Automobiles arrive at the drive$-$through window at the downtown Baton Rouge, Louisiana, post office at

the rate of $4$ every $10$ minutes. The average service time is $2$ minutes. The Poisson distribution is appropriate

for the arrival rate and service times are negative exponentially distributed.

a$)$ What is the average time a car is in the system?

b$)$ What is the average number of cars in the system?

c$)$ What is the average number of cars waiting to receive service?

d$)$ What is the average time a car is in the queue?

e$)$ What is the probability that there are no cars at the window?

f$)$ What percentage of the time is the postal clerk busy?

g$)$ What is the probability that there are exactly $2$ cars in the system? An advertising project manager gathered the precedence relationship and time information for each activity involved in a new advertising campaign. They are shown in the following table.

$\backslash$table$[[$Time Estimates $($weeks$),],[$Activity$,$Optimistic,Most Likely,Pessimistic,Immediate Predecessor$($s$),],[$A$,1,4,7,-,],[$B$,2,6,7,-,],[$C$,3,3,6,$B$,],[$D$,6,13,14,$A$,],[$E$,3,6,12,$A$,$ C$,],[$F$,6,8,16,$B$,],[$G$,1,5,6,$E$,$ F$,]]$

a$)$ Draw the AON diagram for the project.

b$)$ Calculate the expected time and variance for each activity.

c$)$ Determine the critical path, using the expected activity times.

d$)$ What is the probability of completing the project within $23$ weeks?Suzy Jones is trying to decide whether to use one or two suppliers for the motors than go into the chain saws that her company produces. She wants to use local suppliers because her firm runs a JIT operation. Her factory is located in a coastal town that is prone to hurricanes. She estimates that the probability in any year of a "super$-$event" that might shut down all suppliers at the same time for at least two weeks is $5\%.$ Such a total shutdown would cost the company approximately $$100,000.$ She estimates the "unique$-$event" risk for any of the suppliers to be $10\%.$ Assuming that the marginal cost of managing an additional supplier is $$12,000$ per year, should Suzy use one or two suppliers? Justify your answer pragmatically or arithmetically.