Congestion in a Running Race. The following model was developed by Georgia Tech undergraduate students to assess the congestion in the 10-kilometer Atlanta Road Race, which is held every July 4th....

Congestion in a Running Race. The following model was developed by Georgia Tech undergraduate students to assess the congestion in the 10-kilometer Atlanta Road Race, which is held every July 4th. After the pack of elite runners begins the race, the rest of the runners start the race a little later as follows. The runners are partitioned into m groups, with rk runners assigned to group k, 1 ≤ k ≤ m, depending on their anticipated completion times (the runners in each group being about equal in ability). The groups are released every τ minutes, with group k starting the race at time kτ (the groups are ordered so that the faster runners go earlier). Although the group sizes rk are random, assume for simplicity that they are not. Typical numbers are 10 groups of 5000 runners in each group. The aim was to design the race so that the congestion did not exceed a critical level that would force runners to walk. To do this, the students developed a model for computing the probability that the congestion would be above the critical level. (They used this model to determine reasonable group sizes and their start times under which the runners would start as soon as possible, with a low probability of runners being forced to walk.)