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Sub-problem 1b - Page 5 of 9

ID# C201B05

Sub-problem 1b: Maxwell Drive PM Peak Hour - With Conditions

Cycle Length Education
For a pre-timed signal, the standard equation for computing the minimum cycle length is:


where vi is the volume for the critical movement in phase i, si is the saturation flow rate for the critical movement in phase i, N is the number of phases per cycle, and L is the lost time per phase.

For actuated signals with dual-ring controllers like the one we portrayed in Exhibit 2-11, it helps to expand Equation (1) into three equations. The first one computes the v/s ratio for sequential pairs of movements to the left and right of the middle barrier in the A and B rings:


Here pq takes on the values 1&2, 3&4, 5&6, and 7&8 corresponding to the four quadrants of the dual ring pattern.

The second equation computes the maximum v/s ratio for the left and right halves of the dual ring pattern:


(v/s)L = max((v/s)12,(v/s)56)   and (v/s)R = max((v/s)34, (v/s)78)

The third equation takes the two results from Equation (3) and computes a minimum cycle length:


This is the process being followed in the spreadsheet we used for the analysis.

Equations (2) through (4) show that Equation (1) can be obtained easily. First, realize that N equals 4. Second, see that the sequence of critical movements is either: 1,2,3,4; 1,2,7,8; 5,6,3,4; or 5,6,7,8. Third, notice that the sum of the vi/si ratios identified in the denominator of Equation (1) is the same as the left and right half based sum shown in Equation (4).  

In practice, we don't often use equations in the formal style they are presented here. The equations are helpful, though, because they are a shorthand-way of expressing ideas. You might check-out the spreadsheet and see if you can identify how equations (2), (3) and (4) have been implemented.

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