## 5.4 Formulae for Displaced Negative Exponential Headways in Major Traffic

Sections 5.2 and 5.3 have restricted their attention to the case of random arrivals – that is, a negative exponential distribution of headways – in the major traffic flow. As the type of headway distribution directly affects the pattern of acceptable gaps, it is instructive to consider how the key gap acceptance formulae are changed if a different type of distribution applies. As an illustration, the case of a displaced negative exponential distribution of headways in the major traffic flow is considered in Commentary 10 and the key formulae from that analysis are summarised below.

The probability density function for headway size, t, for a displaced negative exponential distribution with a minimum possible headway \(\beta\) is:

\(f(t) = \frac{q}{1 - q\beta } e^{\frac{-q(t-\beta )}{1 - q\beta } } \) for \(t\ge \beta \) | 5.10 |

and

f(t)=0 for \(t<\beta \) | 5.11 |

The probability of a headway of size t or greater (where \(t\ge \beta \)), in the major traffic stream, is:

\[\Pr (\beta \le t\le h) = e^{\frac{-q(t-\beta )}{1-q\beta } } \] | 5.12 |

It follows from Equation 5.12 that, for a critical gap T, the proportion of minor road vehicles that suffer no delay at the stop line is:

Proportion not delayed \(= e^{\frac{-q(T-\beta )}{1-q\beta } } \) | 5.13 |

Conversely, the probability of a major traffic headway of size t or less (where \(t\ge \beta \)) is:

\[\Pr (\beta \le h\le t) = 1 - e^{\frac{-q(t-\beta )}{1-q\beta } } \] | 5.14 |

and, for a critical gap T, the proportion of minor road vehicles that are delayed at the stop line is:

Proportion delayed \(= 1- e^{\frac{-q(T-\beta )}{1-q\beta } } \) | 5.15 |

The average duration of headways greater than or equal to t (where \(t\ge \beta \) ) is:

\[h_{av} (\beta \le t\le h) = \frac{1}{q} + t - \beta \] | 5.16 |

and the average duration of headways less than or equal to t (where \(t\ge \beta \)) is:

\[h_{av} (\beta \le h\le t) = \frac{1}{q} - \frac{(t-\beta ) e^{\frac{-q(t-\beta )}{1-q\beta } } }{1 - e^{\frac{-q(t-\beta )}{1-q\beta } } } \] | 5.17 |

Then the average delay experienced by all minor traffic stream units at the gap acceptance point is:

\[d_{av} (d\ge 0) = \frac{1}{q e^{\frac{-q(T-\beta )}{1-q\beta } } } - \frac{1}{q} - (T-\beta )\] | 5.18 |

The average delay at the gap acceptance point to only those minor traffic stream units that do experience such delay is:

\[d_{av} (d>0) = \frac{1}{q e^{\frac{-q(T-\beta )}{1-q\beta } } } - \frac{(T-\beta )}{(1 - e^{\frac{-q(T-\beta )}{1-q\beta } } )} \] | 5.19 |

Finally, the theoretical absorption capacity for a minor traffic stream requiring a critical gap T and with a follow-up headway T_{0}, giving way to a major traffic stream of volume q, with displaced negative exponential headways each greater than or equal to \(\beta\), is:

\[C = \frac{q e^{\frac{-q(T-\beta )}{1-q\beta } } }{1 - e^{\frac{-qT_{0} }{1-q\beta } } } \] | 5.20 |

Note that Equation 5.8 is also valid for displaced negative exponential distributions of major traffic headways. Recall that this equation addresses a minor traffic stream consisting of n sub-streams i, i=1,....,n, with p_{i} being the proportion of the minor traffic volume in sub-stream i and C_{i} being the theoretical absorption capacity for the approach lane if all its traffic belonged to sub-stream i. For this case, the theoretical absorption capacity for the whole minor traffic stream is:

\[C_{T} = \frac{1}{\sum _{i=1}^{n}\frac{p_{i} }{C_{i} } } \] | 5.8 |

For displaced negative exponential major road headways, however, each C_{i} would be evaluated using Equation 5.20.