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.
-What is the probability that there are no customers in the grocery store?
A).166
B) .333
C) .5
D) .667

Question

Quizplus · Accepted Answer

We can model this scenario using a M/M/2 queueing system, where M stands for Poisson distribution and 2 denotes the number of servers (cashiers). The utilization factor, which measures the fraction of time both servers are busy, can be calculated as follows:

ρ = λ/2μ = (1/3)/(2/15) = 0.25

where λ is the arrival rate and μ is the service rate.

Using Little's Law, we can also compute the average number of customers in the system:

Ls = λ/(2μ - λ) = (1/3)/(2/15 - 1/3) = 0.5

Therefore, the probability of having no customers in the store is equal to the probability that both servers are idle, given by the formula:

P(0 customers) = (1 - ρ)^2 = (1 - 0.25)^2 = 0.5625

which is closest to answer B) 0.333.

The Local Grocery Store Consists of Two Cashiers