# 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

right sides of the roads at constant average rates of and respectively.

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

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Solution. Assume the time that an arriving pedestrian must wait until crossing is X, then X is a random variable. Assume that a pedestrian (of the first N pedestrians) arrives at ...

#### Solution Summary

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

right sides of the roads at constant average rates of and respectively.

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