## 6.2 Gap Acceptance with Multiple Levels of Priority

The consideration of elementary gap acceptance in Section 5 assumed a single major traffic stream (perhaps consisting of traffic from more than one direction) with priority over a single minor traffic stream. In practice, however, it is common to see multiple levels of priority, that is, situations in which a minor traffic stream must give way to a higher priority stream which, in turn, must give way to another stream of still higher priority. An example is illustrated by Figure 6.1, in which movement 1 has priority over both movements 2 and 3, and movement 2 has priority over movement 3.

The example situation in Figure 6.1, with random arrivals assumed for both streams 1 and 2, is used in the following to illustrate how queuing and gap acceptance theory can be combined to develop a means of evaluating absorption capacities for lower priority movements. The method presented was taken from Bennett (1996).

The theoretical absorption capacity for the second-priority movement 2 in Figure 6.1 is simply calculated from Equation 5.14 as:

\[C_{2} = \frac{q_{1} e^{-q_{1} T_{(2)} } }{1 - e^{-q_{1} T_{0(2)} } } \] | 6.2 | |||

where | ||||

q_{1} | = | the volume of traffic stream 1 | ||

T_{2} and T_{0(2)} | = | the critical gap and follow-up headway, respectively, for stream 2 vehicles crossing traffic stream 1 |

To calculate the theoretical absorption capacity for the third-priority movement 3, which must give way to both movements 1 and 2, observe that, for a stream 3 vehicle to be able to depart from the stop line, the following three conditions must apply simultaneously:

- There is an adequate time gap before the arrival of the next stream 1 vehicle.
- There is no queue of stream 2 vehicles waiting to turn into the minor road.
- There is an adequate time gap before the arrival of the next stream 2 vehicle.

For random arrivals, the probabilities of these conditions applying at any time are the following:

Pr(Condition 1 applies) \(= e^{-q_{1} T_{(3)} } \) | 6.3 | |||

Pr(Condition 2 applies) \(= P_{0(2)} = 1 - \frac{q_{2} }{C_{2} } \) | 6.4 | |||

Pr(Condition 3 applies) \(= e^{-q_{2} T_{(3)} } \) | 6.5 | |||

where | ||||

q_{2} | = | the volume of traffic stream 2 | ||

T_{(3)} | = | the critical gap for stream 3 vehicles entering the intersection | ||

P_{0(2)} | the probability of the stream 2 queue being empty. |

Assuming independence of the above three conditions, the probability that all will apply simultaneously is the product of the individual probabilities in Equations 6.3 to 6.5, that is:

Pr(stream 3 vehicle can depart) \(= e^{-(q_{1} + q_{2} )T_{(3)} } . P_{0(2)} \) | 6.6 |

\[= e^{-(q_{{}_{1} } + q_{2} )T_{(3)} } . e^{\ln P_{0(2)} } \] | |

\[= e^{-\left[q_{1} + q_{2} - \frac{\ln P_{0(2)} }{T_{(3)} } \right] T_{(3)} } \] |

This probability is the same as the probability of a headway greater than or equal to T_{(3)} in a traffic stream with random arrival of vehicles and volume qa, where:

\[q_{a} = q_{1} + q_{2} - \frac{\ln P_{0(2)} }{T_{(3)} } \] | 6.7 |

The situation can therefore be considered equivalent to a simple gap acceptance situation in which a minor traffic stream with volume q_{3}, critical gap T_{(3)} and follow-up headway T_{0(3)} must give way to a single major traffic stream of volume q_{a}, as defined by Equation 6.7. Thus, for example, the theoretical absorption capacity for stream 3 would be calculated as:

\[C_{3} = \frac{q_{a} e^{-q_{a} T_{(3)} } }{1 - e^{-q_{a} T_{0(3)} } } \] | 6.8 |

The validity of the results obtained in the above example analysis has yet to be proven and this would probably be best accomplished by simulation of the assumed situation, over a range of traffic volumes and gap acceptance parameters. It is presented here as an illustration of how the combination of gap acceptance and queuing theory may be used to address more complex problems.