Operations Management

Homework $2$

Due January $30,2024$

For all questions, please show all the details of your work.

Students arrive at Administrative Services Office at an average of one every $15$ minutes, and their requests take on average $10$ minutes to be processed. The service encounter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume and exponential interarrival and service times.

a$.$ What percentage of time is Judy occupied?

b$.$ What is the probability that Judy spends less than $20$ minutes processing a request?

c$.$ How much time, on average, does a student spend waiting in line?

d$.$ If the Administrative Services Office decides to bring in an additional staff member to help Judy while maintaining a single waiting line for students, how much time does a student spend waiting in line on average?

Customers arrive to the checkout area of New Navy such that the interarrival rate is exponential with rate equal to $50$ customers per hour. At the checkout area, the service time of each customer is exponentially distributed with a mean of $12$ minutes. The store has hired $11$ cashiers to checkout customers and the checkout area consists of a single pool of cashiers with one common line in front of them.

a$.$ What is the service rate?

b$.$ What is the average number of customers waiting in the queue to be checked out? This does not include the customers being checked out.

c$.$ What is the average total number of customers in the checkout area? This includes the customers who are waiting to be checked out, and the customers being checked out.

d$.$ Suppose now that New Navy decides to replace the single common line for its customers with $11$ separate lines, one line in front of each of the $11$ cashiers. What is now the average number of customers waiting to be checked out? This only includes the customers who are waiting to be checked out $($and not the customers being served$).$

$1$