# BUS 351- Process and Systems Management

Question
BUS 351: Process and Systems Management: HW 3
This HW will require basic knowledge of probability theory (time to brush up on your BUS350 or stats
class). Remember that 1) probability cannot exceed 1 and cannot be negative; and 2) sum of probabilities
for a set of mutually exclusive and collectively exhaustive events is 1.
Do problems 1-3 by hand (with the help of the M/M/s table) and problem 4 with Excel (M/M/s
spreadsheet, both are posted in Canvas).
Problem 1. Calls arrive to a customer service center on average one every 12 minutes, according to the
Poisson process. Customer requests on average take 10 minutes to process, service times exponentially
distributed. The customer service center is staffed by one clerk. Queue discipline is FIFO.
1. (0.5 point) What is the name of this model?
2. (0.5 point) What is the arrival rate? Express as customers per hour.
3. (0.5 point) What is the service rate? Express as customers per hour.
4. (0.5 point) What fraction of time is the clerk idle?
5. (0.5 point) What is the average waiting time in queue (in minutes)?
6. (0.5 point) What is the average number of customers in the system?
7. (0.5 point) What is the probability that there are 3 or more customers in the system?
8. (0.5 points) What is the probability that the number of arrivals during one hour period equals to
5?
9. (1 point) What is the probability that there are no arrivals during a 30 min period? Compare this
to the probability that the interarrival time is greater than 30 min. What is your conclusion?
10. (1 point) Compute the probability that there are 4 or more arrivals per 1 hour period. Problem 2. (4 points) Hilton Garden Inn in Atlanta needs to decide whether to have 2 or 3 operators
answering calls from customers and hotel guests. The hourly wage of a customer service representative is
\$30. There are approximately 40 calls per hour arriving according to the Poisson process and each call
takes on average 2.5 minutes to answer (exponentially distributed). Calls answered in the order they are
received. The wait cost is incurred by all customers in the system (queue and in service). The estimated
wait cost rate \$50 per hour. Compute the cost rate for the system with 2 and 3 servers and decide whether
there should be 2 or 3 operators. Problem 3. Suppose you observe the following sequence of interarrival and service times for 6 arrivals.
Assume that the first arrival finds the system empty. The queue has 1 server.
1. (1 point) What delay was experienced by arrival #3?
2. (1 point) Find the average wait time experienced by these 6 arrivals.
Customer
#
1
2
3
4
5
6 IAT
12
40
34
14
72
3 Service
time
65
15
45
85
15
65 Problem 4. A manager of a medium-sized call center needs to determine the optimal number of operators
that she should hire. The call center serves two types of clients: Business (B) and Leisure (L) customers.
She has determined that B calls arrive according to a Poisson process at a rate of 18 calls per hour. On the
other hand, L calls arrive according to a Poisson process at a rate of 50 calls per hour. She decides to hire
highly skilled operators who can serve both type of customers and the average amount of time that they
need is 9 minutes per call independent of the type (service times are exponentially distributed). The cost
of hiring an H operator is \$35 per hour. The manager has also estimated that the cost of keeping a B
customer waiting in queue is \$80 per customer per hour while the cost of keeping an L customer waiting
in queue is \$20 per customer per hour. The manager is considering having a single queue for both types of
calls and hire only H operators that will serve both types of calls. Calls will be served on a First-ComeFirst-Serve basis independent of their type. Find:
1. (1 points) Minimum feasible number of operators
2. (2 points) Average number of B and L customers in the queue if the call center is staffed by 12
operators
3. (1 point) Total cost incurred by the call center per hour of operating if it is staffed by 12 operators
4. (2 points) Nopt: the optimal number of operators that minimizes the cost incurred by the call center, and
the corresponding cost, Wq, and utilization. Please show all your work. Be sure to include the formula for the wait cost computation.