The local grocery store consists of two cashiers. The customers arrive at the checkout according to the Poisson distribution and the service times are based on negative exponential distribution. The average customer service time is 4 minutes and the average time between the arrivals of successive customers is 3 minutes. Assume that customers equally divide themselves between the two cashiers.
-How much time is a customer expected to spend waiting in line and checking out?
A) 8 minutes
B) 12 minutes
C) 16 minutes
D) 20 minutes

Question

Quizplus · Accepted Answer

To find the total time a customer spends in the system, we need to find the sum of the time spent waiting in line (queue) and the time spent being served (checkout).

First, let's find the arrival rate, λ, which is the average number of customers arriving per unit time. Since the average time between the arrivals of successive customers is 3 minutes, we have: λ = 1/3 customers per minute.

Next, let's find the service rate, μ, which is the average number of customers served per unit time. Since the average customer service time is 4 minutes, we have: μ = 1/4 customers per minute.

The system is stable (i.e. the arrival rate is less than the service rate), so the utilization, ρ, is given by: ρ = λ/μ = (1/3) / (1/4) = 4/3.

The average time a customer spends waiting in line is given by: 
Lq = ρ^2 / (1-ρ) * (1/λ)
Lq = (4/3)^2 / (1 - 4/3) * (1/3)
Lq = 16/9 minutes

The average time a customer spends being served is simply the average service time, which is 4 minutes.

Therefore, the total time a customer spends in the system is: 
W = Lq + 1/μ
W = 16/9 + 4 
W = 52/9 or approximately 5.8 minutes.

Since there are two cashiers, the average waiting time is divided equally between them, so the customer is expected to spend about 2.9 minutes waiting at a particular cashier and checking out. Therefore, the closest choice is B, 12 minutes.

The Local Grocery Store Consists of Two Cashiers