1. A small bank
has one teller in training on duty all the time. This teller has an average service time of 5 minutes. Whenever there are 3 or more customers in
the system, then the manager comes out and opens another service line. The manager’s average service time is 3
minutes. Customers arrive at the rate
of 6 per hour. Service times and
interarrival times are exponentially distributed. (a) Set up the rate diagram.
(b) Give the formulas for the state distributions: P_{0}, P_{n},
and L. (c) Solve for P_{0}.

25 points.

2. You run a small company that depends on internet sales. You have 3 computer servers to manage the internet activity. These servers are very important. Each one breaks down about once a week with an exponentially distributed inter-breakdown time. About 80% of the time you can fix the problem quickly with an average repair time of ½ hour. The remaining 20% of the time, the repair is more difficult and the average repair time is 6 hours. Both repair times are exponentially distributed. (a) Set up the Markov chain rate diagram for this queue. (b) Give the balance equation for one of the states associated to 2 broken machines.

15 points.

3. You run a small
copy center that is not self serve.
Customers get in line for service.
There is only one server. You’ve
noticed that as the line gets longer, the customers tend to renege. In fact when there are 2 or more in line
waiting to be served, the 2^{nd} in line will wait an average of 10
minutes, and the 3^{rd} an average of 5 minutes. No one gets in line if there are already 3
waiting to be served. When customers
move up in line they are willing to wait longer; that is, if the 3rd person in
line moves up to the 2^{nd} position, then this person will wait an
average of 10 minutes in the 2^{nd} position before reneging. Your server averages about 20 customers per
hour. Customers arrive at the rate of
15 per hour. All times are
exponentially distributed. (a) Set up
the rate diagram. (b) Solve for the
state distributions. (c) Compute
L.

25 points.

4. Take the same copy center as in problem 3, only this time there is no reneging and your line can get arbitrarily long (i.e., infinite queue). (a) Compute L in this situation. (b) Suppose each customer that stays spends $5 in your shop, but those that renege (or don’t come in because the line is too long) spend nothing. Can you give an estimate of dollars lost per hour due to reneging? If so how or what is it?

10 points.

5. At a local automobile repair shop has cars coming in at the rate of 1 every 2 hours which need the front brakes replaced (one for each front wheel). Assume the interarrival times are exponentially distributed. There is one mechanic that works on the brakes. It takes her about 20 minutes to replace the brakes on one wheel with an actual repair time per wheel that is exponentially distributed. (a) Set up the rate diagram. (b) Compute L, W for this queueing system.

25 points.