# Commentary 9

At an intersection where gap acceptance applies, where major road traffic arrives randomly from each direction and where a right-turning vehicle from a minor road seeks different sized critical gaps in the traffic flows from the two different directions on the major road, the theoretical absorption capacity can be developed as follows:

Assume that the total major traffic stream is made up of the traffic flow from the left, with volume q_{L}, and the flow from the right, with volume q_{R}. Each gap or headway in the total traffic stream starts with the arrival of a vehicle from one direction and ends with the next arrival, which may be from the same or the opposite direction. Therefore, during a significant period of time H, the number of headways in the total major traffic stream will be:

Total number of headways = \(H (q_{L} + q_{R} )\) | C62 |

Now assume that a queue of minor traffic stream vehicles is waiting to turn right and must give way to this major traffic stream, which has a negative exponential headway distribution in each direction. Let the combination of a gap of at least t_{L,i} in the major traffic from the left with a gap of at least t_{R,i} in the major traffic from the right be the minimum condition required to allow i minor traffic stream vehicles to make the right turn within the one gap, where i = 1, 2, 3,....

The probability of the simultaneous occurrence of a headway greater than or equal to \(t_{L,i} \) in the major road flow from the left and a headway greater than or equal to \(t_{R,i} \) in the major road flow from the right is:

Pr (\(h_{L} \ge t_{L,i} \)| \(h_{R} \ge t_{R,i} \)) \(= e^{-q_{L} t_{L,i} } . e^{-q_{R} t_{R,i} } = e^{-(q_{L} t_{L,i} + q_{R} t_{R,i} )} \) | C63 |

Hence, during the significant period of time H, the number of major traffic stream headways large enough to allow at least \(i\) minor traffic stream vehicles to make the right turn, i = 1, 2, 3,... is:

No. of headways (\(h_{L} \ge t_{L,i} \)| \(h_{R} \ge t_{R,i} \)) \(= H (q_{L} + q_{R} ) e^{-(q_{L} t_{L,i} + q_{R} t_{R,i} )} \) | C64 |

Therefore, the number of major traffic stream headways that allow exactly i minor traffic stream vehicles to turn right, i = 1, 2, 3,... is:

\[\mathrm{n}_{\mathrm{i}}\mathrm{ }\mathrm{=}\mathrm{ }\mathrm{\text{H.}}\left( \mathrm{q}_{\mathrm{L}}\mathrm{+}\mathrm{q}_{\mathrm{R}} \right)\mathrm{.}\left\lbrack \mathrm{e}^{\mathrm{- (}\mathrm{q}_{\mathrm{L}}\mathrm{t}_{\mathrm{L,i}}\mathrm{+}\mathrm{q}_{\mathrm{R}}\mathrm{t}_{\mathrm{R,i}}\mathrm{)}} - \mathrm{e}^{\mathrm{- (}\mathrm{q}_{\mathrm{L}}\mathrm{t}_{\mathrm{L,i + 1}}\mathrm{+}\mathrm{q}_{\mathrm{R}}\mathrm{t}_{\mathrm{R,i + 1}}\mathrm{)}} \right\rbrack\] | C65 |

Hence, the total number of minor stream vehicles able to turn right during the period \(H\) is:

N = | \(\sum _{i=1}^{\infty }i.n_{i} \) | C66 |

= | \(H (q_{L} + q_{R} )\sum _{i=1}^{\infty }i . \left[e^{-(q_{L} t_{L,i} + q_{R} t_{R,i} )} - e^{-(q_{L} t_{L,i+1} + q_{R} t_{R,i+1} )} \right] \) | |

= | \(H (q_{L} + q_{R} )\sum _{i=1}^{\infty }e^{-(q_{L} t_{L,i} + q_{R} t_{R,i} )} \) |

Now assume that a critical gap T_{L} in the major traffic flow from the left is the minimum that will allow one minor stream right-turner to cross that stream, and that a critical gap T_{R} in the major traffic flow from the right is the minimum that will allow one minor stream right-turner to join that stream. Further, assume that an additional follow-up headway T_{0} is sufficient to allow one additional minor stream vehicle to follow in undertaking the manoeuvre. This implies that:

\[t_{L,i} = T_{L} + (i - 1) T_{0} \ and \ t_{R,i} = T_{R} + (i - 1) T_{0} , i = 1, 2, 3, ...\] | C67 |

Substituting from Equations C67 into Equation C66:

N= | \( H (q_{L} + q_{R} )\sum _{i=1}^{\infty }e^{-(q_{L} T_{L} + q_{L} (i - 1)T_{0} + q_{R} T_{R} + q_{R} (i - 1)T_{0} )} \) | |

= | \(H (q_{L} + q_{R} ) e^{-(q_{L} T_{L} + q_{R} T_{R} )} \sum _{i=1}^{\infty }e^{-(q_{L} + q_{R} ) (i - 1) T_{0} } \) | |

= | \(H (q_{L} + q_{R} ) e^{-(q_{L} T_{L} + q_{R} T_{R} )} \sum _{k=0}^{\infty }\left(e^{-(q_{L} + q_{R} ) T_{0} } \right)^{k} \) | |

= | \(\frac{H (q_{L} + q_{R} ) e^{-(q_{L} T_{L} + q_{R} T_{R} )} }{1 - e^{-(q_{L} + q_{R} ) T_{0} } } \) | C68 |

Thus the theoretical maximum rate at which minor stream vehicles can turn right, that is, the theoretical absorption capacity, is obtained as:

\[C = \frac{N}{H} = \frac{(q_{L} + q_{R} ) e^{-(q_{L} T_{L} + q_{R} T_{R} )} }{1 - e^{-(q_{L} + q_{R} ) T_{0} } } \] | C69 |