Sub-problem 4d: Coordinated
Operation With Left-Turn Protection

In order to estimate the proportion of skipped left-turn
phases during the analysis time period, we need to estimate the probability
that no left-turn vehicles will arrive during a cycle. For this purpose, it is common to assume that the
left-turning vehicles will arrive at the intersection according to a Poisson
distribution (a Poisson distribution reflects random
arrivals). Assuming a Poisson distribution of arrivals, the
probability of zero arrivals on any cycle may be computed as:

where *a* represents the average number of arrivals per
cycle, based on the hourly volume.

Exhibit 1-41 shows the results of these computations. If the average length of a left-turn phase is less than 12 seconds, the
difference is added to the phase time for the opposing through movement.

As an example, we will examine the average northbound
phase times. The average cycle length at this intersection is 90-seconds, or
40 cycles per hour. The northbound left-turn volume is thirty-one vehicles
per hour, which produces an average of 0.78 vehicles per cycle (31 vehicles
per hour/40 cycles per hour). From the above equation we estimate zero
arrivals on 46-percent of the cycles *(e*^{-0.78}), giving us
an average phase time of 6.5 seconds (12 sec * (1-0.46)). Because this is
less than the assumed maximum of 12-seconds green, we may add the difference
in time (5.5 seconds) to the conflicting southbound through phase, producing
63.5 seconds of green (58 sec+5.5 sec).