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3.3.3 The Displaced Negative Exponential Distribution

Where vehicle arrivals are essentially random but the minimum possible headway is greater than zero (which would apply, for example, to single lane flow with no overtaking), the displaced negative exponential distribution may be an appropriate representation of headways. As illustrated by Figure 3.3, the frequency distribution for the displaced negative exponential has the same shape as the negative exponential but is displaced to the right by an interval \(\beta \), equal to the minimum possible headway.

Figure 3.3: Displaced negative exponential headway distribution

Mathematical manipulation of the basic negative exponential form to provide a unit area under the graph from \(t=\beta \) to \(t=\infty \) produces the displaced negative exponential distribution:

\(f(t)=\frac{q}{1-q\beta }e^{\frac{-q(t-\beta )}{1 - q\beta } } \) for \(t\ge \beta \)3.26

Derivations paralleling those used above for the standard negative exponential lead to the following results.

The probability of a headway greater than or equal to \(t \) , for \(t\ge \beta \) , is:

\(\Pr (h\ge t)=e^{\frac{-q(t-\beta )}{1 - q\beta } } \) for \(t\ge \beta \)3.27

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

\(h_{av} (h\ge t)=\frac{1}{q}+ t-\beta \) for \(t\ge \beta \)3.28

Note that Equation 3.28 gives an average headway of 1/q for all headways not less than \(\beta \).