OPMGT $301$ Homework # $2$: Queueing Systems

Problem. Trucks are required to pass through a weighing station so that they can be checked for weigh violations. Trucks arrive at the station at the rate of $20$ an hour. Currently there is only one inspector are on duty, who can inspect $25$ trucks an hour.

Part A$.$ How many trucks would you expect to see at the weighing station including those being inspected?

Part B$.$ How much time should a driver expect to spend at the station?

Part C$.$ What is the probability of having a queue?

Hint: try to relate the requirement of "having a queue" to the "minimum number of customers" needed in the system.

Part D$.$ Assuming that each inspector costs $$20$ an hour and each truck driver represents a cost of $$30$ for every hour spent in the system, calculate the optimal number of operators.