Tollbooths
(MCM 2005B)
Heavily-traveled toll roads such as the Garden State Parkway, Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.
Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.
收费亭
(美国竞赛2005年B题)
像Garden State Parkway，Interstate 95等等这样的长途收费公路，通常是多行道的，被分成几条高速公路，在这些高速公路上每隔一定的间隔会设立一个通行税收费广场。因为征收通行税通常不受欢迎，所以应该尽量减少通过通行税收费广场引起的交通混乱给汽车司机带来的烦恼。通常，收费亭的数量要多于进入收费广场的道路的数量。进入通行税收费广场的时候， 流到大量收费亭的车辆呈扇形展开，当离开通行税收费广场的时候，车流将只能按照收费广场前行车道路的数量排队按次序通过！从而，当交通是拥挤的时，拥挤在违背通行税广场上增加。当交通非常拥挤的时候，因为每车辆付通行费的时间要求，阻塞也会出现在通行税收费广场的入口处。
建立一个模型来确定在一个容易造成阻塞的通行税收费广场中应该部署的最优的收费亭的数量。需要保证每一个进入收费广场的交通线路上都仅有一个收费亭。与当今的实践相比较，在什么条件下这或多或少有效？注意："最佳"的定义由你自己决定。