a single-vehicle bunch is then given by:

\[\Pr (1) = \frac{a+1}{a+b+2} \] | 7.16 |

Miller (1961) then noted that the value of **b** often was small and proposed a one-parameter form of the distribution:

\[\Pr (n) = \frac{(a+1) (a+1)! (n-1)!}{(a+n+1)!} \] | 7.17 |

with the parameter **a** fitted such that the mean bunch size, **m**, is given by:

\[m = \frac{a+1}{a} \] | 7.18 |

#### Headway distributions within bunches

Given the relatively close following behind the lead vehicle in a multi-vehicle bunch and the fact that a minimum feasible headway must apply, a within-bunch headway distribution that provides for a small variance about a mean headway is generally appropriate. For some analyses, it may be sufficiently accurate to assume the extreme case of a uniform distribution, that is, equality of all within-bunch headways.

A more realistic representation would be provided by some other type of distribution (normal, log‑normal or gamma, for example) with small variance and with limits that recognise both the minimum headway feasible for car-following and the maximum headway for which a vehicle can be considered to be following within a bunch rather than free flowing. However, the use of such a more realistic representation should be justified by the level of accuracy required by the particular application.

#### Headway distributions between bunches

Given that the headway for any vehicle is taken to be its time separation from the vehicle immediately ahead of it (see the start of Section 3.3) and that a single free flowing vehicle is taken to be a bunch of size 1 (Section 3.3.4), the headway for any free flowing vehicle is its time separation from the last (possibly the only) vehicle in the bunch immediately ahead. This same time separation is defined as the inter-bunch headway, so that the distribution of headways for free flowing vehicles is the same characteristic as the distribution of headways between bunches.

Because these vehicles are free flowing, a headway distribution based on random arrivals (e.g. negative exponential, displaced negative exponential) is usually appropriate. For example, the discussion of the double exponential distribution in Section 3.3.4 notes that it utilises a standard negative exponential distribution of headways for free flowing vehicles (combining them with a displaced negative exponential distribution of headways for vehicles following in bunches).