1. Automobiles arrive at the drive-thru window at the downtown Fort McMurray 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 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?

h) By how much would your answer to part (a) be reduced if a second drive-thru window, with its own server, were added?

2. A proposal has been presented to the government of Newfoundland and Labrador to build a new section of highway that would provide improved access for residents of a remote coastal area near Bonavista. The highway would be 16 kilometres in length. The initial proposal called for 7 toll booths, each staffed by an employee. But a subsequent proposal recommended replacing the employees with machines. Many factors must be considered because the intended employees are unionized. However, one of the governments concerns is the effect that replacing the employees with machines will have on the times the drivers spend in the system. Customers will arrive to any one toll booth at a rate of 10 per minute. In the exact-change lanes with employees, the service time is essentially constant at 5 seconds for each driver. With machines, the average service time would still be 5 seconds, but it would be exponential rather than constant, because it takes time for the coins to rattle around in the machine. Contrast the two systems for a single lane.

3. Sam Certo, a Nanaimo vet, is running a rabies vaccination clinic for dogs at the local grade school. Sam can shoot a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every 6 minutes according to a Poisson distribution. Also assume that Sams shooting times are exponentially distributed. Compute the following:

a) The probability that Sam is idle.

b) The proportion of the time that Sam is busy.

c) The average number of dogs being vaccinated and waiting to be vaccinated.

d) The average number of dogs waiting to be vaccinated.

e) The average time a dog waits before getting vaccinated.

f) The average amount of time a dog spends waiting in line and being vaccinated.