Expected value

The town of Smallville is considering a new pedestrian crossing system
for a particular stretch of road. A red/green light controls when
pedestrians cross the road. Pedestrians arrive on the left and right
sides of the road, and when their light turns green, they cross the
road.

The system works as follows: the pedestrian’s light stays red until
the total number of

waiting pedestrians equals N, and then it turns green and lets all the
pedestrians cross

(this is called a “dump�). Then it turns red and waits for the next
N pedestrians.

The town council wants to know some of the operating characteristics of
this system, to

see if it is worth the cost, and to see what N works best (to be the
least disruptive to both pedestrians and cars). You are hired not to
make these trade-offs but to provide information.

Assume that the pedestrians cross instantly, and that pedestrians arrive
on the left and

respectively.

Calculate the e-time that an arriving pedestrian must wait until
crossing.