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 managerís 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: P0, Pn, and L.† (c) Solve for P0.
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† Ĺ† 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.
3.† You run a small copy center that is not self serve.† Customers get in line for service.† There is only one server.† Youíve 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 2nd in line will wait an average of 10 minutes, and the 3rd 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 2nd position, then this person will wait an average of 10 minutes in the 2nd 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.
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 donít 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?
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.