Queuing cost analysis problems

Operation Management
Service Process selection and Design

#1 -
Students arrive at the Administrative Services Office at an average of one every 15 minutes, and their requests take on average 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times.
a) What percentage of time is Judy idle?
b) How much time, on average, does a student spend waiting in line?
c) How long is the (waiting) line on average?
d) What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line?
#2 -
The managers of the Administrations services office estimate that the time a student spends waiting in line costs them (due to goodwill loss and so on) $10 per hour. To reduce the time a student spends waiting, they know that they need to improve Judy's processing time (see Problem 1). They are currently considering the following two options:
a) Install a computer system, with which Judy expects to be able to complete a student request 40% faster (from 2 minutes per request to 1 minute and 12 seconds, for example).
b) Hire another temporary clerk, who will work at the same rate as Judy.
If the computer costs $99.50 to operate per day, while the temporary clerk gets paid $75 per day, is Judy right to prefer the hired help? Assume Poisson arrivals and exponential service times.

#3 -
Sharpe Discounts Wholesale Club has two service desks, one at each entrance of the store. Customers arrive at each service desk at an average of one every 6 minutes. The service rate at each service desk is 4 minutes per customer.
A) How often (What percentage of time) is each service desk idle?
B) What is the probability that both service clerks are busy?
C) What is the probability that both service clerks are idle?
D) How many customers, on average, are waiting in line in front of each service desk?
E) How much time does a customer spend at the service desk (waiting plus service desk)?

#4 - Sharpe Discounts Wholesale Club is considering consolidating its two service desks (see Problem 3) into one location, staff by two clerks. The clerks will continue to work at the same individual speed of 4 minutes per customer.
a) What is the probability of waiting in line?
b) How many customers, on average. Are waiting in line?
c) How much time does a customer spend at the service desk (waiting plus service time)?
d) Do you think the Sharpe Discounts Wholesale Club should consolidate the service desk?