Question

1.1 (M/M/1) model - use the qtsPlus or other software to answer the questions. In a grocery store, there is one cashier. Customers arrive at the cashier counter according to a Poisson process. The arrival rate is 18 customers per an hour. Customers are served in order of arrival (FCFS: first come first service). The service time (i.e. the time needed for scanning the items and processing payment) is exponentially distributed. The mean service time is 3 minutes. A) Once you arrived at the cashier counter, on average, how many customers are waiting in the queue? B) Once you arrived at the cashier counter, on average, how long should you wait in the queue? C) Once you arrived at the cashier counter, on average, how long should it take to complete the service? D) What is the cashier’s utilization rate?

1.2 (M/M/c) model - use the qtsPlus or other software to answer the questions. Now, you decided to hire one more cashier, so the store has two cashiers. Customers arrive at the cashier counter according to a Poisson process, which is same as the question 1. The arrival rate is 18 customers per an hour. Customers are served in order of arrival (FCFS: first come first service). The service time (i.e. the time needed for scanning and paying) is exponentially distributed. The mean service time is 3 minutes for each cashier. A) Once you arrived at the cashier counter, on average, how many customers are waiting in the queue? B) Once you arrived at the cashier counter, on average, how long should you wait in the queue? C) Once you arrived at the cashier counter, on average, how long should it take to complete the service? D) What are the cashiers’ utilization rate?

1.3 Based on the outputs of the questions 1.1 and 1.2, please discuss about both system’s pros and cons, in terms of customer services and cashiers’ utilization rate and its operation.

Answer 1

1. Average arrival rate ( λ ) = 18 customers per hour = 18 customers per 60 minutes.

Average service rate ( μ ) = 3 minutes for 1 customer = 60 minutes for 20 customers.

A. The average number of customers in the queue = L_q =

=

= 324 / 40

= 8.1

B. The average waiting time for a customer in queue = W_q =

=

= 0.45

C. The average waiting time to complete the service = average waiting time for a customer in a system = W_s =

=

= 0.5

D. Cashier's utilization rate ( ρ ) = λ / μ

= 18 / 20

= 0.9