Customers at an amusement park arrive at the rate of 10 customers per hour.The entry booth is staffed by one employee.The mean service time at the booth to provide service to each customer is 5 minutes.The arrival rate follows a Poisson distribution,and the service time at the booth follows a negative exponential distribution.Determine the probability of exactly five customers in the system.
A)4.50%
B)6.70%
C)5.60%
D)3.80%

Question

Quizplus · Accepted Answer

Given the arrival rate ($\lambda$) is 10 customers per hour and the mean service rate ($\mu$) is 12 customers per hour (since 60 minutes / 5 minutes = 12 customers per hour), we can use the formula for the probability of having exactly $n$ customers in the system in an M/M/1 queue, which is $P_n = (1 - \rho) \rho^n$, where $\rho = \lambda / \mu$. Here, $\rho = 10 / 12 = 5/6$, and for $n = 5$, $P_5 = (1 - 5/6) (5/6)^5 = (1/6) (5/6)^5$, which calculates to approximately 6.70%.

Customers at an Amusement Park Arrive at the Rate of 10