1. (Process Performance Measures) A loan processing office receives 4 loan requests every hour. The office has 2 departments. It takes the first department on average 20 minutes to do a preliminary check on the requests and 20% are rejected directly without going into the second department. The remaining 80% go into the second department and take 60 minutes per request and then either an acceptance or rejection letter is sent to the client. On average how many requests are in the first department (including those who are waiting)? How many are in the second (including those who are waiting)? (Hint: Since this question does not specify how many servers are in each department, please do not assume that there is only one server. Instead, you may use the Little’s Law here.)
2. (Queuing Model) A single public health worker is inoculating children against measles at a local clinic. An average of 8 children per hour is expected to arrive according to a Poisson distribution. He serves the children at a rate of 10 per hour. Assume that the service-time distribution is approximately exponential.
(a) For the ‘inoculation system’, find:
i) the utilization of the health worker.
ii) the average time spent by a child in the system.
iii) the average number of children in the system.
iv) the average number of children in line.
v) the average number of children in the system if another health worker is added to the system who has the same service rate as the existing one.
(b) At what rate must children be served by the single public health worker so that the average time that a child spends in the system can be reduced by 50%?
3. (Queuing Model) The FastFood restaurant on the Main Street has received a lot of complaints during lunch hour regarding its drive-through service. On average 50 cars arrive at the window according to Poisson distribution. A customer is first served by a clerk who takes the customers’ orders. The customer then drives to the window where another clerk collects payments and passes the order to the customer. Both clerks serve the customers at a rate of 1 customer per minute. Assume service time is exponential distributed. Examine the restaurant’s service performance. Specifically, answer the following questions.
(a) Draw a graph and identify the queue(s), server(s), arrival rate (if you have multiple queues, point out rate for each one) and service rate (for each server if there are multiple servers)?
(b) What’s the utilization of the server(s)?
(c) What’s the average number of cars in the system?
(d) What’s the average number of cars waiting in line?