Problem
The federal government is undertaking a massive process analysis of the notoriously chaotic border San Diego and Tijuana. They have selected the San Ysidro station as a sample case and would like your input in doing the following analysis. Automobiles arrive from Mexico at the rate of 20 per hour according to a Poisson distribution. The station has 3 inspection lanes for cars and 1 inspection lane for trucks. All cars join a common queue and go to the first available car inspection lane. About 30% of the incoming vehicles are trucks and the remaining are cars. Average inspection time for a truck is 5 min and average inspection time for a car is 6 min.
Also, authorities have noticed that about 50% of the trucks and 25% of the incoming cars need further scrutiny of contents and documents. For this, they are diverted to a separate lane in the San Diego side of the border, where it takes about 8 min per vehicle to get checked thoroughly. This single "close scrutiny" lane is designed to handle both cars and trucks. For your convenience, a simple representation of the system is shown below.
Suppose inspection times at all stages are distributed exponentially. Answer the following question regarding the flow of vehicles through this Border Station.
The authorities at this border station believe that adding another lane for truck inspections will ensure that only 25% of the trucks that come through would need close scrutiny. However, the government has refused to fund this new lane. The official reason is: "This new lane would not reduce waiting time for cars, which are the majority of vehicles passing through the station". Is the government correct in its reasoning? Why?